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--- abstract: 'The effect of hydrogen adsorption on the magnetic properties of an Fe$_3$ cluster immersed in a Cu(111) surface has been calculated using densifty functional theory and the results used to parametrize an Alexander-Anderson model which takes into account the interaction of d-electrons with itinerant electrons. A number of adatom configurations containing one to seven H-atoms were analyzed. The sequential addition of hydrogen atoms is found to monotonically reduce the total magnetic moment of the cluster with the effect being strongest when the H-atoms sit at low coordinated sites. Decomposition of the charge density indicates a transfer of 0.4 electrons to each of the H-atoms from both the Fe-atoms and from the copper substrate, irrespective of adsorption site and coverage. The magnetic moment of only the nearest neighbor Fe-atoms is reduced and mainly due to increased population of minority spin d-states. This can be modeled by increased indirect coupling of d-states via the conduction s-band in the Alexander-Anderson model.' address: - '$^1$Science Institute and Faculty of Physical Sciences, VR-III, University of Iceland, Reykjavík, Iceland' - '$^2$Department of Physics, St. Petersburg State University, St. Petersburg, 198504, Russia' - '$^3$National Research University of Information Technologies, Mechanics and Optics, St. Petersburg, 197101, Russia ' - '$^4$Dept. of Applied Physics, Aalto University, Espoo, FI-00076, Finland' author: - 'Pavel F. Bessarab$^{1,2}$, Valery M. Uzdin$^{2,3}$, and Hannes Jónsson$^{1,4}$' title: The Effect of Hydrogen Adsorption on the Magnetic Properties of a Surface Nanocluster of Iron --- Introduction {#sec:Introduction} ============ Hydrogen can strongly affect the magnetic properties of metals, in particular metal surfaces. This can be used to intentionally modulate magnetic properties by introducing hydrogen since hydrogen atoms can in many cases be adsorbed and desorbed relatively easily. Hydrogen can also be present inadvertently as an impurity and thereby affect measurements of magnetism. Either way, it is of considerable importance to understand the way in which hydrogen can affect magnetic properties of metals. The thermal stability of magnetic states of nanoscale islands on surfaces is strongly dependent on the size and shape of the islands and preferential adsorption of impurities on the island rim could affect this dependence. In most cases, hydrogen loading has been found to lead to a monotonic change in the magnetic moment of large transition metal clusters and thin films as hydrogen concentration is increased. Surface magneto-optic Kerr effect experiments have been carried out in order to measure the effect of hydrogen on the magnetism of ultrathin films of Fe, Co and Ni on Cu(001) [@mankey_93]. The magnetization of Ni and Co films has been found to be reduced upon hydrogen adsorption, while for the Fe films, hydrogen enhances the magnetization slightly. Theoretical studies are consistent with these results [@siegbahm_84; @granucci_92; @maca_03]. For example, [*ab initio*]{} calculations based on full-potential linearized augmented plane-wave method for a Ni multilayer on Cu(001) substrate have shown that hydrogen adsorption on the Ni film leads to a considerable decrease of magnetic moment in the upper Ni layers. However, magnetic properties of small, free standing iron, nickel and cobalt clusters, containing just a few atoms, respond to hydrogenation in a more complex way, in some cases showing oscillations in the magnetic moment upon successive addition of H atoms [@jones_04; @ashman_03]. Long range effect of hydrogen adsorption on magnetic properties of atoms have also been reported. While a full coverage of hydrogen on Co films was found to decrease the magnetization of surface atoms, partial H-atom coverage led to a significant increase in the magnetic moment of surface Co-atoms not bonded to hydrogen, even leading to an overall enhancement of surface magnetism [@gallego_10]. Another example of such long-range effect comes from studies of Fe/V superlattices [@remhof_07] using element specific X-ray resonant magnetic scattering. A significant increase in the magnetic moment of the iron layers was found when the vanadium layers were loaded with hydrogen even though the induced antiparallel moment of V-atoms at the Fe/V interface remained unaffected. This effect was explained using model Hamiltonian calculations and is due to a redistribution of [*d*]{}-electrons between Fe and V atoms. The introduction of hydrogen causes a shift of the [*d*]{}-band relative to the Fermi level and thus changes the exchange splitting [@remhof_07]. We have chosen to study a system in between thin magnetic layers and small free standing clusters, namely a small iron cluster embedded in the surface of a non-magnetic metallic substrate. An embedded clusters rather than a cluster of adatoms on top of a substrate was chosen because adatom clusters tend to be highly mobile via multi-atom concerted displacements (see for example ref. [@Xu_PdClust]). Density functional theory (DFT) calculations are used to study the effect of hydrogen on the magnetic properties. Various configurations with the number of hydrogen atoms ranging from 1 to 7 at full saturation are considered. The total magnetic moment of the iron cluster as well as the number of d-electrons, magnetic moment of Fe-atoms and the local density of states (LDOS) is calculated as a function of the number of hydrogen atoms at the various types of adsorption sites. The effect of hydrogen adsorption is then reproduced with an Alexander-Anderson tight-binding model [@AAmodel] with parameter values chosen to reproduce the DFT results. The AA model has previously been shown to describe adequately the magnetic states of clusters of 3$d$ transition metals on non-magnetic substrates. For example, calculations of magnetic structure of supported transition metal clusters [@uzdin_99; @uzdin_01; @uzdin_00] have been found to reproduce all the main features obtained within density functional theory [@bergman_07; @antal_08; @lounis_07], including the possibility of non-collinear ordering of magnetic moments and appearance of several stable magnetic states which are close in energy. The motivation for using such a model is to better elucidate the physical picture of the effect of hydrogen adsorption on the magnetic moment. The parametrized model can then also be used to estimate the effect of hydrogen adsorption on larger islands in systems that are too computationally demanding for the DFT calculations. The Simulated System {#sec:SupIrTrim} ==================== The influence of hydrogen adsorption on magnetic properties of an iron cluster containing three atoms was calculated. A trimer was chosen because it is the minimal structure which contains the various types of sites for adatoms. The cluster is embedded into a Cu(111) surface and has the shape of an equilateral triangle. The Fe-Fe bond-lengths are fixed by the substrate and the geometry of the cluster is stable upon hydrogenation. The Cu substrate is represented by a slab consisting of 33 atoms (see Fig. \[fig:1\]). There are three layers in the slab, each containing 12 metal atoms. The top two layers are allowed to move but the bottom one is kept fixed. The DFT calculations made use of the Perdew, Burke and Ernzerhof (PBE) [@PBE] approximation to the exchange and correlation functional. A plane wave basis set with an energy cutoff of 273.25 eV was used to represent the valence electrons using the projector augmented wave (PAW) formalism [@blochl]. The Vienna [*ab-initio*]{} simulation package (VASP) was used in these calculations [@VASP]. ![Fe trimer cluster embedded into a Cu(111) surface. Up to seven H-adatoms can be adsorbed on the cluster.[]{data-label="fig:1"}](fig1b.pdf){width="110.00000%"} Adsorption sites for the H-atoms were determined by placing the atoms at various sites on the cluster and carrying out energy minimization with respect to the atom coordinates. Also, an initial structure with as many as 10 hydrogen atoms was set up and then heated to 750 K and annealed for 1.5 ps (1500 ionic moves) before finally minimizing the energy with respect to atom coordinates. During the annealing, three hydrogen atoms were desorbed from the system. The maximum coverage was thus determined to be 7 H-adatoms in an arrangement shown in Fig. \[fig:1\]. There are three types of adsorption sites differing in the number of nearest neighbor Fe-atoms. These are denoted as X-, Y- or Z-sites, corresponding to one, two and three nearest neighbor Fe-atoms, respectively (see Fig. \[fig:2\]). Up to three X-type, three Y-type and one Z-type adatoms may be present on the Fe trimer. Except for the Z-type adatom, distances between the H-atom and nearest Fe-atoms are nearly the same, 1.75 . The adsorption of hydrogen was indeed found not to affect the geometry of the cluster. The Fe-Fe nearest neighbor distance turned out to be 2.35 for all configurations of H-adatoms. ![On-top view of the Fe trimer cluster and the three possible sites for adsorbed H-atoms: X-, Y- and Z-site. The cluster is embedded in a (001) surface of a copper slab which is not shown. The distance between Fe-H atom pairs is indicated. []{data-label="fig:2"}](fig2.pdf){width="40.00000%"} The DFT Calculated Magnetization ================================ The total spin magnetic moment of the system was calculated using DFT for the various possible numbers and arrangement of the H-adatoms. The calculated total magnetization of the various configurations is listed in Table 1and the change due to H-atom adsorption shown in Fig. 3. \[tab:1\] Two trends are evident: (1) The total magnetic moment decreases as H-adatoms are introduced. (2) For a given number of H-adatoms the strength of the effect depends on the number of Fe-neighbors of the H-adatom, an X-type adatom lowering the magnetic moment more than a Y-adatom, which in turn lowers it more than a Z-type adatom. The effect of X-type adatoms is approximately twice as great as that of Y-type adatoms. The calculated change in the magnetic moment is shown in Fig. 3. This decrease in the value of the total magnetic moment upon hydrogen adsorption is analogous to what has been observed in thin magnetic layers and large magnetic clusters [@mankey_93; @siegbahm_84; @granucci_92; @maca_03] as opposed to the oscillatory change observed for some small free standing transition metal clusters, such as Fe$_n$ and Co$_n$ [@jones_04; @ashman_03]. The electronic structure of bulk materials and surfaces can be significantly different from that of small clusters. Charge decomposition and changes in the LDOS can help gain an understanding of these effects. This type of analysis is presented in the next section. ![DFT calculated change in the total magnetic moment of the Fe$_3$H$_n$ cluster as a function of the number of H-adatoms of type X for a given number of Y-type H-adatoms (see legend and insets). []{data-label="fig:3"}](fig3.pdf){width="70.00000%"} Analysis of charge density and DOS {#sec:LocEffAdsHAto} ================================== In order to gain insight into the changes taking place in the electronic structure upon adsorption of hydrogen, we have carried out a charge decomposition analysis using the Bader definition of atomic regions [@Bader]. A fast, grid based algorithm including LDOS analysis was used [@Henkelman06; @Gudmundsdottir12]. The results, listed in Table 2, show that each hydrogen atom attracts the same number of electrons, about 0.4, irrespective of the site. No spin density is found within Bader regions associated with the H-atoms. The number of electrons associated with Fe-atoms only changes for the nearest neighbors of the H-adatoms. For example, an H-adatom at a Y-site formed by two Fe-atoms neither changes the total number of electrons nor the number of unpaired electrons associated with the third Fe-atom. The same holds for an H-adatom at a X-site, it changes significantly the electronic structure of the nearest Fe-atom. The total number of electrons at the Fe-atom decreases by $\sim$0.1 and magnetic moment decreases by $\sim$0.25$\mu_B$, but the number of electrons at the other two Fe-atoms in the cluster is not affected significantly. An H-adatom at a X-site reduces the number of unpaired electrons more strongly than an adatom at a Y-site. ![image](table2.pdf){width="120.00000%"} \[table:2\] ![DFT calculated local density of states for [*d*]{}-electrons in a Bader region of an Fe-atom in a Fe$_3$ cluster embedded in a Cu(001) surface, with and without H-adatoms (see inset). The black line corresponds to the bare Fe cluster. The red line corresponds to a cluster with full coverage of hydrogen, see insets. The Fermi level is shown with a dashed line. The minority-spin (spin-down) states get filled to a larger extent in the presence of hydrogen, thus reducing the difference in the number of spin-up and spin-down electrons, and thereby the magnetization. []{data-label="fig:4"}](fig4.pdf){width="90.00000%"} The DFT calculated LDOS for [*d*]{}-electrons in the Bader region of an Fe-atom in a cluster with no and full hydrogen coverage is shown in Fig. 4. The interaction of the [*s*]{}-state of the H-atom with the [*d*]{}-band of the Fe-atoms changes the band structure leading to the considerable changes of both majority- and minority-spins. The majority-spin states are almost all below the Fermi energy, and their redistribution with hydrogenation does not lead to significant changes in their occupation. Integration of the majority-spin LDOS up to the Fermi level decreases by only 0.1 in the presence of hydrogen. The integral of minority-spin states up to the Fermi energy is, however, changed much more, it increases from 1.6 to 2.1 upon hydrogenation. This results in reduced magnetic moment of the Fe-atoms. H-adatoms also induce changes in deeper states but these are not relevant for the magnetic properties. The insight gained from this analysis is used in the next section to parametrize a model which can reproduce the observed changes in the magnetization of the cluster. Analysis with a model Hamiltonian {#sec:AnModHam} ================================= In order to analyze the effects of hydrogen adsorption and to enable calculations of much larger systems, the DFT results are used to parametrize an Alexander-Anderson model for the system [@AAmodel]. The model describes two electronic bands: 3[*d*]{}-electrons localized on the Fe-atoms of the cluster and itinerant [*s*]{}([*p*]{}) electrons. The Hamiltonian of the system is written as $$\label{eq:1} \begin{array}{l} H=\sum_{\bf{k},\sigma}E_{\bf k}\hat{n}_{\bf k\sigma}+ \sum_{i,\sigma}E_{0i}\hat{n}_{i\sigma}+ \sum_{{\bf k},i,\sigma}V_{i\bf k}\hat{d}_{i\sigma}^{\dag}\hat{c}_{\bf k\sigma}\vspace{10pt}\\ \hspace{2em}+\sum_{i\ne j,\sigma}V_{ij}\hat{d}_{i\sigma}^{\dag}\hat{d}_{j\sigma}+ \frac{1}{2}\sum_{i,\sigma}U_i\hat{n}_{i\sigma}\hat{n}_{i-\sigma}+ h.c.\\ \end{array}$$ where $h.c.$ stands for Hermitian conjugate. Here $\hat{d}_{i\sigma}^{\dag}(\hat{d}_{i\sigma})$ and $\hat{c}_{\bf k\sigma}^{\dag}(\hat{c}_{\bf k\sigma})$ are creation (annihilation) operators for [*d*]{}-electrons with spin $\sigma=\pm$ localized on site [*i*]{} and itinerant [*s*]{}([*p*]{})-electron with quasimomentum [**k**]{} respectively; $\hat{n}_{i\sigma}=\hat{d}_{i\sigma}^{\dag}\hat{d}_{i\sigma}$, $\hat{n}_{\bf k\sigma}=\hat{c}_{\bf k\sigma}^{\dag}\hat{c}_{\bf k\sigma}$ are corresponding occupation number operators. The energy of non-interacting [*s*]{}([*p*]{}) electrons, $E_{\bf k}$, and [*d*]{}-electrons, $E_{0i}$, hybridization parameters, $V_{i{\bf k}}$, hopping parameters, $V_{ij}$, and Coulomb repulsion on site, $U_i$, are spin independent. The last term in the Hamiltonian in eqn.(\[eq:1\]) is included within a mean field approximation $$\label{eq:2} U_i\hat{n}_{i\sigma}\hat{n}_{i-\sigma} %\to \approx U_i\hat{n}_{i\sigma}\langle\hat{n}_{i-\sigma}\rangle + U_i\langle\hat{n}_{i\sigma}\rangle\hat{n}_{i-\sigma} - U_i\langle\hat{n}_{i\sigma}\rangle\langle\hat{n}_{i-\sigma}\rangle,$$ where $\langle\hat{n}_{i\sigma}\rangle$ denotes the average value of an occupation number. These numbers are found self-consistently within a Green function method. Only the [*d*]{}-electrons are considered explicitly in the calculations. Therefore, only the three last terms of the Hamiltonian are included while the influence of the [*s*]{}([*p*]{})-electrons is taken into account via the renormalization of the phenomenological model parameters. In particular, [*s*]{}-[*d*]{}-hybridization leads to the appearance of a non-zero width, $\Gamma$, of the [*d*]{}-level which is independent of the hopping parameters, $V_{ij}$. The model has adjustable parameters which in the present case are deduced from the results of DFT calculations. The Fe atoms are characterized by two dimensionless phenomenological parameters. The first determines the energy level of non-interacting [*d*]{}-electrons relative to the Fermi energy in units of the width of the [*d*]{}-level, $\Gamma$, which arises due to [*s*]{}-[*d*]{} hybridization, ${E_i^0}/{\Gamma}$. The second parameter represents the Coulomb repulsion on a site, again relative to $\Gamma$, ${U_i}/{\Gamma}$. These parameters are determined by reproducing the DFT calculated magnetic moment and number of [*d*]{}-electrons of a single Fe-atom in the Cu(001) substrate. The values obtained are ${U}/{\Gamma}=13$ and ${E^0}/{\Gamma}=-12$. The model also includes a dimensionless hopping parameter ${V_{ij}}/{\Gamma}$, describing direct coupling of [*d*]{}-electrons of atoms [*i*]{} and [*j*]{} as well as the indirect [*d*]{}-[*s*]{}([*p*]{})-[*d*]{} coupling via the conductivity band. Since both contributions decrease with the distance between the atoms, hopping parameters are a measure of the interatomic distances and relate to the geometry of the cluster. Here, only one value of this parameter needs to be chosen for the hydrogen free island. We have used the value ${V_{ij}}/{\Gamma}=0.39$ in order to reproduce the magnetic properties obtained in the DFT calculation. The model for the embedded Fe$_3$ cluster, where all atoms are equivalent, therefore, contains only three dimensionless parameters: ${E^0}/{\Gamma}$, ${U}/{\Gamma}$ and ${V}/{\Gamma}$. Any one of these three parameters could in principle change upon hydrogenation. We will, however, search for the simplest change in model parameters that can reproduce reasonably well the DFT results discussed above. The effect of hydrogen adsorption in AA model {#sec:IncrInHopPar} --------------------------------------------- The LDOS shown in Fig. 4 shows that there is not an appreciable change in the width of the [*d*]{}-band upon hydrogenation. The parameter $\Gamma$ is, therefore, taken to be unaffected. The position of the [*d*]{}-level relative to the Fermi energy, ${E_i^0}/{\Gamma}$, and the Coulomb repulsion of [*d*]{}-electrons localized on site [*i*]{}, ${U_i}/{\Gamma}$, appear from the DFT calculations also not to be affected much by the presence of H-adatoms. The presence of the hydrogen [*s*]{}-state will, however, change the hopping parameters ${V_{ij}}/{\Gamma}$. Indeed, hydrogen attracts some electrons thus leading to a local increase in electron density at the adsorption site. This in turn increases indirect coupling between [*d*]{}-electrons of neighboring Fe atoms through the conductivity band. Given the localized effect of the H-adatoms, we have chosen to let an H-atom at a Y-site increase the hopping parameter between the two Fe-atoms it is bonded to. An H-atom at an X-site increases the two hopping parameter between the neighboring Fe-atom and each of its two neighbors. An increase in the hopping parameters ${V_{ij}}/{\Gamma}$ is indeed found to decrease the magnetic moment of the Fe-atoms in the cluster. A rough fit to the DFT data gives an increase, $\Delta V$, in the hopping parameter, $V_{ij}$, for each H-atom bound to either one or both Fe-atoms $i$ and $j$, with $\Delta V/V = 0.25$. The calculated magnetization with these parameters in the AA model is compared with the DFT results in Fig. 5. ![The total magnetic moment of Fe$_3$H$_n$ cluster as a function of the number of H-adatoms in X- and Y-sites. Crosses correspond to calculations with the AA-model [@AAmodel] where the adsorption of each H-adatom increases the value of hopping parameters of the neighboring Fe-atoms by $\Delta V$ where $\Delta V/V = 0.25$. Circles show results of DFT calculations.[]{data-label="fig:5"}](fig5.pdf){width="70.00000%"} The correspondence between the results of the AA-model and the DFT calculations is good especially considering the simplicity of the parameter adjustment. However, significant deviations occur when only a few H-adatoms are present. Most likely, other parameters of the model should also be adjusted slightly to reproduce more closely the effect of the hydrogen adsorption. A more elaborate fitting procedure could be carried out, but here we focus on the larger trends and relatively high coverage. Conclusion {#sec:conclusion} ========== The DFT calculations presented here have shown that the adsorption of H-atoms on a Fe-atom cluster embedded in a Cu(111) substrate reduces the total magnetic moment of the cluster. The strength of the effect depends on the adsorption site of the H-atom, in that an H-atom bound to a single Fe-atom leads to a stronger reduction in magnetic moment than an H-atom bound to two Fe atoms. The Bader decomposition of the charge density shows that an H-atom draws ca. 0.4 electrons from the Fe-atom(s) they are bound to as well as the substrate. An analysis of the LDOS of [*d*]{}-electrons for individual Fe-atoms shows that the reduction of magnetic moment is primarily due to an increase in the occupation of minority-spin states, which are near the Fermi level. The DFT results can be reproduced quite well with a simple, AA model where the adsorption of H-atom increases the indirect coupling between [*d*]{}-states via the conduction band. A 25% increase in the relevant $V_{ij}$ hopping parameters of the model upon hydrogen adsorption gives good agreement with the magnetic moments of the Fe-atoms obtained from the DFT calculations. Once the AA model has been parametrized, it can be used to simulate much larger systems than can be studied by DFT calculations, even systems with thousands of atoms. Furthermore, a recently developed harmonic transition state rate theory for magnetic transitions [@bessarab_12] can be used to analyze size and shape dependence of thermal stability of magnetic states of nano-islands on surfaces, as has been demonstrated for Fe islands on W(110) [@bessarab_13]. These kinds of calculations can be performed using the AA model [@bessarab_13b] and the effect of hydrogen adsorption on thermal stability of magnetic states assessed for a wide range in island size and shape. This will be the subject of future work. We thank Dr. Andri Arnaldsson for help with the DFT calculations. This work was supported by the Icelandic Research Fund, the Nordic Energy Research fund and the Academy of Finland. [25]{} G. J. Mankey, M. T. Kief, F. Huang, and R. F. Willis, [*J. Vac. Sci. Technol. A*]{} [**11**]{}, 2034 (1993). P. E. M. Siegbahm, M. R. A. Blomberg, and Jr. C. W. Bauschlicher, [*J. Chem. Phys.*]{} [**81**]{}, 1373 (1984). G. Granucci and M. Perisco, [*Chem. Phys.*]{} [**167**]{}, 121 (1992). F. Máca, A.B. Shick, J. Redinger, R. Podloucky and P. Weinberger, [*Czech. J. Phys.*]{} [**53**]{}, 33 (2003). N. O. Jones, M. R. Beltran, and S. N. Khanna, [*Phys. Rev. B*]{} [**70**]{}, 165406 (2004). C. Ashman, S.N. Khanna, and M.R. Pederson, [*Chem. Phys. 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--- abstract: 'Many systems that exhibit nonmonotonic behavior have been described and studied already in the literature. The general notion of nonmonotonic reasoning, though, has almost always been described only negatively, by the property it does not enjoy, i.e. monotonicity. We study here general patterns of nonmonotonic reasoning and try to isolate properties that could help us map the field of nonmonotonic reasoning by reference to positive properties. We concentrate on a number of families of nonmonotonic consequence relations, defined in the style of Gentzen [@Gent:69]. Both proof-theoretic and semantic points of view are developed in parallel. The former point of view was pioneered by D. Gabbay in [@Gabbay:85], while the latter has been advocated by Y. Shoham in [@Shoham:87]. Five such families are defined and characterized by representation theorems, relating the two points of view. One of the families of interest, that of preferential relations, turns out to have been studied by E. Adams in [@Adams:75]. The [*preferential*]{} models proposed here are a much stronger tool than Adams’ probabilistic semantics. The basic language used in this paper is that of propositional logic. The extension of our results to first order predicate calculi and the study of the computational complexity of the decision problems described in this paper will be treated in another paper.' author: - 'Sarit Kraus[^1],  Daniel Lehmann[^2] and Menachem Magidor[^3]' title: 'Nonmonotonic Reasoning, Preferential Models and Cumulative Logics [^4]' --- =.080in Introduction ============ Nonmonotonic reasoning ---------------------- Nonmonotonic logic is the study of those ways of inferring additional information from given information that do not satisfy the monotonicity property satisfied by all methods based on classical (mathematical) logic. In Mathematics, if a conclusion is warranted on the basis of certain premises, no additional premises will ever invalidate the conclusion. In everyday life, however, it seems clear that we, human beings, draw sensible conclusions from what we know and that, on the face of new information, we often have to take back previous conclusions, even when the new information we gathered in no way made us want to take back our previous assumptions. For example, we may hold the assumption that most birds fly, but that penguins are birds that do not fly and, learning that Tweety is a bird, infer that it flies. Learning that Tweety is a penguin, will in no way make us change our mind about the fact that most birds fly and that penguins are birds that do not fly, or about the fact that Tweety is a bird. It should make us abandon our conclusion about its flying capabilities, though. It is most probable that intelligent automated systems will have to do the same kind of (nonmonotonic) inferences. Many researchers have proposed systems that perform such nonmonotonic inferences. The best known are probably: negation as failure [@Clark:78], circumscription [@McCarthy:80], the modal system of [@McDer:80], default logic [@Reiter:80], autoepistemic logic [@Moo:84] and inheritance systems [@Tour:86]. Each of those systems is worth studying by itself, but a general framework in which those many examples could be compared and classified is missing. We provide here a first attempt at such a general framework, concentrating on properties that are or should be enjoyed by at least important families of nonmonotonic reasoning systems. An up-to-date survey of the field of nonmonotonic reasoning may be found in [@Reiter:87]. Nonmonotonic consequence relations ---------------------------------- The idea that the best framework to study the deduction process is that of consequence relations dates back to A. Tarski [@Tar:30], [@Tar:30a] and [@Tar:35] (see [@Tar:56] for an English translation) and G. Gentzen [@Gent:32] (see [@Gent:69] for an English translation and related papers). For an up-to-date view on monotonic consequence relations, the reader may consult [@Avron:87]. Tarski studied the consequences of arbitrary sets of formulas whereas Gentzen restricted himself to finite such sets. In the presence of compactness, the difference between the two approaches is small for monotonic consequence relations. For nonmonotonic relations, many different notions of compactness come to mind, and the relation between Tarski’s infinitistic approach and Gentzen’s finitistic approach is much less clear. We develop here a finitistic approach in the style of Gentzen. In [@Mak:89], D. Makinson developed, in parallel with and independently from our effort, an infinitistic view of nonmonotonic consequence relations. Later efforts in this direction, by M. Freund and D. Lehmann [@FLM:89], have benefited from the results presented here. D. Gabbay [@Gabbay:85] was probably the first to suggest to focus the study of nonmonotonic logics on their consequence relations. This is a bold step to take since some of the nonmonotonic systems mentioned above were not meant to define a consequence relation, as was soon noticed by D. Israel in [@Isr:80]. D. Gabbay asked the question: what are the minimal conditions a consequence relation should satisfy to represent a bona fide nonmonotonic [*logic*]{}? He proposed three: reflexivity (see equation \[ru:ref\] in section \[subsec:cum\]), cut (see equation \[ru:cut\]) and weak monotonicity (see equation \[ru:caut\]). Weak monotonicity has, since, been renamed cautious monotonicity by D. Makinson [@Mak:89] and we shall follow this last terminology, notwithstanding the fact that D. Makinson has now opted for the term [*cumulative monotony*]{}. D. Gabbay argued for his three conditions on proof-theoretic grounds but provided no semantics against which to check them. He also assumed a $poor$ underlying language for propositions, a language without classical propositional connectives. In [@Mak:89], D. Makinson proposed a semantics for Gabbay’s logic and proved a completeness result, for a [*poor*]{} language. His models have a definitely syntactic flavor, whereas the models presented here seem more truly semantic and more easily suggest rules of inference. Independently, Y. Shoham in [@Shoham:88] and [@Shoham:87] proposed a general model theory for nonmonotonic inference. He suggested models that may be described as a set of worlds equipped with a [*preference*]{} relation: the preference relation is a partial order and a world $v$ is preferable, in the eyes of the reasoner, to some other world $w$ if he considers $v$ to be more $normal$ than $w$. One would then, in the model, on the basis of a proposition $\alpha$, conclude, defeasibly, that a proposition $\beta$ is true if all worlds that satisfy $\alpha$ and are [*most normal*]{} among worlds satisfying $\alpha$ also satisfy $\beta$. Shoham claimed that adequate semantics could be given to known nonmonotonic systems by using such a preference relation. He assumed a [*rich*]{} underlying language for propositions, containing all classical propositional connectives. The idea that nonmonotonic deduction should be modeled by some [*normality*]{} relation between worlds is very natural and may be traced back to J. McCarthy. It appears also in relation with epistemic logic in [@HalpMoses:84]. One of the conclusions of this paper will be that none of the nonmonotonic systems defined so far in the literature, except those based on conditional logic described in [@Del:87], [@Del:88] and [@PearlGeff:88], may represent all nonmonotonic inference systems that may be defined by preferential models. The framework of preferential models, therefore, has an expressive power that cannot be captured by negation as failure, circumscription, default logic or autoepistemic logic. We do not claim that all this expressive power is needed, but will claim that the systems mentioned above lack expressive power. The main point of this work, therefore, is to characterize the consequence relations that can be defined by models similar to Shoham’s in terms of proof-theoretic properties. To this end Gabbay’s conditions have to be augmented. The class of models corresponding exactly to Gabbay’s conditions is also characterized. The elucidation of the relations between proofs and models that is achieved in this paper will allow for the design of decision procedures tuned to different restrictions on the language of propositions or the knowledge bases. Such decision procedures (or heuristics) could be the core of automated engines of sensible inferences. This paper will not propose any specific system of nonmonotonic reasoning. Important steps towards such a system, taken after obtaining the results reported here but before the final redaction of this paper, are reported in [@L:88], [@LMTR:88] and [@Leh:89]. At this point it could be useful to state the philosophy of this paper concerning the relative importance of proof-theory and semantics. We consider, in this paper, the axiomatic systems as the main object of interest (contrary to the point of view expressed in [@Lewis:73] for example). The different families of models described in this paper and that provide semantics to the axiomatic systems are not considered to be an ontological justification for our interest in the formal systems, but only as a technical tool to study those systems and in particular settle questions of interderivability and find efficient decision procedures. Preliminary versions of the material contained in this paper appeared in [@KL:88] and [@KLMTR:88]. Conditional logic ----------------- In this subsection, the relation between our work and conditional logic will be briefly surveyed. Since the link, we claim, is mainly at the level of the formal systems and not at the semantic level, the reader uninterested in conditional logic may easily skip this subsection. This work stems from a very different motivation than the vast body of work concerned with conditional logic and its semantics, (see in particular [@Stal:68],[@Lewis:73] and [@Lewis:71]) which is surveyed in [@Nute:84]. Two main differences must be pointed at. The first difference is that conditional logic considers a binary intensional connective that can be embedded inside other connectives and even itself, whereas we consider a binary relation symbol that is part of the meta-language. The second difference is that the semantics of the conditional implication of conditional logic is essentially different from ours. In conditional logic the formula is interpreted to mean [*if $\alpha$ were (or was) true and the situation were as close as possible, under this hypothesis, to what it really is, then $\beta$ would be true*]{}. For us  means that  is a good enough reason to believe , or that  is a plausible consequence of . The main difference is that conditional logic refers implicitly to the [*actual*]{} state of the world whereas we do not. M. Ginsberg’s [@Gin:86] proposal to harness conditional logic to nonmonotonic reasoning was clearly set with the former semantics in mind, and that explains our disagreements concerning the desirability of certain rules, e.g., the rule of [**Rational Monotonicity**]{} (see equation (\[Rat:mon\])). One of the logical systems, [**P**]{}, studied in this paper turns out to be the flat (i.e. non-nested) fragment of a conditional logic studied by J. Burgess in [@Burgess:81] and by F. Veltman in [@Velt:86]. Because of their richer language, the semantics proposed in those papers are more complex than ours: a ternary relation of accessibility between worlds is used in place of our binary preference relation. Moreover, the semantics of J. Burgess are quite different from ours in some other aspects; our semantics are closer to F. Veltman’s (private communication from J. van Benthem) and to those studied by J. van Benthem in [@vBent:84]. There are some connections between one of our completeness proofs and theirs, but the restricted language considered here simplifies the models and the proof a great deal. Our completeness result cannot be derived from the completeness result of [@Burgess:81] since the latter concerns a extended language and it is not clear that a proof in the extended language may be translated in the restricted one. This very fragment had been considered by E. Adams in [@Adams:75] (see also [@Adams:66] for an earlier version and motivation). E. Adams’ purpose was to propose probabilistic semantics for indicative conditionals and not the study of nonmonotonic logics. Recently J. Pearl and H. Geffner [@PearlGeff:88] have built upon E. Adams’ logics, our system [**P**]{}, and his motivation in an effort to provide a system for nonmonotonic reasoning. For a gentle introduction, see chapter 10 of [@Pearl:88]. The semantics proposed here are not probabilistic. Probabilistic semantics that are equivalent with a restricted family of models (ranked models) will be described elsewhere. The preferential models presented in this paper provide a much sharper understanding of the system [**P**]{} than can obtained by Adams’ methods. Plan of this paper ------------------ This paper first describes the syntax proposed and compares it to more classical nonmonotonic systems. Five logical systems and families of models are then presented in turn and five soundness and completeness results are proven. The first system, [**C**]{}, corresponds to D. Gabbay’s proposal. The second, stronger, system, [**CL**]{}, includes a rule of inference that seems original, and corresponds to models that seem to be more natural. None of those systems above assumes, in any essential way the existence of the classical logical connectives, if one allows a finite set of formulas to appear on the left of our symbol . The systems below assume the classical connectives. The third, stronger, system, [**P**]{}, is the central system of this paper. It has particularly appealing semantics. The fourth system, [**CM**]{}, is stronger than [**CL**]{} but incomparable with [**P**]{}. It provides an example of a monotonic system that is weaker than classical logic. The last one of those systems, [**M**]{}, is stronger than all previous systems and equivalent to classical propositional logic. The language, comparison with other systems {#sec:lan} =========================================== Our language {#subsec:lan} ------------ The first step is to define a language in which to express the basic propositions. We shall assume that a set $L$ of well formed formulas (thereafter formulas) is given. It is very important, from section \[sec:pref\] on, to assume that $L$ is closed under the classical propositional connectives. They will be denoted by $\neg , \vee , \wedge , \rightarrow$ and $\leftrightarrow$. Negation and disjunction will be considered as the basic connectives and the other ones as defined connectives. The connective $\rightarrow$ therefore denotes material implication. Small greek letters will be used to denote formulas. Since no rule relating to the quantifiers will be discussed in this paper, the reader may as well think of $L$ as the set of all propositional formulas on a given set of propositional variables. With the language $L$, we assume semantics given by a set ${\cal U}$, the elements of which will be referred to as worlds, and a binary relation of [*satisfaction*]{} between worlds and formulas. The set ${\cal U}$ is the universe of reference, it is the set of all worlds that we shall consider possible. If $L$ is the set of all propositional formulas on a given set of propositional variables,  is a subset of the set of all assignments of truth values to the propositional variables. We reserve to ourselves the right to consider universes of reference that are strict subsets of the set of all models of $L$. In this way we shall be able to model [*strict*]{} constraints, such as [*penguins are birds*]{}, in a simple and natural way, by restricting  to the set of all worlds that satisfy the material implication . Typical universes of reference are given by the set of all propositional worlds that satisfy a given set of formulas. We shall assume that the satisfaction relation behaves as expected as far as propositional connectives are concerned. If $u\in {\cal U}$ and $\alpha, \beta \in L$ we write $u \models \alpha$ if $u$ satisfies $\alpha$ and assume:\ 1) $u \models \neg \alpha$ iff $u \not \models \alpha$.\ 2) $u \models \alpha \vee \beta$ iff $u \models \alpha$ or $u \models \beta$. The notions of satisfaction of a set of formulas, validity of a formula and satisfiability of a set of formulas are defined as usual. We shall write $\models \alpha$ if $\alpha$ is valid, i.e. iff $\forall u \in {\cal U}$, $u \models \alpha$, and write for . We shall also make the following [**assumption of compactness**]{}[^5]: a set of formulas is satisfiable if all of its finite subsets are. Classical theorems of compactness show that if we take $L$ to be a propositional calculus or a first order predicate calculus and  to be the set of all models that satisfy a given set of formulas, then the assumption of compactness described above is satisfied. Notice that the set of valid formulas, in our sense, is not, in general, closed under substitutions. All that is done in the sequel depends on the choice of $L$ and ${\cal U}$, though we shall often forget this dependence. For this work, the basic language $L$ may well be fixed, but we shall sometimes have to consider different universes of reference. As noticed above, if $\Gamma$ is a set of formulas then the subset of ${\cal U}$ that contains only the worlds that satisfy $\Gamma$ (this set of worlds will be denoted by ${\cal U}_\Gamma$) is a suitable universe. If $\alpha$ and $\beta$ are formulas, then the pair (read [*if $\alpha$, normally $\beta$*]{}, or [* is a plausible consequence of* ]{}) is called a conditional assertion (assertion in short). The formula  is the antecedent of the assertion,  is its consequent. The meaning we attach to such an assertion, and against which the reader should check the logical systems to be presented in the upcoming sections, is the following: if $\alpha$ is true, I am willing to (defeasibly) jump to the conclusion that $\beta$ is true. Our choice, then, is to look at [*normally*]{} as some binary notion. It is clear that efforts to understand [*normally*]{} as some unary notion, e.g. translating [*if , normally* ]{} as or as for some unary modal operator cannot be expressive enough. [*Consequence relations*]{} are sets of conditional assertions. Not all such sets, though, seem to be worthy of that name and our use of the term for any such set is running against a fairly well-established terminology. The term [*conditional assertion*]{} is taken from [@Scott:71] (p. 417). We hope that, by considering nonmonotonic consequence as a meta-notion, but allowing basic propositions on a rich language, we strike at the right language. It allows a new approach of questions about computational complexity (see [@Lewis:74] for some general decidability results), but this is left for future work. Pragmatics ---------- We shall now briefly sketch why we think that the study of nonmonotonic consequence relations will be a benefit to the field of automated nonmonotonic reasoning. The queries one wants to ask an automated knowledge base are formulas (of $L$) and query  should be interpreted as: [*is  expected to be true*]{}? To answer such a query the knowledge base will apply some inference procedure to the information it has. We shall now propose a description of the different types of information a knowledge base has. The first type of information (first in the sense it is the more stable, changes less rapidly) is coded in the universe of reference , that describes both hard constraints (e.g. dogs are mammals) and points of definition (e.g. youngster is equivalent to not adult). Equivalently, such information will be given by a set of formulas defining  to be the set of all worlds that satisfy all the formulas of this set. The second type of information consists of a set of conditional assertions describing the soft constraints (e.g. birds normally fly). This set describes what we know about the way the world generally behaves. This set of conditional assertions will be called the knowledge base, and denoted by [[**K**]{}]{}. The third type of information describes our information about the specific situation at hand (e.g. it is a bird). This information will be represented by a formula. Our decision to consider the first type of information as a separate type is not the only possible way to go. One could, equivalently, treat a formula  of the first type as the conditional assertion . One could also have decided to introduce all information of the third type as information of the first type. Our inference procedure will work in the following way, to answer query . In the context of the universe of reference , it will try to deduce (in a way that is to be discovered yet) the conditional assertion  from the knowledge base [[**K**]{}]{}. This is a particularly elegant way of looking at the inference process: the inference process deduces conditional assertions from sets of conditional assertions. Clearly any system of nonmonotonic reasoning may be considered in this way. So, we may look at circumscription, default logic and other systems as mechanisms to deduce conditional assertions from sets of conditional assertions. We shall now briefly investigate the expressive power of some of those systems in this light. Expressiveness of our language ------------------------------ We shall now compare the expressive power of the language proposed here, i.e. conditional assertions, to that of previous approaches. Our purpose is to show that circumscription, autoepistemic logic and default logic all suffer from fundamental weaknesses, either in their expressive capabilities or in their treatment of conditional information. Let ,  and  be formulas. We shall concentrate on the comparison of two different conditional assertions. The conditional assertion  is . The conditional assertion  is , i.e. . To simplify matters we shall just treat the special case when the formula  is a tautology. In this case  is  and  is . The assertion  expresses that [*if $\alpha$, normally $\beta$*]{}. Assertion  expresses that [*Normally, if $\alpha$ is true then $\beta$ is true*]{}. Those assertions have very different meanings, at least when $\alpha$ is normally false. Assertion  says that in this exceptional case when $\alpha$ is true, one also expects  to be true. Assertion , on the other hand, is automatically verified if $\alpha$ is normally false. In any case it seems that it is perfectly possible that  does not say anything about cases when  is true (if these are exceptional). Take for example $\alpha$ to be [*it is a penguin*]{} and  to be [*it flies*]{}. If we talk about birds, it seems perfectly reasonable to accept  which says that [*normally, either it is not a penguin or it flies*]{}, since normally birds fly (and normally birds are not penguins, but this remark is not necessary). Nevertheless, one should be reluctant to accept  which says [*penguins normally fly*]{}. It seems clear to us, then, that  and  have different meanings and that  does not follow from . We agree with Y. Shoham, and this opinion will be supported in the sequel, to say that  should follow from , but we do not have to argue that case now. In the main system to be presented in this paper, [**P**]{}, the assertion  is strictly stronger than . In the weaker systems [**C**]{} and [**CT**]{},  and  are incomparable. In [**CM**]{},  is strictly stronger than , and this is one of the reasons we shall reject it as a system of nonmonotonic reasoning. It is only in [**M**]{}, which is equivalent to classical logic, that  and  are equivalent. Let us consider now the expression of  and , first using circumscription. For circumscription,  would be expressed as: . In fact, since there would probably be a number of different abnormalities floating around, we probably should have written: , but this is not significant. On the other hand  would be written as: . One immediately notices that the two formulations are logically equivalent. We conclude that circumscription would need some additional mechanism to distinguish between  and . In practise, the user of circumscription would give different priorities (relative to the priorities of abnormalities of the other assertions of the knowledge base), to the two abnormalities considered here; but there is no standard procedure to determine priorities. Let us now use autoepistemic logic. The assertion  would be expressed as: . On the other hand  would be expressed as: . Since the modality  is interpreted as for some epistemic modality  it satisfies: . We immediately notice that, for autoepistemic logic,  is strictly stronger than . This is not what we expect. Let us try default logic now. The natural translation of  in default logic would be the normal default: , whose meaning is [*if $\alpha$ has been concluded to be true and $\beta$ is consistent with what has been concluded so far, conclude that $\beta$ is true*]{}. The assertion  would be expressed as: , which means that in any situation in which $\alpha {\rightarrow}\beta$ is consistent, one should (or could) conclude this last formula to be true. We immediately see that in all situations in which $\alpha$ has been already concluded to be true, both defaults act exactly in the same way, which seems very questionable. In situations in which $\alpha$ has been concluded to be false, the first default is inapplicable, whereas the second default may be applied but yields a trivial result (we do not get any new information from applying it). Again, both defaults are equivalent, but, in this case, this seems to fit our intuition. In situations in which neither $\alpha$ nor its negation have been concluded, the first default cannot be applied. For the second default, in certain situations it cannot be applied either, but in others it may be applied and yields non trivial conclusions. We conclude from this study that in some situations both defaults are equivalent, in others the second is more powerful than the first one. Again this is not what we expected. A particularly spectacular case of this problem occurs when $\beta$ is a logical contradiction. The assertion has a very clear meaning: it says [*if , normally anything*]{}. It expresses the very strong statement that we are willing to disconsider completely the possibility of  being true. To see that this may express very useful information, just think of  as [*I am the Queen of England*]{}. Most people would probably be willing to have in their personal knowledge base. As remarked above, this corresponds to restricting  to those worlds that do not satisfy . Now, the translation, as a normal default, of such an assertion, which is: , is never applicable since [**false**]{} is never consistent with anything. Therefore this default gives no information at all. Somehow, all the strength of our assertion has been lost in the translation. We hope to have convinced the reader that one should look for formalisms in which the distinction between  and  is clear and understandable. Cumulative reasoning ==================== Cumulative consequence relations {#subsec:cum} -------------------------------- We shall, first, study the weakest of our logical systems. It embodies what we think, at this moment, in agreement with D. Gabbay [@Gabbay:85], are the rock-bottom properties without which a system should not be considered a logical system. This appreciation probably only reflects the fact that, so far, we do not know anything interesting about weaker systems. The order of the exposition, roughly from weaker to stronger systems, is aimed at minimizing repetitions: rules that may be derived in weaker systems may also be derived in stronger ones. We shall name this system [**C**]{}, for [*cumulative*]{}. It is closely related to the cumulative inference studied by D. Makinson in [@Mak:89], and seems to correspond exactly, to what D. Gabbay proposed in [@Gabbay:85]. The system [**C**]{} consists of a number of inference rules and an axiom schema. \[def:cumrel\] A consequence relation  is said to be [*cumulative*]{} iff it contains all instances of the [**Reflexivity**]{} axiom and is closed under the inference rules of [**Left Logical Equivalence**]{}, [**Right Weakening**]{}, [**Cut**]{} and [**Cautious Monotonicity**]{} that will be described below. We shall now describe and discuss the axioms and rules mentioned above and some derived rules. The purpose of the discussion is to weight the meaning of the axioms and rules when the relation is interpreted as [*if $\ldots$ , normally $\ldots$*]{}. $$\label{ru:ref} \alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\alpha \hspace {2cm} {\rm ({\bf Reflexivity})}$$ [**Reflexivity**]{} seems to be satisfied universally by any kind of reasoning based on some notion of consequence. Relations that do not satisfy it, probably express some notion of theory change. It corresponds to the axiom ID of conditional logic. The next two rules express the influence of the underlying logic, defined by the universe ${\cal U}$, on the notion of plausible consequence. Their role is similar to that of the rules of consequence of [@Hoare:69]. $$\label{ru:lle} {{\models \alpha \leftrightarrow \beta \ \ , \ \ \alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\gamma} \over {\beta {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\gamma}} \hspace {2cm} {\rm ({\bf Left \ Logical \ Equivalence}) }$$ [**Left Logical Equivalence**]{} expresses the requirement that logically equivalent formulas have exactly the same consequences and corresponds to rule RCEA of conditional logic. The consequences of a formula should depend on its meaning, not on its form. In the presence of the other rules of [**C**]{}, it could have been weakened to: from conclude . The next rule, [**Right Weakening**]{} expresses the fact that one must be ready to accept as plausible consequences all that is logically implied by what one thinks are plausible consequences. In other words, plausible consequences are closed under logical consequences. It corresponds to the rule RCK of conditional logic. $$\label{ru:rweak} {{\models \alpha {\rightarrow}\beta\ \ , \ \ \gamma {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\alpha } \over {\gamma {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\beta}} \hspace {2cm} {\rm ({\bf Right \ Weakening})}$$ [**Right Weakening**]{} obviously implies that one may replace logically equivalent formulas by one another on the right of the  symbol. [**Reflexivity**]{} and [**Right Weakening**]{} already imply that  if . All nonmonotonic systems proposed so far in the literature satisfy [**Reflexivity**]{}, [**Left Logical Equivalence**]{} and [**Right Weakening**]{}. Our next rule is named [**Cut**]{} because of its similarity to Gentzen’s [*Schnitt*]{}. $$\label{ru:cut} {{\alpha \wedge \beta {\mbox{ ${{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim$ }}\gamma \ \ , \ \ \alpha {\mbox{ ${{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim$ }}\beta} \over {\alpha {\mbox{ ${{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim$ }}\gamma \;}} \hspace {2cm} {\rm ({\bf Cut})}$$ It expresses the fact that one may, in his way towards a plausible conclusion, first add an hypothesis to the facts he knows to be true and prove the plausibility of his conclusion from this enlarged set of facts and then deduce (plausibly) this added hypothesis from the facts. This is a valid way of reasoning in monotonic logic, and, as will be seen soon, its validity does not imply monotonicity, therefore it seems to us quite reasonable to accept it. Its meaning, it should be stressed, is that a plausible conclusion is as secure as the assumptions it is based on. Therefore it may be added (this is the origin of the term cumulative) into the assumptions. There is no loss of confidence along the chain of derivations. One may well be unwilling to accept such a principle and think that, on the contrary, no conclusion of a derivation is ever as secure as the assumptions. Indeed, recently, D. Gabbay [@Gab:pers] suggested to replace [**Cut**]{} by a weaker rule. In this paper, we shall study only systems that validate [**Cut**]{}. Our conclusion is that there are many interesting nonmonotonic systems that satisfy [**Cut**]{}. It should be mentioned that some probabilistic interpretations invalidate [**Cut**]{} (Adams’ validates it), e.g. interpreting a conditional assertion  as meaning that the corresponding conditional probability is larger than some $q < 1$. It is easy to see that circumscription satisfies [**Cut**]{}, at least when all models that have to be considered are finite. In [@Mak:89], D. Makinson shows that default logic satisfies [**Cut**]{} too. The following example should help convince the reader to endorse [**Cut**]{}. Suppose we tell you [*we expect it will be raining tonight*]{} and [*if it rains tonight, normally Fireball should win the race tomorrow*]{}. Wouldn’t you conclude that we think that [*normally, Fireball should win the race tomorrow*]{}? Our last rule, named [**Cautious Monotonicity**]{}, is taken from D. Gabbay [@Gabbay:85]. It corresponds to axiom A3 of Burgess’ system ${\cal S}$ in [@Burgess:81]. The same property is named [*triangulation*]{} in [@PearlGeff:88]. $$\label{ru:caut} {{\alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\beta \ \ , \ \ \alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\gamma} \over {\alpha \wedge \beta {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\gamma}}\hspace {2cm} {\rm ({\bf Cautious \ Monotonicity})}$$ [**Cautious Monotonicity**]{} expresses the fact that learning a new fact, the truth of which could have been plausibly concluded should not invalidate previous conclusions. It is a central property of all the systems considered here. The origin of the term [*cautious monotonicity*]{} will be explained in section \[subsec:m\]. The probabilistic semantics that invalidates [**Cut**]{} also invalidates [**Cautious Monotonicity**]{}. In [@Mak:89], D. Makinson showed that default logic, even when defaults are normal, does not always satisfy [**Cautious Monotonicity**]{}. Circumscription, though, satisfies it, at least when all models considered are finite. What are our reasons to accept [**Cautious Monotonicity**]{}? On the general level, D. Gabbay’s argumentation seems convincing: if  is reason enough to believe  and also to believe , then  and  should also be enough to make us believe , since  was enough anyway and, on this basis,  was expected. From a pragmatic point of view [**Cautious Monotonicity**]{} is very important since we typically learn new facts and we would like to minimize the updating we have to make to our beliefs. [**Cautious Monotonicity**]{} and [**Cut**]{} together tell us, as will be made clear in lemma \[le:cum\], that if the new facts learned were expected to be true, nothing changes in our beliefs. This will help minimizing the updating. From a semantic point of view, we want to argue the case for [**Cautious Monotonicity**]{} on the following example. Suppose we tell you [*we expect it will be raining tonight*]{} and [*normally, Fireball should win the race tomorrow*]{}. Wouldn’t you conclude that we think that [*even if it rains tonight, normally Fireball should win the race tomorrow*]{}? \[le:cum\] The rules of [**Cut**]{} and [**Cautious Monotonicity**]{} may be expressed together by the following principle: if  then the plausible consequences of  and of coincide. Let us now consider some rules that may be derived in [**C**]{}. Derived rules of [**C**]{} {#subsec:cumder} -------------------------- The first rule corresponds to CSO of conditional logic and expresses the fact that two propositions that are plausible consequences of each other, have exactly the same plausible consequences. $$\label{ru:equ} {{\alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\beta \ \ , \ \ \beta {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\alpha \ \ , \ \ \alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\gamma} \over {\beta {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\gamma}} \hspace{1.5cm}{\rm ({\bf Equivalence})}$$ The second rule corresponds to CC of conditional logic and expresses the fact that the conjunction of two plausible consequences is a plausible consequence. $$\label{ru:and} {{\alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\beta \ \ , \ \ \alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\gamma} \ \ \ \over {\alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\beta \wedge \gamma}} \hspace{3cm}{\rm ({\bf And})}$$ The third rule amounts to [*modus ponens*]{} in the consequent. $$\label{ru:mpcons} {{\alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\beta {\rightarrow}\gamma \ , \ \alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\beta} \over {\alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\gamma}} \hspace{2cm}{\rm ({\bf MPC})}$$ The fourth rule is perhaps less expected and brought up here to show that [**C**]{} is not as weak as one could think. It will be put to use in section \[subsec:pref.char\]. $$\label{Or:Trans} {{\ \ {{\alpha \vee \beta} {\mbox{ ${{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim$ }}{\alpha}} \ \ , \ \ {\alpha {\mbox{ ${{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim$ }}\gamma}} \ \ \ \over {{\alpha \vee \beta} {\mbox{ ${{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim$ }}\gamma}}$$ \[le:dercum\] [**Equivalence**]{}, [**And**]{}, [**MPC**]{} and (\[Or:Trans\]) are derived rules of the system [**C**]{}. -1000.5 pt[**Proof:** ]{} For [**Equivalence**]{}, use first [**Cautious Monotonicity**]{} to show that , then [**Left Logical Equivalence**]{} to get and then conclude by [**Cut**]{}. For [**And**]{}, first use [**Cautious Monotonicity**]{} to show . Then, since , we have: . Using [**Cut**]{} we conclude that and the desired conclusion is obtained by one more use of [**Cut**]{}. For [**MPC**]{}, use [**And**]{} and [**Right Weakening**]{}. For (\[Or:Trans\]), remark that, since , we have . This, with the hypotheses, enables us to apply [**Equivalence**]{} and conclude. 8.5pt Monotonicity {#subsec:m} ------------ We shall now justify the term [**Cautious Monotonicity**]{} and introduce four new rules. They cannot be derived in [**C**]{}. The first rule is [**Monotonicity**]{}, or [**Left Strengthening**]{}. $$\label{ru:mon} {{\models \alpha {\rightarrow}\beta \ \ , \ \ \beta {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\gamma} \over {\alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\gamma}} \hspace {2cm} {\rm ({\bf Monotonicity})}$$ It is clear that both [**Left Logical Equivalence**]{} and [**Cautious Monotonicity**]{} are special cases of [**Monotonicity**]{}. This explains the name [**Cautious Monotonicity**]{}. The next rule corresponds to the [*easy*]{} half of the deduction theorem. $$\label{ru:HD} {{\alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\beta \rightarrow \gamma} \ \ \ \over {\alpha \wedge \beta {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\gamma}} \hspace{2.2cm}{\rm ({\bf EHD})}$$ The next two rules have been considered by many. $$\label{ru:trans} {{\alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\beta \ \ , \ \ \beta {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\gamma} \ \ \ \over {\alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\gamma}} \hspace{1.3cm}{\rm ({\bf Transitivity})}$$ $$\label{ru:contra} {{\alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\beta} \ \ \ \ \over {\neg \beta {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\neg \alpha}} \hspace{3cm}{\rm ({\bf Contraposition})}$$ It is easy to find apparent counter-examples to each one of the last four rules in the folklore of nonmonotonic reasoning. The next lemmas will explain why. Let us notice that, nevertheless, adding the first three of those rules to the system [**C**]{} leaves us with a system, [**CM**]{}, that is strictly weaker than classical monotonic logic, as will be seen in section \[sec:mon.cum\]. The next lemmas will describe some of the relations between the rules above. \[le:dermon\] In the presence of the rules of [**C**]{}, the rules of [**Monotonicity**]{}, [**EHD**]{}, and [**Transitivity**]{} are all equivalent. -1000.5 pt[**Proof:** ]{} We shall not mention the uses of [**Reflexivity**]{}, [**Left Logical Equivalence**]{} and [**Right Weakening**]{}. [**Monotonicity**]{} implies [**EHD**]{}, using [**And**]{}. [**EHD**]{} implies [**Monotonicity**]{}. [**Monotonicity**]{} implies [**Transitivity**]{}, using [**Cut**]{}. [**Transitivity**]{} implies [**Monotonicity**]{}. 8.5pt \[le:dermon2\] In the presence of [**Left Logical Equivalence**]{} and [**Right Weakening**]{}, [**Contraposition**]{} implies [**Monotonicity**]{}. -1000.5 pt[**Proof:** ]{} Use [**Contraposition**]{}, then [**Right Weakening**]{} and [**Contraposition**]{} again. 8.5pt The results of section \[sec:mon.cum\] show that [**Monotonicity**]{} does not imply [**Contraposition**]{} even in the presence of the rules of [**C**]{}. Since [**Monotonicity**]{} seems counter-intuitive in nonmonotonic systems, the two lemmas above show we should not accept [**EHD**]{}, [**Transitivity**]{} or [**Contraposition**]{}. Cumulative models ----------------- We shall now develop a semantic account of cumulative reasoning, i.e. reasoning using the rules of the system [**C**]{}. We shall define a family of models (without any reference to the rules of [**C**]{}) and show how each model defines a consequence relation. We shall then show that each model of the family defines a cumulative consequence relation (this is a soundness result) and that every cumulative consequence relation is defined by some model of the family (this is a completeness result, or a representation theorem). Let us, first, describe the models informally. A model essentially consists of a set of states (they represent possible states of affairs, including perhaps the state of mind or knowledge of the reasoner) and a binary relation on those states. The relation represents the preferences the reasoner may have between different states: he could for example prefer the states in which he is rich to the ones in which he is poor, and prefer the states in which he knows he is rich to those in which he is rich but does not know about it. More realistically, one could prefer states in which Tweety is a bird and flies to those in which Tweety is a bird but does not fly. The reasoner, described by a model, accepts a conditional assertion  iff all those states that are [*most preferred*]{} among all states satisfying , satisfy . The reader should notice we have not yet said what is a state and what formulas are satisfied by a state. We shall not define further the notion of a state, but suppose that every state is, in a model, labeled with a set of worlds (intuitively the set of all worlds the reasoner thinks are possible in this state). Modal logicians will identify our labels as S5 models. Considering a binary relation on states labeled by sets of worlds, instead of considering a binary relation on sets of worlds, gives us an additional degree of freedom in building models: the same set of worlds may appear at many states (that are not equivalent from the point of view of the binary relation). This additional freedom is vital for the representation theorem to hold, and was missing from Shoham’s account [@Shoham:87]. Some technical definitions are needed first. \[def:asy\] Let $\prec$ be a binary relation on a set $U$. We shall say that $\prec$ is [*asymmetric*]{} iff such that , we have . \[def:min\] Let and $\prec$ a binary relation on $U$. We shall say that is [*minimal*]{} in $V$ iff , . We shall say that is a [*minimum*]{} of $V$ iff such that , . The reader has noticed that, though the last definitions sound familiar in the case the relation $\prec$ is a strict partial order, we intend to use them for arbitrary relations. \[def:smoo\] Let and $\prec$ a binary relation on $U$. We shall say that $P$ is [*smooth*]{} iff , either minimal in $P$, such that or $t$ is itself minimal in $P$. We shall use the following lemma, the proof of which is obvious. \[min:min\] Let $U$ be a set and $\prec$ an asymmetric binary relation on $U$. If $U$ has a minimum it is unique, it is a minimal element of $U$ and $U$ is smooth. \[def:cum.mod\] A [*cumulative*]{} model is a triple , where $S$ is a set, the elements of which are called states, is a function that labels every state with a non-empty set of worlds and $\prec$ is a binary relation on $S$, satisfying the [**smoothness condition**]{} that will be defined below in definition \[def:smoocond\]. The relation $\prec$ represents the reasoner’s preference among states. The fact that $s \prec t$ means that, in the agent’s mind, $s$ is [*preferred*]{} to or more [*natural*]{} than $t$. As will be formally defined below, the agent is willing to conclude $\beta$ from $\alpha$, if all [*most natural*]{} states which satisfy $\alpha$ also satisfy $\beta$. \[def:EM\] Let be as above. If  is a formula, we shall say that $s \in S$ satisfies  and write iff for every world , . The set: of all states that satisfy  will be denoted by . \[def:smoocond\] A triple is said to satisfy the smoothness condition iff, , the set is smooth. The smoothness condition is necessary to ensure the validity of [**Cautious Monotonicity**]{}. It is akin to the [*limit assumption*]{} of Stalnaker [@Stal:68] and Lewis [@Lewis:73], but it is defined in a more general context. Smoothness is the property called, contrary to mathematical usage, [*well-foundedness*]{} in [@Eth:85] and in [@Lif:86]. We shall now describe how a cumulative model defines a consequence relation. \[def:cumcons\] Suppose a cumulative model is given. The consequence relation defined by $W$ will be denoted by  and is defined by: iff for any $s$ minimal in $\widehat{\alpha}$, . \[def:strongcum\] A triple is said to be a [*strong*]{} cumulative model iff 1. the relation $\prec$ is asymmetric 2. for each formula , the set $\widehat\alpha$ has a minimum. It is clear that strong cumulative models are cumulative models, i.e. satisfy the smoothness condition. The definition of cumulative models and the consequence relations they define seems quite natural, i.e. a preference relation on epistemic states, but one should not forget that the preference relation $\prec$ is not required to be a partial order and that in triples (even when the set of states is finite) in which the relation $\prec$ is not a partial order, the smoothness condition is, in general, not an easy thing to check. Characterization of cumulative relations {#subsec:char.cumul} ---------------------------------------- In this section we shall characterize the relation between cumulative consequence relations and cumulative models. The first lemma is obvious. \[remark1\] Let be a cumulative model. For , . \[Cu:Sou\] For any cumulative model $W$, the consequence relation  it defines is a cumulative relation, i.e. all the rules of the system [**C**]{} are satisfied by the relations defined by cumulative models. -1000.5 pt[**Proof:** ]{} The proof is easy and we shall only treat [**Cut**]{} and [**Cautious Monotonicity**]{}. The smoothness condition is needed only for dealing with [**Cautious Monotonicity**]{}. For [**Cut**]{}, suppose all minimal elements of $\widehat\alpha$ satisfy  and all minimal elements of satisfy . Any minimal element of $\widehat\alpha$ satisfies  and therefore satisfies $\alpha \wedge \beta$. Since it is minimal in $\widehat\alpha$ and , it is also minimal in . For [**Cautious Monotonicity**]{}, the smoothness condition is needed. Suppose that and . We have to prove that , i.e., that, for any $s$ minimal in , . Such an $s$ is in $\widehat{\alpha}$. We shall prove that it is minimal in $\widehat{\alpha}$. By the smoothness condition, if it were not minimal, there would be an $s'$ minimal in $\widehat{\alpha}$ such that . But therefore and then . By lemma \[remark1\] we conclude that $s'$ is in , in contradiction with the minimality of $s$ in this set. Therefore $s$ is minimal in and, since , one concludes: . 8.5pt We now intend to show that, given any cumulative relation , one may build a cumulative model $W$ that defines a consequence relation that is exactly . Suppose, therefore, that  satisfies the rules of [**C**]{}. All definitions will be relative to this relation. \[def:normworld\] The world is said to be a [*normal*]{} world for $\alpha$ iff such that , . So, a world is normal for a formula if it satisfies all of its plausible consequences. Obviously, if the consequence relation we start from satisfies [**Reflexivity**]{}, a normal world for $\alpha$ satisfies $\alpha$. \[norm:mod\] Suppose a consequence relation  satisfies [**Reflexivity**]{}, [**Right Weakening**]{} and [**And**]{}, and let . All normal worlds for $\alpha$ satisfy $\beta$ iff . -1000.5 pt[**Proof:** ]{} The [*if*]{} part follows from definition \[def:normworld\]. Let us show the [*only if*]{} part. Suppose , we shall build a normal world for $\alpha$ that does not satisfy $\beta$. Let . It is enough to show that $\Gamma_0$ is satisfiable. Suppose not, then, by the compactness assumption, there exists a finite subset of $\Gamma_0$ that is not satisfiable and therefore a finite set such that . Now, and, by [**Reflexivity**]{} and [**Right Weakening**]{} . But, using [**And**]{} one gets . Then, using [**MPC**]{} (the proof of lemma \[le:dercum\] shows that only [**And**]{} and [**Right Weakening**]{} are needed to derive [**MPC**]{}), one concludes , a contradiction. 8.5pt \[tildeq\] We shall say that $\alpha$ is [*equivalent*]{} to $\beta$ and write iff and . \[le:eqcu\] iff $ \forall \gamma \ \ \alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\gamma \Leftrightarrow \beta {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\gamma$. The relation $\sim$ is therefore an equivalence relation. -1000.5 pt[**Proof:** ]{} The [*if*]{} part follows from [**Reflexivity**]{} and the [*only if*]{} part from [**Equivalence**]{}. 8.5pt The equivalence class of a formula $\alpha$, under $\sim$, will be denoted by $\bar{\alpha}$. \[def:leq\] iff such that . It is clear that the definition of $\leq$ makes sense, i.e. does not depend on the choice of the representatives $\alpha$ and $\beta$. The relation $\leq$ is reflexive but is not in general transitive. \[asymm\] The relation $\leq$ is antisymmetric. -1000.5 pt[**Proof:** ]{} Suppose and . There are formulas and such that: and . By lemma \[le:eqcu\], and . Therefore , and . 8.5pt The cumulative model $W$ will be defined the following way: , where is the set of all equivalence classes of formulas under the relation $\sim$, and iff and (the relation $\leq$ has been defined in definition \[def:leq\]). One easily checks the definition of $l$ does not depend on the choice of the representative $\alpha$ and that $\prec$ is asymmetric. \[le:minimum\] For any the state $\bar{\alpha}$ is a minimum of $\widehat{\alpha}$. -1000.5 pt[**Proof:** ]{} Indeed suppose and . This last assumption implies, by the definition of $\widehat{\alpha}$, that every world of satisfies $\alpha$. Let . By the definition of $l$, every normal world for $\beta$ satisfies $\alpha$. By lemma \[norm:mod\], , and therefore . Since we conclude . 8.5pt It follows from lemma \[le:minimum\] and the fact that $\prec$ is asymmetric that the model $W$ defined above is a strong cumulative model. We may now prove what we wanted to achieve. \[un\] iff . -1000.5 pt[**Proof:** ]{} Lemmas \[le:minimum\] and \[min:min\] imply that the only minimal state of $\widehat{\alpha}$ is $\bar{\alpha}$, therefore iff all normal worlds for $\alpha$ satisfy $\beta$ and lemma \[norm:mod\] implies the conclusion. 8.5pt \[th:repcum\] A consequence relation is a cumulative consequence relation iff it is defined by some cumulative model. -1000.5 pt[**Proof:** ]{} The [*if*]{} part is lemma \[Cu:Sou\]. The [*only if*]{} part follows from the construction of $W$ and lemma \[le:minimum\] (that shows $W$ is a cumulative model) and lemma \[un\]. 8.5pt One may remark that the representation result proved is a bit stronger than what is claimed in theorem \[th:repcum\]: any cumulative consequence relation is the consequence relation defined by a strong cumulative model. It is now easy to study the notion of entailment yielded by cumulative models. \[log:imp3\] Let [**K**]{} be a set of conditional assertions, and , the following conditions are equivalent. In case they hold we shall say that [**K**]{} cumulatively entails . 1. for all cumulative models $V$ such that contains [**K**]{}, $\alpha {\mbox{ ${{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim_V$ }}\beta$ 2. has a proof from [**K**]{} in the system [**C**]{}. -1000.5 pt[**Proof:** ]{} From lemma \[Cu:Sou\] one sees that 2) implies 1). For the other direction, suppose 2) is not true. The smallest consequence relation closed under the rules of [**C**]{} that contains [**K**]{} is a cumulative consequence relation that does not contain . By theorem \[th:repcum\], there is a cumulative model that defines it. This model shows property 1) does not hold. 8.5pt \[co:cum2\] Let [**K**]{} be a set of conditional assertions. There is a cumulative model that satisfies exactly those assertions that are cumulatively entailed by [[**K**]{}]{}. The following compactness result follows. \[comp:cum\] [**K**]{} entails  iff a finite subset of [**K**]{} does. -1000.5 pt[**Proof:** ]{} Proofs are always finite and therefore use only a finite number of assumptions from [**K**]{}. 8.5pt To conclude our study of cumulative reasoning, let us say that the system [**C**]{} provides an interesting general setting in which to study nonmonotonic reasoning. Weaker systems are probably very different from systems that are at least as strong as [**C**]{}. The system [**C**]{} is probably too weak to be the backbone of realistic inference systems and cumulative models are quite cumbersome to manipulate. The next section will propose nicer models and an additional rule of inference. Cumulative reasoning with Loop {#sec:cum.loop.reas} ============================== Cumulative ordered models {#subsec:cum.ord} ------------------------- The original motivation for the study of the system [**CL**]{}, that will be proposed in this section, stems from semantic considerations. Later on, a number of results, including the result that will be described in section \[subsec:horn\], which says that, if one restricts oneself to Horn assertions, then the system [**CL**]{} is as strong as [**P**]{}, seemed to point out that [**CL**]{} is worth studying. Looking back on the cumulative models of definition \[def:cum.mod\], one may wonder why we did not require the binary relation $\prec$ to be a strict partial order. We could have required it to be asymmetric without jeopardizing the representation theorem, but the construction of section \[subsec:char.cumul\] builds a model in which $\prec$ is not always transitive. Nevertheless, preferences could probably be assumed to be transitive and, most important, transitivity of $\prec$ eases enormously the task of checking the smoothness condition: if $\prec$ is a partial order (strict), then all finite models (models in which the set of states is finite) satisfy the smoothness condition, and even all well-founded models (in which there is no infinite descending $\prec$-chain) do. Could we have required $\prec$ to be a partial order? In other terms, are there rules that are not valid for cumulative models in general but are valid for all cumulative models the preference relation of which is a strict partial order? We shall now give a positive answer to this last question and exactly characterize this sub-family of cumulative models. A cumulative [*ordered*]{} model is a cumulative model in which the relation $\prec$ is a strict partial order. The system [**CL**]{} --------------------- The following rule, named [**Loop**]{} after its form, will be shown to be the exact counterpart of transitivity of the preference relation in the models. It says that, if propositions may be arranged in a loop, in a way each one is a plausible consequence of the previous one, then each one of them is a plausible consequence of any one of them, i.e. they are all equivalent in the sense of [**Equivalence**]{}. \[def:CL\] The system [**CL**]{} consists of all the rules of [**C**]{} and the following: $$\label{loop} {{\alpha_0 {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\alpha_1 \ \ , \ \ \alpha_1 {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\alpha_2 \ \ , \ldots ,\ \ \alpha_{k-1} {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\alpha_k \ \ , \ \ \alpha_k {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\alpha_0} \over {\alpha_0 {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\alpha_k}} \hspace{1.5cm}{\bf (Loop)}$$ A consequence relation that satisfies all rules of [**CL**]{} is said to be [*loop-cumulative*]{}. \[le:derloop\] The following is a derived rule of [**CL**]{}, for any . $$\label{ru:derloop} {{\alpha_0 {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\alpha_1 \ \ , \ \ \alpha_1 {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\alpha_2 \ \ , \ldots ,\ \ \alpha_{k-1} {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\alpha_k \ \ , \ \ \alpha_k {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\alpha_0} \over {\alpha_i {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\alpha_j}}$$ -1000.5 pt[**Proof:** ]{} It is clear that, because of the invariance of the assumptions under cyclic permutation, the conclusion of [**Loop**]{}, could as well have been , for any $i = 0 , \ldots , k$ (addition is understood modulo $k+1$). From [**Equivalence**]{} one can then conclude , for any $i , j = 0 , \ldots , k$. 8.5pt It seems the rule [**Loop**]{} has never been considered in the literature. We feel it is an acceptable principle of nonmonotonic reasoning. It is particularly interesting that [**Loop**]{} does not mention any of the propositional connectives. \[val:loop\] [**Loop**]{} is valid in all cumulative ordered models. -1000.5 pt[**Proof:** ]{} Let be a cumulative ordered model such that for (addition is understood modulo $k+1$) and let be a minimal state in $\widehat{\alpha_0}$. We shall show that . Since , the state $s_0$ must be in $\widehat{\alpha_1}$. By the smoothness condition, if $s_0$ is not minimal in $\widehat{\alpha_1}$ then there is a state $s_1$ minimal in $\widehat{\alpha_1}$ such that . Similarly, for every there is a state $s_i$ minimal in $\widehat{\alpha_i}$ such that or . Since $\prec$ is transitive, or . But $s_k$ is minimal in $\widehat{\alpha_k}$ and , we conclude that $s_k \in \widehat{\alpha_0}$. But $s_0$ is minimal in $\widehat{\alpha_0}$, we conclude that and . 8.5pt \[notCvalid\] The rule [**Loop**]{} is not valid in cumulative models. -1000.5 pt[**Proof:** ]{} Let $L$ be the propositional calculus on the propositional variables $p_0 , p_1 , p_2$ and ${\cal U}$ be the set of all propositional models on those variables. We shall build a cumulative model such that for all $i = 0 , \dots , 2$ (addition is modulo 3) but . The set $S$ has four states: $s_i$, for $i = -1 , \ldots , 2$. For every $i = 0 , \dots , 2$ we have and . Notice that $\prec$ is not transitive. Let us now describe $l$. For $i = 0 , \ldots , 2$, $l ( s_i )$ is the set of all worlds satisfying $p_i$ and $p_{i + 1}$, and $l ( s_{-1} )$ is the set of all worlds satisfying at least two out of the three variables. First we want to show that $V$ satisfies the smoothness condition. Clearly all subsets of $S$ that contain $s_{-1}$ are smooth since $s_{-1}$ is a minimum in $S$. A set that contains at most two elements is always smooth. We conclude that the only subset of $S$ that is not smooth is . We must show that there is no formula $\alpha$ such that . Let $\alpha$ be any formula and let $i = 0 , \ldots , 2$. If all worlds of must satisfy $\alpha$ and by definition of $l$, . We conclude that if then any world that satisfies at least two of the variables satisfies $\alpha$. We conclude that $\widehat{\alpha}$ must therefore also include $s_{-1}$. To see that , notice that and that, since , the only minimal state in $\widehat{p_i}$ is $s_i$ that satisfies $p_{i + 1}$. The only thing left to check is that . But we just noticed that the only minimal state of $\widehat{p_0}$ is $s_0$ and clearly . 8.5pt Characterization of loop-cumulative consequence relations --------------------------------------------------------- We now want to show that, given any loop-cumulative relation   one may build a cumulative ordered model $V$ such that  is equal to . Suppose  is such a relation and is the cumulative model built out of  in section \[subsec:char.cumul\]. Let $\prec^+$ be the transitive closure of $\prec$. First we shall show that, since  satisfies [**Loop**]{}, the relation $\prec^+$ is a strict partial order. \[strictpo\] The relation $\prec^+$ is irreflexive and therefore a strict partial order. -1000.5 pt[**Proof:** ]{} Suppose . Since $\prec$ is asymmetric, it is irreflexive and t here must be some such that for , (addition is modulo $n$). From the definitions of $\prec$ and $\leq$, we see that, for , there are formulas $\alpha'_i$ such that and . From lemma \[le:eqcu\], we conclude that for . By [**Loop**]{} we see that and therefore and . But this contradicts the asymmetry of $\prec$. We have shown that $\prec$ is irreflexive. Since it is transitive by construction it is a strict partial order. 8.5pt Let us now define where $S , l $ and $ \prec $ are as in the definition of $W$. \[V:min\] In $V$, for any $\alpha$, the state $\bar{\alpha}$ is a minimum of $\widehat{\alpha}$. Therefore $V$ is a strong cumulative ordered model. -1000.5 pt[**Proof:** ]{} Lemma \[le:minimum\] says $\bar{\alpha}$ is a minimum of $\widehat{\alpha}$ with respect to $\prec$. It is therefore a minimum with respect to any weaker relation and in particular $\prec^+$. Lemma \[strictpo\] implies that $\prec^+$ is asymmetric and, by lemma \[min:min\], $V$ satisfies the smoothness condition. 8.5pt \[iff\]  iff . -1000.5 pt[**Proof:** ]{} Lemma \[V:min\] implies that the only minimal state of $\widehat{\alpha}$ is $\bar{\alpha}$, therefore iff all normal worlds for  satisfy , and lemma \[norm:mod\] implies the conclusion. 8.5pt We may now summarize. \[rep:PC\] A consequence relation is a loop-cumulative relation iff it is defined by some cumulative ordered model. As in the cumulative case one may study the notion of entailment yielded by cumulative ordered models and obtain results that parallel corollaries \[log:imp3\], \[co:cum2\] and \[comp:cum\]. Preferential reasoning {#sec:pref} ====================== The system [**P**]{} {#subsec:P} -------------------- We shall now consider a system that seems to occupy a central position in the hierarchy of nonmonotonic systems. It is strictly stronger than [**CL**]{}, but assumes the existence of disjunction in the language of formulas. We call this system [**P**]{}, for [*preferential*]{}, because its semantics, described in section \[subsec:prefmod\], are a variation on those proposed by Y. Shoham in [@Shoham:87]. The differences (the distinction we make and he does not between states and worlds) are nevertheless technically important, as noticed above just before definition \[def:asy\], and as will be shown at the end of section \[subsec:prefmod\]. This very system has been considered by E. Adams [@Adams:75] and proposed by J. Pearl and H. Geffner [@PearlGeff:88] to serve as the [*conservative core*]{} of a nonmonotonic reasoning system. It is the flat fragment of the system ${\cal S}$ studied by J. Burgess in [@Burgess:81]. \[def:P\] The system [**P**]{} consists of all the rules of [**C**]{} and the following: $$\label{ru:or} {{\alpha {\mbox{ ${{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim$ }}\gamma \ \ , \ \ \beta {\mbox{ ${{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim$ }}\gamma} \over {\alpha \vee \beta {\mbox{ ${{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim$ }}\gamma}} \hspace {2cm} {\rm ({\bf Or})}$$ A consequence relation that satisfies all rules of [**P**]{} is said to be [*preferential*]{}. The rule [**Or**]{} corresponds to the axiom CA of conditional logic. It says that any formula that is, separately, a plausible consequence of two different formulas, should also be a plausible consequence of their disjunction. It is a valid principle of monotonic classical reasoning and does not imply monotonicity, therefore we tend to accept it. Further consideration also seems to support [**Or**]{}: if we think that [*if John attends the party, normally, the evening will be great*]{} and also that [*if Cathy attends the party, normally, the evening will be great*]{} and hear that at least one of Cathy or John will attend the party, shouldn’t we be tempted to join in? There is, though, an [*epistemic*]{} reading of that invalidates the [**Or**]{} rule. If we interpret as meaning: [*if all I know about the world is $\alpha$ then it is sensible for me to suppose that $\beta$ is true*]{}, we must reject the [**Or**]{} rule. Indeed, one may imagine a situation in which $\alpha$ expresses a fact that can very well be true or false but the truth value of which is normally not known to me. If I knew $\alpha$ to be true, that would be quite an abnormal situation in which I may be willing to accept $\gamma$. If I knew $\alpha$ to be false, similarly, it would be an exceptional situation in which I may accept $\gamma$, but the knowledge that is true is essentially void and certainly does not allow me to conclude that anything exceptional is happening. Notice that, in this reading, the left hand side of the symbol involves a hidden epistemic operator (the right hand side may also do so, but need not). We shall therefore defend the [**Or**]{} rule by saying that such a hidden operator should be made explicit and the example just above only invalidates the inference: from and infer . But nobody would defend such an inference anyway. The interplay between [**Or**]{} and the rules of [**C**]{} makes [**P**]{} a powerful system. For example, [**Loop**]{} is a derived rule of [**P**]{}. Since this result will be obvious once we have characterized preferential relations semantically, we shall leave a proof-theoretic derivation of [**Loop**]{} in [**P**]{} for the reader to find. We shall now put together a number of remarks revolving around the rule [**Or**]{}. Our first remark is that we may derive from [**Or**]{} a rule that is similar to the [*hard*]{} half of the deduction theorem. This rule was suggested in [@Shoham:88]. It is a very useful rule and expresses the fact that deductions performed under strong assumptions may be useful even if the assumptions are not known facts. \[S\] In the presence of [**Reflexivity**]{}, [**Right Weakening**]{} and [**Left Logical Equivalence**]{}, the rule of [**Or**]{} implies the following: $${{\alpha \wedge \beta {\mbox{ ${{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim$ }}\gamma} \over {\alpha {\mbox{ ${{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim$ }}\: \beta \rightarrow \gamma}}\hspace {3.8cm} {\rm ({\bf S})}$$ [**S**]{} is therefore a derived rule of [**P**]{}. -1000.5 pt[**Proof:** ]{} Suppose . We have , by [**Right Weakening**]{}. But one has . One concludes by [**Or**]{} and [**Left Logical Equivalence**]{}. 8.5pt Our second remark is that, in the presence of [**S**]{}, the rule of [**Cut**]{} is implied by [**And**]{}. Therefore [**Reflexivity**]{}, [**Left Logical Equivalence**]{}, [**Right Weakening**]{}, [**And**]{}, [**Or**]{} and [**Cautious Monotonicity**]{} are an elegant equivalent axiomatization of the system [**P**]{}. \[le:A.C\] In the presence of [**Right Weakening**]{}, [**S**]{} and [**And**]{} imply [**Cut**]{}. -1000.5 pt[**Proof:** ]{} Use [**S**]{}, [**And**]{} and [**Right Weakening**]{}. 8.5pt D. Makinson [@Mak:88] suggested the following rule. It expresses the principle of proof by cases. $$\label{ru:D} {{\alpha \wedge \neg \beta {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\gamma \ \ , \ \ \alpha \wedge \beta {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\gamma} \ \ \ \over {\alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\gamma}} \hspace{1cm}{\rm ({\bf D})}$$ \[le:D\] In the presence of [**Reflexivity**]{}, [**Right Weakening**]{} and [**Left Logical Equivalence**]{}, 1. [**Or**]{} implies [**D**]{} and 2. [**D**]{} implies [**Or**]{} in the presence of [**And**]{}. Therefore [**D**]{} is a derived rule of [**P**]{}. The proof is left to the reader. The next lemma gathers some more derived rules of the system [**P**]{}. They will be used in the proof of the representation theorem of section \[subsec:pref.char\]. The importance of these rules is mainly technical. The reader should notice that [**P**]{} is a powerful system, in which one may build quite sophisticated proofs. \[lemma:ded\] The following are derived rules of [**P**]{}: $$\label{or2} {{\ \ {\alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\gamma} \ \ , \ \ {\beta {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\delta}} \ \ \ \over {{\alpha \vee \beta} {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}{\gamma \vee \delta}}}$$ $$\label{or:implies1} {{\ \ {\alpha \vee \gamma} {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\gamma \ \ , \ \ \alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\beta} \ \ \ \over {\gamma {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}{\alpha \rightarrow \beta}}}$$ $$\label{Less:Trans} {{\ \ {\alpha \vee \beta} {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\alpha \ \ , \ \ {\beta \vee \gamma} {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\beta} \ \ \ \over {{\alpha \vee \gamma} {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\alpha}}$$ $$\label{new1} {{\ \ {\alpha \vee \beta} {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\alpha \ \ , \ \ {\beta \vee \gamma} {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}\beta} \ \ \ \over {\alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\sim}{\gamma \rightarrow \beta}}}$$ -1000.5 pt[**Proof:** ]{} The uses of [**Left Logical Equivalence**]{} will not always be mentioned any more. For (\[or2\]), use first [**Right Weakening**]{} on each of the two hypotheses and then [**Or**]{}. This seems to be a very intuitive rule that is often useful. For (\[or:implies1\]), from the second hypothesis, using [**Left Logical Equivalence**]{} we have . By [**S**]{} we conclude . But, using the first hypothesis and [**Cautious Monotonicity**]{} one may now conclude. For (\[Less:Trans\]), from both hypotheses and using (\[or2\]) one concludes . Now, using our first hypothesis and (\[Or:Trans\]) we see . Leaving this result for a moment, notice that from the first hypothesis and , using (\[or2\]) we obtain . Now, coming back to the result we left hanging, using [**Cautious Monotonicity**]{}, we may conclude. For (\[new1\]), from the second hypothesis one has . By [**S**]{}: . By [**Right Weakening**]{}, one may then obtain But from the two hypotheses, using (\[or2\]), one obtains: . Using [**Cautious Monotonicity**]{} on those last two results, we obtain: . Using the first hypothesis and [**Cautious Monotonicity**]{} one concludes. 8.5pt Preferential Models {#subsec:prefmod} ------------------- We may now describe our version of preferential models. Preferential models are cumulative ordered models in which states are labeled by single worlds (and not sets of worlds). The reasoner has, then, essentially, a preference over worlds (except that the same world may label different states). We may now define the family of models we are interested in. \[def:prefmod\] A [*preferential*]{} model $W$ is a triple where $S$ is a set, the elements of which will be called states, assigns a world to each state and $\prec$ is a strict partial order on $S$ (i.e. an irreflexive, transitive relation), satisfying the [**smoothness condition**]{} of definition \[def:smoocond\]. Notice that, for a preferential model, iff . The smoothness condition, here, as explained in section \[subsec:cum.ord\], is only a technical condition needed to deal with infinite sets of formulas, it is always satisfied in any preferential model in which $S$ is finite, or in which $\prec$ is well-founded (i.e. no infinite descending chains). The requirement that the relation $\prec$ be a strict partial order has been introduced only because such models are nicer and the smoothness condition is easier to check on those models, but the soundness result of lemma \[soundness:non\] is true for the larger family of models, where $\prec$ is just any binary relation. In such a case, obviously, the smoothness condition cannot be dropped even for finite models. The completeness result of theorem \[Com:pref\] holds obviously, too, for the larger family, but is less interesting. Preferential models, since they are cumulative models, define consequence relations as in definition \[def:cumcons\]. Y. Shoham, in [@Shoham:87], proposed a more restricted notion of preferential models. He required the set of states $S$ to be a subset of the universe  and the labeling function $l$ to be the identity. He also required the relation $\prec$ to be a well-order. Any one of those two requirements would make the representation theorem incorrect. The second point is treated in [@LMTR:88]. For the first point, we leave it as an exercise to the reader to show that the following model has no equivalent model in which no label appears twice. Let $L$ be the propositional calculus on two variables $p$ and $q$. Let $S$ have four states: $s_0 \prec s_2$ and $s_1 \prec s_3$. Let $s_0$ satisfy $p$ and $\neg q$, $s_1$ satisfy $\neg p$ and $\neg q$ and $s_2$ and $s_3$ both satisfy $p$ and $q$. Characterization of preferential consequence relations {#subsec:pref.char} ------------------------------------------------------ Our first lemma is obvious. It does not hold in cumulative models and should be contradistincted with lemma \[remark1\]. \[remark2\] Let be a preferential model. For any , . \[soundness:non\] For any preferential model $W$, the consequence relation  it defines is a preferential relation, i.e. all the rules of the system [**P**]{} are satisfied by the relations defined by preferential models. -1000.5 pt[**Proof:** ]{} Indeed, as we remarked above, the fact that $\prec$ is a partial order is not used at all. Since a preferential model is a cumulative model, in light of lemma \[Cu:Sou\], we only need to check the validity of [**Or**]{}. Suppose a preferential model and are given. Suppose that and . Any state minimal in is, by lemma \[remark2\], minimal in the set , and therefore minimal in any of the subsets it belongs to. 8.5pt We shall now begin the proof of the representation theorem. Let us, first, define a relation among formulas, that will turn out to be a pre-ordering whenever the relation  satisfies [**P**]{}. \[def:order\] We say that $\alpha$ is not less [*ordinary*]{} than $\beta$ and write iff . Indeed, if we would conclude that $\alpha$ is true on the basis that either $\alpha$ or $\beta$ is true, this means that the former is not more out of the ordinary than the latter. Notice that, if  satisfies [**Reflexivity**]{} and [**Left Logical Equivalence**]{}, then for any , . \[lemma:order\] If the relation  is preferential, the relation $\leq$ is reflexive and transitive. -1000.5 pt[**Proof:** ]{} Reflexivity follows from [**Left Logical Equivalence**]{} and [**Reflexivity**]{}. Transitivity follows from (\[Less:Trans\]) of lemma \[lemma:ded\]. 8.5pt From now on, and until theorem \[Com:pref\], we shall suppose that the relation  is preferential. \[le:upnorm\] If and $m$ is a normal world for $\alpha$ that satisfies $\beta$, then $m$ is a normal world for $\beta$. -1000.5 pt[**Proof:** ]{} Suppose . By (\[or:implies1\]) of lemma \[lemma:ded\], we have . If $m$ is normal for $\alpha$ it must satisfy , and since it satisfies $\beta$, it must satisfy $\delta$. 8.5pt \[lemma:chain\] If and $m$ is a normal world for $\alpha$ that satisfies $\gamma$ then it is a normal world for $\beta$. -1000.5 pt[**Proof:** ]{} By lemma \[le:upnorm\], it is enough to show that $m$ satisfies $\beta$. By (\[new1\]) of lemma \[lemma:ded\] we have , but $m$ is a normal world for $\alpha$ that satisfies $\gamma$, therefore it must satisfy $\beta$. 8.5pt We may now describe the preferential model we need for the representation result. Remember that we start from any preferential relation . We then consider the following model: where 1. $S {\stackrel{\rm def}{=}}\{ {< m , \alpha >} \mid {m \mbox{ is a normal world for } \alpha} \}$, 2. and 3. iff and . The first thing we want to show is that $W$ is a preferential model, i.e. that $\prec$ is a strict partial order and that $W$ satisfies the smoothness condition. We shall then show that the relation  is exactly . \[lemma:strict\] The relation $\prec$ is a strict partial order, i.e. it is irreflexive and transitive. -1000.5 pt[**Proof:** ]{} The relation $\prec$ is irreflexive since would imply , but $m$ is a normal world for $\alpha$, and since by [**Reflexivity**]{}, it satisfies $\alpha$. It is left to show that $\prec$ is transitive. Suppose and . By the definition of $\prec$ we have and . From this we may conclude two things. First, by lemma \[lemma:order\] we conclude . Secondly, since $m_0$ is a normal world for $\alpha_0$ that does not satisfy $\alpha_1$, we may conclude by lemma \[lemma:chain\] that it does not satisfy $\alpha_2$. 8.5pt We are now going to characterize all minimal states in sets of the form $\widehat{\alpha}$. \[minimal:or\] In the model $W$, is minimal in $\widehat{\alpha}$ iff and $\beta \leq \alpha$. -1000.5 pt[**Proof:** ]{} For the [*if*]{} part, suppose and . Clearly . Suppose now that and . We would have , $n$ normal for $\gamma$, and and . This stands in contradiction with lemma \[lemma:chain\]. For the [*only if*]{} part, suppose is minimal in $\widehat\alpha$. Clearly . Suppose $n$ is a normal world for that does not satisfy $\beta$ (it is not claimed that such a normal world exists). Since , we must have . But $n$ is a normal world for that does not satisfy $\beta$ and therefore must satisfy $\alpha$. This stands in contradiction with the minimality of in $\widehat\alpha$. We conclude that every normal world for satisfies $\beta$. By lemma \[norm:mod\], . 8.5pt We shall now prove that $W$ satisfies the smoothness condition. \[lemma:smoo\] For any $\alpha \in L$, $\widehat \alpha$ is smooth. -1000.5 pt[**Proof:** ]{} Suppose , i.e., . If then, by lemma \[minimal:or\] is minimal in $\widehat\alpha$. On the other hand, if then by lemma \[norm:mod\] there is a normal world $n$ for such that . But and therefore . But, and therefore . Since , Lemma \[minimal:or\] enables us to conclude that is minimal in $\widehat{\alpha}$. 8.5pt We have shown that $W$ is a preferential model. We shall now show that  is exactly the relation  we started from. \[le:1\] If , then . -1000.5 pt[**Proof:** ]{} We must show that all minimal states of $\widehat{\alpha}$ satisfy $\beta$. Suppose is minimal in $\widehat \alpha$. Then $m$ is a normal world for $\gamma$ that satisfies $\alpha$. By lemma \[minimal:or\], and therefore, by lemma \[le:upnorm\], $m$ is a normal world for $\alpha$. 8.5pt \[le:2\] If , then . -1000.5 pt[**Proof:** ]{} It follows from the definition of the relation $\prec$ (lemma \[minimal:or\] could also be used, but is not really necessary here) that, given any normal world $m$ for $\alpha$, is minimal in $\widehat{\alpha}$. If , $\beta$ is satisfied by all normal worlds for $\alpha$, and we may conclude by lemma \[norm:mod\]. 8.5pt We may now state the main result of this section. \[Com:pref\] A consequence relation is a preferential consequence relation iff it is defined by some preferential model. -1000.5 pt[**Proof:** ]{} The [*if*]{} part is Lemma \[soundness:non\]. For the [*only if*]{} part, let  be any consequence relation satisfying the rules above and let $W$ be defined as above. Lemmas \[lemma:strict\] and \[lemma:smoo\] show that $W$ is a preferential model. Lemmas \[le:1\] and \[le:2\] show that it defines an consequence relation that is exactly . 8.5pt As in the cumulative and cumulative ordered cases we may study the notion of preferential entailment and obtain results similar to Corollaries \[log:imp3\], \[co:cum2\] and \[comp:cum\]. Some rules that cannot be derived in [**P**]{} ---------------------------------------------- Is [**P**]{} a reasonable system for nonmonotonic reasoning? We think a good reasoning system should validate all the rules of [**P**]{}. Notice that all the rules we have considered so far are of the form: from the presence of certain assertions in the consequence relation, deduce the presence of some other assertion. After careful consideration of many other rules of this form, we may say we have good reason to think that there are no rules of this type that should be added. Certain principles of reasoning that seem appealing, though, fail to be validated by certain preferential consequence relations. This means, in our sense, that many agents that reason in a way that is fully consistent with all the rules of [**P**]{}, nevertheless behave irrationally. We shall show that circumscription does not, in general, satisfy even the weakest of the principles we shall present. The reader will notice that the form of these principles is different from that of all the rules previously discussed: from the [*absence*]{} of certain assertions in the relation, we deduce the [*absence*]{} of some other assertion. $$\label{Neg:mon} {{\alpha \wedge \gamma {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\not\sim}\beta \ , \ \alpha \wedge \neg \gamma {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\not\sim}\beta} \over {\alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\not\sim}\beta}} \hspace {1.6cm} {\rm ({\bf Negation \ Rationality}) }$$ $$\label{Or:mon} {{\alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\not\sim}\gamma \ \ , \ \ \beta {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\not\sim}\gamma} \over {\alpha \vee \beta {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\not\sim}\gamma}} \hspace {3cm} {\rm ({\bf Disjunctive \ Rationality}) }$$ $$\label{Rat:mon} {{\alpha \wedge \beta {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\not\sim}\gamma \ \ , \ \ \alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\not\sim}\neg\beta} \over {\alpha {{\hspace{0.28em}}{{\rule[-0.4mm]{.1mm}{3mm}}\hspace{-3.5pt}}\not\sim}\gamma}} \hspace {2cm} {\rm ({\bf Rational \ Monotonicity}) }$$ Each one of those rules is implied by [**Monotonicity**]{} and therefore expresses some kind of restricted monotonicity. Any rational reasoner should, in our opinion, support them, and we shall, now, explain and justify them. The rule of [**Negation Rationality**]{} says that inferences are not made solely on the basis of ignorance. If we accept that  is a plausible consequence of , we must either accept that it is a plausible consequence of or accept that it is a plausible consequence of . Indeed, suppose we hold that [*normally, the party should be great*]{}, but that we do not hold that [*even if Peter comes to the party, it will be great*]{}, i.e. we seriously doubt the party could stand Peter’s presence. It seems we could not possibly hold that we also seriously doubt that the party could stand Peter’s absence. If we do not expect the party to be great if Peter is there and do not expect it to be great if Peter is not there, how could we expect it to be great? After all, either Peter is going to be there or he is not. It is, though, easy to find examples of preferential models that define consequence relations that do not satisfy [**Negation Rationality**]{}. We shall even show, now, that circumscriptive reasoning does not always obeys [**Negation Rationality**]{}. Suppose our language has two unary predicate symbols [*special*]{} and [*beautiful*]{}, and one individual constant $a$. We know that, [*normally an object is not special*]{}, i.e. we circumscribe by minimizing the extension of [*special*]{}, keeping [*beautiful*]{} constant. Take  to be [**true**]{} and  to be . Indeed, without any information, we shall suppose that $a$ is not special. But take  to be . If we had the information that $a$ is beautiful if and only if it is special, we could not conclude that $a$ is not special anymore, since it could well be beautiful, i.e. there are two [*minimal*]{} models that must be considered: the first one with $a$ neither beautiful nor special and the second one with $a$ beautiful and special. On the other hand, had we had the informatiom that either $a$ is beautiful or it is special but not both, we could not have concluded that it is not special either, since it could well not be beautiful. It seems that circumscription may lead to unexpected conclusions. The example presented here is a simplification, due to M. Ginsberg, of an example due to the second author. If we try to understand where circumscription differs from intuitive reasoning, we probably will have to say that, even with the knowledge that $a$ is special if and only if it is beautiful, we would have kept the expectation that it is not special, and therefore gained the expectation that it is not beautiful. Similarly, with the knowledge that $a$ is either special or beautiful but not both, we would have kept the expectation that it is not special and therefore formed the expectation that it is beautiful. The rule of [**Disjunctive Rationality**]{} says that inferences made from a disjunction of propositions must be supported by at least one of the component propositions. Again, this seems like a reasonable requirement. If we do not hold that [*if Peter comes to the party, it will be great*]{} and do not hold that [*if Cathy comes to the party, it will be great*]{}, how could we hold that [*if at least one of Peter or Cathy comes, the party will be great*]{}? In this example, the reader may prefer to read [*even if*]{} instead of [*if*]{}, but the conclusion stands anyway. It is easy to see that [**Disjunctive Rationality**]{} implies [**Negation Rationality**]{}. The second author recently showed that [**Disjunctive Rationality**]{} is strictly stronger than [**Negation Rationality**]{}. The rule of [**Rational Monotonicity**]{} is similar to the axiom CV of conditional logic. It expresses the fact that only additional information the negation of which was expected should force us to withdraw plausible conclusions previously drawn. It is an important tool in minimizing the updating we have to do when learning new information. Suppose we hold that [*normally, the party will be great*]{} but do not hold that [*even if Peter comes, the party will be great*]{}, i.e. we think Peter’s presence could well spoil the party, shouldn’t we hold that [*normally, Peter will not come to the party*]{}? One easily shows that, in the presence of the rules of [**C**]{}, [**Rational Monotonicity**]{} implies [**Disjunctive Rationality**]{}. D. Makinson proved that [**Rational Monotonicity**]{} is strictly stronger than [**Disjunctive Rationality**]{} and conjectured a model-theoretic characterization of preferential relations that satisfy [**Rational Monotonicity**]{}. The second author proved the corresponding representation result in the case the language $L$ is finite. The third author lifted the restriction on $L$. These results will appear in a separate paper. Examples: diamonds and triangles {#subsec:exa} -------------------------------- We shall now show what preferential reasoning may provide in the setting of two toy situations that have become classics in the literature. First the so-called [*Nixon diamond*]{}. Suppose our knowledge base [[**K**]{}]{} contains the four assertions that follow. The reader may read [*teen-ager*]{} for $t$, [*poor*]{} for $p$, [*student*]{} for $s$ and [*employed*]{} for $e$. 1. $t$ $p$ 2. $t$ $s$ 3. $p$ $e$ 4. $s$ $\neg e$ It is easy to see, by describing suitable preferential models, that no assertion that would look like some kind of contradiction is preferentially entailed by [[**K**]{}]{}. In particular neither , nor is preferentially entailed by [[**K**]{}]{}. We cannot conclude, from the information given above, that teen-agers are normally employed, neither can we conclude that they generally are not employed. This seems much preferable than the consideration of multiple extensions. This weakness of the system [**P**]{} seems to be exactly what we want. Nevertheless, preferential reasoning allows for some quite subtle conclusions. For example the following assertions are preferentially entailed by [[**K**]{}]{}: ([*normally, people are not teen-agers*]{}), ([*normally, people are not poor students*]{}). The following assertions are not preferentially entailed: ([*students, normally are not poor*]{}), or ([*poor persons are normally not students*]{}), and we feel indeed that there is not enough information in [[**K**]{}]{} to justify them. An example of an assertion that is not preferentially entailed by [[**K**]{}]{} but we think should follow from [[**K**]{}]{} is: , since $a$ is not mentioned in [[**K**]{}]{}. The reader may consult [@Leh:89] for a possible solution. A second classical example is the [*penguin triangle*]{}. Suppose our knowledge base [[**K**]{}]{} contains the three assertions that follow. The reader may read [*penguin*]{} for $p$, [*flies*]{} for $f$, and [*bird*]{} for $b$. 1. $p$ $b$ 2. $p$ $\neg f$ 3. $b$ $f$ It is easy to see, by describing suitable preferential models, that no assertion that would lead to some kind of contradiction is preferentially entailed by [[**K**]{}]{}. In particular is not preferentially entailed by [[**K**]{}]{}. On the other hand, the following assertions are preferentially entailed by [[**K**]{}]{} and we leave it to the reader to show that they are satisfied by all preferential models that satisfy [[**K**]{}]{}: 1. 2. 3. 4. 5. The reader should remark that no [*multiple extension*]{} problem arises here and that preferential reasoning correctly chooses the most specific information and in effect pre-empts the application of a less specific default. Horn assertions {#subsec:horn} --------------- In this section we shall show that, if we consider only assertions of a restricted type (i.e. Horn assertions), then the system [**P**]{} is no stronger than [**CL**]{}. For this result we shall need the full strength of theorem \[rep:PC\]. To keep notations simple, let us suppose $L$ is a propositional language. \[def:horn\] An assertion  will be called a Horn assertion iff the antecedent  is a conjunction of zero or more propositional variables and the consequent  is either a single propositional variable or the formula [**false**]{}. The crucial remark is the following. \[le:ordcum.constr\] If $W$ is a cumulative ordered model, there is a preferential model $V$ such that  and  coincide as far as Horn assertions are concerned. -1000.5 pt[**Proof:** ]{} Let $W$ be the model . We shall define $V$ to be the model , where $l'$ is defined in the following way. For any and for any propositional variable $p$, iff for every , , in other words iff in $W$. It is clear that, if  is a conjunction of propositional variables then the sets $\widehat\alpha$ in $W$ and $V$ coincide. Therefore, if $W$ satisfies the smoothness condition, so does $V$ and  and  agree on Horn formulas. 8.5pt \[the:PLOOP\] Let [[**K**]{}]{} be a knowledge base containing only Horn assertions, and  a Horn assertion. If the assertion  may be derived from [[**K**]{}]{} in the system [**P**]{}, then it may be derived from [[**K**]{}]{} in the system [**CL**]{}. -1000.5 pt[**Proof:** ]{} Suppose  cannot be derived in [**CL**]{}. By the representation theorem \[rep:PC\], there is a cumulative ordered model $W$ that satisfies all the assertions of [[**K**]{}]{}, but does not satisfy . By lemma \[le:ordcum.constr\], there is a preferential model $V$ that satisfies [[**K**]{}]{}, but does not satisfy . We conclude, by the soundness part of theorem \[Com:pref\], that  cannot be derived in [**P**]{}. 8.5pt Cumulative monotonic reasoning {#sec:mon.cum} ============================== The system [**CM**]{} {#subsec:CM} --------------------- In section \[subsec:m\], three rules were shown equivalent in the presence of the rules of [**C**]{}. We shall now study the system obtained by adding those rules (or one of them) to the system [**C**]{}. One obtains a system that is strictly stronger than [**CL**]{}, but incomparable with [**P**]{}. It is corresponds to some natural family of models. The system [**CM**]{} contains all the rules of [**C**]{} and the rule of [**Monotonicity**]{}, defined in (\[ru:mon\]). A consequence relation that satisfies all the rules of [**CM**]{} is said to be [*cumulative monotonic*]{}. In fact, [**Left Logical Equivalence**]{} and [**Cautious Monotonicity**]{} are now redundant, since they follow from [**Monotonicity**]{}. From lemma \[le:dermon\], one sees that [**EHD**]{} and [**Transitivity**]{} are derived rules of [**CM**]{}. It is obvious that [**Loop**]{} is also a derived rule of [**CM**]{} (by [**Transitivity**]{}). It is not difficult to find preferential models that do not satisfy [**Monotonicity**]{} and we conclude that [**CM**]{} is strictly stronger than [**CL**]{} and not weaker than [**P**]{}. Simple cumulative models {#subsec:simcum.mod} ------------------------ A cumulative model will be called a [*simple*]{} cumulative model iff the binary relation $\prec$ on its states is empty. A simple cumulative model is a cumulative ordered model. The smoothness condition is always satisfied in such a model. It is very easy to see that the consequence relation defined by any simple cumulative model satisfies [**Monotonicity**]{}. It is not difficult to find simple cumulative models that do not satisfy certain instances of the [**Or**]{} rule. We conclude that [**P**]{} and [**CM**]{} are incomparable. It is also easy to find such models that do not satisfy certain instances of [**Contraposition**]{}. Characterization of monotonic cumulative consequence relations -------------------------------------------------------------- \[char:MC\] A consequence relation is cumulative monotonic iff it is defined by some simple cumulative model. -1000.5 pt[**Proof:** ]{} It has been noticed above that the [*if*]{} part is trivial. For the [*only if*]{} part, suppose  is a consequence relation that satisfies the rules of [**CM**]{}. Let , where is the set of all formulas  such that and $l {\stackrel{\rm def}{=}}$ . Lemma \[norm:mod\] implies that all labels are non-empty. By lemma \[norm:mod\], for any formula $\alpha$, . Since all states of $\widehat{\alpha}$ are minimal in $\widehat{\alpha}$, we see that iff for all $\gamma$ such that and all normal worlds $m$ for $\gamma$, . By lemma \[norm:mod\] this last condition is equivalent to and we have: iff for any $\gamma$, . Suppose , take any $\gamma$ such that , we have by [**Transitivity**]{}, a derived rule of [**CM**]{}, . Therefore . Suppose now that , then, by taking one sees that . 8.5pt As in the cumulative, cumulative ordered and preferential cases, one may study the notion of entailment yielded by simple cumulative models and obtain results similar to Corollaries \[log:imp3\], \[co:cum2\] and \[comp:cum\]. Monotonic reasoning {#sec:M} =================== The system [**M**]{} -------------------- The results presented in this section are probably folklore. They are presented here for completeness’ sake. \[def:M\] The system [**M**]{} consists of all the rules of [**C**]{} and the rule of [**Contraposition**]{}. A consequence relation that satisfies all the rules of [**M**]{} is said to be [*monotonic*]{}. Lemma \[le:dermon2\] and the results to come will show that the system [**M**]{} is strictly stronger than [**P**]{} and [**CM**]{}. \[le:derM\] The rule [**Or**]{} is a derived rule of [**M**]{}. -1000.5 pt[**Proof:** ]{} Use [**Contraposition**]{} twice, then [**And**]{} and finally [**Contraposition**]{}. 8.5pt \[le:equivM\] A consequence relation is monotonic iff it satisfies [**Reflexivity**]{}, [**Right Weakening**]{}, [**Monotonicity**]{}, [**And**]{} and [**Or**]{}. -1000.5 pt[**Proof:** ]{} The [*only if*]{} part follows from lemmas \[le:dercum\], \[le:dermon2\] and \[le:derM\]. For the [*if*]{} part, notice, first, that [**Left Logical Equivalence**]{} and [**Cautious Monotonicity**]{} are special cases of [**Monotonicity**]{}. The remark preceding lemma \[le:A.C\] shows that all rules of [**P**]{} may be derived from the rules above. We must now show that [**Contraposition**]{} may be derived from the rules of [**P**]{} and [**Monotonicity**]{}. Suppose . By [**S**]{}, one has . By [**Right Weakening**]{}, we conclude . By [**Monotonicity**]{}, we have . We conclude by [**Reflexivity**]{} and [**MPC**]{}. 8.5pt Simple preferential models {#subsec:spm} -------------------------- The account of monotonic reasoning that we propose is essentially the following. The agent has in mind a set of possible worlds $V$: this is the set of worlds the agent thinks are possible in practice. This set $V$ is a subset of the set ${\cal U}$ of all logically possible worlds. The agent is willing to conclude $\beta$ from $\alpha$ if all worlds of $V$ that satisfy $\alpha$ also satisfy $\beta$. A [*simple*]{} preferential model is a preferential model in which the binary relation $\prec$ is empty. A simple preferential model is a simple cumulative model in which the labeling function $l$ labels each state with a single world. Since repeated labels are obviously useless we could, as well, have considered a model to be a subset of . Characterization of monotonic consequence relations {#subsec:charmon} --------------------------------------------------- \[completeness:mon\] A consequence relation is monotonic iff it is defined by some simple preferential model. -1000.5 pt[**Proof:** ]{} The proof of the [*if*]{} part is trivial. For the [*only if*]{} part we shall build a simple preferential model for any given monotonic consequence relation . Let $V {\stackrel{\rm def}{=}}\{ m \in {\cal U} \: | \: \forall \alpha$,$\beta \in L$, if then and let where $l$ is the identity function. So, iff . We shall prove that iff . If then by the construction of $V$, . Suppose now that , we shall show that there is a world that does not satisfy . Let . Since , $\Gamma_0$ is satisfiable (the full proof is given in a more general case in lemma \[norm:mod\]). Let $m$ be a world that satisfies $\Gamma_0$. We shall prove that if then . If then by [**S**]{} and by [**Monotonicity**]{}. Therefore, by the definition of $\Gamma_0$, and . We conclude that and clearly but . 8.5pt It will now be shown that all the constructions and results described above relativize without problems to a given set of conditional assertions. \[log:imp1\] Let [**K**]{} be a set of conditional assertions, and $\alpha , \beta \in L$. Let $\Delta {\stackrel{\rm def}{=}}$ and let $W$ be the monotonic model , where $l$ is the identity function. The notation ${\cal U}_\Delta$ has been defined in section \[subsec:lan\]. The following conditions are equivalent. If they are satisfied we shall say that [**K**]{} monotonically entails . 1. \[one\] for all monotonic models $V$ such that  contains [**K**]{}, 2. \[two\] 3. \[three\] has a proof from [**K**]{} in the system [**M**]{}. 4. \[four\] follows logically (with respect to ${\cal U}$) from the formulas of $\Delta$. -1000.5 pt[**Proof:** ]{} We shall first show the equivalence of \[one\] and \[two\]. The relation defined in \[one\] is the intersection of all those monotonic consequence relations that contain [**K**]{}. If $V$ is any monotonic model such that  contains [**K**]{} then the labels of its states must be in ${\cal U}_\Delta$ (as defined in \[two\]) and therefore contains . But  contains [**K**]{} and is one of the relations considered in \[one\]. To see the equivalence of \[one\] and \[three\], notice that the relation defined in \[three\] is the intersection of all those monotonic relations that contain [**K**]{}. Theorem \[completeness:mon\] implies that \[one\] and \[three\] define the same relation. The equivalence between \[two\] and \[four\] is immediate. 8.5pt From the equivalence of conditions \[one\] and \[three\] one easily proves the following compactness result: \[comp:mon\] [**K**]{} monotonically entails iff a finite subset of [**K**]{} does. Summary, future work and conclusion =================================== Five families of models and consequence relations have been defined and their relations will be summarized here. Each family has been characterized by a logical system and no two of those systems are equivalent. The family of cumulative models contains all other families and is characterized by the logical system [**C**]{} that consists of [**Logical Left Equivalence**]{}, [**Right Weakening**]{}, [**Reflexivity**]{}, [**Cut**]{} and [**Cautious Monotonicity**]{}. The next largest family is that of cumulative ordered models. It contains all three families not yet mentioned here. It is characterized by the logical system [**CT**]{} that contains, in addition to the rules of [**C**]{}, the rule of [**Loop**]{}. The families of simple cumulative models and of preferential models are two incomparable subfamilies of the family of cumulative ordered models. Simple cumulative models are characterized by the logical system [**CM**]{} that contains, in addition to the rules of [**C**]{}, the rule of [**Monotonicity**]{} (or equivalently, [**Transitivity**]{}). The family of preferential models, probably the most important one, is characterized by the logical system [**P**]{} that contains, in addition to the rules of [**C**]{} the rule of [**Or**]{}. The family of monotonic models is the smallest one of them all. It is contained in all other four. It is characterized by the logical system [**M**]{} that contains, in addition to the rules of [**C**]{}, both rules [**Monotonicity**]{} and [**Or**]{}. Of those families of consequence relations, which is the best suited to represent the inferences of a nonmonotonic reasoner in the presence of a fixed knowledge base? Monotonic and and cumulative monotonic reasoning are too powerful, i.e. simple cumulative and simple preferential models are too restrictive to represent the wealth of nonmonotonic inference procedures we would like to consider. We feel that all bona fide logical systems should implement reasoning patterns that fall inside the framework of cumulative reasoning, but probably not all cumulative models represent useful nonmonotonic systems. The same may probably said about cumulative ordered models. Preferential reasoning seems to be closest to what we are looking for. Nevertheless, many preferential reasoners lack properties that seem desirable, for example [**Rational Monotonicity**]{}. A major problem that is not solved in this paper is to describe reasonable inference procedures that would guarantee that the set of assertions that may be deduced from any conditional knowledge base satisfies the property of [**Rational Monotonicity**]{}. The second author proposed a solution to this problem in [@Leh:89]. Another major problem, not solved here, is to extend the results presented here to predicate calculus and answer the question: how should quantifiers be treated, or what is the meaning of the conditional assertion ? The second and third authors have a solution, still unpublished, to this problem too. We hope the results presented above will convince the reader that the field of artificial nonmonotonic reasoning may benefit from the study of nonmonotonic consequence relations. Acknowledgments =============== Discussions with and remarks from David Makinson, Johan van Benthem, David Israel, Benjamin Grosof and an anonymous referee helped us put this work in perspective and improve its presentation. They are gratefully acknowledged. [10]{} Ernest W. Adams. Probability and the logic of conditional. In J. Hintikka and P. Suppes, editors, [*Aspects of Inductive Logic*]{}. North Holland, Amsterdam, 1966. Ernest W. Adams. . D. Reidel, Dordrecht, 1975. Arnon Avron. 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Johan van Benthem. Foundations of conditional logic. , 13:303–349, 1984. Frank Veltman. . PhD thesis, Filosofisch Instituut, Universiteit van Amsterdam, 1986. [^1]: Institute for Advanced Computer Studies and Department of Computer Science, University of Maryland, College Park, MD 20742 U.S.A. This work was done while the author was at Hebrew University. [^2]: Department of Computer Science, Hebrew University, Jerusalem 91904 (Israel) [^3]: Department of Mathematics, Hebrew University, Jerusalem 91904 (Israel) [^4]: This work was partially supported by the Jean and Helene Alfassa fund for research in Artificial Intelligence [^5]: The compactness assumption is needed only to treat consequence relations defined as the set of all assertions entailed by infinite sets of conditional assertions.
--- abstract: 'We present a novel compact image descriptor for large scale image search. Our proposed descriptor - Geometric VLAD (gVLAD) is an extension of VLAD (Vector of Locally Aggregated Descriptors) that incorporates weak geometry information into the VLAD framework. The proposed geometry cues are derived as a membership function over keypoint angles which contain evident and informative information but yet often discarded. A principled technique for learning the membership function by clustering angles is also presented. Further, to address the overhead of iterative codebook training over real-time datasets, a novel codebook adaptation strategy is outlined. Finally, we demonstrate the efficacy of proposed gVLAD based retrieval framework where we achieve more than $15\%$ improvement in mAP over existing benchmarks.' author: - | Zixuan Wang$^{\star}$, Wei Di$^{\dagger}$, Anurag Bhardwaj$^{\dagger}$, Vignesh Jagadeesh$^{\dagger}$, Robinson Piramuthu$^{\dagger}$\ $^{\star}$ Dept.of Electrical Engineering, Stanford University, CA 94305\ $^{\dagger}$ eBay Research Labs, eBay Inc., San Jose, CA 95125\ [zxwang@stanford.edu, wedi@ebay.com, anbhardwaj@ebay.com, vjagadeesh@ebay.com, rpiramuthu@ebay.com]{} bibliography: - 'gvladbib.bib' title: Geometric VLAD for Large Scale Image Search --- Introduction {#sec:introduction} ============ Proliferation of large-scale image collections on web has made the task of efficient image retrieval challenging. Given a query image or region, the goal is to retrieve images of the same object or scene from a large scale database with high accuracy, efficiency and less memory usage. One of the core problems is how to concisely represent the visual information present in images. A number of methods have been proposed recently that address this issue from both computational efficiency as well as retrieval accuracy perspectives. However, there is a growing need for algorithms that can achieve reasonable trade-offs on both these aspects. Vector of Locally Aggregated Descriptors (VLAD) [@jegou2010aggregating] proposed by J[é]{}gou et al. is one of the seminal contributions in this area as they show that compact and accurate VLAD representation is able to scale to billions of descriptors (by avoiding expensive hard disk operations) and still retain superior retrieval performance. However, one of the limitations of this representation is its inability to incorporate more keypoint level information that can potentially lead to enhanced performance. One such information is the dominant angle of the detected keypoint, also referred to as “Keypoint Angle”, which is often discarded for the sake of obtaining rotational invariance in matches. A toy example is illustrated in Figure \[fig:vlad\_motivation\], in which VLAD is unable to differentiate between the configurations shown in the left and right figures where keypoints (red dots) differ only in their orientations, while having same descriptor representation and distance in the feature space towards the centroid $c_i$. Thus, we hypothesize that keypoint angles provide useful geometric cues which can be very useful for matching images. Integrating this information in a principled way can substantially improve the performance of existing VLAD based representation. In this paper, we present Geometric VLAD (gVLAD) that strengthens the VLAD representation by incorporating weak geometric cue in form of keypoint angles. ![gVLAD motivation: A set of key points (denoted in red dot) locates in the feature space with same distance $r$ towards the centroid $c_i$, assuming they are of same feature descriptor. VLAD is unable to differentiate between the configurations shown in the left and right figures which differ only in the orientations of keypoints (depicted by arrow). However, by separating keypoints into two bins according to their dominant orientation, and measuring distance of points from each bin towards the centroid separately, the proposed gVLAD can successfully differentiate between the two configurations.[]{data-label="fig:vlad_motivation"}](gvlad-motivation.pdf){width="1\linewidth"} Our contributions in this paper are as follows: - **Angle Binning Based VLAD:** We propose a novel formulation of gVLAD that incorporates low level keypoint angles in form of a membership function into the VLAD representation. - **Circular Preserved Angle Membership Learning:** We propose a simple but effective principled technique to learn the membership function of keypoint angles based on trigonometric transform and clustering in a fashion that preserves their circular distribution. - **Codebook Adaptation:** To eliminate the need of iterative codebook training for large scale real-world image collections, a codebook adaptation scheme is presented. - **Z-Score Normalization:** Z-score based normalization technique is proposed that outperforms existing normalization methods for VLAD-based representations. - **Superior New Benchmark Results:** State-of-the-art image retrieval performance of proposed framework over a number of existing retrieval benchmarks are achieved. The paper is organized as follows. In section \[sec:relatedwork\], we outline related work in large-scale image search and strategies of integrating geometric information into image representations. In section \[sec:model\], we describe the geometric VLAD representation in detail. In section \[sec:experiment\], we demonstrate the performance gain on *Oxford*, *Holidays* and *Paris* benchmarks, as well as on extended large scale datasets. We conclude the paper and discuss the future work in section \[sec:conclusion\]. Related Work {#sec:relatedwork} ============ The Bag-of-Words (BoW) representation is one of the most widely used method for image retrieval [@sivic2003video; @philbin2007object]. It quantizes each local descriptor SIFT [@lowe1999object] or SURF [@bay2006surf], to its nearest cluster center and encodes each image as a histogram over cluster centers also known as “Visual Words”. Good retrieval performance is achieved with a high dimensional sparse BOW vector, in which case inverted lists can be used to implement efficient search. However, the search time grows quadratically as the number of images increase [@chum2010large]. To overcome this issue, the Fisher kernel based approach proposed by Perronnin *et al*. [@perronnin2010large] transforms an set of variable-sized independent samples into a fixed size vector representation. The samples are distributed according to a parametric generative model, in this case a Gaussian Mixture Model (GMM) estimated on a training set. A simplified version of Fishers kernels, the VLAD is proposed by J[é]{}gou *et al*. [@jegou2010aggregating; @jegou2012aggregating]. It encodes the difference from the cluster center in a more direct manner, rather than the frequency assigned to the cluster. It requires less computation than Fisher kernels but can achieve comparable retrieval performance. However, most of existing methods ignore the geometric information present in images. Spatial re-ranking [@philbin2007object] is usually used as a geometric filter to remove unrelated images from retrieval results. However, due to its expensive computation it is applied only to top ranked images for re-ranking. The spatial pyramid [@lazebnik2006beyond] is a simple extension of the BOW representation which partitions the image into increasingly fine sub-regions and computes histograms of local features found inside each sub-region. It shows improved performance on scene classification tasks. The weak geometric consistency constraints (WGC) [@jegou2008hamming] uses angle and scale information from key points to verify the consistency of matching descriptors. It can improve the retrieval performance significantly. Recently, Zhang *et al*. [@zhang2011image] propose a technique to encode more spatial information through the geometry-preserving visual phrases (GVP) which requires a pair of images to obtain geometric information. Chum *et al*. [@chum2009geometric] propose geometric min-hash, which extends min-hash by adding local spatial extent to increase the discriminability of the descriptor. It can be used for nearly duplicate image detection but has not achieved the state-of-the art performance in retrieval. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Each figure on the left shows an input image. Each figure on the right shows the detected key points in the image. Keypoints are grouped into four bins based on their angles represented by the direction of the line, and is colored by four unique colors. Length of the line corresponds to scale. Note that each image has a distinct representation based on the orientation of key points which suggests that this information can be potentially useful in image representation.[]{data-label="fig:objExperiments"}](eiffel.png "fig:"){width=".47\linewidth"} ![Each figure on the left shows an input image. Each figure on the right shows the detected key points in the image. Keypoints are grouped into four bins based on their angles represented by the direction of the line, and is colored by four unique colors. Length of the line corresponds to scale. Note that each image has a distinct representation based on the orientation of key points which suggests that this information can be potentially useful in image representation.[]{data-label="fig:objExperiments"}](eiffel_angle.png "fig:"){width=".47\linewidth"} -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Each figure on the left shows an input image. Each figure on the right shows the detected key points in the image. Keypoints are grouped into four bins based on their angles represented by the direction of the line, and is colored by four unique colors. Length of the line corresponds to scale. Note that each image has a distinct representation based on the orientation of key points which suggests that this information can be potentially useful in image representation.[]{data-label="fig:objExperiments"}](temple.png "fig:"){width=".47\linewidth"} ![Each figure on the left shows an input image. Each figure on the right shows the detected key points in the image. Keypoints are grouped into four bins based on their angles represented by the direction of the line, and is colored by four unique colors. Length of the line corresponds to scale. Note that each image has a distinct representation based on the orientation of key points which suggests that this information can be potentially useful in image representation.[]{data-label="fig:objExperiments"}](temple_angle.png "fig:"){width=".47\linewidth"} -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Proposed Framework {#sec:model} ================== In this section, we introduce the Geometric VLAD (gVLAD) to improve retrieval performance by incorporating low level angle information from the key points into VLAD framework. Geometric VLAD {#sec:gvlad} -------------- Let us represent the local descriptor $\mathbf{x}$ to be $d$-dimensional vector (e.g. SURF or SIFT descriptors). Codebook or visual words are denoted as $\mu = \left[\mu_1, \mu_2, \dots, \mu_K\right]$, where $K$ represents the size of the vocabulary. Let $NN(\mathbf{x})$ represent the nearest-neighbor function that maps an input descriptor $\mathbf{x}$ to its nearest visual word $i$ where $1 \leq i \leq K$. In the original VLAD [@jegou2010aggregating; @jegou2012aggregating], to represent a given image, a set of local descriptors are extracted first. Then, the contribution of each visual word $v_i$ is defined by accumulating distances of all the descriptors that belong to the $i^{th}$ visual word $\mu_i$ as: $$v_i = \sum_{\mathbf{x}:NN(\mathbf{x})=i} \mathbf{x} - \mu_i$$ Such representation is further L2-normalized, and concatenated to form a vector representation with size $d \times K$ to represent each image. However, the above formulation suffers from the drawback that it is unable to incorporate extra descriptor level information such as angle which can be of very useful in providing a weak geometrical cue. Thus, we present a gVLAD representation which encodes such angle information of the descriptor into the VLAD framework for efficient image matching. To define gVLAD, we redefine a local descriptor by $\mathbf{x}^{\theta}$, where $\mathbf{x}$ still represents the appearance feature vector of the descriptor and $\theta$ represents the angle of the descriptor, i.e. the dominant angle of the keypoint. For example, in SIFT descriptor, the angle corresponds to the dominant direction of gradient within a local window. To model the distribution of angles, we introduce clustering idea and define a membership function over the angles as: $\psi(\theta(\mathbf{x})):0 \leq \theta < 2\pi \to \{1,2,\dots,M\}$, where $M$ denotes the total number of angular bins. The gVLAD $v_i^j$ for $i^{th}$ of the $K$ visual words (feature bin) and $j^{th}$ of the $M$ angular bins can now be represented as: $$\label{eqn:gvlad} v_i^j = \left\{ \begin{array}{l l} \sum_{ \mathbf{x}^{\theta} : NN(\mathbf{x}) = i} \mathbf{x}^{\theta} - \mu_i & \quad \text{if $\psi(\theta(\mathbf{x}))=j$}\\ \mathbf{0}^d & \quad \text{if $\psi(\theta( \mathbf{x})) \neq j$} \end{array} \right.$$ where $d$ is the dimension of feature vector of local descriptor $\mathbf{x}$. The contribution of each visual word $V_i$ in the geometric VLAD can now be written as combining individual contributions from each angle bin: $$V_i = [v_i^1, v_i^2, \cdots, v_i^{M-1}, v_i^M] \label{eqn:V_i}$$ where $V_i$ is a row vector with size of $d\times M$. Our geometric VLAD (gVLAD) representation $\mathcal{V}$ is defined by accumulating contributions of from all $K$ visual words, and has $D$ dimensions: $D=K\times d \times M$. $$\mathcal{V} = [V_1, V_2, \cdots, V_{K-1}, V_{K}]$$ Learning Membership Function - $\psi(\theta)$ {#sec:membership} --------------------------------------------- One principal way to learn the membership function $\psi(\theta)$ is to apply clustering over the angle distribution and find the appropriate membership assignments for each angle value among $M$ learned clusters. Typically angles have a circular distribution of in the range of $[0, 2\pi )$, whereas existing clustering algorithms that based on L2 distance such as $k$-means assume a Cartesian co-ordinate space for input data, and can not be applied directly. To address this issue, we propose to represent each keypoint as $(r, \theta)$, where $r$ is the radial coordinate. Since we are only interested in angles of key point $\theta$, we fix $r$ as an arbitrary number $r>0$. We now perform a non-linear transform from this polar co-ordinate to 2D Cartesian co-ordinate space using the trigonometric functions: $$\begin{aligned} x &= r \times \cos{\theta} \\ y &= r \times \sin{\theta}\end{aligned}$$ Thus, each angle is mapped to a point $\mathbf{z} (\theta)=(x,y)$ in this 2-d space. To learn the membership of function $\psi(\theta)$, we perform k-means clustering in this space satisfying: $${\arg \min}_{ \lbrace \mathbf{\alpha}_1, \ldots , \mathbf{\alpha}_M \rbrace } \sum_{i=1}^M \sum_{\mathbf{z}_j \in \Xi_i} { \| \mathbf{z}_j - \mathbf{\alpha}_i \| }^2$$ where $\mathbf{\alpha}_i$ is the cluster centroid by averaging all points in cluster set $\Xi_i$. The membership of each angle $\theta$ can be estimated through: $$\psi(\theta) = {\arg \min}_{i \in \lbrace1,2, \ldots, M \rbrace } { \| \mathbf{z}(\theta) - \mathbf{\alpha}_i \| }^2$$ Codebook Adaptation {#sec:adaptation} ------------------- Most real-world image databases grow continuously which leads to frequent codebook training processes that are often desirable. We propose a simple codebook adaptation process that can adapt existing codebooks with incremental dataset and alleviate the need of frequent large-scale codebook training. Secondly, this technique also allows codebook training from diverse datasets as codebook trained on one dataset (i.e. Paris building images) can be adapted to retrieve images from another dataset (i.e. Flickr holidays images). To define our codebook adaptation, let us represent a source dataset of images $S$ where an initial codebook $\mu = \left[\mu_1, \mu_2, \dots, \mu_K\right]$ is trained. Given a new domain specific dataset $T$, our goal is to adapt $\mu$ to another domain specific codebook $\hat{\mu}$ given as: $$\begin{aligned} \hat{\mu_i} &= \frac{1}{N}\sum_{t=1}^N \gamma_i(t), \mathbf{x}^{\theta}(t) \in T \\ \text{where}~\gamma_i(t) &= \left\{ \begin{array}{l l} \mathbf{x}^{\theta}(t) & \quad \text{ if $NN( \mathbf{x}^{\theta}(t))=\mu_i$}\\ \mathbf{0}^d & \quad \text{if $NN(\mathbf{x}^{\theta}(t)) \neq \mu_i$} \end{array} \right.\end{aligned}$$ where $N$ is the total number of descriptors in dataset $T$ and $\mathbf{x}^{\theta}(t)$ represents $t^{th}$ descriptor. In our experiment, the initial codebook $\mu$ is trained using the *Paris* dataset. For all the other experiments on different datasets, $\hat{\mu_i}$ is used in conjunction with Equation \[eqn:gvlad\] to compute the representation of the geometric VLAD. gVLAD Normalization {#sec:z-normalization} ------------------- Normalization is important to effectively and correctly measure distance between vector representation. Here we propose three stages of normalization. First, we use the intra-normalization [@arandjelovicall2013all], where the sum of residuals of each visual word $v_i^j$ is L2 normalized independently, where $1 \leq i \leq K$ and $1 \leq j \leq M$. This step is followed by inter-Z-score normalization across different visual words. Given a vector $X$, its Z-score normalization is computed as: $\frac{X-\mu}{\sigma}$, where $\mu$ and $\sigma$ represent the mean and standard deviation of $X$. Let’s denote the $t^{th}$ entry of $V_i$ as $v_{i,t}$, where $V_i$ is defined in Equation \[eqn:V\_i\]. We apply the inter-Z-score normalization on each $ \left[ v_{1,t}, v_{2,t}, \ldots, v_{i,t}, \ldots, v_{K,t} \right]$, where $1 \leq t \leq {M \times d}$ and $1 \leq i \leq K $. At last, the gVLAD vector $\mathcal{V}$ is L2 normalized $\mathcal{V} := \mathcal{V} / { \| \mathcal{V}\| }_2$. PCA Whitening {#sec:pca-whitening} ------------- Given a large collection of images, the size of representation needs to be carefully considered so as to be feasible for practical real time retrieval. For instance, using only $256$ visual words with $64$ dimensional SURF descriptors and $4$ angle bins generates a feature representation of size $D=64\times 256 \times 4=65,536$. To achieve memory-efficient representation of this vector, we use standard PCA with pre-whitening as described in [@jegou2012negative]. The PCA whitening matrix can be expressed in the form of: $$\textbf{P} = \textbf{D}^{-1/2}\textbf{E}^{T}$$ where $\textbf{E} \textbf{D} \textbf{E}^{T}=E \lbrace \bar{\mathbf{V}} \bar{\textbf{V}}^{T} \rbrace$ is the eigenvector decomposition of the covariance matrix of the (zero mean) data $\bar{\textbf{V}}$, where each row $\bar{\mathcal{V}_l} = \mathcal{V}_l - \mathcal{V}_0$, and $\mathcal{V}_0$ is the mean vector computed from all gVALD representation vectors. $\textbf{D}=\mathrm{diag}[d_1, d_2, \dots, d_D]$ is the $D \times D $ diagonal matrix containing the eigenvalues and $\textbf{E}=[e_1, e_2, \dots, e_D]$ is an orthogonal $N \times D $ matrix having the eigenvectors as columns. The obtained whitened gVLAD representation is: $$\tilde{\mathcal{V}_l} = \textbf{P}(:,1:\rho)^{T} \times \bar{\mathcal{V}_l}$$ where $\rho$ is the number of eigenvectors to keep, i.e. the dimension of reduced feature vectors. $\tilde{V}_l$ is then L2 normalized. The complete algorithm is outlined in Algorithm \[alg:flow\]. \[alg:flow\] ![image](pipeline.pdf){width=".9\textwidth"} Experiments & Evaluations {#sec:experiment} ========================= Benchmark Dataset {#sec:dataset} ----------------- We evaluate the proposed approach on several public available benchmark datasets: *Oxford* buildings, *Paris* and *Holidays*. Large scale experiments are conducted on these datasets by adding $1$M Flickr images as distractors [@jegou2008hamming]. For each of these datasets, performance is measured by mean average precision (mAP) over a set of pre-defined queries and their annotated ground truth matches. **Holidays Dataset:** *Holidays* dataset [@jegou2008hamming] contains $1491$ high resolution personal holiday photos with $500$ annotated queries. For large scale experiments, $1$ million Flickr images are added to it to create *Holidays + Flickr 1M* dataset. About 5%-10% of the images in holiday dataset have orientations which are unnatural for human observer [@perd2009efficient]. We manually rotate these images to create *Rotated Holidays* dataset. **Oxford Dataset:** This dataset, *Oxford 5K* contains $5062$ images of Oxford buildings gathered from Flickr [@philbin2007object]. There are 55 query images each with a rectangular bounding box specifying the region of interest. To test large scale retrieval, it is firstly extended with a 100K Flickr images[^1] to create *Oxford 105K* dataset. We further extend the dataset with 1 million Flickr image[^2] creating *Oxford 5K + Flickr 1M* dataset. **Paris Dataset:** The Paris Dataset [@philbin2008lost] *Paris 6K* consists of $6412$ images collected from Flickr by searching for particular Paris landmarks. There are $60$ query images, each with a rectangular bounding box specifying the region of interest. We found that both 100K Flickr images and Flickr 1M images contains a large number of Paris landmarks, hence we do not extend the Paris dataset with Flickr images. Implementation Details {#sec:implementation} ---------------------- **Descriptor computation:** The pipeline of computing gVLAD descriptor is characterized in Figure \[fig:pipeline\]. First, all images are resized to $1024\times 768$. We find that when using the original resolution of *Holidays* images, the performance is inferior to the down-sampled images. We can also benefit from the speed when using smaller images. In *Oxford* and *Paris* datasets, bounding boxes are provided for queries. We only extract descriptors inside bounding boxes. We use the SIFT and SURF implementations in OpenCV[^3] to detect keypoints and extract descriptors. Each SIFT descriptor has 128 dimensions and each SURF descriptor has 64 dimensions. We find that VLAD based features have better performance using SURF keypoints and descriptors [@bay2006surf] than SIFT keypoints and descriptors [@lowe1999object]. In general, we observed about 10% improvement using SURF as compared to SIFT. More details about the performance difference can be seen by comparing results in Table \[tbl:root\_sift\_vlad\] and Table \[tbl:step-performance\]. **Angle Membership Function:** The angle distribution of SURF keypoints from *Holidays* dataset is shown in Figure \[fig:angleDistribution\] (a). We find that majority keypoints have vertical or horizontal angles as detectors have larger response at these points, resulting in roughly 4 centers ($\pi/2$, $\pi$, $2\pi/3$, $2\pi$). To learn the membership function of each keypoint angle, we apply the proposed approach in  \[sec:membership\]. Because larger number of bins will increase the dimension of the final derived gVLAD feature, to gain a reasonable representation as well as low dimensionality, we set the number of angle bins to be $4$ to fit the distribution. A $\pi/4$ offset and a set of evenly distributed bins: $[-\pi/4, \pi/4)$, $[\pi/4, 3\pi/4)$, $[3\pi/4, 5\pi/4)$ and $[5\pi/4, 7\pi/4)$, are automatically estimated from the algorithm, which is visualized in Figure \[fig:angleDistribution\] (b). We use this angle bin partition in following evaluations. We had also experimented using different number of bins and offset on *Rotated Holidays* dataset. We observe increasing performance as more bins are used, as shown in Figure \[fig:angle-compare\]. This is because that increasing number of bins is equivalent to increasing number of subspaces, by which the distance of descriptor towards centroid can be computed in a more discernible way. However, gains by using 5 or 6 bins as compared to the predicted angle partition ( 4 bins with $\pi/4$ offset) by propose algorithm are marginal, also our learned setting has much smaller dimensions. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![(a) Distribution of keypoint angles from *Holidays* dataset. (b) Learnt 4 angle bins with $\pi / 4$ offset.[]{data-label="fig:angleDistribution"}](holiday_surf.pdf "fig:"){width=".47\linewidth"} ![(a) Distribution of keypoint angles from *Holidays* dataset. (b) Learnt 4 angle bins with $\pi / 4$ offset.[]{data-label="fig:angleDistribution"}](angle_grid.png "fig:"){width=".47\linewidth"} (a) (b) ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- **Vocabulary Generation:** The vocabulary consisting of $K=256$ visual words is computed from all SURF descriptors on Paris dataset. Various different cluster initializations of $k$-means are executed and the best clustering is used as the vocabulary for all evaluations. As the number of extracted descriptors is typically much larger than $K$, e.g. even the smallest Holidays dataset contains 8.3 million SURF descriptors. This vocabulary can be considered independent from all datasets. Such simplification has been used in literature [@arandjelovicall2013all]. For every dataset, this vocabulary is used as a reference vocabulary and a vocabulary adaptation is performed as described in section \[sec:adaptation\]. **Retrieval:** During retrieval, L2 distance is computed to rank images with respect to input query. Since our focus is generating a compact and efficient image descriptor, to illustrate the power of the proposed descriptor, we use brute-force distance computation to report our results. However, our proposed descriptors can in principle be used with approximate distance matching or other hashing based techniques as well, which is beyond the scope of this paper. Performance Evaluation & Analysis {#sec:result} --------------------------------- The performance in all retrieval experiments is evaluated using the mean average precision (mAP), which is defined as the mean of the average precision over all the queries given a dataset. Average Precision is computed as the average of the precision value obtained for the set of top $k$ images after each relevant image is retrieved. We use standard evaluation packages obtained from the data websites. **The Power of Angle:** To illustrate the power of angle, we performed a simple experiment which uses only the angle information from each keypoint to retrieve similar images. After obtaining the keypoints and the angle of each keypoint, we generate an angle histogram for each of image by binning all angles into $Q$ bins. We use L2 distance to compute the similarity between angle descriptors. Table \[tbl:only-angle-bin\] shows the retrieval results on *Rotated Holidays* dataset. It can be seen that surprisingly using only angle information (without any appearance information from SURF or other descriptors), we can still achieve about $26.9\%$ mAP results. Note that the dimension of the angle bin histogram for the best result is only $72$, which is a much smaller number compared to conventional BOW or VLAD descriptors. --------- ------- ------- ------- ------- ------- ------- **$Q$** 2 4 8 18 36 72 **mAP** 0.015 0.037 0.149 0.241 0.261 0.269 --------- ------- ------- ------- ------- ------- ------- : Retrieval performance on *Rotated Holidays* dataset using only Angle binning histogram with varying dimensions, no appearance information is used. Given only $72$ dimension of angle histogram, a surprising $26.9\%$ mAP result is achieved. []{data-label="tbl:only-angle-bin"} **Step-by-Step Performance Evaluation:** To show the performance gain obtained from each of the proposed steps, we performed a step-by-step experiment on *Rotated Holidays* dataset and baselined it with VLAD performance. Results are listed in Table \[tbl:step-performance\]. All results use SURF detector and SURF descriptor. It can be seen that by adding inter-Z-Score normalization to the original VLAD, the performance is increased by $5.4\%$. Performing Angle Binning over VLAD leads to a gain of $7.3\%$. By combining both Angle Binning and Z-Score normalization, we achieve $14.7\%$ improvement over VLAD representation. Performing vocabulary adaptation for *Rotated Holidays* dataset provides additional $3.8\%$ performance gain. Finally, PCA whitening is applied which is able to reduce the dimension significantly with only about $1.1\%$ performance loss, as compared to PCA without whitening having a loss of $3.6\%$. To demonstrate the performance of low-dimensional gVLAD descriptor using PCA whitening, we further plot the mAP performance curve by varying $\rho=2^4$ to $2^{16}$ in Figure \[fig:pca\_compression\]. It can be seen that with only $32$ dimensions, the performance by the proposed descriptor can reach to $mAP=0.737$, which already outperforms the original VLAD descriptor using $1024$ visual words with $65,536$ dimension ($mAP=0.670$) as shown in Table \[tbl:step-performance\]. We also test our proposed method using SIFT detectors and root SIFT [@arandjelovic2012three] descriptors, since most previous published work use SIFT. For fair comparison, we implement VLAD with root SIFT descriptors, which have better performance compared with 0.526 on *Holidays* dataset reported in [@jegou2010aggregating]. Results as shown in Table \[tbl:root\_sift\_vlad\] demonstrate the superior performance of proposed approach over SIFT descriptors as well. As noted, comparing Tabel \[tbl:root\_sift\_vlad\] and Table \[tbl:step-performance\], we observe in general that using SURF descriptors outperforms SIFT based descriptors. ![Dimension reduction on original gVLAD descriptor using PCA whitening. The original feature dimension is 65,536. After compressed to 128-D, the mAP decreases only about 1%.[]{data-label="fig:pca_compression"}](pca_compression.pdf){width=".45\textwidth"} **Dataset** **VLAD**\* **gVLAD** ------------------ ------------ ----------- Holidays 0.548 **0.710** Rotated Holidays 0.550 **0.786** : Comparison of our proposed gVLAD with VLAD on benchmark datasets. Both VLAD and gVLAD use SIFT detectors and root SIFT descriptors. $256$ visual words are used. The feature dimension of VLAD is $256\times 128 = 32768$, and for gVLAD is $256\times 128\times 4 = 131072$. \* denotes our implementation.[]{data-label="tbl:root_sift_vlad"} **Method** **Dimension** **mAP** ------------------------------------------- --------------- ----------- VLAD ($K=256$)\* [@jegou2012aggregating] 16,384 0.662 VLAD ($K=1024$)\* [@jegou2012aggregating] 65,536 0.670 VLAD ($K=256$) + inter-norm 16,384 0.716 VLAD ($K=256$) + Angle Binning 65,536 0.735 + inter-norm 65,536 0.809 + Voc Adaptation 65,536 **0.847** PCA 128 0.811 PCA + whitening 128 **0.836** : mAP on the ***Rotated Holidays*** dataset comparing to start-of-art results. Best performances are in bold. \*VLAD result in this table are based on our implementation. All results use SURF detector and SURF descriptor. \[tbl:step-performance\] **Full Size & Low Dimensional gVLAD Descriptors:** We compared our proposed method with several benchmark results in [@jegou2012aggregating; @sivic2003video; @perronnin2010large; @arandjelovicall2013all] for both full size descriptor and dimension reduced descriptor ($\rho=128$). Experiments are done using both *Holidays* dataset and *Oxford 5K* dataset. Table \[tbl:fullfeature\] shows that the proposed approach significantly outperforms the state-of-the-art performance by approximately $16.6\%$ and $7\%$ on *Holidays* and *Oxford 5K* dataset respectively. For low dimensional case, as shown in Table \[tbl:pcafeature\], our algorithm outperforms the best state-of-art result by $15\%$ on both datasets. Comparing Table \[tbl:fullfeature\] and \[tbl:pcafeature\], results also show that the proposed gVLAD descriptor is quite powerful in the sense that even with PCA whitening and reduced dimension, it can still achieve better result as compare to the best benchmark results with full size descriptors. In addition, PCA whitening based dimension reduction only results in small amount of performance decrease which is about $2.76\%$ in average of both datasets, and $1.1\%$ in the best case (*Rotated Holidays*). **Method** **Dimension** **Holidays** **Oxford** ------------------------------------------------------ --------------- -------------- ------------ BoW 20k-D [@jegou2012aggregating] [@sivic2003video] 20,000 0.452 0.354 BoW 200k-D [@jegou2012aggregating] [@sivic2003video] 200,000 0.540 0.364 Improved Fisher [@perronnin2010large] 16,384 0.626 0.418 VLAD [@jegou2010aggregating] 8,192 0.526 - VLAD + SSR [@jegou2012aggregating] 16,384 0.598 0.378 Improved VLAD + SSR [@arandjelovicall2013all] 32,768 - 0.532 VLAD + intra-norm [@arandjelovicall2013all] 32,768 0.646 0.555 Ours 65,536 **0.812** **0.626** : mAP performance by full size gVLAD descriptors as compared to state-of-the-art results on ***Holidays*** and ***Oxford***. Existing approaches are based on SIFT descriptors, while the proposed gVLAD descriptor uses SURF detector and SURF descriptor. Best performances are in bold. \[tbl:fullfeature\] **Method** **Holidays** **Oxford** ----------------------------------------------- -------------- ------------ GIST [@jegou2012aggregating] 0.365 - BoW  [@jegou2012aggregating; @sivic2003video] 0.452 0.194 Improved Fisher [@perronnin2010large] 0.565 0.301 VLAD [@jegou2010aggregating] 0.510 - VLAD + SSR [@jegou2012aggregating] 0.557 0.287 Multivoc-BoW [@jegou2012negative] 0.567 0.413 Multivoc-VLAD [@jegou2012negative] 0.614 - VLAD + intra-norm [@arandjelovicall2013all] 0.625 0.448 Ours **0.779** **0.600** : mAP performance by gVLAD low dimensional descriptors ($\rho=128$): comparison with state-of-the-art on the *Holidays* and *Oxford 5k* benchmarks. The existing approaches are based on SIFT descriptors, while the proposed gVLAD descriptor uses SURF detector and descriptor. Best performances are in bold. \[tbl:pcafeature\] **Performance on Large Scale Dataset:** We scale the proposed algorithm to large scale image dataset with millions of images, and test on both using full size gVLAD and PCA dimension reduced 128-D descriptors. In total, $4$ large scale datasets are used, including *Holidays + Flickr 1M*, *Rotated Holidays + Flickr 1M*, *Oxford 105K*, and *Oxford 5K + Flickr 1M*. As can be seen from Table \[tbl:large\], our methods outperform all current state-of-the-art methods. For example, using dimension reduced 128-D gVLAD descriptors, on *Holidays + Flickr 1M* dataset, our algorithm outperforms the best result [@arandjelovicall2013all] reported in literature with a significant gain of $22.8\%$. On *Oxford 105K* dataset, we are able to achieve $11.6\%$ better result than  [@arandjelovicall2013all]. Further, same with our previous observation in Table \[tbl:pcafeature\] as compared to Table \[tbl:fullfeature\], Table \[tbl:large\] also shows performance only drops very slightly using the proposed PCA whitening. This implies that the proposed gVLAD descriptor is quite powerful. Also, being combined with proper dimension reduction schema, effective representation with computational efficiency can be achieved. -------------------------------- --------------------------------- --------------------------------- ------------------------ ----------- ----------------- **State of the Art** **State of the Art** **Ours** **Ours** **Ours** **Dataset** **Original Dimension** **128-D** **Original Dimension** **128-D** **Loss in PCA** *Holidays* 0.646 [@arandjelovicall2013all] 0.625 [@arandjelovicall2013all] **0.812** **0.779** 0.033 *Holidays + Flickr 1M* - 0.378 [@arandjelovicall2013all] - **0.607** - *Rotated Holidays* - - **0.847** **0.836** 0.011 *Rotated Holidays + Flickr 1M* - - - **0.654** - *Oxford 5K* 0.555 [@arandjelovicall2013all] 0.448 [@arandjelovicall2013all] **0.626** **0.600** 0.026 *Oxford 105K* - 0.374 [@arandjelovicall2013all] - **0.490** - *Oxford 5K + Flickr 1M* - - - **0.438** - *Paris 6K* 0.494 [@philbin2008lost] - **0.631** **0.592** 0.039 -------------------------------- --------------------------------- --------------------------------- ------------------------ ----------- ----------------- \[tbl:large\] Time Complexity and Memory Footprint {#sec:complexity} ------------------------------------ Each image takes 512 bytes in memory after being converted to 128 dimensional gVLAD feature vector by PCA compression. The largest dataset (*Holidays + Flickr 1M*) in our experiment occupies $0.5$GB of RAM for keeping all features in memory. To evaluate the time complexity of each step in the proposed gVLAD computation, we conduct experiments on this dataset using a Ubuntu machine with two Xeon X5675 CPUs at 3.07GHz, with 12 physical cores and 24 logical cores in total. We rely on multi-threading whenever possible. Table \[tbl:timing\] illustrates the average results on $10$ randomly selected queries. As shown, our proposed technique takes approximately $100$ millisecond to compute gVLAD representation, and $750$ millisecond to perform an end-to-end brute-force retrieval over the entire inventory. Since our proposed descriptors can in principle be used with other approximate distance matching or indexing schema, better retrieval speed can be expected, which will be very useful in practical applications. **Process** **Mean $\pm$ std. (ms)** ------------------------------ -------------------------- SURF detection & description 373.5 $\pm$ 69.1 gVLAD computation 71.7 $\pm$ 20.3 PCA compression 28.0 $\pm$ 3.6 Nearest neighbor search 266.7 $\pm$ 36.3 : Speed analysis based on 10 random query images from *Holidays + Flickr 1M* dataset. \[tbl:timing\] Conclusion {#sec:conclusion} ========== We present gVLAD which is a novel extension of popular VLAD descriptor for large scale image search. Our proposed descriptor extends VLAD by integrating weak geometric cues in form of key point angles. A principled technique to represent this information as membership function over angles is also presented. The vocabulary adaptation and inter-Z-score normalization are also proposed to improve the performance of the system. Extensive experiments are conducted on existing publicly available benchmark datasets which demonstrate the superior performance of our approach. Our future work focuses on exploring efficient indexing strategies to avoid the brute-force matching of images. We are also investigating other related low level information that can be further integrated into gVLAD to make the representation more powerful. [^1]: [http://www.robots.ox.ac.uk/ vgg/data/oxbuildings/flickr100k.html](http://www.robots.ox.ac.uk/~vgg/data/oxbuildings/flickr100k.html) [^2]: <http://press.liacs.nl/mirflickr/> [^3]: <http://opencv.org/>
--- abstract: 'We establish the global well-posedness of overdamped dynamical density functional theory (DDFT): a nonlinear, nonlocal integro-partial differential equation used in statistical mechanical models of colloidal flow and other applications including nonlinear reaction-diffusion systems and opinion dynamics. With no-flux boundary conditions, we determine the well-posedness of the full nonlocal equations including two-body hydrodynamic interactions (HI) through the theory of Fredolm operators. Principally, this is done by rewriting the dynamics for the density $\varrho$ as a nonlocal Smoluchowski equation with a non-constant diffusion tensor $\bm{D}$ dependent on the diagonal part ($\bm{Z}_1$) of the HI tensor, and an effective drift $\bm{A}[\vec{a}]$ dependent on the off-diagonal part ($\bm{Z}_2$). We derive a scheme to uniquely construct the mean colloid flux $\vec{a}(\vec{r},t)$ in terms of eigenvectors of $\bm{D}$, show that the stationary density $\varrho(\vec{r})$ is independent of the HI tensors, as well as proving exponentially fast convergence to equilibrium. The stability of the equilibria $\varrho(\vec{r})$ is studied by considering the bounded (nonlocal) perturbation of the differential (local) part of the linearised operator. We show that the spectral properties of the full nonlocal operator with no-flux boundary conditions can differ considerably from those with periodic boundary conditions. We showcase our results by using the numerical methods available in the pseudo-spectral collocation scheme 2DChebClass.' author: - '$^\dagger$,' - '[^1] ,' - '[^2]' bibliography: - 'bibYear3.bib' title: '****' --- dynamic density functional theory (DDFT), colloids, overdamped limit, hydrodynamic interactions, nonlocal-differential PDEs, interacting particle systems, McKean-Vlasov equation, phase transitions, bifurcation theory. 60F10; 60J75; 62P10; 92C37 [^3] Introduction {#intro} ============ For suspended particles in a viscous fluid, the Navier-Stokes equations are not sufficient to model flows on a spatial scale comparable with the size of the individual particles. Instead, one requires a computationally tractable model that captures meso/macro-scale dynamics whilst also including physical effects driven by particle-level interactions. Dynamic density functional theories (DDFTs) are excellent candidates for modelling such systems [@marconi1999dynamic; @ArcherEvans04]. They are typically applied in condensed matter physics in the colloidal particle regime with particles of typical diameters 1nm$-$1$\mu$m. Recent advances have allowed the inclusion of inertia [@MarconiTarazonaCecconiMelchionna07; @archer2009dynamical], multiple species [@Archer05; @RothRauscherArcher09; @GNK13; @LichtnerArcherKlapp12], hydrodynamic interactions (HI) [@rex2009dynamical; @Rauscher10; @goddard2012general; @goddard2012unification], background flows [@RauscherDominguezKrugerPenna07], temperature gradients [@WittkowskiLowenBrand12; @anero2013functional], hard spheres [@Rosenfeld89; @RothEvansLangKahl02; @roth2010fundamental; @StopperMaroltRothHansen-Goos15], confined geometries [@goddard2016dynamical; @zimmermann2016flow], arbitrary shaped particles [@WittkowskiLowen11], and active microswimmers [@menzel2016dynamical; @hoell2017dynamical]. For equilibrium fluids, there is a rigorous mathematical framework proving the existence of nontrivial fluid densities, different from those found by classical fluid dynamical formalisms, by taking into account both many body effects and external force fields. This is commonly known as (classical) density functional theory (DFT) [@Mermin:1965lo]. It is able to predict effects driven by the microscale, e.g., the non-smooth droplet profiles which are formed at the gas-liquid-solid trijunction in contact line problems [@berim2009simple] and the coexistence of multiple fluid films at critical values of the chemical potential energy in droplet spreading [@pereira2012equilibrium]. It has been used to resolve the paradox of stress and pressure singularities normally found in classical moving contact line problems [@sibley2013contact]. What is more, DFT agrees well with molecular dynamics simulations; see, e.g., [@Lutsko10] and references therein. These advancements motivate more mathematical analysis, in particular, on the well-posedness of the underlying equations being used and on the number and structure of equilibrium states. As a non-equilibrium extension to DFT for classical fluids, dynamic DFT (DDFT) has been applied to a wide range of problems: polymeric solutions [@PennaDzubiellaTarazona03], spinodal decomposition [@ArcherEvans04], phase separation [@Archer05], granular dynamics [@MarconiMelchionna07; @MarconiTarazonaCecconi07], nucleation [@vanTeeffelenLikosLowen08], liquid crystals [@WittkowskiLowenBrand10], and evaporating films [@ArcherRobbinsThiele10]. Recently, a stochastic version of DDFT has been derived [@Lutsko12], which allows the study of energy barrier crossings, such as in nucleation. A crucial point is that the computational complexity of DDFT is (essentially) constant in the number of particles, which allows the treatment of macroscopically large systems, whilst retaining microscopic information. Furthermore, due to the universality of the underlying nonlinear, nonlocal partial differential equations, DDFT may be considered as a generalisation of a wider class of such models used in the continuum modelling of many natural phenomena consisting of complex, many body, multi-agent interparticle effects including: pattern formation [@camazine2003self], the flocking of birds, cell proliferation, the self organising of morphogenetic and bacterial species [@canizo2010collective; @carrillo2009double], nonlocal reaction-diffusion equations [@al2018dynamical] and even consensus modelling in opinion dynamics[@chazelle2017well]. Many of these applications are often described as systems of interacting (Brownian) particles and, in the case of hard particle viscous suspensions, bath-mediated HI effects may be included. The HI are forces on the colloids mediated by the bath flow, generated by the motion of the colloidal particles. This in turn produces a nontrivial particle–fluid–particle hydrodynamic phenomenon, the inclusion of which has been shown to have substantial effects on the physics of many systems; for example, they have been found to be the underlying mechanism for the increased viscosity of suspensions compared to a pure bath [@Einstein06], the blurring of laning that arises in driven flow [@WysockiLowen11], the migration of molecules away from a wall [@HodaKumar07], and are particularly complex in confined systems [@happel2012low; @LaugaSquires05], and for active particles and microswimmers, which result in additional HI [@HuberKoehlerYang11]. Mathematically, the inter-particle forces and HI can be described through the hydrodynamic fields $\varrho$ and $\vec{v}$, the one-body density and one-body velocity fields, respectively. These fields, inherent to a continuum description of a collection of particles, are derived by considering successive moments (density, velocity, heat flux, …) of the underlying kinetic system [@gorban2014hilbert]. In particular, for systems of interacting Newtonian particles, when the momenta are non-negligible, the evolution of the phase space density $f(\vec{r}^N,\vec{p}^N, t)$ for a system of $N$ colloids determining the probability of finding the system in the state $(\vec{r}^N,\vec{p}^N)$ at time $t$ is described by the $N$-body Fokker-Planck equation and the dynamics of the hydrodynamic fields are defined by obtaining closed equations for $\{\varrho, \varrho\times \vec{v}\} := \int \mathrm{d}\vec{r}^{N-1}\,\mathrm{d}\vec{p}^N\, \{1, \vec{p}/m\}f(\vec{r}^N,\vec{p}^N, t)$, where $m$ is the particle mass. Here, $\vec{r}^N$ and $\vec{p}^N$ denote the $3N$-dimensional position and momentum vectors of all $N$ particles. The inclusion of HI leads to a much richer hierachy of fluid equations compared to systems without HI; compare e.g. [@goddard2012unification] and [@archer2009dynamical]. In particular, see e.g. [@goddard2012unification], by integration over all but one particle position, the one-body Fokker-Planck equation may be obtained. If, in addition, two-body HI and interparticle interactions are assumed and the inertia of the colloids is considered small, a high friction limit $\gamma\to \infty$ may be taken [@goddard2012overdamped]. The result is that the velocity distribution converges to a Maxwellian, and one can eliminate the momentum variable through an adiabatic elimination process that is based on multiscale analysis [@pavliotis2008multiscale]. The final one-body Smoluchowski equation for $\varrho$ is a novel, nonlinear, nonlocal PDE shown to be independent of the unknown kinetic pressure term $\int \mathrm{d}\vec{r}\,\mathrm{d}\vec{p}\, m^{-2} \vec{p}\otimes\vec{p} f(\vec{r},\vec{p}, t)$, which normally persists at $\gamma = O(1)$ (see[@goddard2012overdamped], Theorem 4.1). Existence, uniqueness and global asymptotic stability of the novel Smoluchowski equation in this overdamped limit has, until this work, remained unproven. It is the inclusion of HI that provides richness through additional nonlinearities in both the dissipation and convection terms. The inclusion of HI is interesting from both physical and mathematical standpoints. Physically, as above, the HI give rise to a much more complex evolution in the density. Mathematically, the convergence to equilibrium will depend inherently on the spectral properties of the effective diffusion tensor and effective drift vector arising from the HI. What is more, since the full $N$-body Fokker-Planck equation is a PDE in a very high dimensional phase space, well-posed nonlinear, nonlocal PDEs governing the evolution of the one-particle distribution function, valid in the mean field limit, describing the flow of nonhomogeneous fluids are desirable for computational reasons. The equations studied in this paper are related to the McKean-Vlasov equation [@chayes2010mckean], a nonlinear nonlocal PDE of Fokker-Planck type that arises in the meanfield limit of weakly interacting diffusions. The novelty of the present problem lies in the space dependent diffusion tensor and nonlinear, nonlocal boundary conditions. Additionally, the problem that we study in this paper may in general not be written as a gradient flow, with the exception of the modelling assumption that the off-diagonal elements of the friction tensor $\bm{\Gamma}$ are zero. This choice is equivalent to setting $\bm{Z}_2$ to zero, and would be physically relevant for a diffuse system of particles with a strong hydrodynamic interaction with a wall but weak inter-particle hydrodynamic interactions [@goddard2016dynamical]. Description of the Model. ------------------------- In this work we analyse the overdamped partial differential equation (PDE) associated to a system of interacting stochastic differential equations (SDEs) on $U$ an open, bounded subset of $\mathbb{R}^d$ of the following form, governing the positions $\vec{r}_i$ and momenta $\vec{p}_i$ of $i = 1,\dots, N$ colloidal particles immersed in a bath of many more, much smaller and much lighter particles: $$\begin{aligned} \frac{\mathrm{d}\vec{r}_i}{\mathrm{d}t} &= \frac{1}{m}\vec{p}_i,\label{eq:SDE_overdamped_pos}\\ \frac{\mathrm{d}\vec{p}_i}{\mathrm{d}t} &= -\nabla_{\vec{r}_i} V(\vec{r}^N,t)-\sum_{j=1}^{N}\boldsymbol{\Gamma}_{ij}(\vec{r}^N)\vec{p}_j + \sum_{j=1}^N\boldsymbol{B}_{ij}(\vec{r}^N)\vec{f}_{j}(t) \label{eq:SDE_overdamped_mom}\end{aligned}$$ where $\vec{r}^N = (\vec{r}_1,\cdots,\vec{r}_N)$, $\boldsymbol{B} = \left( mk_BT\boldsymbol{\Gamma}\right)^{1/2}$, $\boldsymbol{\Gamma} = \gamma (\bm{1} + \tilde{\boldsymbol{\Gamma}})$ (where the tilde denotes the nondimensional tensor and $\bm{1}$ is the $3N\times 3N$ identity matrix), $V$ is a potential, $k_B$, $\mathrm{T}$, $\gamma$ are Boltzmann’s constant, temperature and friction, respectively, and $\vec{f}_i(t) = (\zeta^x_i(t),\zeta^y_i(t),\zeta^z_i(t))^\top$ is a Gaussian white noise term with mean and correlation given by $\langle \zeta_i^a(t)\rangle = 0$ and $\langle\zeta_i^a(t),\zeta_j^b(t) \rangle = 2\delta_{ij}\delta^{ab}\delta(t-t')$. In $d = 3$ dimensions, the friction tensor $\boldsymbol{\Gamma}$ comprises $N^2$ positive definite $3\times 3$ mobility matrices $\boldsymbol{\Gamma}_{ij}$ for the colloidal particles. These couple the momenta of the colloidal particles to HI forces on the same particles, mediated by fluid flows in the bath. Typically, in the underdamped limit with dense suspensions, the HI may be short range lubrication forces, whereas in disperse systems in the overdamped limit, the HI are taken to be the long range forces given by the Rotne-Prager-Yamakawa tensor [@rotne1969variational]. However, we do not make any such assumptions on the form of the tensors here. We have described a general set of coupled Langevin equations with spatially-dependent friction tensor $\boldsymbol{\Gamma}(\vec{r}^N)$. As we will see, the dynamics – tend towards an equilibrium given by the Gibbs probability measure, which we will show to be independent of the friction tensor. Instead of computing the trajectories of individual particles we consider the evolution of the density of particles $\varrho(\vec{r},t)$ given by the Smoluchowski equation in the high friction limit $\gamma\to \infty$, $$\begin{aligned} \label{eq:mk-eq} \qquad \partial_{t}\varrho(\vec{r},t) =-\tfrac{k_B\mathrm{T}}{m\gamma} \nabla_{\vec{r}}\cdot\vec{a}(\vec{r},[\varrho], t) \qquad \text{ for } \vec{r} \in U,\,t\in [0,T]\end{aligned}$$ where $\vec{a}(\vec{r},[\varrho], t)$ is the flux, $[\varrho]$ denotes functional dependence, $U\subseteq\mathbb{R}^d$ and $T<\infty$. Equation was derived rigorously as a solvability condition of the corresponding Vlasov-Fokker-Planck equation for the one-body density in position and momentum space $f(\vec{r},\vec{p},t)$ by writing $f$ as a Hilbert expansion in a small nondimensional parameter $\epsilon\propto\gamma^{-1}$ [@goddard2012overdamped]. Therein, $\epsilon$ has units length, and therefore a problem specific length scale must be introduced to make it truly nondimensional. We are interested in global existence, uniqueness, positivity and regularity of the weak solution to when $\vec{a}(\vec{r},t)$ is given by the integral equation $$\begin{aligned} \vec{a}(\vec{r},t) + \boldsymbol{H}[\vec{a},\varrho](\vec{r},t)+\frac{\varrho(\vec{r},t)}{k_BT}\bm{D}(\vec{r},[\varrho],t)\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho](\vec{r},t) =0,\label{eq:eqn_for_a}\end{aligned}$$ $$\begin{aligned} \boldsymbol{H}[\vec{a},\varrho](\vec{r},t):= \varrho(\vec{r},t)\bm{D}(\vec{r},[\varrho],t)\int_U\mathrm{d}\vec{r}'\, g(\vec{r},\vec{r}')\boldsymbol{Z}_2(\vec{r},\vec{r}')\vec{a}(\vec{r}',t),\label{eq:eqn_for_H}\end{aligned}$$ $$\begin{aligned} &\frac{\varrho(\vec{r},t)}{k_BT}\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho](\vec{r},t) := [\nabla_{\vec{r}}+\tfrac{1}{k_B\mathrm{T}}\Big(\nabla_{\vec{r}}V_1(\vec{r},t)\nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad+\int_U\mathrm{d}\vec{r}'\varrho(\vec{r}',t)g(\vec{r},\vec{r}')\nabla_{\vec{r}}V_{2}(\vec{r},\vec{r}')\Big) ]\varrho(\vec{r},t),\label{eq:eqn_for_J}\end{aligned}$$ where to ease notation we have suppressed $[\varrho]$ in the argument of $\vec{a}$ and $\mathcal{F}$ is the free energy functional which will be defined in Section \[subsec:free\_energy\_framework\]. The functions $V_1$ and $V_2$ are the external and (two body) interparticle potentials respectively. Additionally, the non-constant diffusion tensor $$\begin{aligned} \label{eq:def_diffusion_tensor} \boldsymbol{D}(\vec{r},[\varrho],t):=\frac{k_{\text{B} }\mathrm{T}}{m \gamma}\Big[\boldsymbol{1}+\int\mathrm{d}\vec{r}'g(\vec{r},\vec{r}')\varrho(\vec{r}',t)\boldsymbol{Z}_1(\vec{r},\vec{r}')\Big]^{-1}\end{aligned}$$ will be considered; this is interesting from a physical point of view. It has been previously shown (see [@goddard2012overdamped]) that for $\boldsymbol{Z}_1$ being positive definite, $\boldsymbol{D}$ is also positive definite and therefore has positive, finite eigenvalues. The term $g(\vec{r},\vec{r}')$ (regarded as known) is the correlation function defined by the two-body density $\varrho^{(2)}(\vec{r},\vec{r}',t) =g(\vec{r},\vec{r}')\varrho(\vec{r},t)\varrho(\vec{r}',t)$ and the operator $\boldsymbol{H}[\cdot]$ describes terms corresponding to HI. We note that if $\boldsymbol{D}$ were positive semidefinite, a zero eigenvalue of $\boldsymbol{D}$ is permitted, which physically-speaking would amount to the colloidal system possessing a zero diffusion rate in some subset of $U$ with nonzero measure. Such systems are interesting (for example, in many biological systems the physical domain $U$ could be a substrate including cuts, voids or interior walls) but are not considered in this paper. Throughout this work the largest and smallest eigenvalues of $\boldsymbol{D}$ will be denoted $\mu_{\max}$ and $\mu_{\min}$, respectively. Furthermore, for two-body HI, $\boldsymbol{Z}_1$, $\boldsymbol{Z}_2$ are the diagonal and off-diagonal blocks respectively of the translational component of the grand resistance matrix originating in the classical theory of low Reynolds number hydrodynamics between suspended particles [@happel2012low], [@jeffrey1984calculation], related to the friction tensor by $$\begin{aligned} \tilde{\boldsymbol{\Gamma}}_{ij}(\vec{r}^N) = \delta_{ij}\sum_{l\neq i}\boldsymbol{Z}_{1}(\vec{r}_i,\vec{r}_l)+(1-\delta_{ij})\boldsymbol{Z}_{2}(\vec{r}_i,\vec{r}_j).\end{aligned}$$ In $d = 3$ dimensions, and for the particular case $N=2$ (where $N$ is the number of particles in the system), $\bm{\Gamma}\in \mathbb{R}^{6\times 6}$ and $\bm{\Gamma}_{ij}$ may be seen as equivalent to the second-rank tensor of the translational part of the resistance matrix as found in [@jeffrey1984calculation] used to model lubrication forces. It should be noted however that the definition of those resistance matrices are formalism dependent, that is, the individual entries are scalar functions arising from the solution of Stokes equations for two-body lubrication interactions using multipole methods. Conversely, $\bm{\Gamma}_{ij}$ are general tensors, independent of the type of HI under consideration, and are therefore a more general representation of hydrodynamic phenomena of colloidal suspensions. Additionally, $\bm{\Gamma}_{ij}$ may be used to model not just lubrication forces between particles but also long range forces, wall effects and more. In the case of inter-particle HI, the diagonal blocks $\bm{\Gamma}_{ii}$ each represent the force exerted on the fluid due to the motion of particle $i$, which is simply the sum of all the pairwise HI from the perspective of particle $i$. The off-diagonal blocks $\bm{\Gamma}_{ij}$ represent the force on particle $i$ due to the motion of particle $j$. The stationary equations for the equilibrium density $\varrho(\vec{r})$ and equilibrium flux $\vec{a}(\vec{r})$ are given by $$\begin{aligned} &\qquad\qquad\qquad\qquad\nabla_{\vec{r}}\cdot \vec{a}(\vec{r}) = 0,\label{eq:div_a_0}\\ &\vec{a}(\vec{r}) + \boldsymbol{H}[\vec{a},\varrho](\vec{r})+\frac{\varrho(\vec{r},t)}{k_BT}\bm{D}(\vec{r},[\varrho],t)\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho](\vec{r}) =0.\label{eq:eqn_for_a_in_equilibrium} \end{aligned}$$ Note that given a finite flux vector $\vec{a}$ solving -, it is not obvious that $\varrho$ is necessarily a minimiser of the free energy $\mathcal{F}-\int_U\mathrm{d}\vec{r}\,\mu_{c}\varrho$ (where $\mu_{c}$ is the chemical potential of the species). However, for the particular choice $\vec{a}\equiv \vec{0}$ (which is a natural and physically realistic solution), $\varrho$ is necessarily a minimiser of $\mathcal{F}-\int_U\mathrm{d}\vec{r}\,\mu_{c}\varrho$, and we will show that under reasonable assumptions these are indeed the only fixed points of the system. Previous well-posedness studies of similar nonlinear, nonlocal PDEs focused on periodic boundary conditions; see, e.g., [@chazelle2017well; @greg_mckean_vlasov_torus]. In contrast, we are interested in the well-posedness of , - subject to no-flux boundary conditions. This choice admits the nontrivial effect of the two body forces generated by the potential $V_2$ interacting with density on the boundary of the physical domain. We also seek to understand the asymptotic stability of stationary states. The motivation for this choice of boundary condition is physical; it corresponds to a closed system of particles in which the particle number is conserved over time. It is clear that most applications of such equations will be in confined systems, rather than a periodic domain and, as such, no-flux boundary conditions are natural. We note that the choice of boundary condition is expected to have significant effects on the dynamics, including the form of the bifurcation diagram. Free Energy Framework. {#subsec:free_energy_framework} ---------------------- Related to the system -, we define the free energy functional $\mathcal{F}:P^+_{\text{ac}}(U)\to \mathbb{R}$ where $P^+_{\text{ac}}$ is the set of strictly positive definite absolutely continuous probability measures on $U$. We define $$\begin{aligned} \label{eq:def_of_F} \mathcal{F}[\varrho]&:=\int_U\mathrm{d}\vec{r}\,\varrho(\vec{r},t)\log\varrho(\vec{r},t)+\int_U\mathrm{d}\vec{r}\,\varrho(\vec{r},t)\,\Big[V_1(\vec{r},t)+\tfrac{1}{2}(gV_2)\star\varrho \Big],\end{aligned}$$ where $\star$ denotes convolution in space. Here we assume the probability measure $\varrho$ has density with respect to the Lebesgue measure. Additionally we define the probability measure on $U$ $$\begin{aligned} \label{eq:def_of_measure} \mu(\mathrm{d}\vec{r}) = \mathrm{d}\vec{r}\,Z^{-1}e^{-\tfrac{(V_1+(gV_2)\star\varrho)}{k_BT}}\end{aligned}$$ where $Z = \int_U \mathrm{d}\vec{r}\,e^{-\tfrac{(V_1+(gV_2)\star\varrho)}{k_BT}}$ and $\varrho$ (when it exists) satisfies the nonlinear equation $$\varrho = Z^{-1}e^{-\tfrac{(V_1+(gV_2)\star\varrho)}{k_BT}}.$$ The existence of a probability density $\varrho$, and therefore a probability measure $\mu$ in , is obtained by Lemma \[thm:exis\_fix\_point\]. The functional $\mathcal{F}$ gives rise to the density minimising the free energy associated to the system - as $\gamma \to \infty$, which will be shown in Theorem \[thm:association \_of\_free\_energy\]. To make the connection between the free energy functional $\mathcal{F}$ in and the theory of non-uniform classical fluids, one may consider the Helmholtz free energy functional, which is the central energy functional of DFT [@evans1979nature] $$\begin{aligned} \label{eq:ddft-helmholtz_func} \mathcal{F}_{H}[\varrho] = \int_U\mathrm{d}\vec{r}\,\varrho(\vec{r},t)V_1(\vec{r},t)+ k_BT\int_U\mathrm{d}\vec{r}\,\varrho(\vec{r},t)[\log(\Lambda^3\varrho(\vec{r},t))-1]+\mathcal{F}_\text{ex}[\varrho]\end{aligned}$$ where $\mathcal{F}_{\text{ex}}$ is the excess over ideal gas term and $\Lambda$ the de Broglie wavelength, which turns out to be superfluous. The term $\mathcal{F}_{\text{ex}}$ is not in general known, the exception being for one dimensional hard rods [@percus1976equilibrium]. Using the free energy functional $\mathcal{F}_H$, the corresponding Euler-Lagrange equation is $$\begin{aligned} \label{eq:chemical_potential_euler_lagrange_eqn} \mu_{c}=V_1(\vec{r}) + k_BT [\log (\Lambda^3\varrho(\vec{r}))-1]+\tfrac{\delta\mathcal{F}_{\text{ex}}}{\delta\rho}[\varrho]\end{aligned}$$ where $\mu_{c}$ is the chemical potential which is constant at equilibrium. Note that $\mu_c$ should not be confused with the measure $\mu$ defined in . After taking the gradient of and multiplying by $\varrho$ we obtain $$\begin{aligned} 0=\varrho(\vec{r})\nabla_{\vec{r}}\tfrac{\delta\mathcal{F}}{\delta\rho}[\varrho]=k_BT\nabla_{\vec{r}}\varrho+\varrho(\vec{r})\nabla_{\vec{r}}\Big(V_1(\vec{r})+\tfrac{\delta\mathcal{F}_{\text{ex}}}{\delta\rho}[\varrho]\Big).\end{aligned}$$ At equilibrium, the sum rule holds (see, e.g. [@goddard2012unification]) $$\begin{aligned} \label{eq:excess_free_energy_equilibrium_sum} \varrho(\vec{r})\nabla_{\vec{r}}\tfrac{\delta\mathcal{F}_{\text{ex}}}{\delta\varrho}[\varrho]=\sum_{n=2}^N\int\mathrm{d}\vec{r}^{n}\nabla_{\vec{r}}V_n(\vec{r}^n)\varrho_n(\vec{r}^n).\end{aligned}$$ where $\varrho_n(\vec{r}^n)$ is the standard $n-$particle configuration distribution function in equilibrium. Limiting the particle interactions to two-body, for example with the approximation $\varrho_2(\vec{r},\vec{r}') = \varrho(\vec{r})\varrho(\vec{r}')g(\vec{r},\vec{r}',[\varrho])$, we take the first term in the above series to obtain the equality $\nabla_{\vec{r}}\mathcal{F}_H[\varrho]= \nabla_{\vec{r}}\mathcal{F}[\varrho]$. In this way wee see that the density minimising $\mathcal{F}_H$ will minimise $\mathcal{F}$. When $\boldsymbol{Z}_2\equiv 0 $, and by using the adiabatic approximation that holds out of equilibrium, we note that PDE simplifies to (cf. [@rex2009dynamical]) $$\begin{aligned} \label{eq:ddft-eq} \partial_{t}\varrho = \nabla_{\vec{r}}\cdot \left[\boldsymbol{D}(\vec{r},t)\varrho(\vec{r},t)\,\nabla_{\vec{r}}\tfrac{\delta \mathcal{F}}{\delta \varrho}[\varrho]\right].\end{aligned}$$ From we conclude that the dynamics under the choice $\bm{Z}_2 \equiv 0$ has a gradient flow structure. When $\bm{Z}_2$ is not necessarily zero, one cannot in general write the full dynamics as a gradient flow and, hence, the inclusion of HI introduces a novel perturbation away from the classical theory of gradient flow structure. Additionally, one sees how the free energy functional gives rise to the concept of a local pressure variation by the term inside the divergence of . In particular, the term $\tfrac{k_\text{B}\mathrm{T}}{m}\varrho(\vec{r},t)\,\nabla_{\vec{r}}\tfrac{\delta \mathcal{F}}{\delta \varrho}[\varrho]$ represents the spatial variation of the energy available to change particle configurations per unit volume at fixed particle number, in other words, it is an analogue of a local pressure gradient for the particle density. We will show that $\mathcal{F}[\varrho]$ is associated to the PDE even when $\bm{Z}_2 \neq 0$, that is $\partial_t \varrho = 0$ implies $\varrho$ is a critical point of $\mathcal{F}$. [0.48]{} ![(a). The bifurcation diagram for (a). $V_2(x,y) = x\cdot y$ and (b). $V_2(x,y) = -\cos\left(\frac{2\pi(x-y)}{\mathrm{L}}\right)$ in Section \[subsec:numericalexperiments\]: the solid [blue]{} line denotes the stable branch of solutions while the dotted [red]{} line denotes the unstable branch of solutions. In (a) the stationary density $e^{-x^2}/Z$ changes stability at the critical interaction energy $\kappa_2 = \kappa_{2\sharp}= -2.4$ and the new stable density is asymmetric adhering to one wall (Figure \[fig:bif\_fig\_right\]). In (b), in the absence of a confining potential, the uniform density becomes unstable at the critical interaction energy $\kappa_2 = \kappa_{2\sharp} = 0.4$ and the density may become multi-modal (Figure \[fig:bif\_stable\_equilibria\_V2\_cosine\]).[]{data-label="fig:bifurcation_diagram"}](bifurcation_diagram_for_Phi2.pdf "fig:"){width="\textwidth"}   [0.48]{} ![(a). The bifurcation diagram for (a). $V_2(x,y) = x\cdot y$ and (b). $V_2(x,y) = -\cos\left(\frac{2\pi(x-y)}{\mathrm{L}}\right)$ in Section \[subsec:numericalexperiments\]: the solid [blue]{} line denotes the stable branch of solutions while the dotted [red]{} line denotes the unstable branch of solutions. In (a) the stationary density $e^{-x^2}/Z$ changes stability at the critical interaction energy $\kappa_2 = \kappa_{2\sharp}= -2.4$ and the new stable density is asymmetric adhering to one wall (Figure \[fig:bif\_fig\_right\]). In (b), in the absence of a confining potential, the uniform density becomes unstable at the critical interaction energy $\kappa_2 = \kappa_{2\sharp} = 0.4$ and the density may become multi-modal (Figure \[fig:bif\_stable\_equilibria\_V2\_cosine\]).[]{data-label="fig:bifurcation_diagram"}](bifurcation_diagram_for_Phi2_cosine.pdf "fig:"){width="\textwidth"} Description of Main Results and Organisation of the Paper. ---------------------------------------------------------- ### Main Results {#main-results .unnumbered} The main results of this work are threefold. 1. We establish existence and uniqueness of weak solutions to DDFTs including two-body HI governed by equations , - with no-flux boundary conditions. 2. We derive *a priori* convergence estimates of the density $\varrho(\vec{r},t)$ to equilibrium in $L^2$ and relative entropy. 3. We study the stability of equilibrium states and construct bifurcation diagrams for two numerical applications. These results are of particular interest for physical applications of colloidal systems where conservation of mass is either a desirable or necessary property of the system. Additionally, the stability theorem contrasts with simpler linear stability analyses of similar systems of gradient flow structure with periodic boundary conditions [@martzel2001mean], [@greg_mckean_vlasov_torus] which may be tackled by means of Fourier analysis. ### Organisation of the Paper {#organisation-of-the-paper .unnumbered} The paper is organised as follows: in Section \[sec:preliminaries\] we present the boundary and initial conditions, introduce the main notation, nondimensionalise the main equations, state the stationary equation for the density, define the weak formulation of the Smoluchowski equation including full HI and provide a list of assumptions. In Section \[sec:statement\_of\_main\_results\] we state the main results of the present work in a precise manner. In Section \[sec:ex\_uni\_full\_HI\] we provide an existence and uniqueness theorem for the flux $\vec{a}$ when full HI are included. In Section \[sec:char\_stationary\_sol\] we characterise solutions of the stationary problem and convergence to equilibrium in $L^2$ as $t \to \infty$. In Section \[sec:global\_asymptotic\_stability\] we obtain results on the global asymptotic stability of the stationary densities by showing that the free energy is a continuous functional for all two-body interaction strengths. Additionally we prove an H- theorem for the equilibria, provide *a priori* convergence estimates in relative entropy, derive an asymptotic expansion of the equilibria for small interaction energy and perform a spectral analysis of the linearised nonlocal Smoluchowski operator. In Section \[sec:bifurcation\_theory\] we provide necessary and sufficient conditions for phase transitions in generalised DDFT-like systems with no-flux boundary conditions. In Section \[sec:manufactured\_bif\] we construct the bifurcation diagram for some example problems. In Section \[sec:existence\_uniqueness\_with\_partial\_HI\] we obtain an existence and uniqueness theorem for the Smoluchowski equation with non-constant diffusion tensor and effective drift vector dependent on the two-body HI tensors $\bm{Z}_1$ and $\bm{Z}_2$. In Section \[sec:discussion\] we present our concluding remarks and state some open problems. In Appendix \[sec:classical\_paraboliv\_pde\] we provide some technical results that are used in the proof of Theorem \[thm:exis\_uniq\_weak\_sol\_rho\]. Finally in Appendix \[app:nomenclature\] we provide a list of nomenclature. Preliminaries {#sec:preliminaries} ============= In this section we specify the nonlinear boundary conditions and initial data for the DDFT . We also nondimensionalise the governing equations and provide the assumptions on the regularity of the potentials, correlation function, diffusion tensor and initial data. Boundary Conditions. {#subsec:boundary_conditions} -------------------- When $U = \mathbb{R}^d$ we take $$\begin{aligned} \label{bc:density_and_flux_decaying} \left\{\begin{aligned} \varrho(\vec{r},t)\to 0 \\ \vec{a}(\vec{r},t)\to \vec{0} \end{aligned}\right. \quad \text{ as } \quad |\vec{r}|\to \infty,\end{aligned}$$ where we require $V_1$ to be growing at least quadratically as $\vec{r}\to \infty$. Physically-speaking this prevents the density from running out to infinity. When $U\subset \mathbb{R}^d$ is open and bounded we impose that the total mass of the system $M$ remains constant, in particular we have $$\begin{aligned} \label{bc:mass_preserving} \vec{a}(\vec{r},t)\cdot\vec{n}\bigg|_{\partial U\times [0,T]} = 0.\end{aligned}$$ The boundary condition may be viewed as a [*nonlinear*]{} Robin condition imposing the flux through the boundary $\partial U$ is zero for all time $t\in [0,T]$. If $\varrho$ is a number density then $\int \mathrm{d}\vec{r}\, \varrho = N$ for all time, however for the analysis in Section \[sec:ex\_uni\_full\_HI\] and onwards we will assume $\varrho$ is a probability density so that $\int \mathrm{d}\vec{r}\, \varrho = 1$. The rescaling between number and probability densities is discussed in the following section. Initial Conditions. ------------------- We will assume that the initial data has finite free energy and is consistent with the imposed boundary conditions. For example, one could prescribe initial data $(\varrho_0,\vec{a}_0)^\top$ such that $$\begin{aligned} \label{eq:initial_data_for_a_and_rho} \frac{\delta\mathcal{F}}{\delta\varrho}[\varrho_0](\vec{r}) = \mu_{c},\qquad \vec{a}_0 = \vec{0}.\end{aligned}$$ where $\mu_{c}$ is the chemical potential, constant at equilibrium. It is straightforward to check that $(\varrho_0,\vec{a}_0)^\top$ is an equilibrium point of the system . Commonly, one then drives the system out of equilibrium via a time-dependent external potential. In principle $\mu_{c}$ may be given and the equations , are well defined. In practice, for complicated particle configurations, $\mu_{c}$ is not known but can be computed by minimising the free energy along with the additional constraint $\int_U\mathrm{d}\vec{r}\, \varrho_0(\vec{r})=N$, where $N$ is the (expected) number of particles for a finite system and $\varrho_0$ is a number density. Note that $\mu_{c}$ is a potential, so by raising it one may force more particles into the system. We will assume that $\mu_c$ is constant to fix the number of particles. To ensure $\varrho$ (and $\varrho_0$) is a probability density one may rescale $\varrho/N = \tilde{\varrho}$, $N g = \tilde{g}$ and $\vec{a}/N^2 = \tilde{\vec{a}}$, where the tilde denotes the new variable, so that $\int_U\mathrm{d}\vec{r}\, \varrho_0(\vec{r})=1$ and equations - become independent of $N$. This provides a method of converting back to the number density which is typically used in numerical modelling of finite colloidal systems [@goddard2016dynamical], [@goddard2012unification], [@goddard2012general]. Throughout however, since we will frequently use the integral of the density, we will assume $\varrho$ and $\varrho_0$ are probability densities to ease notation. With this, one has three equations for three unknowns $\mu_{c}$, $\varrho_0$, $\vec{a}_0$ and the initial density $\varrho_0$ can be computed. For the rest of paper it is convenient to work in dimensionless units. We now nondimensionalise the governing equations. Evolution Equations. -------------------- We now nondimensionalise our equations. Let $\mathrm{L}$, $\tau$, $\text{U}$ be characteristic length, time and velocity scales respectively, then by nondimensionalising $$\begin{aligned} \vec{r}\sim \mathrm{L} \tilde{\vec{r}},\quad t\sim\tau\tilde{t}, \quad \mathrm{U} = \tfrac{\mathrm{L}}{\tau}, \quad \varrho\sim\tfrac{1}{\mathrm{L}^d}\tilde{\varrho},\quad \mathcal{F}\sim k_BT \tilde{\mathcal{F}},\quad \vec{a}\sim \mathrm{A}\tilde{ \vec{a}}.\end{aligned}$$ where $d$ is the physical dimension and $\mathrm{A}$ is a characteristic flux scale. The system becomes (after dropping tildes) $$\begin{aligned} \partial_t\varrho(\vec{r},t) = -\tfrac{1}{Fr}\times \tfrac{\tau^{-1}}{\gamma}\times \mathrm{A}\times \mathrm{L}^{d+1} \nabla_{\vec{r}}\cdot \vec{a}(\vec{r},t),\end{aligned}$$ where we have defined the Froude number $Fr = m\mathrm{U}^2/(k_BT)$. By choosing $Fr = 1$, $\tau = \gamma^{-1}$ and $\mathrm{A} = 1/\mathrm{L}^d$ we simplify the system of equations to the following boundary value problem.\ \[prop:non\_dimensional\_time\_evolving\_flux\_eqn\] The non-dimensional one-body density $\varrho(\vec{r},t)$ and flux $\vec{a}(\vec{r},t)$ evolve according the the boundary value problem $$\begin{aligned} \label{eq:evolution_eqn_for_a_dimensionless} \begin{cases} &\qquad\qquad\partial_t\varrho = -\nabla_{\vec{r}}\cdot \vec{a}(\vec{r},t), \\ &\vec{a}(\vec{r},t) + \bm{H}[\vec{a},\varrho] +\varrho(\vec{r},t)\bm{D}(\vec{r},[\varrho],t)\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho]=0, \\ &\qquad [\bm{H}[\vec{a},\varrho] +\varrho(\vec{r},t)\bm{D}(\vec{r},[\varrho],t)\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho]]\cdot\vec{n}\big|_{\partial U}=0. \end{cases}\end{aligned}$$ We note that when the off-diagonal HI tensor $\bm{Z}_{2}=0$, by using the definitions of $\mathcal{F}$ and $\bm{D}$ , the evolution equations in may be written as a nonlinear Smoluchowski equation (such as ) with non-constant diffusion coefficient. However we observe that even when $\bm{Z}_2 \neq 0$ the dynamics may be recast into a Smoluchowski equation for $\varrho$ under an effective drift vector dependent on $\bm{Z}_2$.\ \[prop:non\_dimensional\_time\_evolving\_rho\_eqn\] The non-dimensional one-body density $\varrho(\vec{r},t)$ evolves according the the boundary value problem $$\begin{aligned} \label{eq:evolution_eqn_for_rho_dimensionless} \begin{cases} &\qquad\partial_t\varrho=\,\nabla_{\vec{r}}\cdot\left[Pe^{-1}\bm{D}\nabla_{\vec{r}}\varrho+\varrho\,\bm{D}\left(\nabla_{\vec{r}} (\kappa _1V_1+\kappa _2\,(gV_2)\star\varrho) + \bm{A}[\bm{a}]\right)\right], \\ &\qquad \qquad \qquad \qquad \qquad \Pi [\varrho]\cdot\vec{n}\big|_{\partial U} = 0,\\ &\qquad\Pi[\varrho]:= \bm{D}\,\left(\nabla_{\vec{r}}\varrho + \varrho\, \nabla_{\vec{r}}(\kappa_1 V_1(\vec{r},t)+ \kappa_2 (gV_2)\star \varrho)+\bm{A}[\bm{a}]\right), \end{cases}\end{aligned}$$ where $\bm{A}[\bm{a}]$ is an effective background flow induced by the hydrodynamic interactions defined by $$\begin{aligned} \label{eq:def_of_V1_eff} \bm{A}[\vec{a}]:=\int_U\mathrm{d}\vec{r}'\, g(\vec{r},\vec{r}') \bm{Z}_2(\vec{r},\vec{r}')\vec{a}(\vec{r'},t),\end{aligned}$$ $\kappa_1$, $\kappa_2$ are non-dimensional constants measuring the strength of confining and interaction potentials respectively, $Pe=\mathrm{L}\text{U}/\alpha$ is the P[é]{}clet number measuring the ratio of advection rates to diffusive rates and $\alpha = k_B\mathrm{T}/(m\gamma)$. Corollary \[prop:non\_dimensional\_time\_evolving\_rho\_eqn\] is the general formulation of the nondimensional equations - when $\bm{Z}_2\neq0$, including a non-constant diffusion coefficient and an effective drift. Throughout this paper, to study the intermediate regime of equally strong advection and diffusion, we set $Pe = 1$. Additionally, we redefine the two-body potential to absorb the correlation function $g$ to ease notation, $V_2(\vec{r},\vec{r}'):= g(\vec{r},\vec{r}') V_2(\vec{r},\vec{r}')$. In practice, there are many choices for $g$, for example the hard sphere approximation takes $g(|\vec{r}- \vec{r}'|) = 0$ for $|\vec{r}-\vec{r}'| < 1$ and unity otherwise. Alternatively $g$ may be obtained numerically from microscopic dynamics. We consolidate the choices for equations , in Section \[subsec:assumptions\_definitions\]. The effective drift $\bm{A}[\vec{a}]$, dependent on $\bm{Z}_2$ and $\vec{a}$ may be determined once $\vec{a}(\vec{r},t)$ is solved from the second equation in . Note that the evolution equation in may be viewed as a generalised McKean-Vlasov equation with a non-constant diffusion tensor and confining potential. In particular the McKean-Vlasov equation may be recovered in the special case $\bm{Z}_1 = \bm{Z}_2 = V_1 = 0$, see for example [@greg_mckean_vlasov_torus], [@chayes2010mckean]. We will use Corollary \[prop:non\_dimensional\_time\_evolving\_rho\_eqn\], to write the full dynamics including full HI, to obtain our results on weak solutions for $\varrho(\vec{r},t)$ (see Theorem \[thm:eigenfn\_expansion\_of\_flux\], Section \[sec:ex\_uni\_full\_HI\] and Theorem \[thm:existence\_and\_uniqueness\], Section \[sec:existence\_uniqueness\_with\_partial\_HI\]). We continue to the next section by stating the stationary boundary value problem for equilibrium states $\varrho(\vec{r})$. Stationary Equations. --------------------- For general $\bm{Z}_2$ we will show in Theorem \[thm:association \_of\_free\_energy\] that the stationary density $\varrho(\vec{r})$ satisfies $$\begin{aligned} \label{eq:stationary_eqn_for_rho_dimensionless} \begin{cases} &\qquad 0=\,\nabla_{\vec{r}}\cdot[\bm{D}\nabla_{\vec{r}}\varrho+\varrho\,\bm{D}\nabla_{\vec{r}}(\kappa _1V_1+\kappa _2\,V_2\star\varrho)], \\ &\qquad \qquad \qquad \qquad \qquad \Pi [\varrho]\cdot\vec{n}\big|_{\partial U} = 0,\\ &\qquad\Pi[\varrho]:= \bm{D}\,(\nabla_{\vec{r}}\varrho + \varrho\, \nabla_{\vec{r}}(\kappa_1 V_1(\vec{r},t)+ \kappa_2 \,V_2\star \varrho)). \end{cases}\end{aligned}$$ We now discuss regularity on the potentials and diffusion tensor. Assumptions & Definitions. {#subsec:assumptions_definitions} -------------------------- Typically for long range HI the $\boldsymbol{Z}_i$ exhibit singularities at the origin (particle centres) so the correlation function $g$ is a necessary inclusion and provides a way of smoothing $\boldsymbol{D}$ and we assume $g \in L^\infty(U)$. For $\varrho \geq 0$ the diffusion tensor $\boldsymbol{D}$ as a convolution with the density will then be a weakly differentiable function. For the existence and uniqueness theory in Appendix \[sec:classical\_paraboliv\_pde\] and Section \[sec:existence\_uniqueness\_with\_partial\_HI\] we require that first derivatives of $\bm{D}_{ij}$ to be bounded in $L^\infty(U)$ so that all coeeffiecients of the PDE are uniformly bounded. Out of equilibirum, we will suppress the time dependence on $\bm{D}$, $V_1$ simply to ease notation. However at equilibrium $\bm{D}$, $V_1$ are assumed to be independent of time, indeed in order for equilibrium states of the density and flux to be well defined. We note that is $\bm{D}$ positive definite and symmetric, as it has been rigorously shown to be [@goddard2012overdamped]. In summary we have the following notational choices and assumptions for the evolution problem . #### **[Notation]{}** Throughout we ease notation on the two-body interaction potential. - The two-body interaction potential is redefined to absorb the correlation function $g$ $$\begin{aligned} \label{ass:V2_redef} V_2\stackrel{\text{redef}}{:=} g V_2. \tag{N1}\end{aligned}$$ For the dynamics we assume: #### **[Assumptions D]{}** - The diffusion tensor $\bm{D}$ is symmetric, positive definite, and the first derivatives of $\bm{D}_{ij}$ are bounded in $L^\infty(U)$ $$\begin{aligned} \label{ass:D_pos_def_weak_diffable} \bm{D}_{ij}\in W^{1,\infty}(U). \tag{D1}\end{aligned}$$ - The diagonal and off-diagonal blocks of the HI tensors are uniformly bounded in the sense $$\begin{aligned} \label{ass:Z2_uniformly_bd} \|gZ_2\|_{L^\infty(U)}<\infty, \quad \|gZ_1\|_{L^\infty(U)}<\infty\tag{D2}\end{aligned}$$ - The initial data $\varrho_0$ is a non-negative, square-integrable, absolutely continuous probability density $$\begin{aligned} \label{ass:rho_in_L2_P_ac} \varrho_0\in P_{ac}(U)\cap L^2(U). \tag{D3}\end{aligned}$$ - The potentials each have two bounded derivatives $$\begin{aligned} \label{ass:V1_V2_in_W_1_inf} V_1,V_2\in W^{2,\infty}(U). \tag{D4}\end{aligned}$$ The functions $V_1$ and $V_2$ are the confining and two-body interaction potentials respectively, the former having explicit time dependence ($V_1 = V_1(\vec{r},t)$) only when we intend to drive - and out of equilibrium, and $V_1 = V_1(\vec{r})$ when we are concerned with the equilibrium properties of - and . This distinction will be important for the H Theorem and equilibrium theory in Section \[sec:global\_asymptotic\_stability\]. For the equilibrium problem we will assume: #### **[Assumptions E]{}** - The potentials have first order weak derivatives in $L^2(U)$ $$\begin{aligned} \label{ass:V1_V2_in_H_1} V_1,V_2\in H^{1}(U). \tag{E1}\end{aligned}$$ In particular, Assumption will permit us to establish smooth stationary densities. Note that typical inter-particle potentials, such as Morse or Coulomb, are unbounded as the particle separation goes to zero. This is once again mitigated by the choice of $g$, which we recall has been absorbed into $V_2$ by assumption . In general we admit non-convex $V_1$ and $V_2$, for example multi-well potentials, except for in the convergence result of Theorem \[thm:rel\_entropy\_convergence\] where $V_1$ must be convex in order to invoke a log-Sobolev inequality on the measure $\mu$ given by . The assumption that $\varrho_0\in P_{\text{ac}}(U)$ is included in order to cover a wider set of physically relevant scenarios. In particular we permit initial data such that $\varrho_0|_{A}=0$ for some $A\subset U$ where $A$ is non-empty. Physically speaking this system could correspond to, at time $t=0$, a box partitioned into closed regions with at least one region containing no particles. Then, instantaneously as soon as $t>0$, the partition is removed allowing the particles to move freely. At the end of Section \[sec:existence\_uniqueness\_with\_partial\_HI\] we will show by simple application of Harnack’s inequality, that we obtain strictly positive densities $\varrho(\vec{r},t_1)>0$ after an arbitrarily small time $t_1>0$. Principally this is provided by the property , since $\bm{D}$ is positive definite, the diffusion of density in the system is everywhere propagating in $U$. Additionally, by the positive definite property in we may uniquely define the square root of $\bm{D}$ denoted $\bm{D}^{1/2}$ such that $$\bm{D}^{1/2}\bm{D}^{1/2}=\bm{D}$$ for every $\vec{r}\in U$, $t\in [0,T]$ and each $\varrho$. We also define the eigenvalues $\mu_i\in \mathbb{R}^+$ of $\bm{D}$ and eigenvectors $\vec{e}_i(\vec{r},[\varrho],t)\in L^2(U)$ such that $$\begin{aligned} \label{eq:mu_i_eigenvalue_eqn_D} \bm{D}\vec{e}_i = \mu_i\vec{e}_i.\end{aligned}$$ Note that $\{\vec{e}_i\}_{i=1}^d$ forms an orthonormal basis of $\mathbb{R}^d$ (since $\bm{D}$ is a bounded, symmetric operator) for $i = 1,\cdots, d$ such that $$\begin{aligned} \label{eq:orthogonality_of_ei} \langle\vec{e}_i(\vec{r},[\varrho],t),\vec{e}_j(\vec{r},[\varrho],t)\rangle = \int_U\mathrm{d}\vec{r}\, \vec{e}_i(\vec{r},[\varrho],t)\cdot \vec{e}_j(\vec{r},[\varrho],t) = \delta_{ij}.\end{aligned}$$ We continue to the next section by defining a weak formulation of the dynamics .\ Weak Formulation. ----------------- We provide the weak formulation of the full dynamics including HI for $\bm{Z}_1, \bm{Z}_2$ not necessarily zero.\ Let $\vec{a}(\vec{r},t)$ be a given flux. We say $\varrho\in L^2([0,T]; H^1(U))\cap L^\infty([0,T]; L^2(U))$ and $\partial_t\varrho\in L^2([0,T]; H^{-1}(U))$ is a weak solution to if for every $\eta\in L^2([0,T]; H^1(U))$ $$\begin{aligned} \label{eq:weak_formulation_including_V1eff} \int_0^T\mathrm{d}t\, \langle \partial_t \varrho(t), \, \eta(t) \rangle +\int_0^T\mathrm{d}t\, \int_U \mathrm{d}\vec{r}\, \nabla_{\vec{r}}\eta\cdot \bm{D}\,[\nabla_{\vec{r}}\varrho +\varrho\,\nabla_{\vec{r}}(\kappa _1V_1+\kappa _2\,V_2\star\varrho + \bm{A}[\vec{a}])]=0\end{aligned}$$ where $\varrho_0 = \varrho(0)$. Here, $\bm{A}[\vec{a}]$ is the effective drift induced by $\bm{Z}_2$ and is defined by equation . It will be shown in the following sections (in particular Corollary \[cor:a\_is\_zero\_at\_equilibrium\]) that $\bm{A}[\vec{a}] \to \vec{0}$ as $t \to \infty$. We now state our main results in a precise manner.\ Statement of Main Results {#sec:statement_of_main_results} ========================= Our main results concern existence, uniqueness and convergence to equilibrium of the density of colloids $\varrho$ and flux $\vec{a}$ on $U$ a compact subset of $\mathbb{R}^d$. The first result concerns existence of the flux $\vec{a}(\vec{r},t)$ with non-zero hydrodynamic interactions, the convergence of $\vec{a}(\vec{r},t)$ to zero at equilibrium and existence and uniqueness of fixed points of .\ \[them:exis\_uni\_of\_flux\] Let $\bm{Z}_1,\bm{Z}_2$ be real, symmetric and $\mu_{\max}\|g\bm{Z}_2\|_{L^\infty(U)}<1$. Then\ 1. There exists a unique $\vec{a}(\vec{r},t)\in L^2(U)$ solving the evolution equation for each $\varrho(\vec{r},t)$. In particular $$\begin{aligned} \vec{a}(\vec{r},t) = \sum_{n=1}^d\delta_n\sum_{i=1}^d\frac{\psi_i}{\phi_n-\mu_i^{-1}} \vec{e}_i(\vec{r},[\varrho],t)\end{aligned}$$ where $\vec{e}_i(\vec{r},[\varrho],t)$ are eigenvectors of the diffusion tensor $\bm{D}(\vec{r},[\varrho],t)$ and $\delta_n$, $\phi_n$, $\psi_i$, $\mu_i^{-1}\in \mathbb{R}$. 2. In addition, every stationary density $\varrho(\vec{r})$ and stationary flux $\vec{a}(\vec{r})$ are independent of the HI tensors and satisfy $$\begin{aligned} \varrho(\vec{r})\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho(\vec{r})] = \vec{0}, \quad \vec{a}(\vec{r}) = \vec{0}\end{aligned}$$ and consequently $\varrho(\vec{r})$ minimises the free energy $\mathcal{F}[\varrho](r)-\mu_{c} \int_U\mathrm{d}\vec{r}\,\varrho$, where $\mu_{c}$ is the chemical potential. 3. If, in addition, $|\kappa_2|< \|V_2\|_{L^\infty(U)}^{-1}$ then $(\vec{a}_\star(\vec{r}),\varrho_\star) = (\vec{0}, \varrho_\infty)$ are the unique fixed points of and $\varrho_\infty(\vec{r})$ is given by the self-consistency equation $$\begin{aligned} \varrho_\infty(\vec{r}) = \frac{e^{-(\kappa_1 V_1(\vec{r})+\kappa_2 V_2\star \varrho_\infty)}}{Z(\varrho_\infty)}\end{aligned}$$ for $Z(\varrho_\infty) = \int_U\mathrm{d}\vec{r}\,e^{-(\kappa_1 V_1(\vec{r})+\kappa_2 V_2\star \varrho_\infty)}$. For the evolution system we present the following second main result of the paper.\ \[thm:exis\_uniq\_weak\_sol\_rho\] Let $\bm{Z}_1,\bm{Z}_2$ be real, symmetric and $\mu_{\max}\|g\bm{Z}_2\|_{L^{\infty}(U)}<1$, where $\mu_{\max}$ is the largest eigenvalue of $\bm{D}$, with $\varrho_0 \in C^\infty(U)$, $\varrho\geq 0$ and $\int_U \mathrm{d}\vec{r} \varrho_0(\vec{r})=1$. Then there exists a unique weak solution $\varrho\in L^\infty([0,T];L^2(U))\cap L^2([0,T]; H^1(U))$, with $\partial_t\varrho\in L^2([0,T]; H^{-1}(U))$ for , in the sense , and the following energy estimate holds $$\begin{aligned} \|\varrho\|_{L^{\infty}([0,T];L^2(U))}+\|\varrho\|_{L^{2}([0,T];H^1(U))}+\|\partial_t\varrho\|_{L^2([0,T]; H^{-1}(U))}\leq C(T) \|\varrho_{0}\|_ {L^2(U)},\end{aligned}$$ where $C(T)$ is a constant dependent on $T$, $U$ and $\mu_{\max}$. The existence and uniqueness is proved in Theorem \[thm:existence\_and\_uniqueness\], whilst the bound is shown in Lemma \[lem:weak\_conv\_results\]. Furthermore, we prove existence and uniqueness of the stationary density, and exponentially fast convergence in relative entropy.\ Let $\bm{Z}_1,\bm{Z}_2$ be real, symmetric and $\varrho$ be a solution to the DDFT with smooth initial data and smooth $V_1$, $V_2$. Then there exists stationary density $\varrho(\vec{r},t) = \varrho_0(\vec{r})$. If $|\kappa_2| \leq 1/4\times \|V_2\|_{L^\infty}^{-1}$ then the stationary solution is unique and is denoted by $\varrho_\infty$. The proof of this result is standard, see [@dressler1987stationary]. The third main result of this paper concerns *a priori* estimates for exponential convergence of the density to stationarity.\ Let $\bm{Z}_1,\bm{Z}_2$ be real, symmetric and $\varrho$ be a solution to the DDFT with smooth initial data and smooth $V_1$, $V_2$. If $\kappa_2 \leq 1/4\times \|V_2\|_{L^\infty}^{-1}$ then\ 1. [**[Convergence in ]{}**]{}$\bm{L^2(U):}$ For $\kappa_1=0$ (in the absence of a confining potential) if $$\begin{aligned} \kappa_2^2<\frac{\mu_{\min}c^{-2}_{pw}\|\nabla_{\vec{r}}V_2\|^{-2}_{L^\infty(U)}}{2(1+e)\mu_{\max}},\end{aligned}$$ where $\mu_{\min}$ and $\mu_{\max}$ are the smallest and largest eigenvalues of the diffusion tensor $\bm{D}$, then $\varrho\to \varrho_\infty$ in $L^2(U)$ exponentially fast as $t\to \infty$. For $\kappa_1\neq 0$ the convergence criteria is modified to $$\begin{aligned} \mu_{\max}(\kappa_1^2\|\nabla_{\vec{r}}V_1\|_{L^\infty(U)}^2+2\kappa_2^2(1+e)\|\nabla_{\vec{r}}V_2\|_{L^\infty(U)}^2)<\frac{\mu_{\min}}{c_{pw}^2}.\end{aligned}$$ 2. [**[Convergence in Relative Entropy:]{}**]{} For any fixed confining potential $V_1$ such that the measure $\mu'(\mathrm{d}\vec{r}) = \mathrm{d}\vec{r}\,e^{-\kappa_1V_1}/Z$ satisfies a log-Sobolev inequality and provided $$\begin{aligned} \kappa_2^2<\frac{c^{-1}_{ls}}{2\|\nabla V_2\|^2_{L^{\infty(U)}}}\end{aligned}$$ then the measure $\mu$ in satisfies a log-Sobolev inequality and $\mathcal{H}(\varrho |\varrho_\infty) \to 0$ exponentially fast as $t\to \infty$ where $$\begin{aligned} \mathcal{H}(\varrho |\varrho_\infty) = \int_U\mathrm{d}\vec{r}\, \varrho \log \left(\tfrac{\varrho}{\varrho_\infty}\right)\end{aligned}$$ denotes the relative entropy. For part 1, see Theorem \[lem:exp\_conv\_in\_L2\] and Proposition \[prop:general\_kappa1\_convergence\]. Theorem \[thm:rel\_entropy\_convergence\] gives the result for part 2. The log-Sobolev inequality for $\mu$ is established by the Holley-Stroock perturbation lemma [@holley1986logarithmic]. The constants $c_{pw}$, $c_{ls}$ are the Poincar[é]{}-Wirtinger and log-Sobolev constants respectively. Nowhere do we assume parity on the two-body potential nor $V_2$ have zero mean. Additionally the optimal $c_{pw}$ is the inverse square root of the smallest eigenvalue of the Laplacian on the domain $U$ with no-flux boundary conditions. We have the following conditions for the existence of bifurcating branches of steady states $\varrho(\vec{r})$.\ Fix $\kappa_1$ and let $\kappa_2\in(-\infty,\infty)$. Let $\mathcal{L}_{1} = \mathcal{A}_{\kappa_2}+\kappa_2\mathcal{B}$ denote the linearised operator to the stationary problem with eigenvalues $\lambda(\kappa_2)$ and eigenfunctions $w^{(\kappa_2)}(\vec{r})$. Denote by $\mathcal{A}_{\kappa_2}$, $\mathcal{B}$ the local and nonlocal parts of $\mathcal{L}_{1}$ respectively. Denote by $\gamma_k^{(\kappa_2)}$ the eigenvalues of $\mathcal{A}_{\kappa_2}$ with eigenvectors $v^{(\kappa_2)}$. If the solution $\kappa_2^\star(\lambda)$ of the equation $\lambda = \lambda_{k^\star}(\kappa_2^\star) $ exists, then it is unique and is given by the nonlinear equation $$\begin{aligned} \kappa_2^\star(\lambda) = \left(\sum_{i=1}^\infty\tfrac{\theta_i^{(\kappa_2)}\gamma _i^{(\kappa_2)}\beta_i^{(\kappa_2)}}{\lambda-\gamma_i^{(\kappa_2)}}\right)^{-1}. \end{aligned}$$ As a corollary we can determine the necessary condition on the interaction strength for a bifurcation of stable equilibrium densities solving .\ Provided that the spectral gap of $\mathcal{A}_{\kappa_2}$ is sufficiently large, that is, $$\begin{aligned} |\kappa_2|<\frac{\min_{i,j\in\mathbb{N}}|\gamma_i^{(\kappa_2)}-\gamma_{j}^{(\kappa_2)}|}{2\|\mathcal{B}\|}\end{aligned}$$ then $\lambda(\kappa_2)\in\mathbb{R}$ and the point of critical stability $\kappa_{2_\sharp}$ occurs at the solution of the nonlinear equation $$\begin{aligned} \kappa_{2_{\sharp}} = -\left(\sum_{i=1}^\infty\theta_i^{(\kappa_2)}\beta_i^{(\kappa_2)}\right)^{-1},\end{aligned}$$ where $\theta_i\beta_i$ are coefficients of the two-body potential expanded in the orthonormal basis of eigenvectors $\{v_k^{(\kappa_{2_\sharp})}\}_{k = 1}^\infty$. The proofs of these results are given by Theorem \[thm:epsilon\_as \_a\_fn\_of\_lambda\] and the discussion immediately following it. We also obtain the following theorem for existence of bifurcations for the stationary equation .\ Let $\kappa_2\in (-\infty,\infty)$and let $\{\beta_{n}^{-1}\}_{n=1}^{\infty}$ be the eigenvalues of $\mathcal{R}$ with eigenfunctions $\{u_n\}_{n=1}^{\infty}$ where $$\begin{aligned} \mathcal{R}[u_n] = -\varrho_{\kappa_2}(\vec{r})\int_U\mathrm{d}\vec{r}'\, V_2(\vec{r},\vec{r}')u_n(\vec{r}')\end{aligned}$$ and $\varrho_{\kappa_2}$ is a stationary solution to . If $|\kappa_2|\geq |\beta_1|$ then $\varrho_{\kappa_2}$ is unstable with respect to $\{u_n\}_{n=1}^{\infty}$ with $(\beta_1,u_1)$ a bifurcation point of where $\beta_1$ is the smallest eigenvalue of $\mathcal{R}^{-1}$ and $w_1$ is the eigenfunction of $\mathcal{R}$ associated to $\beta_1^{-1}$. There exists $\varrho_\ast>0$ such that $\mathcal{F}[\varrho_\ast]<\mathcal{F}[\varrho_{\kappa_2}]$. We now give our arguments for Theorem \[them:exis\_uni\_of\_flux\].\ Existence & Uniqueness of Flux With Full HI {#sec:ex_uni_full_HI} =========================================== We return to the full formulation of the overdamped DDFT with HI. The contraction condition $\mu_{\max}\|g\bm{Z}_2\|_{L^{\infty}(U)}<1$ can be seen as a necessary condition on the invertibility of an operator closely related to the positive definite grand friction tensor $\boldsymbol{\Gamma}$. Note that the flux equation in may be written more generally as $$\begin{aligned} \label{eq:matrix_operator_friction_eqn} (\bm{1}+ \mathcal{Z}_1^\varrho + \mathcal{Z}_2^\varrho)[\vec{a}(\vec{r},t)]=-\varrho(\vec{r},t)\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho]\end{aligned}$$ where the actions of the integral operators $\bm{1}+ \mathcal{Z}_1^\varrho$ and $\mathcal{Z}_2^\varrho$ are defined by $$\begin{aligned} (\bm{1}+ \mathcal{Z}_1^\varrho)[\vec{a}_1]&=\vec{a}_1(\vec{r},t)+\int_U\mathrm{d}\vec{r}' g(\vec{r},\vec{r}')\varrho(\vec{r}',t)\boldsymbol{Z}_1(\vec{r},\vec{r}')\times \vec{a}_1(\vec{r},t),\label{eq:friction_like_system_1plusZ1}\\ \mathcal{Z}_2^\varrho[\vec{a}_2]&=\varrho(\vec{r},t)\int_U\mathrm{d}\vec{r}'\, g(\vec{r},\vec{r}')\bm{Z}_2(\vec{r},\vec{r}')\vec{a}_2(\vec{r}',t).\label{eq:friction_like_system_Z2}\end{aligned}$$ Notice how the integral-matrix operators in - resemble the operators in the first row of the grand resistance matrix for a two particle system from classical hydrodynamics [@happel2012low]. The following lemma establishes a solvability condition for the flux equation in equation . \[thm:cond\_converg\_fred\_det\] Let $\bm{1}+ \mathcal{Z}_1^\varrho$ and $\mathcal{Z}_2^\varrho$ be bounded linear operators. Suppose $\mathcal{A}_\varrho:=(\bm{1}+ \mathcal{Z}_1^\varrho)^{-1}\mathcal{Z}_2^\varrho$ is compact in $L^2(U,\varrho^{-1}(\vec{r},t))$. If $\mu_{\max}\|g\bm{Z}_2\|_{L^\infty(U)}<1$ then the matrix integral operator $\bm{1}+ \mathcal{Z}_1^\varrho + \mathcal{Z}_2^\varrho $ is invertible and the system is well-posed. Since $\bm{1}+ \mathcal{Z}_1^\varrho$ is positive definite it is invertible, therefore may be rewritten $$\begin{aligned} \label{eq:matrix_integral_op_eqn_rewrite} (\bm{1} + (\bm{1}+ \mathcal{Z}_1^\varrho )^{-1}\mathcal{Z}_2^\varrho) [\vec{a}(\vec{r},t)] = -(\bm{1}+ \mathcal{Z}_1^\varrho)^{-1} \varrho(\vec{r},t)\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho].\end{aligned}$$ We note that $\mathcal{A}^\varrho$ is a trace-class operator and that the left hand side of is an operator of the form $\bm{1}-\lambda\mathcal{A}_\varrho$. By classical theory [@fredholm1900nouvelle], [@lax2014functional] we have the identity $$\begin{aligned} \label{eq:det(1+A)_in_terms_of_exp} \det(\bm{1}-\lambda\mathcal{A}_\varrho) = \exp\Big\{-\sum_{n=1}^\infty\tfrac{\text{Tr}(\mathcal{A}_\varrho^n)}{n}\lambda^n \Big\}.\end{aligned}$$ When $\lambda = -1$ we recover the determinant for the Fredholm operator on the left hand side of . Particularly, since for our consideration $|\lambda|=1$, the convergence of the infinite summation inside the argument of the exponential in will depend on the size of $\text{Tr}\mathcal{A}_\varrho$, and when $\lambda = -1$, the summand is an alternating sequence so we demand absolute convergence for the sum in to converge. We obtain results in $L^2(U, \varrho^{-1})$. By definition of the trace we have $$\begin{aligned} \text{Tr}\mathcal{A}^n_\varrho = \sum_{k=1}^d\langle \mathcal{A}^n_\varrho \vec{e}_k(\vec{r}), \vec{e}_k(\vec{r})\rangle_{L^2(U,\varrho^{-1})},\end{aligned}$$ where $\{\vec{e}_k\}_{k}$ are vectors such that their components form an orthonormal basis of $L^2(U,\varrho^{-1})$, in particular we choose the eigenvectors of the diffusion tensor $\bm{D}$. Since $\mathcal{A}$ is an integro-matrix operator the inner product is given by, for $n=1$ $$\begin{aligned} \text{Tr}\mathcal{A}_\varrho &= \sum_{k=1}^d\langle \mathcal{A}_\varrho \vec{e}_k(\vec{r}), \vec{e}_k(\vec{r})\rangle_{L^2(U,\varrho^{-1})}\\ &=\sum_{k=1}^d\int_U\mathrm{d}\vec{r}\,\vec{e}_{k}(\vec{r})\cdot\bm{D}(\vec{r})\int_U\mathrm{d}\vec{r}'\,g(\vec{r},\vec{r}')\bm{Z}_2(\vec{r},\vec{r}')\vec{e}_{k}(\vec{r}')\\ &\leq \sum_{k=1}^d\mu_{\max}\|g \bm{Z}_2\|_{L^\infty(U)}\int_U\mathrm{d}\vec{r}\,\vec{e}_{k}(\vec{r})\cdot\int_U\mathrm{d}\vec{r}'\,\vec{e}_{k}(\vec{r}')\leq d\mu_{\max}\|g \bm{Z}_2\|_{L^\infty(U)}|U|\end{aligned}$$ where we have used $\int_U\mathrm{d}\vec{r}'\,\vec{e}_{k}(\vec{r}')\leq \|\vec{e}_k\|_{L^1(U)}\leq |U|^{1/2}\|\vec{e}_k\|_{L^2(U)} = |U|^{1/2}$ by orthonormality of the basis. Now for $n=2$ we have $$\begin{aligned} \text{Tr}\mathcal{A}^2_\varrho &= \sum_{k=1}^d\langle \mathcal{A}^2_\varrho \vec{e}_k(\vec{r}), \vec{e}_k(\vec{r})\rangle_{L^2(U,\varrho^{-1})}\\ &=\sum_{k=1}^d\int_U\mathrm{d}\vec{r}\,\vec{e}_{k}(\vec{r})\cdot\varrho(\vec{r})^{-1}\int_U\mathrm{d}\vec{r}_1\,\varrho(\vec{r})\bm{D}(\vec{r})g(\vec{r},\vec{r}_1)\bm{Z}_2(\vec{r},\vec{r}_1)\\ &\quad \times\int_U\mathrm{d}\vec{r}_2\,\varrho(\vec{r}_1)\bm{D}(\vec{r}_1)g(\vec{r}_1,\vec{r}_2)\bm{Z}_2(\vec{r}_1,\vec{r}_2)\vec{e}_{k}(\vec{r}_2)\\ &\leq \sum_{k=1}^d\mu_{\max}^2\|g\bm{Z}_2\|_{L^\infty(U)}^2\int_U\mathrm{d}\vec{r}\,\vec{e}_{k}(\vec{r})\cdot\int_U\mathrm{d}\vec{r}_2\,\vec{e}_{k}(\vec{r}_2)\\ &\leq d\mu_{\max}^2\|g\bm{Z}_2\|_{L^\infty(U)}^2|U|,\end{aligned}$$ where we have used the fact that $\int_U\mathrm{d}\vec{r}_1\,\varrho(\vec{r}_1,t) = 1$ for $t\geq 0$ by the no-flux boundary condition (see Section \[subsec:boundary\_conditions\]). Iterating this argument one may obtain $$\begin{aligned} \text{Tr}\mathcal{A}^n_\varrho &= \sum_{k=1}^d\langle \mathcal{A}^n_\varrho \vec{e}_k(\vec{r}), \vec{e}_k(\vec{r})\rangle_{L^2(U,\varrho^{-1})}\\ &\leq \mu_{\max}^n\|g\bm{Z}_2\|_{L^\infty(U)}^n\sum_{k=1}^d\int_U\mathrm{d}\vec{r}\,\vec{e}_{k}(\vec{r})\cdot\int_U\mathrm{d}\vec{r}_n\,\vec{e}_{k}(\vec{r}_n)\\ & = |U|d\times \mu_{\max}^n\|g\bm{Z}_2\|_{L^{\infty}(U)}^n.\end{aligned}$$ We observe that absolute convergence of requires $\mu_{\max}\|g\bm{Z}_2\|<1$. In particular, the sum of the absolute values of the terms is given by $$\begin{aligned} \sum_{n=1}^\infty\Big|\tfrac{\text{Tr}(\mathcal{A}_\varrho^n)}{n}\Big| \leq d|U|\sum_{n=1}^\infty\tfrac{\mu_{\max}^n\|g\bm{Z}_2\|_{L^\infty(U)}^n}{n} = d|U|\log\left(\frac{1}{1-\mu_{\max}\|g\bm{Z}_2\|_{L^\infty(U)}}\right).\end{aligned}$$ Thus for $\mu_{\max}||g \bm{Z}_2||_{L^\infty(U)}<1$ the logarithm is finite and the determinant is positive, otherwise for the boundary case $\mu_{\max}||g \bm{Z}_2||_{L^\infty(U)} =1$ it may vanish, thus making $(\bm{I} + \mathcal{Z}_1^\varrho + \mathcal{Z}_2^\varrho)$ singular. We now provide a scheme for computing solutions of equation for each time dependent $\varrho(\vec{r},t)$. The existence and uniqueness of $\varrho(\vec{r},t)$ is given in Section \[sec:existence\_uniqueness\_with\_partial\_HI\]. First we establish that $\bm{1}+\mathcal{Z}^\varrho_1-\lambda\mathcal{Z}^\varrho_2$ is a compact self-adjoint operator in $L^2(U,\varrho^{-1})$.\ \[lem:flux\_operator\_is\_compact\_self\_adjoint\] Let $\lambda\in (-\infty,\infty)$ and assumption hold. Then $\bm{1}+\mathcal{Z}^\varrho_1-\lambda\mathcal{Z}^\varrho_2$ is a compact and self-adjoint operator. We let $\vec{a}\in L^1(U)$ and calculate $\|(\bm{1}+\mathcal{Z}^\varrho_1-\lambda\mathcal{Z}^\varrho_2)[\vec{a}]\|_{L^1(U)}$. In particular we have $$\begin{aligned} \|(\bm{1}+\mathcal{Z}^\varrho_1-\lambda\mathcal{Z}^\varrho_2)[\vec{a}]\|_{L^1(U)} &= \int\mathrm{d}\vec{r}\,\Big|(\bm{1}+\mathcal{Z}^\varrho_1-\lambda\mathcal{Z}^\varrho_2)[\vec{a}]\Big|\nonumber\\ &\leq \int\mathrm{d}\vec{r}\,\Big|(\bm{1}+\mathcal{Z}^\varrho_1)\vec{a}\Big|+|\lambda|\Big|\mathcal{Z}^\varrho_2[\vec{a}]\Big|\nonumber\\ &\leq (1+\|g \bm{Z}_1\|_{L^\infty(U)}+|\lambda|\|g\bm{Z}_2\|_{L^\infty(U)})\|\vec{a}\|_{L^1(U)}<\infty.\end{aligned}$$ Hence $\text{Im}(\bm{1}+\mathcal{Z}^\varrho_1-\lambda\mathcal{Z}^\varrho_2)$ is bounded in $\mathbb{R}^3$. Now by Heine–Borel, the closure of $\text{Im}(\bm{1}+\mathcal{Z}^\varrho_1-\lambda\mathcal{Z}^\varrho_2)$ is compact and hence $\bm{1}+\mathcal{Z}^\varrho_1-\lambda\mathcal{Z}^\varrho_2$ is a compact operator. We now show that $\bm{1}+\mathcal{Z}^\varrho_1-\lambda\mathcal{Z}^\varrho_2$ is self-adjoint. The local part $\bm{1}+\mathcal{Z}^\varrho_1$ is a real, symmetric matrix and it is therefore self-adjoint, and in particular self-adjoint in $L^2(U,\varrho^{-1})$ . All that remains is to study the nonlocal part $\mathcal{Z}^\varrho_2$. By direct calculation we see that for $\vec{b}\in L^2(U)$ $$\begin{aligned} \langle \vec{b},\mathcal{Z}_2^\varrho[\vec{a}]\rangle_{L^2(U,\varrho^{-1})} &= \int_U\mathrm{d}\vec{r}\,\vec{b}(\vec{r})\cdot \int_U\mathrm{d}\vec{r}'\,g(\vec{r},\vec{r}')\bm{Z}_2(\vec{r},\vec{r}')\vec{a}(\vec{r}')\\ &=\int_U\mathrm{d}\vec{r'}\,\vec{b}(\vec{r})^\top \int_U\mathrm{d}\vec{r}\,g(\vec{r},\vec{r}')\bm{Z}_2(\vec{r},\vec{r}')^\top\vec{a}(\vec{r}')\\ &=\int_U\mathrm{d}\vec{r'}\,\int_U\mathrm{d}\vec{r}\,\left(g(\vec{r},\vec{r}')\bm{Z}_2(\vec{r},\vec{r}') \vec{b}(\vec{r})\right)^\top \vec{a}(\vec{r}')\\ &=\int_U\mathrm{d}\vec{r'} \vec{a}(\vec{r}')\cdot\int_U\mathrm{d}\vec{r}\,g(\vec{r},\vec{r}')\bm{Z}_2(\vec{r},\vec{r}') \vec{b}(\vec{r})\\ &=\langle \mathcal{Z}_2^\varrho[\vec{b}],\vec{a}\rangle_{L^2(U,\varrho^{-1})}\end{aligned}$$ where we have used the symmetry of $\bm{Z}_2$, and on the last line used Fubini’s theorem to interchange the order of the integration between the $\vec{r}'$ and $\vec{r}$ variables. Hence the lemma is proved. Since we have now established that $\bm{1}+\mathcal{Z}^\varrho_1-\lambda\mathcal{Z}^\varrho_2$ is a compact and self-adjoint operator we may use its eigenvectors as a complete basis of $\mathbb{R}^3$ to expand the flux $\vec{a}(\vec{r},t)$.\ \[thm:eigenfn\_expansion\_of\_flux\] Let $\bm{Z}_2$ be symmetric and real and $\mu_{\max}\|g\bm{Z}_2\|_{L^\infty(U)}<1$ and let $\vec{e}_i(\vec{r},[\varrho],t)$ and $\mu_i^{-1}$ be the eigenvectors and eigenvalues of $\bm{D}^{-1}(\vec{r},[\varrho],t)$ where $[\cdot]$ denotes functional dependence and $i = 1,\cdots, d$. Then there is a unique $\vec{a}(\vec{r},t)\in L^2(U)$ solving given by the eigenfunction expansion $$\begin{aligned} \vec{a}(\vec{r},t) = \sum_{n=1}^d\delta_n\vec{w}_{n}(\vec{r},t)\label{eq:a_expanded_in_wn}.\end{aligned}$$ Here, $\vec{w}_n$ are eigenfunctions of $(\bm{1}+\mathcal{Z}_1^\varrho -\lambda\mathcal{Z}_2^\varrho)$ obtained by a second expansion in $\vec{e}_i(\vec{r},[\varrho],t)$ of the form $$\begin{aligned} \label{eq:wn_expanded_in_es} \vec{w}_n(\vec{r},t) = \sum_{i=1}^d\frac{\psi_i}{\mu_i^{-1}-\phi_n} \vec{e}_i(\vec{r},[\varrho],t).\end{aligned}$$ Additionally, the expansion coefficients $\delta_n$ are given by the formula $$\begin{aligned} \label{eq:eqn_for_deltans} \delta_n = \frac{1}{\phi_n}\sum_{i=1}^d \frac{\psi_i}{\phi_n-\mu_i^{-1}}\int_U\mathrm{d}\vec{r}\, \varrho(\vec{r},t)\vec{e}_i(\vec{r},[\varrho],t)\cdot\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho]\end{aligned}$$ where $\{\phi_n\}_{n=1}^d$ are the discrete set of eigenvalues of $(\bm{1}+\mathcal{Z}_1^\varrho -\lambda\mathcal{Z}_2^\varrho)$ given by roots of the equation $\lambda(\phi_n) = -1$, where the function $\lambda(\cdot)$ is defined by $$\begin{aligned} \lambda(\phi_n) := \left[\sum_{l=1}^d\frac{\eta_l \,\psi_l}{\mu_l^{-1}-\phi_n}\right]^{-1}.\end{aligned}$$ Finally, $\psi_k$ and $\eta_l$ are the expansion coefficients defined by $$\begin{aligned} \varrho(\vec{r},t)g(\vec{r},\vec{r}')\bm{Z}_2(\vec{r},\vec{r}') = \sum_{k=1}^d\sum_{l=1}^d \psi_k\eta_l \vec{e}_k(\vec{r},[\varrho],t)\otimes\vec{e}_l(\vec{r}',[\varrho],t)\label{eq:rho_g_Z2_expanded_in_e}\end{aligned}$$ and each $\varrho$ is obtained from the continuity equation and no-flux condition $$\begin{aligned} \partial_t \varrho &= -\nabla_{\vec{r}}\cdot \vec{a},\\ 0 & =\Pi[\varrho]\cdot \vec{n}|_{\partial U}.\end{aligned}$$ The scalars $\mu_i^{-1}$, $\psi_k$, $\eta_l$ (and by proxy $\delta_n$) each have functional dependence on $\varrho$ since they are obtained by integrals involving $\vec{e}_j(\vec{r},[\varrho],t)$, for $i,j,k,l = 1,\cdots, d$. The eigenvalues $\phi_n$ are so called ‘moving eigenvalues’ of $(\bm{1}+\mathcal{Z}_1^\varrho -\lambda\mathcal{Z}_2^\varrho)$ (cf. [@davidson2006spectral]). If $\bm{Z}_2 = 0$ then $\phi_i = \mu_i^{-1}$ for each $i=1,\cdots, d$. In general, for $\bm{Z}_2 \neq 0$, an eigenvalue of $\bm{D}^{-1}$ may also be an eigenvalue of $(\bm{1}+\mathcal{Z}_1^\varrho -\lambda\mathcal{Z}_2^\varrho)$ and this occurs on the line $\lambda = 0$. Since $\bm{Z}_2$ is symmetric it can be diagonalised, and therefore the kernel of the operator $\mathcal{Z}_2$ can be decomposed into a finite (of length $d$) sum of products of continuous functions and has at most $d$ eigenvalues. The equation $\lambda(\phi_n) = -1$ may be rearranged into a characteristic polynomial equation in $\phi_n$ with coefficients dependent on $\eta_l$, $\psi_l$ and $\mu_l$ and since $(\bm{1}+\mathcal{Z}_1^\varrho -\lambda\mathcal{Z}_2^\varrho)$ is assumed to be real and symmetric, each $\phi_n\in\mathbb{R}$. Finally, the condition $\mu_{max}\|g\bm{Z}_2\|_{L^\infty(U)}<1$ ensures $\phi_n \neq 0$ for any $n\in \mathbb{N}$. We consider the more general operator $(\bm{1}+\mathcal{Z}_1^\varrho-\lambda\mathcal{Z}_2^\varrho)$ where $\lambda \in \mathbb{R}$. One may think of this operator as a nonlocal matrix operator where $(\bm{1}+\mathcal{Z}_1^\varrho)$ is the local part and $\mathcal{Z}_2^\varrho$ is the nonlocal part. Here $\lambda$ is a perturbation parameter measuring the distance of the full operator $(\bm{1}+\mathcal{Z}_1^\varrho-\lambda\mathcal{Z}_2^\varrho)$ from locality. Since $\bm{Z}_1$ and $\bm{Z}_2$ are real and symmetric and $\lambda\in \mathbb{R}$ , $\bm{1}+\mathcal{Z}_1^\varrho-\lambda\mathcal{Z}_2^\varrho$ coincides with its adjoint in $L^2(U,\varrho^{-1})$. For the homogeneous adjoint equation $$\begin{aligned} \label{eq:adjoint_eqn} (\bm{1}+\mathcal{Z}_1^\varrho-\lambda\mathcal{Z}_2^\varrho)^\dagger\bm{z} = 0\end{aligned}$$ we know from Lemma \[thm:cond\_converg\_fred\_det\] that when $\mu_{\max}\|g\bm{Z}_2\|_{L^\infty(U)}<1$ there is no $\lambda\in \mathbb{R}$ satisfying $\det((\bm{1}+\mathcal{Z}_1^\varrho-\lambda\mathcal{Z}_2^\varrho)^\dagger) = 0$ and therefore the only solution to the homogeneous adjoint equation is $\bm{z} = \vec{0}$. Therefore by the Fredholm alternative there is a unique solution to . Now consider the eigenvalue problem $$\begin{aligned} \label{eq:eigenvalue_problem_(1+lambdaA)} (1+\mathcal{Z}_1^\varrho-\lambda \mathcal{Z}_2^\varrho)[\vec{w}_n(\vec{r},t)] = \phi_n\vec{w}_n(\vec{r},t)\end{aligned}$$ for eigenvalues $\phi_n\in\mathbb{R}$ and eigenvectors $\vec{w}_n\in \mathbb{R}^d$. We write $$\begin{aligned} \label{eq:wn_expanded_in_ejs} \vec{w}_n = \sum_{j=1}^d\alpha_{j,n}\vec{e}_j(\vec{r},[\varrho],t).\end{aligned}$$ By inserting into we obtain $$\begin{aligned} \label{eq:eigenvalue_problem_expanded} (\bm{1}+\mathcal{Z}_1^\varrho)\sum_{j=1}^d\alpha_{j,n}\vec{e}_j(\vec{r},[\varrho],t) - \lambda \mathcal{Z}_2^\varrho\left[ \sum_{j=1}^d\alpha_{j,n}\vec{e}_j(\vec{r},[\varrho],t)\right] = \phi_n\sum_{j=1}^d\alpha_{j,n}\vec{e}_j(\vec{r},[\varrho],t).\end{aligned}$$ Now by inserting the expansion into we obtain $$\begin{aligned} \sum_{j=1}^d\alpha_{j,n}(\mu_j^{-1}-\phi_n)\vec{e}_{j}(\vec{r},[\varrho],t) - \lambda\sum_{k,l=1}^d\psi_k\,\eta_l\int_U\mathrm{d}\vec{r}'\vec{e}_k(\vec{r},[\varrho],t)\otimes \vec{e}_l(\vec{r}',[\varrho],t)\vec{w}_n(\vec{r}',t) = 0.\end{aligned}$$ Taking the inner product of this equation with $\vec{e}_i(\vec{r},[\varrho],t)$ and integrating we obtain $$\begin{aligned} \alpha_{i,n}(\mu^{-1}_i-\phi_n) -\lambda\,\psi_i\sum_{l=1}^d\eta_l\int_U\mathrm{d}\vec{r}'\vec{e}_l(\vec{r},[\varrho],t)\cdot \vec{w}_n(\vec{r}',t)=0,\end{aligned}$$ which may be rearranged to obtain $$\begin{aligned} \label{eq:separtion_eqn_for_lambda} \lambda = \frac{\alpha_{i,n}(\mu_i^{-1}-\phi_n)}{\psi_i\,\sum_{l=1}^d\eta_l\int_U\mathrm{d}\vec{r}'\vec{e}_l(\vec{r},[\varrho],t)\cdot \vec{w}_n(\vec{r}',t)}.\end{aligned}$$ Since both the left hand side of and $\sum_{l=1}^d\eta_l\int_U\mathrm{d}\vec{r}'\vec{e}_l(\vec{r},[\varrho],t)\cdot \vec{w}_n(\vec{r}',t)$ are independent of the index $i$ it must be that $$\begin{aligned} \frac{\alpha_{i,n}(\mu_i^{-1}-\phi_n)}{\psi_i} = K\end{aligned}$$ for some constant $K$ for which, without loss of generality, we choose $K=1$. With this we obtain an expression for the coefficients $\alpha_{i,n}$ $$\begin{aligned} \label{eq:eqn_for_alpha_in} \alpha_{i,n} = \frac{\psi_i}{\mu_i^{-1}-\phi_n}.\end{aligned}$$ We may also obtain a scheme to determine the $\phi_n$. In particular by and we have $$\begin{aligned} \lambda &= \left(\sum_{l=1}^d\eta_l\int_U\mathrm{d}\vec{r}'\vec{e}_l(\vec{r},[\varrho],t)\cdot \vec{w}_n(\vec{r}',t)\right)^{-1}\nonumber\\ & = \left(\sum_{l=1}^d\eta_l\int_U\mathrm{d}\vec{r}'\vec{e}_l(\vec{r},[\varrho],t)\cdot \sum_{j=1}^d\frac{\psi_j}{\mu_j^{-1}-\phi_n}\vec{e}_j(\vec{r}',[\varrho],t)\right)^{-1} = \left(\sum_{l=1}^d\frac{\eta_l\,\psi_l}{\mu_l^{-1}-\phi_n}\right)^{-1}\end{aligned}$$ hence we have that the eigenvalues of $(\bm{1}+\mathcal{Z}_1^\varrho+\mathcal{Z}_2^\varrho)$ are given by the roots of the equation $\lambda(\phi_n) = -1$. We now return to the inhomogeneous problem and expand $\vec{a}(\vec{r},t)$ in eigenfunctions $\vec{w}_n(\vec{r},t)$. We propose an expansion of the form and insert into to obtain $$\begin{aligned} \sum_{n=1}^d \delta_n\phi_n\vec{w}_n(\vec{r},t) = -\varrho(\vec{r},t)\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho].\end{aligned}$$ Now by taking the inner product with some $\vec{w}_k(\vec{r},t)$ and integrating we obtain $$\begin{aligned} \delta_k\phi_k = -\int_U\mathrm{d}\vec{r}\,\varrho(\vec{r},t)\vec{w}_k(\vec{r},t)\cdot \nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho].\end{aligned}$$ By inserting the definition of $\vec{w}_k$ from we deduce $$\begin{aligned} \label{eq:eqn_for_delta_k_coeffs} \delta_k\phi_k = \sum_{i=1}^d \frac{\psi_i}{\phi_k-\mu_i^{-1}}\int_U\mathrm{d}\vec{r}\, \vec{e}_i(\vec{r},[\varrho],t)\cdot\varrho(\vec{r},t)\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho].\end{aligned}$$ Now we would like to divide through by $\phi_k$ but must check that no $\phi_k$ is zero for each $k = 1,\cdots, d$. This is a consequence of the condition $\mu_{\max}\|g \bm{Z}_2\|<1$. In particular, using properties of the determinant, we have that $$\begin{aligned} \det(\bm{1}+\mathcal{Z}_1^\varrho-\lambda\mathcal{Z}_2^\varrho) = \det(\bm{1}+\mathcal{Z}_1^\varrho) \times\det(\bm{1}-\lambda(\bm{1}+\mathcal{Z}_1^\varrho)^{-1}\mathcal{Z}_2^\varrho).\end{aligned}$$ Now since $\bm{D}$ is positive definite, so is $\bm{1}+\mathcal{Z}_1^\varrho$ and therefore $\det(\bm{1}+\mathcal{Z}_1^\varrho) >0$ because the determinant is simply the product of its (strictly positive) eigenvalues. Additionally, since $\mu_{\max}\|g \bm{Z}_2\|<1$, we have by Lemma \[thm:cond\_converg\_fred\_det\] $\det(\bm{1}-\lambda(\bm{1}+\mathcal{Z}_1^\varrho)^{-1}\mathcal{Z}_2^\varrho)>0$ therefore $\det(\bm{1}+\mathcal{Z}_1^\varrho-\lambda\mathcal{Z}_2^\varrho)>0$ and $\phi_k \neq 0$ for all $k\in \mathbb{N}$. We may now divide by $\phi_k$ to obtain . Finally $\vec{a}(\vec{r},t)\in L^2(U)$ may be seen by squaring , integrating over $\mathrm{d}\vec{r}$ and using . Theorem \[thm:eigenfn\_expansion\_of\_flux\] provides a scheme for computing the unique flux $\vec{a}(\vec{r},t)$, given $\varrho$ satisfying $\partial_t\varrho = -\nabla_{\vec{r}}\cdot \vec{a}$ over time. We now use this result to show that the free energy functional $\mathcal{F}[\varrho]$ may be associated to the full system even when $\bm{Z}_2 \neq 0$. In particular, that $\varrho(\vec{r},t)$ solving implies $\varrho$ is a critical point of the free energy $\mathcal{F}[\varrho]$.\ \[thm:association \_of\_free\_energy\] Let $\mu_{\max}\|g\bm{Z}_2\|_{L^\infty(U)}<1$ $V_1 = V_1(\vec{r})$ be a time independent confining potential so that $\varrho(\vec{r})$ is a stationary density to the system then $\varrho(\vec{r})$ is a critical point of $\mathcal{F}[\varrho]$. Let $\varrho(\vec{r},t) = \varrho(\vec{r})$ be a stationary density. Then by equation one has $$\begin{aligned} (\bm{1}+(\bm{1}+\mathcal{Z}_1^\varrho)^{-1}\mathcal{Z}_2^\varrho)[\vec{a}(\vec{r})]&=-\varrho(\vec{r})\bm{D}(\vec{r},[\varrho],t)\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho],\label{eq:stationary_eqn_for_a_dimensionless}\\ \nabla_{\vec{r}}\cdot \vec{a}(\vec{r})&=0.\label{eq:incomp_a_at_equilibrium}\end{aligned}$$ We have that for each $\lambda$, the operator $\bm{1}+\mathcal{Z}_1^\varrho-\lambda\mathcal{Z}_2^\varrho$ is compact self-adjoint in $L^2(U,\varrho^{-1}(\vec{r},t))$ (by Lemma \[lem:flux\_operator\_is\_compact\_self\_adjoint\]). We also have that $\bm{1}+\mathcal{Z}_1^\varrho-\lambda\mathcal{Z}_2^\varrho$ is positive definite for $\mu_{\max}\|g Z_2\|_{L^\infty(U)}<1$. In particular, $\phi_n\neq 0$ for every $n = 1,\cdots d $ and $\phi_n(\lambda)$ is continuous function of $\lambda$ such that $\phi_n(0) = \mu_{n}^{-1}>0$ for each $n$. Hence we may invert $\bm{1}+\mathcal{Z}_1^\varrho+\mathcal{Z}_2^\varrho$ given $\mu_{\max}\|g Z_2\|_{L^\infty(U)}<1$. With this, by using equations , we have $$\begin{aligned} \label{eq:div_stationary_eqn_in_terms_of_F} 0 = \nabla_{\vec{r}}\cdot \vec{a} = \nabla_{\vec{r}}\cdot \left(\varrho(\vec{r})(\bm{1}+\mathcal{Z}_1^\varrho+\mathcal{Z}_2^\varrho)^{-1}\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho]\right). \end{aligned}$$ Now, assuming $\varrho$ is stationary we see that $$\begin{aligned} 0 = \Big\langle \frac{\delta\mathcal{F}}{\delta\varrho}[\varrho],\partial_t\varrho\Big\rangle = -\int_U\mathrm{d}\vec{r}\, \frac{\delta\mathcal{F}}{\delta\varrho}[\varrho]\nabla_{\vec{r}}\cdot \vec{a} = \int_U\mathrm{d}\vec{r}\, \nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho]\cdot \vec{a}\end{aligned}$$ where we have used the no-flux boundary condition. Now since $(\bm{1}+\mathcal{Z}_1^\varrho+\mathcal{Z}_2^\varrho)^{-1}$ is strictly positive definite and self-adjoint in $L^2(U,\varrho^{-1}(\vec{r},t))$ it possesses a unique strictly positive definite self-adjoint square root in $L^2(U,\varrho^{-1}(\vec{r},t))$ (see [@wouk1966note]). We define $\mathcal{X}_\varrho = (\bm{1}+\mathcal{Z}_1^\varrho+\mathcal{Z}_2^\varrho)^{-1}$ and $\mathcal{X}_\varrho^{1/2}\mathcal{X}_\varrho^{1/2} = \mathcal{X}_\varrho$. Then we find $$\begin{aligned} 0 &= \int_U\mathrm{d}\vec{r}\, \nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho]\cdot \vec{a} = \int_U\mathrm{d}\vec{r}\, \nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho]\cdot \mathcal{X}_\varrho \left[\varrho\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho]\right]\nonumber\\ &= \Big\langle \varrho \nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho],\mathcal{X}_\varrho \left[\varrho\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho]\right]\Big\rangle_{L^2(U,\varrho^{-1})}\nonumber\\ &=\Big\langle \mathcal{X}_\varrho^{1/2} \left[\varrho\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho]\right],\mathcal{X}_\varrho^{1/2} \left[\varrho\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho]\right]\Big\rangle_{L^2(U,\varrho^{-1})}\nonumber\\ &=\Big\|\mathcal{X}_\varrho^{1/2} \left[\varrho\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho]\right]\Big\|_{L^2(U,\varrho^{-1})}^2\label{eq:X_sqrt_energy_argument}\end{aligned}$$ where we have used the self-adjoint property of $\mathcal{X}_\varrho^{1/2} $. From the above we deduce that, since the integrand in the last line of is positive, that the stationary density $\varrho(\vec{r})$ satisfies $$\begin{aligned} \label{eq:grad_F_rho_is_zero} \varrho(\vec{r})\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho(\vec{r})] = \vec{0}.\end{aligned}$$ Therefore we obtain that $\varrho$ is a critical point the free energy $\mathcal{F}[\varrho]$. Let $\mu_{\max}\|g\bm{Z}_2\|_{L^\infty(U)}<1$ $V_1 = V_1(\vec{r})$ be a time independent confining potential. If $\varrho$ is a stationary density then it is a critical point of the free energy $\mathcal{F}[\varrho]-\int\mathrm{d}\vec{r}\mu_c\varrho$. From Theorem \[thm:association \_of\_free\_energy\] we obtain the following two corollaries. In particular, the proof shows rigorously how the diffusion tenor decouples from the stationary density.\ \[cor:stationary\_density\_independent\_of\_D\] Let $\bm{Z}_1$, $\bm{Z}_2$ be real and symmetric. Then the stationary density $\varrho$ is independent of $\bm{Z}_1$, $\bm{Z}_2$ and, as a consequence, of $\bm{D}$. Additionally since holds in equilibrium even when $\bm{Z}_2 \neq 0$ and the condition $\mu_{\max}\|g\bm{Z}_2\|_{L^ \infty(U)}<1$ implies that the operator $(\bm{1}+(\bm{1}+\mathcal{Z}_1^\varrho)^{-1}\mathcal{Z}_2^\varrho)$ has no zero eigenvalue, the homogeneous problem $(\bm{1}+(\bm{1}+\mathcal{Z}_1^\varrho)^{-1}\mathcal{Z}_2^\varrho) = \vec{0}$ (i.e. at equilibrium) must have only the trivial solution $\vec{a}(\vec{r}) = \vec{0}$. In addition by equation , at equilibrium one has $$\begin{aligned} \label{eq:stationary_eqn_rigorously_derived} \nabla_{\vec{r}}\cdot \left(\varrho(\vec{r})\bm{D}(\vec{r},[\varrho])\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho]\right)=0\end{aligned}$$ where $\bm{D}(\vec{r},[\varrho])$ is the time limiting diffusion tensor.\ \[cor:a\_is\_zero\_at\_equilibrium\] Let $\bm{Z}_1$, $\bm{Z}_2$ be real and symmetric. Then $\vec{a}(\vec{r}) = \vec{0}$ is the unique stationary flux. In particular, there do not exist stationary densities which are advected by the existence of some non-zero flux, hence the only stationary states are equilibrium states. We remark that Corollary \[cor:stationary\_density\_independent\_of\_D\] is related to the well-known result that for finite dimensional reversible diffusions, i.e. Langevin dynamics of the form $dX_t = - (D((X_t)) \nabla V((X_t))) \, dt + \nabla \cdot D(X_t) \, dt + \sqrt{2 D(X_t)} \, dW_t$ for an arbitrary strictly positive definite mobility matrix $D$, $V$ a confining potential and Wiener process $W_t$, the invariant measure $\mu(dx) = \frac{1}{Z} e^{-V(x)} \, dx$ is independent of $D$. We refer to [@pavliotis2014stochastic Sec 4.6]. To our knowledge, this is the first instance where such a result is proved in the context of DDFT. In the following Sections \[sec:char\_stationary\_sol\], \[sec:global\_asymptotic\_stability\], \[sec:bifurcation\_theory\], we consider the global asymptotic stability of the stationary equation (equivalently ) for which, we have shown by Corollary \[cor:stationary\_density\_independent\_of\_D\], that is the equation determining the equilibrium density the dynamics driven to equilibrium by the HI tensors $\bm{Z}_1,\bm{Z}_2$.\ \[rem:final\_assumptions\] Out of equilibrium, the effective drift is augmented by $\bm{A}[\vec{a}]$ (as defined in ), the flow induced by the HI. In order to simplify the presentation of the calculations needed for the proofs of several results presented later on, (Theorem \[lem:exp\_conv\_in\_L2\], Proposition \[prop:general\_kappa1\_convergence\], all results in Sections \[sec:existence\_uniqueness\_with\_partial\_HI\] and \[sec:classical\_paraboliv\_pde\]) we suppress $\bm{A}[\vec{a}]$ because it may trivially included as a linear contribution which is bounded in $L^1(U)$: $$\begin{aligned} \|\bm{A}[\vec{a}]\|_{L^1(U)} = \int_U\mathrm{d}\vec{r}\,\Big| \int_U\mathrm{d}\vec{r}'\, \bm{Z}_2(\vec{r},\vec{r}')\vec{a}(\vec{r}',t)\Big|\leq \|\bm{Z}_2\|_{L^\infty(U)}\|\vec{a}\|_{L^1(U)}<\infty\end{aligned}$$ where we have used and the fact that, by Theorem \[thm:eigenfn\_expansion\_of\_flux\], $\|\vec{a}\|_{L^1(U)}^2\leq |U|\|\vec{a}\|^2_{L^2(U)}<\infty$. Hence all the coefficients of remain uniformly bounded and the existence and uniqueness results of Section \[sec:existence\_uniqueness\_with\_partial\_HI\] may be easily obtained with $\bm{A}[\vec{a}]$ included. Additionally, since we have shown that at equilibrium $\bm{A}[\vec{a}]= \vec{0}$ uniquely, the results of Sections \[sec:global\_asymptotic\_stability\], \[sec:bifurcation\_theory\] hold for the dynamics tending to equilibrium including the effects of the HI. Given this remark, we now discuss the existence of stationary solutions to .\ Characterisation of Stationary Solutions {#sec:char_stationary_sol} ======================================== We now define the stationary problem. \[sec:stationary\_problem\] We seek classical solutions $\varrho\in C^2(\bar{U})$ of $$\begin{aligned} \nabla_{\vec{r}}\cdot \left[\bm{D}\left( \nabla_{\vec{r}}\varrho+\varrho\nabla_{\vec{r}}[\kappa_1 V_1+\kappa_2V_2\star\varrho]\right) \right] &= 0 \qquad \vec{r}\in U, \label{eq:stationary_problem}\\ \Pi[\varrho]\cdot \vec{n} &= 0 \qquad \vec{r} \text{ on } \partial_U. \label{eq:stationary_problem_bc}\end{aligned}$$ where $$\begin{aligned} \Pi[\varrho]:=\bm{D}\left( \nabla_{\vec{r}}\varrho+\varrho\nabla_{\vec{r}}[\kappa_1 V_1+\kappa_2V_2\star\varrho]\right).\end{aligned}$$ The existence and uniqueness for the stationary problem is based on a fixed point argument for the nonlinear map, defined by integrating equation . In particular we find the stationary distribution satisfies the nonlinear map (the self-consistency equation) $$\begin{aligned} \label{eq:self_cons_eq} \varrho(\vec{r}) = \frac{e^{-(\kappa_1V_1(\vec{r})+ \kappa_2V_2\star\varrho(\vec{r}))}}{Z},\end{aligned}$$ where $Z = \int_U\mathrm{d}\vec{r}\, \exp\{-(\kappa_1V_1(\vec{r})+ \kappa_2V_2\star\varrho(\vec{r}))\}$. Note that the stationary distribution is independent of the diffusion tensor (see Corollary \[cor:stationary\_density\_independent\_of\_D\]). We now present our first result concerning the existence and uniqueness of the solutions to the self-consistency equation.\ \[thm:exis\_fix\_point\] The stationary equation with boundary condition has a smooth, non-negative solution with $\|\varrho\|_{L^1(U)}=1$. When the interaction energy is sufficiently small, $| \kappa_2 | \leq 1/4\times \|V_2\|_{L^\infty}^{-1}$, the solution is unique. The proof follows Dressler et al. [@dressler1987stationary]. The main idea is to show that the right hand side of equation is a contraction map on $C^2(U)$, and for sufficiently small interaction energy $\kappa_2$, $\varrho_\infty\in L^1(U)$ is the unique invariant measure which is a non-negative function with unit mean. Consider the stationary problem such that Assumption holds. Then we have that 1. There exists a weak solution $\varrho\in H^1(U)\cap P_{ac}(U)$ to as a fixed point of the equation . 2. Any weak solution $\varrho\in H^1(U)\cap P_{ac}(U)$ is smooth and strictly positive, that is $\varrho\in C^{\infty}(\bar{U})\cap P^+_{ac}(U)$. The proof is similar to [@greg_mckean_vlasov_torus Theorem 2.3] but one must check the conclusions of the theorem hold with no flux boundary conditions and a confining potential $V_1$. This result is similar to arguments in [@tamura1984asymptotic] but here we consider a compact domain $U$. The weak formulation of is $$\begin{aligned} \label{eq:weak_form_of_stationary_eqn} -\int_U\mathrm{d}\vec{r}\,\nabla_{\vec{r}}\eta\cdot \bm{D}\nabla_{\vec{r}}\varrho -\kappa_1 \int_U\mathrm{d}\vec{r}\,\nabla_{\vec{r}}\eta\cdot \varrho\bm{D}\nabla_{\vec{r}}V_1 - \kappa_2 \int_U\mathrm{d}\vec{r}\,\nabla_{\vec{r}}\eta\cdot \varrho\bm{D}\nabla_{\vec{r}}V_2\star \varrho = 0,\end{aligned}$$ for $\eta \in H^1(U)$ where we have used the no-flux boundary condition in on $\varrho$ and we seek solutions $\varrho\in H^1(U)\cap P_{ac}(U)$. Now define $F:P_{ac}(U)\to P_{ac}(U)$ by $$\begin{aligned} \label{eq:def_of_T_rho} F \varrho = \frac{1}{Z(\varrho,\kappa_2)}e^{-(\kappa_1V_1+ \kappa_2V_2\star \varrho)}, \quad Z(\varrho,\kappa_2) = \int_U\mathrm{d}\vec{r}\,e^{-(\kappa_1V_1+ \kappa_2V_2\star \varrho)}.\end{aligned}$$ By we see that $$\begin{aligned} \label{eq:T_on_E} \|F\varrho\|_{L^2(U)}^2 \leq \frac{1}{|U|} e^{4(|\kappa_1|\|V_1\|_{L^\infty(U)}+|\kappa_2|\|V_2\|_{L^\infty(U)})} =: E_0,\end{aligned}$$ and therefore we seek solutions to in the set $E := \{\varrho\in L^2(U) \,:\, \|\varrho\|_{L^2(U)}^2 \leq E_0\}$. Note that $E$ is a closed, convex subset of $L^2(U)$ and therefore we may redefine $T$ to act on $E$. Additionally we see that for $\varrho\in E$ $$\begin{aligned} \|F\varrho\|_{H^1(U)}^2 &= \|F\varrho\|_{L^2(U)}^2 + \|\nabla_{\vec{r}}T\varrho\|_{L^2(U)}^2\nonumber\\ &\leq E_0\left(1+2|\kappa_1|^2\|\nabla V_1\|_{L^2(U)}^2+|\kappa_2|^2\|\nabla V_2\|_{L^2(U)}^2E_0\right),\label{eq:H1_estimate_of_rho}\end{aligned}$$ where we have used that $\varrho\in L^1(U)$ by Lemma \[thm:exis\_fix\_point\] and $V_1,V_2\in H^1(U)$. Similarly to [@greg_mckean_vlasov_torus Theorem 2.3] we have by that $F(E)\subset E$ and by $F(E)$ is uniformly bounded in $H^1(U)$. Therefore by Rellich’s compactness theorem, $F(E)$ is relatively compact in $L^2(U)$, and therefore in $E$, since $E$ is closed. We may show using similar calculations to [@dressler1987stationary Theorem 1] that the non-linear map in is Lipschitz continuous in $E$, and by Schauder fixed point theorem there exists $\varrho\in E$ solving which by is in $H^1(U)$. By inserting the expression for $F\varrho$ into we obtain (1). Also note that solutions $\varrho\in E$ to are bounded below by $E_0^{-1}/|U|^2$ giving positivity of solutions. We now show that every weak solution in $\varrho\in H^1(U)\cap P_{ac}(U)$ is a fixed point of the nonlinear map in . Consider the frozen weak formulation $$\begin{aligned} \label{eq:weak_form_of_stationary_eqn_frozen} -\int_U\mathrm{d}\vec{r}\,\nabla_{\vec{r}}\eta\cdot \bm{D}\nabla_{\vec{r}}\theta -\kappa_1 \int_U\mathrm{d}\vec{r}\,\nabla_{\vec{r}}\eta\cdot \bm{D}\nabla_{\vec{r}}V_1\theta - \kappa_2 \int_U\mathrm{d}\vec{r}\,\nabla_{\vec{r}}\eta\cdot \bm{D}\nabla_{\vec{r}}V_2\star \varrho\, \theta = 0.\end{aligned}$$ This is the weak formulation of the PDE (for the unknown function $\theta$) $$\begin{aligned} \nabla_{\vec{r}}\cdot\left(\bm{D}\nabla_{\vec{r}}\theta + \theta\bm{D}(\nabla_{\vec{r}}V_1+\nabla_{\vec{r}}V_2\star\varrho)\right) = 0, \quad \text{ s.t. } \nabla_{\vec{r}}\left((F\varrho)^{-1}\theta\right)\cdot\vec{n}|_{\partial U} = 0.\end{aligned}$$ We note that we may rewrite the weak formulation as $$\begin{aligned} \label{eq:weak_form_of_stationary_eqn_frozen_rewrite} -\int_U\mathrm{d}\vec{r}\,\nabla_{\vec{r}}\eta\cdot \bm{D}\nabla_{\vec{r}}h\,F\varrho = 0\end{aligned}$$ for every $\eta\in H^1(U)$ and where $h = \theta/(F\varrho)$. This holds true for any $\eta$, in particular $\eta = h$ hence we find $$\begin{aligned} -\int_U\mathrm{d}\vec{r}\,\Big| (F\varrho)^{1/2}\bm{D}^{1/2}\nabla_{\vec{r}}h\Big|^2 = 0\end{aligned}$$ where we have used that $\bm{D}$ is positive definite by and $F\varrho$ is strictly positive. All in all we obtain $\nabla_{\vec{r}}h = 0$ a.e. and hence $\theta = F\varrho$ up to normalisation. But if $F\varrho$ is a probability density we must have $\theta \equiv F\varrho$ and we conclude that since $\varrho = F\varrho$, any weak solution $\varrho\in H^1(U)\cap P^+_{ac}(U)$ of must be such that $\varrho = F\varrho$. The regularity of $\varrho$ follows from the same bootstrapping argument of [@greg_mckean_vlasov_torus Theorem 2.3]. We can also obtain an estimate on the rate of convergence to the equilibrium density in $L^2(U)$ as $t\to \infty$ with the following theorem. In order to forgo additional assumptions on the initial data $\varrho_0$ we restrict ourselves to the case where the equilibrium density is unique and given by $\varrho_\infty$.\ \[lem:exp\_conv\_in\_L2\] Let $\varrho\in C^1([0,\infty];C^2(U)) $ be a solution of with initial data $\varrho_0\in L^2(U)$ a probability density. For $\kappa_1=0$, if $$\begin{aligned} \kappa_2^2 < \min \Big\{ \frac{\mu_{\min}c_{pw}^{-2}\|\nabla_{\vec{r}}V_2\|^{-2}_{L^\infty}}{2(1+e)\mu_{\max}}, \frac{1}{4\|V_2\|_{L^\infty}} \Big\},\end{aligned}$$ where $c_{pw}$ is a Poincar[' e]{}$-$Wirtinger constant on the domain $U$ and $\mu_{\max}$ and $\mu_{\min}$ are the largest and smallest eigenvalues of $\bm{D}$, then $\varrho\to \varrho_\infty$ in $L^2(U)$ exponentially as $t\to \infty$. In particular the convergence in $L^2(U)$ is given by $$\begin{aligned} \|\varrho(\cdot, t)-\varrho_\infty(\cdot)\|^2_{L^2(U)}\leq \|\varrho_0(\cdot)-\varrho_\infty(\cdot)\|_{L^2(U)}^2 e^{-r_{\kappa_2}t}\end{aligned}$$ as $t \to \infty$ where $r_{\kappa_2} = \mu_{\min}c^{-2}_{pw}- 2\mu_{\max}|\kappa_2|^2(e+1)\|\nabla_{\vec{r}}V_2\|_{L^{\infty}(U)}^2$ is the rate of convergence. Let $\psi = \varrho-\varrho_\infty$, then the evolution equation for $\psi$ may be written $$\begin{aligned} \label{eq:exp_conv_in_l2_bound_1} \partial_t\psi - \nabla_{\vec{r}}\cdot[\bm{D}\,\nabla_{\vec{r}}\psi] = \kappa_2 \nabla_{\vec{r}}\cdot[\bm{D}\,(\varrho_\infty\nabla_\vec{r}V_2\star \psi +\psi\nabla_{\vec{r}}V_2\star \varrho)].\end{aligned}$$ Multiplying by $\psi$, integrating and using the boundary condition $\Pi[\psi]\cdot\vec{n} = 0$ on $\partial U\times [0,T]$ we obtain $$\begin{aligned} &\tfrac{1}{2}\der[]{t}\|\psi (t)\|^2_{L^2(U)} +\|\bm{D}^{1/2}\nabla_{\vec{r}}\psi \|_{L^2(U)}^2 \nonumber\\ &\leq \int_U\mathrm{d}\vec{r}\,|\bm{D}^{1/2}\nabla_{\vec{r}}\psi |\times |\kappa_2 \bm{D}^{1/2}(\varrho_\infty\nabla_\vec{r}V_2\star \psi +\psi\nabla_{\vec{r}}V_2\star \varrho)|. \end{aligned}$$ Using H[ö]{}lder’s inequality on the right hand side this becomes $$\begin{aligned} &\tfrac{1}{2}\der[]{t}\|\psi (t)\|^2_{L^2(U)} +\|\bm{D}^{1/2}\nabla_{\vec{r}}\psi \|_{L^2(U)}^2 \nonumber\\ &\leq \|\bm{D}^{1/2}\nabla_{\vec{r}}\psi \|_{L^2(U)}\times \|\kappa_2 \bm{D}^{1/2}(\varrho_\infty\nabla_\vec{r}V_2\star \psi +\psi\nabla_{\vec{r}}V_2\star \varrho)\|_{L^2(U)}. \end{aligned}$$ Now using Young’s inequality twice on the right hand side we obtain $$\begin{aligned} &\tfrac{1}{2}\der[]{t}\|\psi (t)\|^2_{L^2(U)} +\|\bm{D}^{1/2}\nabla_{\vec{r}}\psi \|_{L^2(U)}^2\nonumber\\ & \leq \tfrac{1}{2}\|\bm{D}^{1/2}\nabla_{\vec{r}}\psi \|_{L^2(U)}^2 + \tfrac{1}{2} \|\bm{D}^{1/2}(\varrho_\infty\nabla_\vec{r}V_2\star \psi +\psi\nabla_{\vec{r}}V_2\star \varrho)\|_{L^2(U)}^2 \nonumber \\ & \leq \tfrac{1}{2}\|\bm{D}^{1/2}\nabla_{\vec{r}}\psi \|_{L^2(U)}^2 + |\kappa_2|^2 \|\varrho_\infty\bm{D}^{1/2}\nabla_{\vec{r}}V_2\star \psi \|^2_{L^2(U)} + |\kappa_2|^2 \|\psi\bm{D}^{1/2}\nabla_{\vec{r}}V_2\star \varrho \|^2_{L^2(U)}.\label{eq:l2_trend_inq_1}\end{aligned}$$ From the positive definiteness and boundedness of the diffusion tensor, we have $\mu_{\min}\leq \|\bm{D}\|_{L^\infty(U)}\leq \mu_{\max}$. We also have the following bounds in terms of $\|\psi\|_{L^2(U)}^2$ $$\begin{aligned} &\|\psi\bm{D}^{1/2}\nabla_{\vec{r}}V_2\star \varrho \|^2_{L^2(U)}\leq\mu_{\max}\|\nabla_{\vec{r}}V_2\|_{L^\infty(U)}^2\|\psi\|_{L^2(U)}^2\label{eq:l2_trend_formula_1a}\\ &\|\varrho_\infty\bm{D}^{1/2}\nabla_{\vec{r}}V_2\star \psi \|^2_{L^2(U)}\leq |U|\mu_{\max}\|\varrho_\infty\|_{L^2(U)}^2\|\nabla_{\vec{r}}V_2\|^2_{L^\infty(U)}\|\psi \|_{L^2(U)}^2\label{eq:l2_trend_formula_1b}\end{aligned}$$ where $|U|$ denotes the size of $U$ and in we have used that $\nabla V_2 \star \varrho \leq \| \nabla V_2\|_{L^\infty(U)} \| \varrho \|_{L^1(U)}$ and the fact that $\varrho$ is a probability density with $\| \varrho \|_{L^1} =1$ (see Corollary \[cor:L\_1\_varrho\_is\_1\]). To obtain we use that $$\|\varrho_\infty\bm{D}^{1/2}\nabla_{\vec{r}}V_2\star \psi \|^2_{L^2(U)}\leq \mu_{\max}\|\varrho_\infty\|_{L^2(U)}^2\|\nabla_{\vec{r}}V_2\|^2_{L^\infty(U)} \int_U \mathrm{d}\vec{r}\, \Big| \rho_\infty(\vec{r}) \int \mathrm{d}\vec{r} ' \, \psi(\vec{r}') \Big|^2.$$ We then note that, by Hölder’s inequality, $ \int \mathrm{d}\vec{r} ' \, \psi(\vec{r}') \leq \|\psi\|_{L^2}\|1\|_{L^2(U)} = |U|^{1/2} \|\psi\|_{L^2}$, which gives the result. For it remains to bound the non explicit stationary distribution $\varrho_\infty$ in $L^2(U)$, to do this we observe that by the self-consistency equation $$\begin{aligned} \|\varrho_\infty\|_{L^2(U)}^2\leq \frac{|U|\times e^{2|\kappa_2\||V_2\|_{L^\infty}}}{|U|^2\times e^{-2|\kappa_2\||V_2\|_{L^\infty}}}.\label{eq:l2_trend_formula_2}\end{aligned}$$ Using , and the bounds on $\bm{D}$, inequality becomes $$\begin{aligned} \tfrac{1}{2}\der[]{t}\|\psi (t)\|^2_{L^2(U)}&\leq -\tfrac{\mu_{\min}}{2}\|\nabla_{\vec{r}}\psi \|_{L^2(U)}^2\nonumber\\ &\quad+ \mu_{\max}|\kappa_2|^2(e^{4|\kappa_2\||V_2\|_{L^\infty}}+1)\|\nabla_{\vec{r}}V_2\|_{L^{\infty}(U)}^2\|\psi \|^2_{L^2(U)}. \end{aligned}$$ Now since $\psi$ has mean zero we may use the Poincar[' e]{}–Wirtinger inequality to write $$\begin{aligned} \der[]{t}\|\psi (t)\|^2_{L^2(U)}&\leq -\mu_{\min}c^{-2}_{pw}\|\psi \|_{L^2(U)}^2\nonumber\\ &\quad+ 2\mu_{\max}|\kappa_2|^2(e^{4|\kappa_2\||V_2\|_{L^\infty}}+1)\|\nabla_{\vec{r}}V_2\|_{L^{\infty}(U)}^2\|\psi \|^2_{L^2(U)}.\end{aligned}$$ Finally, by Gr[ö]{}nwall’s lemma [@evans2002partial], we obtain $$\begin{aligned} &\|\psi (t)\|^2_{L^2(U)}\nonumber\\ &\leq \|\psi(0)\|_{L^2(U)}^2\exp\left\lbrace -(\mu_{\min}c^{-2}_{pw}- 2\mu_{\max}|\kappa_2|^2(e^{4|\kappa_2\||V_2\|_{L^\infty}}+1)\|\nabla_{\vec{r}}V_2\|_{L^{\infty}(U)}^2)t\right\rbrace. \label{eq:l2_trend_gronwall_1}\end{aligned}$$ Therefore for any $\varrho_\ast$ a stationary density the necessary condition for exponential convergence $\varrho\to\varrho_\ast$ in $L^2(U)$ as $t\to\infty$ is $$\label{eq:gen_kappa_2_ineq_for_exp_convergence} \mu_{\min}c^{-2}_{pw}- 2\mu_{\max}|\kappa_2|^2(e^{4|\kappa_2\||V_2\|_{L^\infty}}+1)\|\nabla_{\vec{r}}V_2\|_{L^{\infty}(U)}^2>0.$$ It will now be seen that, under the assumption that $\varrho_\infty$ is the unique stationary density with $\kappa_2 \leq \|V_2\|_{L^\infty}^{-1}/4$, we may obtain an explicit condition for $|\kappa_2|$. In particular becomes $$\begin{aligned} \|\psi (t)\|^2_{L^2(U)}\leq \|\psi(0)\|_{L^2(U)}^2 \exp\left\lbrace -(\mu_{\min}c^{-2}_{pw}- 2\mu_{\max}|\kappa_2|^2(e+1)\|\nabla_{\vec{r}}V_2\|_{L^{\infty}(U)}^2)t\right\rbrace. \end{aligned}$$\[eq:l2\_trend\_gronwall\_2\] Then to ensure the argument in the exponential remains negative, we require $$\begin{aligned} |\kappa_2|^2<\frac{\mu_{\min}c_{pw}^{-2}\|\nabla_{\vec{r}}V_2\|^{-2}_{L^\infty}}{2(1+e)\mu_{\max}}.\end{aligned}$$ This completes the proof of the theorem. We remark that $\psi\in \left\lbrace u\in H^1(U)\, | \,\int_U\mathrm{d}\vec{r}\,u = 0 \right\rbrace $, therefore, we may determine that the sharpest value of $c_{pw}$ conincides with the Poincar[é]{} constant as found by Steklov [@kuznetsov2015sharp], equal to $\nu_1^{-1/2}$ where $\nu_1$ is the smallest eigenvalue of the problem $$\begin{aligned} \Delta u &= -\nu u \quad \text{ in } U,\\ \partial_{\vec{n}}u & = 0 \qquad \,\,\,\, \text{ on } \partial U.\end{aligned}$$ Here $\partial_{\vec{n}}$ is the directional derivative along the unit vector $\vec{n}$ pointing out of the domain $U$. Additionally Payne and Weinberger [@payne1960optimal] proved that for convex domains in $\mathbb{R}^n$ one has $c_{pw}\leq \tfrac{\mathrm{diam}(U)}{\pi}$. One may obtain a similar convergence result including a confining potential as given by the following corollary.\ \[prop:general\_kappa1\_convergence\] Let $\kappa_1\neq 0$ and let $\varrho\in C^1([0,\infty];C^2(U)) $ be a solution of with initial data $\varrho_0\in L^2(U)$ a probability density. Then the exponential convergence $\varrho\to\varrho_\infty$ in $L^2$ criteria is modified to $$\begin{aligned} \mu_{\max}\kappa_1^2\|\nabla_{\vec{r}}V_1\|_{L^\infty(U)}^2<r_{\kappa_2}\end{aligned}$$ along with $| \kappa_2 | \leq 1/4\times \|V_2\|_{L^\infty(U)}^{-1}$. In particular the convergence in $L^2$ is given by $$\begin{aligned} \|\varrho(\cdot, t)-\varrho_\infty(\cdot)\|^2_{L^2(U)}\leq \|\varrho_0(\cdot)-\varrho_\infty(\cdot)\|_{L^2(U)}^2 e^{-(r_{\kappa_2}-\mu_{\max}\kappa_1^2\|\nabla_{\vec{r}}V_1\|_{L^\infty(U)}^2)t}.\end{aligned}$$ Since the inclusion of an external field is linear in the PDEs , the proof is similar to Lemma \[lem:exp\_conv\_in\_L2\], the only term to resolve for the evolution equation for $\psi$ first occurring at being $$\begin{aligned} \kappa_1^2\|\psi\boldsymbol{D}^{1/2}\nabla_{\vec{r}}V_1\|_{L^2(U)}^2 \leq \kappa_1^2\mu_{\max}\|\nabla V_1\|_{L^\infty(U)}^2\|\psi\|_{L^2(U)}^2.\end{aligned}$$ The remainder of the calculations to derive a Gr[ö]{}nwall type inequality including this term are similar. Global Asymptotic Stability {#sec:global_asymptotic_stability} =========================== In this section we study the stability properties of stationary states. We start by showing the free energy is a strictly convex functional, provided $\kappa_2$ is sufficiently small, and that $\mathcal{F}$ is bounded below. Recall the free energy functional $\mathcal{F}:P^+_{\text{ac}}(U)\to \mathbb{R}$ is given by $$\begin{aligned} \mathcal{F}[\varrho]&:=\int_U\mathrm{d}\vec{r}\,\varrho(\vec{r})\log\varrho(\vec{r})+\kappa_1\int_U\mathrm{d}\vec{r}\,V_1(\vec{r})\varrho(\vec{r})+\frac{\kappa_2}{2}\int_U\mathrm{d}\vec{r}\,\varrho(\vec{r})V_2\star\varrho(\vec{r}). \nonumber\end{aligned}$$\ \[prop:F\_is\_strictly\_convex\] For $|\kappa_2|\in [0,\|V_2\|_{L^\infty(U)}^{-1})$ the free energy functional $\mathcal{F}$ is strictly convex. Additionally there exists a positive constant $B_0<\infty$ for every $\varrho\in P_{\text{ac}}^+$ such that $|\mathcal{F}[\varrho]| \geq B_0$. Suppose $\varrho_1$ and $\varrho_2$ satisfy with $\Pi[\varrho_1]\cdot\vec{n} = \Pi[\varrho_2]\cdot\vec{n} = 0$ on $\partial_U$ for all $t\in [0,\infty)$. Letting $\zeta = \varrho_2-\varrho_1$ and $\varrho_s = (1-s)\varrho_1+s\varrho_2$ we compute $\tfrac{\mathrm{d}^2}{\mathrm{d}s^2}\mathcal{F}_H[\varrho_s]$ by direct calculation $$\begin{aligned} \frac{\mathrm{d}^2}{\mathrm{d}s^2}\mathcal{F}_H[\varrho_s] &= \frac{\mathrm{d}}{\mathrm{d}s}\frac{\mathrm{d}}{\mathrm{d}s}\left[\int_U\mathrm{d}\vec{r}\,\varrho_s\log\varrho_s +\kappa_1 \int_U\mathrm{d}\vec{r}\,\varrho_s V_1 +\frac{\kappa_2}{2} \int_U\mathrm{d}\vec{r}\,\varrho_s V_2\star\varrho_s\right]\nonumber\\ &= \frac{\mathrm{d}}{\mathrm{d}s}\left[\int_U\mathrm{d}\vec{r}\,\zeta \log\varrho_s +\zeta \right.\nonumber\\ & \qquad \left. + \kappa_1\int_U\mathrm{d}\vec{r}\,\zeta V_1 + \frac{\kappa_2}{2}\int_U\mathrm{d}\vec{r}\,\zeta V_2\star\varrho_s + \frac{\kappa_2}{2}\int_U\mathrm{d}\vec{r}\,\varrho_s V_2\star\zeta\right]\nonumber\\ &=\int_U\mathrm{d}\vec{r}\,\frac{\zeta^2}{\varrho_s} + \kappa_2\int_U\mathrm{d}\vec{r}\,\zeta V_2\star \zeta.\end{aligned}$$ Now using the measure $\mathrm{d}\mu = \varrho_s\mathrm{d}\vec{r}$ we have, by Jensen’s inequality, $$\begin{aligned} \int_U\mathrm{d}\vec{r}\,\frac{\zeta^2}{\varrho_s} = \int_U\mathrm{d}\mu\,\frac{\zeta^2}{\varrho_s^2} \geq \left(\int_U\mathrm{d}\vec{r}\,|\zeta|\right)^2.\end{aligned}$$ We also have that $V_2$ is bounded below by the negative of its its essential supremum from . Combining these facts we find $$\begin{aligned} \label{eq:convexity_condition_of_F} \frac{\mathrm{d}^2}{\mathrm{d}s^2}\mathcal{F}_H[\varrho_s] \geq (1-|\kappa_2\|V_2\|_{L^\infty(U)}\})\left(\int_U\mathrm{d}\vec{r}\,|\zeta|\right)^2\end{aligned}$$ and we therefore find that, for $\kappa_2$ such that $|k_2|\leq \tfrac{1}{4}\|V_2\|_{L^\infty(U)}^{-1}$, the free energy functional $\mathcal{F}$ is strictly convex. Now let $\varrho\in P^+_{\text{ac}}$ and observe that $$\begin{aligned} \mathcal{F}[\varrho]\geq -\Big|\int_U\mathrm{d}\vec{r}\,\varrho\log\varrho\Big| -|\kappa_1|\|V_1\|_{L^\infty(U)}-\tfrac{|\kappa_2|}{2}\|V_2\|_{L^\infty(U)}.\end{aligned}$$ The entropy $\varrho\log\varrho$ is continuous and bounded below on $U$ and therefore we have that $$\begin{aligned} \mathcal{F}[\varrho]\geq \int\mathrm{d}\vec{r}\,|\varrho\log\varrho|-|\kappa_1|\|V_1\|_{L^\infty(U)}-\tfrac{|\kappa_2|}{2}\|V_2\|_{L^\infty(U)}>-\infty.\end{aligned}$$ where we have used the assumptions on the potentials in . Hence $\mathcal{F}[\cdot]$ is bounded below. Note that the convexity condition in Proposition \[prop:F\_is\_strictly\_convex\] holds independently of the confining potential $V_1$. We therefore have the following Corollary for the total free energy $\mathcal{F}-\int_U\mathrm{d}\vec{r}\,\mu_{c}\varrho$.\ \[cor:the\_free\_energy\_is\_convex\_and\_bounded\_below\] The total free energy $\mathcal{F}-\int\mathrm{d}\vec{r}\,\mu_{c}\varrho$ is strictly convex for $|\kappa_2|\in [0,\|V_2\|_{L^\infty(U)}^{-1})$ and bounded below. We now provide a useful Lemma which will be used eventually to show that $\mathcal{F}$ always has a minimiser, for any $\kappa_2$ (see Lemma \[lem:minimisers\_always\_exist\]).\ \[lem:can\_pick\_a\_smaller\_free\_energy\_density\] Let $V_1$, $V_2$ satisfy the assumptions then there exists a positive constant $B_0$ such that for every $\varrho\in P_{ac}(U)$ with $\|\varrho\|_{L^\infty(U)}>B_0$ there exists some $\varrho^\dagger \in P_{ac}(U)$ with $\|\varrho^{\dagger}\|_{L^\infty(U)}\leq B_0$ such that $$\begin{aligned} \mathcal{F}(\varrho^\dagger)<\mathcal{F}(\varrho).\end{aligned}$$ For a proof see [@greg_mckean_vlasov_torus Lemma 2.5] or [@chayes2010mckean Lemma 2.1], the only modification required is to include $V_1$ which by assumption is bounded below and the proof follows a similar argument. We now show that minimisers of $\mathcal{F}$ exist for all $\kappa_2$. First we define the integral operator $\mathcal{R}$ which will be useful for the following calculations.\ \[def:def\_of\_R\_op\] Let $\mathcal{R}:L^1(U)\to L^1(U)$ be given by $$\begin{aligned} \label{eq:def_of_R_operator} \mathcal{R}u = -\varrho V_2\star u. \end{aligned}$$ We note that $\mathcal{R}$ is a compact (since $V_2$ is uniformly bounded in $L^\infty(U)$) self-adjoint operator in $L^2(U,\varrho^{-1})$. We label its eigenvalues $\{\beta_n^{-1}\}_{n=1}^\infty$ and eigenfunctions $\{u_n\}_{n=1}^\infty$ satisfying $$\begin{aligned} \label{eq:eigenvalue_problem_for_R} \mathcal{R}u_n = \beta_n^{-1}u_n.\end{aligned}$$ \[lem:minimisers\_always\_exist\] Let $\kappa_2\in (-\infty,\infty)$ and let $V_1$, $V_2$ satisfy the assumptions . Then there exists a $\varrho\in P_{ac}(U)$ that minimizes $\mathcal{F}$. Since $\mathcal{F}$ is bounded below there exists a minimising sequence $\{\varrho_j\}_{j=1}^\infty \in P_{ac}(U)$ so that $\mathcal{F}(\varrho_{j})<\mathcal{F}(\varrho_{j+1})$. Therefore, by Lemma \[lem:can\_pick\_a\_smaller\_free\_energy\_density\] $\{\varrho_j\}_{j = 1}^\infty$ may be chosen such that $\|\varrho_j\|_{L^2}(U)\leq \|\varrho_j\|^2_{L^\infty(U)}|U|$. Now by the Eberlein-Smuljan theorem, since $\{\varrho_j\}_{j = 1}^\infty$ is bounded, there exists a subsequence (which we will denote again by $\{\varrho_j\}_{j = 1}^\infty$) such that $\varrho_j \rightharpoonup \varrho_\ast$ weakly in $L^2$ to some $\varrho_\ast$. Therefore $\lim_{j\to \infty}\int_U\mathrm{d}\vec{r}\,\eta(\varrho_j-\varrho_\ast) = 0$ for every $\eta\in L^2(U)$, so in particular for $\eta = 1$ we obtain $\lim_{j\to \infty}\int_U\mathrm{d}\vec{r}\, \varrho_j = 1 = \int_U\mathrm{d}\vec{r}\, \varrho_\ast$. Additionally we note that $|\varrho_j| \rightharpoonup |\varrho_\ast|$ in $L^2(U)$, and therefore $\|\varrho_\ast\|_{L^1(U)} = 1$, which is enough to show that $\varrho_\ast\geq 0 $ a.e. by standard arguments (see, for example, the proof of Corollary \[cor:L\_1\_varrho\_is\_1\]). We define $\Lambda:P_{ac}\to \mathbb R$ such that $$\begin{aligned} \Lambda(z) := \int_U\mathrm{d}\vec{r}\,zV_2\star z.\end{aligned}$$ Now let $\varrho_{\beta_{n}}\in L^1(U)$ be a solution to , which is known to exist by Lemma \[thm:exis\_fix\_point\]. Note that $\varrho_{\beta_{n}}$ need not be a minimiser of $\mathcal{F}$ and may be an inflection point or local maximum. Additionally since $\varrho_{\beta_{n}}\in L^1(U)$ solves , we have that $\varrho_{\beta_{n}}>e^{-(|\kappa_1|\|V_1\|_{L^\infty(U)}+|\beta_{n}|\|V_2\|_{L^\infty(U)})}/Z>0$ (where $Z$ is a normalisation constant) and therefore there exists $\delta\in \mathbb{R}^+$ such that $\varrho_{\beta_{n}}>\delta$ for every $\vec{r}\in U$. Now we estimate the interaction energy difference by $$\begin{aligned} |\Lambda(\varrho_j)-\Lambda(\varrho_\ast)| &\leq \sum_{n=1}^N|\beta_n^{-1}|\Big|\langle \varrho_j,w_n\rangle_{L^2(U,\varrho^{-1}_{\beta_{n}})}-\langle \varrho_\ast,w_n\rangle_{L^2(U,\varrho^{-1}_{\beta_{n}})}\Big| + 2 |\beta_N^{-1}|\delta^{-1}B_0\nonumber\\ & \leq 2\delta^{-1} B_0 \sum_{n=1}^N \langle\varrho_j -\varrho_\ast, w_n\rangle_{L^2(U)} + 2 |\beta_N^{-1}|\delta^{-1} B_0\end{aligned}$$ where we have used the fact that the integrand of $\Lambda(z)$ is equal to $\mathcal{R}$ acting on $z \in P_{ac}$. Additionally we have used that $\mathcal{R}$ is self-adjoint in $L^2(U,\varrho^{-1}_{\beta_{n}})$, to write $\mathcal{R}$ as a projection onto its eigenvectors $\{w_n\}_{n=1}^{\infty}$ and bounded the tail of the infinite sum using Bessel’s inequality. Now since $\mathcal{R}$ is self-adjoint in $L^2(U, \varrho_{\beta_n}^{-1})$ we have that (after reordering) $|\beta_n^{-1}|\to 0$ as $n \to \infty$ so the second term may be made arbitrarily small. The first term may be made arbitrarily small by taking the limit $j \to \infty$ inside the finite sum and using that $\varrho_j\rightharpoonup \varrho_{\ast}$ weakly in $L^2(U)$. This shows that $\Lambda(\cdot)$ is continuous in $\varrho$. Additionally, for the external energy, we have $$\begin{aligned} \Big|\int_U\mathrm{d}\vec{r}\, V_1(\vec{r})\varrho_j(\vec{r})-\int_U\mathrm{d}\vec{r}\, V_1(\vec{r})\varrho_\ast(\vec{r})\Big| & = \Big|\int_U\mathrm{d}\vec{r}\, V_1(\vec{r})(\varrho_j(\vec{r})-\varrho_\ast(\vec{r}))\Big| \nonumber\\ & \leq \Big|\int_U\mathrm{d}\vec{r}\, V_1(\vec{r})(\varrho_j(\vec{r})-\varrho_\ast(\vec{r}))\Big| \to 0\end{aligned}$$ as $j\to \infty$. The lower semicontinuity of the entropy term in follows from standard results [@jost1998calculus Lemma 4.3.1]. Therefore the free energy $\mathcal{F}[\varrho]$ has a minimiser $\varrho$ over $P_{ac}(U)$. We may refine this result to show that minimisers are attained in $P^+_{ac}(U)$ with the following lemma.\ \[lem:minimisers\_are\_positive\] Let $\varrho\in P_{ac}(U)\backslash P^+_{ac}(U)$. Then there exists $\varrho^\dagger\in P^+_{ac}(U)$ such that $\mathcal{F}[\varrho^\dagger]<\mathcal{F}[\varrho]$. The proof is similar to [@greg_mckean_vlasov_torus Lemma 2.6]. One must show that the potential energy for a $P^+_{ac}(U)$ density may be bounded by the potential energy of a $ P_{ac}(U)$ density. We let $\epsilon>0$ and define the competition state $\varrho_\epsilon$ such that $$\begin{aligned} \varrho_\epsilon(\vec{r}) = \frac{(\varrho(\vec{r})+\epsilon\mathbb{I}_{\mathbb{B}_0}(\vec{r}))}{1+\epsilon |\mathbb{B}_0|}\end{aligned}$$ where $\mathbb{B}_0 = \{\vec{r}\in U \, :\, \varrho(\vec{r}) = 0\}$ and since by assumption $\varrho \notin P^+_{ac}(U)$ one has $|\mathbb{B}_0|>0$ and $\varrho_\epsilon\in P^+_{ac}(U)$. Then we obtain that $$\begin{aligned} \int_U\mathrm{d}\vec{r}\, V_1 \varrho_\epsilon \leq \int_U\mathrm{d}\vec{r}\, V_1 \varrho + \epsilon |\mathbb{B}_0|.\end{aligned}$$ Using this bound, together with the result [@greg_mckean_vlasov_torus Lemma 2.6] we obtain the required result. Exponential Convergence to Equilibrium in Relative Entropy. ----------------------------------------------------------- In this section we derive an H-theorem which guarantees that the time evolution of the dynamics converges to the equilibrium distribution given by the self-consistency equation. First consider the time derivative of the integral of the free energy $$\begin{aligned} &\der[]{t} \mathcal{F}[\varrho] = \int_U \mathrm{d}\vec{r}\, \partial_t \varrho\, \tfrac{\delta\mathcal{F}[\varrho] }{\delta \varrho} = \int_U \mathrm{d}\vec{r}\,\nabla\cdot\left[\bm{D}(\vec{r},t)\varrho(\vec{r},t)\nabla_{\vec{r}}\tfrac{\delta\mathcal{F}[\varrho] }{\delta \varrho} \right]\tfrac{\delta\mathcal{F}[\varrho] }{\delta \varrho} \nonumber\\ &=-\int_U \mathrm{d}\vec{r}\,\Big|\bm{D}(\vec{r},t)^{1/2}\varrho(\vec{r},t)^{1/2}\nabla_{\vec{r}}\tfrac{\delta\mathcal{F}[\varrho] }{\delta \varrho} \Big|^2,\end{aligned}$$ where we have integrated by parts and used the boundary condition $\Pi\varrho \cdot\vec{n}\big|_{\partial_U}= 0$ or $\varrho\to 0$ as $|\vec{r}|\to\infty$ for bounded and unbounded domains respectively. Here we see that as long as both $\bm{D}(\vec{r},t)$ and $\varrho(\vec{r},t)$ remain positive definite then the free energy is monotonically decreasing in time. Indeed the diffusion tensor $\bm{D}$ is positive definite as proven in [@goddard2012overdamped] and we will show strict positivity of $\varrho(\vec{r},t)$ in Section \[subsec:strict\_pos\_rho\]. We now introduce the relative entropy functional $$\begin{aligned} \label{eq:def_of_rel_entropy_func} \mathcal{H}[\varrho |\varrho_\infty]:=\int_{U}\mathrm{d}\vec{r}\, \varrho \log\left(\frac{\varrho}{\varrho_\infty}\right),\end{aligned}$$ and obtain the following theorem for convergence to equilibrium in relative entropy. \[thm:rel\_entropy\_convergence\] Let $V_1$ be convex, $|\kappa_2|<\tfrac{1}{4}\|V_2\|_{L^\infty(U)}^{-1}$ and $\varrho\in C^1([0,\infty];C^2(U))$ be a classical solution to equation . If $\kappa_2^2 < \frac{c_{ls}^{-1}}{2 \| \nabla V_2 \|_{L^\infty(U)}^2}$ then $\varrho$ is exponentially stable in relative entropy and it holds that $$\begin{aligned} \mathcal{H}[\varrho |\varrho_\infty]\leq \mathcal{H}[\varrho_0|\varrho_\infty] e^{-\tfrac{1}{2}(c_{ls}^{-1}-2|\kappa_2|^2 \|\nabla V_2\|_{L^\infty(U)}^2)t},\end{aligned}$$ where $c_{ls}>0$ is the log-Sobolev constant for the measure $\mu$. By direct calculation we find $$\begin{aligned} \label{eq:H_time_der} \frac{\mathrm{d} \mathcal{H}[\varrho |\varrho_\infty]}{\mathrm{d}t} &= \int_U\mathrm{d}\vec{r}\, \partial_t\left( \varrho\log\left(\frac{\varrho}{\varrho_\infty}\right)\right) = \int_U\mathrm{d}\vec{r}\, \partial_t\varrho \log\left(\frac{\varrho}{\varrho_\infty}\right) + \int_U\mathrm{d}\vec{r}\,\partial_t \varrho\nonumber\\ & = \int_U\mathrm{d}\vec{r}\, \partial_t\varrho \log\left(\frac{\varrho}{\varrho_\infty}\right) + \frac{\mathrm{d}M}{\mathrm{d}t} = - \int_U\mathrm{d}\vec{r}\, \varrho\nabla \frac{\delta \mathcal{F}[\varrho]}{\delta \varrho}\cdot \nabla \log\left(\frac{\varrho}{\varrho_\infty}\right) + 0 \nonumber\\ &= - \int_U\mathrm{d}\vec{r}\, \varrho \left(\nabla \log\varrho + \kappa_1 \nabla V_1 +\kappa_2\nabla V_2\star \varrho\right)\cdot \nabla \log\left(\frac{\varrho}{\varrho_\infty}\right)\nonumber\\ & = - \int_U\mathrm{d}\vec{r}\, \varrho \left(\nabla \log\varrho +\kappa_2\nabla V_2\star \varrho-\left( \nabla \log \varrho_\infty +\kappa_2\nabla V_2\star \varrho_\infty\right)\right)\cdot \nabla \log\left(\frac{\varrho}{\varrho_\infty}\right)\nonumber\\ & =- \int_U\mathrm{d}\vec{r}\, \varrho \left(\nabla \log\left(\frac{\varrho}{\varrho_\infty}\right) +\kappa_2\nabla V_2\star (\varrho-\varrho_\infty)\right)\cdot \nabla \log\left(\frac{\varrho}{\varrho_\infty}\right)\end{aligned}$$ where we have used the no-flux boundary condition and the self-consistency equation $\nabla \log\varrho_\infty + \kappa_1 \nabla V_1 +\kappa_2\nabla V_2\star \varrho_\infty = 0$. Note that the contribution from the $V_1$ term is constant, independent of $\rho$, and so cancels after using the self-consistency equation. Continuing by expanding out the integrand and using H[ö]{}lder’s inequality we obtain $$\begin{aligned} \frac{\mathrm{d} \mathcal{H}[\varrho |\varrho_\infty]}{\mathrm{d}t} &= - \int_U\mathrm{d}\vec{r}\, \varrho \Big| \nabla \log\left(\frac{\varrho}{\varrho_\infty}\right) \Big|^2 + \kappa_2 \int_U\mathrm{d}\vec{r}\, \varrho \nabla\log \left(\frac{\varrho}{\varrho_\infty}\right) \cdot \nabla V_2\star (\varrho- \varrho_\infty)\nonumber\\ &\leq -\int_U\mathrm{d}\vec{r}\, \varrho \Big| \nabla \log\left(\frac{\varrho}{\varrho_\infty}\right)\Big|^2\nonumber\\ &\qquad + \left[\int_U\mathrm{d}\vec{r}\, \varrho \Big|\nabla\log \left(\frac{\varrho}{\varrho_\infty}\right)\Big|^2\right]^{1/2} \times \left(\kappa_2^2 \int_U\mathrm{d}\vec{r}\, \varrho |\nabla V_2\star (\varrho-\varrho_\infty)|^2\right)^{1/2}.\end{aligned}$$ Now, by Young’s inequality, $$\begin{aligned} \frac{\mathrm{d} \mathcal{H}[\varrho |\varrho_\infty]}{\mathrm{d}t} \leq -\tfrac{1}{2}\int_U\mathrm{d}\vec{r}\, \varrho \Big| \nabla \log\left(\frac{\varrho}{\varrho_\infty}\right)\Big|^2 +\tfrac{\kappa_2^2}{2} \int_U\mathrm{d}\vec{r}\, \varrho |\nabla V_2\star (\varrho-\varrho_\infty)|^2\end{aligned}$$ and we may estimate the second term on the right hand side (in particular using that $\int_U \rho = 1$ from Corollary \[cor:L\_1\_varrho\_is\_1\]), giving $$\begin{aligned} \label{eq:rel_entropy_proof_first_bound} \frac{\mathrm{d} \mathcal{H}[\varrho |\varrho_\infty]}{\mathrm{d}t} \leq -\tfrac{1}{2}\int_U\mathrm{d}\vec{r}\, \varrho \Big| \nabla \log\left(\frac{\varrho}{\varrho_\infty}\right)\Big|^2 + \tfrac{\kappa_2^2}{2}\|\nabla V_2\|_{L^\infty(U)}^2\|\varrho-\varrho_\infty \|_{L^1(U)}^2\end{aligned}$$ We bound the first term as follows. Since $V_1$ is convex, we have $$\begin{aligned} \nabla_{\vec{r}}^2V_1\geq \theta_1>0\end{aligned}$$ for some $\theta_1\in \mathbb{R}^+$. Now by the Bakry–[É]{}mery criterion (see [@menz2014poincare Sec 3, Theorem 3.1], and [@malrieu2001logarithmic]) the measure $\mu'(\mathrm{d}\vec{r}) = \mathrm{d}\vec{r}\,e^{-\kappa_1 V_1}/Z$ where $Z$ is a normalisation constant satisfies a log-Sobolev inequality (LSI) with constant $c_{ls}'$ such that $$\begin{aligned} \frac{1}{c_{ls}'}\geq \theta_1\kappa_1.\end{aligned}$$ However since $V_2$ is not general a convex function, we cannot use the Bakry–[É]{}mery criterion for $\mu$ as defined in . However we may deduce a LSI using the Holley–Stroock perturbation lemma [@menz2014poincare Sec 3, Theorem 3.2] since $V_1+V_2\star\varrho_\infty$ is a bounded perturbation of $V_1$, in particular $$\begin{aligned} \Big|V_1+V_2\star\varrho_\infty\Big| \leq \Big|V_1\Big|+ \|V_2\|_{L^\infty}\|\varrho_\infty\|_{L^1(U)} <\infty.\end{aligned}$$ Therefore $\mu$ as defined in with $\varrho = \varrho_\infty$ (after appropriate nondimensionalisation) is unique and satisfies a LSI with constant $$\begin{aligned} c_{ls}^{-1} \geq \exp\left(-\kappa_1\kappa_2\,\text{Osc}\left[V_2\star \varrho_\infty\right]\right)\frac{1}{c_{ls}'}\end{aligned}$$ where $$\begin{aligned} \text{Osc}\left[V_2\star \varrho_\infty\right] = \sup V_2\star \varrho_\infty - \inf V_2\star \varrho_\infty.\end{aligned}$$ The constant $c_{ls}$ is such that such that for each $f:U\to \mathbb{R}^+$ one has $$\begin{aligned} \label{eq:LSI} \int_U f^2\log f^2\mathrm{d}\mu -\int_U f^2 \log \left(\int_U f^2 \mathrm{d}\mu\right)\mathrm{d}\mu \leq c_{ls}\int_U |\nabla f|^2\mathrm{d}\mu = c_{ls} \int_U f^2 | \nabla \log f^2 |^2 \mathrm{d} \mu.\end{aligned}$$ We let $f = \sqrt{\varrho/\varrho_\infty}$ and $\mathrm{d}\mu = \varrho_\infty \mathrm{d}\vec{r}$ and the second term on the left hand side of is zero (since, again $\int_U \rho = 1$). Hence this shows that $$\mathcal{H}[\varrho |\varrho_\infty] = \int_U f^2\log f^2\mathrm{d}\mu \leq c_{ls} \int_U\mathrm{d}\vec{r}\, \varrho \Big| \nabla \log\left(\frac{\varrho}{\varrho_\infty}\right)\Big|^2.$$ We combine the LSI with Pinsker’s inequality [@bolley2005weighted] to deduce $$\begin{aligned} \frac{\mathrm{d} \mathcal{H}[\varrho |\varrho_\infty]}{\mathrm{d}t}\leq -\tfrac{1}{2}(c_{ls}^{-1}-2 \kappa_2^2\|\nabla V_2\|_{L^\infty(U)}^2) \mathcal{H}[\varrho|\varrho_\infty]. $$ Thus we obtain, by Gr[ö]{}nwall’s inequality, $$\begin{aligned} \mathcal{H}[\varrho |\varrho_\infty]\leq \mathcal{H}[\varrho_0|\varrho_\infty] \exp[ -\tfrac{1}{2}(c_{ls}^{-1}-2\kappa_2^2 \|\nabla V_2\|_{L^\infty(U)}^2) t ]\end{aligned}$$ and the theorem is proved. The constant $c_{ls}$ is not known explicitly but may be estimated in terms of the convexity of $V_1$, $V_2$ and the curvature of $U$ [@chen1997estimates]. We now consider asymptotic expansions of the steady states for small interaction energy $\kappa_2$.\ Asymptotic Expansion of the Steady States For Weak Interactions. ---------------------------------------------------------------- We begin this section by recalling that steady states satisfy the self-consistency equation $$\begin{aligned} \label{eq:self_consis} \varrho = \frac{e^{-(\kappa_1 V_1+\kappa_2 V_2\star \varrho)}}{Z},\end{aligned}$$ where $Z = \int_U \mathrm{d}\vec{r}\,e^{-(\kappa_1V_1+\kappa_2V_2\star\varrho)}$. We know from Lemma \[thm:exis\_fix\_point\] that for sufficiently weak interactions, i.e. $|\kappa_2|<1/4\|V_2\|_{L^\infty(U)}^{-1}$, the stationary distribution is unique; equivalently, the nonlinear equation has a unique fixed point. Let $\kappa_2 \ll 1$, then the stationary solution $\varrho(\vec{r}) = \varrho_\infty(\vec{r})$ has the form $$\begin{aligned} \varrho(\vec{r}) = \frac{e^{-\kappa_1 V_1(\vec{r})}}{Z(\varrho)}(1 + O(\kappa_2)),\end{aligned}$$ where the first order correction may be obtained explicitly as follows. Recall the stationary equation for $\varrho$: $$\begin{aligned} \nabla_{\vec{r}} \cdot [\bm{D} ( \nabla_{\vec{r}}\varrho+\kappa_1\varrho\,\nabla_{\vec{r}} V_1(\vec{r}) +\kappa_2\varrho\,\nabla_{\vec{r}} V_2\star \varrho)]=0 & \text{ on } U, \label{eq:stat_pde_for_rho}\\ \bm{D}(\nabla_{\vec{r}}\varrho+\kappa_1\varrho\,\nabla_{\vec{r}}V_1 +\kappa_2\varrho\,\nabla_{\vec{r}} V_2\star \varrho)\cdot\vec{n}=0 & \text{ on } \partial_U. \label{eq:stat_pde_for_rho_bc}\end{aligned}$$ Fix $\kappa_1=1$ and insert the perturbation expansion $$\begin{aligned} \varrho(\vec{r}) = \sum_{k=0}^\infty \kappa_2^k \varrho_k(\vec{r}).\end{aligned}$$ We find at the first order of $\kappa_2$ $$\begin{aligned} \mathcal{L}_0\varrho_0:=\nabla_{\vec{r}} \cdot (\bm{D}\nabla_{\vec{r}}\varrho_0+\bm{D}\left(\varrho_0\nabla_{\vec{r}}V_1)\right) = 0 & \text{ on } U,\label{eq:fredholm_1d_eqn}\\ \bm{D}(\nabla_{\vec{r}}\varrho_0+\varrho_0\nabla_{\vec{r}}V_1) \cdot \vec{n} =0 & \text{ on } \partial_U,\label{eq:fredholm_1d_eqn_bc}\end{aligned}$$ from which we deduce $$\begin{aligned} \varrho_0(\vec{r}) = \frac{e^{-V_1(\vec{r})}}{Z_0}\end{aligned}$$ for $Z_0 = \int_{U}\mathrm{d}\vec{r}\,e^{-V_1(\vec{r})}$. Note that $\mathcal{L}_0$ is self-adjoint in the space $L^2(U,\varrho_0^{-1})$. We may also show that the resolvent of $\mathcal{L}_0$ is compact in $L^2(U,\varrho^{-1}_0)$.\ \[lem:L0\_has\_compact\_resolvent\] The operator $\mathcal{L}_0$ has a compact resolvent in $L^2(U,\varrho_0^{-1})$. We let $\phi\in C^2(U)$, by direct calculation we have that $$\begin{aligned} \mathcal{L}_0\phi = [\nabla_{\vec{r}}\cdot \bm{D}]\cdot \nabla_{\vec{r}}\phi + \text{Tr}\left[\bm{D}\nabla_{\vec{r}}^2\phi \right] + [\bm{D}\nabla_{\vec{r}}V_1]\cdot \nabla_{\vec{r}}\phi + \phi\left[[\nabla_{\vec{r}}\cdot \bm{D}]\cdot \nabla_{\vec{r}}V_1+\text{Tr}[\bm{D}\nabla_{\vec{r}}^2V_1]\right] \end{aligned}$$ then we have that $$\begin{aligned} \|\mathcal{L}_0\phi\|_{L^2(U,\varrho_0^{-1})}^2 &= \int_U\mathrm{d}\vec{r}\, \Big|\nabla_{\vec{r}} \cdot (\bm{D}\nabla_{\vec{r}}\phi+\bm{D}\left(\phi\nabla_{\vec{r}}V_1)\right)\Big|^2\varrho_0^{-1}\nonumber\\ &\leq C(U;\bm{D};V_1)\sum_{n = 0}^2\sup_{\vec{r}\in U}\Big|\phi^{(n)}(\vec{r})\Big|<\infty\end{aligned}$$ where the constant $C(U;\bm{D};V_1)$ is dependent on $U$, the diffusion tensor $\bm{D}$ and the first weak derivatives of its entries (bounded in $L^{\infty}(U)$ by ), and the confining potential $V_1$ and its first two weak derivatives (bounded in $L^{\infty}(U)$ by ). Therefore there exists $C\in \mathbb{R}^+$ such that $\|\mathcal{L}_0\|_{L^2(U,\varrho^{-1}_0)}<C$ and the spectrum of $\mathcal{L}_0$ is bounded. Now let $z\in \rho(\mathcal{L}_0)$ with $|z|>C$, where $\rho(\cdot)$ denotes the resolvent set, then we may write the resolvent $R(z;\mathcal{L}_0)$ of the operator $\mathcal{L}_0$ as $$\begin{aligned} R(z;\mathcal{L}_0) = -z^{-1}\sum_{k = 0}^\infty z^{-k} \mathcal{L}_0^k.\end{aligned}$$ We now show that $R$ is compact. First consider the sequence $(R^N)_{N\geq 1}$ defined by $$\begin{aligned} R^N(z;\mathcal{L}_0) := -z^{-1}\sum_{k = 0}^N z^{-k} \mathcal{L}_0^k,\end{aligned}$$ then let $(\phi_j)_{j\geq 1}$ be a sequence in $C^2(U)$. We have that $(\phi_j)_{j\geq 1}$ is a bounded sequence in $C^2(U)$ and $$\begin{aligned} \|R^N(z;\mathcal{L}_0)[\phi_j]\|_{L^2(U,\varrho_0^{-1})} &\leq |z|^{-1}\sum_{k = 0}^N |z|^{-k} \|\mathcal{L}_0^k[\phi_j]\|_{L^2(U,\varrho_0^{-1})}\nonumber\\ &\leq |z|^{-1}\sum_{k = 0}^N |z|^{-k} C^{K}.\end{aligned}$$ Hence, as long as $|z|>C$ then, $\|R^N(z;\mathcal{L}_0)[\phi_j]\|_{L^2(U,\varrho_0^{-1})}$ converges for all $N$ and $\text{Im}\left(R^N\right)$ is relatively compact in $L^2(U,\varrho^{-1}_0)$. It is then a standard result that the limit of a sequence of compact operators is compact, hence $R$ is compact. Thus we have a complete set of orthonormal basis functions $\{v^{(0)}_k \}_{k=0}^{\infty}$ and corresponding eigenvalues $\{\gamma^{(0)}_n\}_{n\geq 1}$. Note that $v^{(0)}_0 = \varrho_0$ and $\gamma_0^{(0)} = 0$. At the next order of $\kappa_2$ we obtain $$\begin{aligned} \label{eq:lin_stab_pert_equilib_order_eps} \mathcal{L}_0\varrho_1+f(\varrho_0) = 0,\end{aligned}$$ where $$\begin{aligned} f(\varrho_0):=-\nabla_{\vec{r}} \cdot (\bm{D}\varrho_0\,\nabla_{\vec{r}}V_2\star \varrho_0),\end{aligned}$$ subject to $$\begin{aligned} \bm{D}(\nabla_{\vec{r}}\varrho_1+\varrho_1\,\nabla_{\vec{r}}V_1 +\varrho_0\,\nabla_{\vec{r}} V_2\star \varrho)\cdot\vec{n}=0 & \text{ on } \partial_U.\end{aligned}$$ The solvability condition for then becomes $$\begin{aligned} \label{eq:solvability_condition_V2_star_rho0_dot_n} 0=\langle f(\varrho_0),\,v_0^{(0)}\rangle_{L^2(U,\varrho_0^{-1})} & =\int_U\mathrm{d}\vec{r}\,\nabla_{\vec{r}} \cdot (\varrho_0\,\bm{D}\nabla_{\vec{r}}V_2\star \varrho_0) = \int_{\partial U}\mathrm{d}S\,\vec{n}\cdot\varrho_0\,\bm{D}\nabla_{\vec{r}}V_2\star \varrho_0.\end{aligned}$$ If the solvability condition is satisfied then, by the Fredholm alternative, there exists a solution to . We may then write $\varrho_1$ in an eigenfunction expansion $$\begin{aligned} \varrho_1(\vec{r}) = \sum_{j=0}^{\infty}\alpha_j v_{j}^{(0)} \quad \text{ where } \quad \alpha_j = -\frac{1}{\gamma_j\|v_j^{(0)}\|^2_{L^2_{\varrho_0^{-1}}}}\langle f(\varrho_0),\,v_j^{(0)} \rangle_{L^2_{\varrho_0^{-1}}}.\end{aligned}$$ This yields that $$\begin{aligned} \varrho(\vec{r}) = \frac{e^{-V_1(\vec{r})}}{Z_0} + \kappa_2 \sum_{j=0}^{\infty} \frac{\langle \nabla_{\vec{r}} \cdot (\bm{D} \varrho_0 \,\nabla_{\vec{r}} V_2 \star \varrho_0),\,v_j^{(0)} \rangle_{L^2_{\varrho_0^{-1}}} v_{j}^{(0)} (\vec{r})}{\gamma_j\|v_j^{(0)}\|^2_{L^2_{\varrho_0^{-1}}}} + O(\kappa_2^2).\end{aligned}$$\ We now consider a linear stability analysis of the equilibrium density solving .\ Linear Stability Analysis. -------------------------- We first investigate the spectrum of the linearised operator $\mathcal{L}_1$ in terms of the eigenspace of its local part. We determine a scheme for computing the eigenvalues of $\mathcal{L}_1$ explicitly. Writing $\varrho = \varrho +\epsilon\,\omega + O(\epsilon^2)$ where $\epsilon \ll 1$ is an arbitrary parameter and not equal to $\kappa_2$, we obtain\ #### $O(\epsilon^{0})$: $$\begin{aligned} \mathcal{L}\varrho = 0 \end{aligned}$$ where we have set $\varrho = \varrho_\infty$ (the unique stationary state) to ease notation and $$\begin{aligned} \mathcal{L}\varrho = \nabla\cdot (\bm{D}\,\nabla \varrho) +\kappa_1\nabla\cdot(\bm{D}\,\varrho\nabla V_1) +\kappa_2\nabla\cdot (\varrho\bm{D}\,\nabla V_2\star \varrho). \end{aligned}$$ #### $O(\epsilon^{1})$: $$\begin{aligned} \label{eq:dynamical_eqn_for_omega} \dot{\omega} = \mathcal{L}_1w\end{aligned}$$ where $$\begin{aligned} \label{eq:def_of_linearised_L} \mathcal{L}_1\omega:=\nabla\cdot (\bm{D}\,\nabla \omega) +\kappa_1\nabla\cdot(\bm{D}\,\omega\nabla V_1) +\kappa_2\nabla\cdot (\varrho\bm{D}\,\nabla V_2\star \omega) + \kappa_2\nabla\cdot (\omega\bm{D}\,\nabla V_2\star \varrho).\end{aligned}$$ We remark that the operator $\mathcal{L}_1$ is different to the one found in the linear stability analysis of [@greg_mckean_vlasov_torus Sec 3.3] due to the difference in boundary conditions. Perturbations must be mean zero, that is $\int_U\mathrm{d}\vec{r}\,\omega=0$, which may be determined by observing that $$\begin{aligned} 1 = \int\mathrm{d}\vec{r}\,\varrho + \epsilon \int\mathrm{d}\vec{r}\,w + O(\epsilon^2).\end{aligned}$$ Equally, all higher order perturbations must have mean zero. Physically speaking this is a compatibility condition with the no-flux boundary condition in to ensure that perturbations do not change the mass of the system. Additionally by linearising the self-consistency equation we find that mean zero perturbations $w$ satisfy the integral equation $$\begin{aligned} \label{eq:int_eqn_for_omega} w = -\varrho_\infty \kappa_2 V_2\star w.\end{aligned}$$ We linearise the nonlinear boundary condition to find that $$\begin{aligned} \Pi_1[\omega]\cdot \vec{n}|_{\partial_U}:= 0 \end{aligned}$$ where $$\begin{aligned} \label{bc:linearised_bc_for_omega} \Pi_1[\omega] = \bm{D}\,(\nabla_{\vec{r}}\omega + \omega\nabla_{\vec{r}}(\kappa_1 V_1(\vec{r},t)+ \kappa_2 \,V_2\star \varrho) + \kappa_2 \varrho\,\nabla_{\vec{r}}V_2\star \omega).\end{aligned}$$ We note that if any such $\omega$ exist for , then trivially holds, and equation is underdetermined. In order to properly determine $\omega$ we let $\omega \in L^2_{c}(U,\varrho^{-1})$ where $$\begin{aligned} L^2_{c}(U,\varrho^{-1}) := \Big\{ u \in L^2(U,\varrho^{-1}) \, : \, \nabla_{\vec{r}}(\varrho^{-1}u)\cdot \vec{n}|_{\partial U} = 0\Big\}.\end{aligned}$$ The choice $\omega\in L^2_{c}(U,\varrho^{-1})$ preserves the boundary condition $\Pi_1[\varrho]\cdot \vec{n} |_{\partial U} =0$ and we will show in Lemma \[lem:self\_adjoint\_A\] that it is the most general restriction to ensure that the local part of $\mathcal{L}_1$ is self-adjoint in $L^2(U,\varrho^{-1})$. With this we write $$\begin{aligned} \mathcal{L}_{1} &= \mathcal{A}_{ \kappa_2} + { \kappa_2}\mathcal{B},\label{eq:def_of_L1_op}\\ \mathcal{A}_{\kappa_2}w &: =\nabla_{\vec{r}}\cdot[ \bm{D} (\nabla_{\vec{r}}w+ w \nabla_{\vec{r}}\varphi_{\kappa_2} )] ,\label{eq:def_of_A_op}\\ \mathcal{B}w&: = \nabla_{\vec{r}}\cdot\left(\varrho\bm{D}\,\nabla_{\vec{r}}V_2\star w\right),\label{eq:def_of_B_op}\\ \varphi_{\kappa_2} &:= \kappa_1 V_1+{\kappa_2} V_2\star \rho.\label{eq:def_of_varphi_k2}\end{aligned}$$ Here, $\mathcal{A}_{\kappa_2}$ and $\mathcal{B}$ are the local and nonlocal parts of $\mathcal{L}_{1}$, respectively. Note however that $\mathcal{A}_{\kappa_2}\neq \mathcal{L}_0$ by definition since $\kappa_2$ is no longer small. All operators $\mathcal{A}_{\kappa_2}$, $\mathcal{B}$, $\mathcal{L}_{1}$ are maps $H^2(U, \varrho_\infty^{-1})\to L^2(U)$. We now show that $A_{\kappa_2}$ is a self-adjoint operator in the space $L^2_{c}(U,\varrho^{-1})$. [0.485]{} ![Plots of a) The eigenfunctions of $A_{\kappa_2}$ in $L^2([-1,1],\varrho_\infty^{-1})$ as computed with pseudospectral methods for $\kappa_2 = 1$ and $N=100$ spectral points, b) the inner product between pairs of eigenfunctions showing orthogonality of the $\{v_k^{(\kappa_2)}\}$, c) the eigenfunction expansion of $V_2(r) = 1/2(-\tanh((r-1/2)/.05)+\tanh((r+1/2)/.05))$ and d) the absolute error between the expansion $V_{2e}$ and $V_{e}$. The $L^2$ error between $V_2$ and its expansion in eigenfunctions $V_{2e}$ was found to be 5.761e-9.[]{data-label="fig:eigenfunction_figs"}](eigenfuns_k2_1.pdf "fig:"){width="\textwidth"}   [0.485]{} ![Plots of a) The eigenfunctions of $A_{\kappa_2}$ in $L^2([-1,1],\varrho_\infty^{-1})$ as computed with pseudospectral methods for $\kappa_2 = 1$ and $N=100$ spectral points, b) the inner product between pairs of eigenfunctions showing orthogonality of the $\{v_k^{(\kappa_2)}\}$, c) the eigenfunction expansion of $V_2(r) = 1/2(-\tanh((r-1/2)/.05)+\tanh((r+1/2)/.05))$ and d) the absolute error between the expansion $V_{2e}$ and $V_{e}$. The $L^2$ error between $V_2$ and its expansion in eigenfunctions $V_{2e}$ was found to be 5.761e-9.[]{data-label="fig:eigenfunction_figs"}](inner_prod_eigenfunctions_on_interval.pdf "fig:"){width="\textwidth"}   [0.485]{} ![Plots of a) The eigenfunctions of $A_{\kappa_2}$ in $L^2([-1,1],\varrho_\infty^{-1})$ as computed with pseudospectral methods for $\kappa_2 = 1$ and $N=100$ spectral points, b) the inner product between pairs of eigenfunctions showing orthogonality of the $\{v_k^{(\kappa_2)}\}$, c) the eigenfunction expansion of $V_2(r) = 1/2(-\tanh((r-1/2)/.05)+\tanh((r+1/2)/.05))$ and d) the absolute error between the expansion $V_{2e}$ and $V_{e}$. The $L^2$ error between $V_2$ and its expansion in eigenfunctions $V_{2e}$ was found to be 5.761e-9.[]{data-label="fig:eigenfunction_figs"}](V2_expanded.pdf "fig:"){width="\textwidth"}   [0.485]{} ![Plots of a) The eigenfunctions of $A_{\kappa_2}$ in $L^2([-1,1],\varrho_\infty^{-1})$ as computed with pseudospectral methods for $\kappa_2 = 1$ and $N=100$ spectral points, b) the inner product between pairs of eigenfunctions showing orthogonality of the $\{v_k^{(\kappa_2)}\}$, c) the eigenfunction expansion of $V_2(r) = 1/2(-\tanh((r-1/2)/.05)+\tanh((r+1/2)/.05))$ and d) the absolute error between the expansion $V_{2e}$ and $V_{e}$. The $L^2$ error between $V_2$ and its expansion in eigenfunctions $V_{2e}$ was found to be 5.761e-9.[]{data-label="fig:eigenfunction_figs"}](error_V2_and_expanded.pdf "fig:"){width="\textwidth"} \ \[lem:self\_adjoint\_A\] $\mathcal{A}_{\kappa_2}$ is self-adjoint in $L^2_{c}(U,\varrho^{-1})$. First note that from and we have that $\nabla_{\vec{r}} \varphi_{\kappa_2} = \rho \nabla_{\vec{r}} \rho^{-1}$ and so $ \mathcal{A}_{\kappa_2}w = \nabla_{\vec{r}} \cdot [ \rho \bm{D} \nabla_{\vec{r}} ( \rho^{-1} w )]. $ Let $u\in L^2_c(U,\varrho^{-1})$ then $$\begin{aligned} \langle u, \, \mathcal{A}_{\kappa_2} w\rangle_{L^2(U, \varrho^{-1})} &= \int_U\mathrm{d}\vec{r}\, \varrho^{-1} u \mathcal{A}_{\kappa_2} w \nonumber\\ & = \int_U\mathrm{d}\vec{r}\, \varrho^{-1} u \nabla_{\vec{r}} \cdot [ \rho \bm{D} \nabla_{\vec{r}} ( \rho^{-1} w )] \nonumber \\ &=\int_{\partial U}\mathrm{d}S\vec{n}\cdot u\bm{D}\nabla_{\vec{r}}(\varrho^{-1}\omega)-\int_U \mathrm{d}\vec{r}\, \nabla_{\vec{r}}\left[ \varrho^{-1}u \right]\cdot\left[\varrho \bm{D}\nabla_{\vec{r}}\left(\varrho^{-1}w\right) \right] \nonumber\\ &= -\int_{\partial U}\mathrm{d}S\vec{n}\cdot w\bm{D}\nabla_{\vec{r}}(\varrho^{-1}u) + \int_U \mathrm{d}\vec{r}\, \nabla_{\vec{r}} \cdot [ \rho \bm{D} \nabla_{\vec{r}} ( \rho^{-1} u )] \varrho^{-1}w\nonumber\\ &= \langle A_{\kappa_2} u, \, w\rangle_{L^2(\mathbb{R}, \varrho^{-1})} \nonumber\end{aligned}$$ where we have integrated by parts twice and used that $\bm{D}$ is symmetric and the fact that $u, w\in L^2_c(U,\varrho^{-1})$ to eliminate the boundary terms. We have established that $\mathcal{A}_{\kappa_2}$ is self-adjoint in $L^2_c(U,\varrho^{-1})$. Additionally we observe that $\mathcal{A}_{\kappa_2}$ has a compact resolvent in $L^2_c(U,\varrho^{-1})$ by a similar result to Lemma \[lem:L0\_has\_compact\_resolvent\]. The spectral theorem therefore provides a complete basis of orthonormal eigenfunctions $v_k^{({\kappa_2})}$ spanning $L^2_c(U,\varrho^{-1})$ with corresponding eigenvalues $\gamma_k^{({\kappa_2})}$ such that $$\begin{aligned} \label{eq:A_k2_eigenvalue_problem} A_{\kappa_2} v_{k}^{({\kappa_2})} = \gamma_k^{({\kappa_2})}v_{k}^{(\kappa_2)}.\end{aligned}$$ We note that the operator $\mathcal{B}$ as defined in is, in general, not self-adjoint. From now on we assume that the set of eigenfunctions $\{v_{k}^{({\kappa_2})}\}_{k=1}^\infty$ are normalised to form an orthonormal basis. [0.485]{} ![Moving eigenvalues for various differential nonlocal operators.[]{data-label="fig:lambda_paths"}](davidson_dodds_example_1.pdf "fig:"){width="\textwidth"}   [0.485]{} ![Moving eigenvalues for various differential nonlocal operators.[]{data-label="fig:lambda_paths"}](davidson_dodds_example_2.pdf "fig:"){width="\textwidth"}   [0.485]{} ![Moving eigenvalues for various differential nonlocal operators.[]{data-label="fig:lambda_paths"}](tanh_pot_epsilon_lambda.pdf "fig:"){width="\textwidth"}   [0.485]{} ![Moving eigenvalues for various differential nonlocal operators.[]{data-label="fig:lambda_paths"}](gaussian_pot_epsilon_lambda.pdf "fig:"){width="\textwidth"} The stability of the equilibrium density will depend on the spectrum of the operator $\mathcal{L}_1$ so that perturbations evolving according to either grow or decay. We now study the spectrum of $\mathcal{L}_1$. We fix $\kappa_1$ and consider $\kappa_2$, not necessarily small, as a perturbation parameter from the differential part of $\mathcal{L}_1$. The following theorem establishes the parametrisation of the eigenvalues $\lambda$ by $\kappa_2$.\ \[thm:epsilon\_as \_a\_fn\_of\_lambda\] Suppose that $\lambda \neq \gamma_k^{(\kappa_2)}$ for all $k \in \mathbb{N}$. If the solution $\kappa_2^\star(\lambda)$ of the equation $\lambda = \lambda_{k^\star}(\kappa_2^\star) $ exists, then it is given by $$\begin{aligned} \label{eq:kappa2_parametrisation} \kappa_2^\star(\lambda) = \left(\sum_{i=0}^\infty\tfrac{\theta_i^{(\kappa_2)}\gamma _i^{(\kappa_2)}\beta_i^{(\kappa_2)}}{\lambda-\gamma_i^{(\kappa_2)}}\right)^{-1},\end{aligned}$$ where $\theta_j^{(\kappa_2)}$ and $\beta_j^{(\kappa_2)}$ are given by $$\begin{aligned} \label{eq:expansion_coeffs_of_V2} \theta_k^{(\kappa_2)}\beta_l^{(\kappa_2)} = \int_U\mathrm{d}\vec{r}\,v_l^{(\kappa_2)}(\vec{r})V_2\star v_{k}^{(\kappa_2)}(\vec{r}).\end{aligned}$$ Let $$\begin{aligned} V_2(\vec{r}-\vec{r}') &= \varrho^{-1}(\vec{r}) \varrho^{-1}(\vec{r'}) \sum_{j,k=0}^\infty \beta_j^{(\kappa_2)} v_j^{(\kappa_2)}(\vec{r})\theta_k^{(\kappa_2)} v_k^{(\kappa_2)}(\vec{r}'),\nonumber\\ w(\vec{r}) &= \sum_{i=0}^\infty \alpha_iv_i^{(\kappa_2)}(\vec{r}).\end{aligned}$$ Inserting these expressions into the eigenvalue problem $\mathcal{L}_1 w = \lambda w$ we find $$\begin{aligned} \sum_{i=1}^\infty \alpha_i(\gamma^{(\kappa_2)}_i-\lambda)v_i^{(\kappa_2)}(\vec{r}) + \kappa_2\nabla_{\vec{r}}\cdot\left(\varrho\bm{D} \nabla_{\vec{r}} V_2\star w\right) =0.\end{aligned}$$ Multiplying this equation by $v^{(\kappa_2)}_n$ and integrating against the weight function $\varrho^{-1}$ we obtain $$\begin{aligned} 0 &= \alpha_n(\gamma^{(\kappa_2)}_n-\lambda)+\kappa_2\int_U\mathrm{d}\vec{r}\, \varrho^{-1}(\vec{r}) v^{(\kappa_2)}_n(\vec{r})\nabla_{\vec{r}}\cdot\left(\varrho(\vec{r}) \bm{D} \nabla_{\vec{r}} V_2\star w\right) \nonumber\\ & =\alpha_n(\gamma^{(\kappa_2)}_n-\lambda) - \kappa_2 \int_U\mathrm{d}\vec{r}\, \left[\nabla_{\vec{r}}v^{(\kappa_2)}_n(\vec{r}) + v^{(\kappa_2)}_n(\vec{r})\nabla_{\vec{r}}\varphi_{\kappa_2} \right] \cdot \bm{D}\, \nabla_{\vec{r}} V_2\star w,\end{aligned}$$ where there is no boundary term since $\nabla_{\vec{r}} V_2\star w = -\kappa_2^{-1}\nabla_{\vec{r}}(\varrho^{-1}w)$ is zero on the boundary of $U$ because $w\in L^2_c(U,\varrho^{-1})$. Continuing by integrating by parts we find $$\begin{aligned} 0 &= \alpha_n(\gamma^{(\kappa_2)}_n-\lambda)+\kappa_2\int_U\mathrm{d}\vec{r}\,\nabla_{\vec{r}} \cdot [\bm{D} ( \nabla_{\vec{r}}v^{(\kappa_2)}_n(\vec{r}) + v^{(\kappa_2)}_n(\vec{r})\nabla_{\vec{r}}\varphi_{\kappa_2})] V_2\star w\nonumber\\ &= \alpha_n(\gamma^{(\kappa_2)}_n-\lambda)-\kappa_2\int_U\mathrm{d}\vec{r}\, \nabla_{\vec{r}}\cdot(\varrho\bm{D}\nabla_{\vec{r}}(\varrho^{-1}v^{(\kappa_2)}_n(\vec{r}))) V_2\star w\nonumber\\ &= \alpha_n(\gamma^{(\kappa_2)}_n-\lambda)+\kappa_2\gamma^{(\kappa_2)}_n\int_U\mathrm{d}\vec{r}\, v^{(\kappa_2)}_n(\vec{r}) V_2\star w\end{aligned}$$ where we have used $\nabla_{\vec{r}}(\varrho^{-1}v_n^{\kappa_2}) = 0$ on $\partial_U$ to eliminate the boundary term, and in the last line used the fact that $v_n^{\kappa_2}$ is an eigenfunction of $\mathcal{A}_{\kappa_2}$. Inserting the expansion for $V_2$ and using the orthonormality of the $v_i^{(\kappa_2)}$ gives $$\begin{aligned} \kappa_2 = \frac{(\lambda-\gamma^{(\kappa_2)}_n)\alpha_n}{\gamma_n^{(\kappa_2)}\theta_n^{(\kappa_2)}\sum_{j=0}^\infty \int_U\mathrm{d}\vec{r}'\, \varrho_\infty^{-1}(\vec{r}') \beta_j^{\kappa_2} v_j^{(\kappa_2)}(\vec{r}') w(\vec{r}')}.\end{aligned}$$ This holds for all $V_2$ and, in particular, for all $\theta_n^{(\kappa_2)}\neq 0$ so it be must be the case that $$\begin{aligned} \frac{(\lambda-\gamma^{(\kappa_2)}_n)\alpha_n}{\gamma_n^{(\kappa_2)}\theta_n^{(\kappa_2)}}=K,\end{aligned}$$ for some constant $K$, independent of $n$. Without loss of generality we can take $K=1$. Hence we have $$\begin{aligned} w(x) = \sum_{i=0}^\infty \tfrac{\gamma_n^{(\kappa_2)}\theta_n^{(\kappa_2)}}{\lambda-\gamma^{(\kappa_2)}_n}v_n^{(\kappa_2)}(x)\end{aligned}$$ and it follows that $$\begin{aligned} \label{eq:lem_lambda_eqn_fin_line} \kappa_2 = \left( \sum_{j=0}^\infty \int_U\mathrm{d}\vec{r}'\, \varrho^{-1}(\vec{r}') \beta_j^{(\kappa_2)} v_j^{(\kappa_2)}(\vec{r}') w(\vec{r}')\right)^{-1} =\left(\sum_{i=0}^\infty\tfrac{\theta_i^{(\kappa_2)}\gamma^{(\kappa_2)} _i\beta_i^{(\kappa_2)}}{\lambda-\gamma^{(\kappa_2)}_i}\right)^{-1}.\end{aligned}$$ Hence the theorem is proved. $k$ $-\gamma_k^{(\epsilon)}\cdot 10^3$ $\nabla_{\vec{r}}(\varrho^{-1}v_k^{(\kappa_2)})\cdot\vec{n}\big|_{x = -1}$ $\nabla_{\vec{r}}(\varrho^{-1}v_k^{(\kappa_2)})\cdot\vec{n}\big|_{x = 1}$ ----- ------------------------------------ ---------------------------------------------------------------------------- --------------------------------------------------------------------------- 1 0.042470931917315 -0.326629834290770e-10 -0.049430114879555e-10 2 0.161343622578368 0.177791115163473e-11 -0.638172005587933e-11 3 0.359053918066979 -0.532729416136135e-11 -0.705155659306588e-11 4 0.635777464488092 0.040225600628219e-10 -0.10593000196636e-10 5 0.991543834922971 -0.143929312912405e-11 -0.982251087217608e-11 6 1.426361079097481 -0.421040979858844e-11 0.332564751050043e-11 7 1.940232081076136 0.130651045537888e-11 0.966037681231493e-11 8 2.533158075256826 -0.417399448338074e-11 -0.401360089043934e-11 9 3.205139660109592 -0.019828583219805e-11 -0.919929559744713e-11 10 3.956177153938488 -0.687472301308389e-11 0.423840601914807e-11 : The first 10 eigenvalues $-\gamma_k^{(\kappa_2)}\cdot 10^3$ and boundary condition values of the corresponding eigenvectors $v_{k^{(\kappa_2)}}$ for $\kappa_2 = .05$.[]{data-label="tab:eigenstat_table"} The expression for $\kappa_2$ allows the paths of the eigenvalues $\lambda_k(\kappa_2)$ to be computed. For practical purposes, it may be sufficient to use a truncation of the series or, if $w(\vec{r})$ can be computed explicitly, the first expression in can be used. As shown in [@kato2013perturbation Section II-5.1], the eigenvalues of $\mathcal{L}_1$ will remain real as long as $$\begin{aligned} \label{eq:kappa2_bound_spectral_gap} |\kappa_2|<\frac{\min_{i,j\in\mathbb{N}}|\gamma_i^{(\kappa_2)}-\gamma_{j}^{(\kappa_2)}|}{2\|\mathcal{B}\|}.\end{aligned}$$ We see from that the point of critical stability (if it exists) $\kappa_{2\sharp}$ occurs at $$\begin{aligned} \label{eq:critical_kappa_2} \kappa_{2\sharp} = -\left(\sum_{i=0}^\infty\theta_i^{(\kappa_2)}\beta_i^{(\kappa_2)}\right)^{-1}\end{aligned}$$ and is independent of $\gamma_n^{(\kappa_2)}$ (the eigenvalues of the local operator $A_{\kappa_2}$). The critical point of stability will have implicit dependence on $\bm{D}$, $V_1$ and $V_2$ through . As long as $\kappa_2$ remains sufficiently small, Lemma \[thm:epsilon\_as \_a\_fn\_of\_lambda\] provides a nonlinear map to compute $\kappa_2$ parametrised by $\lambda$ therefore permitting the paths of the moving eigenvalues to be calculated. In particular by fixing $\lambda\in\mathbb{R}$ we have the iterative problem $$\begin{aligned} \label{eq:iteration_for_the_mving_lambdas} \begin{cases} \frac{1}{\kappa_2^{l+1}} =\sum_{i=1}^\infty\tfrac{\theta_i^{(\kappa_2^l)}\gamma^{(\kappa_2^l)} _i\beta_i^{(\kappa_2^l)}}{\lambda-\gamma^{(\kappa_2^l)}_i},\\ \gamma_{n}^{(\kappa_2^0)} = \gamma_{n}^{(0)}. \end{cases}\end{aligned}$$ We note that the eigenvalues $\lambda_{k}^{(\kappa_2)}$ are implicitly dependent on the diffusion tensor $\bm{D}$ and confining potential $V_1$. Figure \[fig:eigenfuns\_k2\_1\] shows typical eigenfunctions $v_k^{(\kappa_2)}$ of the local part of the linearised operator $\mathcal{L}$. Figure \[fig:inner\_prod\_eigenfunctions\_on\_interval\] shows the pairwise $L^2_c(U,\varrho^{-1})$ inner product of the $v_k^{(\kappa_2)}$ demonstrating orthogonality of the basis functions. Figure \[fig:V2\_expanded\] shows the expansion of the two-body function $V_2$ (here a Morse like potential) in terms of the eigenfunctions $v_k$ meanwhile Figure \[fig:error\_V2\_and\_expanded\] shows the error between the expansion and $V_2$. We also demonstrate the accuracy of the collocation scheme in computing eigenvalues and eigenfunctions of $\mathcal{A}_{\kappa_2}$ in Table \[tab:eigenstat\_table\]. In particular, $\mathcal{A}_{\kappa_2}$ is composed of dense first and second order differentiation matrices and the value $\nabla_{\vec{r}}(\varrho^{-1}v_k^{(\kappa_2)})\cdot\vec{n}$ is very small on the boundary using only 100 collocation points. In Figures \[fig:tanh\_pot\_moving\_eigenvals\], \[fig:gaussian\_pot\_epsilon\_lambda\], we plot various paths $\kappa_2^{\star}(\lambda)$ as solutions to the equation $\lambda = \lambda_{k^\star}(\kappa_2^*)$ for $k$ the wave number by numerically solving for different two-body potentials. We also reproduce figures from Davidson & Dodds [@davidson2006spectral] in Figures \[fig:davidson\_dodds\_example\_1\], \[fig:davidson\_dodds\_example\_2\], verifying our numerical procedure for computing the spectra of similar nonlocal differential operators. Note however that operators in [@davidson2006spectral] do not contain convolution type integral operators, and, with Dirichlet boundary conditions, their spectra differ substantially from those considered here (for example Figures \[fig:tanh\_pot\_moving\_eigenvals\], \[fig:gaussian\_pot\_epsilon\_lambda\]). The intersection through the $\lambda$ axis in each Figure \[fig:davidson\_dodds\_example\_1\]–\[fig:gaussian\_pot\_epsilon\_lambda\] gives the local eigenvalues $\gamma^{(0)}_k$ for the corresponding nonlocal differential operator. Note that it is not necessary for $\gamma^{(0)}_k$ to lie on the moving path for every $k$. The numerical solution of involves both a truncation of the infinite series and a numerical tolerance for the zeros of the nonlinear function $f(\kappa_2) = \kappa_2-\kappa_2(\lambda_k)$. Note that $\mathcal{L}$ is self-adjoint in $L^2_c(U, \varrho^{-1})$ (with real eigenvalues) only for $\kappa_2 = 0$. The $\lambda$’s are otherwise complex and the curves plotted show when the paths drop to the real plane. When $|\kappa_2|$ is sufficiently large, that is when is violated, the $\lambda$’s have non-zero imaginary part.\ \ We now investigate the spectrum of the linearised operator $\mathcal{L}$ in terms of the eigenspace of its nonlocal part. We determine necessary conditions for bifurcations. Bifurcation Theory {#sec:bifurcation_theory} ================== We now provide our first result of the section which relates the stability of equilibrium density to the two-body interaction potential.\ \[thm:stability\_of\_rho\_inf\] Let $\kappa_2\in(-\infty,\infty)$ and suppose $\varrho$ is a solution to the self-consistency equation . Let $\mathcal{R}$ be given by $$\begin{aligned} \label{eq:R_int_operator_defn} \mathcal{R}w = -\varrho V_2\star w, \end{aligned}$$ where $w\in L^2(U,\varrho^{-1})$ is mean zero. If $\mathcal{R}$ is positive definite and $\kappa_2<\beta_1$ where $\beta_1$ is the smallest eigenvalue of $\mathcal{R}^{-1}$, then equilibrium densities formed from repulsive two–body kernels $V_2$ are stable. Conversely if $\mathcal{R}$ is negative definite and $\kappa_2>\beta_1$ where $\beta_1$ is the largest eigenvalue of $\mathcal{R}^{-1}$, then equilibrium densities formed from attractive two–body kernels $V_2$ are stable. We observe that $\mathcal{L}_1$ is self-adjoint in $L^2(U,\varrho^{-1})$ only when there is no interaction ($\kappa_2 = 0$). We may however expand the eigenfunctions of $\mathcal{L}_1$ in the eigenfunctions of $\mathcal{R}$, $\{u_n\}_{n = 1}^\infty$ which form an orthonormal basis of $L^2(U,\varrho^{-1})$ (see Definition \[def:def\_of\_R\_op\]). We write $w_n = \sum_{i = 1}\alpha_{n_i}u_i$. By the definition of the eigenvalue problem for $\mathcal{L}_1$ $$\begin{aligned} \mathcal{L}_1w_n = \lambda_nw_n.\end{aligned}$$ Now inserting the expansion in $u_i$’s we obtain $$\begin{aligned} \lambda_n\sum_{i = 1}\alpha_{n_i}u_i &=\mathcal{L}_1\sum_{i = 1}\alpha_{n_i}u_i \\ &=\left[\mathcal{A}_{\kappa_2}+\kappa_2\mathcal{B}\right] \sum_{i = 1}\alpha_{n_i}u_i\\ &=\sum_{i = 1}\alpha_{n_i}\left\lbrace\mathcal{A}_{\kappa_2}u_i+\kappa_2\mathcal{B}u_i \right\rbrace \\ &=\sum_{i = 1}\alpha_{n_i}\left\lbrace \nabla_{\vec{r}}\cdot \left(\bm{D}\varrho\left(\nabla_{\vec{r}}(\varrho^{-1}u_i)\right) \right)-\kappa_2\nabla_{\vec{r}}\cdot \left(\bm{D}\varrho\left(\nabla_{\vec{r}}(\varrho^{-1}(\varrho\mathcal{R}u_i))\right) \right)\right\rbrace\\ &=\sum_{i = 1}\alpha_{n_i}\left\lbrace\nabla_{\vec{r}}\cdot \left(\bm{D}\varrho\left(\nabla_{\vec{r}}(\varrho^{-1}u_i)\right) \right)-\kappa_2\nabla_{\vec{r}}\cdot \left(\bm{D}\varrho\left(\nabla_{\vec{r}}(\varrho^{-1}(\varrho\mathcal{R}u_i))\right) \right)\right\rbrace \\ &=\sum_{i = 1}\alpha_{n_i}\left\lbrace 1-\frac{\kappa_2}{\beta_i}\right\rbrace \nabla_{\vec{r}}\cdot \left(\bm{D}\varrho\left(\nabla_{\vec{r}}(\varrho^{-1}u_i)\right)\right),\end{aligned}$$ where we have used the definitions , and and that each $u_i$ is an eigenfunction of $\mathcal{R}$. Now by multiplying my $\varrho^{-1}u_j$ and integrating we obtain $$\begin{aligned} \lambda_n\alpha_{n_j}\|u_j\|_{L^2(U,\varrho^{-1})}^2 = \sum_{i = 1}\alpha_{n_i} \left\lbrace 1-\frac{\kappa_2}{\beta_i}\right\rbrace\int_{U}\mathrm{d}\vec{r}\,u_j \nabla_{\vec{r}}\cdot \left(\bm{D}\varrho\left(\nabla_{\vec{r}}(\varrho^{-1}u_i)\right)\right)\end{aligned}$$ Now by integrating by parts, using Gauss’s theorem and the condition that $\nabla_{\vec{r}}(\varrho^{-1}u_i)$ is zero on the boundary of $U$, we obtain $$\begin{aligned} \lambda_n =\alpha_{n_j}^{-1} \sum_{i = 1}\alpha_{n_i}\left(\frac{\kappa_2}{\beta_i}-1\right)\int_U\mathrm{d}\vec{r}\, \Big|\varrho^{1/2}\bm{D}^{1/2}\nabla_{\vec{r}}\left(\varrho^{-1}u_i\right)\Big|^2 \end{aligned}$$ for every $j = 1,\cdots $. Hence, a bifurcation from the equilibrium density $\varrho$ may occur when $\kappa_2$ coincides with $\beta_j$, for some $j = 1,\cdots $ and perturbations $w_n$ are linear combinations of $u_j$. To ensure $\varrho$ is stable one must have, for every $j\in \mathbb{N}$ $$\begin{aligned} \begin{cases} \kappa_2<\beta_j \quad \text{ if } \mathcal{R} \text{ is positive definite,}\\ \beta_j<\kappa_2 \quad \text{ if } \mathcal{R} \text{ is negative definite.} \end{cases}\end{aligned}$$ Now by the spectral theorem, the $\{\beta_n^{-1}\}_{n\geq 1}$ are discrete, countable and may be ordered such that $|\beta_n^{-1}|\to 0$. Therefore to ensure the stability of $\varrho$ we require $\kappa_2<\beta_1$ if $\mathcal{R}$ is positive definite and $\beta_1<\kappa_2$ if $\mathcal{R}$ is negative definite. This completes the proof of the theorem. We now relate theorem \[thm:stability\_of\_rho\_inf\] to the H-stability result in [@greg_mckean_vlasov_torus].\ \ \[rem:estimate\_betans\] We remark on the consistency with the H-stability condition of [@greg_mckean_vlasov_torus] with periodic boundary conditions, the equilibrium density may bifurcate if the interaction kernel has a negative Fourier mode. In the present work, the distribution of the eigenvalues of the operator $\mathcal{R}$ determines whether the equilibrium density is stable with respect to $\{u_j\}_{j = 1}^{\infty}$. In particular, if $\mathcal{R}$ has a negative eigenvalue then equilibrium densities formed from repulsive $V_2$ may become unstable. We may obtain an estimate for the eigenvalues $\beta_n^{-1}$ in terms of $V_2$ and $\varrho$ in the following way, by the eigenvalue problem we have $$\begin{aligned} |\beta_n^{-1}| &= |\beta_n^{-1}|\langle u_n,u_n\rangle_{L^{2}(U,\varrho^{-1})} = |-\langle u_n,V_2\star u_n\rangle_{L^{2}(U)}| \nonumber\\ &\leq \|V_2\|_{L^\infty(U)} \|u_n\|^2_{L^1(U)}= \|V_2\|_{L^\infty(U)}\|\varrho^{1/2}(\varrho^{-1/2})u_n\|_{L^1}^2\nonumber\\ &\leq \|V_2\|_{L^\infty(U)}\|\varrho\|_{L^1(U)}^2\|(\varrho^{-1/2})u_n\|_{L^2(U)}^2= \|V_2\|_{L^\infty(U)},\end{aligned}$$ where we have used the Cauchy-Schwarz inequality and the fact that the $\{u_n\}_{n = 1}^\infty$ are orthonormal in $L^2(U,\varrho^{-1})$. From this we obtain the lower bound $\|V_2\|_{L^\infty(U)}^{-1}\leq |\beta_n|$, this lower bound shows that the bifurcation point coincides with the boundary of the interval in which free energy $\mathcal{F}$ is convex (c.f. Proposition \[prop:F\_is\_strictly\_convex\]). \[thm:necessary\_bifurcation\_conditions\] Let $\{\beta_{n}^{-1}\}_{n=1}^{\infty}$ be the ordered eigenvalues of $\mathcal{R}$. If $|\kappa_2|\geq |\beta_1|$ then $(\beta_1,w_1)$ is a bifurcation point of where $w_1$ is the eigenfunction of $\mathcal{R}$ associated to $\beta_1^{-1}$ and there exists $0<\varrho_\ast\neq \varrho_{\infty}$ solving . Let $\varrho_{\kappa_2}$ denote the solution to for a given $\kappa_2$ which is known to exist by Theorem \[thm:exis\_fix\_point\]. Since $\varrho_{\kappa_2}$ is continuous in $\kappa_2$ and $\mathcal{F}[\varrho]$ is continuous in $\varrho$, then $\mathcal{F}$ is continuous in $\kappa_2$. By Lemma \[lem:minimisers\_always\_exist\] we know that a minimiser of $\mathcal{F}$ exists for each $\kappa_2$ and by Lemma \[lem:minimisers\_are\_positive\] the minimiser is strictly positive. Given $|\kappa_2|\geq \|V_2\|_{L^\infty(U)}^{-1}$ then by Proposition \[prop:F\_is\_strictly\_convex\], $\mathcal{F}$ is no longer convex and $\varrho_{\kappa_2}$ is either an inflection point or a local maximum of $\mathcal{F}$. Hence $\varrho_{\kappa_2}$ is unstable and by Lemma \[lem:minimisers\_always\_exist\] there exists $\varrho_{\ast}$ such that $\mathcal{F}[\varrho_\ast]<\mathcal{F}[\varrho_{\kappa_2}]$. Additionally by the self-adjointness and compactness of $\mathcal{R}$, one has that $\beta_n^{-1}\to 0$ as $n\to \infty$ and hence $\beta_n\to \infty$ as $n\to \infty$ and $\beta_1$ is the smallest of the $\{\beta_{n}\}_{n=1}^{\infty}$. If $\mathcal{R}$ is positive definite, there are no negative $\beta_n$ and the only solution to is $u_n\equiv 0$ and $\varrho_{\kappa_2}$ will be stable for all $\kappa_2<\beta_1$. Similarly, if $\mathcal{R}$ is negative definite, there are no positive $\beta_n$ and the only solution to is $u_n\equiv 0$ and $\varrho_{\kappa_2}$ will be stable for all $\kappa_2>\beta_1$. If $\mathcal{R}$ is indefinite, by Remark \[rem:estimate\_betans\], for $|\beta_n|<\|V_2\|^{-1}_{L^\infty(U)}$ there are no solutions (other than $w_n\equiv 0$) to $\mathcal{R}[w_n] = \beta_n^{-1}w_n$, and once again for $|\kappa_2|<|\beta_1|$, $\varrho_{\kappa_2} = \varrho_{\infty}$ is stable. For $\kappa_2\geq \|V_2\|^{-1}_{L^\infty(U)}$ there are infinitely many non-trivial solutions to $\mathcal{R}[w_n] = \beta_n^{-1}w_n$ and $\kappa_2 = \beta_1$ is the first. Hence if $|\kappa_2|\geq|\beta_1|$ then the unique stationary density $\varrho_\infty$ is unstable and by Lemma \[lem:minimisers\_always\_exist\] there must exist $\varrho_{\ast}$ such that $\mathcal{F}[\varrho_\ast]<\mathcal{F}[\varrho_{\kappa_2}]$. We define the $\mathcal{W}:L^2(U)\to \mathbb{R}$ transform such that $$\begin{aligned} \mathcal{W}[f](n) = \int_U\mathrm{d}\vec{r}'\varrho^{-1}_{\beta_n}w_n(\vec{r})f(r)\end{aligned}$$ where $\varrho_{\beta_n}$ solves with $\kappa_2 = \beta_n$. With this we may plot the bifurcation diagram for the stability of the unique equilibrium state $\varrho = \varrho_\infty$, see for example Figure \[fig:bifurcation\_diagram\]. Application To Nonlinear Diffusion Equations {#sec:manufactured_bif} ============================================ In this section we consider sufficient conditions for bifurcations under particular forms of nonlocal operators. We will show that, by use of numerical examples, there may be more than one stationary solution under additional assumptions on the two-body potential by making use of the bifurcation theory developed in Section \[sec:bifurcation\_theory\]. We fix $\kappa_1$ and consider boundary value problems where the nonlocal term is not of convolution type. Let $V_2(\vec{r},\vec{r}')$ be a two-body function and consider $$\begin{aligned} \begin{cases}\label{eq:bif_stationary_eqn} \mathcal{P}[\varrho]:=\nabla \cdot \Big[ \bm{D} \Big(\nabla\varrho + \varrho \kappa_1 \nabla V_1+\kappa_2 \varrho \nabla \int_U\mathrm{d}\vec{r}' V_2(\vec{r},\vec{r}') \varrho(\vec{r'}) \Big) \Big]= 0 & \text{ in } U,\\ \Omega [\varrho]\cdot\vec{n} := \bm{D} \Big(\nabla\varrho + \varrho \kappa_1 \nabla V_1+\kappa_2 \varrho \nabla \int_U\mathrm{d}\vec{r}' V_2(\vec{r},\vec{r}') \varrho(\vec{r'}) \Big) \cdot \vec{n}= 0 & \text{ on } \partial U. \end{cases}\end{aligned}$$ Solutions of $\mathcal{P}\varrho=0$ with $\Omega [\varrho]\cdot\vec{n} = 0 $ on the boundary are denoted by $\varrho=\varrho_{\kappa_2}$ and satisfy the self-consistency equation $$\begin{aligned} \label{eq:bifur_self_con_eqn} \varrho_{\kappa_2} = \frac{e^{-(\kappa_1V_1+\kappa_2\int_U\mathrm{d}\vec{r}'\,V_2(\vec{r},\vec{r}')\varrho_{\kappa_2}(\vec{r}'))}}{Z}.\end{aligned}$$ The linear stability of the steady state may be studied implicitly by examining the properties of the linearised self-consistency map. By linearising equation , by writing $\varrho_{\kappa_2} = \phi_0+\epsilon\phi_1$, for some small $\epsilon$, we obtain the original nonlinear problem $$\begin{aligned} \label{eq:bifur_self_con_eqn_O_0} \phi_{0} = \frac{e^{-(\kappa_1V_1+\kappa_2\int_U\mathrm{d}\vec{r}'\,V_2(\vec{r},\vec{r}')\phi_0(\vec{r}'))}}{Z_0} \quad \text{ s.t } \quad \Omega[\phi_{0}]\cdot \vec{n}=0\end{aligned}$$ where $Z_0 = \int_U\mathrm{d}\vec{r}\, e^{-(\kappa_1V_1+\kappa_2\int_U\mathrm{d}\vec{r}'\,V_2(\vec{r},\vec{r}')\phi_0(\vec{r}'))}$, along with the linearised equation $$\begin{aligned} \label{eq:rho_pert_series_u1} \phi_1 = - \kappa_2\,\phi_0\int_U\mathrm{d}\vec{r}'\,V_2(\vec{r},\vec{r}')\phi_1(\vec{r}') \quad \text{ s.t } \quad \int_U\mathrm{d}\vec{r}\, \phi_1(\vec{r}) = 0.\end{aligned}$$ The integral condition in comes from the fact that higher order perturbations to $\phi_0$ must possess zero mean to preserve the mass in the system. We define the linear operator $\mathcal{T}$ in $L_1(U)$ by $$\begin{aligned} \label{eq:def_of_T_operator} \mathcal{T}\phi(\vec{r}) := \phi_0(\vec{r})\int_U\mathrm{d}\vec{r}'\,V_2(\vec{r},\vec{r}')\phi(\vec{r}').\end{aligned}$$ We also define the mapping from $\mathcal{G}:\left(L_1(U), \mathbb{R}\right) \to L_1(U)$ by $$\begin{aligned} \mathcal{G}(v,\kappa) := \phi-f(\phi,\kappa)\end{aligned}$$ where $f(\phi,\kappa) := \tfrac{e^{-(\kappa_1V_1+\kappa\mathrm{d}\vec{r}'\,V_2(\vec{r},\vec{r}')\phi(\vec{r}'))}}{\int_U\mathrm{d}\vec{r}e^{-(\kappa_1V_1+\kappa\mathrm{d}\vec{r}'\,V_2(\vec{r},\vec{r}')\phi(\vec{r}'))}}$. To construct the bifurcation diagram, we will use the following result from [@tamura1984asymptotic Tamura (1984)], or [@greg_mckean_vlasov_torus Carrillo et al. 2019], which is a direct consequence of the Crandall-Rabinowitz theorem, see, e.g. [@crandall1971bifurcation].\ \[thm:tamura\_bifurcations\] Let $V_2(x,y) = V_2(y,x)$. Also let $(\psi_0,\mu_0)$ be a fixed point in $L^1(U)\times \mathbb{R}$ such that: 1. $\mathcal{G}(\psi_0,\mu_0)=0$, 2. $\mu_0^{-1}$ is an eigenvalue of $\mathcal{T}$, 3. $\int_U\mathrm{d}\vec{r}\,V_2(\vec{r},\vec{r}')\psi_0(\vec{r})=0$, 4. $\dim \{\phi\in L^1(U)\,:\, v = \mu_0 \mathcal{T}\phi\}=1$. Then $(\psi_0,\mu_0)$ is a bifurcation point of $\mathcal{G}=0$. That is, for any neighbourhood $B$ of $(\psi_0,\mu_0)$ in $L^1(U)\times \mathbb{R}$ there exists $(\psi_1,\mu_1)\in B$ such that $\psi_1\neq \psi_0$ and $\mathcal{G}(\psi_1,\mu_1)=0$. \[thm:bifurcations\] The proof relies on checking the conditions of the Crandall-Rabinowitz Theorem and is equivalent to Tamura’s proof [@tamura1984asymptotic]. [0.48]{} ![Stable densities bifurcating from (a). $\psi_0 = \exp\{-\kappa_1V_1(x)\}/Z$ and (b). $\psi_0 = \frac{N}{2\mathrm{L}}$ where $2L$ is the length of the interval, which solve for different two-body functions (a). $V_2(x,y) = xy$ and (b). $V_2(x,y) = -\cos\left(\frac{2\pi(x-y)}{\mathrm{L}}\right)$ . Insets show the shape of perturbation function.[]{data-label="fig:bifurcations"}](xy_bifurcation_perturb_2.pdf "fig:"){width="\textwidth"}   [0.48]{} ![Stable densities bifurcating from (a). $\psi_0 = \exp\{-\kappa_1V_1(x)\}/Z$ and (b). $\psi_0 = \frac{N}{2\mathrm{L}}$ where $2L$ is the length of the interval, which solve for different two-body functions (a). $V_2(x,y) = xy$ and (b). $V_2(x,y) = -\cos\left(\frac{2\pi(x-y)}{\mathrm{L}}\right)$ . Insets show the shape of perturbation function.[]{data-label="fig:bifurcations"}](stable_equilibria_V2_cosine.pdf "fig:"){width="\textwidth"} Note that $\psi_0$ is, by construction, the background density given by $\psi_0 = \tfrac{e^{-V_1(\vec{r})}}{\int_U\mathrm{d}\vec{r}e^{-V_1(\vec{r})}}$. Theorem \[thm:tamura\_bifurcations\] presents sufficient conditions to permit bifurcations from $v_0$ with stationary equations of the form . In particular it will be sufficient that the two-body potential satisfies the normality condition (condition 3. of Theorem \[thm:tamura\_bifurcations\]). Then bifurcations occur at discrete eigenvalues of the nonlocal operator $\mathcal{T}$ as defined in . We remark that these conditions are consistent with Theorem \[thm:necessary\_bifurcation\_conditions\]. Numerical Experiments. {#subsec:numericalexperiments} ---------------------- In this section we compute the branches of solutions that may evolve in the DDFT-like example considered in Section \[sec:manufactured\_bif\] with nonlinear, nonlocal boundary conditions. Given simple interaction kernels we show that symmetry-breaking systems may be constructed quite easily given sufficiently high interaction strength. For the numerical examples presented here, $\varrho$ is a number density and hence $\int_U\mathrm{d}\vec{r}\,\varrho = 0$. We consider numerical solutions to $$\begin{aligned} \begin{cases}\label{eq:bif_evolve_eqn} \partial_t \varrho = \nabla \cdot \Big[ \bm{D} \Big(\nabla\varrho + \varrho \kappa_1 \nabla V_1+\kappa_2 \varrho \nabla \int_U\mathrm{d}\vec{r}' V_2(\vec{r},\vec{r}') \varrho(\vec{r'}) \Big) \Big] & \text{ in } U,\\ \Omega [\varrho]\cdot\vec{n} := \bm{D} \Big(\nabla\varrho + \varrho \kappa_1 \nabla V_1+\kappa_2 \varrho \nabla \int_U\mathrm{d}\vec{r}' V_2(\vec{r},\vec{r}') \varrho(\vec{r'}) \Big) \cdot \vec{n}= 0 & \text{ on } \partial U, \\ \varrho(\vec{r},0) = \tfrac{e^{-\kappa_1V_1(\vec{r})+\kappa_2\int_U\mathrm{d}\vec{r}'\,V_2(\vec{r},\vec{r}')\varrho(\vec{r}',0)}}{Z} \quad &\text{ at } t=0. \end{cases}\end{aligned}$$\ The nonlocal terms in , both in the evolution equation and the boundary condition, mean that numerical implementations require efficient and accurate quadrature. We demonstrate the power with which the pseudo-spectral collocation scheme 2DChebClass [@DDFTCode] may compute solutions with such efficiency and accuracy. For a more detailed explanation of pseudospectral methods for DDFT problems, particularly the efficient computation of convolution integrals, see [@nold2017pseudospectral]. Some numerical experiments were performed by solving in 1D with the choice $V_2 = xy$ and $V_1 = \kappa_1 x^2$ on $U = [-1/2,1/2]$. Under this choice of confining and two-body potentials the normality condition (3) of Theorem \[thm:tamura\_bifurcations\] holds. Additionally, for $|\kappa_2|$ sufficiently small, the unique stationary density is $v_0 = e^{-\kappa_1V_1}/Z$. Upon increasing $\kappa_2$ and perturbing with a mean zero function $\eta(x,\theta)$ the stability of $v_0$ breaks and transitions may be observed to non symmetric equilibria. The asymmetry of the equilibria depends on the sign of $\eta$ as seen in Figure \[fig:bifurcations\]. Figure \[fig:bifurcations\] shows long time numerical solutions to the IBVP subject to a mean zero perturbation for different interaction strengths $\kappa_2$, with $V_2$ fixed. In Figure \[fig:bif\_fig\_right\], the symmetric solution $\psi_0 = \exp(-\kappa_1 V1)/Z$ was shown to be unstable as interaction strength $\kappa_2$ was made ever negative. In particular by perturbing with a sinusoidal function with positive or negative sign, the stationary density can be shown to adhere to one boundary, thereby bifurcating from the previously symmetric solution $\psi_0$. The skewness of the density is controlled by the sign of the perturbation function $\eta$ and $\eta\in \text{Span}{\mathcal{T}}$, hence densities which adhere to the left boundary may be obtained by changing the sign of $\eta$. We predict the stable and symmetric branch to bifurcate at the critical interaction energy $\kappa_2 = -2.4$ (to 1 decimal place) which is the negative inverse of smallest eigenvalue of $\psi_0^{-1}\mathcal{T}$ in $U = [-1/2,1/2]$. This is verified in Figure \[fig:bifurcation\_diagram\] and the transition between a stable symmetric density and a stable nonsymmetric one is observed in Figure \[fig:bif\_fig\_right\] for the curves labelled $\kappa_2 = -2$ and $\kappa_2 = -3$. In Figure \[fig:bif\_stable\_equilibria\_V2\_cosine\], we see how the uniform density may become unstable. Here $\psi_0 = N/(2\mathrm{L})$ where $2L$ is the length of the interval. We perturb with eigenvectors of $\mathcal{T}$ at increasing interaction strengths. The critical strength was $\kappa_{2\sharp} = 0.4$ (to 1 decimal place), the negative inverse of smallest eigenvalue of $\psi_0^{-1}\mathcal{T}$. This is verified in Figure \[fig:bifurcation\_diagram\] and the transition between a stable uniform density and a stable multi-modal one is observed in Figure \[fig:bif\_stable\_equilibria\_V2\_cosine\] for the curves labelled $\kappa_2 = 0$ and $\kappa_2 = 0.5$. Existence & Uniqueness of Weak Solutions to Density with Full HI {#sec:existence_uniqueness_with_partial_HI} ================================================================ In this section we determine the existence and uniqueness of the weak density $\varrho(\vec{r},t)$ solving in the sense . To ease notation we suppress $\bm{A}[\vec{a}]$ as it may be trivially added (see Remark \[rem:final\_assumptions\]). We begin by determining some useful results: first, that $\varrho(\vec{r},t)$ is bounded above in $L^1(U)$ for all time by initial data $\varrho_0$ and second, the $L^1(U)$ norm of $\varrho$ is unity for all time and $\varrho(\vec{r},t)$ is non-negative. We will strengthen the non-negativity to strict positivity of $\varrho(\vec{r},t)$ in Section \[subsec:strict\_pos\_rho\]. The results in this section are analogous to those in [@chazelle2017well], [@greg_mckean_vlasov_torus] with the difference that the boundary conditions we consider are no-flux the diffusion tensor is non-constant. Useful Results. --------------- We identify the expansion of the absolute value function. \[def:def\_of\_chi\_approx\_abs\] Let $\epsilon>0$ and define the convex $C^2$ approximation of $|\cdot|$ by $$\begin{aligned} \chi_\epsilon(\psi) = \begin{cases} |\psi| \quad \text{ for } \quad \psi>\epsilon,\\ -\tfrac{\psi^4}{8 \epsilon^3 }+\tfrac{3 \psi^2}{4 \epsilon}+ \tfrac{3 \epsilon}{8} \quad \text{ for } \quad \psi\leq \epsilon. \end{cases}\end{aligned}$$ We now present our first result concerning the boundedness of the the $L^1$ norm of $\varrho$ in terms of the initial data $\varrho_0$.\ If $ \varrho\in C^1([0,\infty);C^2(U))$ is a solution of with $\varrho_0\in L^1(U)$ then $\|\varrho(t)\|_{L^1(U)}\leq \|\varrho_0\|_{L^1(U)}$ for all time $t\geq 0$. Multiplying by $\chi_\epsilon'(\varrho)$, integrating and using the divergence theorem and chain rule, we have $$\begin{aligned} &\der[]{t}\int_U\mathrm{d}\vec{r}\,\chi_\epsilon(\varrho)+\|\bm{D}^{1/2}\nabla_{\vec{r}}\varrho\,[\chi_\epsilon''(\varrho)]^{1/2}\|_{L^2(U)}^2\nonumber\\ &=-\int\mathrm{d}\vec{r}\, \nabla_{\vec{r}}\varrho\, \chi_\epsilon''(\varrho)\cdot[\varrho\,\bm{D}(\vec{r})\nabla_{\vec{r}}(\kappa _1V_1(\vec{r})+\kappa _2[V_2\star\varrho](\vec{r}) )].\end{aligned}$$ Now by H[ö]{}lder’s inequality and then Young’s inequality $$\begin{aligned} &\der[]{t}\int_U\mathrm{d}\vec{r}\,\chi_\epsilon(\varrho)+\|\bm{D}^{1/2}\nabla_{\vec{r}}\varrho\,[\chi_\epsilon''(\varrho)]^{1/2}\|_{L^2(U)}^2\nonumber\\ &\leq \|\bm{D}^{1/2}\nabla_{\vec{r}}\varrho [\chi_\epsilon''(\varrho)]^{1/2}\|_{L^2(U)} \times \| [\chi_\epsilon''(\varrho)]^{1/2}\varrho \,\bm{D}^{1/2}\nabla_{\vec{r}}(\kappa _1V_1+\kappa _2[V_2\star \varrho] )\|_{L^2(U)}\nonumber\\ &\leq \tfrac{1}{2}\|\bm{D}^{1/2}\nabla_{\vec{r}}\varrho [\chi_\epsilon''(\varrho)]^{1/2}\|_{L^2(U)}^2 +\tfrac{1}{2}\| [\chi_\epsilon''(\varrho)]^{1/2}\varrho\, \bm{D}^{1/2}\nabla_{\vec{r}}(\kappa _1V_1+\kappa _2[V_2\star \varrho] )\|_{L^2(U)}^2.\end{aligned}$$ Note there are no boundary terms due to the condition $\Pi[\varrho]\cdot\vec{n}=0$ on $\partial U$. All together this implies the inequality $$\begin{aligned} &\der[]{t}\int_U\mathrm{d}\vec{r}\,\chi_\epsilon(\varrho)+\tfrac{1}{2}\|\bm{D}^{1/2}\nabla_{\vec{r}}\varrho\,\chi_\epsilon''(\varrho)^{1/2}\|_{L^2(U)}^2\nonumber\\ &\leq \tfrac{1}{2} \| [\chi_\epsilon''(\varrho)]^{1/2}\varrho\, \bm{D}^{1/2}\nabla_{\vec{r}}(\kappa _1V_1+\kappa _2[V_2\star\varrho] )\|_{L^2(U)}^2\nonumber\\ &\leq \tfrac{1}{2}\| \bm{D}^{1/2}\nabla_{\vec{r}}(\kappa _1V_1+\kappa _2[V_2\star\varrho] )\|_{L^\infty}^2 \| [\chi_\epsilon''(\varrho)]^{1/2} \varrho \|_{L^2}^2\nonumber\\ &\leq c_0 \| [\chi_\epsilon''(\varrho)]^{1/2} \varrho \|_{L^2}^2 (1+\|\varrho\|_{L^1(U)}^2)\label{eq:ddt_chi_rho_bound}\end{aligned}$$ for the constant $c_0 = 2\mu_{\max} \max\{|\kappa_1|^2 \|\nabla_{\vec{r}}V_1\|_{L^\infty}^2,|\kappa_2|^2 \|V_2\|_{L^\infty}^2\}$. It is an elementary calculation to show that $$\begin{aligned} \varrho^2\chi_\epsilon''(\varrho) = \tfrac{3\varrho^2}{2\epsilon} - \tfrac{3\varrho^4}{2\epsilon^3}\end{aligned}$$ for $\varrho\leq \epsilon$. With this, and the fact that $\chi''(\varrho) = 0$ for $\varrho>\epsilon$, we have $$\begin{aligned} \| [\chi_\epsilon''(\varrho)]^{1/2} \varrho \|_{L^2}^2 &= \int_U \mathrm{d}\vec{r}\, \varrho^2 \chi_\epsilon''(\varrho) \mathbb{I}_{\varrho\leq \epsilon} + \int_U \mathrm{d}\vec{r}\, \varrho^2 \chi_\epsilon''(\varrho) \mathbb{I}_{\varrho > \epsilon} \nonumber \\ & = \int_U \mathrm{d}\vec{r}\, \frac{3 \varrho^2(\epsilon^2 - \varrho^2)}{2 \epsilon^2} \mathbb{I}_{\varrho\leq \epsilon} \leq \int_U \mathrm{d}\vec{r}\, \frac{3 \epsilon}{2}\mathbb{I}_{\varrho\leq \epsilon} \leq c_1\epsilon \label{eq:sqrt_chi''_rho_L2}\end{aligned}$$ for some constant $c_1$ dependent on $U$. Applying Gr[ö]{}nwall’s lemma to $\eta(\cdot)$ a non-negative, absolutely continuous function on $[0,T]$ which satisfies for a.e. $t$ $$\begin{aligned} \eta'(t)\leq \phi(t)\eta(t) + \psi(t)\end{aligned}$$ where $\phi$, $\psi$ non-negative and integrable functions on $[0,T]$ gives $$\begin{aligned} \label{eq:statement_gronwall} \eta(t)\leq e^{\int\mathrm{d}s_0^t\,\phi(s)}\eta(t)\Big[ \eta(0)+ \int_0^t\mathrm{d}s\, \psi(s)\Big].\end{aligned}$$ Observe that $\|\varrho\|_{L^1(U)}\leq \int_U\mathrm{d}\vec{r}\,\chi_\epsilon(\varrho)$. Using this with , and with $\eta(t) = \phi(t) =c_1\epsilon \int_U\mathrm{d}\vec{r}\,\chi_\epsilon(\varrho)$ and $\psi(t) = c_1\epsilon$ we obtain $$\begin{aligned} \int_U\mathrm{d}\vec{r}\,\chi_\epsilon(\varrho)\leq \left(\int_U\mathrm{d}\vec{r}\,\chi_\epsilon(\varrho_0)+c_1\epsilon \,t\right)\, e^{c_1\epsilon\int_0^t\mathrm{d}s\,\int_U\mathrm{d}\vec{r}\,\chi_\epsilon(\varrho(\vec{r},s))}.\end{aligned}$$ Now since $\varrho$ is assumed to be continuous in time on $[0,\infty)$ the integral in the exponential is finite. Therefore taking $\epsilon \to 0$ one obtains $$\begin{aligned} \|\varrho\|_{L^1}\leq \|\varrho_0\|_{L^1}\end{aligned}$$ for every $t>0$. \[cor:L\_1\_varrho\_is\_1\] If $ \varrho\in C^1([0,\infty);C^2(U))$ is a solution of with $\varrho_0$ a probability density, that is $\varrho_0\geq 0$ and $\int_U \mathrm{d}\vec{r}\,\varrho_0(\vec{r})=1$, then $\|\varrho(t)\|_{L^1(U)}=1$ and $\varrho(t)\geq 0$ in $U$ for all time $t\geq 0$. The argument is a standard one. Since, due to no-flux boundary conditions, $ \der[]{t} \int \mathrm{d}\vec{r} \, \rho(\vec{r},t) = 0$, we have $$\begin{aligned} 1 = \int_U\mathrm{d}\vec{r}\, \varrho_0(\vec{r}) = \int_U\mathrm{d}\vec{r}\, \varrho(\vec{r},t) \leq \|\varrho(t)\|_{L^1(U)} \leq \|\varrho(0)\|_{L^1(U)} = \int_U\mathrm{d}\vec{r}\, \varrho_0(\vec{r}) = 1,\end{aligned}$$ so $\|\varrho(t)\|_{L^1(U)} = 1$. Also observe the two equalities $$\begin{aligned} &1 = \int_U\mathrm{d}\vec{r}\, \varrho(\vec{r},t) = \int_U\mathrm{d}\vec{r}\, \varrho(\vec{r},t)\mathbb{I}_{\varrho\geq 0} + \int_U\mathrm{d}\vec{r}\, \varrho(\vec{r},t)\mathbb{I}_{\varrho< 0},\nonumber\\ &1 = \int_U\mathrm{d}\vec{r}\, |\varrho(\vec{r},t)| = \int_U\mathrm{d}\vec{r}\, \varrho(\vec{r},t)\mathbb{I}_{\varrho\geq 0} - \int_U\mathrm{d}\vec{r}\, \varrho(\vec{r},t)\mathbb{I}_{\varrho< 0},\nonumber\end{aligned}$$ where in the second line we have used the definition of the absolute value function. Subtracting these equalities we obtain $$\begin{aligned} 2\int_U\mathrm{d}\vec{r}\, \varrho(\vec{r},t)\mathbb{I}_{\varrho< 0} = 0\end{aligned}$$ which implies $ \varrho(\vec{r},t)\geq 0$ almost everywhere in $U$. Non-negativity of $\varrho$ on all of $U$ follows from continuity. With these results we may continue to determine the existence and uniqueness of weak densities solving in the sense . The method we use follows [@chazelle2017well] but here we must include calculations for the confining potential $V_1^{\text{eff}}$ (which for ease of notation is written $V_1$ for each $\vec{a}(\vec{r},t)$) and a much wider class of two-body potentials $V_2$ which are not necessarily step functions. To start we introduce , the frozen version of , indexed by $n\in \mathbb{N}$, by substituting $\varrho = u_n$ everywhere except in the convolution term where we substitute $\varrho = u_{n-1}$. Each equation is parametrised by $n$, a linear parabolic PDE for the unknown $u_n$ in terms of the solution $u_{n-1}$ at the previous index, for which we have existence and uniqueness of weak solutions for each $n$. The remainder of the argument is to show $\lim_{n\to\infty} u_{n}$ exists and is a limit point solving the weak problem . In this section we will make references to Appendix \[sec:classical\_paraboliv\_pde\] for results and definitions required for $u_{n}\in H^1(U)$, which differ slightly from the standard arguments found in textbooks for classical linear PDE theory (e.g. [@evans2002partial]). Energy Estimates. ----------------- The results of Appendix \[sec:classical\_paraboliv\_pde\] are that the initial boundary value problem $$\begin{aligned} \label{eq:well_posed_un_ibvp} \begin{cases} \quad\partial_t u_n-\nabla_{\vec{r}}\cdot[\bm{D}\nabla_{\vec{r}}u_n]=\nabla_{\vec{r}}\cdot[u_n\,\bm{D}\nabla_{\vec{r}}(\kappa _1V_1+\kappa _2V_2\star u_{n-1})],\\ \qquad \qquad \qquad \qquad \Xi [u_n]\cdot\vec{n} = 0 \quad \text{ on } \partial U \times [0, T],\\ \qquad\Xi[u_n]:= \bm{D}\,(\nabla_{\vec{r}}u_n+ u_n\, \nabla_{\vec{r}}(\kappa_1 V_1(\vec{r},t)+ \kappa_2 V_2\star u_{n-1})),\\ \qquad \qquad \qquad \qquad \quad u_n = \varrho_0 \quad \text{ on } U\times \left\lbrace t=0\right\rbrace \end{cases}\end{aligned}$$ is well posed, and there exists weak solutions $u_{n}$ for each $n\in\mathbb{N}$ in the sense . All that remains is to take the limit $n\to \infty$ to recover the original Smoluchowski equation . We start by deriving our first estimate on energy of $u_n$. To ease notation we derive all results with the time dependence on $\bm{D}$ suppressed since time may be trivially added to the exposition. Additionally, for a stationary density one has $$\begin{aligned} \lim_{t\to\infty}\bm{D}(\vec{r},t) = \left(\bm{1} + \int \mathrm{d}\vec{r}' g(\vec{r},\vec{r}')\bm{Z}_{1}(\vec{r},\vec{r}')\varrho(\vec{r}')\right)^{-1} \end{aligned}$$ which is a positive definite tensor and hence diagonalisable, and may be bounded by its smallest and largest eigenvalues which are positive and finite for $t\to\infty$. Hence energy estimates remain valid for $0<t\leq T$ when provided in terms of $\mu_{\min}$ and $\mu_{\max}$, both eigenvalues which depend on time but always remain positive and finite. It will be seen that a natural dual space to $H^1(U)$ is provided by the no-flux condition. In particular we denote by $H^{-1}(U)$ the dual space of $H^{1}(U)$, this is due to the divergence theorem and the boundary condition $\Xi[u_n]\cdot\vec{n}=0$ on $\partial U \times [0,T]$, there is no boundary term, and the normal characterisation of $H^{-1}=(H^1_0)^\ast$ carries over to $H^{1}(U)$. We now obtain uniform estimates on $u_n$ in terms of the initial data $\varrho_0$ in all the required energy norms. The detailed calculations follow [@chazelle2017well] but take into account the confining potential and non-constant diffusion tensor $\boldsymbol{D}$. The explicit calculations can be found in RDMW’s PhD thesis [@rdmwthesisddft]. The first estimate is in $L^{\infty}([0,T];L^2(U))$ and $L^{2}([0,T];H^1(U))$ norms.\ \[prop:bound\_rhon\] Let $T>0$ and suppose $\{u_n\}_{n\geq 1}$ satisfies with $\varrho_0\in C^{\infty}(U)$ a probability density. Then there exists a constant $C(T)$, dependent on time and $\mu_{\max}$, such that $$\begin{aligned} \label{eq:L2_H_1_bound_for_rhon} \|u_n\|_{L^{\infty}([0,T];L^2(U))}+\|u_n\|_{L^{2}([0,T];H^1(U))}\leq C(T,\mu_{\max}) \|\varrho_{0}\|_{L^2(U)}.\end{aligned}$$ The second estimate is for $L^{\infty}([0,T]; H^1(U))$ and $L^2([0,T];L^2(U))$ norms.\ \[prop:un\_H1\_rho0\_bound\] Let $T>0$ and suppose $\{u_n\}_{n\geq 1}$ satisfies with $\varrho_0\in C^{\infty}(U)$ a probability density. Then there exists some constant dependent on time $C(T)$ such that $$\begin{aligned} &\|u_n\|_{L^{\infty}([0,T]; H^1(U))}+\|\nabla_{\vec{r}}\cdot[\bm{D}\,\nabla_{\vec{r}}u_n]\|_{L^2([0,T];L^2(U))}^2\nonumber\\ &\leq C(T)(\|\varrho_0\|_{H^1(U)}^2+(1+\|\varrho_0\|_{L^2(U)})\|\varrho_0\|_{L^2(U)})^{1/2}.\end{aligned}$$ We now have strong convergence of $\left(u_n\right)_{n=1}^{\infty}$, by showing it is a Cauchy sequence in a complete metric space.\ \[lem:rhon\_is\_cauchy\] Let $T>0$ and suppose $\{u_n\}_{n\geq1}$ satisfies with $\varrho_0\in C^\infty(U)$. Then there exists $\varrho\in L^1([0,T];L^1(U))$ such that $u_n\to \varrho$ in $L^1([0,T];L^1(U))$. Lastly we have the uniform estimate on the limit point $\varrho(\vec{r},t)$ in terms of the initial data $\varrho_0$.\ \[lem:weak\_conv\_results\] One has $\varrho\in L^2([0,T]; H^1(U))\cap L^\infty([0,T]; L^2(U))$ and that $\partial_t\varrho\in L^2([0,T]; H^{-1}(U))$ with the uniform bound $$\begin{aligned} \label{eq:total_energy_bound} \|\varrho\|_{L^{\infty}([0,T];L^2(U))}+\|\varrho\|_{L^{2}([0,T];H^1(U))}+\|\partial_t\varrho\|_{L^2([0,T]; H^{-1}(U))}\leq C(T) \|\varrho_{0}\|_{L^2(U)}.\end{aligned}$$ Additionally there exists a subsequence $\{u_{n_k}\}_{k\geq 1}$ such that $$\begin{aligned} u_{n_k}\rightharpoonup \varrho& \quad \text{ in } L^2([0,T]; H^1(U)),\label{lim:rho_nk_in_H1_c}\\ \partial_tu_{n_k}\rightharpoonup \partial_t\varrho& \quad \text{ in } L^2([0,T]; H^{-1}(U)).\label{lim:partial_trho_nk_in_H_minus_1}\end{aligned}$$ where $\rightharpoonup$ denotes weak convergence. The nature of convergence of the sequence $\{\varrho_n\}_{n\geq 1}$ as $n\to\infty$ are consolidated into the following result.\ \[cor:summary\_of\_convergence\] There exists a subsequence $\{u_{n_k}\}_{k\geq 1}\subset \{u_{n}\}_{n\geq 1}$ and a function $\varrho\in L^2([0,T];H^1(U))$ with $\partial_t\varrho\in L^2([0,T];H^{-1}(U))$ such that $$\begin{aligned} u_n\to \varrho& \quad \text{ in } L^1([0,T]; L^1(U)),\\ u_{n_k}\rightharpoonup \varrho& \quad \text{(weakly) in } L^2([0,T]; H^1(U)),\\ \partial_t u_{n_k}\rightharpoonup \partial_t\varrho& \quad \text{(weakly) in } L^2([0,T]; H^{-1}(U)).\end{aligned}$$ We are now in the position to obtain the existence and uniqueness of weak solutions $\varrho(\vec{r},t)$. First we state a calculus result which will be useful when working with the weak formulation .\ \[lem:calculus\_of\_inner\_product\] Suppose $\varrho\in L^2([0,T]; H^1(U))$ and $\partial_t\varrho\in L^2([0,T]; H^{-1}(U))$ then the mapping $$\begin{aligned} t\to \|\varrho(t)\|_{L^2(U)}^2\end{aligned}$$ is absolutely continuous with $$\begin{aligned} \der[]{t}\|\varrho(t)\|_{L^2(U)}^2 = 2\langle \partial_t\varrho(t),\, \varrho(t)\rangle\end{aligned}$$ for a.e. $t\in [0,T]$. Since the condition $\Pi[\varrho]\cdot \vec{n} = 0$ on $\partial U \times [0,T]$ guarantees integration by parts without extra terms the proof is identical to the textbook one [@evans2002partial]. We are now in the position to prove existence of the weak solution to . Existence and Uniqueness. ------------------------- By using Propositions \[prop:bound\_rhon\], \[prop:un\_H1\_rho0\_bound\] and Lemmas \[lem:rhon\_is\_cauchy\], \[lem:weak\_conv\_results\], \[lem:calculus\_of\_inner\_product\] we may obtain the following theorem.\ (Existence and Uniqueness of Weak Density)\[thm:existence\_and\_uniqueness\] Let $\varrho_0 \in C^\infty(U)$, $\varrho\geq 0$ and $\int_U \mathrm{d}\vec{r} \varrho_0(\vec{r})=1$. Then there exists a unique weak solution $\varrho\in L^\infty([0,T];L^2(U))\cap L^2([0,T]; H^1(U))$, with $\partial_t\varrho\in L^2([0,T]; H^{-1}(U))$, to equation in the sense with the estimate . Multiply by $\eta\in L^2([0,T]; H^1(U))$ after setting $n=n_k\in \mathbb{N}$ and integrate over $U_T$ to obtain $$\begin{aligned} &\int_0^T\mathrm{d}t\, \langle \partial_tu_{n_k},\, \eta(t) \rangle\nonumber\\ & + \int_0^T\mathrm{d}t\,\int_U\mathrm{d}\vec{r}\, \nabla_{\vec{r}}\eta \cdot\bm{D}\,[\nabla_{\vec{r}}u_{n_k}+ u_{n_k}\nabla_{\vec{r}}(\kappa_1V_1+\kappa_2\,V_2\star u_{n_k-1})]=0.\end{aligned}$$ For the transport term we write $$\begin{aligned} &\int_0^T\mathrm{d}t\, \nabla_{\vec{r}}\eta \cdot u_{n_k}\bm{D}\,\nabla_{\vec{r}}[\kappa_1V_1+\kappa_2\,V_2\star u_{n_k-1}]\nonumber\\ & = \int_0^T\mathrm{d}t\, \nabla_{\vec{r}}\eta \cdot\,(u_{n_k}-\varrho)\bm{D}\,\nabla_{\vec{r}}[\kappa_1V_1+\kappa_2\,V_2\star u_{n_k-1}]\nonumber\\ &\quad +\int_0^T\mathrm{d}t\, \nabla_{\vec{r}}\eta \cdot \,\varrho \bm{D}\,\nabla_{\vec{r}}[\kappa_1V_1+\kappa_2\,V_2\star (u_{n_k-1}-\varrho)] +\int_0^T\mathrm{d}t\, \nabla_{\vec{r}}\eta\cdot \,\varrho \bm{D}\,\nabla_{\vec{r}}\kappa_2\,V_2\star \varrho.\end{aligned}$$ Note that $u_{n_k}\rightharpoonup\varrho$ in $L^2([0,T];H^1(U))\subset L^2([0,T];L^2(U))$ and $(\nabla_{\vec{r}}\cdot\bm{D})\cdot\nabla_{\vec{r}}[\kappa_1V_1(\vec{r})+\kappa_2\,V_2\star (\varrho_{n_k-1})]$ is uniformly bounded and so $$\begin{aligned} \int_0^T\mathrm{d}t\, \int_U\mathrm{d}\vec{r}\,\nabla_{\vec{r}}^\top\eta \, (\varrho_{n_k}-\varrho)\bm{D}\,\nabla_{\vec{r}}[\kappa_1V_1+\kappa_2\,V_2\star \varrho_{n_k-1}]\to 0\end{aligned}$$ as $k\to \infty$. Now by H[ö]{}lder’s inequality one has $$\begin{aligned} & \int_0^T\mathrm{d}t\, \nabla_{\vec{r}}\eta\cdot \,\varrho\,\bm{D}\,\nabla_{\vec{r}}\kappa_2(V_2\star (u_{n_k-1}-\varrho)) \leq \mu_{\max}\|\nabla_{\vec{r}}\eta \|_{L^2([0,T]; L^2(U))}\|\nabla_{\vec{r}}V_2\|_{L^{\infty}(U)}\nonumber\\ &\times \left( \int_0^T\mathrm{d}t\,\|u_{n_k-1}(t)-\varrho(t)\|_{L^1(U)}^2\right)^{1/2}\to 0.\end{aligned}$$ Now note that by Lemma \[lem:rhon\_is\_cauchy\], $\|\phi_n\|_{L^1(U)}$ is bounded and therefore $$\begin{aligned} \int_0^T\mathrm{d}t \|u_{n_k-1}(t)-\varrho(t)\|_{L^1(U)}^2\leq C\int_0^T\mathrm{d}t \|u_{n_k-1}(t)-\varrho(t)\|_{L^1([0,T];L^1(U))} \to 0.\end{aligned}$$ Therefore we have $$\begin{aligned} \int_0^T\mathrm{d}t\, \nabla_{\vec{r}}\eta\, \cdot \,u_{n_k}\bm{D}\,\nabla_{\vec{r}}[\kappa_1V_1+\kappa_2\,V_2\star u_{n_k-1}] \to\int_0^T\mathrm{d}t\, \nabla_{\vec{r}}\eta \cdot\varrho \bm{D}\,\nabla_{\vec{r}}[\kappa_1V_1+\kappa_2\,V_2\star \varrho]\end{aligned}$$ as $k\to \infty$. By the weak convergence results of Lemma \[lem:weak\_conv\_results\] we have $$\begin{aligned} \int_0^T\mathrm{d}t\, \langle \partial_t u_{n_k},\, u_{n_k} \rangle &\to \int_0^T\mathrm{d}t\, \langle \partial_t\varrho,\, \varrho \rangle,\nonumber\\ \int_0^T\mathrm{d}t\,\int_U\mathrm{d}\vec{r}\, \nabla_{\vec{r}}\eta \cdot\bm{D}\,\nabla_{\vec{r}}u_{n_k} &\to \int_0^T\mathrm{d}t\,\int_U\mathrm{d}\vec{r}\, \nabla_{\vec{r}}\eta \cdot\bm{D}\,\nabla_{\vec{r}}\varrho\end{aligned}$$ as $k\to \infty$. This establishes existence of weak solution to in the sense . Establishing $\varrho(0)=\varrho_0$ is a routine argument (see [@evans2002partial]). To prove uniqueness we set $\xi = \varrho_1-\varrho_2$ where $\varrho_1,\varrho_2$ are weak solutions then we have $$\begin{aligned} &\int_0^T\mathrm{d}t\, \langle \partial_t \xi(t), \, \eta(t) \rangle\nonumber\\ & +\int_0^T\mathrm{d}t\, \int_U \mathrm{d}\vec{r}\, \nabla_{\vec{r}}\eta\cdot \bm{D}\,[\nabla_{\vec{r}}\xi +\xi\,\nabla_{\vec{r}}\kappa _1V_1+\kappa _2\varrho_1\nabla_{\vec{r}}\,V_2\star\varrho_1-\kappa_2\varrho_1\nabla_{\vec{r}}\,V_2\star\varrho_2]=0\end{aligned}$$ Adding and subtracting $\int_0^T\mathrm{d}t\,\int_U\mathrm{d}\vec{r}'\,\nabla_{\vec{r}}\eta\cdot \kappa_1\varrho_2\nabla_{\vec{r}}V_2\star \varrho_1$ we find $$\begin{aligned} &\int_0^T\mathrm{d}t\, \langle \partial_t \xi(t), \, \eta(t) \rangle + \int_0^T\mathrm{d}t\, \int_U \mathrm{d}\vec{r}\, \nabla_{\vec{r}}\eta\cdot \bm{D}\,\nabla_{\vec{r}}\xi \nonumber\\ &=-\int_0^T\mathrm{d}t\, \int_U \mathrm{d}\vec{r}\, \nabla_{\vec{r}}\eta\cdot \bm{D}\,[\xi\,\nabla_{\vec{r}}\kappa _1V_1+\kappa _2\xi\nabla_{\vec{r}}\,V_2\star\varrho_1-\kappa_2\varrho_2\nabla_{\vec{r}}\,V_2\star\xi]\nonumber\\ &\leq \int_0^T\mathrm{d}t\, \int_U \mathrm{d}\vec{r}\, |\nabla_{\vec{r}}\eta\cdot \bm{D}^{1/2} \bm{D}^{1/2}\,[\xi\,\nabla_{\vec{r}}\kappa _1V_1+\kappa _2\xi\nabla_{\vec{r}}\,V_2\star\varrho_1-\kappa_2\varrho_2\nabla_{\vec{r}}\,V_2\star\xi]|.\label{eq:uniqueness_bound_1}\end{aligned}$$ By Young’s inequality we have $$\begin{aligned} &\int_0^T\mathrm{d}t\, \int_U \mathrm{d}\vec{r}\, |\nabla_{\vec{r}}\eta\cdot \bm{D}^{1/2} \bm{D}^{1/2}\,[\xi\,\nabla_{\vec{r}}\kappa _1V_1+\kappa _2\xi\nabla_{\vec{r}}\,V_2\star\varrho_1-\kappa_2\varrho_2\nabla_{\vec{r}}\,V_2\star\xi]|\nonumber\\ &\leq \int_0^T\mathrm{d}t\, \int_U \mathrm{d}\vec{r}\, |\bm{D}^{1/2} \nabla_{\vec{r}}\eta |^2\nonumber\\ & +\tfrac{1}{4}\int_0^T\mathrm{d}t\, \int_U \mathrm{d}\vec{r}\, | \bm{D}^{1/2} [\xi\,\nabla_{\vec{r}}\kappa _1V_1+\kappa _2\xi\nabla_{\vec{r}}\,V_2\star\varrho_1-\kappa_2\varrho_2\nabla_{\vec{r}}\,V_2\star\xi ] |^2.\end{aligned}$$ Using the triangle inequality and Young’s inequality we expand the absolute value inside the integral $$\begin{aligned} &\tfrac{1}{4}\int_0^T\mathrm{d}t\, \int_U \mathrm{d}\vec{r}\, |\bm{D}^{1/2} [ \xi\,\nabla_{\vec{r}}\kappa _1V_1+\kappa _2\xi\nabla_{\vec{r}}\,V_2\star\varrho_1-\kappa_2\varrho_2\nabla_{\vec{r}}\,V_2\star\xi ]|^2\nonumber\\ &\leq \tfrac{1}{4}\int_0^T\mathrm{d}t\, \int_U \mathrm{d}\vec{r}\, |\bm{D}^{1/2} \xi\,\nabla_{\vec{r}}\kappa _1V_1|^2 +\kappa _2^2|\bm{D}^{1/2} [ \xi\nabla_{\vec{r}}\,V_2\star\varrho_1-\varrho_2\nabla_{\vec{r}}\,V_2\star\xi ] |^2 \nonumber\\ &\leq \tfrac{1}{4}\int_0^T\mathrm{d}t\, \int_U \mathrm{d}\vec{r} \left( |\bm{D}^{1/2}\xi\,\nabla_{\vec{r}}\kappa _1V_1|^2 +2\kappa _2^2|\bm{D}^{1/2}\xi\nabla_{\vec{r}}\,V_2\star\varrho_1|^2 +2\kappa _2^2|\bm{D}^{1/2}\varrho_2\nabla_{\vec{r}}\,V_2\star\xi|^2\right)\nonumber\\ &\leq \tfrac{\mu_{\max}}{4}\int_0^T\mathrm{d}t\, \int_U \mathrm{d}\vec{r}\, |\xi\,\nabla_{\vec{r}}\kappa _1V_1|^2+2\kappa _2^2|\xi\nabla_{\vec{r}}\,V_2\star\varrho_1|^2+2\kappa _2^2|\varrho_2\nabla_{\vec{r}}\,V_2\star\xi|^2. \label{eq:uniqueness_bound_2}\end{aligned}$$ Estimating each of these terms, first $$\begin{aligned} \label{eq:uniqueness_bound_3} \int_0^T\mathrm{d}t\, \int_U \mathrm{d}\vec{r}\, |\xi\,\nabla_{\vec{r}}\kappa _1V_1|^2\leq \kappa_1^2\|\nabla_{\vec{r}}V_1\|_{L^\infty(U)}^2\|\xi\|_{L^2([0,T];L^2(U))}.\end{aligned}$$ Second, $$\begin{aligned} \label{eq:uniqueness_bound_4} 2\kappa _2^2\int_0^T\mathrm{d}t\, \int_U \mathrm{d}\vec{r}\, |\xi\nabla_{\vec{r}}\,V_2\star\varrho_1|^2 \leq 2\kappa _2^2|U\||\nabla_{\vec{r}}V_2\|_{L^\infty(U)}^2\|\xi\|_{L^2([0,T];L^2(U))},\end{aligned}$$ and third $$\begin{aligned} &2\kappa _2^2\int_0^T\mathrm{d}t\, \int_U \mathrm{d}\vec{r}\,|\varrho_2\nabla_{\vec{r}}\,V_2\star\xi|^2\nonumber\\ &\leq 2\kappa _2^2|U| \|\varrho_2\|_{L^\infty([0,T];L^2(U))} \|\nabla_{\vec{r}}V_2\|_{L^\infty(U)}^2\|\xi\|_{L^2([0,T];L^2(U))}.\label{eq:uniqueness_bound_5}\end{aligned}$$ Combining , , , , we obtain, after setting $\eta = \xi$, and using boundedness of $\varrho_2$ in terms of its initial data $$\begin{aligned} \int_0^T\mathrm{d}t\, \langle \partial_t \xi(t),\, \xi(t)\rangle \leq (C_1(T)+C_2(T)\|\varrho_0\|_{L^2(U)}^2)\|\xi \|_{L^2([0,T];L^2(U))}^2\end{aligned}$$ for some constants $C_1(T)$, $C_2(T)$ dependent on $U$. This holds for all $T$ so it must be the case that $$\begin{aligned} \der[]{t}\|\xi(t)\|_{L^2(U)}^2\leq (C_1(T)+C_2(T)\|\varrho_0\|_{L^2(U)}^2)\|\xi(t)\|_{L^2(U)}^2\end{aligned}$$ implying by Gr[ö]{}nwall’s lemma that $$\begin{aligned} \|\xi(t)\|_{L^2(U)} \leq (C_1(T)+C_2(T)\|\varrho_0\|_{L^2(U)}^2)\|\xi(0)\|_{L^2(U)}\end{aligned}$$ a.e. $t\in [0,T]$. However, $\xi(0)\equiv 0$ hence $\|\varrho_1(t)-\varrho_2(t)\|_L^2(U)=0$ for all $t\in [0,T]$. Strict Positivity of $\varrho$. {#subsec:strict_pos_rho} ------------------------------- With the existence of weak solutions we may establish positivity of $\varrho$ solving with reference to [@bogachev2015fokker]. In particular since $\boldsymbol{D}$ is positive definite and $\vec{b}$ is uniformly bounded and $$\begin{aligned} \sup_{\vec{r}\in U}\varrho(\vec{r},t_1)<C \inf_{\vec{r}\in U}\varrho(\vec{r},t_2)\end{aligned}$$ for $0<t_1<t_2<\infty$ and $C$ is a constant depending on $d$ (the dimension) and $\mu_{\max}$. Since $\varrho$ is non-negative for all time we must have $\inf_{\vec{r}\in U}\varrho(\vec{r},t)$ is positive and hence $\varrho$ is positive.\ Discussion & Open Problems {#sec:discussion} ========================== In this paper, the global asymptotic stability and well-posedness of overdamped DDFT with two-body HI was studied. It was shown that bifurcations occur in DDFT systems with no-flux boundary conditions at an infinite and discrete set of critical energies equal to eigenvalues of the two-body interaction integral operator $\mathcal{R}$. Additionally we have shown that a weak solution to the density with no-flux boundary conditions and strong solution to the flux equation exist and are unique under sensible assumptions on the confining and interaction potentials and initial data $V_1$, $V_2$ and $\varrho(\vec{r},0)$ respectively. Assuming a classical solution to the DDFT we also derived *a priori* convergence estimates in $L^2$ and relative entopy, the latter restricted to convex two-body potentials. Well-posedness and global asymptotic stability of the phase space equation for the time evolution of $f(\vec{r},\vec{p},t)$ remains open (see [@goddard2012overdamped Proposition 2.1] for the evolution equation for $f(\vec{r},\vec{p},t)$). It is of similar form to the Vlasov equation considered by [@degond1986global] but with Hermite dissipative term and modified nonlocal term in the momentum variable $\vec{p}$ dependent on the HI tensors. To progress further some maximum principles on $f$ solving the linearised version of the phase space equation must be found. Additionally, the existence results on the overdamped equations considered here may be made more regular by routine arguments. We also note that the present analysis is based on the Smoluchowski equation rigorously derived from the phase space Fokker-Planck equation using homogenisation methods [@goddard2012overdamped]. As an alternative to this, assuming inertia is small altogether, or if one is interested only in very short times to begin with, the system of interacting particles maybe considered solely in configuration space. Only the positions (and not the momenta) of a system of interacting Brownian particles are then taken into account with Smoluchowski equation as in [@rex2009dynamical], and, the underlying Langevin dynamics contain only velocity equations for each particle which are usually written down *a posteriori*. The justification for this is that the momentum distribution is assumed to have a minor role in the dynamical description of the fluid density, and indeed is taken to be irrelevant at the microscopic level. This Brownian approximation may also hold for highly dense suspensions, since in dense Newtonian systems there is a fast transfer of momentum and kinetic energy from the particle collisions, and this effect may be accounted for most efficiently by the bath in the Brownian dynamics with a non constant diffusion tensor. It is known however that the one-body Smoluchowski equation in [@rex2009dynamical] does not equate to equations - which are obtained in the rigorous overdamped limit starting from the Newtonian dynamics. Intuitively this is because the two-body assumption for the HI ($\boldsymbol{\Gamma}$) and mobility ($\boldsymbol{D}$) tensors and the matrix inversion $\boldsymbol{D} = \boldsymbol{\Gamma}^{-1}$ are not commutable operations; even if $\boldsymbol{D}$ is two-body then $\text{det}(\boldsymbol{D})$ is not. A flow chart demonstrating the permitted commutations between various formalisms is included in [@goddard2012overdamped]. The nonequivalence of the two Smoluchowski equations is not considered here, and therefore a natural extension for future work would be to determine the existence, uniqueness and regularity of of the density starting from [@rex2009dynamical] as well as the corresponding conditions for linear stability. Finally we remark that a well-posedness analysis of DDFT equations of the form to include a hard-sphere contribution to the free energy by fundamental measure theory (FMT) e.g. Rosenfeld [@rosenfeld1989free] or Roth [@roth2010fundamental] would be very interesting.\ ————————————- Classical Linear Parabolic PDE {#sec:classical_paraboliv_pde} ============================== The first goal is to derive a similar set of estimates as [@chazelle2017well Lemma 3.5, Lemma 3.7]. The standard argument is to set up a sequence of linear parabolic PDEs. Let $U$ be a bounded and open subset of $\mathbb{R}^d$ and set $U_T=U\times (0,\,T]$ for some time $T>0$. Now consider the linear parabolic equation $$\begin{aligned} \label{eq:linear_classical_parabolic_eqn} \partial_t u_n-\nabla_{\vec{r}}\cdot[\bm{D}\nabla_{\vec{r}}u_n]=\nabla_{\vec{r}}\cdot[u_n\,\bm{D}\nabla_{\vec{r}}(\kappa _1V_1+\kappa _2V_2\star u_{n-1})].\end{aligned}$$ In general $d$ dimensions we are in the divergence form of the parabolic PDE $$\begin{aligned} \begin{cases} \label{eq:ibvp_pde_for_rho_lin} \qquad\partial_t u_n+Lu_n = 0 \quad \text{ in } U_T,\\ \Xi [u_n]\cdot\vec{n} = 0 \quad \text{ on } \partial U \times [0, T],\\ \qquad u_n = \varrho_0 \quad \text{ on } U\times \left\lbrace t=0\right\rbrace \end{cases}\end{aligned}$$ where $\partial U$ is a $C^1$ boundary with unit normal $\vec{n}$. We define $L$ to be the linear differential operator given by $$\begin{aligned} &Lu_n := -\sum_{ij=1}^d\partial_{r_j}(\bm{D}_{ij}(\vec{r},t)\partial_{r_i}u_n)+\sum_{i=1}^d b_i(\vec{r})\partial_{r_i}u_n+c(\vec{r})u_n,\label{eq:def_of_linear_parabolic_op}\\ & \vec{b}(\vec{r}):=-\bm{D}(\vec{r},t)\nabla_{\vec{r}}(\kappa_1V_1(\vec{r})+\kappa _2 [V_2\star u_{n-1}]), \\ & c(\vec{r}):=-\nabla_{\vec{r}}\cdot (\bm{D}(\vec{r},t)\nabla_{\vec{r}}(\kappa_1V_1(\vec{r})+\kappa _2[V_2\star u_{n-1}]),\\ & \Xi[u_n]:= \bm{D}(\vec{r},t)\,(\nabla_{\vec{r}}u_n + u_n\, \nabla_{\vec{r}}(\kappa_1 V_1(\vec{r},t)+ \kappa_2 \,V_2\star u_{n-1})).\end{aligned}$$ Since $\bm{D}(\vec{r},t)$ is assumed to positive definite, there exists $\theta$ for every $\vec{r}$, $\xi$ such that $\xi^\top D(\vec{r},t) \xi\geq \theta |\xi |^2$, therefore the operator $\partial_t + L$ is uniformly parabolic. The Sobolev space of functions that permit the no-flux condition $\Xi [u_n]\cdot \vec{n}$ on $\partial U\times [0,T]$ is $H^1(U)$ which is reflexive, so that $\partial_t u$ interpreted as a bounded linear functional can be paired to an element in $H^1(U)$, and further by the Riez-Representation theorem there exists a unique element from $H^1(U)$ for the pairing. Additionally $H^1(U)$ is separable so that the (unique) weak solution may be approximated by a sequence of smooth functions coming from a countably dense subset. Weak Formulation. ----------------- Equation may be recast into weak form. We first introduce the bilinear operator, defined by $$\begin{aligned} B[u,v;t] := \int_U\mathrm{d}\vec{r}\, \nabla_{\vec{r}}v\cdot\bm{D}\,\nabla_{\vec{r}}u+\int_U\mathrm{d}\vec{r}\,\vec{b}(\vec{r})\cdot\nabla_{\vec{r}}u\, v+\int_U\mathrm{d}\vec{r}\,c(\vec{r})u\, v\end{aligned}$$ for $u,v\in H^1(U)$ and a.e.  $0\leq t \leq T$. We regard $u$ as a mapping $[\mathfrak{u}(t)](\vec{r}):=u(\vec{r},t)$ from the time interval $[0,T]$ to the function space $H^1(U)$. Now fixing $v \in H^{1}(U)$ we multiply by $v$ and integrate by parts to obtain the weak formulation $$\begin{aligned} \label{eq:weak_form_for_un} (\partial_t\mathfrak{u},\,v)+B[\mathfrak{u},\,v;t] = 0\end{aligned}$$ for each $0\leq t\leq T$ with $(\,,\,)$ denoting inner product in $L^2(U)$. Existence. ---------- The method to establish weak solution for the indexed problem is a textbook one. The method is described as follows. Fix $n$ then the evolution equation is a uniformly parabolic PDE for the unknown $u_n = u$. One now expands $\mathfrak{u} = \mathfrak{u}^m$ in a linear combination of $m$ eigenvectors of the operator $-\nabla_{\vec{r}}\cdot( \bm{D}(\vec{r})\nabla_{\vec{r}}w_k)$ for finite dimensional approximation to $u$. Since $\bm{D}_{ij}(\cdot)$ is a compact and symmetric operator then the eigenfunctions $w_k$ form an orthonormal basis of $L^2(U)$ with $w_k\in H^1(U)$. Thus $\mathfrak{u}^m$ is projected onto the finite dimensional subspace spanned by $\left\lbrace w_k\right\rbrace_{k=1}^m$. The standard existence theory of ODEs (the Carath[é]{}odory conditions with the Cauchy–Picard theorem) gives existence of weak solutions $\mathfrak{u}^m$ as expanded in the functions $\left\lbrace w_k\right\rbrace_{k=1}^m$ on a finite dimensional subspace of $H^1(U)$. All that remains is to pass to the limit $m\to \infty$ to realise the result in $H^1(U)$. To do this energy estimates are required on $\mathfrak{u}^m$, these are routine calculations except in the textbooks they are done for simpler boundary condition choices (homogeneous Dirichlet or periodic) and make use of Poincare’s inequality (holding only for $H^1_0$ functions). The calculations are similar and for the present boundary condition choice, a weaker Poincar[é]{}$-$Wirtinger inequality is used through out to obtain $$\begin{aligned} &\max_{0\leq t\leq T}\|\mathfrak{u}^m(t)\|_{L^2(U)}+\|\mathfrak{u}^m\|_{L^2([0,T]; H^{1} (U))}\nonumber\\ &+\|\pder[]{t}\mathfrak{u}^m\|_{L^2([0,T]; H^{-1}(U))} \leq c_1\|u_{0}\|_{L^2(U)}+ c_2.\label{eq:uniform_bound_on_three_norms}\end{aligned}$$ where $c_1$, $c_2$ are constants dependent on $T$ and $U$ and $\mu_{\min},\mu_{\max}$. Note that the left hand side of forms a bounded sequence in $\mathbb{R}$ and by the Bolzano–Weierstrass theorem there exists a convergent subsequence $\{\mathfrak{u}^{ m_l}\}_{l\geq 1}\subset\{\mathfrak{u}^{ m}\}_{m\geq 1}$. In particular there exists $\mathfrak{u}$ such that $$\begin{aligned} \begin{split} \mathfrak{u}^{ m_l} \rightharpoonup \mathfrak{u}\quad &\text{ weakly in } L^2([0,T]; H^1(U)),\\ \partial_t\mathfrak{u}^{ m_l} \rightharpoonup \mathfrak{u}' \quad &\text{ weakly in } L^2([0,T]; H^{-1}(U)). \end{split}\end{aligned}$$ Note of course that $\mathfrak{u} = \mathfrak{u}_n$, but we have not yet established existence of weak solution to the full nonlinear Smoluchowski equation . This result establishes existence of weak solution for the parabolic equation for every index $n$. Now since $L^2([0,T]; H^1(U))$ is separable, and weak solutions currently only exist in a finite dimensional subspace of $H^1(U)$, it makes sense to choose a test function $\bm{\phi}\in C^1([0,T]; H^1(U))\subset L^2([0,T]; H^1(U))$. We may therefore write $$\begin{aligned} \int_0^T\mathrm{d}t\,\langle\partial_t\mathfrak{u}^{ m} ,\,\bm{\phi}^N\rangle +B[\mathfrak{u}^m,\,\bm{\phi}^N;t] =0\end{aligned}$$ for $\bm{\phi}^N=\sum_{k=1}^Nd^k(t)w_k$. Making the choice $N\leq m$ and letting $N\to\infty$ one obtains $$\begin{aligned} \int_0^T\mathrm{d}t\,\langle\partial_t\mathfrak{u} ,\,\bm{\phi}^{\infty}\rangle +B[\mathfrak{u},\,\bm{\phi}^\infty;t] =0\end{aligned}$$ for any function $\bm{\phi}^\infty\in L^2([0,T]; H^1(U))$ since $\phi^N$ are dense in $L^2([0,T]; H^1(U))$. Now since $\bm{\phi}^\infty$ is arbitrary we obtain $$\begin{aligned} \langle\partial_t\mathfrak{u} ,\,\phi\rangle +B[\mathfrak{u},\,\phi;t] = 0\end{aligned}$$ for an arbitrary $\phi\in H^1(U)$. Hence the criteria of weak solution is satisfied. Uniqueness. ----------- To show uniqueness we argue by contradiction that there exists two weak solutions solutions. By linearity, their difference $\bm{\chi}$ is a weak solution of with $\chi_0\equiv 0$, for $\chi_0$ initial data. Then as it is a weak solution, we may test $\bm{\chi}$ against itself $$\begin{aligned} \langle\partial_t\bm{\chi} ,\,\bm{\chi}\rangle +B[\bm{\chi},\,\bm{\chi};t] \equiv 0\end{aligned}$$ giving $$\begin{aligned} \tfrac{1}{2}\der[]{t}(\|\bm{\chi}(t)\|^2_{L^2(U)})+B[\bm{\chi},\,\bm{\chi};t] =0\end{aligned}$$ but $B[\bm{\chi},\,\bm{\chi};t]\geq -c_{7}\|\bm{\chi}(t)\|^2_{L^2(U)}$ which may be obtained by the following estimate $$\begin{aligned} \label{eq:second_bound_on_B[u,u]} c_5\|\mathfrak{u}^m-c\|_{H^{1}(U)}^2\leq B[\mathfrak{u}^m,\mathfrak{u}^m]+c_6\|\mathfrak{u}^m\|^2_{L^2(U)}.\end{aligned}$$ and hence by Gr[ö]{}nwall $$\|\bm{\chi}(t)\|^2_{L^2(U)}\leq c_{7}(t)\|\chi_0\|^2_{L^2(U)}=0$$ and $\bm{\chi} =0 $ for a.e. $\vec{r}\in U$ for every $0\leq t \leq T$. We have established the existence and uniqueness of the weak solution to the linear parabolic equation and may apply this to an iteration problem on .\ Nomenclature {#app:nomenclature} ============ ------------------------------------------------- -- ----------------------------------------------------------------------------------------------------------------------------------- $\alpha$ Mass diffusivity $\beta_i^{(\kappa_2)}$, $\theta_i^{(\kappa_2)}$ Expansion coefficients used in Theorem \[thm:epsilon\_as \_a\_fn\_of\_lambda\] $\beta_{n}^{-1}$ Eigenvalues of $\mathcal{R}$ from Definition \[def:def\_of\_R\_op\] $\gamma$ Friction coefficient $\gamma^{(\kappa_2)}_k$ Eigenvalues of $\mathcal{A}_{\kappa_2}$ defined in $\delta_n$, $\phi_n$, $\psi_n$ Expansion coefficients used in Theorem \[thm:eigenfn\_expansion\_of\_flux\] $\epsilon$ Small nondimensional parameter $\zeta^a_i$ Gaussian white noise process $\kappa_1$ Nondimensional external potential strength $\kappa_2$ Nondimensional two body potential strength $\kappa_{2_\sharp}$ Point of critical stability defined in $\lambda$ Scalar value of $\mathbb{C}$ used in Lemma \[lem:flux\_operator\_is\_compact\_self\_adjoint\] $\lambda(\kappa_2)$ Eigenvalue of $\mathcal{L}_{1}$ used in Theorem \[thm:epsilon\_as \_a\_fn\_of\_lambda\] $\mu$ Probability measure $\mu_c$ Chemical potential $\mu_i$ Eigenvalues of $\bm{D}$ defined in $\mu_{\max}$, $\mu_{\min}$ Largest and smallest eigenvalues of $\bm{D}$ respectively $\varrho$ Density $\varrho_0$ Initial density $\varrho_n$ $n$-particle configuration space distribution function, for $n\geq 2$ $\varrho_\infty$ Unique equilibrium density $\tau$ Characteristic time scale $\phi_n$ Eigenvalues of $\bm{1}+\mathcal{Z}_1^\varrho -\lambda \mathcal{Z}_2^\varrho$ used in Theorem \[thm:eigenfn\_expansion\_of\_flux\] $\chi_\epsilon$ Convex approximation to $|\cdot |$ in Definition \[def:def\_of\_chi\_approx\_abs\] ------------------------------------------------- -- ----------------------------------------------------------------------------------------------------------------------------------- : [**[Upper Case Greek]{}**]{} \ \ -------------------- -- --------------------------------------------------------- $\bm{\Gamma}$ $3N\times 3N$ friction tensor $\bm{\Gamma}_{ij}$ $3\times 3$ block matrices of $\bm{\Gamma}$ $\Pi$, $\Pi_1$ Nonlinear and linear boundary operators respectively, , -------------------- -- --------------------------------------------------------- : [**[Upper Case Greek]{}**]{} --------------------------------- -- ------------------------------------------------------------------------------------------------------------------------------------ $\vec{a}$ Flux $c_{ls}$ Log–Sobolev constant $c_{pw}$ Poincar[é]{}–Wirtinger constant $d$ Dimension number $\vec{e}_i$ Eigenvectors of $\bm{D}$ defined in $f(\vec{r},\vec{p},t)$ Phase space density $\vec{f}_i$ Gaussian white noise vector $g(\vec{r},\vec{r}',[\varrho])$ Correlation function $k_B$ Boltzmann constant $m$ Mass of particle $i$ $\vec{p}_i$ Momentum vector of particle $i$ $\vec{r}_i$ Position vector of particle $i$ $t$ Time $u_n$ Eigenfunctions of $\mathcal{R}$ from Definition \[def:def\_of\_R\_op\] $v^{(\kappa_2)}_k$ Eigenfunctions of $\mathcal{A}_{\kappa_2}$ defined in $w^{(\kappa_2)}$ Eigenfunction of $\mathcal{L}_1$ used in Theorem \[thm:epsilon\_as \_a\_fn\_of\_lambda\] $\vec{w}_n$ Eigenfunctions of $\bm{1}+\mathcal{Z}_1^\varrho-\lambda\mathcal{Z}_2^\varrho$ used in Theorem \[thm:eigenfn\_expansion\_of\_flux\] --------------------------------- -- ------------------------------------------------------------------------------------------------------------------------------------ : [**[Upper Case Roman]{}**]{} \ \ ---------------------------------- -- --------------------------------------------------------------------------- $\bm{1}$ $3\times 3$ identity matrix $\bm{1} + \mathcal{Z}_1^\varrho$ Local operator on acting on $\vec{a}$ defined in $\mathrm{A}$ Characteristic flux scale $\bm{A}([\vec{a}],t)$ Advection tensor defined in $\mathcal{A}_\varrho$ The operator $(\bm{1} + \mathcal{Z}_1^\varrho)^{-1}\mathcal{Z}_2^\varrho$ $\mathcal{A}_{\kappa_2}$ Differential operator defined in $\boldsymbol{B}$ $\left(m k_B T \boldsymbol{\Gamma}\right)^{1/2}$ $\mathcal{B}$ Nonlocal operator $C(T)$ Constant dependent on the final time $T$ $\bm{D}(\vec{r},[\varrho],t)$ Diffusion tensor defined in $F$ Nonlinear map defined in $\mathcal{F}$ Free energy functional defined in $Fr$ Froude number $\mathcal{F}_H$ Helmholtz free energy functional defined in $\mathcal{H}$ Relative entropy functional defined in $\mathrm{L}$ Characteristic length scale $\mathcal{L}_1$ Linearised nonlocal differential operator defined in ---------------------------------- -- --------------------------------------------------------------------------- : [**[Upper Case Roman]{}**]{} --------------------------------- -- ------------------------------------------------------------------------------------------------------------------------------------- $N$ Number of particles $Pe$ P[é]{}clet number $\mathcal{R}$ Nonlocal operator from Definition \[def:def\_of\_R\_op\] $\mathrm{T}$ Temperature $T$ Final time $\mathcal{T}$ Nonlocal operator defined in $U$ Spatial domain $U_T$ Space-time cylinder $U\times [0,T]$ $\mathrm{U}$ Characteristic velocity scale $V$ Potential $V_1$, $V_2$, $V_n$ External, two body and $n$-body potentials respectively $\mathcal{X}_\varrho$ Inverse operator $(\bm{1}+\mathcal{Z}_1^\varrho+\mathcal{Z}_2^\varrho)^{-1}$ used in Theorem \[thm:association \_of\_free\_energy\] $Z$ Normalisation constant $\bm{Z}_1(\vec{r}_1,\vec{r}_2)$ Diagonal two body HI tensor $\bm{Z}_2(\vec{r}_1,\vec{r}_2)$ Off-diagonal two body HI tensor $\mathcal{Z}_2^\varrho$ Nonlocal operator acting on $\vec{a}$ defined in --------------------------------- -- ------------------------------------------------------------------------------------------------------------------------------------- : [**[Sets and Mathematical Symbols]{}**]{} \ \ ------------------------- -- ---------------------------------------------------------------------------------------- $L^2(U,\varrho^{-1})$ Weighted $L^2(U)$ space $P_{ac}(U)$ Space of absolutely continuous probability densities supported on $U$ $P_{ac}^{+}(U)$ $P_{ac}(U)$ restricted to strictly positive functions $\text{Tr}$ Trace operator $\top$ Transpose $u\star v$ Convolution of two functions $\int_U\mathrm{d}\vec{r}\,u(\vec{r}-\vec{r}')v(\vec{r}')$ $\vec{u}\otimes\vec{v}$ Outer product / dyadic of two vectors $\vec{u}\vec{v}^\top$ ------------------------- -- ---------------------------------------------------------------------------------------- : [**[Sets and Mathematical Symbols]{}**]{} \[bib:References\] [^1]: School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh EH9 3FD, UK. Corresponding author: R. D. Mills-Williams (r.mills@ed.ac.uk). [^2]: Department of Mathematics, Imperial College London, London SW7 2AZ, UK. [^3]: BDG would like to acknowledge support from EPSRC EP/L025159/1. RDMW is grateful to EPSRC for PhD funding. GAP was supported by the EPSRC through grant numbers EP/P031587/1, EP/L024926/1, and EP/L020564/1. This research was funded in part by JPMorgan Chase & Co. Any views or opinions expressed herein are solely those of the authors listed, and may differ from the views and opinions expressed by JPMorgan Chase & Co. or its affiliates. This material is not a product of the Research Department of J.P. Morgan Securities LLC. This material does not constitute a solicitation or offer in any jurisdiction.
--- abstract: 'Tomograms are obtained as probability distributions and are used to reconstruct a quantum state from experimentally measured values. We study the evolution of tomograms for different quantum systems, both finite and infinite dimensional. In realistic experimental conditions, quantum states are exposed to the ambient environment and hence subject to effects like decoherence and dissipation, which are dealt with here, consistently, using the formalism of open quantum systems. This is extremely relevant from the perspective of experimental implementation and issues related to state reconstruction in quantum computation and communication. These considerations are also expected to affect the quasiprobability distribution obtained from experimentally generated tomograms and nonclassicality observed from them.' author: - 'Kishore Thapliyal$^{a,}$[^1], Subhashish Banerjee$^{b,}$[^2], Anirban Pathak$^{a,}$[^3]' title: 'Tomograms for open quantum systems: in(finite) dimensional optical and spin systems' --- [Introduction]{} ================ A quantum state can be characterized by a number of probability and quasiprobability distribution functions [@st-est]. The quasiprobability distributions are not true probability distributions as most of them can have nonpositive values. Interestingly, this nonpositivity can be viewed as a signature of nonclassicality or quantumness. Specifically, the negative values of the Wigner [@wig] and $P$ [@glaub; @sudar] functions serve as witnesses of nonclassicality. Further, zeros of $Q$ function [@comment1] also serve as witness of nonclassicality. As there does not exist any straight forward prescription for direct measurement of these quasiprobability distributions, several efforts have been made to construct measurable probability distributions that can be used to uniquely construct either all or some of these quasiprobability distributions. Such measurable probability distributions are referred to as tomograms [@tom-review; @with-adam; @manko-fss; @manko-spin-tom]. In other words, the tomogram is a scheme for measuring a quantum state by using a representation in one to one correspondence with the true probability distribution rather than with a quasidistribution [@comment13-ref1]. A relationship between a tomogram and a quasidistribution function, such as the Wigner function, can be established for both continuous and discrete systems [@op-tom; @fss-tom]. Specifically, in Ref. [@op-tom] it was shown that quasiprobability distributions ($P$, $Q$, and Wigner functions) can be uniquely determined in terms of probability distributions for the rotated quadrature phase which can be viewed as an optical tomogram of the state. Similarly, in Ref. [@fss-tom] it was shown that for finite dimensional phase states, discrete Wigner functions and tomograms are connected by a discretization of the continuous variable Radon transformation and was referred to as the *Plato transformation*. In the recent past, a few successful attempts have been made to measure Wigner function directly in experiments [@wig-exp; @wig-exp2], but the methods adopted are state specific. The same limitation is also valid for the theoretical proposals [@wig-diff] for the measurement of Wigner function. Further, optical homodyne tomography has been employed for the experimental measurement of the Wigner functions of vacuum and squeezed states in [@comment5exp-ref1; @comment5exp-ref2], while distributions corresponding to Pegg-Barnett and Susskind-Glogower phase operators were also obtained in [@comment5exp-ref2]. An experimental measurement of the $P$, $Q$ and Wigner quantum phase distributions for the squeezed vacuum state has been reported in [@comment5exp-ref3]. Precision of homodyne tomography technique was compared with conventional detection techniques in [@comment5theory-ref1]. A number of alternative methods of tomography have also been proposed [@comment5theory-ref2; @comment5theory-ref4; @comment5theory-ref5], and exploited to obtain phase distributions like Wigner and $Q$ functions [@comment5theory-ref3]. In [@CV-rev] continuous variable quantum state tomography was reviewed from the perspective of quantum information. In brief, there does not exist any general prescription for direct experimental measurement of the Wigner function and other quasidistribution functions. In practice, to detect the nonclassicality in a system the Wigner function is obtained either by photon counting or from experimentally measured tomograms [@wig-diff]. Thus, tomograms are very important for the identification of nonclassical character of a physical system. In another line of studies, simulation of quantum systems have been performed using tomography. For example, tomograms were used for simulation of tunneling [@comment9-ref1; @comment9-ref2; @comment9-ref4] and multimode quantum states [@comment9-ref3]. Attempts have also been made to understand the tomogram via path integrals [@comment9-ref5; @comment9-ref6]. Furthermore, how to reconstruct a quantum state from experimentally measured values is of prime interest for both quantum computation [@wig-exp] and communication [@tel-143]. Specifically, in Ref. [@wig-exp] it is strongly established that tomography and spectroscopy can be interpreted as dual forms of quantum computation, and in Ref. [@tel-143], quantum teleportation was experimentally performed over a distance of 143 km and the quality of teleportation was verified with the help of quantum process tomography (QPT) of quantum teleportation without feed-forward. Here it would be apt to note that QPT is an aspect of quantum state tomography in which a quantum process is obtained as a trace preserving positive linear map [@QPT]. In the recent past, quantum process tomography has been discussed from the perspective of open quantum system effects [@QPT-open; @marzolino1; @marzolino2]. A novel method of complete experimental characterization of quantum optical processes was introduced in [@comment8-ref1]. It was further developed in [@comment8-ref2; @comment8-ref3] and extended to characterization of $N$-modes in [@comment8-ref7]. In [@comment8-ref4], QPT was applied to the characterization of optical memory based on electromagnetically induced transparency while [@comment8-ref5] and [@comment8-ref6] were devoted to QPT of the electromagnetic field and conditional state engineering, respectively. Quantum state tomography has its applications in quantum cryptography as well [@tom-cryp]. Specifically, in Ref. [@tom-cryp] an interesting protocol of quantum cryptography was proposed in which eavesdropping in the quantum channel was checked by requiring consistency of outcome of the tomography with the unbiased noise situation. Keeping these facts in mind, we aim to construct tomograms for a number of physical systems of practical relevance (mostly having applications in quantum computation and communication) and investigate the effects of various types of noise on them. From the experimental perspective, a quantum state always interacts with its surroundings. Hence, the evolution of the corresponding tomogram after taking into account the interaction of the quantum state with its environment should be considered. This can be achieved with the open quantum system formalism [@louis; @bp; @pi; @sbqbm]. Specifically, both purely dephasing (QND) [@QND] and dissipative [@SGAD] open quantum system effects have been studied here. Interestingly, both these effects have also been experimentally realized in the recent past [@haroche; @turchette]. In Ref. [@our-qd-paper], a systematic study of quasidistribution functions was made for a host of interesting states under general open system evolutions. Here, we set ourselves the task of obtaining the tomograms for various finite and infinite dimensional quantum systems in different open quantum system scenarios. For finite dimensional spin states, the tomogram is the distribution function of the projections of the spin on an arbitrary axis, characterized by Euler angles, and can be obtained from the diagonal elements of the rotated density matrix, while for continuous variable systems, such as the radiation field, the analog would be the homodyne probability. It follows from general group theoretical arguments that, making use of unitary irreducible square integrable representation of the tomographic group under consideration, a unified tomographic prescription can be developed for both finite dimensional and continuous variable systems [@spin-ariano]. Tomograms for spin states have been developed both as projections on an arbitrary axis [@comment6-ref1] as well as by using a discrete variable analog of symplectic tomography [@comment6-ref2]. Tomograms of optical systems have been well studied in the past [@op-tom; @dodonov-manko; @tom-review; @CV-rev; @paris-rev]. In Ref. [@mar-non-mar] quantum state tomography was used to determine the degree of non-Markovianity in an open system. Further, thermal noise is used in tomography (for reconstruction of photon number distributions) as a probe [@tom-by-noise]. The paper is organized as follows. In Section \[sec:Tomograms-of-single-half\], tomograms of single spin-$\frac{1}{2}$ (qubit) atomic coherent state under purely dephasing (QND) and dissipative evolution are obtained. Further, the tomogram of two spin-$\frac{1}{2}$ (qubit) quantum state is studied in Section \[sec:Tomogram-of-two-spin\] under the influence of a vacuum bath. This is followed by a tomogram for a general spin-1 pure state in Section \[sec:Tomogram-of-spin-1\]. The tomograms of finite dimensional number-phase states under open quantum system evolution are discussed in Section \[sec:Tomogram-of-qutrit\]. This is illustrated by a specific example of a three-level quantum (qutrit) system evolving under a spontaneous emission channel. In Section \[sec:Optical-tomogram\], we discuss the tomogram of an infinite dimensional system, the ubiquitous dissipative harmonic oscillator. We conclude in Section \[sec:Conclusion\]. Tomograms of single spin-$\frac{1}{2}$ states \[sec:Tomograms-of-single-half\] ============================================================================== In this section, we study the tomograms for single spin-$\frac{1}{2}$ (qubit) atomic coherent state evolving under two general noise models, i.e., pure dephasing (QND) and dissipative squeezed generalized amplitude damping (SGAD) evolution, incorporating the effects of dissipation, decoherence and bath squeezing. [QND Evolution]{} ----------------- The master equation of a quantum state under QND evolution [@QND] is $$\begin{array}{lcl} \dot{\rho}{}_{nm}^{s}\left(t\right) & = & \left[-\frac{i}{\hbar}\left(E_{n}-E_{m}\right)+i\dot{\eta}\left(t\right)\left(E_{n}^{2}-E_{m}^{2}\right)-\left(E_{n}-E_{m}\right)^{2}\dot{\gamma}\left(t\right)\right]\rho_{nm}^{s}\left(t\right), \end{array}\label{eq:master-eq-QND}$$ where $E_{n}$’s are the eigenvalues of the system Hamiltonian in the system eigenbasis $|n\rangle$, $$\eta\left(t\right)=-\sum_{k}\frac{g_{k}^{2}}{\hbar^{2}\omega_{k}^{2}}\sin\left(\omega_{k}t\right),$$ and $$\begin{array}{lcl} \gamma\left(t\right) & = & \frac{1}{2}\underset{k}{\sum}\frac{g_{k}^{2}}{\hbar^{2}\omega_{k}^{2}}\coth\left(\frac{\beta\hbar\omega_{k}}{2}\right) \left|\left(e^{i\omega_{k}t}-1\right)\cosh\left(r_{k}\right)+\left(e^{-i\omega_{k}t}-1\right)\sinh\left(r_{k}\right)e^{2i\Phi_{k}}\right|^{2}, \end{array}$$ with $\beta=\frac{1}{k_{B}T}$. Here, $k_{B}$ is the Boltzmann constant, $r_{k}$ and $\Phi_{k}$ are the bath squeezing parameters and $g_{k}$ is the system-bath coupling coefficient. The initial density matrix for the atomic coherent state is given by $$\rho^{s}\left(0\right)=|\alpha,\beta\rangle\langle\alpha,\beta|,\label{eq:initial-state-QND}$$ where the atomic coherent state is given by $$\begin{array}{lcl} |\alpha,\beta\rangle & = & \stackrel[m=-j]{j}{\sum}\left(\begin{array}{c} 2j\\ j+m \end{array}\right)^{1/2}\sin\left(\frac{\alpha}{2}\right)^{j+m} \cos\left(\frac{\alpha}{2}\right)^{j-m}|j,m\rangle e^{-i\left(j+m\right)\beta}. \end{array}\label{eq:atomic-coherent-state}$$ The different elements of the density matrix in Eq. (\[eq:initial-state-QND\]) at time $t$ under QND evolution becomes $$\begin{array}{lcl} \rho_{jm,jn}^{s}\left(t\right) & = & e^{-i\omega\left(m-n\right)t}e^{i\left(\hbar\omega\right)^{2}\left(m^{2}-n^{2}\right)\eta\left(t\right)} e^{-\left(\hbar\omega\right)^{2}\left(m-n\right)^{2}\gamma\left(t\right)}\rho_{jm,jn}^{s}\left(0\right), \end{array}\label{eq:QND-densitymatirix}$$ with $\rho_{jm,jn}^{s}\left(0\right)=\langle j,m|\rho^{s}\left(0\right)|j,n\rangle$. Considering the initial state of the system as atomic coherent state, i.e., using Eq. (\[eq:atomic-coherent-state\]), different elements of the density matrix in Eq. (\[eq:QND-densitymatirix\]) at time $t=0$ are $$\begin{array}{lcl} \rho_{jm,jn}^{s}\left(0\right) & = & \left(\begin{array}{c} 2j\\ j+m \end{array}\right)^{1/2}\left(\begin{array}{c} 2j\\ j+n \end{array}\right)^{1/2}e^{i\left(n-m\right)\beta} \sin\left(\frac{\alpha}{2}\right)^{2j+m+n}\cos\left(\frac{\alpha}{2}\right)^{2j-m-n}. \end{array}\label{eq:at_t0_with_acs}$$ Using Eq. (\[eq:at\_t0\_with\_acs\]) as the initial density matrix elements in Eq. (\[eq:QND-densitymatirix\]), we can write all the elements of the density matrix at time $t$ as $$\begin{array}{lcl} \rho_{jm,jn}^{s}\left(t\right) & = & \left(\begin{array}{c} 2j\\ j+m \end{array}\right)^{1/2}\left(\begin{array}{c} 2j\\ j+n \end{array}\right)^{1/2}e^{-i\omega\left(m-n\right)t} e^{i\left(\hbar\omega\right)^{2}\left(m^{2}-n^{2}\right)\eta\left(t\right)}\\ & \times & e^{-\left(\hbar\omega\right)^{2}\left(m-n\right)^{2}\gamma\left(t\right)}\sin\left(\frac{\alpha}{2}\right)^{2j+m+n}\cos\left(\frac{\alpha}{2}\right)^{2j-m-n}e^{i\left(n-m\right)\beta}. \end{array}\label{eq:at_t_with_acs}$$ To obtain a tomogram of a spin-$\frac{1}{2}$ atomic coherent state under QND evolution, we can express the density matrix in terms of Wigner-Dicke states as $$\rho^{\left(j\right)}\equiv\rho^{\left(j\right)}\left(t\right)=\sum_{m,m^{\prime}=-j}^{j}\rho_{m,m^{\prime}}^{\left(j\right)}|j,m\rangle\langle j,m^{\prime}|.\label{eq:wigner_dicke_state}$$ The different elements of this density matrix $\rho_{m,m^{\prime}}^{\left(j\right)}=\langle m|\rho^{\left(j\right)}|m^{\prime}\rangle$ can be obtained using Eq. (\[eq:at\_t\_with\_acs\]), with $m,\,n\rightarrow m,\,m^{\prime},$ for $j=\frac{1}{2}$, $m,\,m^{\prime}=\pm\frac{1}{2}$. Subsequently, the density matrix is obtained as $$\rho^{\left(1/2\right)}=\left[\begin{array}{cc} \sin^{2}\left(\frac{\alpha}{2}\right) & \frac{1}{2}e^{-i\omega t}e^{-\left(\hbar\omega\right)^{2}\gamma\left(t\right)}\sin\alpha e^{-i\beta}\\ \frac{1}{2}e^{i\omega t}e^{-\left(\hbar\omega\right)^{2}\gamma\left(t\right)}\sin\alpha e^{i\beta} & \cos^{2}\left(\frac{\alpha}{2}\right) \end{array}\right].\label{eq:density-matrix-QND}$$ We can easily check that the trace of the density matrix $\left(\rho^{\left(1/2\right)}\right)$ is one, i.e., $\stackrel[m=-1/2]{1/2}{\sum}\rho_{m,m}^{\left(1/2\right)}=1.$ Further, the tomogram of this state can be expressed as [@manko-spin-tom] $$\begin{array}{lcl} \omega\left(m_{1},\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right) & = & \stackrel[m=-j]{j}{\sum}\stackrel[m^{\prime}=-j]{j}{\sum}D_{m_{1},m}^{\left(j\right)}\left(\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right) \rho_{m,m^{\prime}}^{\left(j\right)}D_{m_{1},m^{\prime}}^{\left(j\right)*}\left(\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right), \end{array}\label{eq:tomogram}$$ where $D_{m,m^{\prime}}^{\left(j\right)}\left(\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right)$ is the Wigner $D$-function $$\begin{array}{lcl} D_{m,m^{\prime}}^{\left(j\right)}\left(\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right) & = & e^{-im\widetilde{\alpha}}d_{m,m^{\prime}}^{\left(j\right)}\left(\widetilde{\beta}\right)e^{-im^{\prime}\widetilde{\gamma}}\end{array}, \label{eq:wigner_D-function}$$ and the notation used here is consistent with that in Ref. [@varshalo]. Here, $\widetilde{\alpha}$, $\widetilde{\beta}$, and $\widetilde{\gamma}$ are Euler angles $\equiv\phi$ , $\theta$, and $\psi$, with $\phi,\psi\in\left[0,2\pi\right]$, and $\theta\in\left[0,\pi\right]$, and $$\begin{array}{lcl} d_{m,m^{\prime}}^{\left(j\right)}\left(\widetilde{\beta}\right) & = & \left[\frac{\left(j+m\right)!\left(j-m\right)!}{\left(j+m^{\prime}\right)!\left(j-m^{\prime}\right)!}\right]^{1/2}\left(\cos\frac{\widetilde{\beta}}{2}\right)^{m+m^{\prime}} \left(\sin\frac{\widetilde{\beta}}{2}\right)^{m-m^{\prime}}P_{j-m}^{\left(m-m^{\prime},m+m^{\prime}\right)}\left(\cos\widetilde{\beta}\right), \end{array}\label{eq:D-function-d}$$ where $P_{n}^{\left(a,b\right)}\left(x\right)$ are Jacobi polynomials. A tomogram is the spin projection onto an arbitrary, rotated, axis. The physical significance of the $D$-function is its connection to the process of rotation and can be illustrated by $$\begin{array}{lcl} \langle j,m_{1}|R\left(\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right)|j,m_{1}^{\prime}\rangle & = & D_{m_{1},m_{1}^{\prime}}^{\left(j\right)}\left(\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right),\\ \langle j,m_{2}^{\prime}|R^{\dagger}\left(\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right)|j,m_{1}\rangle & = & D_{m_{1},m_{2}^{\prime}}^{*\left(j\right)}\left(\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right). \end{array}\label{eq:Meaning-of-D}$$ Here, $R\left(\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right)$ stands for the operation of rotation about an axis whose orientation is specified by $\widetilde{\alpha},\,\widetilde{\beta},$ and $\widetilde{\gamma}.$ Using the different values of $m$ and $m^{\prime}$, we can obtain various Wigner $D$-functions as $$\begin{array}{lcl} D_{\frac{1}{2},-\frac{1}{2}}^{\left(1/2\right)}\left(\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right) & = & -\sin\left(\frac{\widetilde{\beta}}{2}\right)e^{-\frac{i}{2}\left(\widetilde{\alpha}-\widetilde{\gamma}\right)},\\ D_{\frac{1}{2},\frac{1}{2}}^{\left(1/2\right)}\left(\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right) & = & \cos\left(\frac{\widetilde{\beta}}{2}\right)e^{-\frac{i}{2}\left(\widetilde{\alpha}+\widetilde{\gamma}\right)},\\ D_{-\frac{1}{2},-\frac{1}{2}}^{\left(1/2\right)}\left(\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right) & = & \cos\left(\frac{\widetilde{\beta}}{2}\right)e^{\frac{i}{2}\left(\widetilde{\alpha}+\widetilde{\gamma}\right)},\\ D_{-\frac{1}{2},\frac{1}{2}}^{\left(1/2\right)}\left(\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right) & = & \sin\left(\frac{\widetilde{\beta}}{2}\right)e^{\frac{i}{2}\left(\widetilde{\alpha}-\widetilde{\gamma}\right)}. \end{array}\label{eq:values-of-Ds}$$ Using the first two relations of Eq. (\[eq:values-of-Ds\]) and Eq. (\[eq:density-matrix-QND\]), the first component of the tomogram can be obtained from Eq. (\[eq:tomogram\]) as $$\begin{array}{lcl} \omega\left(\frac{1}{2},\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right)\equiv\omega_{1} & = & \cos^{2}\left(\frac{\widetilde{\beta}}{2}\right)-\cos\widetilde{\beta}\cos^{2}\left(\frac{\alpha}{2}\right) - \frac{1}{2}\sin\widetilde{\beta}\sin\alpha\cos\left(\omega t+\beta+\widetilde{\gamma}\right) e^{-\left(\hbar\omega\right)^{2}\gamma\left(t\right)}. \end{array}\label{eq:tomogram1-QND}$$ From Eq. (\[eq:tomogram1-QND\]), it can be inferred that the tomogram is free from Euler angle $\widetilde{\alpha}$, and consequently is a function of $\widetilde{\beta}$ and $\widetilde{\gamma}$ only, or $f(\widetilde{\beta},\widetilde{\gamma})$. It is worth mentioning here that $\widetilde{\gamma}$ and $\gamma\left(t\right)$ are two different parameters, the former being an Euler angle while the latter is responsible for decoherence. The variation of the tomogram is given in Fig. \[fig:qnd-single\] with time, for the different temperatures. For the second component of the tomogram with $m_{1}=-\frac{1}{2}$, using last two relations of Eq. (\[eq:values-of-Ds\]) and substituting Eq. (\[eq:density-matrix-QND\]) in Eq. (\[eq:tomogram\]), we obtain $$\begin{array}{lcl} \omega\left(-\frac{1}{2},\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right)\equiv\omega_{2} & = & \cos^{2}\left(\frac{\widetilde{\beta}}{2}\right)-\cos\widetilde{\beta}\cos^{2}\left(\frac{\alpha}{2}\right) + \frac{1}{2}\sin\widetilde{\beta}\sin\alpha\cos\left(\omega t+\beta+\widetilde{\gamma}\right) e^{-\left(\hbar\omega\right)^{2}\gamma\left(t\right)}. \end{array}\label{eq:tomogram-2-QND}$$ We can check the validity of the tomogram obtained by verifying that $\sum\omega_{i}=\begin{array}{lcl} \omega_{1}+\omega_{2} & = & 1.\end{array}$ Interestingly, we can see that the knowledge of one of the components of the tomogram is enough to reconstruct the whole state. Keeping this in mind, we have only shown the variation of $\omega_{1}$ in Fig. \[fig:qnd-single\] as $\omega_{2}=1-\omega_{1}$. In Fig. \[fig:qnd-single\], we can easily see the expected behavior of tomogram with increase in temperature for zero bath squeezing. Specifically, with increase in temperature, the tomogram tends to randomize more quickly towards probability $1/2$. Fig. \[fig:qnd-single-3D\] further establishes the effect of the environment on the tomogram. Particularly, Fig. \[fig:qnd-single-3D\] b brings out the oscillatory nature of tomogram with time while temperature tends to randomize it. Similarly, Figs. \[fig:qnd-single-3D\] a and c show the dependence of the tomogram on Euler angles and the atomic coherent state parameters, respectively. ![\[fig:qnd-single\](Color online) The variation of the tomogram with time ($t$) for single spin-$\frac{1}{2}$ atomic coherent state in the presence of QND noise with bath parameters $\gamma_{0}=0.1,\,\omega_{c}=100,$ squeezing parameters $r=0,\,a=0,$ and $\omega=1.0$ and $\alpha=\frac{\pi}{2},\,\beta=\frac{\pi}{3},\,\widetilde{\beta}=\frac{\pi}{3},\,\widetilde{\gamma}=\frac{\pi}{4},$ in the units of $\hbar=k_{B}=1$. The smooth (blue) line, dashed (red) line and dot-dashed (magenta) line correspond to the tomogram with time for different temperatures $T=0,\,1$ and $2$, respectively.](figure1) [Dissipative SGAD channel]{} ---------------------------- Master equation for the dissipative evolution of a given state in the squeezed generalized amplitude damping (SGAD) channel is given by [@SGAD] $$\begin{array}{lcl} \frac{d}{dt}\rho^{s}\left(t\right) & = & -\frac{i\omega}{2}\left[\sigma_{z},\rho^{s}\left(t\right)\right]+\gamma_{0}\left(N+1\right) \left\{ \sigma_{-}\rho^{s}\left(t\right)\sigma_{+}-\frac{1}{2}\sigma_{+}\sigma_{-}\rho^{s}\left(t\right)-\frac{1}{2}\rho^{s}\left(t\right)\sigma_{+}\sigma_{-}\right\} \\ & + & \left\{ \sigma_{+}\rho^{s}\left(t\right)\sigma_{-}-\frac{1}{2}\sigma_{-}\sigma_{+}\rho^{s}\left(t\right)-\frac{1}{2}\rho^{s}\left(t\right)\sigma_{-}\sigma_{+}\right\} \gamma_{0}N-\gamma_{0}M\sigma_{+}\rho^{s}\left(t\right)\sigma_{+}-\gamma_{0}M^{*}\sigma_{-}\rho^{s}\left(0\right)\sigma_{-}. \end{array}\label{eq:master_eq-SGAD}$$ The density matrix for a quantum state under a dissipative SGAD channel at time $t$ can be obtained, from the above equation, as $$\begin{array}{lcl} \rho^{s}\left(t\right) & = & \frac{1}{4}\rho^{s}\left(0\right)f_{+}+\frac{1}{4}\sigma_{z}\rho^{s}\left(0\right)\sigma_{z}f_{-}-\frac{1}{4}\rho^{s}\left(0\right)\sigma_{z}g_{-} - \frac{1}{4}\sigma_{z}\rho^{s}\left(0\right)g_{+}-\gamma_{0}\frac{\sinh\left(\alpha^{\prime}t\right)}{\alpha^{\prime}}e^{-\frac{\gamma^{\beta}t}{2}}\\ & \times & \left\{ M\sigma_{+}\rho^{s}\left(0\right)\sigma_{+}+M^{*}\sigma_{-}\rho^{s}\left(0\right)\sigma_{-}\right\} + \left(1-e^{-\gamma^{\beta}t}\right)\left\{ \frac{\gamma_{+}}{\gamma^{\beta}}\sigma_{-}\rho^{s}\left(0\right)\sigma_{+}+\frac{\gamma_{-}}{\gamma^{\beta}}\sigma_{+}\rho^{s}\left(0\right)\sigma_{-}\right\}, \end{array}\label{eq:density-matrix-SGAD}$$ where $f_{\pm}=\left\{ 1+e^{-\gamma^{\beta}t} \pm 2\cosh\left(\alpha^{\prime}t\right)e^{-\frac{\gamma^{\beta}t}{2}} \right\},$ $g_{\pm}=\left\{ \frac{\gamma}{\gamma^{\beta}}\left(1-e^{-\gamma^{\beta}t} \right) \pm \frac{2i\omega}{\alpha^{\prime}}\sinh\left(\alpha^{\prime}t\right)e^{-\frac{\gamma^{\beta}t}{2}} \right\},$ $\gamma_{+}=\gamma_{0}\left(N+1\right)$, $\gamma_{-}=\gamma_{0}N$, $\gamma^{\beta}=\gamma_{+}+\gamma_{-}$, $\gamma=\gamma_{+}-\gamma_{-}=\gamma_{0}$, $\alpha^{\prime}=\sqrt{\gamma_{0}^{2}\left|M\right|^{2}-\omega^{2}}$; and $$\begin{array}{lcl} \sigma_{+} & = & |1\rangle\langle0|,\,\,\sigma_{-}=|0\rangle\langle1|,\\ \sigma_{z} & = & \sigma_{+}\sigma_{-}-\sigma_{-}\sigma_{+}\\ & = & |1\rangle\langle1|-|0\rangle\langle0|\\ & = & |e\rangle\langle e|-|g\rangle\langle g|. \end{array}$$ Also, $$\begin{array}{lclccc} \sigma_{z}|g\rangle & = & -|g\rangle, & \sigma_{z}|e\rangle & = & |e\rangle;\\ \sigma_{+}|g\rangle & = & |e\rangle, & \sigma_{+}|e\rangle & = & 0;\\ \sigma_{-}|g\rangle & = & 0, & \sigma_{-}|e\rangle & = & |g\rangle. \end{array}$$ Here, $\gamma_{0}$ is the spontaneous emission rate, $M=-\frac{1}{2}\left\{ 2N_{th}+1\right\} \exp\left(i\phi\right)\sinh\left(2r\right)$, and $N=N_{th}\left\{ \cosh^{2}\left(r\right)+\sinh^{2}\left(r\right)\right\} +\sinh^{2}\left(r\right),$ where $N_{th}=1/\left\{ \exp\left(\hbar\omega/k_{B}T\right)-1\right\} $ being the Planck distribution, and $r$ and the bath squeezing angle ($\phi$) are the bath squeezing parameters. The initial state, as for the tomogram of a quantum state under QND evolution, is the atomic coherent state given in Eq. (\[eq:initial-state-QND\]). Using Eq. (\[eq:density-matrix-SGAD\]), the density matrix can be written as $$\rho^{s}\left(t\right)=\left[\begin{array}{cc} \langle\frac{1}{2}|\rho^{s}\left(t\right)|\frac{1}{2}\rangle & \langle\frac{1}{2}|\rho^{s}\left(t\right)|-\frac{1}{2}\rangle\\ \langle-\frac{1}{2}|\rho^{s}\left(t\right)|\frac{1}{2}\rangle & \langle-\frac{1}{2}|\rho^{s}\left(t\right)|-\frac{1}{2}\rangle \end{array}\right],\label{eq:final_density-matrix-SGAD}$$ where the various terms are $$\begin{array}{lcl} \langle\frac{1}{2}|\rho^{s}\left(t\right)|\frac{1}{2}\rangle & = & \sin^{2}\left(\frac{\alpha}{2}\right)e^{-\gamma^{\beta}t}+\frac{\gamma_{-}}{\gamma^{\beta}}\left(1-e^{-\gamma^{\beta}t}\right),\\ \langle\frac{1}{2}|\rho^{s}\left(t\right)|-\frac{1}{2}\rangle & = & \frac{1}{2}\sin\alpha\left[\left\{ \cosh\left(\alpha^{\prime}t\right)-\frac{i\omega}{\alpha^{\prime}}\sinh\left(\alpha^{\prime}t\right)\right\} e^{-i\beta}-\frac{\gamma_{0}M}{\alpha^{\prime}}\sinh\left(\alpha^{\prime}t\right)e^{i\beta}\right]e^{-\frac{\gamma^{\beta}t}{2}},\\ \langle-\frac{1}{2}|\rho^{s}\left(t\right)|\frac{1}{2}\rangle & = & \frac{1}{2}\sin\alpha\left[\left\{ \cosh\left(\alpha^{\prime}t\right)+\frac{i\omega}{\alpha^{\prime}}\sinh\left(\alpha^{\prime}t\right)\right\} e^{i\beta}-\frac{\gamma_{0}M^{*}}{\alpha^{\prime}}\sinh\left(\alpha^{\prime}t\right)e^{-i\beta}\right]e^{-\frac{\gamma^{\beta}t}{2}},\\ \langle-\frac{1}{2}|\rho^{s}\left(t\right)|-\frac{1}{2}\rangle & = & \cos^{2}\left(\frac{\alpha}{2}\right)e^{-\gamma^{\beta}t}+\frac{\gamma_{+}}{\gamma^{\beta}}\left(1-e^{-\gamma^{\beta}t}\right), \end{array}$$ and the density matrix can be seen to be normalized as $\stackrel[m=-1/2]{1/2}{\sum}\langle m|\rho^{s}\left(t\right)|m\rangle=1.$ The tomogram of a state evolving in a dissipative SGAD channel, in analogy to the QND case, can be obtained by expanding the density matrix in the basis of the Wigner-Dicke states, as in Eq. (\[eq:wigner\_dicke\_state\]). Using Eq. (\[eq:tomogram\]), the first two relations of Eq. (\[eq:values-of-Ds\]) and Eq. (\[eq:final\_density-matrix-SGAD\]), the first component of the tomogram is $$\begin{array}{lcl} \omega\left(\frac{1}{2},\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right)\equiv\omega_{1} & = & \sin^{2}\left(\frac{\widetilde{\beta}}{2}\right)\left\{ \cos^{2}\left(\frac{\alpha}{2}\right)e^{-\gamma^{\beta}t}+\frac{\gamma_{+}}{\gamma^{\beta}}\left(1-e^{-\gamma^{\beta}t}\right)\right\} +\cos^{2}\left(\frac{\widetilde{\beta}}{2}\right)\left\{ \sin^{2}\left(\frac{\alpha}{2}\right)e^{-\gamma^{\beta}t}+\frac{\gamma_{-}}{\gamma^{\beta}}\left(1-e^{-\gamma^{\beta}t}\right)\right\} \\ & - & \frac{1}{2}\sin\widetilde{\beta}\left\{ e^{-i\widetilde{\gamma}}\left[\frac{1}{2}\sin\alpha e^{-i\beta}e^{-\frac{\gamma^{\beta}t}{2}}\left\{ \cosh\left(\alpha^{\prime}t\right)-\frac{i\omega}{\alpha^{\prime}}\sinh\left(\alpha^{\prime}t\right)\right\} -\frac{\gamma_{0}M}{2\alpha^{\prime}}\sin\alpha\sinh\left(\alpha^{\prime}t\right)e^{i\beta}e^{-\frac{\gamma^{\beta}t}{2}}\right]\right.\\ & + & \left.{\rm c.c.}\right\} . \end{array}\label{eq:tomogram1-SGAD}$$ Again, we can check the validity of the analytic expression of the tomogram in the absence of open system effects, i.e., by considering $\gamma_{0}=\gamma=0$, $\gamma_{+}=\gamma_{-}=0=\gamma^{\beta}$, which leads to $\alpha^{\prime}=i\omega$, we have $$\begin{array}{lcl} \omega\left(\frac{1}{2},\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right) & = & \cos^{2}\left(\frac{\widetilde{\beta}}{2}\right)-\cos\widetilde{\beta}\cos^{2}\left(\frac{\alpha}{2}\right) - \frac{1}{2}\sin\widetilde{\beta}\sin\alpha\cos\left(\omega t+\beta+\widetilde{\gamma}\right), \end{array}\label{eq:check-tomogram1-SGAD}$$ which is identical to the QND case, i.e., Eq. (\[eq:tomogram1-QND\]), with $\gamma\left(t\right)=0$. Similarly, using Eq. (\[eq:tomogram\]), the last two relations of Eq. (\[eq:values-of-Ds\]), and Eq. (\[eq:final\_density-matrix-SGAD\]), we obtain the second component as $$\begin{array}{lcl} \omega\left(-\frac{1}{2},\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right)\equiv\omega_{2} & = & \cos^{2}\left(\frac{\widetilde{\beta}}{2}\right)\left\{ \cos^{2}\left(\frac{\alpha}{2}\right)e^{-\gamma^{\beta}t}+\frac{\gamma_{+}}{\gamma^{\beta}}\left(1-e^{-\gamma^{\beta}t}\right)\right\} +\sin^{2}\left(\frac{\widetilde{\beta}}{2}\right)\left\{ \sin^{2}\left(\frac{\alpha}{2}\right)e^{-\gamma^{\beta}t}+\frac{\gamma_{-}}{\gamma^{\beta}}\left(1-e^{-\gamma^{\beta}t}\right)\right\} \\ & + & \frac{1}{2}\sin\widetilde{\beta}\left\{ e^{-i\widetilde{\gamma}}\left[\frac{1}{2}\sin\alpha e^{-i\beta}e^{-\frac{\gamma^{\beta}t}{2}}\left\{ \cosh\left(\alpha^{\prime}t\right)-\frac{i\omega}{\alpha^{\prime}}\sinh\left(\alpha^{\prime}t\right)\right\} -\frac{\gamma_{0}M}{2\alpha^{\prime}}\sin\alpha\sinh\left(\alpha^{\prime}t\right)e^{i\beta}e^{-\frac{\gamma^{\beta}t}{2}}\right]\right.\\ & + & \left.{\rm c.c.}\right\} . \end{array}\label{eq:tomogram2-SGAD}$$ Similar to the first tomogram of the dissipative SGAD channel, we can check the solution in the absence of the open system effects which leads to $\alpha^{\prime}=i\omega$. This can be seen to be the same as the corresponding QND case, i.e., Eq. (\[eq:tomogram-2-QND\]), with $\gamma\left(t\right)=0$ $$\begin{array}{lcl} \omega\left(-\frac{1}{2},\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right) & = & \cos^{2}\left(\frac{\widetilde{\beta}}{2}\right)-\cos\widetilde{\beta}\sin^{2}\left(\frac{\alpha}{2}\right) + \frac{1}{2}\sin\widetilde{\beta}\sin\alpha\cos\left(\omega t+\beta+\widetilde{\gamma}\right). \end{array}\label{eq:check-tomogram-2-SGAD}$$ We can also check the validity of the tomogram as in the QND case by $\sum\omega_{i}=\omega_{1}+\omega_{2}=1.$ Hence, as before, one component of the tomogram would be enough to recover all the information. This is why, in the plots, we only show the first component of the tomogram. The other component can be easily obtained from it. The variation of tomogram with different parameters is shown in Figs. \[fig:SGAD-ACS\] and \[fig:SGAD-3D-ACS\]. Fig. \[fig:SGAD-ACS\] exhibits the randomization of the tomogram with increase in temperature. This fact can be observed in the smooth (blue) and dashed (red) lines. However, an interesting behavior is observed here with respect to bath squeezing. It can be seen that it takes relatively longer to randomize the tomogram in presence of squeezing than in its absence, temperature remaining same, as illustrated by a comparison between the dot-dashed (magenta) and dashed (red) line. This fact, in turn, establishes that squeezing is a useful quantum resource. This behavior is further elaborated in Fig. \[fig:SGAD-3D-ACS\], where the effect of bath squeezing can be observed and is consistent with the quadrature behavior of squeezing. This beneficial effect of squeezing is not observed for evolution under QND channel. From the present analysis, it could be envisaged that a tomographic connection could be established between the state, under consideration and the generic open system interaction evolving it. ![\[fig:SGAD-ACS\](Color online) The tomogram varying with time ($t$) is shown for a single spin-$\frac{1}{2}$ atomic coherent state in the presence of the SGAD noise for bath squeezing angle $\phi=\pi$ in the units of $\hbar=k_{B}=1$, with $\omega=1.0,\,\gamma_{0}=0.25,$ and $\alpha=\frac{\pi}{2},\,\beta=\frac{\pi}{3},\,\widetilde{\beta}=\frac{\pi}{3},\,\widetilde{\gamma}=\frac{\pi}{4}$. The smooth (blue) line, dashed (red) line and dot-dashed (magenta) line correspond to the tomogram with time for different temperatures and squeezing parameters $T=1,\,10$ and $10$, and $r=0,\,0$ and $1$, respectively.](figure3) ![\[fig:SGAD-3D-ACS\](Color online) The tomogram for single spin-$\frac{1}{2}$ atomic coherent state in the presence of SGAD noise is shown as a function of squeezing parameters $r$ and $\phi$ with $\alpha=\frac{\pi}{2},\,\beta=\frac{\pi}{3},\,\widetilde{\beta}=\frac{\pi}{3},\,\widetilde{\gamma}=\frac{\pi}{4},$ and $\omega=1.0,\,\gamma_{0}=0.25,$ in the units of $\hbar=k_{B}=1$ for $T=1$ at time $t=1$.](figure4) Tomogram of two spin-$\frac{1}{2}$ (qubit) states \[sec:Tomogram-of-two-spin\] ============================================================================== Various two-qubit tomography schemes have been proposed in the recent past [@Manko-1/2; @Manko-2spin; @adam-2qubit-tom; @comment12-ref1; @comment12-ref2]. Specifically, the tomogram for two spin-$\frac{1}{2}$ (qubit) states can be obtained using the star product scheme [@Manko-1/2; @Manko-2spin]. In [@comment12-ref1], two-qubit states were analyzed from the perspective of tomographic causal analysis, while in [@comment12-ref2], an interesting connection between tomographic construction of two-qubit states to aspects of quantum correlations such as discord and measurement induced disturbance was developed. For a two qubit state $\rho$ one can obtain the tomogram as $$\omega\left(m_{1},m_{2}\right)={\rm Tr}\left[\rho\left\{ Q_{1}\left(m_{1}\right)\otimes Q_{2}\left(m_{2}\right)\right\} \right],\label{eq:2qubitTomogram}$$ where $Q_{i}\left(m_{i}\right)=U_{i}^{\dagger}\left|m_{i}\right\rangle \left\langle m_{i}\right|U_{i},$ and $m_{i}=\pm\frac{1}{2},$ while the unitary matrices $U_{i}$ are $$U_{i}=\left[\begin{array}{cc} \cos\frac{\widetilde{\beta}_{i}}{2}\exp\left\{ \frac{i\left(\widetilde{\alpha}_{i}+\widetilde{\gamma}_{i}\right)}{2}\right\} & \sin\frac{\widetilde{\beta}_{i}}{2}\exp\left\{ \frac{i\left(\widetilde{\alpha}_{i}-\widetilde{\gamma}_{i}\right)}{2}\right\} \\ -\sin\frac{\widetilde{\beta}_{i}}{2}\exp\left\{ -\frac{i\left(\widetilde{\alpha}_{i}-\widetilde{\gamma}_{i}\right)}{2}\right\} & \cos\frac{\widetilde{\beta}_{i}}{2}\exp\left\{ -\frac{i\left(\widetilde{\alpha}_{i}+\widetilde{\gamma}_{i}\right)}{2}\right\} \end{array}\right]$$ for $i\in\left\{ 1,2\right\}.$ Hence, the tomogram of the two qubit state can be written as the diagonal elements of $\widetilde{\rho},$ where $\widetilde{\rho}=\left(U_{1}\otimes U_{2}\right)\rho\left(U_{1}\otimes U_{2}\right)^{\dagger}.$ Tomogram of two qubits under dissipative evolution in a vacuum bath ------------------------------------------------------------------- Now, we construct the tomogram of a two qubit state in a vacuum bath under dissipative evolution, as discussed in Ref. [@squ-ther-bath]. The initial state of the system is considered with one qubit in the excited state $\left|e_{1}\right\rangle $ and the other in the ground state $\left|g_{2}\right\rangle $, i.e., $\left|e_{1}\right\rangle \left|g_{2}\right\rangle $. The reduced density matrix of the system of interest, here the two qubits, is $$\begin{array}{lcl} \rho\left(t\right) & = & \left[\begin{array}{cccc} \rho_{ee}\left(t\right) & \rho_{es}\left(t\right) & \rho_{ea}\left(t\right) & \rho_{eg}\left(t\right)\\ \rho_{es}^{*}\left(t\right) & \rho_{ss}\left(t\right) & \rho_{sa}\left(t\right) & \rho_{sg}\left(t\right)\\ \rho_{ea}^{*}\left(t\right) & \rho_{sa}^{*}\left(t\right) & \rho_{aa}\left(t\right) & \rho_{ag}\left(t\right)\\ \rho_{eg}^{*}\left(t\right) & \rho_{sg}^{*}\left(t\right) & \rho_{ag}^{*}\left(t\right) & \rho_{gg}\left(t\right) \end{array}\right],\end{array}\label{eq:densitymatrix-vaccumbath}$$ where the analytic form of all the elements of the density matrix is given in Appendix 1. The tomogram can be thought of as a tomographic-probability vector $\omega=\left[\omega_{1},\omega_{2},\omega_{3},\omega_{4}\right]^{T}$ (here $T$ corresponds to transpose of the vector), where each component can be expressed analytically as $$\begin{array}{lcl} \omega_{1}\left(t\right) & = & \frac{1}{4}\left[4\rho_{ee}\cos^{2}\frac{\widetilde{\beta}_{1}}{2}\cos^{2}\frac{\widetilde{\beta}_{2}}{2}+4\rho_{gg}\sin^{2}\frac{\widetilde{\beta}_{1}}{2}\sin^{2}\frac{\widetilde{\beta}_{2}}{2}\right. + \left(\rho_{aa}+\rho_{ss}\right)\left(1-\cos\widetilde{\beta}_{1}\cos\widetilde{\beta}_{2}\right)\\ & - & \left(\rho_{aa}-\rho_{ss}\right)\sin\widetilde{\beta}_{1}\sin\widetilde{\beta}_{2}\cos\left(\widetilde{\gamma}_{1}-\widetilde{\gamma}_{2}\right) + \left\{ \rho_{sa}\left(\cos\widetilde{\beta}_{1}-\cos\widetilde{\beta}_{2}-i\sin\widetilde{\beta}_{1}\sin\widetilde{\beta}_{2}\right.\sin\left(\widetilde{\gamma}_{1}-\widetilde{\gamma}_{2}\right)\right)\\ & + & \sqrt{2}\left[\left(\left(-\rho_{ea}+\rho_{es}\right)\cos^{2}\frac{\widetilde{\beta}_{2}}{2}+\left(\rho_{ag}+\rho_{sg}\right)\sin^{2}\frac{\widetilde{\beta}_{2}}{2}\right)\right.\sin\widetilde{\beta}_{1}\exp\left(i\widetilde{\gamma}_{1}\right)+\sin\widetilde{\beta}_{2}\exp\left(i\widetilde{\gamma}_{2}\right)\\ & \times & \left.\left(\left(\rho_{ea}+\rho_{es}\right)\cos^{2}\frac{\widetilde{\beta}_{1}}{2}-\left(\rho_{ag}-\rho_{sg}\right)\sin^{2}\frac{\widetilde{\beta}_{1}}{2}\right)\right]+ \left.\left.\exp\left(i\widetilde{\gamma}_{1}+i\widetilde{\gamma}_{2}\right)\rho_{eg}\sin\widetilde{\beta}_{1}\sin\widetilde{\beta}_{2}+{\rm c.c.}\right\} \right], \end{array}\label{eq:2qubit-tomo1}$$ $$\begin{array}{lcl} \omega_{2}\left(t\right) & = & \frac{1}{4}\left[4\rho_{ee}\cos^{2}\frac{\widetilde{\beta}_{1}}{2}\sin^{2}\frac{\widetilde{\beta}_{2}}{2}+4\rho_{gg}\sin^{2}\frac{\widetilde{\beta}_{1}}{2}\cos^{2}\frac{\widetilde{\beta}_{2}}{2}\right. + \left(\rho_{aa}+\rho_{ss}\right)\left(1+\cos\widetilde{\beta}_{1}\cos\widetilde{\beta}_{2}\right)\\ & + & \left(\rho_{aa}-\rho_{ss}\right)\sin\widetilde{\beta}_{1}\sin\widetilde{\beta}_{2}\cos\left(\widetilde{\gamma}_{1}-\widetilde{\gamma}_{2}\right) +\left\{ \rho_{sa}\left(\cos\widetilde{\beta}_{1}+\cos\widetilde{\beta}_{2}+i\sin\widetilde{\beta}_{1}\sin\widetilde{\beta}_{2}\right.\sin\left(\widetilde{\gamma}_{1}-\widetilde{\gamma}_{2}\right)\right)\\ & + & \sqrt{2}\left[\left(\left(\rho_{ag}+\rho_{sg}\right)\cos^{2}\frac{\widetilde{\beta}_{2}}{2}-\left(\rho_{ea}-\rho_{es}\right)\sin^{2}\frac{\widetilde{\beta}_{2}}{2}\right)\sin\widetilde{\beta}_{1}\exp\left(i\widetilde{\gamma}_{1}\right)\right.\\ & + & \sin\widetilde{\beta}_{2}\exp\left(i\widetilde{\gamma}_{2}\right)\left.\left(-\left(\rho_{ea}+\rho_{es}\right)\cos^{2}\frac{\widetilde{\beta}_{1}}{2}+\left(\rho_{ag}-\rho_{sg}\right)\sin^{2}\frac{\widetilde{\beta}_{1}}{2}\right)\right] - \left.\left.\exp\left(i\widetilde{\gamma}_{1}+i\widetilde{\gamma}_{2}\right)\rho_{eg}\sin\widetilde{\beta}_{1}\sin\widetilde{\beta}_{2}+{\rm c.c.}\right\} \right], \end{array}\label{eq:2qubit-tomo2}$$ $$\begin{array}{lcl} \omega_{3}\left(t\right) & = & \frac{1}{4}\left[4\rho_{ee}\sin^{2}\frac{\widetilde{\beta}_{1}}{2}\cos^{2}\frac{\widetilde{\beta}_{2}}{2}+4\rho_{gg}\cos^{2}\frac{\widetilde{\beta}_{1}}{2}\sin^{2}\frac{\widetilde{\beta}_{2}}{2}\right. + \left(\rho_{aa}+\rho_{ss}\right)\left(1+\cos\widetilde{\beta}_{1}\cos\widetilde{\beta}_{2}\right)\\ & + & \left(\rho_{aa}-\rho_{ss}\right)\sin\widetilde{\beta}_{1}\sin\widetilde{\beta}_{2}\cos\left(\widetilde{\gamma}_{1}-\widetilde{\gamma}_{2}\right) + \left\{ -\rho_{sa}\left(\cos\widetilde{\beta}_{1}+\cos\widetilde{\beta}_{2}-i\sin\widetilde{\beta}_{1}\sin\widetilde{\beta}_{2}\right.\sin\left(\widetilde{\gamma}_{1}-\widetilde{\gamma}_{2}\right)\right)\\ & + & \sqrt{2}\left[\left(-\left(\rho_{ag}+\rho_{sg}\right)\sin^{2}\frac{\widetilde{\beta}_{2}}{2}+\left(\rho_{ea}-\rho_{es}\right)\cos^{2}\frac{\widetilde{\beta}_{2}}{2}\right)\sin\widetilde{\beta}_{1}\exp\left(i\widetilde{\gamma}_{1}\right)\right.\\ & + & \sin\widetilde{\beta}_{2}\exp\left(i\widetilde{\gamma}_{2}\right) \left.\left(\left(\rho_{ea}+\rho_{es}\right)\sin^{2}\frac{\widetilde{\beta}_{1}}{2}-\left(\rho_{ag}-\rho_{sg}\right)\cos^{2}\frac{\widetilde{\beta}_{1}}{2}\right)\right] - \left.\left.\exp\left(i\widetilde{\gamma}_{1}+i\widetilde{\gamma}_{2}\right)\rho_{eg}\sin\widetilde{\beta}_{1}\sin\widetilde{\beta}_{2}+{\rm c.c.}\right\} \right], \end{array}\label{eq:2qubit-tomo3}$$ and $$\begin{array}{lcl} \omega_{4}\left(t\right) & = & \frac{1}{4}\left[4\rho_{ee}\sin^{2}\frac{\widetilde{\beta}_{1}}{2}\sin^{2}\frac{\widetilde{\beta}_{2}}{2}+4\rho_{gg}\cos^{2}\frac{\widetilde{\beta}_{1}}{2}\cos^{2}\frac{\widetilde{\beta}_{2}}{2}\right. + \left(\rho_{aa}+\rho_{ss}\right)\left(1-\cos\widetilde{\beta}_{1}\cos\widetilde{\beta}_{2}\right)\\ & - & \left(\rho_{aa}-\rho_{ss}\right)\sin\widetilde{\beta}_{1}\sin\widetilde{\beta}_{2}\cos\left(\widetilde{\gamma}_{1}-\widetilde{\gamma}_{2}\right) + \left\{ -\rho_{sa}\left(\cos\widetilde{\beta}_{1}-\cos\widetilde{\beta}_{2}+i\sin\widetilde{\beta}_{1}\sin\widetilde{\beta}_{2}\right.\sin\left(\widetilde{\gamma}_{1}-\widetilde{\gamma}_{2}\right)\right)\\ & + & \sqrt{2}\left[\left(-\left(\rho_{ag}+\rho_{sg}\right)\cos^{2}\frac{\widetilde{\beta}_{2}}{2}+\left(\rho_{ea}-\rho_{es}\right)\sin^{2}\frac{\widetilde{\beta}_{2}}{2}\right)\sin\widetilde{\beta}_{1}\exp\left(i\widetilde{\gamma}_{1}\right)\right.\\ & + & \sin\widetilde{\beta}_{2}\exp\left(i\widetilde{\gamma}_{2}\right)\left.\left(-\left(\rho_{ea}+\rho_{es}\right)\sin^{2}\frac{\widetilde{\beta}_{1}}{2}+\left(\rho_{ag}-\rho_{sg}\right)\cos^{2}\frac{\widetilde{\beta}_{1}}{2}\right)\right] - \left.\left.\exp\left(i\widetilde{\gamma}_{1}+i\widetilde{\gamma}_{2}\right)\rho_{eg}\sin\widetilde{\beta}_{1}\sin\widetilde{\beta}_{2}+{\rm c.c.}\right\} \right]. \end{array}\label{eq:2qubit-tomo4}$$ Here, $\rho_{ij}$ are the elements of the matrix in Eq. (\[eq:densitymatrix-vaccumbath\]) and are given in Appendix 1. For simplicity of notations the time dependence in the arguments of matrix elements is omitted. Similar to the tomograms for single spin-$\frac{1}{2}$ states the tomogram obtained here is also free from $\widetilde{\alpha}.$ As in the cases of single qubit tomograms, we can again verify that the tomogram obtained here satisfies the condition $\sum\omega_{i}=\rho_{ee}+\rho_{gg}+\rho_{aa}+\rho_{ss},$ which is the trace of the density matrix given in Eq. (\[eq:densitymatrix-vaccumbath\]), and hence equal to one. For the case of identical qubits considered here, we take the wave-vector and mean frequency to be $k_{0}=\omega_{0}=1$, the spontaneous emission rate $\Gamma_{j}=0.05$ and $\hat{\mu}\cdot\hat{r}_{ij}=0$. Here, $\hat{\mu}$ is the unit vector along the atomic transition dipole moment and $\hat{r}_{ij}$ is the inter-atomic distance. Further, the initial state of the system is taken to be $\rho_{ee}\left(0\right)=\rho_{gg}\left(0\right)=\rho_{es}\left(0\right)=\rho_{ea}\left(0\right)=\rho_{eg}\left(0\right)=\rho_{sg}\left(0\right)= \rho_{ag}\left(0\right)=0$, and $\rho_{ss}\left(0\right)=\rho_{aa}\left(0\right)=\rho_{sa}\left(0\right)=0.5$. The variation of all four components of the tomogram is shown with different parameters in Figs. \[fig:Vacuum-bath\] and \[fig:Vacuum-bath-r\]. In Fig. \[fig:Vacuum-bath\], large oscillations can be observed for small interqubit spacing, which is consistent with the earlier observations in a plethora of scenario [@our-qd-paper; @squ-ther-bath; @GP]. Fig. \[fig:Vacuum-bath-r\] further demonstrates similar behavior for small interqubit spacing. For small interqubit spacing the ambient environment opens up a channel between the qubits resulting in enhancement of oscillations. Tomogram of single spin-1 state \[sec:Tomogram-of-spin-1\] ========================================================== The tomograms for finite spin states have been considered, among others, in Refs. [@manko-spin-tom; @spin-j-tom; @manko-spin; @comment6-ref1]. In continuation with the theme of this work, we take up an arbitrary spin-1 state $$\begin{array}{lcl} \psi^{\left(1\right)} & = & N\left[\begin{array}{c} a\\ b\\ c \end{array}\right]\end{array},\label{eq:spin-1_state}$$ where $N=\frac{1}{\sqrt{\left|a\right|^{2}+\left|b\right|^{2}+\left|c\right|^{2}}}$ is the normalization factor. The corresponding density matrix is $$\begin{array}{lcl} \rho^{\left(1\right)} & = & \left|N\right|^{2}\left[\begin{array}{ccc} \left|a\right|^{2} & ab^{*} & ac^{*}\\ a^{*}b & \left|b\right|^{2} & bc^{*}\\ a^{*}c & b^{*}c & \left|c\right|^{2} \end{array}\right]\end{array}.\label{eq:density_mat-spin-1}$$ Here, we restrict ourselves to obtaining the tomogram for the state (\[eq:density\_mat-spin-1\]), without considering open system effects. Using Eqs. (\[eq:tomogram\])-(\[eq:D-function-d\]), all the Wigner $D$-functions for the tomogram can be calculated as before. Using Eqs. (\[eq:tomogram\]) and (\[eq:density\_mat-spin-1\]), $\omega\left(1,\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right)$ can be written as $$\begin{array}{lcl} \omega\left(1,\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right)\equiv\omega_{1} & = & \left|N\right|^{2}\left[\left\{ \frac{\left|a\right|^{2}}{4}\left(1+\cos\widetilde{\beta}\right)^{2}+\frac{\left|b\right|^{2}}{2}\sin^{2}\widetilde{\beta}+\frac{\left|c\right|^{2}}{4}\left(1-\cos\widetilde{\beta}\right)^{2}\right\} \right.\\ & + & \left.\left\{ \left(-\frac{ab^{*}e^{i\widetilde{\gamma}}}{2\sqrt{2}}\right)\sin\widetilde{\beta}\left(1+\cos\widetilde{\beta}\right)+\frac{ac^{*}e^{2i\widetilde{\gamma}}}{4}\sin^{2}\widetilde{\beta}-\frac{bc^{*}e^{i\widetilde{\gamma}}}{2\sqrt{2}}\sin\widetilde{\beta}\left(1-\cos\widetilde{\beta}\right)+{\rm c.c.}\right\} \right]. \end{array}\label{eq:spin-1_tomgram1}$$ Similarly, $\omega\left(0,\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right)$ can be obtained as $$\begin{array}{lcl} \omega\left(0,\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right) & \equiv\omega_{0}= & \left|N\right|^{2}\left[\left\{ \frac{\left|a\right|^{2}}{2}\sin^{2}\widetilde{\beta}+\left|b\right|^{2}\cos^{2}\widetilde{\beta}+\frac{\left|c\right|^{2}}{2}\sin^{2}\widetilde{\beta}\right\} \right.\\ & + & \left.\left\{ \frac{ab^{*}e^{i\widetilde{\gamma}}}{2\sqrt{2}}\sin2\widetilde{\beta}-\frac{ac^{*}e^{2i\widetilde{\gamma}}}{2}\sin^{2}\widetilde{\beta}-\frac{bc^{*}e^{i\widetilde{\gamma}}}{2\sqrt{2}}\sin2\widetilde{\beta}+{\rm c.c.}\right\} \right]. \end{array}\label{eq:spin-1_tomogram2}$$ Also, $\omega\left(-1,\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right)$ is $$\begin{array}{lcl} \omega\left(-1,\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right)\equiv\omega_{-1} & = & \left|N\right|^{2}\left[\left\{ \frac{\left|a\right|^{2}}{4}\left(1-\cos\widetilde{\beta}\right)^{2}+\frac{\left|b\right|^{2}}{2}\sin^{2}\widetilde{\beta}+\frac{\left|c\right|^{2}}{4}\left(1+\cos\widetilde{\beta}\right)^{2}\right\} \right.\\ & + & \left.\left\{ \frac{ab^{*}e^{i\widetilde{\gamma}}}{2\sqrt{2}}\sin\widetilde{\beta}\left(1-\cos\widetilde{\beta}\right)+\frac{ac^{*}e^{2i\widetilde{\gamma}}}{4}\sin^{2}\widetilde{\beta}+\frac{bc^{*}e^{i\widetilde{\gamma}}}{2\sqrt{2}}\sin\widetilde{\beta}\left(1+\cos\widetilde{\beta}\right)+{\rm c.c.}\right\} \right]. \end{array}\label{eq:spin-1_tomogram3}$$ Interestingly, the tomogram obtained for a general spin-1 quantum state is also free from $\widetilde{\alpha}$ as for spin-$\frac{1}{2}$ cases discussed above. Further, it can be checked here that the tomogram satisfies the condition $\omega_{1}+\omega_{0}+\begin{array}{lcl} \omega_{-1} & = & 1.\end{array}$ For $a=1,b=0=c$, the tomogram, obtained here, is seen to be consistent with the results reported earlier [@manko-spin], $$\begin{array}{lcl} \omega\left(1,\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right) & = & \frac{\left(1+\cos\widetilde{\beta}\right)^{2}}{4},\\ \omega\left(0,\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right) & = & \frac{\left(1-\cos^{2}\widetilde{\beta}\right)}{2},\\ \omega\left(-1,\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\right) & = & \frac{\left(1-\cos\widetilde{\beta}\right)^{2}}{4}. \end{array}\label{eq:Manko's-spin-1}$$ The variation of all three components of the tomogram with Euler angles is given in Fig. \[fig:spin1-3D\]. The peaks in one component have corresponding valleys in other components of the tomogram, which are manifestations of normalization of the tomogram to one. Tomogram of a finite dimensional state \[sec:Tomogram-of-qutrit\] ================================================================= In this section, we discuss tomography of finite dimensional states. The tomogram of a finite dimensional state can be defined as [@fss-tom] $$\omega\left(m,t,q\right)=\stackrel[\chi=0]{d-1}{\sum}W\left(tm-q\chi,qm+t\chi\right),\label{eq:fss-tom}$$ where $W\left(\chi,m\right)$ is the discrete Wigner function with integers $t$ and $q$, depicting complementarity between number $m$ (denoting angular momentum) and phase $\chi$ of finite dimensional states. Here, the discrete Wigner function is expressed as $$W\left(\chi,m\right)=\frac{1}{d}\stackrel[\Theta=0]{d-1}{\sum}\exp\left(\frac{4\pi i}{d}m\Theta\right)\left\langle \chi-\Theta\right|\rho\left|\chi+\Theta\right\rangle .\label{eq:wig-fss}$$ It would be apt here to mention that the phase states $\left|\chi\right\rangle $ are periodic, such that $\left|\chi+d\right\rangle =\left|\chi\right\rangle $. A general $d$ dimensional density matrix can be written in Weyl operator basis as [@gen-bloch-vector] $$\rho=\frac{1}{d}\mathbb{I}+\stackrel[n,m=0]{d-1}{\sum}b_{nm}U_{nm}\label{eq:rho-weyl-op}$$ with $b_{00}=0$ and $U_{nm}=\stackrel[\alpha=0]{d-1}{\sum}\exp\left(\frac{2\pi i}{d}{\alpha}n\right)\left|\alpha\right\rangle \left\langle \alpha+m\right|$. We make use of the periodicity of phase states in computations involving $U_{nm}$. Up to now the treatment is applicable to any generic finite dimensional system. Here, for concreteness, we concentrate on an important finite dimensional system, viz. a qutrit with $d=3$. We study the effect of spontaneous emission (SE) channel [@noisy-qutrit] on the qutrit. SE is a dissipative process which can be modeled by the following Kraus operators $$\begin{array}{lcl} K_{0} & = & \left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & e^{-\frac{\eta_{1}t}{2}} & 0\\ 0 & 0 & e^{-\frac{\eta_{2}t}{2}} \end{array}\right],\end{array}$$ $$\begin{array}{lcl} K_{1} & = & \left[\begin{array}{ccc} 0 & \sqrt{1-e^{-\eta_{1}t}} & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right],\end{array}$$ $$\begin{array}{lcl} K_{2} & = & \left[\begin{array}{ccc} 0 & 0 & \sqrt{1-e^{-\eta_{2}t}}\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right],\end{array}\label{eq:krauss-spon}$$ where $\eta_{1}$ and $\eta_{2}$ are two Einstein coefficients which control the population of the excited states. Thus, we can write the density matrix of an arbitrary three dimensional state at time $t$ evolving under the spontaneous emission channel as $$\rho\left(t\right)=\stackrel[j,k=0]{2}{\sum}A_{jk}\left(t\right)\left|j\right\rangle \left\langle k\right|=\stackrel[i=0]{2}{\sum}K_{i}\rho\left(0\right)K_{i}^{\dagger}.\label{eq:rhot-qutrit}$$ Using this we can obtain $$W\left(\chi,m,t\right)=\frac{1}{3}\stackrel[\Theta=0]{2}{\sum}A_{\chi-\Theta,\chi+\Theta}\left(t\right)\exp\left(\frac{4\pi i}{3}m\Theta\right).\label{eq:wig-qutrit}$$ As mentioned above $\chi-\Theta$ and $\chi+\Theta$ are mod $d$ operations. Here, we have considered an initial density matrix given by Eq. (\[eq:rho-weyl-op\]) with $b_{01}=b_{10}=\frac{1}{4}$ and $b_{12}=b_{21}=\frac{1}{5}$, while the remaining coefficients can be obtained from these values. Hence, the obtained tomogram with $t=0$ and $q=1$ has three components as $$\begin{array}{lcl} \omega\left(0,0,1\right)\equiv\omega_{0} & = & \frac{1}{30}\left[10+7\left(e^{-\frac{\eta_{1}t}{2}}+e^{-\frac{\eta_{2}t}{2}}\right) + e^{-\frac{1}{2}\left(\eta_{1}+\eta_{2}\right)t}\right],\\ \omega\left(1,0,1\right)\equiv\omega_{1} & = & \frac{1}{60}\left[20-\left(e^{-\frac{\eta_{1}t}{2}}+e^{-\frac{\eta_{2}t}{2}}\right)- 13e^{-\frac{1}{2}\left(\eta_{1}+\eta_{2}\right)t}\right],\\ \omega\left(2,0,1\right)\equiv\omega_{2} & = & \frac{1}{60}\left[20-13\left(e^{-\frac{\eta_{1}t}{2}}+e^{-\frac{\eta_{2}t}{2}}\right)+ 11e^{-\frac{1}{2}\left(\eta_{1}+\eta_{2}\right)t}\right]. \end{array}\label{eq:tom-qutrit}$$ It can be easily seen here that the tomogram obtained is normalized as $\stackrel[m=0]{2}{\sum}\omega_{m}=1$ in Eq. (\[eq:tom-qutrit\]). For specific values of parameters of the spontaneous emission channel, i.e., Einstein coefficients, the evolution of tomogram for the qutrit state is shown in Fig. \[fig:qutrit\]. The tomogram shows that the noisy channel tends to randomize all the components of the tomogram to one-third. We have already noted that once a tomogram is obtained for a finite dimensional system, it is possible to transform it to obtain the Wigner function for the system, and vice verse, but in an experiment we obtain tomograms. A lot of work has been devoted to the study of Wigner functions for finite dimensional systems [@adam-wig1; @adam-wig2; @with-adam; @with-jay]. Some efforts have also been made to study tomograms for finite dimensional coherent states [@with-adam; @with-jay; @manko-fss]. However, to the best of our knowledge, no such efforts had yet been made to study evolution of tomograms in noisy environment. ![\[fig:qutrit\](Color online) The variation of different components of tomogram for a qutrit state with time in the presence of a spontaneous emission channel with Einstein coefficients $\eta_{1}=2$ and $\eta_{2}=4$. The smooth (blue), dashed (red) and dot-dashed (magenta) lines correspond to $\omega_{m}$ with $m=0,\,1,$ and 2, respectively.](figure8) Optical tomogram for a dissipative harmonic oscillator \[sec:Optical-tomogram\] =============================================================================== In the end, we come to the tomogram of an infinite dimensional system, the harmonic oscillator. This is typical of a plethora of oscillatory and optical systems [@louis; @perina-book]. In Ref. [@comment13-ref1], the quantum mechanics of the damped harmonic oscillator was examined, from the perspective of a classical description of quantum mechanics [@tombesi96]. Use was made of the generating function method, resulting in the avoidance of the need to evaluate the Wigner function as an intermediary step for obtaining the tomogram. Further, in [@comment13-ref2] the density matrix, state tomogram and Wigner function of a parametric oscillator were studied. Here, we construct the tomogram of the dissipative harmonic oscillator evolving under a Lindbladian evolution, in a phase sensitive reservoir [@op-tom-rho]. It would be pertinent to mention that tomographic reconstruction of Gaussian states evolving under a Markovian evolution has also been considered in [@marzolino1]. The dissipative harmonic oscillator can be described by the Hamiltonian $$H=H_{S}+H_{R}+H_{SR},\label{eq:hamil-opt-tom}$$ where the system Hamiltonian $H_{s}$ of a harmonic oscillator is described as $$H_{S}=\frac{p^{2}}{2m}+\frac{1}{2}m\omega^{2}x^{2},$$ while the reservoir Hamiltonian $H_{R}$ is given by $$H_{R}=\sum_{j}\frac{p_{j}^{2}}{2m_{j}}+\frac{1}{2}m_{j}\omega_{j}^{2}x_{j}^{2},$$ with the system-reservoir interaction Hamiltonian $H_{SR}$ as $$H_{SR}=\sum_{j}c_{j}xx_{j}.$$ Here, the reservoir is modeled as a bath of harmonic oscillators with $c_{j}$ as the coupling constant. The dynamics of the system harmonic oscillator is obtained by tracing over the reservoir degrees of freedom. The optical tomogram from the Wigner function can be obtained using [@op-tom] $$\begin{array}{l} \omega\left(X,\theta\right)=\int W\left(X\cos\theta-p\sin\theta,X\sin\theta+p\cos\theta\right)dp,\end{array}\label{eq:tomogram_from_wigner}$$ where $W\left(x,y,t\right)$ is the Wigner function. Similarly, the corresponding Wigner function can also be reconstructed from the tomogram by inverse Radon transformation. The analytic expression of the tomogram for the system, initially in the coherent state $\left|\beta\right\rangle,$ is $$\begin{array}{lcl} \omega(X,\theta,t) & = & \sqrt{\frac{2}{\pi}}\frac{1}{\sqrt{\left(2NM+1\right)-\left(rMe^{-2i\theta}+{\rm c.c.}\right)}} \exp\left(-\frac{2\left(Re[\beta e^{i\theta}]e^{-kt}-X\right)^{2}}{\left(2NM+1\right)-\left(rMe^{-2i\theta}+{\rm c.c.}\right)}\right). \end{array}\label{eq:optical_tomogram}$$ Here, $k_{B}$ is the Boltzmann constant and $N=\frac{1}{\exp(\hbar \omega_{k}/k_{B}T) - 1}$ is the average thermal photon number of the environment at temperature $T$. Also, $r$ is the bath squeezing parameter, $Re[u]$ denotes the real part of $u$ and $M=1-\exp\left(-2kt\right),$ where $k$ is the dissipation coefficient, analogous to the spontaneous emission term. ![\[fig:op-tom\](Color online) The tomogram of a dissipative harmonic oscillator varying with time is shown for the initial coherent state parameter $\beta=2$ and $\theta=\frac{\pi}{3}$. The smooth (blue) and dashed (red) lines correspond to the tomogram for $N=5,\,r=1$ for $X=1$ and 2, respectively. Similarly, the dot-dashed (cyan) and dotted (magenta) lines correspond to the tomogram for $N=10,\,r=4$ for $X=1$ and 2, respectively.](figure10) In the corresponding figures for the tomogram with specific values of different parameters, we can observe the decay of the tomogram. Specifically, Fig. \[fig:op-tom-3D\]-a shows the tomogram of an initial coherent state, where we can see a beautiful valley like shape surrounded by a mountain. Interestingly, a similar tomogram has been observed for a binomial state of large dimension (cf. Fig. 2 in [@bin-tom]). However, in Fig. \[fig:op-tom-3D\]-b and c, we can see this sharp structure gradually fade away due to interaction with its surrounding. Thus, with increase in temperature, the effect of decoherence and dissipation, due to the ambient environment, deteriorates the obtained tomogram. Further, Fig. \[fig:op-tom\] illustrates the effect of change of average thermal photon number and squeezing parameter, where in smooth (blue) and dot-dashed (cyan) lines we can observe the enhancement of decay. Similarly, dashed (red) and dotted (magenta) lines also show the effects due to changes in bath parameters for another set of parameters. Conclusion \[sec:Conclusion\] ============================= Tomography is a powerful quantum state reconstruction tool. Its wide applicability in obtaining quasidistribution functions, quantum process tomography and density matrix reconstruction in quantum computation and communication is already established. However, these properties can get effected by the influence of the ambient environment. Here, an effort has been made to study the evolution of tomograms for different quantum systems, both finite and infinite dimensional, under general system-reservoir interactions, using the formalism of open quantum systems. The effect of the environment on the finite dimensional quantum systems, both spin and number-phase states, is to randomize the tomogram. For spin quantum states, single and two spin-$\frac{1}{2}$ states are considered with open quantum system effects. For the number-phase states, a general expression is obtained and is illustrated through the example of a three level quantum (qutrit) system in a spontaneous emission channel. The increase in temperature tends to decohere the tomograms while squeezing is shown to be a useful quantum resource. Besides this, a tomogram for a spin-1 pure quantum state is also obtained. Further, the tomogram for an infinite dimensional system, the ubiquitous dissipative harmonic oscillator, is also studied. The results obtained here are expected to have an impact on issues related to quantum state reconstruction in quantum computation, communication and information processing. Acknowledgments {#acknowledgments .unnumbered} =============== Authors thank anonymous referee for constructive comments and for drawing their attention to a number of extremely relevant papers. Appendix 1 {#appendix-1 .unnumbered} ========== The elements of the density matrix (\[eq:densitymatrix-vaccumbath\]) are $$\begin{array}{lcl} \rho_{ee}\left(t\right) & = & e^{-2\Gamma t}\rho_{ee}\left(0\right),\\ \rho_{ss}\left(t\right) & = & e^{-\left(\Gamma+\Gamma_{12}\right)t}\rho_{ss}\left(0\right) + \frac{\left(\Gamma+\Gamma_{12}\right)}{\left(\Gamma-\Gamma_{12}\right)}\left(1-e^{-\left(\Gamma-\Gamma_{12}\right)t}\right)e^{-\left(\Gamma+\Gamma_{12}\right)t}\rho_{ee}\left(0\right),\\ \rho_{aa}\left(t\right) & = & e^{-\left(\Gamma-\Gamma_{12}\right)t}\rho_{aa}\left(0\right) + \frac{\left(\Gamma-\Gamma_{12}\right)}{\left(\Gamma+\Gamma_{12}\right)}\left(1-e^{-\left(\Gamma+\Gamma_{12}\right)t}\right)e^{-\left(\Gamma-\Gamma_{12}\right)t}\rho_{ee}\left(0\right),\\ \rho_{gg}\left(t\right) & = & \rho_{gg}\left(0\right)+\left(1-e^{-\left(\Gamma+\Gamma_{12}\right)t}\right)\rho_{ss}\left(0\right) + \left(1-e^{-\left(\Gamma-\Gamma_{12}\right)t}\right)\rho_{aa}\left(0\right)\\ & + & \left[\frac{\left(\Gamma+\Gamma_{12}\right)}{2\Gamma}\left\{ 1-\frac{2}{\left(\Gamma-\Gamma_{12}\right)}e^{-\left(\Gamma+\Gamma_{12}\right)t}\right. \left[\frac{\left(\Gamma+\Gamma_{12}\right)}{2}\left(1-e^{-\left(\Gamma-\Gamma_{12}\right)t}\right)+\frac{\left(\Gamma-\Gamma_{12}\right)}{2}\right]\right\} \\ & + & \frac{\left(\Gamma-\Gamma_{12}\right)}{\left(\Gamma+\Gamma_{12}\right)}\left\{ \left(1-e^{-\left(\Gamma-\Gamma_{12}\right)t}\right) - \left.\frac{\left(\Gamma-\Gamma_{12}\right)}{2\Gamma}\left(1-e^{-2\Gamma t}\right)\right\} \right]\rho_{ee}\left(0\right),\\ \rho_{es}\left(t\right) & = & e^{-i\left(\omega_{0}-\Omega_{12}\right)t}e^{-\frac{1}{2}\left(3\Gamma+\Gamma_{12}\right)t}\rho_{es}\left(0\right),\\ \rho_{ea}\left(t\right) & = & e^{-i\left(\omega_{0}+\Omega_{12}\right)t}e^{-\frac{1}{2}\left(3\Gamma-\Gamma_{12}\right)t}\rho_{ea}\left(0\right),\\ \rho_{eg}\left(t\right) & = & e^{-2i\omega_{0}t}e^{-\Gamma t}\rho_{eg}\left(0\right),\\ \rho_{sa}\left(t\right) & = & e^{-2i\Omega_{12}t}e^{-\Gamma t}\rho_{sa}\left(0\right),\\ \rho_{sg}\left(t\right) & = & e^{-i\left(\omega_{0}+\Omega_{12}\right)t}e^{-\frac{1}{2}\left(\Gamma+\Gamma_{12}\right)t} \left[\rho_{sg}\left(0\right)+\frac{\left(\Gamma+\Gamma_{12}\right)}{\left(\Gamma^{2}+4\Omega_{12}^{2}\right)}\left(\left\{ 2\Omega_{12}e^{-\Gamma t}\sin\left(2\Omega_{12}t\right)\right.\right.\right.\\ & + & \Gamma\left.\left(1-e^{-\Gamma t}\cos\left(2\Omega_{12}t\right)\right)\right\} + i\left\{ 2\Omega_{12}\left(1-e^{-\Gamma t}\cos\left(2\Omega_{12}t\right)\right) - \left.\Gamma e^{-\Gamma t}\sin\left(2\Omega_{12}t\right)\right\}\rho_{es}\left(0\right)\right],\\ \rho_{ag}\left(t\right) & = & e^{-i\left(\omega_{0}-\Omega_{12}\right)t}e^{-\frac{1}{2}\left(\Gamma-\Gamma_{12}\right)t}\left[\rho_{ag}\left(0\right)-\frac{\left(\Gamma-\Gamma_{12}\right)}{\left(\Gamma^{2}+4\Omega_{12}^{2}\right)}\right. \left(\left\{ 2\Omega_{12}e^{-\Gamma t}\sin\left(2\Omega_{12}t\right)\right.\right.\\ & + & \left.\Gamma\left(1-e^{-\Gamma t}\cos\left(2\Omega_{12}t\right)\right)\right\} - i\left\{ 2\Omega_{12}\left(1-e^{-\Gamma t}\cos\left(2\Omega_{12}t\right)\right) - \left.\left.\Gamma e^{-\Gamma t}\sin\left(2\Omega_{12}t\right)\right\} \right)\rho_{ea}\left(0\right)\right]. \end{array}\label{eq:matrix-elements}$$ Here, all the matrix elements are written in the dressed state basis, which is connected with the bare state basis by $$\begin{array}{lcl} \left|g\right\rangle & = & \left|g_{1}\right\rangle \left|g_{2}\right\rangle ,\\ \left|s\right\rangle & = & \frac{1}{\sqrt{2}}\left(\left|e_{1}\right\rangle \left|g_{2}\right\rangle +\left|g_{1}\right\rangle \left|e_{2}\right\rangle \right),\\ \left|a\right\rangle & = & \frac{1}{\sqrt{2}}\left(\left|e_{1}\right\rangle \left|g_{2}\right\rangle -\left|g_{1}\right\rangle \left|e_{2}\right\rangle \right),\\ \left|e\right\rangle & = & \left|e_{1}\right\rangle \left|e_{2}\right\rangle . \end{array}$$ Further, $$\begin{array}{lcl} \Omega_{ij} & = & \frac{3}{4}\sqrt{\Gamma_{i}\Gamma_{j}}\left[-\left[1-\left(\hat{\mu}\cdot\hat{r}_{ij}\right)^{2}\right]\frac{\cos\left(k_{0}r_{ij}\right)}{k_{0}r_{ij}} + \left[1-3\left(\hat{\mu}\cdot\hat{r}_{ij}\right)^{2}\right]\left(\frac{\sin\left(k_{0}r_{ij}\right)}{\left(k_{0}r_{ij}\right)^{2}}+\frac{\cos\left(k_{0}r_{ij}\right)}{\left(k_{0}r_{ij}\right)^{3}}\right)\right], \end{array}$$ where $\hat{\mu}=\hat{\mu}_{1}=\hat{\mu}_{2}$ are the unit vectors along the atomic transition dipole moments, $\hat{r}_{ij}=\hat{r}_{i}-\hat{r}_{j}$, and $k_{0}=\frac{\omega_{0}}{c}$ with $\omega_{0}=\frac{\omega_{1}+\omega_{2}}{2}$; the spontaneous emission rate is $$\begin{array}{lcl} \Gamma_{i} & = & \frac{\omega_{i}^{3}\mu_{i}^{2}}{3\pi\epsilon\hbar c^{3}},\end{array}$$ and the collective incoherent effect due to the dissipative multi-qubit interaction with the bath is $$\begin{array}{lcl} \Gamma_{ij} & = & \Gamma_{ji}=\sqrt{\Gamma_{i}\Gamma_{j}}F\left(k_{0}r_{ij}\right),\end{array}$$ for $i\neq j$ with $$\begin{array}{lcl} F\left(k_{0}r_{ij}\right) & = & \frac{3}{2}\left[\left[1-\left(\hat{\mu}\cdot\hat{r}_{ij}\right)^{2}\right]\frac{\sin\left(k_{0}r_{ij}\right)}{k_{0}r_{ij}} + \left[1-3\left(\hat{\mu}\cdot\hat{r}_{ij}\right)^{2}\right]\left(\frac{\cos\left(k_{0}r_{ij}\right)}{\left(k_{0}r_{ij}\right)^{2}}-\frac{\sin\left(k_{0}r_{ij}\right)}{\left(k_{0}r_{ij}\right)^{3}}\right)\right]. \end{array}$$ Further, for the case of identical qubits, as considered here, $\Omega_{12}=\Omega_{21}$, $\Gamma_{12}=\Gamma_{21}$, and $\Gamma_{1}=\Gamma_{2}=\Gamma$. 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--- bibliography: - 'refs.bib' - 'refs\_icml.bib' --- Theory ======
[**** ]{}\ Luca Canale, Axel Laborieux, Agasthya Aroul Mogane, Laetitia Jubin, Jean Comtet, Antoine Lainé, Lydéric Bocquet, Alessandro Siria, Antoine Niguès^\*^.\ Laboratoire de Physique Statistique de l’Ecole Normale Superiéure, UMR CNRS 8550, PSL Research University, 24 Rue Lhomond 75005 Paris, France\ antoine.nigues@lps.ens.fr Abstract {#abstract .unnumbered} ======== Atomic Force Microscopy (AFM) allows to reconstruct the topography of surface with a resolution in the nanometer range. The exceptional resolution attainable with the AFM makes this instrument a key tool in nanoscience and technology. The core of the set-up relies on the detection of the mechanical properties of a micro-oscillator when approached to a sample to image. Despite the fact that AFM is nowadays a very common instrument for research and development applications, thanks to the exceptional performances and the relative simplicity to use it, the fabrication of the micrometric scale mechanical oscillator is still a very complicated and expensive task requiring a dedicated platform. Being able to perform atomic force microscopy with a macroscopic oscillator would make the instrument more versatile and accessible for an even larger spectrum of applications and audiences. We present for the first time atomic force imaging with a centimetric oscillator. We show how it is possible to perform topographical images with nanometric resolution with a grams tuning fork. The images presented here are obtained with an aluminum tuning fork of centimeter size as sensor on which an accelerometer is glued on one prong to measure the oscillation of the resonator. In addition to the stunning sensitivity, by imaging both in air and in liquid, we show the high versatility of such oscillator. The set up proposed here can be extended to numerous experiments where the probe needs to be heavy and/or very complex as well as the environment. Introduction {#introduction .unnumbered} ============ Atomic Force Microscope (AFM) is a powerful instrument to both reconstruct topography at the nanoscale of a sample surface and measure interactions at nanoscale. Since its invention in 1986 by Binning and Rohrer [@Binnig1986], lots of efforts have been dedicated to this instrument to improve its capacities [@Giessibl1995; @Giessibl2003; @Extance2018] and to make it affordable; nowadays AFM is an essential tool for a large spectrum of application ranging from condensed matter and soft matter to biological science [@McGraw2017; @Comtet2016; @Dufrene2017]. In the mostly used configuration, a tiny mechanical oscillator, externally excited at the resonant frequency, is scanned over a surface: the interaction forces between a sharp tip at the apex of the oscillator and the sample induce a change in the mechanical properties of the oscillator itself. Keeping constant the interactions between the tip and the surface during an image allows then to reconstruct the sample topography with a resolution in the nanometer range. While the spatial resolution is only limited by the size of the tip, the ability to detect interaction forces relies totally on the oscillator that is the force probe of AFM. The standard and most common force probe is a cantilever with micro- and sub-micrometer dimensions. The quest to ultimate force sensitivities has pushed the development of alternative kind of probes such as unidimensional objects like nanowires and nanotubes and suspended membranes made of graphen and other 2D materials [@Nigues2014a; @Gloppe2014; @Poncharal1999; @Miao2014; @DeAlba2016]. While the sensitivity is actually being pushed down to impressive values of zepto-Newton [@Chaste2012], these new probes present important constraints due to the challenges in detection and working conditions and it is not possible to easily move them outside laboratory applications [@Nigues2017; @verlot2017]. On the other hand, and somehow in contrast to this, it is important to develop force probes that couple high sensitivities together with versatility: in this work we present a new atomic sensor, named MicroMegascope (MiMes), based on a centimetric harmonic oscillator. The advantages of using macroscopic probes is twofold : first, due to its dimensions, it is possible to change the specificity of the probe at convenience: this allows then to study interactions in a variety of geometries ranging from nanometer size tips up to macroscopic spheres or more complex shapes ; Secondly, due to its mass ($\approx$ 100 g) the coupling with macroscopic devices for position measurements doesn’t affect the mechanical properties of the tuning fork enough to substantially decrease the force detection performances of the set-up. In addition to a study of the force detection performances of the MiMes and to demonstrate the potentiality of this new sensor, we perform in this work images at the nanoscale of a sample in air and totally immersed in a highly viscous liquid. MiMes: experimental set-up and force sensor properties {#mimes-experimental-set-up-and-force-sensor-properties .unnumbered} ====================================================== The MiMes is presented in Figure 1a. The core of the microscope is a centimeter-sized tuning fork made of aluminum. The tuning fork has been designed and realized to reproduce the same geometry and dimension ratio between the different elements as in quartz tuning forks widely used in AFM but with a rescaling factor of 20 [@Karrai1995]. The prong of the tuning fork is $l=7.5cm$ long, $w=6,8mm$ wide and $t=12mm$ thick. The prong oscillations are detected using an accelerometer directly glued at the extremity of one prong. The oscillation amplitude $A$ is directly proportional to the acceleration $a_{acc}$ measured by the accelerometer such that $A=a_{acc}/f_0^{2}$. The tuning fork coupled to the accelerometer alone represents the MiMes. A similar device has been presented by Bosma et al [@Bosma2010], showing that the topography of a coin surface could be reconstructed with a resolution in the micron range. By increasing the mechanical properties of the force sensor and the displacement detection resolution, we can apply the technique to the field of Atomic Force Microscopy. ![image](fig1.jpg){width="\textwidth"} \ The tuning fork and its accelerometer are attached to a XYZ micrometric translation stage for the coarse approach. A piezo-actuator glued to the base of the tuning fork ensures the mechanical excitation. Further, to perform images of sample surfaces, a chemically etched tungsten wire with a radius at the apex of $\approx$ 50 nm is glued at the extremity of one prong . Finally the sample is placed on a three axis piezoscanner with sub-nanometric resolution in displacement (Tritor101 Piezosystemjena).\ The spring constant $k$ of the tuning fork is given by : $$\begin{aligned} k=\frac{Ewt^3}{4l^3}\end{aligned}$$ where $E$ is the Young modulus for Aluminum, $E=69$ GPa leading to $k=480$ kN/m. The resonance frequency of the fundamental mode is given by : $$\begin{aligned} f_0=\frac{\sqrt{k/m_{eff}}}{2\pi}\end{aligned}$$ where $m_{eff}=0.24\ \rho\times t\times w\times l=3,8$ g is the effective mass of that mode, $\rho=2600$ kg/m$^3$ the mass density of Aluminum. We then obtain $f_0=1788Hz$. A finite element study performed with COMSOL enables to test the mechanical parameters and characteristics of the tuning fork. ![image](fig2.jpg){width="\textwidth"} \ In figure 2, we show the mechanical response of the tuning fork around the fundamental resonant frequency. Despite its size, the macroscopic tuning fork is characterized by a low intrinsic dissipation and a large quality factor up to $\approx$ 10000 in air, allowing the detection of the oscillation amplitude of the tuning fork down to its thermal motion and to oscillation amplitudes of the order of 15 pm. It is worth to compare this value to the thermal variance expected for such tuning fork, $\Delta x^2_{th}=k_BT/m_{eff}\omega^{2}_{0}$, with $k_B$ Boltzmann’s constant and $T=300K$ the ambient temperature. We find $\Delta x^2_{th}=(10pm)^2$, in very good agreement with the experimental value measured above.\ At this point it is now important to determine the force sensitivity of our tuning fork. The force sensitivity of an oscillator in a certain bandwidth $B$ is given by [@rugar1997]: $$\begin{aligned} F_{min}=\sqrt{\frac{wt^2}{lQ}}(E\rho)^{(1/4)}(k_BTB)^{(1/2)}\end{aligned}$$ Inserting the parameters of the MiMes in eq. 3 we obtain a minimal force detection of 21 pN$/\sqrt{Hz}$. This minimum achievable force can be improved by a factor of one to two by improving the intrinsic dissipation of our tuning fork, by changing its manufacturing material and/or by working under vacuum conditions, as shown in figure 2a. However the force sensitivity obtained for the chosen configuration is already compatible with near field force measurement and atomic force microscopy.\ The operating principle of a tuning fork as a force sensor is as follows: when excited by an external sinusoidal force $F_{ext}(\omega)=F_{ext}e^{i\omega t}$, the tuning fork behaves in first approximation as a spring-mass system with oscillation amplitude and phase with respect to the excitation given by: $$\begin{aligned} A(\omega)=\frac{F_{ext}}{\sqrt{m^2_{eff}(\omega^{2}_{0}-\omega^{2})^2+\gamma^2\omega^{2}}} \\ \phi(\omega)=\arctan\left(\frac{\gamma\omega}{m_{eff}(\omega^{2}_{0}-\omega^{2})}\right)\end{aligned}$$ with $\gamma$ the damping factor. As the interaction of the oscillator with its environment is modified, one observes a change in both the frequency and the amplitude at resonance. The shift in resonance frequency $\delta f$ is related to the conservative force response, whereas the broadening of the resonance (change of quality factor $Q_0\rightarrow Q_1$) is related to dissipation: $$\begin{aligned} \frac{\partial F}{\partial r}= 2 k \frac{\delta f}{f_{0}} \\ F_{D} = \frac{kA}{\sqrt{3}} \left( \frac{1}{Q_0} -\frac{1}{Q_1} \right)\end{aligned}$$ During the experiments, measurements and controls are performed in Real-time by a complete Specs-Nanonis package (RT5, SC5 and OC4) and two feedback loops enable to work at the resonance and maintain constant the oscillation amplitude $A$ by changing the voltage amplitude applied to the piezo actuator.\ From equations 6 and 7, the ability to detect the interaction forces with a sample is determined by the spring constant of the force sensor. Even if the stiffness of the tuning fork is an order of magnitude larger than the classical quartz tuning fork and above the range of optimal stiffness values for frequency modulation microscopy [@Giessibl2013; @Bosma2010], we will show in the following that this is not a limiting factor for obtaining nanometrically resolved image of the surface of a sample. ![image](fig3.jpg){width="\textwidth"} Results {#results .unnumbered} ======= To demonstrate the potential of MiMes for force microscopy we initially performed a series of approach-retract curves on a Silicon Dioxide flat surface. The interaction between the apex of a sharp tungsten tip glued at the extremity of one prong and the sample surface are detected by measuring the shift in the resonant frequency. In Figure 3, we present the measurement for the fundamental frequency of the normal mode, the first harmonics of the same mode as well as the fundamental frequency of the tangential mode. These measurements prove that the macroscopic force sensor can detect the near field interaction forces demonstrating that the technique can be applied for AFM applications and friction studies.\ The AFM images in Figure 4 have been performed in the so-called FM-AFM. In this mode the substrate is scanned with constant frequency shift, i.e. constant force gradient. The amplitude of vibration $A$ of the tuning fork is kept constant at $10nm$. Figure 4a shows a nanometricaly resolved standard calibration grating with a pitch of $5\mu m$ and depth $180nm$. Notwithstanding the effects inherent in the piezoelectricity of open-loop scanners (creep, hysteresis...), this first image obtained with a centimetric oscillator corresponds in every aspect to the criteria expected with a conventional AFM probe. This irrefutably shows the exceptional sensitivity that our centimetric mechanical oscillator coupled with MEMS detection can achieve. In order to push the nail even deeper and prove the great versatility of MiMes, we proceeded to the imaging of the same surface but completely immersed in a highly viscous liquid, silicone oil (10000 cst). Indeed, it is no longer necessary to specify that AFM imaging in liquid media is still a challenge to this day. When fully immersed, the quality factor of conventional levers decreases drastically, laser detection is deteriorated by beam reflection on the liquid surface and multi-peaks resonances appear making it difficult to distinguish the natural frequency of the lever. Similar problems appear with quartz tuning forks. In our case, our sensor does not interact directly with the liquid, so its properties and sensitivity are not deteriorated. The image shown in Figure 4b does not show any more defects than the one obtained in the air and its realization did not require more technical means than in the air. This completes the demonstration of the versatility, sensitivity and ease of use of the MiMes. ![image](fig4.jpg){width="\textwidth"} Conclusions and discussions {#conclusions-and-discussions .unnumbered} =========================== In conclusion we have demonstrated that a macroscopic mechanical oscillator can be used as force sensor for atomic force microscopy. Despite the size and mass, the macroscopic tuning fork presents the force sensitivity needed to probe near field surface interactions and image surface topography with nanometer resolution, provided a suitable tip is attached at the extremity of one prong. We have performed atomic force measurements and imaging in air and in high viscous liquid showing no remarkable effect of the environment on the image quality. 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--- abstract: 'We analyze the scaling of avalanche precursors in the three dimensional random fuse model by numerical simulations. We find that both the integrated and non-integrated avalanche size distributions are in good agreement with the results of the global load sharing fiber bundle model, which represents the mean-field limit of the model.' address: - 'INFM UdR Roma 1 and SMC, Dipartimento di Fisica, Università “La Sapienza”, P.le A. Moro 2, 00185 Roma, Italy' - 'Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6359, USA' author: - Stefano Zapperi - 'Phani Kumar V.V. Nukala' - Sran Šimunović title: Crack avalanches in the three dimensional random fuse model --- fracture ,random fuse model ,avalanches 46.50.+a ,64.60.Ak Introduction ============ Understanding the scaling properties of fracture in disordered media represents an intriguing theoretical problem with some technological implications [@breakdown]. Of particular interest is the acoustic emission (AE) recorded in a stressed material before failure. The noise is a consequence of micro-cracks forming and propagating in the material and provides an indirect measure of the damage accumulated in the system. For this reason, AE is often used as a non-destructive tool in material testing and evaluation. The distribution of crackle amplitudes follows a power law, suggesting an interpretation in terms of scaling theories. This behavior has been observed in several materials such as wood [@ciliberto], cellular glass [@strauven], concrete [@ae] and paper [@paper]. The statistical properties of fracture in disordered media are captured qualitatively by lattice models, describing the medium as a discrete set of elastic bonds with randomly distributed failure thresholds [@breakdown; @fuse; @duxbury]. In the simplest approximation of a scalar displacement, one recovers the random fuse model (RFM) where a lattice of fuses with random thresholds are subject to an increasing external current [@fuse; @duxbury]. Fracture of the RFM is preceded by avalanches of failure events [@hansen; @zrvs; @alava] which are reminiscent of the acoustic emission activity observed in experiments. The distribution of avalanche sizes (i.e. the number of bonds participating in an avalanche) follows a power law. Initially two dimensional simulations yielded an exponent close to $\tau=5/2$ [@zrvs], the value expected in the fiber bundle model (FBM) [@dfbm; @hansen1]. In that model load is redistributed equally to all the fibers, representing thus a sort of mean-field limit of the RFM [@zrvs]. More recent large scale simulations, however, displayed significant (non-universal) deviations from the mean-field result [@cond-mat]. Only some preliminary results are reported in the literature for three dimensions [@rai-98]. Here we show that avalanches in the three dimensional RFM follow quite closely the mean-field predictions. This is partly expected since normally scaling exponents tend to the mean-field limit as the lattice dimensionality increases, although the exact value for the upper critical dimension is not known for this problem. The random fuse model ===================== In the random thresholds fuse model [@fuse; @duxbury], the lattice is initially fully intact with bonds having the same conductance, but the bond breaking thresholds, $t$, are randomly distributed based on a thresholds probability distribution, $p(t)$. The burning of a fuse occurs irreversibly, whenever the electrical current in the fuse exceeds the breaking threshold current value, $t$, of the fuse. Periodic boundary conditions are imposed in both of the horizontal directions to simulate an infinite system and a constant voltage difference, $V$, is applied between the top and the bottom of the lattice system bus bars. Numerically, a unit voltage difference, $V = 1$, is set between the bus bars and the Kirchhoff equations are solved to determine the current flowing in each of the fuses. Subsequently, for each fuse $j$, the ratio between the current $i_j$ and the breaking threshold $t_j$ is evaluated, and the bond $j_c$ having the largest value, $\mbox{max}_j \frac{i_j}{t_j}$, is irreversibly removed (burnt). The current is redistributed instantaneously after a fuse is burnt implying that the current relaxation in the lattice system is much faster than the breaking of a fuse. Each time a fuse is burnt, it is necessary to re-calculate the current redistribution in the lattice to determine the subsequent breaking of a bond. The process of breaking of a bond, one at a time, is repeated until the lattice system falls apart. In this work, we assume that the bond breaking thresholds are distributed based on a uniform probability distribution, which is constant between 0 and 1. Numerical simulation of fracture using large fuse networks is often hampered due to the high computational cost associated with solving a new large set of linear equations every time a new lattice bond is broken. Although the sparse direct solvers presented in [@nukalajpamg] are superior to iterative solvers in two-dimensional lattice systems, for 3D lattice systems, the memory demands brought about by the amount of fill-in during the sparse Cholesky factorization favor iterative solvers. Hence, iterative solvers are in common use for large scale 3D lattice simulations. The authors have developed an algorithm based on a block-circulant preconditioned conjugate gradient (CG) iterative scheme [@nukalajpamg2] for simulating 3D random fuse networks. The block-circulant preconditioner was shown to be superior compared with the [*optimal*]{} point-circulant preconditioner for simulating 3D random fuse networks [@nukalajpamg2]. Since the block-circulant and [*optimal*]{} point-circulant preconditioners achieve favorable clustering of eigenvalues (in general, the more clustered the eigenvalues are, the faster the convergence rate is), in comparison with the Fourier accelerated iterative schemes used for modeling lattice breakdown [@bat-98], this algorithm significantly reduced the computational time required for solving large lattice systems. Using the algorithm presented in [@nukalajpamg2], we have performed numerical simulations on 3D cube lattice networks. For many lattice system sizes, the number of sample configurations, $N_{config}$, used are extremely large to reduce the statistical error in the numerical results. In particular, we used $N_{config}=40000,3840,512,128,32$ for $L=10,16,24,32,48$ respectively. Avalanches ========== When the current is increased at an infinitesimal rate failure events cluster in the form of avalanches. The typical avalanche size increases with the current up to the last catastrophic failure event. The avalanche size distribution is a power law followed by an exponential cutoff at large sizes. The cutoff size $s_0$ is increasing with the lattice size, so that we can describe the distribution by a scaling form $$P(s,L)=s^{-\tau} g(s/L^D),$$ where $D$ represents the fractal dimension of the avalanches. To confirm this finite size scaling assumption, we perform a data collapse imposing the mean-field exponent $\tau=5/2$ and choosing $D=1.5$ (see Fig. \[fig:1\]). A direct power law fit of the distribution yields instead $\tau=2.55$. We have considered avalanche statistics integrating the distribution over all the values of the current, but the avalanche signal is not stationary: as the current increases so does the avalanche size. In Fig. \[fig:2\] we report the distribution of avalanche sizes sampled at different values of the current $I$. For each sample, we normalize the current by its maximum value $I_c$ and divide the $I^*=I/I_c$ axis in 20 bins. We then compute the avalanche size distribution $p(s,I^*)$ for each bin and average over different realization of the disorder. The distribution follows a law of the type $$p(s,I^*)= s^{-\gamma}\exp(-s/s^*),\label{eq:binsize}$$ with $\gamma \simeq 1.5$ and $s^*$ is an increasing function of $I^*$, in good agreement with mean-field results. Conclusions =========== We have performed numerical simulations of the random fuse model in three dimensions, focusing on the avalanche distributions. The scaling of the distributions is well captured by mean-field theory. This is in contrast with the behavior in two dimensions that shows larger deviations [@cond-mat]. This can be expected on general grounds since typically the mean-field limit is approached as the dimensions are increased. [**Acknowledgment**]{} PKVVN and SS are sponsored by the Mathematical, Information and Computational Sciences Division, Office of Advanced Scientific Computing Research, U.S. Department of Energy under contract number DE-AC05-00OR22725 with UT-Battelle, LLC. H. J. Herrmann and S. Roux (eds.), [ *Statistical Models for the Fracture of Disordered Media*]{}, (North-Holland, Amsterdam, 1990). B. K. Chakrabarti and L. G. Benguigui, [ *Statistical physics of fracture and breakdown in disordered systems*]{} (Oxford Univ. Press, Oxford, 1997). A. Garcimartín, A. Guarino, L. Bellon and S. Ciliberto, Phys. Rev. Lett. [**79**]{}, 3202 (1997); A. Guarino, A. Garcimartín and S. Ciliberto, Eur. Phys. J. 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--- abstract: 'In this contribution we review the large body of work carried out over the past two decades to probe the dark matter in the local universe using redshift survey and peculiar velocity data. While redshift surveys have evolved rapidly over the years, gathering suitable peculiar velocity data and understanding the short-comings of different analyses have proven to be a difficult task. These difficulties have led to conflicting results which have casted some doubts on the usefulness of cosmic flows to constrain cosmological models. Recently, however, a consistent picture seems to be finally emerging with various methods of analyses applied to different data sets yielding concordant results. These favor a low-density universe, with constraints which are in good agreement with those obtained from LSS, high-redshift supernovae and CMB studies.' author: - 'L. da Costa' title: Matter in the Local Universe --- \#1 Introduction ============ LSS studies of the nearby universe are arguably ideal to address the question posed by the title of this conference. Indeed, if all the mass in the universe were locked into galaxies, complete redshift surveys of galaxies would provide the data required to fully characterize the matter distribution. However, we have learned that the luminous matter associated to galaxies represents a small fraction of the mass density of the universe, and that galaxies may be biased relative to the underlying distribution of matter. Still, if structures grow as a result of gravity alone, observation of the peculiar velocity of galaxies provides the means to probe the distribution of the total matter. In the standard picture for the formation of cosmic structures via gravitational instability the peculiar velocity of a galaxy is generated by fluctuations in the mass distribution. For galaxies outside virialized systems, linear perturbation theory predicts $$\vvec (\rvec) \approx {\Omega^{0.6}H_o \over 4\pi} \int{ d^3r^\prime \delta_m {(\rvec^\prime - \rvec) \over \vert \rvec^\prime - \rvec \vert ^3}} \; . \label{lingrav}$$ This can also be expressed in the following differential form $$\nabla \cdot {\bf v} = -\Omega^{0.6} \delta_m, \label{divv}$$ where $\Omega$ is the mass density parameter, $H_o$ is the Hubble constant and $\delta_m$ is the mass density fluctuation field. If galaxies are fair tracers of the underlying mass distribution and galaxy biasing is linear then $\delta_g = b \delta_m$, where $\delta_g$ is the galaxy density contrast and $b$ is the bias parameter for a given population of mass tracers. The above equations show that by mapping the peculiar velocity field one can determine the distribution of mass and measure the parameter $\beta=\Omega^{0.6}/b$ by comparing the reconstructed density field with that observed for galaxies or by comparing the measured velocity field with the predicted gravity field generated by fluctuations of the galaxy density field. These simple ideas have been the underlying motivation for the major wide-angle redshift surveys of optical and infrared galaxies and the Tully-Fisher (TF) and $D_n-\sigma$ redshift-distance surveys conducted over the past two decades. In this contribution we review all of these efforts, giving special emphasis to the results obtained from recently completed redshift-distance surveys. In section \[z\], we briefly mention the redshift surveys that have contributed to our understanding of the local galaxy distribution and those which have played a major role in the analysis of peculiar velocity data. In Section \[zv\], we review the redshift-distance surveys and the peculiar velocity catalogs that have been used to map the peculiar velocity field in the nearby universe. In section \[results\], the most recent surveys are used to reconstruct the velocity and density fields and to measure $\beta$. These results are also compared with those obtained in earlier works. Finally, in Section \[summary\] we briefly summarize the current status of the field. Galaxy Distribution {#z} =================== Over the past two decades the number of redshift surveys and redshift data has greatly increased and a complete review is beyond the scope of the present contribution and can be found elsewhere [@strauss-willick][@dacostampaeso]. Here we point out two classes of surveys that have had a strong bearing on some of the issues discussed here. The first class consists of wide-angle, dense sampling surveys such as the CfA2 [@gellerhuchra] and SSRS2 [@dacostassrs] which revealed for the first time the full complexity of the galaxy distribution. The discovery of extended, coherent wall-like structures and of large regions devoid of luminous matter with scales comparable to the survey depth represented a serious challenge to the prevailing theories of structure formation and evolution. Furthermore, these surveys probed relatively large volumes which allowed for reasonable estimates of the power-spectrum of the galaxy density fluctuations to be made for the first time [@gott][@vogeley][@dacostaps]. Even though unable to reach very large scales, when COBE normalized, comparison with N-body simulations demonstrated that the results were consistent with a low-$\Omega$ cosmological model and an unbiased galaxy distribution. The PS was well described by a shape parameter $\Gamma=0.2$, consistent with other determinations [@peakcockdodds]. Attempts were also made to study the small-scale velocity field by analyzing the redshift-space distortions. However, the small number of independent structures within the sampled volume made the results extremely sensitive to shot-noise [@marzke]. These early surveys were followed by the considerably deeper LCRS [@lcrs] which demonstrated unambiguously that the largest scales of inhomogeneities had finally been reached. Quantitative analyses of the LCRS, by and large confirmed earlier statistical results, albeit with considerably smaller errors. More recently, the first results of the 2dFGRS project have become available. The survey consists of over 100,000 galaxies to a depth comparable to the LCRS, allowing for precise measurements of redshift-space distortions and large-scale power spectrum. Analysis of the redshift distortions caused by large-scale infall velocities yields a value of $\beta_o=0.43$ [@peakcock2df], where $\beta_o$ refers to optical galaxies. Assuming a relative bias $b_o/b_I\sim 1.3$ between optical and galaxies this implies $\beta_I\sim0.56$. The derived galaxy power-spectrum [@percival] was found to be well-represented by a shape parameter $\Gamma=0.2$, in good agreement with previous determinations. These results provide important constraints on the mass power-spectrum which can later be compared with those obtained using cosmic flows to test for consistency. The second class of redshift surveys worth mentioning in the present context is that involving galaxy samples extracted from the [*IRAS*]{} survey such as the 1.9 Jy[@strauss1.9], the 1.2 Jy[@fisher1.2] and the  [@saunderspscz] redshift surveys. While sparsely sampling the galaxy distribution, these surveys provide a sky coverage unmatched by optical surveys. This all-sky coverage allows a more reliable determination of the gravity field induced by fluctuations of the galaxy density field which can be compared to the measured peculiar velocity field to estimate the parameter $\beta$. From the above discussion, it is clear that by themselves redshift surveys are more useful for studying the properties of galaxies than as cosmological probes. However, combining them with redshift independent distances to map out the peculiar velocity field of galaxies and to predict the peculiar velocity field from galaxy density fluctuations provide powerful tools to probe the nature of the matter distribution and its relation with the galaxy distribution. Mapping the Peculiar Velocity Field {#zv} =================================== The radial component of the peculiar velocity is given by $$U = cz - d$$ where $d$ is an estimate of the galaxy distance derived from a secondary distance indicator. Most of the available samples rely on the TF and Fundamental Plane relations for spirals and early-type galaxies, respectively, with typical errors in distance of $\sim 20\%$. However, samples based on distance indicators with significant smaller errors, such as those based on surface brightness fluctuations and nearby Type Ia supernovae, are slowly growing and have already been successfully used to measure $\beta$[@tonry][@riess]. In contrast to the rapid growth of samples with complete redshift information, redshift-distance samples have been difficult to gather. There are various practical reasons for that. First, to ensure the uniformity of the data and of the sky coverage requires coordinated observations in both hemispheres. Second, TF distances require the measurement of the rotational velocity of the galaxy either from the HI line width, which can only be efficiently measured in the northern hemisphere, or from measurements of optical rotation curves, a challenging observation. Third, for early-type galaxies, high signal-to-noise spectra are required for accurate measurements of the velocity dispersion. Finally, both distance indicators require high-quality photometric data. Table \[rd\] summarizes the redshift-distance surveys conducted to date. The table includes only wide-angle redshift-distance surveys and the number of objects is just indicative of the sample size. Not included are the various surveys conducted to measure distances and peculiar velocities of clusters of galaxies which have been used to constrain the amplitude of the bulk flow on very large scales. [lrll]{}\ Survey & $ N_{obj}$ & Type & Coverage\ \ Aaronson & 300 & spirals & all-sky\ Tonry & Davis & 300 & early & north\ 7 Samurai & 400 & early & all-sky\ Willick & 320 & spirals & Perseus-Pisces\ Courteau & 380 & Sb-Sc & north\ Mathewson & 2000+ & spirals & south\ SFI & 1300 & Sbc-Sc & all-sky\ ENEAR & 1600 & early & all-sky\ Shellflow & 300 & Sb-Sc & all-sky\ \ \ Early studies [@tonry-davis][@aaronson] focused on the properties of the flow field near Virgo. However, it was soon realized that the assumption of a spherical infall was too restrictive and that Virgo alone could not explain the motion of the Local Group relative to the CMB. A major contribution to the field was the work of the 7 Samurai[@lynden-bell], the first to probe well beyond the local supercluster, albeit sparsely. Analyses of this sample led to startling results such as the measurement of a large amplitude bulk flow, and the discovery of the Great Attractor, a large mass concentration associated with the Hydra-Centaurus complex. Among the main conclusions of this work was that the large peculiar velocities measured implied large values of $\Omega$, a result that placed cosmic flows at odds with several other analyses. By the end of the 80’s the first attempt to produce a homogeneous catalog by merging different peculiar velocity data set was made (Mark II) and used to obtain the first map of the dark matter in the nearby universe [@potent]. Even though providing an important first glimpse of the dark matter distribution, this early map showed that important regions of the sky were severely undersampled. In subsequent years a major effort was made to expand the available sample to confirm the conclusions of the 7S and to improve the mass maps. These efforts included small surveys of specific areas of the sky [@courteausurvey][@willickdata] and major TF surveys of spirals, such as those carried out by Matthewson and collaborators [@mat1][@mat2] and the SFI survey [@haynes1][@haynes2], and FP surveys of early-type galaxies, such as the recently completed ENEAR survey [@enear] Trying to capitalize as much as possible on all of the available data, Willick and collaborators[@markiii] assembled the data from these different surveys into a catalog (Mark III) consisting of about 3000 galaxies, predominantly spirals, with measured peculiar velocities. The Mark III catalog, which does not include the more recent all-sky SFI and ENEAR surveys, has been extensively used in the analyses of peculiar velocity data. While considerable effort was made to ensure uniformity, it is a compilation of heterogeneous data sets. As illustrated in figure 11 of Kollat  [@kollat], it lacks uniformity in sky coverage due to the uneven coverage of the main data sets included in the compilation. While the availability of this catalog prompted the development of several techniques to analyze peculiar velocity data and efforts to understand possible bias, its use has led to conflicting results. The reasons for the discrepancies are not understood and could indicate limitations of the data or of the methods used. Efforts to re-calibrate this catalog using new observations [@shellflow] are still underway. In this context the completion of the SFI I-band TF survey of late spirals and the ENEAR $D_n-\sigma$ survey of early-type galaxies are important additions, providing homogeneous samples of comparable sizes. Figure \[fig:skydist\] shows the projected distribution of galaxies in these two surveys. In contrast to the Mark III compilation, the sky coverage of both surveys is remarkably uniform and nearly all of the data consist of new measurements reduced in a uniform way. Also note that the surveys nicely complement each other: ENEAR galaxies probe high density regions and delineate large-scale structures more sharply; SFI galaxies probe lower density regimes and are more uniformly distributed across the sky. Another important point is that the peculiar velocities in these catalogs are measured using distinct distance estimators based on different observed quantities. Therefore, to take full advantage of these characteristics these samples have been analyzed separately, to test the reproducibility of the results, and combined into the SEcat catalog to produce a fair sample probing a wide range of density regimes. The results of analyses based on these new catalogs of peculiar velocity data are reviewed below and compared to those obtained using Mark III. Results ======= Reconstructed density and velocity fields ----------------------------------------- An underlying assumption of all methods used in estimating $\beta$ is that galaxies, even though biased, are fair tracers of the mass distribution. This hypothesis can be, in principle, directly tested by comparing the galaxy density field as derived from redshift surveys and the mass density field reconstructed from peculiar velocity data. POTENT, Wiener Filter[@wf] and more recently the Unbiased Minimal Variance estimator (UMV)[@umv] are examples of methods developed to reconstruct the three-dimensional velocity and density fields from the observed radial component of the peculiar velocity. All methods assume that on the scales of interest the perturbations are small and non-linear effects can be neglected. The various methods have also been extensively tested using mock catalogs drawn from simulations that mimic the nearby universe. Recently, the UMV method has been applied to the SEcat catalog of peculiar velocities and to the redshift survey data. Figure \[fig:mass\] shows the map of the PSCz galaxy density field (left panel) and the mass density field (right panel) along the Supergalactic plane, the latter obtained from the SEcat data using a Gaussian smoothing radius of 1200 . The main features of our local universe are easily identified in these maps, including the Great Attractor (GA) on the left and the Perseus-Pisces supercluster (PP) in the lower right. There is also a hint of the Coma cluster, which lies just outside the volume probed by SEcat, in the upper part on the map. The similarity between the mass and galaxy density fields is striking, especially considering the limitations to the peculiar velocity data imposed by the Zone of Avoidance. Furthermore, even though different in details, the gross features of the mass density field are similar to those obtained by applying either the same or the POTENT formalism to the Mark III catalog[@wfmark][@potentm3] and SFI catalogs[@dacostasc]. This is an outstanding result considering the different ways these catalogs were constructed and the peculiar velocities measured. Current results are also a remarkable improvement over those obtained from earlier catalogs. In particular, it is worth mentioning the prominence of the Perseus-Pisces region, completely absent in the earlier maps, and the well-defined voids, well-known features in redshift surveys which are now clearly seen in the reconstructed mass distribution. The reconstructed three-dimensional velocity field can also be used to measure other quantities of interest. For instance, the amplitude of the bulk flow is found to vary from $V_{B} =300 \pm$ 70  for a sphere of $R=20\hmpc$ to $160 \pm$ 60  for $R=60\hmpc$. This value is in good agreement with that obtained from a direct fit to the radial peculiar velocities for the SFI[@sdipole] and the ENEAR[@edipole] samples. This result disagrees with the bulk flow determined for the Mark III survey, which has an amplitude of roughly twice this value[@wfmark]. The small amplitude of the bulk flow recently measured is in marked contrast to earlier claims of large amplitude coherent motions over scales of the order of 100$h^{-1}$ Mpc[@courteausurvey], which at face value would imply excess power on very large scales. This result is in line with the results of recent redshift surveys which have not detected inhomogeneities on very large scales. Greater insight on the characteristic of the flowfield can be obtained by decomposing the 3-D velocity field into two components, one which is induced by the local mass distribution and a tidal component due to mass fluctuations external to the volume considered[@hoffman][@wfmark][@wfenear]. Figure \[fig:tidal\] shows the results of this decomposition applied to the ENEAR survey[@wfenear], where the local volume is a sphere of $ 80\hmpc$ centered on the Local Group. The plots show the full velocity field (upper left panel), the divergent (upper right panel) and the tidal (lower left panel) components. To further understand the nature of the tidal field, its bulk velocity component has been subtracted and the residual is shown in the lower right panel. This residual is clearly dominated by a quadrupole component. In principle, the analysis of this residual field can shed light on the exterior mass distribution. For the ENEAR catalog we find that the local dynamics is hardly affected by structure on scales larger than its depth. For this sample not only the bulk velocity at large radii is small but so is the $rms$ value of the tidal field, estimated to be of the order of 60 . This is in marked contrast to the results obtained from the analysis of the Mark III survey which yields a much stronger tidal field, pointing (in the sense of its quadrupole moment) towards the Shapley concentration. For Mark III, the tidal field contributes $\sim$ 200 to the total bulk velocity. Estimates of $\beta$ -------------------- Equations (\[lingrav\]) and (\[divv\]) show that there are two alternative ways for estimating $\beta$ - velocity-velocity or density-density comparisons. In the first case, the observed galaxy distribution is used to infer a mass density field from which peculiar velocities can be predicted and compared to the observed ones. In the second case, the three-dimensional velocity field is obtained from the observed radial velocities and used to infer a self-consistent mass density field and thus a galaxy distribution, via linear biasing. The latter is then compared to the one obtained from large all-sky redshift surveys. A particularly useful method for performing a velocity-velocity comparison is the modal expansion method[@nd]. This method expands the velocity fields by means of smooth functions (Bessel and spherical harmonics) defined in redshift space, thus alleviating the Malmquist biases inherent in real space analysis. Furthermore, the modal expansion smooths the observed and predicted velocities in the same way, so that the smoothed fields can be compared directly. Because the number of modes is substantially smaller than the number of data points, the method also provides the means of estimating $\beta$ from a likelihood analysis carried out on a mode-by-mode basis, instead of galaxy-by-galaxy. The similar smoothing and the mode-by-mode comparison substantially simplify the error analysis. The modal expansion method has been used in comparisons between the 1.2 Jy predicted velocities and observed velocities inferred from TF measurements[@dnw] in the Mark III catalog, yielding $\beta_I\sim$ 0.4. However, examination of the residual field showed a strong dipole signature suggesting a significant mismatch between the Mark III and the fields. The reasons for the mismatch are still not well-understood. More recently, the same method has been employed in the comparison of the 1.2 Jy and SFI[@dacostanusser] and of the and ENEAR velocity fields[@nusserdacosta]. Figure \[fig:12sfi\] shows the smoothed velocity field predicted from the 1.2 Jy survey (left), adopting the best-fit value of $\beta_I=0.6$, and the measured SFI field (right). The infall to Virgo ($l =284^\circ, b = 74^\circ$) dominates the nearby SFI flow. In the middle panel, the field exhibits a dipole pattern corresponding to the reflex motion of the Local Group with infalling galaxies in the Hydra-Centaurus direction and an outward flow in the Perseus-Pisces direction, as seen in the LG restframe. Comparing the two fields one immediately sees that the general pattern of the velocity fields is remarkably similar with excellent agreement in the location of outflows and inflows and with only a few nearby galaxies having large residuals. This result gives confidence in the determination of $\beta_I$. Most encouraging is the absence of large regions of coherent residuals such as the dipole signature seen in the Mark III analysis at intermediate and distant redshift shells. Similar analysis has been performed using the survey and the ENEAR catalog of peculiar velocities. Figure \[fig:psczenear\] shows the corresponding smoothed velocity fields, for an adopted value of $\beta_I=0.5$. Comparison of the right-side of Figures \[fig:12sfi\] and \[fig:psczenear\] shows that the general flow pattern of the SFI and ENEAR velocity fields is remarkably similar. In the ENEAR case, very few prominent structures are probed by bright ellipticals in the innermost shell. However, in the next two shells a strong dipole pattern can be easily recognized, having an amplitude comparable to that observed in SFI. The agreement between the and ENEAR velocity fields is also very good with only a few more distant galaxies having large residuals. The above results demonstrate that the velocity fields of both SFI and ENEAR are similar and well described by the gravity fields of the 1.2 Jy and surveys, yielding comparable values of $\beta_I$. Consistent values of $\beta_I$ have also been obtained from similar analysis of the SBF survey of galaxy distances ($\beta_I=0.42$)[@sbfbeta] and from the peculiar velocities measured for a sample of nearby Type Ia supernovae ($\beta_I=0.4$) [@riess]. Another method to carry out a velocity-velocity comparison considered is VELMOD, a maximum likelihood method which takes as input the distance indicator observables and galaxy redshifts and determines the parameters describing the distance relation and the velocity model adopted. The method does not require smoothing and it is constructed for high-resolution analysis. The method has been used to analyze sub-samples of spiral galaxies extracted from the Mark III [@velmod][@velmod1] and the SFI data[@branchini], yielding $\beta_I=0.49$ and $\beta_I=0.42$, respectively. These results show that the value of $\beta_I$ obtained from velocity-velocity comparisons is independent not only of the data set considered but also of the method used, with all estimates being in the range $0.4\lsim \beta_I \lsim 0.6$. Unfortunately, until recently there has been a disparity between the results obtained from velocity-velocity comparisons and other methods such as density-density comparisons and maximum-likelihood estimates of the power-spectrum (PS) of mass fluctuations derived from peculiar velocity data [@zaroubips]. For instance, density-density comparisons using different data sets have invariably led to high values of $\beta$[@sigad], consistent with unity. In particular, comparison of the 1.2 Jy and POTENT reconstructed density field, based on the Mark III catalog, yields $\beta_I=0.89$. Similarly, estimates based on the PS derived from peculiar velocity data using Mark III[@zaroubips], SFI[@freudling] and ENEAR[@wfenear] have yielded values of $\beta$ in the range 0.82-1.1. The nature of this discrepancy is unknown. Both density-density and velocity-velocity methods have been carefully tested using mock catalogs extracted from N-body simulations and have been shown to provide unbiased estimates of $\beta$. Possible reasons for the discrepancy are non-linear effects, scale dependence of the biasing, poorly understood errors and/or problems with the data. However, attempts to evaluate their impact have so far failed to explain the discrepancy. In general, velocity-velocity comparisons are considered more robust as they depend more on redshift data, while density-density comparisons uses less reliable peculiar velocity data. Recently, a new attempt to carry out a density-density comparison has been made using the SEcat catalog mentioned earlier and the UMV method to reconstruct the 3-D velocity and density fields. These reconstructed fields were then used to determine the value of $\beta$ from direct velocity-velocity and density-density comparisons with the corresponding fields predicted from the redshift survey[@secat]. Figure \[fig:umv\] shows the results of the density-density (left) and velocity-velocity (right) comparisons, which give $\beta_I=0.56 \pm 0.1$ and $\beta_I=0.51 \pm 0.05$, respectively. This result is remarkable since it is the first time a good agreement is found for $\beta$ values derived from these two methods. This encouraging new result, which apparently resolves a long-standing dispute, may be due either to the new method used in the reconstruction of the fields or to the more homogeneous peculiar velocity data used or a combination of both. High values of $\beta$ have also been derived from applying a maximum-likelihood technique to the peculiar velocity data to derive the power-spectrum of mass density fluctuations. These results are summarized in Figure \[fig:ps\]. In the left panel the PS obtained from the ENEAR sample[@wfenear] with those measured for Mark III[@zaroubips] and SFI[@freudling]. The right panel shows the contour map of the likelihood (in the $\Gamma-\eta_8$ plane) for a $\Gamma$ model fit to the ENEAR data, where $\eta_8=\sigma_8\Omega^{0.6}$ and $\sigma_8$ is the $rms$ fluctuation amplitude within a sphere of $8~h^{-1}$ Mpc radius. It is clear from the figure that all data sets lead to similar high-amplitude PS, equivalent to high values of $\beta$. From the figure one also can see that while the likelihood analysis poses a strong constraint on $\eta_8$, the value of $\Gamma$ is poorly determined. Note, however, that low values of $\Gamma$, such as those required from the analyses of redshift survey data ($\Gamma \sim0.2$), are excluded at about the $3\sigma$ level. It is important to recall that the likelihood method used in estimating the PS involves the use of model power-spectra to compute the velocity correlation tensor which is then compared to that computed from the peculiar velocity data to determine the fit parameters. An equivalent way of exploring the same information is to use the scalar velocity correlation function, computed under the assumption of a homogeneous and isotropic flow. The results of this analysis can then be compared directly to model predictions using linear theory and an ensemble average of cosmic flow realizations for different cosmological models. The statistics of the model velocity field is parameterized by the amplitude, $\eta_8$, and by the shape parameter, $\Gamma$, of a CDM–like power spectrum. Applying the velocity correlation statistics to the SFI[@borgani1] and ENEAR[@borgani2] data sets one finds $\eta_8=0.34$ (SFI) and $\eta_8=0.51$ (ENEAR) for $\Gamma=0.25$. These values translate to $\beta_I=0.45-0.67$, assuming $b_I/b_o\sim 1.3$, results which agree within the uncertainties with the lower values of $\beta$ obtained by other methods presented above. More importantly, in contrast to the PS analyses, the region of acceptable solutions comfortably overlaps with other constraints on $\eta_8$ derived from the $rms$ of cluster peculiar velocities and cluster abundances, and on $\Gamma$ as determined for the galaxy power spectrum. One possible explanation for the discrepancy between the results of the PS analysis and the velocity correlation statistics is the different way the errors in the distance measurements are taken into account. An important clue is the weak constraint imposed on the shape parameter by the PS analysis. This suggests that the available samples may not be sufficiently deep for this type of analysis, making the method insensitive to the effect that large scale power may have in inducing velocities on small scales. 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--- address: 'Department of Mathematics,Texas A $\&$ M University-Commerce,Commerce, TX 75429, U S A.E-mail:yelin$\_$ou@tamu-commerce.edu' author: - 'Ye-Lin Ou$^{*}$' date: '08/18/08' title: - On conformal biharmonic immersions - On Conformal biharmonic immersions --- [^1] Abstract {#abstract .unnumbered} ======== > [This paper studies conformal biharmonic immersions. We first study the transformations of Jacobi operator and the bitension field under conformal change of metrics. We then obtain an invariant equation for a conformal biharmonic immersion of a surface into Euclidean $3$-space. As applications, we construct a $2$-parameter family of non-minimal conformal biharmonic immersions of cylinder into $\r^3$ and some examples of conformal biharmonic immersions of $4$-dimensional Euclidean space into sphere and hyperbolic space thus provide many simple examples of proper biharmonic maps with rich geometric meanings. These suggest that there are abundant proper biharmonic maps in the family of conformal immersions. We also explore the relationship between biharmonicity and holomorphicity of conformal immersions of surfaces.]{} Introduction ============ This paper works on the smooth objects, so we assume that manifolds, maps, vector fields, etc, are smooth unless it is stated otherwise.\ A biharmonic map is a map $\varphi:(M, g)\longrightarrow (N, h)$ between Riemannian manifolds that is a critical point of the bienergy functional $$\nonumber E^{2}\left(\varphi,\Omega \right)= \frac{1}{2} {\int}_{\Omega} \left|\tau(\varphi) \right|^{2}{\rm d}x$$ for every compact subset $\Omega$ of $M$, where $\tau(\varphi)={\rm Trace}_{g}\nabla {\rm d} \varphi$ is the tension field of $\varphi$. The Euler-Lagrange equation of this functional gives the biharmonic map equation ([@Ji1]) $$\label{BI1} \tau^{2}(\varphi):={\rm Trace}_{g}(\nabla^{\varphi}\nabla^{\varphi}-\nabla^{\varphi}_{\nabla^{M}})\tau(\varphi) - {\rm Trace}_{g} R^{N}({\rm d}\varphi, \tau(\varphi)){\rm d}\varphi =0,$$ which states the fact that the map $\varphi$ is biharmonic if and only if its bitension field $\tau^{2}(\varphi)$ vanishes identically. In the above equation we have used $R^{N}$ to denote the curvature operator of $(N, h)$ defined by $$R^{N}(X,Y)Z= [\nabla^{N}_{X},\nabla^{N}_{Y}]Z-\nabla^{N}_{[X,Y]}Z.$$ Harmonic maps are clearly biharmonic, so it is more interesting to study [*proper*]{} (meaning non-harmonic) biharmonic maps as far as one seeks to pursuit a new study. However, apart from the maps between Euclidean spaces defined by polynomials of degree less than four (a class of maps that seems so wild to exhibit any characteristic property) not many examples of proper biharmonic maps between Riemannain manifolds have been found (see, e.g., [@MO], [@LO2], [@Ou1], and the bibliography of biharmonic maps [@LMO]). So, currently, one priority and a practical thing to do seems to be finding more examples of proper biharmonic maps between certain model spaces or studying biharmonic maps under some geometric constraints. For example, one can study biharmonic isometric immersions which lead to the concept of biharmonic submanifolds (see e.g., [@Ji2], [@CH], [@CI], [@CMO1], [@CMO2], [@MO] and [@BMO]); one can also study, as in [@BK], [@BFO], [@LO2], horizontally weakly conformal biharmonic maps which generalize both the notion of harmonic morphisms (maps that are both horizontally weakly conformal and harmonic) and that of biharmonic morphisms (maps that are horizontally weakly conformal biharmonic with other constraints, see [@OU1], [@LO1], [@Ou1], and [@LO2] for details).\ The interesting link between harmonicity and conformality has a long history. It was known to Weierstrass that a conformal immersion $\varphi: M^2\longrightarrow \r^{3}$ is harmonic if and only if $\varphi(M)$ is a minimal submanifold of $\r^3$. It is also well known that [**conformal harmonic immersions**]{} of surfaces are precisely [**conformal minimal immersions**]{} of surfaces of which there has been a rich theory exhibiting a beautiful interplay among geometry, topology, and real and complex analysis. So it would be interesting to know if we can generalize (or use the tools of) the theory on conformal minimal immersions to conformal biharmonic immersions. On the other hand, Jiang and Chen-Ishikawa independently proved that an isometric immersion $\varphi: M^2\longrightarrow \r^{3}$ is biharmonic if and only if $\varphi$ is harmonic. It would also be interesting to know whether this result can be generalized to the case of conformal biharmonic immersions. Motivated by these, we study conformal biharmonic immersions in this paper. First, we study the transformations of Jacobi operator and bitension field under conformal change of metrics. We then obtain an invariant equation for a conformal biharmonic immersion of a surface into Euclidean $3$-space, and using this, we construct a $2$-parameter family of non-minimal conformal biharmonic immersions of cylinder into $\r^3$ and some examples of conformal immersions of $4$-dimensional Euclidean space into sphere and hyperbolic space, thus provide many simple examples of proper biharmonic maps with rich geometric meanings. We also explore Weierstrass type representations for conformal biharmonic immersions. Jacobi operators and the bitension fields under conformal change of metrics =========================================================================== For a map $\varphi : (M^{m},g) \longrightarrow (N^{n},h)$, the Jacobi operator is defined as $$\label{JO} J^{\varphi}_{g}(X)=-\{{\rm Trace}_{g}(\nabla^{\varphi}\nabla^{\varphi}-\nabla^{\varphi}_{\nabla^{M}})X - {\rm Trace}_{g} R^{N}({\rm d}\varphi, X){\rm d}\varphi\}$$ for any vector field $X$ along the map $\varphi$. Thus, by (\[BI1\]) and (\[JO\]), the relationship between the Jacobi operator and the bitension field of $\varphi$ is explained by $J^{\varphi}_{g}(\tau(\varphi))=-\tau^2(\varphi)$. \[confC\] Let $\varphi : (M^{m},g) \longrightarrow (N^{n},h)$ be a map. Then, under the conformal change of metrics ${\bar g}=F^{-2}g$, we have - the transformation of the Jacobi operators $J^{\varphi}_{g}$ and $J^{\varphi}_{\bar g}$ of $\varphi$ is given by $$\label{JF} J^{\varphi}_{\bar g}(X)= F^2J^{ \varphi}_{g}(X)+F^2(m-2)\nabla^{ \varphi}_{{\rm grad\,ln}F}X,$$ and - the transformation of the bitension fields $\tau^{2}(\varphi, g)$ and $\tau^{2}(\varphi,{\bar g})$ of $\varphi$ is given by $$\begin{aligned} \label{tao2} &&\tau^{2}(\varphi,{\bar g})= F^4\{\tau^{2}(\varphi, g)+(m-2)J^{ \varphi}_{g}({\rm d}{\varphi}({\rm grad\,ln}F))\\\notag && + 2(\Delta {\rm ln}F-(m-4)\left|{\rm grad\,ln}F\right|^2)\tau(\varphi, g)-(m-6)\nabla^{ \varphi}_{{\rm grad\,ln}\,F}\,\tau(\varphi, g)\\\notag && -2(m-2)(\Delta {\rm ln}F-(m-4)\left|{\rm grad\,ln}F\right|^2){\rm d}{\varphi}({\rm grad\,ln}F)\\\notag &&+(m-2)(m-6)\nabla^{ \varphi}_{{\rm grad\,ln}\,F}\,{\rm d}{\varphi}({\rm grad\,ln}F)\},\end{aligned}$$ where ${\rm grad}$ and $\Delta$ denote the gradient and the Laplacian taken with respect to the metric $g$. Choose a local orthonormal frames $\{e_i\}$ with respect to $g$ on $M$, then $\{{\bar e}_i=Fe_i\}$ is a local orthonormal frames with respect to ${\bar g}$.\ A direct computation gives the transformation of the tension fields under the conformal change of a metric as $$\begin{aligned} \notag \tau(\varphi,{\bar g})&=& F^2\{\tau (\varphi, g)-(m-2){\rm d}{\varphi}({\rm grad\,ln}F)\}.\end{aligned}$$ Also, a straightforward computation (see, e.g., [@BK]) yields $$\begin{aligned} \label{J2} &&{\rm Trace}_{\bar g}(\nabla^{\varphi}\nabla^{\varphi}-\nabla^{\varphi}_{{\bar\nabla}^{M}})X\\\notag &=&F^2\{{\rm Trace}_{g}(\nabla^{\varphi}\nabla^{\varphi}-\nabla^{\varphi}_{\nabla^{M}})X-(m-2)\nabla^{\varphi}_{{\rm grad\,ln}F}X\}.\end{aligned}$$ On the other hand, $$\begin{aligned} \label{J3} {\rm Trace}_{\bar g} R^{N}({\rm d}\varphi, X){\rm d}\varphi & =& \sum_{i=1}^{m}R^{N}({\rm d}\varphi (Fe_i), X){\rm d}\varphi(Fe_i)\\\notag & = & F^2{\rm Trace}_{g} R^{N}({\rm d}\varphi, X){\rm d}\varphi.\end{aligned}$$ Using Equations (\[J2\]) and (\[J3\]) we have $$\begin{aligned} \notag &&-J^{\varphi}_{\bar g}(X)={\rm Trace}_{\bar g}(\nabla^{\varphi}\nabla^{\varphi}-\nabla^{\varphi}_{{\bar\nabla}^{M}})X- {\rm Trace}_{\bar g} R^{N}({\rm d}\varphi, X){\rm d}\varphi\\\notag &=& F^2\{{\rm Trace}_{g}(\nabla^{\varphi}\nabla^{\varphi}-\nabla^{\varphi}_{\nabla^{M}})X- {\rm Trace}_{g} R^{N}({\rm d}\varphi, X){\rm d}\varphi\}\\\notag &&-(m-2)F^2\nabla^{\varphi}_{{\rm grad\,ln}F}X\\\notag &&= - F^2J^{ \varphi}_{g}(X)-(m-2)F^2\nabla^{ \varphi}_{{\rm grad\,ln}F}X,\end{aligned}$$ from which we obtain part (I) of the theorem.\ To prove the second part of the Theorem, we compute $$\begin{aligned} \notag\label{J100} J^{\varphi}(f X)&=&-\{{\rm Trace}_{g}(\nabla^{\varphi}\nabla^{\varphi}-\nabla^{\varphi}_{\nabla^{M}})(f X)- {\rm Trace}_{g} R^{N}({\rm d}\varphi, fX){\rm d}\varphi\}\\\label{J100} &=& fJ^{\varphi}(X)-(\Delta f)X-2 \nabla^{\varphi}_{{\rm grad}f}X, \\\label{LO0} &&\Delta F^2=2F^2\Delta {\rm ln}F+4F^2\left|{\rm grad\,ln}F\right|^2,\\\label{LO1} &&\nabla^{ \varphi}_{{\rm grad}\,F^2}\,{\rm d}{\varphi}({\rm grad\,ln}F)=2F^2\nabla^{ \varphi}_{{\rm grad\,ln}\,F}\,{\rm d}{\varphi}({\rm grad\,ln}F),\\\label{LO2} && \nabla^{ \varphi}_{{\rm grad\,ln}F}(F^2{\rm d}{\varphi}({\rm grad\,ln}F)) =2F^2\left| {\rm grad\,ln}F\right|^2{\rm d}{\varphi}({\rm grad\,ln}F)\\\notag &&+F^2\nabla^{ \varphi}_{{\rm grad\,ln}F}{\rm d}{\varphi}({\rm grad\,ln}F), \\ &&\nabla^{ \varphi}_{{\rm grad}\,F^2}\,\tau (\varphi, g)=2F^2\nabla^{ \varphi}_{{\rm grad\,ln}\,F}\,\tau (\varphi, g), {\rm and}\\\label{J200} && \nabla^{\varphi}_{{\rm grad\,ln}F}(F^2\tau(\varphi, g))= 2F^2 \left|{\rm grad\,ln}F \right|^2\tau(\varphi, g) +F^2\nabla^{ \varphi}_{{\rm grad\,ln}F}\tau(\varphi, g).\end{aligned}$$ Substituting $X= \tau(\varphi,{\bar g})= F^2\{\tau (\varphi, g)-(m-2){\rm d}{\varphi}({\rm grad\,ln}F)\}$ into (\[JF\]) and using Equations (\[J100\])$-$(\[J200\]) we have $$\begin{aligned} \notag &&\tau^{2}(\varphi,{\bar g})=-J^{\varphi}_{\bar g}(\tau(\varphi,{\bar g}))\\\notag &=&-F^2\{J^{ \varphi}_{g}(F^2\tau(\varphi, g)- (m-2)F^2{\rm d}{\varphi}({\rm grad\,ln}F))\\\notag &&+(m-2)\nabla^{ \varphi}_{{\rm grad\,ln}F}(F^2\tau(\varphi, g)- (m-2)F^2{\rm d}{\varphi}({\rm grad\,ln}F))\}\\\notag &=&-F^2\{F^2J^{ \varphi}_{g}(\tau(\varphi, g))-(\Delta F^2)\tau(\varphi, g))-2\nabla^{ \varphi}_{{\rm grad}\,F^2}\,\tau(\varphi, g)\\\notag &&-(m-2)F^2J^{ \varphi}_{g}({\rm d}{\varphi}({\rm grad\,ln}F))+(m-2)(\Delta F^2){\rm d}{\varphi}({\rm grad\,ln}F)\\\notag &&+2(m-2)\nabla^{ \varphi}_{{\rm grad}\,F^2}\,{\rm d}{\varphi}({\rm grad\,ln}F)\\\notag &&+(m-2)\nabla^{ \varphi}_{{\rm grad\,ln}F}(F^2\tau(\varphi, g)- (m-2)F^2{\rm d}{\varphi}({\rm grad\,ln}F))\}\\\notag &=& F^4\{\tau^{2}(\varphi, g)+(m-2)J^{ \varphi}_{g}({\rm d}{\varphi}({\rm grad\,ln}F))\\\notag && + 2(\Delta {\rm ln}F-(m-4)\left|{\rm grad\,ln}F\right|^2)\tau(\varphi, g)-(m-6)\nabla^{ \varphi}_{{\rm grad\,ln}\,F}\,\tau(\varphi, g)\\\notag&& -2(m-2)(\Delta {\rm ln}F-(m-4)\left|{\rm grad\,ln}F\right|^2){\rm d}{\varphi}({\rm grad\,ln}F)\\\notag &&+(m-2)(m-6)\nabla^{ \varphi}_{{\rm grad\,ln}\,F}\,{\rm d}{\varphi}({\rm grad\,ln}F)\}.\end{aligned}$$ This gives the second part of the theorem. Let $\varphi : (M^{2},g) \longrightarrow (N^{n},h)$ be a map and ${\bar g}=F^{-2}g$ be a conformal change of the metric $g$. Let $\tau^{2}(\varphi, g)$ and $\tau^{2}(\varphi,{\bar g})$ be the bitension fields of $\varphi$ with respect to the metrics $g$ and ${\bar g}$ respectively. Then, $$\begin{aligned} \label{tao10} \tau^{2}(\varphi,{\bar g})&=& F^4\{\tau^{2}(\varphi, g) +2 (\Delta {\rm ln}F+2\left|{\rm grad\,ln}F\right|^2)\tau(\varphi, g))\\\notag && +4\nabla^{ \varphi}_{{\rm grad}\,\ln F}\,\tau(\varphi, g)\}.\end{aligned}$$ Substituting $m=2$ into the Equation (\[tao2\]) we get $$\begin{aligned} &&\tau^{2}(\varphi,{\bar g})=F^4\tau^{2}(\varphi, g) +F^2\{ (\Delta F^2)\tau(\varphi, g))+2\nabla^{ \varphi}_{{\rm grad}\,F^2}\,\tau(\varphi, g)\}\\\notag && =F^4\{\tau^{2}(\varphi, g) +2 (\Delta {\rm ln}F+2\left|{\rm grad\,ln}F\right|^2)\tau(\varphi, g))+4\nabla_{{\rm grad}\,\ln F}\,\tau(\varphi, g)\}.\end{aligned}$$ \[bk\] Let $\varphi : (M^{m},g) \longrightarrow (N^{n},h)$ be a harmonic map with $m\neq 2$, and let ${\bar g}=F^{-2}g$ be a conformal change of the metric $g$. Then, the map $\varphi : (M^{m},{\bar g}) \longrightarrow (N^{n},h)$ is a biharmonic map if and only if, $$\begin{aligned} \label{tao20} &&J^{ \varphi}_{g}({\rm d}{\varphi}({\rm grad\,ln}F))+(m-6)\nabla^{ \varphi}_{{\rm grad\,ln}F}{\rm d}{\varphi}({\rm grad\,ln}F)\\\notag &&-2\left(\Delta {\rm\,ln}F-(m-4)\left| {\rm grad\,ln}F\right|^2\right){\rm d}{\varphi}({\rm grad\,ln}F)=0.\end{aligned}$$ The corollary is obtained by applying Theorem \[confC\] with $\tau(\varphi, g)=\tau^2(\varphi, g)=0$ and $m\neq 2$. Let $\gamma =-\ln F$, then Corollary \[bk\] recovers Proposition 2.1 in [@BK] after taking into account that their convention for Laplacian on functions is $\Delta f=-{\rm trace} \nabla {\rm d} f$ which is different from ours by a negative sign. \[E1\] The conformal immersion from Euclidean space into the hyperbolic space $$\label{cfi} \varphi : (\r^3\times \r^{+},\bar{g}=\delta_{ij}) \longrightarrow (H^5=\r^4\times \r^{+},h=y_5^{-2}\delta_{\alpha\beta})$$ given by $\varphi(x_1,\ldots,x_4)=(1,x_1,\ldots,x_4)$ is a proper biharmonic map. In fact, the associated isometric immersion $$\varphi : (\r^3\times \r^{+},g=x_4^{-2}\delta_{ij}) \longrightarrow (H^5=\r^4\times \r^{+},h=y_5^{-2}\delta_{\alpha\beta})$$ is totally geodesic and hence harmonic. Here, $\bar{g}=F^{-2}g$ with $F=x_4^{-1}$. By Corollary \[bk\], the conformal immersion (\[cfi\]) is biharmonic if and only if Equation (\[tao20\]) holds, which is equivalent to $J^{ \varphi}_{\bar{g}}({\rm d}{\varphi}({\rm grad_{\bar{g}}\,ln}F))=0$. A straightforward computation yields $$\begin{aligned} \notag J^{ \varphi}_{\bar{g}}({\rm d}{\varphi}({\rm grad_{\bar{g}}\,ln}F))=-x_4^{-1}J^{ \varphi}_{\bar{g}}({\rm d}{\varphi}(\partial_4))+\Delta (x_4^{-1}){\rm d}{\varphi}(\partial_4)+2\nabla^{ \varphi}_{{\rm grad}\,(x_4^{-1})}{\rm d}{\varphi}(\partial_4),\end{aligned}$$ which is identically zero as one can check that $$\notag -x_4^{-1}J^{ \varphi}_{\bar{g}}({\rm d}{\varphi}(\partial_4))=-4x_4^{-3}\partial y^5,\;\;\Delta (x_4^{-1}){\rm d}{\varphi}(\partial_4)=2x_4^{-3}\partial y^5=2\nabla^{ \varphi}_{{\rm grad}\,(x_4^{-1})}{\rm d}{\varphi}(\partial_4).$$ The conformal immersion from Euclidean space into the sphere $$\notag \varphi : (\r^4,\bar{g}=\delta_{ij}) \longrightarrow (S^5\setminus \{N\}\equiv \r^5,h=\frac{4\delta_{\alpha\beta}}{(1+|y|^2)^2})$$ given by $\varphi(u_1,\ldots,u_4)=(u_1,\ldots,u_4,0)$, where $(u_1,\ldots,u_5)$ are conformal coordinates on $S^5\setminus \{N\}\equiv \r^5$, is a proper biharmonic map. In fact, the map is the inverse stereographic projection that maps $\r^4$ into a great hypersphere in $S^5$. The associated isometric immersion $$\notag \varphi : (S^4\setminus \{P\}\equiv\r^4,\bar{g}=\frac{4\delta_{ij}}{(1+|u|^2)^2}) \longrightarrow (S^5\setminus \{N\}\equiv \r^5,h=\frac{4\delta_{\alpha\beta}}{(1+|y|^2)^2})$$ is totally geodesic and hence harmonic. A computation similar to those in Example \[E1\] shows that the conformal immersion is indeed a proper biharmonic map. Conformal biharmonic immersions =============================== Let $\varphi : (M^{m},g) \longrightarrow (N^{n},h)$ be a conformal immersion with $\varphi^{*}h=\lambda^{2}g$. Let $\varphi : (M^{m},{\bar g}) \longrightarrow (N^{n},h)$ be the associated isometric immersion with mean curvature vector $\eta$, where ${\bar g}=\varphi^{*}h=\lambda^{2}g$. Then, the conformal immersion $\varphi : (M^{m},g) \longrightarrow (N^{n},h)$ is biharmonic if and only if $$\begin{aligned} \notag \lambda^{4}\tau^{2}(\varphi,{\bar g})&=& -(m-2)J^{ \varphi}_{g}({\rm d}{\varphi}({\rm grad\,ln}\lambda)) + 2m\lambda^2(-\Delta {\rm ln}\lambda-2\left|{\rm grad\,ln}\lambda\right|^2)\eta\\\label{Confi} &&+m(m-6)\lambda^2\nabla^{ \varphi}_{{\rm grad\,ln}\,\lambda}\, \eta.\end{aligned}$$ Substituting $F=\lambda^{-1}$ and $\ln F=-\ln \lambda$ into the Equation (\[tao2\]) we have $$\begin{aligned} \label{gd10} &&\tau^{2}(\varphi,{\bar g})= \lambda^{-4}\{\tau^{2}(\varphi, g)-(m-2)J^{ \varphi}_{g}({\rm d}{\varphi}({\rm grad\,ln}\lambda))\\\notag && + 2(-\Delta {\rm ln}\lambda-(m-4)\left|{\rm grad\,ln}\lambda\right|^2)\tau(\varphi, g)+(m-6)\nabla^{ \varphi}_{{\rm grad\,ln}\,\lambda}\,\tau(\varphi, g)\\\notag&& +2(m-2)(-\Delta {\rm ln}\lambda-(m-4)\left|{\rm grad\,ln}\lambda\right|^2){\rm d}{\varphi}({\rm grad\,ln}\lambda)\\\notag &&+(m-2)(m-6)\nabla^{ \varphi}_{{\rm grad\,ln}\,\lambda}\,{\rm d}{\varphi}({\rm grad\,ln}\lambda)\}.\end{aligned}$$ Note that the tension field of the conformal immersion $\varphi$ is given by $$\begin{aligned} \label{TCI} \tau(\varphi)=m\lambda^2 \eta+(2-m){\rm d}\varphi \left( {\rm grad}\, {\rm ln} \lambda\right).\end{aligned}$$ Substituting (\[TCI\]) into (\[gd10\]) we have $$\begin{aligned} &&\tau^{2}(\varphi,{\bar g})= \lambda^{-4}\{\tau^{2}(\varphi, g)-(m-2)J^{ \varphi}_{g}({\rm d}{\varphi}({\rm grad\,ln}\lambda))\\\notag && + 2m\lambda^2(-\Delta {\rm ln}\lambda-2\left|{\rm grad\,ln}\lambda\right|^2)\eta\\\notag &&+m(m-6)\lambda^2\nabla^{ \varphi}_{{\rm grad\,ln}\,\lambda}\, \eta\}.\end{aligned}$$ From this we obtain the proposition. The conformal immersion $\varphi : (M^{2},g) \longrightarrow (\r^3,\langle,\rangle_{0})$ into Euclidean space with $\varphi^{*}\langle,\rangle_{0}=\lambda^{2}g$ is biharmonic if and only if $$\label{R3} \begin{cases} A_{\xi}({\rm grad} H)+ \frac{1}{2}{\rm grad} (H^2)+2H\,A_{\xi}({\rm grad\;ln} \lambda)=0\\\Delta H -H\,|B|^2+2H(\Delta {\rm ln}\lambda+2\left|{\rm grad\,ln}\lambda\right|^2)+4g({\rm grad\;ln} \lambda,{\rm grad} H)=0, \end{cases}$$ where $\xi$ is the unit normal vector field of the surface $\varphi(M)\subset \mathbb{R}^3$ and $A_{\xi}$ and $H$ are the shape operator and the mean curvature function of the surface respectively. It follows from (\[Confi\]) with $m=2$ that the conformal immersion $\varphi$ is biharmonic if and only if $$\begin{aligned} \label{GD12} &&\lambda^{2}\tau^{2}(\varphi,{\bar g})= -4(\Delta {\rm ln}\lambda+2\left|{\rm grad\,ln}\lambda\right|^2)\eta-8\nabla^{ \varphi}_{{\rm grad\,ln}\,\lambda}\,\eta,\end{aligned}$$ where $\tau^{2}(\varphi,{\bar g})$ denotes the bitension field of the associated isometric immersion $\varphi : (M^{2},\bar{g}=\lambda^{2}g) \longrightarrow \mathbb{R}^3$ with mean curvature vector $\eta=H\xi$, where $\xi$ and $H$ are the unit normal vector field and the mean curvature function of the surface $\varphi (M)$ respectively. Then, we have (see, e.g., [@Ji2], [@CH] and [@CMO2]) $$\begin{aligned} \notag \tau^{2}(\varphi, \bar{g}) = 2(\Delta_{\bar{g}} H -H\,|B|_{\bar{g}}^2 )\xi - 2[ 2A_{\xi}({\rm grad}_{\bar{g}} H)+ {\rm grad}_{\bar{g}} (H^2)],\end{aligned}$$ substitute this into (\[GD12\]) we have $$\begin{aligned} \label{gd20} &&\lambda^{2}(\Delta_{\bar{g}} H -H\,|B|^2 )\xi - \lambda^{2}[ 2A_{\xi}({\rm grad}_{\bar{g}} H)+ {\rm grad}_{\bar{g}} (H^2)]\\\notag &=& -2(\Delta {\rm ln}\lambda+2\left|{\rm grad\,ln}\lambda\right|^2)H\xi-4\nabla^{ \varphi}_{{\rm grad\,ln}\,\lambda}\,H\xi.\end{aligned}$$ Notice that the transformations of Laplacian and the gradient operators under a conformal change of metrics $\bar{g}=\lambda^{2}g$ in two dimensional manifold are given by $$\begin{aligned} \label{La} \Delta_{\bar{g}} u=\lambda^{-2}\Delta u,\;\;\;{\rm grad}_{\bar{g}}u=\lambda^{-2}{\rm grad}u.\end{aligned}$$ On the other hand, we have $$\begin{aligned} \label{La1} -4\nabla^{ \varphi}_{{\rm grad\,ln}\,\lambda}\,H\xi=-4g({\rm grad\,ln}\lambda,{\rm grad}H) \xi +4H\,A_{\xi}({\rm grad\;ln}\lambda).\end{aligned}$$ Using (\[La\]) and substituting (\[La1\]) into (\[gd20\]) and comparing the tangential and the normal components we obtain equation (\[R3\]) which completes the proof of the theorem. \[EZHU\] For $\lambda^2=\big(C_2e^{\pm z/R}-C_1C_2^{-1}R^2e^{\mp z/R}\big)/2$ with constants $C_1, C_2$, the maps $\phi:( D, g=\lambda^{-2}(R^2 d\theta^2+dz^2))\longrightarrow (\r^3,d\sigma^2=d\rho^2+\rho^2\,d\theta^2+dz^2)$ with $\phi(\theta, z)=(R, \theta, z)$ is a family of proper biharmonic conformal immersions of a cylinder of radius $R$ into Euclidean space $\r^3$. Let $\phi:\r^2\supseteq D\longrightarrow \r^3$, $\phi(\theta,z)=(R\cos\,\theta, R\sin\,\theta, z)$ be the isometric immersion with the image $\phi(D)$ being a cylinder of radius $R$ in $3$-space. Using cylindrical coordinates $(\rho, \theta, z)$ on $\mathbb{R}^3$ we can represent the isometric immersion of the cylinder as $\phi:\r^2\supseteq D\longrightarrow \r^3$ with $\phi(\theta, z)=(R, \theta, z)$. It is easy to check that$ E_1=\frac{\partial}{\partial \rho},\;\; E_2=\frac{1}{\rho}\frac{\partial}{\partial \theta},\;\;E_3=\frac{\partial}{\partial z}$ constitute a local orthonormal frame of $\r^3$ and that $e_1=E_2,\;\; e_2=E_3, \xi=E_1$ is an adapted orthonormal frame along the cylinder with $\xi$ being unit normal vector field. We can check that the induced metric on the cylinder is ${\bar g}=R^2d\theta^2+dz^2$. Let $g=\lambda^{-2}(R^2d\theta^2+dz^2)$ be a conformal change of the metric on the cylinder. Then, we have a conformal immersion $\phi:( D, g)\longrightarrow (\r^3, d\sigma^2=d\rho^2+\rho^2\,d\theta^2+dz^2)$ with $\phi^{*}d\sigma^2=\lambda^2g={\bar g}$. A straightforward computation gives $$\begin{aligned} \label{gd30} \begin{cases} A_{\xi}e_1=-\frac{1}{R}e_1,\;\;A_{\xi}e_2=0,\\ H=\frac{1}{2}(\langle A_{\xi}e_1,e_1\rangle+\langle A_{\xi}e_2,e_2\rangle)=-\frac{1}{2R}\ne 0\\ |B|^2=\lambda^2|B|^2_{{\bar g}}=\lambda^2\sum_{i=1}^2|A(e_i)|^2=\lambda^2\frac{1}{R^2},\\ {\rm grad}\,H=0,\\ \Delta H=0. \end{cases}\end{aligned}$$ Substituting (\[gd30\]) into (\[R3\]) we conclude that conformal immersion $\phi$ is biharmonic if and only if $$\notag \begin{cases} A_{\xi}({\rm grad\;ln} \lambda)=0\\ \lambda^2-2R^2(\Delta {\rm ln}\lambda+2\left|{\rm grad\,ln}\lambda\right|^2)=0. \end{cases}$$ It is not difficult to check that this system is equivalent to $$\notag \begin{cases} \lambda(\theta, z)=\lambda(z)\\ 1-2R^2[({\rm ln}\lambda)''+2({\rm ln}\lambda)'^2)]=0 \end{cases}$$ or, $$\notag (\lambda^2)''=\frac{1}{R^2}\lambda^2.$$ It follows that $\lambda^2$ is a solution of the ordinary differential equation $$\notag y''=\frac{1}{R^2}y,$$ which has (see e.g., [@Cu]) the first integral $$\label{GD10} y'^2=y^2/R^2+C_1.$$ Solving Equation (\[GD10\]) we have $$y=\big(C_2e^{\pm z/R}-C_1C_2^{-1}R^2e^{\mp z/R}\big)/2.$$ Notice that the conformal immersion has nonzero constant mean curvature $H$ so it is not harmonic. Therefore, we complete the proof of the proposition. It follows from Proposition \[EZHU\] that the biharmonic conformal immersions of the cylinder in $\r^3$ are not minimal, thus the well-known fact that a conformal harmonic immersion of a surface must be a minimal surface fails to generalize to conformal biharmonic immersion of a surface. Our proposition also shows that if B. Y. Chen’s conjecture [@CH] about biharmonic isometric immersions into Euclidean space is generalized to biharmonic conformal immersions, then the answer is negative. Biharmonicity and holomorphicity of conformal immersions ======================================================== Let $\varphi : (M^2 ,g) \longrightarrow \r^n $ be a conformal immersion of a Riemann surface. Let $(u, v)$ be the local coordinates on $M$ and we write $z=u+iv$ in the local complex parameter. We also use the usual notations $$\notag \frac{\partial}{\partial z}=\frac{1}{2}(\frac{\partial}{\partial u}-i \frac{\partial}{\partial v}),\;\; {\rm and}\;\; \frac{\partial}{\partial {\bar z}}=\frac{1}{2}(\frac{\partial}{\partial u}+i \frac{\partial}{\partial v}).$$ Then, the well-known Weierstrass representation theorem for conformal harmonic immersions can be stated as: Let $\varphi : (M^2 ,g) \longrightarrow (\r^n, \langle ,\rangle_{0})$ be a harmonic conformal immersion. Then, the section $ \phi=\frac{\partial \varphi}{\partial z}=\frac{1}{2}( \varphi_{u}-i \varphi_{v})=\phi^\alpha(z)\frac{\partial}{\partial y^\alpha}$ is holomorphic and satisfies $$\begin{aligned} \label{W1} \sum _{\alpha=1}^{n}(\phi^\alpha)^2=0,\\\label{W2} \sum _{\alpha=1}^{n}|\phi^\alpha|^2\ne 0.\end{aligned}$$ Conversely, given any holomorphic section $ \phi=\phi^\alpha\frac{\partial}{\partial y^\alpha}:M\longrightarrow \mathbb{E}$ satisfying (\[W1\]) and (\[W2\]) and the periodic condition: $$\notag \mathfrak{Re}\int_{\gamma}(\phi^1,\ldots, \phi^n){\rm d}z=0,$$ for any closed path in $M$. Then, the map $$\notag \varphi(z)=2\,\mathfrak{Re}\int_{z_0}^{z}(\phi^1,\ldots, \phi^n){\rm d}z$$ defines a harmonic conformal immersion of a Riemann surface into the Euclidean space.\ For conformal biharmonic immersions of surfaces into Euclidean space, we have \[Weie\] $\varphi : (M^2 ,g) \longrightarrow (\r^n, \langle ,\rangle_{0})$ is a conformal biharmonic immersion with $\varphi^{*}\langle,\rangle=\lambda^2g$ if and only if the section $ \phi=\frac{\partial \varphi}{\partial z}=\frac{1}{2}( \varphi_{u}-i \varphi_{v})=\phi^\alpha(z)\frac{\partial}{\partial y^\alpha}$ satisfies Equations (\[W1\]), (\[W2\]) and $$\begin{aligned} \label{W3} \frac{\partial}{\partial {\bar z}}\frac{\partial}{\partial z}\left(\lambda^{-2}\frac{\partial \phi^\sigma}{\partial {\bar z}}\right)=0\end{aligned}$$ Using the local conformal parameter $z=u+iv$ and the characteristic property that $g=\varphi^{*}\langle ,\rangle_{0}$ of the conformal immersion $\varphi : (M^2 ,g) \longrightarrow (\r^n, \langle ,\rangle_{0}) $ we can be write the metric $g$ as $g=\lambda^2({\rm d}u^2+{\rm d}v^2)=\lambda^2|{\rm d} z|^2$, where $\lambda^2=\langle \varphi_{u} ,\varphi_{u}\rangle_{0}=\langle \varphi_{v} ,\varphi_{v}\rangle_{0}$. The Laplacian operator on $(M, g)$ can be written as $$\notag \Delta=\lambda^{-2}(\frac{\partial^2}{\partial u^2}+\frac{\partial^2}{\partial v^2})=4\lambda^{-2}\frac{\partial}{\partial {\bar z}}\frac{\partial}{\partial z}.$$ Let $\{y^\alpha, \frac{\partial}{\partial y^\alpha}\}$ be local coordinates in a neighborhood $U$ of $\r^n$ such that $U\cap \varphi (M)$ is nonempty. We can write the local expression $\varphi(z)=(\varphi^1(z),\ldots,\varphi^n(z))$ as $\varphi(z)=\varphi^\alpha(z)\frac{\partial}{\partial y^\alpha}$. If we define the section $ \phi=\frac{\partial \varphi}{\partial z}=\frac{1}{2}( \varphi_{u}-i \varphi_{v})$. Then, it is well-known (see, e.g., [@ES]) the tension field of $\varphi$ can be written as $$\notag \tau (\varphi)=(\Delta\varphi^1,\ldots,\Delta\varphi^n)= 4\lambda^{-2}\frac{\partial^2 \varphi^\sigma}{\partial {\bar z}\partial z}\frac{\partial}{\partial y^\sigma}=4\lambda^{-2}\frac{\partial \phi^\sigma}{\partial {\bar z}}\frac{\partial}{\partial y^\sigma},$$ and hence the bitension field is given by $$\begin{aligned} \label{bih49} \tau^2 (\varphi)=(\Delta^2\varphi^1,\ldots,\Delta^2\varphi^n) =4\lambda^{-2}\frac{\partial}{\partial {\bar z}}\frac{\partial}{\partial z}\left(4\lambda^{-2}\frac{\partial \phi^\sigma}{\partial {\bar z}}\right)\frac{\partial}{\partial y^\sigma}.\end{aligned}$$ It is easy to see that Equations (\[W1\]) and (\[W2\]) is equivalent to $\varphi$ being a conformal immersion whilst Equation (\[W3\]) is equivalent to the biharmonicity of $\varphi$ by (\[bih49\]). We can use Weierstrass representation to prove that the map $\varphi:(\r^2,g=e^{y/R}(dx^2+dy^2)\longrightarrow \r^3$, $\varphi(x,y)=(R\cos\,\frac{x}{R}, R\sin\,\frac{x}{R}, y)$ is a proper biharmonic conformal immersion of $\r^2$ into Euclidean space $\r^3$.\ Indeed, in this case, $\varphi_x=(-\sin \frac{x}{R}, \cos \frac{x}{R}, 0),\;\;\varphi_y=(0, 0, 1)$ and $\varphi$ is a conformal immersion with $\varphi^{*}\langle,\rangle_{0}={\bar g}=dx^2+dy^2=\lambda^2g$ for $\lambda^2=e^{-y/R}$. The section $ \phi=\frac{\partial \varphi}{\partial z}=\frac{1}{2}( \varphi_{x}-i \varphi_{y})=\frac{1}{2}(-\sin \frac{x}{R}, \cos \frac{x}{R}, -i)$ with components $$\phi^1=-\frac{1}{2}\sin\,\frac{z+{\bar z}}{2R},\;\;\phi^2=\frac{1}{2}\cos\,\frac{x}{R}=\frac{1}{2}\cos\,\frac{z+{\bar z}}{2R},\;\;\phi^3=-i/2.$$ A straightforward computation yields $$\begin{cases} \lambda^{-2}\frac{\partial \phi^1}{\partial \bar{z}}=-\frac{1}{4R}e^{-(z-{\bar z})i/(2R)}\cos\,\frac{z+{\bar z}}{2R} =-\frac{i}{8R}(e^{\bar{z}i/R}+e^{-zi/R}),\\ \lambda^{-2}\frac{\partial \phi^2}{\partial \bar{z}}=-\frac{1}{4R}e^{-(z-{\bar z})i/(2R)}\sin\,\frac{z+{\bar z}}{2R} =-\frac{i}{8R}(e^{e^{-zi/R}-\bar{z}i/R}),\\ \lambda^{-2}\frac{\partial \phi^3}{\partial \bar{z}}=0. \end{cases}$$ Clearly, we have $\frac{\partial}{\partial {\bar z}}\frac{\partial}{\partial z}\big(\lambda^{-2}\frac{\partial \phi^\sigma}{\partial {\bar z}}\big)=0$ for $\sigma=1, 2, 3$ and hence, by Theorem \[Weie\], $\varphi$ is a biharmonic conformal immersion which is not harmonic as the section $\phi$ is not holomorphic. We can also easily check that the map $\varphi:(\r^2,g=e^{y/R}(dx^2+dy^2)\longrightarrow \r^6$, $\varphi(x,y)=(R\cos\,\frac{x}{R}, R\sin\,\frac{x}{R}, y,R\cos\,\frac{x}{R}, R\sin\,\frac{x}{R}, y)$ is a proper biharmonic conformal immersion of $\r^2$ into Euclidean space $\r^6$. [99]{} P. Baird and D. Kamissoko, [*On constructing biharmonic maps and metrics*]{}, Ann. Global Anal. Geom. 23 (2003), no. 1, 65–75. P. Baird, A. Fardoun and S. Ouakkas, [*Conformal and semi-conformal biharmonic maps*]{}, Ann. Glob. Anal. Geom., to appear, 2008. A. Balmus, S. Montaldo and C. Oniciuc, [*Classification results for biharmonic submanifolds in spheres*]{}, Preprint 2007, arXiv:math/0701155. R. Caddeo, S. Montaldo and C. Oniciuc, [*Biharmonic submanifolds of $S\sp 3$*]{}, Internat. J. Math. 12 (2001), no. 8, 867–876. R. Caddeo, S. Montaldo and C. Oniciuc, [*Biharmonic submanifolds in spheres*]{}, Israel J. Math. 130 (2002), 109–123. B. Y. Chen, [*Some open problems and conjectures on submanifolds of finite type*]{}, Soochow J. Math. 17 (1991), no. 2, 169–188. B. Y. Chen and S. Ishikawa, [*Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces*]{}, Kyushu J. Math. 52 (1998), no. 1, 167–185. W. J. Cunningham, [*Introduction to nonlinear analysis*]{}, McGraw-Hill Book Company, Inc. New York, 1958. J. Eells and J.H. Sampson, [*[H]{}armonic mappings of [R]{}iemannian manifolds.*]{} Amer. J. Math. [**86**]{} (1964), 109–160. G. Y. Jiang, [*$2$-harmonic maps and their first and second variational formulas*]{}. Chinese Ann. Math. Ser. A, 7 (1986), 389–402. G. Y. Jiang, [*Some non-existence theorems of $2$-harmonic isometric immersions into Euclidean spaces* ]{}, Chin. Ann. Math. Ser. 8A (1987) 376-383. E. Loubeau, S. Montaldo, and C. Oniciuc, The bibliograph of biharmonic maps, http://beltrami.sc.unica.it/biharmonic/ E. Loubeau and Y.-L. 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--- abstract: 'We continue to study the problems of discovering new temporal and spatial properties of neutrinos from the point of the possible multi-dimensional extension the D=(3+1)- special theory of relativity. It is neutrino that can connect our Universe with new types of the matter, the new Universe. The possible discovery with neutrino new structure of the Time can confirm these ideas. However, the neutrino experiments aimed at new phenomena search can lead us to paradoxes related with the limits of applicability of the theory of relativity which demands special studies. This new phenomenon one can call by “Neutrino Paradoxes in Theories of Relativity”. As examples of such paradoxes one can illustrate on the neutrino experiments MINOS,OPERA et al devoted to the measurements of neutrino velocity. As the interpretation of their results are model-dependent, by our opinion, the main goal of the observation a possible new time structure in these experiments is not reached. As the solutions to this problem it seems to us the way of holding cycle of long base neutrino experiments at high(super-high) energies which requires further debate in the literature. In addition to our discussions one can remark that the questions of neutrino physics ($M_S \rightarrow O(1-20\, TeV- range)$) could be directly related to the collider physics at the LHC.' --- [$\beta\beta$]{}[$\beta\beta~ $]{} 0 [$0^+\to 0^+$]{} ł[$\lambda$]{} [ To the memory of\ E.P.Kuznetsov, A. V. Samoilov,\ Yu.P. Nikitin, V. V. Ammosov]{}\ [**Neutrino On The Possible New Time Structure.**]{}\ [**D.S. Baranov$*$ and G.G.Volkov$**$**]{}\ [*$*$ Joint Institute of High Temperatures Moscow, Russia*]{}\ [*$**$ St. Petersburg Nuclear Physics Institute Gatchina, Russia*]{}\ [*Neutrino Light Collaboration*]{}\ [E-mail:baranovd@rambler.ru; ge.volkov@yandex.ru\ ]{} Introduction. ============= The neutrino speed measurement experiments are the continuations of the classic light speed measurement experiments have been done in range of the solar planet system(Ole Roemer-1676), in star system (James Bradely,1728) and, at last, on the Earth( Lois Fizeau, 1849),.... The finite light speed measurement has led to the revolution in the humanity consciousness and eventually led to a new understanding of the visible universe. In 1998-2005, there were a lot of excited discussions at CERN [@AV],[@AV2],[@Paris] about the possibilities to perform the neutrino experiments to test the super-luminal neutrino hypothesis and to find new phenomena beyond the . From one hand the idea of such experiments was associated with the hope to understand the role of the $V-A$- weak interactions, the quark-lepton family symmetry, the neutrino space-time properties and to observe some indications on a new vacuum structure existence outside of the Weak Scale, [*i.e.*]{} in the region $1/R \sim (0.1-20)TeV$. From another hand the general trends of this idea has been related to the possible existence some extra space-time non-compact dimensions of the universe. In this context it would be first serious encounter with the dual conception between the physical phenomena of micro-cosmos and of universe. One of the main goals is to find with neutrino some new spatial and temporal structures that might explain the formation of our visible $D=(3+1)$-universe with all its space-time and internal symmetries which could be only a part of a vast Universe filled with other kinds of matter. The main difficulties of such experiments related to the possible relativity principle paradoxes have been discussed. Light in the Evolution of the Time Structure. ============================================== Up to date there are several reasons for considering the possible extension of the observed $D = (3 +1)$- space-time and its symmetries. In the first instance a number of fundamental problems in the field of high energy physics encountered in the Standard Model as well as in modern approaches $D = 10$ super-string theories / D-branes, 11-dimensional M-theory and 12-dimensional F-theory have to be taken into account. The further great advance in Physics could be related to the progress of a modern mathematics: multidimensional Riemannian geometry, new theories of numbers, algebras and symmetries. Especially, we expect the powerful influence of this progress in the understanding of the basic Standard Model symmetries and beyond, in mysterious neutrino physics and its possible relation to the dark matter and dark energy, in high dimensional Gravity and Cosmology. It could lead to the further development in the understanding of our ambient space-time, the origin of the Poincar$\acute{e}$-Lorentz symmetry with the matter-antimatter asymmetry, the geometrical basis of the fundamental physical characteristics: $EM$-charge, color, spin, mass. To date it seems naturally expect that there is necessity to expand our knowledge about new geometrical Riemann and tensor structures in the multi-dimensional spaces to achieve the better understanding of the Standard Model dynamic approaches. These new geometric objects could be associated with some new types of external symmetries (symmetry vacuum), which could allow to create a “reasonable” (renormalized) quantum field theories in multidimensional spaces with $D> 4$ and to construct the multidimensional generalization of the D-dimensional pseudo-Lorentz groups, what is an essential feature of the progress in the understanding the principles of the general relativity theory. The differential equations for the propagation of waves in a hypothetical multi-dimensional space-time could have the third- or higher degree with some exotic properties as a result of observing new symmetries. There have been presented enough experimental arguments that the Special Theory of Relativity is being related to the electromagnetic charged matter what can be applied only in $D = (3 +1) $ Minkowski space-time. The special relativity theory was formulated on the basis of axioms comprise the relativity principle, absolutism and the finite speed of light. Galilean symmetry group has been extended to the group of Lorentz transformations, and Poincar$\acute{e}$ translation group, and the absolutism of time transformed into absolutism of light. Due to light synchronization in stationary system one can determine time globally. The link time and spatial coordinates between two inertial systems moving relatively to each other at a constant speed is defined by Lorentz transformations. These transformations can be built on the principle of maximal and constant speed of light and, therefore, locally determine the geometric structure of the electromagnetic vacuum, which is reflected in the fact that these transformations leave the four-dimensional interval $$ds^2=g_{\mu\nu}x^{\mu}x^{\nu}=c^2dt^2-dx_1^2-dx_2^2-dx_3^2$$ invariant. The progress with understanding the light speed axiom was gone in the direct accordance with the progress in the study the Euclidean plane axioms where changing the axiom of parallel lines had led after very long period to the discovery of Lobachevsky spaces and Riemann geometry, and eventually had led to the discovery of the special theory of relativity in Minkowsky $D=(3+1)$-space-time. It worth to note that the light speed maximum axiom can be interpreted primarily in close connection with the properties of electro-magnetic vacuum of our visible Universe. In Maxwell theory the absolutism of light speed is confirmed by identification the velocity of e.m. waves with the basic fundamental constants characterizing the internal electromagnetic vacuum structure: $$c=(\mu_0 \epsilon_0)^{-1/2}$$ The concept of the light speed absolutism in the observable Universe was especially emphasized in the analysis of Einstein’s fundamental ideas of the special relativity theory. The question of the new forms of the matter existence other than electromagnetic did not arise at those days ! The mysteries neutrino was embedded in physics later. Attempts of solving the problem of the Standard Model incompleteness were converted into multi-dimensional geometry where there could be a hypothetical sterile Matter (Dark Matter) with his “invisible” radiation in addition to the observed electromagnetic-charged matter and for the description of which there can appear the necessity to generalize some $D = (3 +1)$ axioms of the relativity theory. The basic idea of the such new phenomena discovering could be associated with the neutrino (or dark matter) since their unique properties could also spread in the space-time with one or two extra dimensions, $D = (4 +1)$ or $D = (4 +2)$, respectively. The corresponding extension of the $D=(3+1)$- special theory of relativity can lead to the possibility that for neutrino the 4-interval $ds^2$ is not invariant, what could be happened if neutrino can expand in $D>4$-space-time with the extra ’topological" cycle [@AV]. This cycle could be related with some new spatial as well as with some temporal neutrino properties. To the contrast of the spatial and temporal properties of neutrinos with respect to the similar properties of charged quarks and charged leptons there is a room to consider the observed three neutrino states as a single quantum field in the space of dimension 6, that is, with 2 additional non-compact dimensions, and, in accordance with ternary complexity, one can imagine three implementations of neutrinos as a “particle” - “anti-particle” and “anti-anti-particle” (ternary neutrino model) [@AV],[@AV2], [@Paris],[@D6] in analogy with the 4-dimensional Dirac electron-anti-electron(positron) theory( see [@Dirac], [@Majorana], [@Weyl] and for review [@Bog],[@Pal]). The EM-Charge Matter in the $D=(3+1)$ Space-Time ================================================= . Relativistic quantum electrodynamics was formulated on the basis on the internal $U (1_{EM})$- and external Lorentz $SO (3,1) $+ Poincar$\acute{e}$ (translation) group symmetries of the gauge boson and fermion fields- photon and electron/positron, respectively. The internal symmetry is related to the local and global conservation of electric charge $Q_{EM}$. The external symmetries reflect the fact that our space is isotropic and homogeneous what we observe in the form of the law’s conservations of such fundamental parameters as the angular momentum, momentum, energy, mass and life time at rest [*etc*]{} in the $D=(3 +1)$ universe. In theory of electron+positron there can be some duality links between the space-time geometrical structure and fundamental properties of the particles [@Paris],[@D6]. For example, if one knows the fundamental properties of the particles one can get the information such as the ambient space-time dimensions. So, the four-dimensional $D=(3+1)$ space-time with external Lorentz/Poincar$\acute{e}$ quantum electrodynamics symmetry correctly corresponds to the possible quantum states - electrons+positrons - having the following internal properties: two spin states plus two charge conjugated states, electron/positron. The finite discrete group symmetries related to the $C$-,$P$-,$T$- transformations make this link more subtle putting it finally to the fundamental theorem of $CPT$-invariance (see for example [@Bog]. The $CPT$-invariance proved in such spaces for local quantum theory gives the very important results such as the equality of the particle and antiparticle masses(and life time): m() = m([|]{}) binary CPT -invariance. Similar to the role of the axiom of constant speed of light in the definition of the global time the conservation laws of these symmetries allow us to determine such fundamental parameters globally in the whole space-time. CPT-invariance allows to correctly define globally the concept of a particle and its antiparticle in the whole $D=(3+1)$-space-time. In this approach the $CPT$-invariance and $Q_{em}$-conservation law can be the prerogatives for Minkowski $R^{3,1}$- space-time where the $SO(3,1)$ Lorentz group (Poincar$\acute {e}$) symmetry and $U(1_{em})$ gauge symmetry[@AV],[@AV2],[@Paris] are valid. So, we want to emphasize that the proposal about the duality between the electric charge conservation and CPT-invariance can be valid in our Minkowski space $D=(3 +1)$ only, but for the hypothetical interactions of the $Q_{em}$-charged matter with the new exotic matter, these arguments are not valid any more. In this approximation the observation of effects with CPT invariance violation and/or with $Q_{em}$ charge non conservation could indicate some new exotic geometrical vacuum structures at the smaller distances beyond the weak interaction region or/and the existence of some global extra dimensions in Universe. This observation will help us to extend the concepts of particles and antiparticles in the ternary case with 3-neutrino specie, for which a new type of hypothetical conjugation $ C_{D=6}$ can extend the concept of anti-world to the high-dimensional analogue of $D=(3+1)$-CPT-theorem [@Paris], [@D6]. The observable violation of the conservation laws in $D=(3+1)$ must be associated with some additional geometry and tensor structures of vacuum and can be linked to the appearance some hypothetical phenomena like new interactions, new particles,... In Standard Model this was vividly illustrated by physics of $K^0-\bar K^0$, $D^0-\bar D^0$, $B^0-\bar B^0$,....mixing. The possible Majorano neutrino nature (see for review [@KLAP]) among the all other kinds Dirac charged fermions prompts another dynamics of based on the composite fermion Dirac matter structure created from more simple pra-fermions like Majorano neutrino sterile matter filling the extra dimensional world- Meta-Universe. The fundamental conception such as idea is related with attempts to figure out a common the $Q_{em}$ charge Dirac matter creation mechanism with enough reasonable assumption that mechanism is such as must give a duality between the $Q_{em}$ conservation and $CPT$ invariance: $$\begin{aligned} CPT-\,invariance \qquad \leftrightarrow \qquad (Q_{em})\, \,charge\,\, conservation,\end{aligned}$$ [*i.e.*]{} the invariance of $CPT$ in $D=4$ space-time means the $Q_{em}$ charge conservation and vice versa. Thus, if this kind duality exists, the CPT-invariance violating processes should accompanied by the electromagnetic charge violation too. May be, it could be one of the reasons why we have not saw some rare decays such as the proton decay. In this case, the idea of grand unification symmetries without the electromagnetic charge origin explanation is not enough to solve the proton stability problem. Also a similar problem could be related with searches for the rare flavour-changing decay channels such as $\mu \rightarrow e + \gamma$, $\mu \rightarrow 3e$ etc . First, one must solve the origin of the quark-lepton families problem. In such approach one can propose a mechanism of the geometrical electromagnetic charge $Q_{em}$ origin and the Dirac complex matter from more fundamental pra-matter [@Paris]. The Majorano neutrinos with $m_{Dirac}(D=3+1)=0$ could be some representatives of a new matter (sterile or dark matter?). The idea can be applied to the further attempts to solve the baryon asymmetry of universe problem linking such question with an origin $Q_{em}$ charge matter in $D=(3+1)$-space-time. There is one very remarkable fact $$\begin{aligned} |Q_p+Q_e|<10 ^{-21}\end{aligned}$$ which can indicate the unique origin of the proton(quarks) and electron. It suggests an existence of a hypothetical interaction into high dimensional space connecting the Dirac-charged fermions to the pra-matter. This interaction could be based on a new symmetry beyond Lie groups and can provide the universal electron/proton non-stability mechanism [@Paris]. We add two extra dimensions to illustrate a possible mechanism of this kind interaction with a mass scale near $M_S \sim 10-20 TeV$- region [@Paris]. Note, that the 3-color “up”- and “down”- quarks states interacting via $SU(3^C)$-gauge color bosons at the corresponding distances embedded into $D=(3+1)$-space-time is connected to the problems of a new quantum charge “color” and fractional magnitudes of electromagnetic charge $Q=\pm 1/3,\pm 2/3$ origin explanations. We suppose that these problems could be closely related to a possible extension of the electromagnetic vacuum substructure and its link to the origin of the 3-quark-lepton families. One could consider some extra compactified dimensions what could change the foam structure of electromagnetic vacuum to find a new quantum number geometrical sense due to its confinement property. Thus,one should to produce the integer values of the charged leptons and the fractional values of quarks by unify way to find electromagnetic charge creation mechanism in universe. Neutrino About New Time Structures of Universe. ================================================ Exclusive properties of three neutrinos could point out the existence of a new vacuum, with properties different from the properties of the electromagnetic and color vacua. Moreover, it can give some information about the symmetry of this hypothetical vacuum that might be associated with the exceptional properties of the three neutrino states-ternary symmetry [@Paris],[@D6] in addition to the spin. This new ternary symmetry could shed light on the SM dark symmetry: $$N(Color)=N(Family)=N(dim. R^3)=3.$$ So the three types of neutrinos can be described by a single 6-dimensional wave function and it would imply the existence of two additional dimensions with appearance in $D=6$ the fundamental discrete space-time symmetries. It must be emphasized that the assumed charged matter ternary symmetry must be broken in $D=(3+1)$-space-time with all the attendant circumstances. Opposite to the 3-neutrino masses one can see the charged leptons and charged quarks grand mass hierarchies increasing with the number of the families from one to the third: $$\begin{aligned} \begin{array}{ccc} m(e)\approx0.5 MeV & m(\mu)\approx 106 MeV & m(\tau) \approx 1.7 GeV \\ m(up)\approx3.5 MeV & m(c) \approx 1250 MeV & m(T) \approx 175 Gev \\ m(down)\approx5.5 Mev & m(s) \approx 150 MeV & m(B) \approx 4.5 GeV \\ \end{array}\end{aligned}$$ Also one can see the reverse hierarchy of life times of the electromagnetic charge particles according to increasing the number of the generations: $1\rightarrow 2\rightarrow 3\rightarrow ...$. There are some peculiarities related to the mass limits what could be important for our interpretation of the weak interaction region as a boundary between two vacua: electromagnetic and new hypothetical in $D=6$-space-time. The first peculiarity requires to postulate the minimal possible mass in EM-vacuum: electron mass ? Then the “up”- and “down”- quark masses could be expressed through the electron mass and the number of colors : $$1/2(m(up)+m(down)) \approx N_c^2 m(e)$$. Then the next peculiarity is related to a trend of upper limits on the masses with increasing number of the generation. Under this circumstance it will be important to clarify the following problem: does the fourth generation exist or doesn’t? Some superstring models possessing the hypothetical family symmetry expect the fourth quark generation having some exceptional properties[@MNV3],[@MNV4] In this case one could consider the quaternary extension of ternary hidden symmetry: $$N(Q-3Color +L-1Color )=N(Fam.-3 + Ex.Fam.-1)=N(Dim. R^{3,1})=4.$$ The experimental observation of the fourth quark generation could support the idea about real role of weak interactions in the and in universe: “screening” at the very small distances beyond the weak interaction region $r \leq 10^{-17} cm$. Thus one can suggest that the electromagnetic vacuum could be defined by the light speed and by the minimal Dirac mass possible for the stable electromagnetic Universe. One of the our space-time extension possibilities could be due to a new “topological cycle”($\tau,\xi$) existence and it could be described by independent component such as new “time” coordinate ($\tau$)[@AV],[@AV2]. In this scenario, the question what is the real time raises again. These ideas implementation should require the construction of the Universe new geometric representations and, in particular, to find the Riemann metric tensor and might be another geometrical and tensor structure invariants of extended space-time. In fact, the hypothesis of the second “time coordinate” might be considered as a convenient way to describe the possible extension of the neutrino spread laws different from those projected by special relativity, for example, the light speed maximality principle. One of the main difficulty of the study the neutrino intrinsic and space-time properties connected with the considerable discrepancy between the huge experimental data for the processes with neutrinos as products of hadron’s decays and very small amount of the processes where the space-time properties of neutrinos clearly manifested . If the analysis of the myriad of the neutrino channel meson decays restore the energy and angular spectrum of the neutrino collapse the further motion evolution of this collapse can contain significant uncertainty ( see for example series of the articles devoted to the study for the formation of neutrino beams in accelerators [@BAR],[@BAR2],[@SAM],[@AFON],[@K2K],[@NuMI], [@SPS],[@SPS2]). Further only a tiny fraction of neutrinos in these collapses observed in neutrino channels can be identified via the interaction of neutrinos with detector. The ratio of the accelerator produced neutrinos in the collapses to those could be observed in the neutrino detector can be, depending on the experimental conditions, the order of $\sim 10^{7-10}$. This is especially important to review the samples of long base-line neutrino experiments in K2K-SuperKamiokande [@SKAM], FNAL NuMi-MINOS [@MINOS] and SPS CNGS-LNGS OPERA [@SPS],[@OPERA], [@BRUN]. The CNGS beam is obtained by accelerating protons to 400 GeV/c and ejecting ones into neutrino channel as two spills, each lasting $10.5 \mu s$ and separated by $50ms$.The SPS CNGS cycle is 6 s long. Each spill contains from 2100 bunches with the time substructure $3+2=5 nsec$ and intensity $\sim 10^{10}POT$. The resulting neutrino collapse is formed at the neutrino channel along a distance of $\sim 1000 m $[@SPS], [@SPS2]. The total statistics used for analysis reported in this paper [@OPERA] was $\sim 15 000$ events (from $\sim 60 000$ total events) detected in rock and in detector, corresponding to about $10^{20}$- protons on target collected during the 2009, 2010, 2011 CNGS runs and the estimation of the total work-time is about $5 \times {10}^7$ sec. So the total number of spills could be about $< \sim 10^6$ and each spill produces $< \sim 0.01 \nu $-event in detector or in rock ( the exact numbers one can see in [@BRUN]). It gives very complex problem to restore the total information about the all parameters of neutrino collapse spectrum. Naturally the question Whether is there a chance to synchronize neutrino events in such experiments to within less than time of extraction [*i.e.*]{} $\leq 10.5 \mu sec$ ? [@AV],[@AV2]. Sufficiently, Do we know well the spatial and temporal properties of neutrinos to achieve such synchronization accuracy? This occasion can bring the any kind paradoxes caused by incorrect experimentalist understanding of the space-time behaviour dynamics of the neutrino collapses based only on extremely small recorded statistics of the detected neutrino events. The main paradox of such experiments is that the results of long term studies become to be equivalent to the following inference: what had been assumed that it was received. The opportunity of a wrong interpretation of the ambiguous experimental results makes the modern experimental neutrino physics is very complex and raises such experiments at the level of art. It is well known that neutrino experiments consist of three phases: the neutrino collapse production process, its space-time spread through the matter and the possible interaction of the collapses with the detector material. The neutrino collapse dynamics moving in space-time is another major challenge because of the proposal that the neutrino is an another kind of matter representative significantly different from the electromagnetic matter. This suggests that neutrinos could spread in accordance with the new vacuum structure kinematics. Despite the existence of three quark-lepton generations the three states forming a single wave field of $D=6$-space-time evolution might be assumed and would be described by the corresponding wave equation. In a ternary model the neutrino wave field could have the own charge - “neutrino light” [@Paris],[@D6]. In this approach the neutrino field could be distributed according to the motion equations different from the equations used in $D=(3+1)$- geometry defined by the Lorentz group symmetry. It can give some new additional interpretations of the processes related to the well known neutrino oscillations. The possible extra dimensional geometry existence can lead to the circumstance that the neutrino waves could spread by geodesic lines different from the geodesic lines of the electromagnetic charged particles (see for example, [@Koka]). Appears from the above the neutrino flow cannot conserve in the $D=(3+1)$-space-time. It could be a reason of disappearance of neutrino flows at a distance. In the article [@AV] some neutrino experiments were proposed to observe the possible our space-time expansion comprising another cycle characterized by its fundamental speed which could be much faster than electromagnetic light. The last assumption was supported by some arguments to solve the horizon problem in cosmological models [@Mof]. Neutrinos due to their outstanding properties available in both cycles and the electromagnetic light speed maximality principle does not work any more. In particular, the new multi-dimensional geometrical spaces have the projective symmetries the understanding of them could help us to visualize new universes. Another factor is that the space-time expansion can carry out the introduction of new topological cosmic cycles. It means that these topological cosmic cycles may have own fundamental parameters such as “speed”, “mass”, “charge”,... Therefore, to check the hypothesis that neutrinos spread different than the light the experiment based on the possibility of measuring the neutrino speed depends on the parameters that might be related to the fundamental another cycle properties has been suggested, and we expected that dependence on such parameter one could get the neutrino speed: $v_{\nu}>c_{light}$ ( see the interesting discussions in [@MA],[@MA2],[@NA], [@GIA], [@Wolf]). Such experiments could prove the existence the new vacuum and extra dimensions directly but this way involves a very delicate element associated with synchronization “almost invisible” neutrino. In the classical experiments to measure the new fundamental constant the validity limits of the special relativity theory need to be understood. For our approach it was necessary to examine on what setting might have changed the neutrino velocity value if it really has a link to the new vacuum. The latter implies that there should be in minds some method of the possible $D=(3+1)$ space-time expansions with the corresponding metric tensor forms for such models. Otherwise such experiments can lead to the logical paradoxes. In fact, such experiments can meet the challenge of measuring the absolute velocity or absolute motion or something else. Non-Compact Large Extra Dimensions and Emergence of EM-Charge. ============================================================== The main experience we have got from , /D- branes, $D=11M$, $D=12F$- theories and from the study of the Riemann and tensor structures of the high dimensional Cartan symmetric and Berger-non-symmetric spaces is the compact small and the non-compact large dimensions are closely connected to the origin of internal and external space-time symmetries, respectively, in corresponding theories (see discussion in [@VOL3], [@VOL4]). The compact small dimensions are connected with the origin of internal symmetries. The role of the compact Calabi-Yau spaces was perfectly illustrated in the 5-superstring dual theories. Correspondingly, non compact large dimensions are related to the extra space-time symmetries of the our ambient world. For the this circumstance could be very important since we suppose that the problem of three neutrino species could be solved by adding the some global non-compact dimensions to our $D=3+1$ space-time[@AV],[@Paris]. So the family symmetry appearance can be related to the large non-compact extra dimensions like it was happened with two “families”, particle-antiparticle, and was proved by Dirac relativistic equations for the $D=3+1$- Minkowski space-time. In the past a lot of publications has been devoted to the possibility to solve the three family problems through the internal gauge family symmetries introduction. Let note that in super-string approach the $N=1SUSY$ $SU(3)_H\times U(1)_H$ gauge family symmetry appears with $3+1$ quark-lepton family [@MNV3],[@MNV4]. The possible fourth family must have the exceptional properties since this family is singlet under $SU(3)_H$-symmetry in this approach . This broken family gauge symmetry could be responsible for the mechanism of $CP$-violation in $K-$,$D-$,$B$- meson decays [@MNV], [@MNV2]. Thus one can see the common grand problem of the flavour mass hierarchy of quarks and charged leptons, family mixing, $CP$-violation that cannot be solved without understanding the role of the $(V-A)$-weak interactions. There is also the very important difference between the three charged quarks/leptons and three neutrino states: Dirac-Majorano space-time nature [@Dirac],[@Majorana], [@Weyl], their masses and etc. We can expect that for Majorano neutrino species the global family symmetry is exact... To explain the ambient geometry of our world with some extra infinite dimensions one can consider the our visible world (Universe) as just a subspace of the Meta-Universe with the pra-matter having new quantum numbers different from already known in our world. The visibility of a new world phenomena is determined by the our understanding of the SM structure and its consequences for the Cosmology processes. The Majorano neutrino can travel in this Universe![@AV] To make it available we should introduce a new space time-symmetry with the usual D-Lorentz symmetry generalization. In this case the region $ \leq M_S \sim (10-20) Tev$ could be considered as a “boundary” of a new world [@Paris]. This proposal suggests an existence in high dimensional space a class of prs-fermions connected with our fermions through a new interaction which is based on a new symmetry beyond Lie. Such interaction could predict a non-stability of the electrons or protons. We can illustrate a scale of such an interaction, taking for example, two extra dimensions. To make estimations we are in the situation in which occured E.Fermi. So we can follow to his ideas which he used for construction the four-fermion weak theory with coupling constant $G_F$, which dimension is $[G_F]=[M^{-2}]$. So we can start from multi-fermion $D=6$ Fermi Lagrangian the corresponding Fermi constant $G_{F_S}$, which should have a the dimension[@Paris]: $$[G_{F_S}]=[M^{-4}]$$ In our opinion this coupling constant dimension corresponds to a new interaction that propagator could have a form like $1/[P(q,M_S)]$, where $P(q)$ could be a polynomial of $4$-th degree. Such as propagator form corresponds to a new $D=6$-metric tensor. So, for the tree level calculations of the quark or charged lepton decays into pra-fermions $\nu_S$ $$\begin{aligned} q \mapsto n \, \nu_S, \qquad e^{\pm} \mapsto n \, \nu_S\end{aligned}$$ can get the following estimation for the partial width $$\begin{aligned} \Gamma(e/q \rightarrow n \nu_S) \sim O(g_S) \cdot\frac{m_{e/q}^9}{M_S^8},\end{aligned}$$ where $m_{e/q}$ is the electron mass, and $M_S$ is a new mass scale related to the hypothetical interaction in extra dimensional world what could be associated to the some new symmetries. What is very interesting that we can construct the universal mechanism of the decays for the all known quarks -$u,d,s,c,b,t$- and charged leptons- electron, muon, $\tau$lepton- into the $EM$-invisible matter. To get the lower boundary for $M_S$ let’s compare the partial width for electron decay with the life time of muon in frame of $D_4$-Fermi interactions: $$\begin{aligned} \frac{\Gamma(e \rightarrow 3 \nu_s)}{\Gamma(\mu \rightarrow e\nu\bar \nu)} =O(g_S/g)\frac{m_e^9}{m_{\mu}^5}\frac{M_W^4}{M_S^8} \nonumber\\\end{aligned}$$ From the lower boundary on the electron life time one can get the following upper boundary for $M_S$: $$M_S \geq O(g_S/g)\cdot (10-20) \cdot M_W.$$ This boundary has the universal magnitude what one can check from searching the baryon violation processes of neutron $$N \mapsto 3 \nu \qquad or \qquad N \mapsto \,n \nu_S .$$ Apart from charge violation decays we can expect also the the $CPT$-invariance violation processes. For example, the $M_S$ magnitude estimation can be get from the $K^0$-$ \bar K^0$- mass difference: $$\begin{aligned} \delta_m=|m-\bar m| \sim \frac{m^5}{M_S^4} < 10^{-15} GeV.\end{aligned}$$ This estimation show that the $M_S $ can be also in $1-10 TeV$ region. The Paradoxes of Theory of Relativity in Long Base Neutrino Experiments. ======================================================================== The measurements of neutrino speed on the accelerator experiments can be associated with Fizeau experiment to measure the speed of Light. Opposite to Fizeau ideology ($\{2\times 8.5 \times 720 \times 24.5\} km/sec$) in the neutrino projects [@AV] there must be studied three main discrepancies what are related to the some experimental and theoretical ambiguities. In contrast to the Fizeau experiment to measure the speed of neutrinos was a very daunting task to identify and synchronize the departure of neutrinos or neutrino wave collapse formed during the release of the accelerator proton bunch on the target. The second discrepancy was related to the understanding the structure of neutrino collapse formed in the neutrino channel and getting all its parameters. The third important discrepancy was linked to the driving dynamics of neutrino fronts propagating over long distances, [*i.e.*]{} to represent its evolution during the flight from accelerator till detector . To solve the third problem one should have the information about possible some new spatial - temporal properties of neutrinos,[*i.e.*]{} that is to construct or have some sorts of models explaining the physical reasons why the neutrinos properties could be beyond some principles of special theory of relativity, in particularly to overcome the speed of light. If you do not accept this ideology the experiments of measurements the neutrino velocity will involve with attempts to measure the “absolute motion”( Aristotle, Galileo Galilei). The main conclusion from this discussion is how to measure correctly speed of the objects with the properties completely different from the electromagnetic media (new time structure, synchronization). To make such experiments one can coincide themselves to the paradoxes of measurement the absolute movement. In electromagnetic media- universe and vacuum- only light can have the property of absolutism, $c$ is invariant fundamental constant . Between energy and wave length there exists the quantum link: $E_{\gamma}\cdot \lambda_{\gamma}=c \cdot h$, where $h$ is Plank constant. If neutrino is not related to the supersymmetry [@DVV] or some unknown yet now phenomena in the frames of the special theory of relativity there appear the ambiguous situation. Absolutism of neutrinos? In special theory we know how to synchronize the events in electromagnetic media. Now there can appear grand question how to synchronize the events in a new vacuum media. In this regard, in our projects, [@AV] we were looking for those neutrino observable parameters that could link the neutrino with a new vacuum in which the laws of the $D=(3+1)$- electromagnetic universe could not be valid any more. Other vacuums different than electromagnetic can have new fundamental properties what we suggested to check in the experiments related to the measurements the propagator properties of neutrino like its velocity. The extra dimensional (non trivial) generalization of the Lorentz group could imply the existence of another boost and possible extensions the concept of the time, even its structure. To find the confirmation of existing of exotic vacuum structure beyond the electromagnetic one should look for the experimental measuring parameters of neutrinos what could connected to the other hypothetical world. Or we can meet to the Aristotle - Galilei absolute movement problem. We know enough well the ways to synchronize electromagnetic events on the long distances using the light. But the many neutrino phenomena we don’t know yet well to be sure that we can synchronize correctly the neutrino events. In neutrino experiments made during some last years there was realized only the first part of the projects [@AV] what was based on conclusions of the the first measurement the neutrino velocity in 1977 FNAL[@KAL] and in 1987 $[{\bf 24}]+[{\bf 5]=[{\bf 11}KII+{\bf 8}IMB+{\bf 5}B]}+[{\bf 5}M]$)-neutrino events getting from SN87A and detected in KAMIOKANDAII, IMB, BAXAN, Mont-Blanc [@SN87A]. From that observations there has been done the main conclusion that the magnitude of the speed of neutrino should be very closed to the corresponding magnitude of the light speed. Since neutrino could link both worlds for neutrino the principle of absolutism of light is not valid more and we can take the energy as possible such parameter. The other possible parameter could be related to the sources of neutrinos since it gives the information about the region of neutrino production on the smaller and smaller distances. In this choice it will be important to study the possible spatial and temporal structures of neutrino fluxes having the wide energy spectrum, producing from certain sources and moving on the different distances. We can compare the ideology of neutrino measurement velocity experiments [@KAL], [@SN87A],[@MINOS],[@OPERA], [@ICARUS] to the our conceptions [@AV],[@AV2], [@Paris]. As examples we can consider the productions of neutrino fluxes in SPS CERN from regular extractions of proton bunches in which the energy of proton beams equal to $E_p=400GeV$ with regular extractions during $3nsec$ with separation $500 nsec$ and intensity about and $10^{11}POT $. For each extraction one can estimate the corresponding neutrino energy spectrum of $10^{8,9}$ neutrinos with the primary period about $3 nsec$. For us it will be important to make the analysis of time and spatial expanding of this bunch on some different distances: $83 km,366km,732 km,1464 km$. In line with the hypothesis about the existence of $(1+1)$- extra dimensions we suggested that the expanding of neutrino fluxes depends on some parameters, $\{ t,L,E_{\nu},k_i(r),c_i(y)\}$, where $t,L,E_{\nu}$ are ordinary parameters what we can measure in the experiments, parameters $k_i$ are related to the type of sources of neutrino (muon, $\pi$- and $K$- mesons, $charm$, beauty, - quarks ,..., parameters $c_{i}$ should be directly linked to the fundamental characteristics of a new vacuum, depending on the type of a new hypothetical metric tensor. As a result of this approach the spread of the neutrino collapses will be completely different from the flow of electromagnetically charged particles, [i.e.]{} the expansions of neutrinos must be beyond the laws determined by standard Lorentz $D=(3+1)$-metrics[@AV]. As we approached the speed of the neutrino is not absolute characteristic for the electro-magnetic vacuum there was made the assumption that the neutrino energy could one of the parameters binding our vacuum with a new vacuum. We can propose that the speed of neutrino could effectively depend on the neutrino energy what we were searching through ternary or quaternary extensions of $D=(3+1)$-metrics what can be constructed in $R^n$-spaces, ($n \geq 5$) [@D6], [@VOL]: $$ds^2=f_1(y_a)ds_{3,1}^2+f_2(x_{\mu})ds_y^2.$$ As a possible variant, one can consider that the speed of neutrino is the product of electromagnetic charged particles could have some deviations from the speed of light: $ (v_{\nu}/c-1)_{i} \approx k_i \cdot (E_{nu}/M_S)^2$, where parameter $k_i$ is determining by the region on neutrino production, for example, in our examples we consider two cases, neutrino from $\pi$ - and $K$-meson decay for which $r\sim 1/m_{\pi}$ and $r\sim 1/m_K$, correspondingly. In our scenario, the behaviour of the neutrino velocity at super-high energies could be in accordance with the formula $v_{\nu}\rightarrow c_{new} [E^2/(E^2+M_S^2)]$, where new hypothetical velocity constant could be much more than light speed, $c_{new}>>c $ ($c_{new}/c\geq 10^{7-8}$?). The proposed dependence of neutrino speed from the energy leads to the substantial change of the spatial and temporal picture of the neutrino collapses. Thus according to articles [@AV] we consider two variants of taking parameters what could help to observe the effects of existing a new space-time vacuum structure: - [1. energy of neutrinos]{},\ - [2. parents of neutrino production]{}.\ For illustrations in the first case we consider the neutrinos formed only from $\pi$-meson decays in CNGS neutrino channel formed as a result of the discharge of protons on target with the energy of 400 $GeV$ in certain time intervals(see fig.[(1),fig.[2]{},fig.[3]{},fig.[4]{}]{} and fig.[(5)]{}. For this case the coefficients $k_{\pi}$ is related to the magnitude of the wave function distribution of $up$- and $down$- quarks inside the $\pi$-meson structure region. For the second case we considered the possible parents of neutrino- kaons(see fig.[6]{}.,fig.[7]{},fig.[8]{},fig.[9]{} In this case we take into account both variants-energy and origin of production . The distribution of the neutrino energy formed from lepton- and semi-lepton- decays K-meson decays for which we take the following coefficients for $ k_{K}/k_{\pi}\sim (m_{K}/m_{\pi})^2$ . According to [@AV] the neutrino velocity effect should more significant for neutrinos produced from the heavy quark decays. This one can see also on the figures fig.[(10)]{} where the temporal spread of the $\nu_{K}$- neutrino collapses can be much more than it was in $\nu_{\pi}$- cases. Combining both cases one can consider for neutrino velocity the more strongest energy dependence like as $v_{\nu}/c \sim (E_{\nu}/M_S)^4$ ( see the fig.[11]{} and fig.[12]{}). For illustration we can give the distributions of $\nu$-fluxes from $SN87A$ at $M_S=0.1-,0.2-,0.5-,1 TeV$(see fig[(9)]{}). This cosmic experiment may be the first time gives us a hint about what the neutrinos cross the huge spatial and time intervals according to other laws. To draw concrete conclusions from this experiment is difficult to do. Any predictions depend on the theory of stellar evolution, and the structure of the tremendous medium through which the neutrino waves swept generated in the depths of supernova [@SN87A]. ![The temporal distributions of the intensity of the neutrino fluxes for the only one bunch of the output proton beam at energies $E=400 GeV$ [@SPS]. The duration of one bunch is equal to 3 nanoseconds, the gap between the neighboring bunches equals 500 nanoseconds. Consider the case of formation of the neutrino fluxes from $\pi$-meson decays. Four distributions $\nu_{\pi}$- fluxes at the different scales: $M_S=1-,2-,5-,10- TeV$ on the distance $183km$.[]{data-label="overflow"}](183ms-14.eps){width="120mm"} ![The temporal distributions of the intensity of the neutrino fluxes for the only one bunch of the output proton beam at energies $E=400 GeV$ [@SPS]. The duration of one bunch is equal to 3 nanoseconds, the gap between the neighboring bunches equals 500 nanoseconds. Consider the case of formation of the neutrino fluxes from $\pi$-meson decays. Four distributions for $\nu_{\pi}$- fluxes at the different scales: $M_S=1-,2-,5-,10- TeV$ on the distance $366km$.[]{data-label="overflow"}](366ms-24.eps){width="120mm"} ![The temporal distributions of the intensity of the neutrino fluxes for the only one bunch of the output proton beam at energies $E=400 GeV$ [@SPS]. The duration of one bunch is equal to 3 nanoseconds, the gap between the neighboring bunches equals 500 nanoseconds. Consider the case of formation of the neutrino fluxes from $\pi$-meson decays. Four distributions for $\nu_{\pi}$- fluxes at the different scales: $M_S=1-,2-,5-,10- TeV$ on the distance$732km$.[]{data-label="overflow"}](732ms-24.eps){width="120mm"} ![The temporal distributions of the intensity of the neutrino fluxes for the only one bunch of the output proton beam at energies $E=400 GeV$ [@SPS]. The duration of one bunch is equal to 3 nanoseconds, the gap between the neighboring bunches equals 500 nanoseconds. Consider the case of formation of the neutrino fluxes from $\pi$-meson decays. Four distributions for $\nu_{\pi}$- fluxes at the different scales: $M_S=1-,2-,5-,10- TeV$ on the distance $1464km$.[]{data-label="overflow"}](1464ms-224.eps){width="120mm"} ![The temporal distributions of the intensity of the neutrino fluxes for 5-bunches from the output proton beam with energy $E=400 GeV$ at a distances of 366km and 732 km. Ms= 1 Tev, 2 TeV.The duration of one bunch is equal to 3 nanoseconds, the gap between the neighboring bunches equals 500 nanoseconds. Consider the case of formation neutrino fluxes from $\pi$-meson decays. []{data-label="overflow"}](5bunch1.eps){width="120mm"} ![The temporal distributions of the intensity of the neutrino fluxes for the only one bunch of the output proton beam at energies $E=400 GeV$ [@SPS]. The duration of one bunch is equal to 3 nanoseconds, the gap between the neighboring bunches equals 500 nanoseconds. Consider the case of formation of the neutrino fluxes from $K$-meson decays. Four distributions for $\nu_{K}$- fluxes at the different scales: $M_S=2-,5-,10-,20- TeV$ on the distance $183km$.[]{data-label="overflow"}](183msk-14.eps){width="120mm"} ![The temporal distributions of the intensity of the neutrino fluxes for the only one bunch of the output proton beam at energies $E=400 GeV$ [@SPS]. The duration of one bunch is equal to 3 nanoseconds, the gap between the neighboring bunches equals 500 nanoseconds. Consider the case of formation of the neutrino fluxes from $K$-meson decays. Four distributions for $\nu_{K}$- fluxes at the different scales: $M_S=2-,5-,10-,20- TeV$ on the distance $366km$.[]{data-label="overflow"}](366msk-14.eps){width="120mm"} ![The temporal distributions of the intensity of the neutrino fluxes for the only one bunch of the output proton beam at energies $E=400 GeV$ [@SPS]. The duration of one bunch is equal to 3 nanoseconds, the gap between the neighboring bunches equals 500 nanoseconds. Consider the case of formation of the neutrino fluxes from $K$-meson decays. Four distributions for $\nu_{K}$- fluxes at the different scales: $M_S=2-,5-,10-,20- TeV$ on the distance $732km$.[]{data-label="overflow"}](732msk-14.eps){width="120mm"} ![The temporal distributions of the intensity of the neutrino fluxes for the only one bunch of the output proton beam at energies $E=400 GeV$ [@SPS]. The duration of one bunch is equal to 3 nanoseconds, the gap between the neighboring bunches equals 500 nanoseconds. Consider the case of formation of the neutrino fluxes from $K$-meson decays. Four distributions for $\nu_{K}$- fluxes at the different scales: $M_S=2-,5-,10-,20- TeV$ on the distance $1464km$.[]{data-label="overflow"}](1464msk-14.eps){width="120mm"} ![Distributions of $\nu$-fluxes from $SN87A$ at $M_S=0.2-,0.3-,05-,1 TeV$[]{data-label="overflow"}](parseceps-24.eps){width="150mm"} ![The temporal distributions of the intensity of the neutrino fluxes for the only one bunch of the output proton beam at energies $E=400 GeV$ [@SPS]. The duration of one bunch is equal to 3 nanoseconds, the gap between the neighboring bunches equals 500 nanoseconds. Consider the case of formation of the neutrino fluxes from $\pi$-meson decays. Four distributions for $\nu_{\pi}$- fluxes at the different scales: $M_S=0.1-,0.2-,0.5-,1- TeV$ on the distance $732km$.[]{data-label="overflow"}](732Ms445.eps){width="120mm"} ![The temporal distributions of the intensity of the neutrino fluxes for the only one bunch of the output proton beam at energies $E=400 GeV$ [@SPS]. The duration of one bunch is equal to 3 nanoseconds, the gap between the neighboring bunches equals 500 nanoseconds. Consider the case of formation of the neutrino fluxes from $\pi$-meson decays. Four distributions for $\nu_{\pi}$- fluxes at the different scales: $M_S=0.1-,0.2-0.5-,1- TeV$ on the distance $1464km$.[]{data-label="overflow"}](1464Ms45.eps){width="120mm"} Conclusions. Towards the Super-High Energy Neutrino Experiments =============================================================== Comparing with our ideas in [@AV] the conclusions obtained from the data analysis of experiments [@MINOS], [@OPERA], [@ICARUS] demand a comprehensive reconsideration. From these experiments it is not clear what they measured- neutrino or light velocity? The conclusions depend on some proposals of neutrino spatial and temporal properties. In such kind of experiments the main accent should been done on the attempts of the synchronization problem solution for two “neutrino” events: production and detection. However, to solve this problem in these experiments it was assumed that the spatial-temporal behaviour of neutrinos is determined by the Lorentz-Poincar${\acute e}$ group symmetry $\psi_{\nu}\sim (\exp{i k \cdot x}$). It wasn’t taken into account that neutrinos can have extraordinary spatial and temporal properties different from the charged matter properties. For example, in ternary or in quaternary description the following plane wave generalization can be taken in the following form [@D6],[@VOL], [@VOL2]: $$\psi_{\nu}\sim \exp\{{\bf I_1} (\hat k \cdot \hat x) + {\bf I_2} (\hat p \cdot \hat y)+{\bf I_3}({\hat m}(p,k)\cdot {\hat s}(x,y))\}.$$ Therefore, the reached conclusions on the neutrino velocity measurement cann’t be interpreted unambiguously and it was illustrated in the previous section. It can be taken into account that these experiments collide with the absolute movement paradoxes. The results of such kind experiments correspond to common supposition that the neutrino collapses propagate through the space similar to the light fluxes. The good time resolution was achieved in these experiments to measure the speed of neutrinos but the energy resolution was not sufficiently precise for the proposed synchronization problem solution ideas. The energy has been measured with poor accuracy at least 20 percent. This is due to the several reasons: - [1.The bad identification of the charged and neutral currents. In the neutral current neutrino takes a lot of energy]{}\ - [2. The carried away energy of neutrinos is poorly defined.]{}\ - [3. The $\pi$-meson and the proton are often not distinguished.]{}\ - [4. The observation of several events, secondary events are mixed up. ]{}\ the detector containing passive elements can increase the number of problems with the neutrino energy definition. There is also the separation problem of the rock neutrino events. It would be important to study the dependence of registered neutrino fluxes on the energy and distance for the correct neutrino spectra normalization. In fact, the experimental limits on the speed of neutrinos obtained in these experiments can not be proven as the maintained synchronization depends on the specific assumptions. In our opinion, the problem that was posed in the works [@AV] requires a whole series of serious studies under different conditions: the wide energy range of the proton spills till some TeV, the short and the long distances, different time length of proton spills, different neutrino specie, etc. There might be very important the questions connected to the decrease of the flux intensity dependent on the distance what can be occur due existence of extra dimensions and it is naturally expected in the ternary model neutrino. So the neutrino flux evolution with the distance and the energy can also test the hypothesis of non-compact extra dimensions. The considered scenarios showed that the solution of the neutrino phenomena mysteries still needs new and the more detail experiments. The our examples show that it is better to carry out the neutrino measurements at the more higher energies. The proton/electron stability provides the upper bound of $M_S\sim 20 TeV$ [@Paris]. Since the other models can be developed the investigation of this class of neutrino experiments needs to be continued. In our projects [@AV], the suggestions on measurement of some global neutrino flux motion characteristics in spatial-temporal picture have been done. We knew that in the case of neutrinos the successful implementation of these projects is determined by the our knowledge of or the correct ideas on the actual neutrino properties as a particle and a wave or some its generalization. Since it was suggested that the effects of advancing the speed depends on the neutrino energy is natural to consider the neutrino programs at the high or the extremely high energies. The experiments at $\sim O(100-1000)GeV$ [@AV] could be already done in the nearest future at the accelerator modern complexes of FNAL and of CERN. One can say that such kind of experiments are linked to the another class of neutrino experiments at high energies opposite to the ideology of the experiments going to measure the neutrino oscillations. We should note that this class of neutrino experiments at super-high energy and on different bases including very long distances can be very important for the future of the SM-physics and beyond. For example, the experiments on the total cross section measurement of neutrino interactions in dependence on the distance between the source of neutrino beams and detectors might be done The neutrino experiments at the high or the super-high energies will be able to expand the progress possibilities in discovering the secrets of neutrinos significantly if they will n’t not play a decisive role. In our opinion, the two programs of the low-energy neutrino physics associated with the study of the oscillations,[@SKAM],[@MACRO],[@SOUDAN],[@MINOS2], [*etc*]{} and the neutrino physics at the ultra-high energies [@BAR2],[@AV] will only complement each other. Acknowledgements {#acknowledgements .unnumbered} ================ We would like to express his acknowledgements to colleagues in TH-PNPI Ya. Azimov, G. Danilov, A. Erikalov, L. Lipatov, S. Trojan and others for very useful discussions on this subject. Also we thank F. Anselmo, N. Budanov, M.Chernecov, A. Kisselev, A. Liparteliani, E. Slobodyuk for many interesting discussions and support. We thank D.I. Patalakha for reading English version of manuscript and making some useful comments. This is the extension and more precise version of our talk on the SHQCD12-Conference, Gatchina, St-Peterburg,12-16 July 2012. [99]{} P.A.M. Dirac, [*The quantum theory of electron*]{}, Proc. Roy. Soc. [**A117**]{} (1928) 610.\ E. 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--- abstract: 'Quantum state discrimination for two coherent states with opposite phases as measured relative to a reference pulse is analyzed as functions of the intensities of both the signal states and of the reference pulse. This problem is relevant for Quantum Key Distribution with phase encoding. We consider both the optimum measurements and simple measurements that require only beamsplitters and photodetectors.' author: - | S.J. van Enk\ Bell Labs, Lucent Technologies, Room 2C-401\ 600-700 Mountain Ave, Murray Hill NJ 07974 title: Phase measurements with weak reference pulses --- Introduction ============ Suppose we are given a light pulse in one of two possible coherent states $|\alpha\rangle$ or $|-\alpha\rangle$ (with $\alpha$ real and positive) and we are to guess which one of the two we have in our possession. The minimum error probability is [@helstrom] $$\label{min} P_{{\rm min}}=\frac{1-\sqrt{1-|\langle \alpha|-\alpha\rangle |^2}}{2} =\frac{1-\sqrt{1-\exp(-4\alpha^2)}}{2}.$$ A measurement that achieves this minimum error probability using linear optics, photon counters and feedback, but which is hard to implement, was given by Dolinar in [@dolinar]. An alternative optimum scheme, not requiring feedback, but still complicated, was considered in [@sasaki] (see also [@osaki]). A much simpler “near-optimum” scheme using linear optics, photon counters but no feedback was presented by Kennedy in [@kennedy], which achieves an error probability of $$\label{Ken} P_{{\rm Ken}}=\frac{\exp(-4\alpha^2)}{2}.$$ This scheme is near optimal in the sense that in the limit of $\alpha\rightarrow\infty$ (the relevant limit for [*classical*]{} communication) $P_{{\rm Ken}}\rightarrow 2P_{{\rm min}}$. The standard technique of homodyne detection [@yuen] would lead to an error probability (for more details, see below) $$\label{hom} P_{{\rm hom}}=\frac{1}{\sqrt{2\pi}}\int_{2\alpha}^{\infty}\! {\rm d}\tau \exp(-\tau^2/2),$$ which is clearly inferior to the Kennedy measurement for large amplitudes, but which is superior (although not optimal) for small amplitudes $\alpha$, which is the relevant limit for [*quantum*]{} communication. In particular, experimental implementations of Quantum Key Distribution protocols (see for example [@QKD]), rely on the use of weak laser pulses with average numbers of photons of $\alpha^2\approx 0.1-0.2$ with the express goal of producing nonorthogonal and hence not perfectly distinguishable quantum states. Now lasers actually do not produce coherent states but mixtures of coherent states described by density matrices of the form[@moelmer] $$\rho_{|\alpha|}=\int \frac{{\rm d}\phi}{2\pi} |\alpha e^{i\phi}\rangle \langle \alpha e^{i\phi}|.$$ The “phase” of a laser field can only be defined relative to another laser beam. The problem of distinguishing two phases $\phi_0=0$ and $\phi_1=\pi$ of a faint laser beam with amplitude $\alpha$ in the presence of a reference pulse with amplitude $\beta$ can be formulated as distinguishing two mixed states $\rho_0$ and $\rho_1$ given by[@enkfuchs] $$\begin{aligned} \label{ba} \rho_k=\int \frac{{\rm d}\phi}{2\pi} |\beta e^{i\phi}\rangle \langle \beta e^{i\phi}| \otimes |\alpha e^{i(\phi+\phi_k)}\rangle \langle \alpha e^{i(\phi+\phi_k)}|,\end{aligned}$$ for $k=0,1$. The detection schemes mentioned above assume that an absolute phase standard is present and indeed explicitly require an in principle infinite amount of auxiliary light with a known phase (i.e., $\beta\rightarrow\infty$). But suppose one has at one’s disposal only a phase reference pulse of finite amplitude $\beta$. How well can one distinguish the two phases given this restricted resource? This problem shows up in QKD where phase is used to encode information. In such a case a reference pulse is sent along with the signal pulse, typically over the same fiber. But the reference pulse may not be chosen arbitrarily strong as some of that light may cross over to and thus contaminate the signal. Interestingly, even when polarization is the degree of freedom encoding the information the light pulses are properly described by states of the form (\[ba\]) with $\beta=\alpha$ [@norbert]. In Section \[LO\] we consider simple measurements that require linear optics and photon counters but no feedback. We generalize Kennedy’s measurement and homodyne detection to setups with finite reference pulses. We also construct a whole class of measurements that includes those two measurements and other improved measurements. In Section \[OPT\] we consider the optimum measurement for quantum state discrimination of the states (\[ba\]). Measurements with linear optics {#LO} =============================== Generalized Kennedy measurement ------------------------------- Kennedy’s measurement combines the unknown state $|\pm \alpha\rangle$ on a beamsplitter with a reference beam with amplitude $\beta=r\alpha/t$, where $r$ and $t$ are the absolute values of the reflection and transmission coefficients of the beamsplitter. In the limit $t\rightarrow 0$ one of the output ports will either have a coherent state of amplitude $2\alpha$ or the vacuum, depending on the phase of the unknown coherent state (the other output is useless). If a photon is detected in that port, one is certain to have the state $|-\alpha\rangle$, if no photon is registered one guesses that the unknown state is $|\alpha\rangle$. The probability of a wrong guess is then given by (\[Ken\]). The limit of $t\rightarrow 0$, however, implies $\beta\rightarrow\infty$. Given a finite amount of light, we can generalize the Kennedy measurement by requiring that in the useful output port the amplitudes cancel if the state is $|\alpha\rangle$. This requires we keep the relation $\beta=r\alpha/t$. One output port will then either contain the vacuum or a coherent state with amplitude $2r\alpha$. The corresponding error probability is then $$\label{Ken2} \tilde{P}_{{\rm Ken}}=\frac{\exp(-4r^2\alpha^2)}{2}.$$ For consistency the reflection coefficient for finite $\beta$ has to be chosen as $$\label{t} r^2=\frac{\beta^2}{\alpha^2+\beta^2}.$$ The error probability reduces to (\[Ken\]) in the limit $\beta\rightarrow\infty$ and reduces to 1/2 for $\beta\rightarrow 0$, as it should. Viewed as a function of $\alpha$ and $\beta$, (\[Ken2\]) is symmetric in its arguments because the measurement procedure is symmetric in the signal and reference states: the phase of one is defined only relative to the other. For the small values of $\alpha$ we are interested in, all error probabilities are close to 1/2. A better measure to compare different probabilities may be generically defined as $$D=1-2P,$$ in terms of the corresponding error probability. $D$ may be interpreted as a measure of distinguishability and ranges between 0 for identical states and 1 for orthogonal states. See Fig. \[PKEN1\] for plots of both the error probabilities and the corresponding measure of distinguishability for the (generalized) Kennedy measurement. If we consider signal states with $\alpha^2=0.1$ the Figure shows that 10 photons in the reference pulse are sufficient to be within 1% of the best achievable (for either measure) for the Kennedy measurement, and even just 1 photon brings one within 7%. Generalized homodyne detection ------------------------------ In a homodyne detection scheme one splits the unknown coherent state on a 50/50 beamsplitter with a reference light beam of amplitude $\beta$ and measures the difference in photon number between the two output ports. The two output modes are in coherent states with amplitudes $(\beta\pm \alpha) /\sqrt{2}$. In the limit of $\beta\rightarrow\infty$ the difference between the expected photon numbers $|\beta\pm \alpha|^2/2$ becomes linear in $\alpha$ and thus homodyne detection directly measures the amplitude. For finite $\beta$ we use a similar strategy: we guess that the output port with the larger number of detected photons is associated with an amplitude $(\beta+\alpha)/\sqrt{2}$ (in the case of an equal number of photons we make a random guess). The probability to detect $n$ photons in a coherent state with amplitude $(\beta\pm\alpha)/\sqrt{2}$ is $$P_\pm(n)=\exp(-N_\pm)\frac{N_\pm^n}{n!},$$ in terms of the expected number of photons $N_\pm=|\beta\pm\alpha|^2/2$. Our procedure gives a wrong result if the larger coherent state is found to contain fewer photons than the smaller one. Moreover, if we find an equal number of photons we will have to make a random guess. The total error probability, therefore, is $$\begin{aligned} \label{Pd} \tilde{P}_{{\rm hom}}&=&\sum_{n=0}^\infty \sum_{m=n+1}^\infty P_+(n)P_-(m) +\frac{1}{2}\sum_{n=0}^\infty P_+(n)P_-(n).\end{aligned}$$ Just as for the generalized Kennedy measurement, this probability function is symmetric in $\alpha$ and $\beta$. It is plotted in Fig. \[Phom1\]. Just as for the Kennedy measurement a reference pulse containing 10 photons is sufficient to reach a distinguishability of 99% of the best a homodyne measurement can achieve, while a reference pulse with just 1 photon on average brings one to within about 15% at $\alpha^2=0.1$. For completeness we note that in the limit $\beta\rightarrow\infty$ both probability distributions $P_+(n)$ and $P_-(n)$ approach Gaussian distributions. That is, defining continuous variables $x_{\pm}=(n-N_\pm)/\sqrt{2N_\pm}$ we get $$P_{\pm}(x_{\pm})\rightarrow \frac{1}{\sqrt{2\pi}}\exp(-x_\pm^2),$$ with the variable $x_{\pm}$ ranging from $-\infty$ to $\infty$. The probability (\[Pd\]) reduces then to $$\begin{aligned} \label{Pc} \tilde{P}_{{\rm hom}}&\rightarrow& \frac{1}{2\pi}\int_{-\infty}^\infty {\rm d}x_- \exp(-x_-^2) \int_{x_--2\alpha}^\infty {\rm d}x_+ \exp(-x_+^2)\nonumber\\ &=&\frac{1}{\sqrt{2\pi}}\int_{2\alpha}^\infty {\rm d}x_+ \exp(-x_+^2/2),\end{aligned}$$ which confirms (\[hom\]). Class of generalized measurements --------------------------------- The generalized Kennedy and homodyne measurements are special cases of a whole class of similarly straightforward measurements. We can take [*any*]{} beamsplitter with arbitrary transmission and reflection coefficients whose absolute values $t,r$ can be parametrized without loss of generality as $$r=\cos(\phi);\,\,t=\sin(\phi)\,\,0\leq \phi\leq\pi/4.$$ The expected numbers of photons in the two output ports, $$\begin{aligned} N^{(1)}_{\pm}&=&\big|r\beta\pm t\alpha\big|^2\nonumber\\ N^{(2)}_{\pm}&=&\big|t\beta\mp r\alpha\big|^2,\end{aligned}$$ respectively, depend on the phase of the unknown state. If we found $n$ photons in the first detector, $m$ in the second, we calculate the joint probabilities $$P_{\pm}(n,m)=P^{(1)}_{\pm}(n)P^{(2)}_{\pm}(m),$$ where $$P^{(k)}_{\pm}(n)= \exp(-N_\pm^{(k)} )\frac{\left(N_\pm^{(k)}\right)^n}{n!}\,\,\,(k=1,2)$$ If $P_+(n,m)>P_-(n,m)$ we guess that we had the state $|\alpha\rangle$, if $P_+(n,m)<P_-(n,m)$ we guess that we had the state $|-\alpha\rangle$, if the two probabilities happen to be equal we make a random guess. This corresponds to maximizing the [*conditional*]{} probabilities for the unknown state to be $|\pm \alpha\rangle$ given $n$ and $m$ clicks in the respective photodetectors. The error probability is $$\begin{aligned} \tilde{P}(\phi)&=&\frac{1}{2}\sum_{P_->P_+} P_{+}(n,m)+\frac{1}{2}\sum_{P_+>P_-} P_{-}(n,m) \nonumber\\&&+\frac{1}{2}\sum_{P_+=P_-}P_+(n,m),\end{aligned}$$ where the first summation runs over all pairs $n,m$ such that $P_-(n,m)>P_+(n,m)$, etcetera. The generalized homodyne and Kennedy measurements are special cases of this general measurement for angles $\phi_{{\rm hom}}=\pi/4$ and $\phi_{{\rm Ken}}=\arctan (\alpha/\beta)$. In Figures \[phi1\] and \[phi10\] we plot the error probability $\tilde{P}(\phi)$ as a function of $\phi$ for the case of $\alpha^2=0.1$ and $\beta^2=1$ and 10, respectively. For this small value of $\alpha$ homodyne detection is better than Kennedy’s measurement and for certain values of $\phi$ the error probability $\tilde{P}(\phi)$ is even smaller. For $rt\neq 0$ the probability distributions again approach Gaussian distribution functions in the limit $\beta\rightarrow\infty$. Remarkably, $\tilde{P}(\phi)$ approaches $P_{{\rm hom}}$ for any nonzero $\phi$ in that same limit, but the limit is not uniform. Although an arbitrary beamsplitter with nonzero reflection and transmission coefficients does not improve upon standard homodyne detection in the limit $\beta\rightarrow \infty$, for finite values of $\beta$ improvement is in fact possible as illustrated in Fig. \[phi10\]. Optimum measurement {#OPT} =================== The measurements considered in the previous Section were chosen for their simplicity but do not allow one to achieve the minimum error probability. For the case of pure coherent states the minimum error probability was given in (\[min\]), its generalization to mixed states is[@fuchs] $$\label{err} P_{{\rm err}}=\frac{1}{2}-\frac{1}{4}{\rm Tr} |\rho_1-\rho_0|,$$ which gives the minimum error probability for distinguishing two mixed states that are [*a priori*]{} equally likely. This expression reduces to (\[min\]) for $\beta\rightarrow\infty$. The density matrix $\rho_1-\rho_0$ can be written in the number-state basis as $$\begin{aligned} \rho_1-\rho_0&=&\int \frac{{\rm d}\phi}{2\pi} |\beta e^{i\phi}\rangle \langle \beta e^{i\phi}| \nonumber\\ &&\otimes \big[|\alpha e^{i\phi}\rangle \langle \alpha e^{i\phi}|- |-\alpha e^{i\phi}\rangle \langle -\alpha e^{i\phi}|\big]\nonumber\\ &=&e^{-\alpha^2-\beta^2} \sum_{n,m,p,q}\delta(n-m+p-q)\big(1-(-1)^{p+q}\big)\nonumber\\ &&\frac{\beta^{n+m}\alpha^{p+q}}{\sqrt{n!m!p!q!}}|n\rangle\langle m|\otimes |p\rangle\langle q|.\end{aligned}$$ Here we are mostly interested in the limit of small $\alpha$. The lowest-order term in an expansion in powers of $\alpha$ of the density matrix $\rho_1-\rho_0$ is linear in $\alpha$ and contains two terms: $$\begin{aligned} \rho_1-\rho_0&\approx& e^{-\beta^2}\sum_{n\geq 0}\frac{2\beta^{2n+1}\alpha}{\sqrt{n!(n+1)!}}\nonumber\\ &&|n\rangle\langle n+1|\otimes |1\rangle\langle 0|+ |n+1\rangle\langle n|\otimes |0\rangle\langle 1|.\end{aligned}$$ Its eigenvectors are then of the form $$\label{eig} |\psi_n^{\pm}\rangle=\frac{|n\rangle\otimes|1\rangle\pm |n+1\rangle\otimes|0\rangle}{\sqrt{2}},$$ with eigenvalues $$\lambda_n^{\pm}=\pm\frac{2\beta^{2n+1}\alpha e^{-\beta^2} }{\sqrt{n!(n+1)!}},$$ where $n\geq 0$ an integer. The optimum measurement achieving the minimum error probability is then a projective measurement onto the eigenstates (\[eig\]). It is an open question how this measurement can be implemented. In the limit that $\alpha=\beta\rightarrow 0$ the optimum measurement is equivalent to the generalized Kennedy and homodyne detection schemes. We can now evaluate $D_{{\rm err}}=1-2P_{{\rm err}}$ as $$D_{{\rm err}}=\frac{1}{2}\sum|\lambda_n^{\pm}| \approx 2\alpha e^{-\beta^2}\sum_{n\geq 0}\frac{\beta^{2n+1}}{\sqrt{n!(n+1)!}}.$$ In Fig. \[Trrho\] we plot the ratio $D_{{\rm err}}/D_{{\rm min}}$ as a function of $\beta^2$, where $D_{{\rm min}}=2\alpha$ corresponds to the limit of $\beta\rightarrow\infty$ and $\alpha$ small. Conclusions =========== We considered the question how well one can distinguish two faint laser pulses with opposite phases as a function of the intensity of the phase reference pulse. It turns out that even with a reference pulse containing just 1 photon on average one does reasonably well. For example, take a signal state that possesses 0.1 photons on average. Then, depending on what measurement one considers, the distinguishability is somewhere between 75% and 95% of the best achievable with an infinite reference pulse. We considered generalizations of Kennedy’s measurement and homodyne detection, and the optimum measurement. Acknowledgements {#acknowledgements .unnumbered} ================ I thank Chris Fuchs for useful discussions. [99]{} , Academic Press, New York, (1976). S.J. Dolinar, MIT Res. Lab. Electron. Quart. Prog. Rep. No. 111, 115 (1973); For a formulation of this measurement in terms of POVMs, see A. S. Holevo, Izvestiya VUZ. Matematika (Sov. Math.), [**26**]{}, 3 (1982). M. Sasaki and O. Hirota, Phys. Rev. A [**54**]{}, 2728 (1996). M. Osaki, M. Ban, O. Hirota, Phys. Rev. A [**54**]{}, 1691 (1996). R.S. Kennedy, MIT Res. Lab. Electron. Quart. Prog. Rep. No. 108, 219 (1973). H. P. Yuen and J. H. Shapiro, IEEE Trans. Inform. Theory, IT-26,(1980); H. P. Yuen and V. W. S. Chan, Opt. Lett., [**8**]{}, 177 (1983). Z. Y. Ou and L. Mandel, J. Opt. Soc. Am. [**B4**]{}, 1574 (1987). P.D. Townsend, Opt. Fiber. Tech. [**4**]{}, 345 (1998); R.J. Hughes, G.L. Morgan, and C.G. Peterson, J. Mod. Optics [**47**]{}, 533 (2000); N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, [quant-ph/0101098]{}. K. Mølmer, Phys. Rev. A [**55**]{}, 3195 (1997); J. Mod. Optics [**44**]{}, 1937 (1997). S.J. van Enk and C.A. Fuchs, Phys. Rev. Lett. [**88**]{}, 027902 (2002); Quantum Information and Computation, [**2**]{}, 151 (2002). M. Dusek, M. Jahma, and N. Lütkenhaus, Phys. Rev. A [**62**]{}, 022306 (2000). C.A. Fuchs, quant-ph/9611010.
--- abstract: 'A significant accumulation of matter in solid 4 observed during the superflow events, dubbed as the giant isochoric compressibility (or the syringe effect), is discussed within the model of dislocations with superfluid core. It is shown that solid 4 in a contact with superfluid reservoir can develop a bistability with respect to the syringe fraction, with the threshold for the bias by chemical potential determined by a typical free length of dislocations with superfluid core. The main implications of this effect are: hysteresis and strongly non-linear dynamical behavior leading to growth, proliferation and possibly exiting from a crystal of superclimbing dislocations. Three major channels for such dynamics are identified: i) injection and inflation of the prismatic loops from the boundary; ii) Bardeed-Herring generation of the loops in the bulk; iii) helical instability of the screw dislocations. It is argued that the current experiments are likely to be well in this regime. Several testable predictions for the time and the bias dependencies of the dynamics are suggested.' author: - 'A. B. Kuklov' title: 'Giant isochoric compressibility of solid 4: the bistability of superlcimbing dislocations' --- Introduction ============ The superflow through solid 4 observed first in the UMASS group [@Hallock] and then confirmed by other groups [@Beamish; @Moses] is now at the focus of the experimental and theoretical efforts in the field of superfluidity and quantum crystals. One of the striking features is the syringe effect (or the giant isochoric compressibility) [@sclimb]. In its essence, a solid exhibits the response on external chemical potential applied at a point, practically, the same way as liquid does – absorbs or expels a macroscopic fraction of atoms. As it has been suggested in Ref.[@sclimb], this effect can be associated with the so called superclimb of edge dislocations – the climb supported by the superfluid transport along dislocation core. The unusual feature of this scenario is that the linearized isochoric compressibility of a solid permeated by a network of dislocations with superfluid core is independent of density of the superclimbing dislocations and is, instead, determined by the dimensionless parameter – the asymmetry between lengths of superclimbing and non-superclimbing parts. This implies that the effect is strongly non-perturbative, that is, it cannot be treated as a small correction with respect to dislocation density. In particular, as shown in Ref.[@sclimb], the linear isochoric compressibility of a symmetric network is essentially the same as that of a liquid. Here we show that, even if the initial network is strongly asymmetric in favor of the non-superclimbing (superfluid) dislocations, there are scenarios still leading to the giant isochoric compressibility. ![(Color online) A forest of screw dislocations containing edge superclimbing segments. Dashed and solid lines indicate pure screw and pure edge segments, respectively, all characterized by the Burgers vector along the [*hcp*]{} axis.[]{data-label="fig_s"}](figur5.pdf){width="0.9\columnwidth"} ![(Color online) A prismatic loop (solid line) generated by an edge segment AB, from Fig. \[fig\_s\], according to the Bardeen-Herring mechanism: An originally straight edge segment AB (solid horizontal line) bows under the bias (dashed-double-dotted line). Then, further bowing results in the overhangs (dashed-dotted line). The points C,C’ in the overhangs approach each other and eventually the whole prismatic loop detaches from the points A,B. []{data-label="figFR"}](Frank_reed.pdf){width="0.6\columnwidth"} According to the suggestion [@sclimb] the vycor “electrodes” are creating a contact between superfluid reservoirs and a preexisting static network of dislocations with superfluid cores. Such dislocations are characterized by Burgers vector along the main symmetry axis and can be of two basic types – screw [@screw] and edge [@sclimb] (as well as of the mixed type). The edge part of the network can execute superclimb responsible for the syringe effect. An example of such a network with combined segments is shown in Fig. \[fig\_s\]. There is an alternative to the “preexisting static network” scenario – a dynamical network which is created and disrupted by the external bias $\mu$. Here the analysis of the superclimb is extended beyond the linear response considered in Ref.[@sclimb], and it is shown that a segment of a rough superclimbing dislocation is unstable with respect to its unlimited growth, if the bias by chemical potential $\mu$ exceeds a threshold $\mu_c$ which is inversely proportional to a length $L$ of the segment. In high quality crystals typical value of $L$ can be as large as several $\mu$m or a fraction of mm or even reach a sample size. Thus, the threshold can be [*macroscopically*]{} small, so that it may well be exceeded by several orders of magnitude in the experiments [@Hallock; @Beamish; @Moses]. The reason for such a generic situation is that a number of the conducting pathways is $\propto 1/L^2$, so that, in order to detect the superflow, the bias $\mu$ needs to be increased at least as $\mu \propto L^2$ which leads to $\mu >> \mu_c \propto L^{-1}$. It will also be shown that a straight screw dislocation (which cannot perform superclimb) with superfluid core [@screw] can develop a helical instability under the bias so that the edge-like rim is formed and, accordingly, the syringe effect will also be induced. The threshold for this instability has the same dependence $\propto 1/L$ on length of the screw dislocation.\[Helical screw dislocations have been first observed in silicon at high temperatures [@Dash]\]. The instability has two important consequences: First, rough superclimbing segments of dislocations pinned inside solid 4 bulk can generate prismatic loops upon bias $\mu$ in a manner very similar to the Frank-Reed source of gliding dislocation loops under shear stress [@FrankReed]. The difference is that the generated loops during the superclimbing instability carry extra matter or vacancies. \[In this case the instability should rather be called as Bardeen-Herring [@Bardeen]\]. A typical diameter of the loop is determined by the original length of the edge segment $L=L_0$. This process is schematically shown in Fig. \[figFR\]. Second, superclimbing dislocations existing at a solid-vycor boundary can proliferate into the bulk upon applying the bias, so that a percolating network of superfluid pathways is created even if it didn’t exist originally. This process is shown in Figs. \[fig1\],\[fig2\]. Both effects are symmetric with respect to the sign of the bias $\mu$. In one case an additional matter is injected into the solid in the form of parts of extra basal planes, and in the second – existing basal planes are being dissolved, that is, vacancies are being injected instead. The presented analysis is conducted at the level of a single dislocation. It ignores how pinning by 3 impurities or crosslinks with other dislocations may affect the dynamics of the instability. It is clear that the instability may also result in the dislocations exiting the solid from its edges. Several growing segments may also merge or recombine. These processes as well as the interaction between superclimbing and basal plane gliding dislocations are also not considered. In some sense the analysis presented here is limited by low density of superclimbing dislocations so that there is some reasonably long time during which the dynamics can be treated within an approximation of a single dislocation segment. This situation is different from the linearized approach [@sclimb] where the main assumption was that a typical distance between superclimbing segments is of the same order as a typical length of the each segment. The single loop strongly non-linear dynamics considered here relies on a different limit – that is, a typical distance between superclimbing segments is much larger than $L_0$. The injection of the dislocations from the vycor-solid boundary and the Bardeen-Herring type loop generation as well as the helical instability of screw dislocations result in the syringe effect. The dynamics of the instabilities, however, turn out to be different: while in the case of the boundary instability the injected dislocation can grow to sizes far exceeding its original length $L=L_0$, in the case of the Bardeen-Herring type instability the generated loop radius $R$ is of the order of $L_0$ – it is the number of loops that is changing. The helical screw dislocation can also generate loops in a manner similar to that proposed in Ref.[@Dash]. \[The detailed study of this effect in the context of the superfluid core will be conducted elsewhere\]. ![(Color online) Sketch of the growing superclimbing dislocation (dashed line). The area to its left indicates either building of extra plane of atoms or dissolving of the existing plane (not shown) between the upper and lower ones. []{data-label="fig1"}](figures1_1.pdf){width="1.0\columnwidth"} ![(Color online) Geometry of the growing superclimbing dislocation (dashed line) sketched in Fig. \[fig1\]. The endpoints of the dislocation are in a contact with the superfluid reservoir.The (solid) area under the curve indicates either the injected extra matter leading to the formation of new basal layer (actually two of them in [*hcp*]{}) or a removed part of the existing one.[]{data-label="fig2"}](figures4.pdf){width="0.95\columnwidth"} Growth instability of rough superclimbing dislocation ===================================================== Let’s consider one segment of a superclimbing dislocation of some initial length $L=L_0$. Such a segment can be at a crystal boundary or be a part of the superfluid network. If the bias $\mu$ is applied, the segment will bow due to an extra matter delivered through its ends. Such bowing occurs in the basal plane while the Burgers vector $b$ is perpendicular to the plane, that is, along the high symmetry axis. This process is schematically shown in Fig. \[fig1\] as an arc protruding between two basal layers. At this point, let’s specify what the bias $\mu$ is. An increase of chemical potential of the superfluid reservoir $\mu_l$ either by applying pressure [@Hallock; @Beamish; @Moses] or through the Fountain effect [@Hallock] creates a difference $\mu = \mu_s - \mu_l <0$ between chemical potentials of the solid $\mu_s$ and the liquid. As a result, an additional matter can be injected into the solid in a form of growing pairs of basal plane layers. The boundary of one pair of layers is the superclimbing dislocation. If $\mu >0$, an existing pair of layers is being dissolved. Its boundary is also a superclimbing dislocation. The added or removed part of the planes is shown by a colored solid area under the arc in Fig.\[fig1\]. It is important to realize that imposing any finite value of $\mu$ (that is deviation from the equilibrium value) results, strictly speaking, in the instability. This can be understood from simple energy balance: the energy gain due to the bowing $\delta E_b = |\mu \delta N|$, with $\delta N$ being a number of atoms delivered through the core to support the bowing, always exceeds the energy $\delta E_{\rm cr} \propto (L-L_0)$ due to the core length increase from $L_0$ to $L$ for large enough $L$ because $\delta N$ is given by the area swept during the bowing. Thus, for large enough $L$ the energy gain due to the bowing always dominates the energy loss due to the length increase. However, in the limit $\mu \to 0$ the (meta)stability is protected by a macroscopic energy barrier. This barrier vanishes if $|\mu| $ exceeds a threshold $ \sim 1/L_0$ so that the absolute instability develops. As the estimates provided later show, the actual experiments [@Hallock; @Beamish; @Moses] appear to be well in this regime. Let’s estimate the threshold value $\mu_c$ for the bias. The energy of the dislocation per its unit length is given by shear modulus $G$ and Burgers vector $b$ as $\epsilon_c \approx Gb^2 /4\pi$, so that $\delta E_{\rm cr} \approx \epsilon_c (L-L_0)$. The energy gain $\delta E_b$ scales by the area $\sim |\mu| (L/b)^2$ swept by the bowing dislocation. Thus, equating one to the other gives $\mu_c$ as \_c \~. \[inst\] The value of $b$ in Eq.(\[inst\]) is the Burgers vector along the [*hcp*]{} axis, that is, $b=\sqrt{8/3} a$ where $a$ is the interatomic distance $a \sim 3.5$Å. As was found in the simulations [@screw], this dislocation splits into two partials with $b \to b/2$ and the fault in between. Thus, effectively, $b$ is reduced by a factor of 2 so that the actual threshold (\[inst\]) becomes lower by a factor of about $2^4$. In the following discussions I will ignore this peculiarity of the structure, which can only modify the numerical coefficient without affecting the dependence $ \mu_c \propto 1/L_0$. Accordingly, in all the following estimates the value of $b$ will be taken as $b \approx a \approx 3.5$Å  and the core splitting will be ignored. The above relation can be supported by a more quantitative analysis. In the quasi-static situation, so that $\mu$ is the same over the whole dislocation length, the bowing dislocation takes the shape of circular arc characterized by the base $L_0$ and the angle $\alpha$ as indicated in Fig. \[fig2\]. At small $|\mu|$ the center of the circle is outside the crystal so that the circle radius $R>> L_0$ and $\alpha_0 \to 0$. As the arc grows, $\alpha_0$ eventually reaches $\pi$ and $R$ decreases to $R=L_0/2$ and then $\alpha_0 \to 2\pi $ so that the center of the circle enters the crystal and $R$ starts growing to become $R>>L_0$ . The total energy of such a configuration can be represented as E=R\_0 - R\^2(\_0 - \_0) \[arc\] where the radius $R$ of the arc is determined by $L_0$ and the angle $\alpha_0$ as $L_0=2 \sin(\alpha_0/2) R$. This expression indicates that the dislocation is absolutely unstable toward inflation $R \to \infty, \, \alpha_0 \to 2\pi$ for arbitrary small $|\mu|$ simply because the second term is negative and can dominate at large enough $R$. There is, however, an energy barrier to overcome before the instability develops unless $|\mu|$ exceeds some critical value. Let’s consider a specific situation when the end points of the dislocation are pinned by a contact with a superfluid reservoir so that $L_0$ is fixed. Then, the energy (\[arc\]) becomes a variable of the angle $\alpha_0$ only: E(L\_0, \_0)= E\_0 , \[Ed\] where $E_0 \approx Gb^2L_0/8\pi$, $\tilde{g}\equiv \pi |\mu| L_0/( b^4 G)$. As the analysis of the function (\[Ed\]) indicates, for $\tilde{g}<<g_c=0.5$ there is a metastable minimum at $\alpha \approx 4\tilde{g} <<1$ followed by a maximum at larger $\alpha_0$. At $\tilde{g}=g_c$ both,the minimum and the maximum, coincide at $\alpha_0=\pi$ which is the inflection point indicating ending of the metastability domain, so that at $\tilde{g}>g_c$ the dislocation becomes unstable toward unlimited inflation $\alpha_0 \to 2\pi$. As can be seen, the threshold $\tilde{g}=0.5$, that is, \_c= \[inst2\] is consistent with the estimate (\[inst\]). In a general situation one should expect a distribution of the lengths $L_0$ so that some segments remain in a metastable equilibrium and some are overcritical. In what follows such a distribution will be ignored and it will be considered that there are $M$ segments of some length $L_0$ in a solid affected by the bias $\mu$. Typical values utilized in the flow experiments [@Hallock], when the syringe effect was observed, are in the range $\mu =5\cdot 10^{-4} - 5\cdot 10^{-3} $K (which corresponds to $0.001-0.01$J/g in units of Ref.[@Hallock]). This implies that the lengths of the critical segment $L_0\geq 1-5\mu$m (for a typical $G\approx 100$bar and $b\sim 3.5$Å). The expected density of dislocations in high quality 4 crystals is at the level of $\sim 10^4-10^6$cm$^{-2}$ as found in Ref.[@Balibar], which implies that the actual lengths $L_0$ of free segments are about a factor of 10-100 longer than the above estimate. In other words, the experimental range of $\mu$ used in Ref. [@Hallock] appears to be well above the threshold (\[inst\]) (or (\[inst2\])). At this point it should be mentioned that the dislocation density values [@Balibar] are more relevant to the glide effect than to the superclimb. Nevertheless, this order of magnitude estimate can be used as a figure of merit. Helical instability of the screw dislocation with superfluid core ----------------------------------------------------------------- As found in [*ab initio*]{} simulations [@screw], screw dislocation in solid 4 with Burgers along the [*hcp*]{} axis has a superfluid core. If this dislocation is straight, there are no edge-type segments on it, and, therefore, it cannot perform the superclimb. Here it will be shown that, if such a dislocation is biased by chemical potential similarly to the edge segment discussed above, it will become unstable toward forming a helix with its axis parallel to the original orientation of the dislocation. Such a helix has the edge-type rim and, thus, it can be a cause of the syringe effect. Let’s consider a screw dislocation of length $L$ oriented along the z-axis (that is, the [*hcp*]{} axis). Then, a position of the core can be described in the cylindrical coordinates by the radial distance $r(z)$ from its original position $r=0$ (in units of $b$) as well as by the azimuthal angle $\theta (z)$. The energy consists of two terms: the work done $ \mu \Delta N$ by the bias $\mu$ to accumulate some amount of matter $\Delta N$ due to creating the edge-type rim and the energy $\sim \epsilon_c$ due to the core length increase : E\_s= \_0\^[L]{}dz {\_z + \[ (\_z r)\^2 + r\^2 (\_z )\^2 \]}, \[Lscr\] where it was taken into account that the additional matter per unit length of the core is $d \Delta N/dz = \gamma_s r^2 \partial_z \theta /2$, with $\gamma_s=\pm 1$ being the chirality (handedness) of the dislocation. In other words, the total amount $\Delta N$ is given by the projection of the helix on the basal plane times the number of the complete turns. The “sign” of the matter accumulation depends on $\gamma_s$: if the screw is right handed, $\gamma_s=1$, and the helix is right handed, $ \partial_z \theta >0$, the solid mass increases, that is, $\Delta N >0$. Similarly, $\Delta N>0$ for both the screw and the helix being left handed ($\gamma_s =-1,\, \partial_z \theta <0$). Conversely, the amount of the syringe matter becomes negative if the chiralities are opposite to each other. Eventually, it will be seen that the sign of the syringe fraction does not depend on the screw chirality and is solely determined by the sign of the bias $\mu$ as $\Delta N \sim - \mu$. As a specific choice of the boundary condition, let’s presume that this dislocation is pinned at its both ends, that is, $r(0)=r(L)=0$. Then, the variation with respect to $\theta$ gives the equation \_z =- , \[hel\] where the boundary condition is taken into account. Its substitution back to Eq.(\[Lscr\]) results in the effective energy of the dislocation as E= \_0\^[L]{}dz {- + (\_z r)\^2 }. \[Lscr3\] This expression features an instability toward unlimited growth of $r$. At small $\mu$, similarly to the case of the edge dislocation, the solution $r=0$ is a metastable one. As Eq.(\[Lscr3\]) indicates, there is a difference with the edge dislocation case – the screw does not show any linear response of bowing in the limit $\mu \to 0$. There is, however, a threshold $\mu_s$ such that at $|\mu | > \mu_s$ the absolute instability toward $r \to \infty $ develops. In order to find how $\mu_s$ depends on $L$ and $\epsilon_c$ it is enough to perform elementary estimates: $r <<L$ changes on the scale of $L$ so that the total elastic energy is $ \sim \epsilon_c r^2 /L $. As long as the bias energy $ \sim \mu^2 L r^2 /\epsilon_c $ becomes of the same order, the solution $r=0$ becomes absolutely unstable. Thus, \_s , \[mu\_s\] which is essentially the same condition as (\[inst\]). At the threshold the helix is described by the total angle $|\alpha | \approx \mu_s L/\epsilon_c \approx 1$, and at $|\mu| >>\mu_s$ this angle becomes $|\mu|/\mu_s >>1$. Collective elastic effect in a bulk network of superfluid dislocations ---------------------------------------------------------------------- The injection of extra matter (or vacancies) under the bias $\mu$ is limited by the compression elastic modulus $K$ of a sample, so that the system stabilizes at some finite density of extra matter delivered through superclimb. The above estimate (\[inst\]) (or (\[inst2\])) obtained for a single dislocation does not take into account this effect and implies that an inflating loop can reach a sample size. In reality, the generation must stop after the overall density change compensates for the bias $\mu$. Let’s presume that the extra matter resides in $M$ dislocation segments of length $L_0$ which bowed by an amount $y $ each. Such bowing results in the extra matter $\Delta N \sim L y M/b^2$ added to (or subtracted from) the solid. There is the corresponding compression energy increase $E_e \approx K (\Delta N/N)^2 \Omega/2$, where $N$ stands for the total number of particles in the bulk of the volume $\Omega$ affected by the injection. The chemical potential energy gain and the energy loss due to the core length increase are $\sim \mu L y M/b^2 $ and $\sim Gb^2 y^2/L$, respectively. Thus, the total energy change as a function of $y,M,L$ becomes E- y + , \[Etot\] where the dimensionless numerical coefficients are omitted.If $|y| <<L$, the value of $L$ can be set to $L\approx L_0$ and the minimization in $y$ gives y. \[y\] As mentioned above, this solution is actually a metastable one, which, however, is protected by exponentially long waiting time in the limit $ \mu \to 0$. The bowing determined by (\[y\]) corresponds to the syringe fraction $\Delta N/N \sim MLy b/\Omega$ which for the case depicted in Fig. \[fig\_s\], where $M \sim \Omega/(L_s^2 L_z)$ can be written as . \[frac\] If $L_s \sim L_z \sim L$, that is, for a uniform network of the superclimbing dislocations, this fraction becomes , \[comp\] which constitutes the giant isochoric compressibility [@sclimb]. In the limit $L^3_0 << L^2_s L_z$, the syringe fraction becomes reduced within the linearized approach as &lt;&lt; , \[comp3\] If the bias $\mu$ exceeds the threshold (\[inst\]), the bowing of the edge segments cannot be treated in the linear approximation anymore.  In order to find the syringe fraction in a generic situation one can use Eq.(\[Etot\]) where the substitute $y \sim L$ is made and, instead of $L$, the fraction $ N_1 \sim L^2/b^2$ generated by one segment is used as a variable. Then, Eq.(\[Etot\]) becomes: E- || M N\_1 + Gb\^3 M + N\_1\^2 , \[Etot2\] where $N_1$ is taken as a positive value featuring either extra matter delivered to ($\mu <0$) or taken out from ($\mu > 0$) the solid. For small enough $M$ this function of $N_1$ features a maximum at $N_1=N_{mx} \sim (Gb^3/|\mu|)^2$ and then a stable minimum at $N_1=N_{eq} $ where N\_[eq]{} . \[Lm\] This minimum corresponds to the syringe fraction $\Delta N/N = N_1 M b^3/\Omega$, that is, , \[comp2\] where the condition $N_{mx} << N_{eq}$, that is, || \_b (G\^2 K )\^[1/3]{} b\^4 \[Nmxmn\] must hold. In the case of $M$ bulk segments distributed uniformly in, e.g., the situation depicted in Fig. \[fig\_s\], the condition (\[Nmxmn\]) becomes $|\mu| > \left(G^2 K/L_s^2L_z\right)^{1/3} b^4$. Thus, if $L_0$ is the smallest length scale and $|\mu|$ obeys (\[inst\]), the system is guaranteed to be unstable, with the equilibrium (\[Lm\]) to be determined by the bulk elastic energy $E_e$. The fraction (\[Lm\]) corresponds to the limit $|\mu| >> \mu_b$. In the case of the bulk structure shown in Fig. \[fig\_s\] this fraction $N_1$ is residing in several prismatic loops $N_{lp}$ of a radius $R \sim L_0$ generated by each edge segment. This number can be estimated as $N_{eq}b^2/L_0^2$, that is, N\_[lp]{} &gt; ()\^[2/3]{} &gt;&gt; 1, \[Req\] where the condition (\[Nmxmn\]) as well as that $L=L_0$ is the smallest distance among $L_s,L_z,L$ in Fig. \[fig\_s\] are taken into account. Thus, in the overcritical regime the equilibrium fraction (\[comp2\]) is always of the same order as in the liquid – even if the linearized response predicts much smaller values (\[comp3\]). Collective elastic effects due to the boundary instability ---------------------------------------------------------- In the case of the vycor-solid boundary, the analysis should be performed separately because seeds of the unstable dislocations are residing at the boundary. Accordingly, $M$ in this case is rather a surface than the bulk quantity. Then, in the estimate of the bulk deformation energy the affected volume becomes $\Omega \sim L S$, where $S$ stands for the area of the vycor-solid boundary. Accordingly, the extra fraction of the injected matter/vacancies is $\Delta N\sim L^2 M/b^2$ and $N\sim \Omega/b^3$ so that \~, \[surfra\] and the elastic energy takes the form E\_e . \[vs\] This dependence $\propto L^3$ should be contrasted with the elastic term $\propto N_1^2 \propto L^4$ in the case of the bulk instability as represented in Eq.(\[Etot2\]). The total energy in terms of $N_1$ (that is, the extra matter due to one segment) becomes E- || M N\_1 + Gb\^3 M + . \[Etot3\] This form as a function of $N_1$ (or the segment length $L$) can also have two extrema – a maximum followed by the minimum as $N_1$ grows. It happens when || \_s ()\^[1/2]{} b\^4. \[Nmxmn2\] This condition for the surface bistability should be compared with the bulk one (\[Nmxmn\]). The value of $M/S$ is determined by typical distances between the boundary segments along the basal plane $r_b$ and along the hcp axis $r_z$ as $ M/S \approx 1/r_b r_z$. Thus, if $L_0 < \sqrt{r_b r_z}$, the condition (\[Nmxmn2\]) is guaranteed to be satisfied as long as the instability condition (\[inst\]) holds. If $|\mu| >>\mu_s$, the equilibrium is determined by the first and the last terms in the energy (\[Etot3\]). It corresponds to the typical equilibrium length (obtained from the minimization of $E$ in (\[Etot3\]) with respect to $N_1 \sim L^2$) as L=L\_[eq]{} . \[Leq\] It is interesting to note that this length becomes of the order of a sample size ($\sim $ 1 cm) for the smallest values of the bias $\sim 5\cdot 10^{-4}$K used in Ref.[@Hallock] ( or $\mu \sim 10^{-3}$J/g in units of Ref.[@Hallock]) , if $r_b,r_z$ are of the order of the vycor diameter $\sim 1$mm. This value, however, drops quite fast with the product $r_br_z << 1$mm$^2$. In other words, if there are only few seeds of superclimbing dislocations at the solid-vycor boundary, the instability guarantees that the new pathways will reach the other electrode. Conversely, if there are many such seeds, the elastic energy increase due to the injection will stop the syringe effect close to the boundary. Generically, a system of superclimbing dislocations can feature two minima with respect to the syringe fraction. These minima are separated by a barrier, and, therefore, the hysteresis phenomenon should be anticipated with respect to the bias as long as the condition (\[Nmxmn\]) (or (\[Nmxmn2\])) is satisfied. Renormalization of the chemical potential ----------------------------------------- The instability of a single superclimbing dislocation is eventually stabilized by the increase of the bulk elastic energy due to finite density of the injected matter (or vacancies). This corresponds to the renormalization of the difference of chemical potentials $\mu$ between solid and liquid from its initial value to zero. This renormalized value $\tilde{\mu}$ can be obtained from the expressions of the total energy, Eqs.(\[Etot2\],\[Etot3\]), as $\tilde{\mu} = \partial (E/M) /\partial N_1$. Keeping in mind the symmetry $\mu \to - \mu$ let’s consider $\mu < 0$, that is, that the potential of the liquid in vycor is higher than in the solid so that extra atoms enter the solid. In the case of the bulk segments (as in Fig. \[fig\_s\]) the differentiation of the energy (\[Etot2\]) results in = (N\_1)=+ N\_1\^[-1/2]{} + N\_1 , \[mu\_bu\] and in the case of the vycor-solid boundary Eq.(\[Etot3\]) gives =(N\_1)= + N\_1\^[-1/2]{} + N\^[1/2]{}\_1 . \[mu\_b\] There are two roots of $\tilde{\mu}=0$. At small enough $M$ ( as presented in Eqs.(\[Nmxmn\],\[Nmxmn2\]) and guaranteed by the generic condition (\[inst\])) the first one $N_1 \approx (Gb^3/(2 \mu))^2$ corresponds to unstable equilibrium, and the second one describes stable one. It should be mentioned that the equilibrium characterized by small bowings (that is, $N_1 \to 0$) is not captured by Eqs.(\[mu\_bu\],\[mu\_b\]) written for already large bowings $y \sim L_0$. Thus, for all practical purposes this minimum can be viewed as corresponding to the energy $E=0$ reached at $N_1=0$. As discussed above, the practical values of $\mu$ corresponds to the situation when $N_1$ is to evolve from the unstable toward the stable equilibrium. This what is called above as the overcritical regime $|\mu| > \mu_c$, Eq.(\[inst\]). As will be seen, the overcritical dynamics exhibit strongly non-linear features before $N_1$ approaches the vicinity of the stable equilibrium. Liquid-gas type transition and hysteresis ----------------------------------------- It is useful to look on Eqs.(\[Etot2\],\[Etot3\]) from a different perspective. Small bowing of dislocations corresponds to $N_1 \to 0$, that is to zero energy $E=0$. There can exist another equilibrium solution $\partial E/\partial N_1=0$ characterized by finite $N_1$. Thus, there is a value of chemical potential $|\mu|=\mu_I$ at which two phases $N_1\approx 0$ and $N_1$ finite have the same energies. This can be interpreted as a point of first order phase transition. For the case of the bulk system,Eq.(\[Etot2\]), \_I = 1.5 (G\^2 K )\^[1/3]{} b\^4. \[Ist\] At smaller values of $|\mu|$ the second solution for $N_1$ becomes metastable and at $|\mu| =\mu_{sp}$, where \_[sp]{} = 2\^[-1/3]{}\_I 0.794 \_I, \[sp\] it vanishes. Thus, $|\mu |=\mu_{sp}$ corresponds to the spinodal, that is, to the point where the bistability (and hysteresis) vanishes. Similar situation occurs for the case of the interface, Eq.(\[Etot3\]). The transition occurs at $|\mu|=\mu_{Is}$, where \_[Is]{} = (G K )\^[1/2]{} b\^4, \[Ist2\] and the hysteresis vanishes at $|\mu|=\mu_{sps}$, where \_[sps]{} = \_[Is]{} 0.866 \_[Is]{}. \[sp2\] The transition is not characterized by any underlying symmetry, and, to some extent, resembles Ist order liquid-gas transition, where density exhibits a jump. However, there is also a significant difference. The energies (\[Etot2\],\[Etot3\]) contain essentially a non-analytical term $\sim \sqrt{N_1}$ determined by the geometrical nature of dislocations. This term is always dominant at small $N_1$ and is the reason for the energy barrier. Thus, in contrast to the standard liquid-gas transition, the syringe effect does not have a critical point where the first order transition ends. Dynamics of the instability at the vycor-solid boundary ======================================================= Now let’s consider the dynamical aspects of the evolution of the syringe fraction. In this work, the focus is on the dynamics of the edge segments. The dynamics of the helical instability will be analyzed elsewhere. A single loop dynamics during the instability stage is characterized by a short ballistic period during which phase slips and dissipation can be ignored. Then, as the superfluid velocity along cores reaches some terminal value, phase slips induce dissipation which is strongly non-linear in the velocity. Finally, once length of a growing segment approaches equilibrium value (determined by the largest root of $\tilde{\mu}=0$ in Eq.(\[mu\_b\])), the linearized dynamics sets in. The analysis is conducted within the assumption that distance between dislocations is large enough so that there is enough time for a segment to grow to a length $L$ which is much larger than the initial length $L_0$. Within this approach other growing loops are taken into account self-consistently through the renormalization of the chemical potential given by the last term in Eqs.(\[mu\_bu\],\[mu\_b\]). Practically, this means distances $\sim 10^{-3}-10^{-2}$cm for dislocation densities $10^6-10^4$cm$^{-2}$. Ballistic growth from the boundary ---------------------------------- The analysis will be conducted for an almost circular loop of radius $R$ (that is, $\alpha_0 \approx 2\pi$ in Fig. \[fig2\]). In general, the parametrization of the loop should include deviations of its shape from circle. This can be achieved by introducing $R$ as a function of polar angle $\alpha$ (as defined in Fig. \[fig2\]) and time $t$. The Lagrangian can be written as = \_0\^[2]{} d - E\_1 \[L\] where the first term is the Berry contribution in units $\hbar=1, b=1$ , and $\sigma = \pm 1$ is to be chosen depending on whether the matter is injected to ($\sigma =+1$) or from ($\sigma =-1$ ) the solid; the term $ E_1 $ is the energy of one loop which takes into account the elastic energy, that is, the last term in Eq.(\[Etot3\]), attributed to one loop. In addition, the kinetic energy of the flow along the core $ \frac{\rho_s(\partial_{\alpha} \phi)^2 }{2 dl /d\alpha}$, where $\phi$ is the superfluid phase and $\rho_s$ stands for the superfluid stiffness and $ \frac{dl}{d\alpha}=\sqrt{R^2 + (\partial_{\alpha} R)^2}$, should be taken into account. Thus, E\_1 = \_0\^[2]{} d+ N\^[3/2]{}\_1, \[Le\] where $\kappa \approx Kb^5 M/S, \, \epsilon_c= Gb^3/4\pi $ and $N_1$ as the amount of extra matter absorbed by one loop is $N_1 = \int_0^{2\pi} d\alpha R^2/2$. As discussed above, the system is invariant with respect $\mu \to - \mu$. Thus, without loss of generality the value of $\mu$ can be taken negative and $\sigma =+1$ so that $N_1$ describes extra matter added to the solid. There is also the boundary condition indicating that the dislocation is in a contact with the superfluid reservoir at its two endpoints, that is, $\phi(\alpha=0)=\phi(\alpha=2\pi)=\phi_R$, where $\phi_R$ is the phase of the reservoir, which can be set to zero. Within the simplified approach deviations from the circular shape can be ignored, that is, $\partial_{\alpha} R=0$. Then, a certain minimal assumption must be made about the spatial dependence of the phase $\phi$ along the dislocation core. Given the symmetry of the problem, Fig. \[fig2\], and because the total current through the loop must be zero in the syringe regime, the current along the core $ \sim \partial \phi /\partial \alpha$ must be antisymmetric with respect to $\alpha=\pi $ .Thus, the lowest non-trivial angular harmonic satisfying this requirement as well as the boundary condition is = \_0(t) (). \[ans\] Then, a substitution of this ansatz into Eqs.(\[L\],\[Le\]), after performing explicit integration and variation in $\phi_0$, gives = . \[var\_phi\] This equation is essentially the statement of the continuity: the flux of matter through two ends of the growing loop controls the rate of the loop area change. The variation of the action in $R$ gives = |, \[R\] where |= + - - 3 \^[3/2]{} R. \[barmu\] For $|\mu|$ exceeding the threshold (\[inst\]) and when $R$ is yet far from the equilibrium, the dominant time dependence is determined by the first two terms in $\bar{\mu}$. Then, the solution of Eqs.(\[var\_phi\],\[R\],\[barmu\]) can be looked for in the form $\phi_0= A t, R= B t^\nu$ with some unknown parameters $A,B,\nu$. This gives \_0(t)= t, R(t)=()\^[1/3]{} t\^[2/3]{}. \[phi\] These dependencies describe the balistic stage of the syringe effect far from the equilibrium. The accumulated fraction is given by Eq.(\[surfra\]) where the role of $L$ is played by $R$ from Eq.(\[phi\]), that is, (\_s ||)\^[1/3]{} t\^[2/3]{}, \[DNbal\] which corresponds to the local pressure variation $ \sim K |\Delta N|/N $. These fractional powers are specific to the dislocation superclimb and, thus, their experimental observation would be a “smoking gun” for the effect. The question, though, is how long this ballistic stage can last. During the ballistic stage the flow velocity grows in time until some terminal velocity $V_T$ is reached (at the dislocation end points). Then, frequent phase slips take place which convert kinetic energy of superflow into excitations. Thus, as an order of magnitude estimate, a typical time for the phase slip $t_{ps}$ can be taken from the ballistic stage — how long it takes to accelerate the flow from zero to, say, $V_T \sim 100 $m/s. The velocity profile along the growing loop is determined by the phase (\[ans\]) as $V(t,\alpha) = \partial_\alpha \phi /(mR)$, that is, V(t,)= (/2), \[vel\] where $m$ is 4 atomic mass. Using the solution for $\phi_0$ and $R(t)$ in this equation, one finds V(t, )= V\_0(/2), V\_0= ()\^[1/3]{}. \[vel2\] in the standard units, where the time scale $\tau_b$ is given by \_b\^[-1]{}=2.4 , \[tau\] and $ n^{(1d)}_s\approx m\rho_s $ is the superfluid linear density along the dislocation core. Using its value $ n^{(1d)}_s \sim 1$Å$^{-1}$ found in simulations of screw dislocation [@screw] and $b\sim 3.5$Å(so that $\hbar/mb \approx 50$m/s), the terminal speed $V_T$ (at $\alpha=0,2\pi$) is reached at time $ t_T \approx \tau_b$, which for the lowest $\mu$ value used in Ref.[@Hallock] gives $\sim 1$ ms. At this moment it looks unlikely that the available time resolution allows observing the time dependence during this stage. There is, however, an initial portion of the syringe effect which is on the experimental time scale (of minutes) [@Hallock; @Beamish] is accumulated in a jump-like manner right after the bias $\mu$ is imposed (or removed). This quantity will be discussed later. Close to the stable equilibrium point $R\to R_{eq}$ (determined by $\dot{\phi}=0$), where R\_[eq]{}=, \[Req\_b\] which is essentially Eq.(\[Leq\]), Eqs.( \[var\_phi\],\[R\]) can be linearized $R=R_{eq} + \xi $ with $|\xi| << R_{eq}$. The corresponding dynamics is oscillatory: + \_\^2=0, \^2\_= . \[osc\] Thus, $\omega_\mu $ scales as $\propto 1/|\mu|$. These oscillations, however, can take place only if their amplitude is small so that the velocity of the flow along the core remains much smaller than the terminal velocity. Thus, detecting them presents a significant challenge. It should also be mentioned that the phase slips in the ohmic regime at finite temperature $T$ (see below) may make the oscillations overdamped. Dissiptaive stage ----------------- At the experimental time scale of minutes [@Hallock; @Beamish] the dynamics is dominated by diffusive (dissipative) processes. The nature of these processes is not exactly known. One possible scenario is that quantum phase slips assisted by thermal processes in the superflow along dislocation cores are responsible for the dissipation. The dependence of the flow rate vs bias [@Hallock] is consistent with the picture of the phase slips in Luttinger liquid containing a weak link [@Kane; @Borya_Kolya]. At the same time, the origin of a significant temperature dependence [@Hallock; @Beamish] observed in the non-linear regime is not fully understood. In this situation, a phenomenological approach should be used in order to describe the loop inflation. Specifically, in the ballistic dynamical equation (\[R\]) the l.h.s. can be rewritten in terms of the velocity amplitude $V_0$ in Eq.(\[vel\]) as $\phi_0 \to mR V_0$. Then, the effective friction rate $ \gamma(T,V_0) V_0$, with some friction coefficient $\gamma$ depending on $V_0,T$, should be added to the flow acceleration $dV_0/dt$ in Eq.(\[R\]). This transforms Eq.(\[R\]) into & &+ V\_0 \[RV\]\ |&& + - - 3 \^[3/2]{} R, \[Vdis\] where the Bernoulli pressure is now expressed in terms of the velocity rather than the phase. The growth rate of the loop area $\sim \dot{R^2}$ is determined by $V_0$. The actual relation (stemming from the continuity equation) is exactly the same as in Eq.(\[var\_phi\]) where, however, the phase is now expressed in terms of the velocity $V_0$: = 2 \_s V\_0, \[Rate2\] where it is understood that $V=V(\alpha =2\pi)= - V(\alpha=0)$ so that the matter is delivered symmetrically from the both ends of the growing loop of radius $R >>L_0$. Let’s now make a choice for $\gamma$. According to the quantum phase slip scenarios [@Kane; @Borya_Kolya] energy of 1D superflow is converted into excitations of the Luttinger liquid. Generically, the quantum effects assisted by thermal excitations are characterized by power law dependencies of the phase slip rate $\tau_{ps}^{-1} \propto T^\zeta$ with some $\zeta > 0$ determined by the Luttinger liquid parameter $g$. At zero $T$ the flow velocity $V$ controls the dissipation. Within the weak link situation a phase jump by $\sim \pi$ occurs at microscopic distances across the link which is reasonable to take as $\sim b$. This jump is accompanied by energy transfer $\sim V$ between the link and the Luttinger liquid. Thus, $V_0$ plays the role of temperature so that the dependence on $V_0$ should be characterized by the same exponent $\tau_{ps}^{-1} \propto |V_0|^\zeta$, with the crossover taking place at some $V_0=V_T \approx Tb$. Thus, as a single equation the friction rate in Eq.(\[RV\]) can be represented as \^[-1]{}\_[ps]{} \~V\_0 = \_0 [Im]{}\[bT+ i V\_0\]\^, \[gamma\] with some coefficient $\gamma_0$. In the weak-link scenario $\gamma_0$ is determined by frequent phase slips occurring at the location of the link. Thus, the associated time constant $ \propto \gamma_0^{-1}$ can be much shorter than a typical time-scale set by the period of Debye frequency in solid 4. This may essentially eliminate the ballistic stage for practical durations of the experiments. So, below the estimate for the loop inflation will be obtained under this assumption, that is, that the ballistic stage is too short to produce any significant syringe effect. The power $\zeta$ can be empirically related to the power $p$ observed in the flow rate vs bias dependence $ |V| \propto |\mu|^p$ in Ref. [@Hallock] as $ \gamma V_0 \sim |\mu|^p$ so that = p\^[-1]{}. \[zeta\] According to Ref.[@Kane] $\zeta = 2/g -1 $ and the self-consistent result [@Borya_Kolya_prv] gives $\zeta = 2g -1$ . In what follows the power $p$ will be used as a quantity measured directly in the experiment [@Hallock]. This power was found to vary in the range $0.25<p < 0.5$ . For $|V|<<V_T=Tb$ Eq.(\[gamma\]) implies ohmic regime V\_0 = \_0 [Im]{}\[bT+ i V\_0\]\^[1/p]{} p\^[-1]{} \_0 (bT)\^[p\^[-1]{} -1]{} V\_0. \[gammaT\] Eqs.(\[gamma\],\[gammaT\]) will be used below in the analysis of the loop dynamics in the long-time limit where the inertial part in Eq.(\[RV\]) can be omitted. Then, far from the equilibrium at low $T$ the dynamics is dominated by the first term in the brackets of Eq.(\[Vdis\]). Thus, V\_0()\^[p]{} , \[VR\] in the non-linear regime (\[gamma\]), and V\_0 ||, \[VR\_ohm\] in the ohmic regime (\[gammaT\]). These expressions must be used in Eq.(\[Rate2\]). Accordingly, in the non-linear regime the loop radius obeys 2 \_s ()\^[p]{}, \[non-lin\] which implies $R \propto |\mu/\gamma_0|^{p/(2+p)} (t\rho_s)^{1/(2+p)}$, or for the syringe fraction R ()\^ ( \_s t)\^. \[DN\_b\] As Eq.(\[VR\]) indicates, the flow speed actually drops with time as $V \propto t^{- 1/(2+p)}$. Thus, eventually, the non-linear regime must change to the ohmic one characterized by 2 \_s , \[ohm\] which gives R T\^ ()\^. \[ohmDN\] Eqs.(\[DN\_b\],\[ohmDN\]) are obtained under the assumption that the system is far from the equilibrium. If, however, it approaches the equilibrium, the last term in the brackets of Eq.(\[Vdis\]) becomes important (with the second and the third ones still being irrelevant). This term stabilizes the system at the equilibrium radius $R_{eq}$, Eq.(\[Req\_b\]). Close to the equilibrium the dynamics becomes linear in the deviation $|R_{eq} - R| << R_{eq}$. As mentioned above, the time dependence would become either dissipative at high $T$ or oscillatory as in Eq.(\[osc\]). At this point it should be mentioned that the stabilization of the instability may also happen due to the dynamical rather than due to static equilibrium. Specifically, if $R_{eq}$ exceeds a system size, the stabilization is to be achieved by the balance of growing new loops and the loops exiting the sample. This picture essentially depends on sample geometry and size and will not be discussed here. The jump in the syringe fraction due to the ballistic inflation --------------------------------------------------------------- As discussed above, the ballistic stage may lead to a jump of the accumulated syringe fraction right after the bias is applied (or removed). Let’s estimate this fraction, first, for $T=0$. In the dynamical equation (\[RV\]) the dissipative part can be ignored as long as $|\dot{V}| >> \gamma_0 V^{1/p}$. Using the ballistic solution (\[vel2\]) in this estimate, the limiting time becomes t\_[bal]{} , \[tbal0\] where the definition (\[tau\]) of $\tau_b$ is used. A substitution of it into the ballistic fraction, Eq.(\[DN\_b\]), gives the jump as , || &gt;\_c. \[max\_0\] The value of $p$ was found in Ref.[@Hallock] to be below 0.5. Thus, the ballistic jump is a [*decreasing*]{} function of the bias, provided it exceeds the threshold for the instability and the jump itself does not exceed the equilibrium syringe fraction $\propto |\mu|$. \[In this case, the last term in $\bar{\mu}$, Eq.(\[Vdis\]), should be taken into account which will change the ballistic solution (\[vel2\])\]. However, as mentioned earlier, the friction “amplitude” $\gamma_0$ in the denominator may actually suppress the jump below the experimental resolution. Let’s now consider finite $T$. Comparing the acceleration rate with the thermal phase slips in Eq.(\[RV\]) the ballistic evolution takes place (before it is interrupted by the ohmic regime) as long as $t$ is shorter than the smallest of either the ohmic dissipation time $\gamma^{-1}_0 T^{1-1/p}$ or the time when the terminal velocity $V=bT$ is reached. Clearly, at very small $T$ and $p<1$ the latest dominates, which from Eq.(\[vel2\]) follows as $t\approx t_{T} \propto \tau_b T^3$. Then, at longer times the evolution becomes essentially the same as that at $T=0$ and leads to the estimates (\[tbal0\],\[max\_0\]). However, at the experimental values of $T$ and large $\gamma^{-1}_0$, the estimate $t< t_{T} \approx \tau_b (\gamma^{-1}_0 T^{1-1/p} )^3 $ is more appropriate for the time limiting the ballistic evolution. Then, a substitution of $t_{T}$ into Eq.(\[DNbal\]) gives the jump as . \[max\_bal\] This dependence should be considered in the limit $\gamma_0 \to \infty$, that is, that the maximum typical time for the phase slips $\sim \gamma^{-1}_0$ is below $(T/T_0)^{2+1/p}\tau_b$, where $T_0\sim 1K$ is a typical temperature corresponding to the velocities $\sim 100$m/s. At this point, it is worth mentioning that the jumps of the syringe fraction have been observed in Ref.[@Beamish]. To what extent these can be interpreted in terms of the ballistic stage remains to be seen. The Bardeen-Herring type instability ==================================== While a dislocation injected from crystal edge to the bulk can grow up to $R_{eq}$ which is much larger than its initial size ( or even as large as sample size), a finite superclimbing segment inside a solid, e.g., in the case shown in Fig. \[fig\_s\], can generate loops only of a size of the order of its initial length. According to the Bardeen-Herring mechanism [@Bardeen] originally considered for gliding dislocations and known as Frank-Reed instability [@FrankReed], an initially straight segment bows under the bias, and eventually the overhangs are created, Fig. \[figFR\]. These overhangs merge together so that a circular (prismatic) loop of a radius $R_L$, which is of the order of initial length $L_0$ of the straight segment, is created. This process is cyclic and is characterized by time $t_{FR}$ needed for the loop to grow until the overhangs (C,C’ in Fig. \[figFR\]) merge together so that the loop becomes separated from the main network. At this point what happens to this loop is not important – it can,e.g., diffuse away or merge with newly created loops. An estimate for this time can be obtained from Eq.(\[non-lin\]) in the non-linear regime or from Eq.(\[ohm\]) in the ohmic regime, where the time $t_{FR} $ is found as a function of the loop radius $R $ reaching the length of the order of the original segment length $L_0$. It is natural to assume that this time $t_{FR}$ is much shorter than the experimental time $t$, so that many loops are generated by one segment before the equilibrium $\tilde{\mu}=0$ is reached. Thus, the accumulated fraction (far from the equilibrium) can be written as $\Delta N/N \propto t/t_{FR}>>1$. Thus, the syringe rate $d \Delta N/dt$ becomes \[rate\_nn\] in the non-linear regime and \[rate\_ohm\] in the ohmic one. After the bulk accumulated fraction approaches the equilibrium value (\[comp2\]) the constant rates (\[rate\_nn\],\[rate\_ohm\]) transform into the exponential diffusive slowing down. In contrast to the boundary instability in this case, the Bardeen-Herring mechanism is inherently dissipative and no oscillations are to be anticipated. For short loops the Bardeen-Herring cycle can occur in the ballistic regime. In this case the time $t_{FR}$ can be estimated from Eq.(\[phi\]) as $t_{FR} \propto L_0^{3/2} (\rho_s |\mu|)^{-1/2}$. Thus the rate becomes . \[rate\_ohm\] Discussion ========== Solid 4 with finite density of superclimbing dislocations in a contact with superfluid reservoir is found to be, in general, characterized by bistability with respect to the syringe fraction. This feature is due to the interplay between three contributions: i) the chemical potential energy gain due to a transfer of atoms between two phases – solid and superfluid; ii) the energy of the deformation of dislocations needed to accommodate the transfer; and iii) the collective elastic energy. The control parameter of the system are chemical potential and the density of the dislocations. At low densities of dislocations the two fractions are very different from each other and, therefore, the transition between them can be viewed as strongly Ist order transition with significant hysteresis. There is a similarity between this and liquid-gas transitions, with the exception of no critical point in the first case. It is highly likely that the syringe effect observed in Refs.[@Hallock; @Beamish] is essentially in the overcritical regime. In this regime the equilibrium syringe fraction is given by the liquid type isochoric compressibility despite that the linearized response may predict much smaller values. The are two major scenarios for the instability. First, the vycor-solid boundary can be a source of the superclimbing dislocation loops entering the bulk. Its dynamics is characterized by the ballistic and dissipative stages which can be ohmic or strongly non-linear in the flow velocity. Each regime is characterized by specific powers of the bias and time at the initial stages of the evolution, Eqs.(\[DNbal\],\[DN\_b\],\[ohm\]). Second, there is also an option for the bulk syringe effect where the accumulated fraction is distributed evenly through out the bulk. In its turn, the bulk scenario can proceed in two ways – through the Bardeen-Herring generation of the prismatic loops or through the helical instability of screw dislocations. The accumulated fraction in the case of the Bardeen-Herring instability is determined by constant rate dependencies (\[rate\_nn\]), (\[rate\_ohm\]) in the non-linear and ohmic regimes, respectively. The non-linear regime (\[rate\_nn\]) turns out to be showing the same type of the dependence of the syringe rate on the bias as the flow rate through the sample observed in the Ref.[@Hallock]. In this regard, it is worth mentioning a possibility that the flow through solid may not actually be taking place through a static network of dislocations percolating between both vycor electrodes. Instead, the loops generated during the Bardeen-Herring cycles may eventually be moving between two vycor electrodes. These loops are mobile due to the superflow along their rim and can serve as “vehicles” transporting the mater across a sample. Center of mass speed $V_{cm}$ of such a loop is locked to the speed of the superflow $V$ along its rim by a simple geometrical relation $V_{cm} \sim V b/R$ stemming from the matter conservation. Experimental studies of the actual bias-time-temperature dependencies of the syringe fraction and rates are needed to see if any of the above scenarios take place. One of the key questions to answer is about the nature of the $T$-dependence observed in the non-linear regime of the flow rate [@Hallock]. Similar dependence was also observed in a different setup [@Beamish]. Eqs.(\[DN\_b\],\[ohmDN\],\[rate\_nn\],\[rate\_ohm\]) contain the superfluid density $\rho_s$ in the corresponding powers as overall factors. To what extent the observed temperature dependence can be attributed to these factors remains to be seen. One possibility could be that the superfluidity along the cores is strongly affected by structural excitations of dislocation – kinks [@Aleinikava_2012] and jogs – so that as $T$ increases these excitations suppress the overall superfluid density $\rho_s$ in the cores and, thus, reduce the total flow rate. It should also be mentioned that superclimbing dislocation does not fit exactly into the paradigm of Luttinger liquid because its excitation spectrum is not linear in the momentum [@sclimb]. To what extent this feature may modify the results (\[gamma\],\[gammaT\]) is an open question too. The “smoking gun” evidence for truly superfluid flow would be the detection of the ballistic jump in the syringe fraction which is a [*decreasing* ]{} function of the bias. Some jumps have been observed in Ref.[@Beamish]. Thus, their detailed study is of crucial importance. The main assumption of this work is that density of superclimbing dislocations is low and a sample size is large enough so that the equilibrium for the generated loops is achieved at typical sizes smaller than sample size. If this condition is not satisfied, as it could be the case for very small samples [@Moses], a completely different scenario may take place: the conducting network may be created by dislocations proliferating directly between the electrodes. In this case the actual dynamics may be controlled by changing number of the conducting pathways, that is, balanced by the pathways creation and exiting from a sample. Acknowledgements ================ I thank Boris Svistunov and Nikolay Prokof’ev for fruitful discussions. This work was supported by the NSF grant PHY1314469. [99]{} M. W. Ray and R. B. Hallock, Phys. Rev. Lett. [**100**]{}, 235301(2008); Ye. Vekhov and R. B. Hallock, Phys. Rev. Lett. [**109**]{},045303 (2012); Ye. Vekhov and R. B. Hallock, Phys. Rev. [**B 90**]{}, 134511 (2014). Z. G. Cheng, J. Beamish, A. D. Fefferman, F. Souris, S. Balibar, Phys. Rev. Lett. [**114**]{}, 165301 (2015). A. Haziot, Duk Young Kim, M. Chan,Abstract A22.00015, APS March Meeting, March 2–6, 2015; San Antonio, Texas. Abstract A22.00015. S. G. Söyler, A. B. Kuklov, L. Pollet, N. V. Prokof’ev, and B. V. Svistunov, Phys. Rev. Lett. [**103**]{}, 175301 (2009). M. Boninsegni, A. B. Kuklov, L. Pollet, N. V. Prokof’ev, B. V. Svistunov, and M. Troyer, Phys. Rev. Lett. [**99**]{}, 035301 (2007) . W. C. Dash, Phys. Rev. Lett. [**1**]{}, 400 (1958). F.C. Frank, W. T. Read,Jr., Phys. Rev. [**79**]{}, 722 (1950). J. Bardeen and C. Herring, [*Imperfections in Nearly Perfect Crystals*]{}, Wiley, New York, 1952. A.l Haziot, A. D. Fefferman, J. R. Beamish, S. Balibar, Phys. Rev. [**B 87**]{}, 060509(R) (2013). C. L. Kane, M. P. A. Fisher, Phys. Rev. Lett. [**68**]{}, 1220 (1992). V. A. Kashurnikov, A. I. Podlivaev, N. V. Prokof’ev, B. V. Svistunov,Phys. Rev. [**B 53**]{}, 13091 (1996); N. V. Prokof’ev and B. V. Svistunov, Phys.Rev. [**B 61**]{}, 11282 (2000). N. V. Prokof’ev and B. V. Svistunov, private communication. D. Aleinikava, E. Dedits, A. B. Kuklov, D. Schmeltzer, arXiv:0812.0983; D. Aleinikava and A. B. Kuklov, J. Low Temp. Phys. [**169**]{}, 133 (2012).
--- author: - | Pooya Davoodi\ New York University, Polytechnic School of Engineering\ `pooyadavoodi@gmail.com` - | Rajeev Raman\ [University of Leicester]{}\ `r.raman@leicester.ac.uk`\ - | [Srinivasa Rao Satti]{}\ [Seoul National University]{}\ `ssrao@cse.snu.ac.kr` title: 'On Succinct Representations of Binary Trees[^1]' --- [^1]: An abstract of some of the results in this paper appeared in *Computing and Combinatorics: Proceedings of the 18th Annual International Conference COCOON 2012*, Springer LNCS 7434, pp. 396–407, 2012.
--- abstract: 'Pressure-dependent transmittance and reflectance spectra of TiOBr and TiOCl single crystals at room temperature suggest the closure of the Mott-Hubbard gap, i.e., the gap is filled with additional electronic states extending down to the far-infrared range. According to pressure-dependent x-ray powder diffraction data the gap closure coincides with a structural phase transition. The transition in TiOBr occurs at slightly lower pressure ($p$=14 GPa) compared to TiOCl ($p$=16 GPa) under hydrostatic conditions, which is discussed in terms of the chemical pressure effect. The results of pressure-dependent transmittance measurements on TiOBr at low temperatures reveal similar effects at 23 K, where the compound is in the spin-Peierls phase at ambient pressure.' author: - 'C. A. Kuntscher' - 'A. Pashkin' - 'H. Hoffmann' - 'S. Frank' - 'M. Klemm' - 'S. Horn' - 'A. Schönleber' - 'S. van Smaalen' - 'M. Hanfland' - 'S. Glawion' - 'M. Sing' - 'R. Claessen' title: 'Mott-Hubbard gap closure and structural phase transition in the oxyhalides TiOBr and TiOCl under pressure' --- Introduction ============ The layered compounds TiO$X$, where $X$=Br or Cl, are low-dimensional systems which show interesting magnetic and electronic properties. Regarding the spin degree of freedom, at high temperature the system can be well described by a one-dimensional spin-1/2 nearest-neighbor Heisenberg model with a Bonner-Fisher type magnetic susceptibility.[@Seidel03; @Kataev03] Below the transition temperature T$_{c1}$, where T$_{c1}$=27 K for TiOBr and T$_{c1}$=67 K for TiOCl, TiO$X$ undergoes a first-order phase transition to a spin-Peierls state with a dimerization of the chains of Ti atoms along the $b$ axis and a doubling of the unit cell.[@Seidel03; @Caimi04; @Shaz05] Furthermore, an intermediate phase for the temperature range T$_{c1}$$<$T$<$T$_{c2}$ was found (with T$_{c2}$=47 K for TiOBr and T$_{c2}$=91 K for TiOCl), whose nature is now well established as an incommensurately modulated structure with a one-dimensional modulation in monoclinic symmetry. [@Smaalen05] Regarding the charge degree of freedom, the Ti ions have the electronic configuration $3d^1$. The $3d$ electrons are localized due to strong electronic correlations, and hence TiOBr and TiOCl are Mott-Hubbard insulators, with a charge gap of $\approx$2 eV. [@Ruckkamp05; @Kuntscher06; @Kuntscher07] It was predicted that these materials exhibit a resonating valence bond state and high-temperature superconductivity upon doping.[@Beynon93; @Craco06] However, up to now a metallization of TiO$X$ upon doping could not be achieved.[@Klemm08] Recently it was shown that the optical response of both compounds changes drastically under pressure: Above a critical pressure, the transmittance is suppressed and the reflectance increases in the infrared range. The changes could be attributed to additional electronic states filling the Mott-Hubbard gap and they suggest a closure of the gap at elevated pressures.[@Kuntscher06; @Kuntscher07] Under hydrostatic conditions the transition pressures are 14 and 16 GPa for TiOBr and TiOCl, respectively. Concurrent with the closure of the Mott-Hubbard gap a structural phase transition is observed.[@Kuntscher07] This paper is a follow-up of the earlier, short publication[@Kuntscher07] and provides details of the changes in the electronic properties and crystal structure of TiOBr and TiOCl induced by external pressure. The manuscript is organized as follows: After describing the experimental details in Sec.\[sectionexperiment\], we present in Sec. \[transmittance\] the experimental results obtained at room temperature, which suggest the closure of the Mott-Hubbard gap under pressure. We also include low-temperature transmittance spectra of TiOBr at ambient and high pressure in Sec. \[low-temperature results\]. Sec. \[x-ray\] focuses on the pressure-induced changes of the crystal structures for TiOBr and TiOCl. In Sec. \[comparisontransitionpressures\] we comment about a possible chemical pressure effect in the system TiO$X$. In Sec. \[gap closure-structure\] the relation between the closure of the Mott-Hubbard gap and the structural phase transition is discussed. Finally, we summarize our results in Sec. \[summary\]. ![Crystal structure of TiO$X$ ($X$=Br,Cl), viewed along the $a$, $b$, and $c$ crystal axes, consisting of Ti-O bilayers parallel to the $ab$-plane and separated by layers of $X$ ions stacked along the $c$ direction.[@Schaefer58] The black lines mark the unit cell. Also shown is the main building block of the crystal structure, namely the distorted TiO$_4$$X$$_2$ octahedron.[]{data-label="fig:crystalstructure"}](cryst.eps){width="0.9\columnwidth"} Experiment {#sectionexperiment} ========== Single crystals of TiO$X$ ($X$=Br,Cl) were synthesized by chemical vapor transport technique. The TiO$X$ compounds crystallize in the space group $Pmmn$ at ambient conditions and consist of distorted TiO$_4$$X_2$ octahedra.[@Schaefer58; @Schnering72] The octahedra are arranged such that buckled Ti-O bilayers parallel to the $ab$-plane are formed, which are separated by layers of Br/Cl ions stacked along the $c$ direction. Fig. \[fig:crystalstructure\] shows the crystal structure viewed along the crystal axes $a$, $b$, and $c$. TiO$X$ crystals grow in the form of thin platelets with the surface parallel to the $ab$-plane. This is convenient for studies of the optical response of the $ab$-plane. In the pressure-dependent studies diamond anvil cells (DACs) were used for the generation of pressures. The applied pressures $p$ were determined with the ruby luminescence method.[@Mao86] For the transmittance measurements several pressure transmitting media were used; this leads to small differences in the observed values of the critical pressure of phase transition, as expected.[@Frank06; @Kuntscher06] For the reflectance measurements finely ground CsI powder was chosen as pressure medium to insure direct contact of the sample with the diamond window. Pressure-dependent transmittance and reflectance experiments were conducted at room temperature using a Bruker IFS 66v/S FT-IR spectrometer with an infrared microscope (Bruker IRscope II). For the generation of pressure we used a Syassen-Holzapfel DAC[@Huber77] equipped with type IIA diamonds suitable for infrared measurements. Part of the measurements were carried out at the infrared beamline of the synchrotron radiation source ANKA, where the same equipment is installed. Further information on the pressure-dependent transmittance and reflectance measurements conducted at room temperature was included in the earlier publication.[@Kuntscher07] For TiOBr the transmittance measurements under pressure were also conducted at 23 K for the frequency range 3100 - 15000 cm$^{-1}$ (0.38 - 1.9 eV). At 23 K TiOBr is in the spin-Peierls phase at ambient pressure. As pressure medium argon was used. The transmittance measurements on the sample in the DAC placed in the optical cryostat (CryoVac KONTI cryostat) were performed using a home-built infrared microscope with a large working distance. This infrared microscope can be directly coupled to the FT-IR spectrometer and maintained at the same pressure ($\approx$ 3 mbar), i.e., no window between the two devices is needed. Pressure-dependent x-ray powder diffraction measurements at room temperature were carried out at beamline ID09A of the European Synchrotron Radiation Facility at Grenoble. Details about the experiments were described elsewhere (Ref. ). ![Charge gap $\tilde{\Delta}$ (see text for definition) of TiOBr as a function of pressure for [**E**]{}$||$$a$ (full symbols) and [**E**]{}$||$$b$ (open symbols) (pressure medium: CsI). The dashed lines are guides to the eye. Inset: Absorbance spectrum $A$($\omega$) of TiOBr for the lowest pressure (1.5 GPa), calculated according to $A$($\omega$)=log$_{10}$\[1/$T$($\omega$)\], together with the linear extrapolation of the absorption edge (dashed gray line) used to estimate $\tilde{\Delta}$.[]{data-label="fig:gap"}](gap.eps "fig:"){width="0.9\columnwidth"}\ Results and analysis {#sectionresults} ==================== Pressure-dependent transmittance and reflectance at room temperature {#transmittance} -------------------------------------------------------------------- Pressure-dependent transmittance measurements on TiOBr and TiOCl were carried out for several pressure transmitting media. In Refs.  we already showed the spectra of TiOBr and TiOCl for argon and CsI as pressure media, respectively. The transmittance spectra reveal the characteristic excitations in the materials, namely the electronic transitions between the lower and upper Hubbard gap, resulting in a strong suppression of the transmittance above $\approx$2 eV. Furthermore, absorptions occur due to excitations across the crystal-field split Ti$3d$ energy levels (called orbital excitations in the following) located for TiOBr (TiOCl) at 0.63 eV (0.66 eV) for [**E**]{}$||$$a$ and at 1.35 eV (1.53 eV) for [**E**]{}$||$$b$ at ambient conditions. First, one notices that in TiOBr the orbital excitations are slightly redshifted compared to TiOCl. This can be explained by the chemical pressure effect in the system TiO$X$: Based on the g tensors measured by electron spin resonance[@Kato05] the crystal field splittings in TiOBr and TiOCl were obtained. The smaller crystal field splitting in TiOBr could be attributed to the larger size of the Br$^{-}$ ion compared to the Cl$^{-}$ ion, causing a larger volume of the TiO$_4$$X$$_2$ octahedra (see Fig. \[fig:crystalstructure\]) and hence a weaker crystal field.[@Kato05] ![(Color online) Frequency of the orbital excitations in TiOBr as a function of pressure: at room temperature for the polarization (a) [**E**]{}$||$$a$ and (b) [**E**]{}$||$$b$; at 23 K for the polarization (c) [**E**]{}$||$$a$ and (d) [**E**]{}$||$$b$ (pressure medium: argon). The full symbols denote the results with increasing pressure; open symbols denote the results upon pressure release. Lines are guides to the eye. []{data-label="fig:orbital"}](orbital.eps "fig:"){width="1\columnwidth"}\ With increasing pressure one observes the following changes for both compounds along the two studied polarization directions: (i) a blueshift of the orbital excitations; (ii) the absorption edge due to excitations across the charge gap shifts to smaller energies with increasing pressure, and above 11 GPa (12 GPa) the overall transmittance is strongly suppressed in TiOBr (TiOCl). These results were obtained with CsI as pressure medium; when a more hydrostatic pressure medium is used (see Table \[tab:comparison\] and the results in Ref.  ), the suppression of the transmittance occurs at somewhat higher pressure ($\Delta$$p$$\approx$4 GPa). We estimated the charge gap, $\tilde{\Delta}$, by a linear extrapolation of the steep absorption edge. This is illustrated in the inset of Fig. \[fig:gap\], where we show the absorbance spectrum $A$($\omega$) of TiOBr for the lowest pressure (1.5 GPa), calculated from the transmittance $T$($\omega$) according to $A$($\omega$)=log$_{10}$\[1/$T$($\omega$)\], together with the linear extrapolation of the absorption edge. The intersection of the linear extrapolation with the horizontal axis was taken as an estimate of the charge gap. Starting from the lowest applied pressure, $\tilde{\Delta}$ initially slightly decreases with increasing pressure, and above $\approx$10 GPa it rapidly drops to zero (Fig. \[fig:gap\]). Similar observations were made earlier for TiOCl,[@Kuntscher06] with the onset of rapid decrease of $\tilde{\Delta}$ at $p$$\approx$12 GPa. The pressure dependence of the frequencies of the orbital excitations in TiOBr were obtained by fitting the absorption features in the transmittance spectra with Gaussian functions. The results are depicted in Fig. \[fig:orbital\]. With increasing pressure the orbital excitations shift to higher frequencies in a linear fashion. This shift could be attributed to a monotonically increasing strength of the crystal field related to the decreasing volume of the TiO$_4$Br$_2$ octahedra. External pressure could also induce a change in the octahedral distortion and related alterations of the crystal field. One furthermore notices a small difference in the frequency of the orbital excitations for pressure increase and decrease observed in the direction [**E**]{}$||$$b$ \[Fig. \[fig:orbital\](b)\], which suggests that the pressure-induced octahedral volume decrease and/or octahedral distortion are not completely reversible. This non-reversibility is more obvious at low temperatures and will be discussed in Sec. \[low-temperature results\]. ![(Color online) Far-infrared reflectance $R_{\rm s-d}$ of TiOBr and TiOCl at room temperature as a function of pressure, for the polarization [**E**]{}$||$$a$ \[(a) and (c), resp.\] and [**E**]{}$||$$b$ \[(b) and (d), resp.\] (pressure medium: CsI).[]{data-label="fig:FIR-reflectivity"}](FIR-ref.eps){width="1\columnwidth"} ![(Color online) Real part of the optical conductivity of TiOBr as a function of pressure for the polarization (a) [**E**]{}$||$$a$ and (b) [**E**]{}$||$$b$ obtained by Drude-Lorentz fitting of the pressure-dependent reflectance data $R_{\rm s-d}$. Inset: Total effective carrier density, $n_{eff}$, and spectral weight calculated by integrating the real part of the optical conductivity (see text) up to $\omega_0$=8000 cm$^{-1}$ for [**E**]{}$||$$a$ (filled circles) and [**E**]{}$||$$b$ (open circles). Lines are guides to the eye.[]{data-label="fig:cond-all"}](condBr-a.eps){width="0.85\columnwidth"} Additional information about the pressure-induced changes in the optical response were obtained by reflectance measurements on TiOBr and TiOCl at high pressures. The most drastic changes occur in the far-infrared range, as illustrated for both compounds in Figs. \[fig:FIR-reflectivity\] (a) and (b) (for pressure-dependent reflectance spectra over a broader frequency range, see Refs. ). In case of TiOBr, the shape of the spectrum changes drastically for [**E**]{}$||$$b$: At 10 GPa the spectrum consists of a peak-like feature between 300 and 450 cm$^{-1}$, whereas for pressures $\geq$11 GPa it is almost flat with a peak at around 520 cm$^{-1}$. The pressure-induced changes in the far-infrared reflectance spectra $R_{\rm s-d}$ of TiOCl are very similar to those of TiOBr \[see Figs. \[fig:FIR-reflectivity\] (c) and (d)\]. However, for TiOCl the changes occur at somewhat higher pressure, as discussed in Sec. \[comparisontransitionpressures\]. For higher frequencies the overall reflectance increases for both compounds and saturates.[@Kuntscher06; @Kuntscher07] The suppression of the transmittance in TiO$X$ at high pressures suggests the occurrence of new excitations in the infrared frequency range. More information about these additional excitations were obtained by fitting the high-pressure ($p$$>$10 GPa) reflectance spectra $R_{\rm s-d}$ with the Drude-Lorentz model combined with the normal-incidence Fresnel equation, taking into account the diamond-sample interface: $$R_{s-d} =\left| \frac{n_{\rm dia}-\sqrt{\epsilon_s}}{n_{\rm dia}+\sqrt{\epsilon_s}}\right|^2 , \epsilon_s = \epsilon_\infty + \frac{i \sigma}{\epsilon_0 \omega} \quad ,$$ where $\epsilon_s$ is the complex dielectric function of the sample and $\epsilon_\infty$ is the background dielectric constant (here $\epsilon_\infty$$\approx$3). From the function $\epsilon_s$($\omega$) the real part of the optical conductivity, $\sigma_1$($\omega$), can be calculated. Notice that only reflectance data above 10 GPa can be analyzed quantitatively because of the partial transparency of the sample below this critical pressure. The evolution of the optical conductivity of TiOBr with pressure is shown in Fig. \[fig:cond-all\]. We find additional excitations in the infrared range, extending down to the far-infrared. These additional excitations include broad excitations, which cannot be attributed to phonon excitations, in contrast to the optical conductivity spectrum in the insulating phase.[@Caimi04a] Thus, the Mott-Hubbard gap is gradually filled with additional electronic states down to at least 200 cm$^{-1}$ (24 meV). This finding suggests the closure of the Mott-Hubbard gap above $p$=10 GPa. With increasing pressure the spectral weight of the pressure-induced features increases, with a saturation setting in at around 13 GPa. From the spectral weight analysis one can extract the effective density of carriers, $n_{eff}$, involved in the excitations up to $\omega_0$ according to $$\label{carrierdensity} n_{eff}(\omega_0)=(2m_0 / \pi e^2)\int_0^{\omega_0}\sigma_1(\omega)d\omega \quad ,$$ with the free electron mass, $m_0$. In the inset of Fig. \[fig:cond-all\](b) $n_{eff}$($\omega_0$=8000 cm$^{-1}$) is plotted as a function of pressure $p$. $n_{eff}(p)$ illustrates the saturation of the spectral features at high pressures. Also for TiOCl the spectral weight of the pressure-induced excitations increases with increasing pressure and saturates, as presented in the earlier publication.[@Kuntscher06] The saturation, however, happens at somewhat higher pressure ($\approx$15 GPa) compared to TiOBr. In fact, all the pressure-induced effects occur in TiOCl at slightly higher pressures ($\Delta$$p$$\approx$2 GPa) compared to TiOBr. This pressure difference will be discussed in more detail in Sec. \[comparisontransitionpressures\]. ![(Color online) Transmittance $T$($\omega$)=I$_{s}$($\omega$)/I$_{r}$($\omega$) (see text for definitions) of TiOBr as a function of temperature for the lowest pressure (0.8 GPa), for the polarization (a) [**E**]{}$||$$a$ and (b) [**E**]{}$||$$b$ (pressure medium: argon). Insets: Frequency of orbital excitations as a function of temperature. Lines are guides to the eye.[]{data-label="fig:transmittance-T"}](trans-T.eps){width="0.8\columnwidth"} ![(Color online) Transmittance $T$($\omega$)=I$_{s}$($\omega$)/I$_{r}$($\omega$) (see text for definitions) of TiOBr as a function of pressure at 23 K, for the polarization (a) [**E**]{}$||$$a$ and (b) [**E**]{}$||$$b$ (pressure medium: argon).[]{data-label="fig:transmittance-P"}](trans-PT.eps){width="0.8\columnwidth"} Pressure-dependent transmittance of TiOBr at low temperatures {#low-temperature results} ------------------------------------------------------------- We have furthermore checked the stability of the insulating spin-Peierls phase of TiOBr by pressure-dependent transmittance measurements at low temperatures in the near-infrared frequency range (3100 - 15000 cm$^{-1}$). As mentioned in the introduction, TiOBr undergoes two phase transitions as a function of temperature: Upon temperature increase, a first order transition takes place at $T_{c1}$=27 K from the spin-Peierls ground state into an intermediate phase with an incommensurate superstructure.[@Smaalen05] An additional, second-order phase transition is found at $T_{c2}$=47 K, where the material changes from the intermediate phase to the one-dimensional antiferromagnetic phase at high temperature. Starting from room temperature and low-pressure (0.8 GPa) conditions, transmittance measurements on TiOBr were carried out upon temperature decrease. Fig. \[fig:transmittance-T\] shows the temperature-dependent transmittance spectra for the polarizations [**E**]{}$||$$a$ and [**E**]{}$||$$b$. The oscillations in the spectra are Fabry-Perot resonances due to multiple reflections within the thin sample platelet. With decreasing temperature one notices a small but significant shift of the orbital excitations to higher frequencies, as illustrated in the insets of Fig. \[fig:transmittance-T\]. The most pronounced changes occur between 295 and 200 K and can be attributed to the thermal contraction of the lattice while cooling down, leading to a smaller volume of the TiO$_4$Br$_2$ octahedra and thus to a stronger crystal field.[@Kato05] Below 100 K the orbital excitations hardly shift with temperature, which suggests that the structural changes occuring at the phase transitions at $T_{c1}$=27 K and $T_{c2}$=47 K have only a small effect on the TiO$_4$Br$_2$ octahedra and hence on the crystal field. This in agreement with an earlier work showing that the orbital degree of freedom is irrelevant for the low-energy physics, in particular the exotic spin-Peierls behavior with two successive phase transitions.[@Ruckkamp05] ![(Color online) Lattice parameters of TiOBr at room temperature as a function of pressure (pressure medium: helium). (a)-(c) Lattice parameters $a$, $b$, $c$. (d) Lattice parameters $a$, $b$, $c$ normalized to their respective zero-pressure values. Lines are guides to the eye.[]{data-label="fig:lattice-TiOBr"}](lat-Br.eps){width="0.8\columnwidth"} At 23 K, where the sample is in the spin-Peierls state for ambient pressure, transmittance spectra were recorded for several pressures (see Fig. \[fig:transmittance-P\]). Similar to the room-temperature results, the transmittance is suppressed over the whole studied frequency range above a certain pressure; however, at 23 K the suppression occurs only above $\approx$16 GPa, compared to the room-temperature transition pressure of 14 GPa. We also followed the pressure-induced shifts of the orbital excitations at 23 K \[Figs. \[fig:orbital\] (c) and (d)\]. With increasing pressure the frequencies of the orbital excitations increase linearly with increasing pressure. Like for the room-temperature results, we relate this shift to a pressure-induced decrease of the octahedral volume and a possible change in octahedral distortion, causing a change in the crystal field (see Sec. \[transmittance\]). At room-temperature we noticed a small difference in the frequencies of the orbital excitations for pressure increase and decrease. This difference is much more pronounced at 23 K. For example, for [**E**]{}$||$$a$ already at around 4 GPa during pressure release the ambient-pressure excitation energy of $\approx$5370 cm$^{-1}$, and thus the ambient-pressure crystal field strength, has been reached \[see Fig. \[fig:orbital\](c)\]. ![(Color online) Room-temperature x-ray powder diffraction diagrams of TiOCl at high pressures ($\lambda$= 0.4128 Å) together with the LeBail fits (pressure medium: helium). For the lowest applied pressure (1.8 GPa) the difference curve ($I_{obs}-I_{calc}$) between the diffraction diagram and the LeBail fit is shown. Markers show the calculated peak positions for the ambient-pressure phase. Above 15.5 GPa the diffraction diagram can no longer be described by the ambient-pressure crystal symmetry. Arrows indicate the diffraction peaks with the most obvious discrepancy between the data and the LeBail fitting curve.[]{data-label="fig:x-ray-TiOCl"}](x-rayCl.eps){width="0.8\columnwidth"} ![(Color online) Lattice parameters of TiOCl at room temperature as a function of pressure (pressure medium: helium). (a)-(c) Lattice parameters $a$, $b$, $c$. (d) Lattice parameters $a$, $b$, $c$ normalized to their respective zero-pressure values. Lines are guides to the eye.[]{data-label="fig:lattice-TiOCl"}](lat-Cl.eps){width="0.8\columnwidth"} Pressure-induced structural phase transition at room temperature {#x-ray} ---------------------------------------------------------------- For the understanding of the drastic changes in the optical response under pressure, we carried out x-ray powder diffraction measurements on TiOBr and TiOCl at room temperature as a function of pressure. A typical diffraction pattern (not shown) does not consist of concentric rings as expected for powder diffraction data, but it contains separate spots. This is due to the fact that it was not possible to produce good TiO$X$ powders with homogenous grain size distributions and random orientations because of the platelet-like habits and the softnesses of the crystallites. Instead, the crystallites inside the DAC orient their $c$ crystal axis preferentially perpendicular to the diamond anvil surface, i.e., along the direction of incidence of the x-radiation. Therefore, Rietveld refinements of the diffraction patterns could not be carried out. Nevertheless LeBail fits of the diffraction patterns could be accomplished, in order to determine the unit cell volume and the lattice parameters as a function of pressure. The room-temperature diffraction diagrams of TiOBr can be well fitted with the ambient-pressure crystal structure (space group $Pmmn$) at low pressures, as demonstrated in Ref. . The lattice parameters of TiOBr as a function of pressure, as obtained from the LeBail fitting, are presented in Fig. \[fig:lattice-TiOBr\]. The changes of the lattice parameters $a$ and $b$ with pressure are linear over a large pressure range. The behavior of the lattice parameters $c$ rather follows a sublinear fashion. In Fig. \[fig:lattice-TiOBr\] we also show the lattice parameters $a$, $b$, $c$ normalized to their respective zero-pressure values as a function of pressure \[Fig. \[fig:lattice-TiOBr\] (d)\]. According to these results, TiOBr has a very anisotropic compressibility, with the largest compressibility along the $c$ axis, i.e., the stacking axis of the buckled Ti-O bilayers. At around 14 GPa the diffraction diagram of TiOBr undergoes pronounced changes and is no longer compatible with the ambient-pressure crystal structure symmetry.[@Kuntscher07] We can therefore conclude that TiOBr undergoes a structural phase transition at 14 GPa. We also include the corresponding results from the pressure-dependent x-ray powder diffraction on TiOCl, namely the room-temperature diffraction diagrams for selected pressures together with the LeBail fits (Fig. \[fig:x-ray-TiOCl\]) as well as the lattice parameters as a function of pressure extracted by the LeBail fits (Fig. \[fig:lattice-TiOCl\]). The sublinear dependence on pressure is obvious for all three lattice parameters. Pronounced changes of the diffraction diagram occur at 15.5 GPa indicating a pressure-induced structural phase transition in TiOCl, similar like in TiOBr. For both compounds the pressure-induced changes are reversible in terms of the positions of the diffraction peaks. From the lattice parameters the pressure dependence of the unit cell volume $V$ for both compounds was obtained. In Fig. \[fig:volume\] we plot $V(p)$ together with a fit according to the Murnaghan equation [@Murnaghan44] $$\label{Murnaghan} V(p) = V_0 [(B'/B_0)p+1]^{-1/B'}$$ with the bulk modulus $B_0=-dp/dlnV$ and its derivative $B'$ at zero pressure. The ambient-pressure unit cell volume $V_0$ was kept fixed at the experimental value of 112.4(5) Å$^3$ \[102.7(6) Å$^3$\] for TiOBr (TiOCl). [@Sasaki05] The bulk moduli $B_0$ evaluated according to the Murnaghan equation are 33.7$\pm$ 0.8 GPa and 31.0$\pm$ 0.9 GPa for TiOBr and TiOCl, and the derivatives $B'$ are 6.9$\pm$ 0.3 and 6.7$\pm$ 0.3, respectively. The bulk modulus of TiOBr is slightly larger than that of TiOCl, i.e., TiOBr is slightly less compressible than TiOCl. Furthermore, one notices that the pressure derivative $B'$ of both compounds is significantly larger compared to the value $B'$$\approx$4 typically found for three-dimensional solids with isotropic elastic properties. The enhanced value of $B'$ thus suggests anisotropic compression properties of TiO$X$. It is interesting to note that the bulk modulus $B_0$ and its derivative $B'$ of TiO$X$ are close to the corresponding values found for graphite ($B_0$=33.8 GPa, $B'$=8.9).[@Hanfland89] ---------- --------------- ---------------- ------------------- ------------- ------------------- material transmittance transmittance transmittance reflectance x-ray diffraction (CsI) (argon) (alcohol mixture) (CsI) (helium) TiOCl 12 GPa 16 GPa $\approx$16 GPa 12 GPa 15.5 GPa TiOBr 10-11 GPa 14 GPa (295 K) not measured 10-11 GPa 14 GPa 16 GPa (23 K) ---------- --------------- ---------------- ------------------- ------------- ------------------- ![(Color online) Unit cell volume $V$ of TiOCl and TiOBr as a function of pressure $P$. The full, red (gray) lines are fits according to Eq. (\[Murnaghan\]). []{data-label="fig:volume"}](volume.eps){width="0.85\columnwidth"} Discussion {#sectiondiscussion} ========== Comparison of transition pressures: TiOBr and TiOCl {#comparisontransitionpressures} --------------------------------------------------- In Table \[tab:comparison\] we compare the transition pressures of TiOBr and TiOCl at room temperature obtained by different experimental techniques (transmittance, reflectance, x-ray powder diffraction) and for different pressure media. First, comparing the corresponding results for the two compounds, one notices a pressure difference of $\approx$2 GPa. This suggests the existence of some sort of chemical pressure effect in TiO$X$. A starting point for the understanding of this finding could be a comparison of the ambient-pressure lattice parameters. The lattice parameters $b$ and $c$ of TiOBr ($a$=3.785 Å, $b$=3.485 Å, $c$=8.525 Å) are significantly larger than those of TiOCl ($a$=3.789 Å, $b$=3.365 Å, $c$=8.060 Å). [@Sasaki05; @Kataev03] The difference is most pronounced for the $c$ axis; here, the larger value in TiOBr can be attributed to the larger size of the Br$^-$ ions, which form layers separating the buckled Ti-O bilayers. Naively, one would then expect a [*higher*]{} pressure to induce the transition in TiOBr compared to TiOCl, which is in contradiction to our findings. Thus, not the distance between the Ti-O bilayers but the pressure-induced crystal structure changes [*within*]{} the bilayers seem to be the crucial parameter for inducing the closure of the Mott-Hubbard gap in TiO$X$. This furthermore suggests that the high-pressure phase has a dimensionality of less than three, being mainly confined to the buckled Ti-O bilayers. ![Pressure-dependent ratio of the lattice parameters $a$ and $b$ for the TiOBr and TiOCl at room temperature. The ratio $a$/$b$ as a function of pressure follows a linear behavior.[]{data-label="fig:ratio"}](ratio.eps){width="0.85\columnwidth"} A two-dimensional character of the high-pressure phase could indeed explain the difference in the critical pressures for TiOBr and TiOCl: At ambient conditions the one-dimensional character of TiOBr is weaker than in TiOCl, since the lattice parameter ratio ($a$/$b$) in TiOBr ($a$/$b$=1.086) is smaller than in TiOCl ($a$/$b$=1.126).[@Sasaki05] This is consistent with magnetic susceptibility measurements showing a larger deviation from the Bonner-Fisher type behavior above the spin-Peierls transition for TiOBr compared to TiOCl.[@Kato05] The more two-dimensional character of TiOBr was also demonstrated by recent photoemission experiments and supported by density-functional calculations.[@Hoinkis07] Hence, in the case of TiOBr less pressure would be needed to drive the system into a (prospective) two-dimensional, high-pressure phase. Our pressure-dependent crystal structure data can provide a test of this picture (two-dimensional character of the high-pressure phase): In Fig. \[fig:ratio\] we plot the ratio $a/b$ for both studied compounds as a function of pressure. This ratio should decrease towards the value 1 under pressure in the case of a two-dimensional high-pressure phase. Instead, the ratio $a/b$ [*increases*]{} with increasing pressure for both compounds: TiOBr and TiOCl become more one-dimensional under pressure. Obviously, a different criterium regarding the changes of the crystal structure with applied pressure has to be used, in order to explain the difference of $\approx$2 GPa for the critical pressures of TiOBr and TiOCl. At this point we can only speculate about possible criteria for the Mott-Hubbard gap closure – like a critical Ti-Ti distance along $b$ or $a$ direction – since information about the shifts of the atomic coordinates under pressure is not available. Mott-Hubbard gap closure and structural phase transition {#gap closure-structure} -------------------------------------------------------- Under hydrostatic conditions, i.e., for argon as pressure medium, the closure of the Mott-Hubbard gap in TiOBr (TiOCl) occurs at 14 GPa (16 GPa) (Table \[tab:comparison\]). Under less hydrostatic conditions, i.e., using CsI as pressure medium, the gap closure happens at somewhat lower pressure ($\Delta$$P$$\approx$4 GPa). This offset in the transition pressure for different types of pressure media has been reported earlier for TiOCl and LaTiO$_{3.41}$.[@Frank06; @Kuntscher06] The important finding is that under similar hydrostatic conditions the closure of the Mott-Hubbard gap in TiO$X$ coincides with a structural phase transition, as demonstrated by our pressure-dependent x-ray powder diffraction data (Table \[tab:comparison\]). Therefore, the gap closure in TiO$X$ is not of purely electronical origin, but the lattice degree of freedom has to be taken into account too. In this regard it is interesting to compare the results for TiOBr and TiOCl with typical examples of bandwidth-controlled Mott transitions under external pressure, which are discussed in literature. One finds the general observation that the Mott transition coincides with volume discontinuities or even changes of the crystal symmetry. This applies, for example, to the canonical Mott-Hubbard systems VO$_2$ (Ref. ) and vanadium sesquioxide doped with chromium, (V$_{0.95}$Cr$_{0.04}$)$_2$O$_3$,[@McWhan70; @McWhan71; @McWhan73] and also to MnO,[@Yoo05] YNiO$_3$,[@Garcia-Munoz03] Fe$_2$O$_3$,[@Rozenberg02] or FeI$_2$.[@Rozenberg03] It was even suggested that as a rule the Mott transition coincides with a structural phase transition and volume collapse.[@Rozenberg03] Our finding of a pressure-induced structural phase transition in TiOBr and TiOCl at the same pressure where the Mott-Hubbard gap closes, is in agreement with such an interpretation of the Mott-Hubbard transition. The importance of electronic correlations for the underlying mechanism of the observed gap closure in TiO$X$ is suggested by the effective mass of the charge carriers, as estimated from the spectral weight analysis. As demonstrated in Sec. \[transmittance\], for both TiOBr and TiOCl the spectral weight becomes pressure-independent above a certain pressure \[see inset of Fig. \[fig:cond-all\] and Fig. 4(b) in Ref. \]. From the high-pressure value of the spectral weight one can estimate an effective density of charge carriers according to Eq. (\[carrierdensity\]),[@comment1] averaged over the two studied crystal directions, to $n_{eff}$=$(0.6 \pm 0.2)$$\cdot$$10^{21}$cm$^{-3}$ for TiOBr and $n_{eff}$=$(1.3 \pm 0.2)$$\cdot$$10^{21}$cm$^{-3}$ for TiOCl for the same frequency range.[@Kuntscher06] Based on these values the effective number of charge carriers per Ti atom, $N_{eff}$, can in principle be calculated, if the unit cell volume and the number of formula units per unit cell are known. For an estimate of $N_{eff}$ we assumed a high-pressure volume of 93 Å$^3$ (82 Å$^3$) and a number of formula units per unit cell of $Z$=2 ($Z$=2) for TiOBr (TiOCl). Hereby, we neglected the change of the crystal symmetry and a possible collapse of the unit cell volume at the insulator-to-metal transition; the latter effect usually ranges between 1 and 10 %. [@McWhan70; @Yoo05; @Rozenberg02; @Rozenberg03] Under these assumptions we obtained $N_{eff}$=0.03 $\pm$0.01 for TiOBr and $N_{eff}$=0.05 $\pm$0.01 in the case of TiOCl. I. e., $N_{eff}$ is much lower than the expected value of 1. One possible explanation for the reduced value of $N_{eff}$ could be that the charge carriers only partly contribute to the excitations in the specified frequency range. In addition, the reduction might be related to an enhanced effective mass of the charge carriers, typically found in materials with strong electronic correlations. The mass enhancement in TiO$X$ might get stronger when the system approaches the Mott insulating state, as suggested by the suppressed carrier density with decreasing pressure \[see inset of Fig. \[fig:cond-all\] and Fig. 4(b) in Ref. \]. A mass enhancement in the vicinity of a transition to a Mott insulator was theoretically predicted [@Brinkman70] and observed in some cases. [@Qazilbash06; @Merino08] In order to understand the main mechanism driving the observed closure of the Mott-Hubbard gap, the crystal structure of the high-pressure phase might be an important piece of information. However, up to now we could not resolve the symmetry of the crystal structure at high pressures. In this regard, density-functional calculations [@Valenti08] might provide predictions which could then be tested on our x-ray diffraction data. Finally, we would like to comment on the possibility of the metallic character of the high-pressure phase in TiO$X$. Based on our data we cannot prove the existence of a Drude term in the optical response related to coherent quasiparticles at high pressures.[@Rozenberg95] It was, however, demonstrated theoretically and experimentally in various cases, that above a certain temperature the absence of a Drude term in a correlated system located on the metallic side of the Mott transition is to be expected: A lot of theoretical work has been devoted to the transport properties of systems close to the first-order Mott transition at low temperatures and in the crossover regime at elevated temperatures. Optical conductivity spectra for different interaction strengths and different temperatures were obtained in a dynamical mean-field theory (DMFT) treatment of the Hubbard model.[@Georges96] It was shown that only below a certain temperature $T_{coh}$ a quasiparticle peak involving coherent excitations appears at the Fermi energy and the Fermi liquid description applies. As a result, only at low temperatures ($T$$<$$T_{coh}$) a Drude term should be present in the optical conductivity spectrum. With increasing temperature, the quasiparticle peak is gradually destroyed and disappears above the temperature $T_{coh}$. Such a behavior was demonstrated for the two-dimensional organic charge-transfer salts $\kappa$-(BEDT-TTF)$_2$Cu\[N(CN)$_2$\]Br$_x$Cl$_{1-x}$:[@Merino08] Even for a high Br content, i.e., on the metallic side of the Mott transition, no Drude-like peak is present down to approx. 50 K. Only below this temperature a Drude-like feature appears, which can be described with an extended Drude model, with a frequency-dependent scattering rate and effective mass. The optical conductivity spectra of TiO$X$ as a function of pressure were obtained at room temperature. According to the findings for organic salts mentioned above and in other cases, [@Georges96] the seeming absence of a Drude-like contribution in the optical response of TiO$X$ at high pressures could be explained by the elevated measurement temperature. Still, the metallic state appears to be the most plausible high-pressure phase for TiO$X$ based on our experimental results. The shape of the optical conductivity spectra at high pressures is, however, an open issue. Furthermore, a direct proof of the Drude response might be obtainable by pressure-dependent reflectance measurements carried out at low temperatures. Conclusions {#summary} =========== In conclusion, we have studied the pressure-dependent optical response of TiOBr and TiOCl at room temperature by transmittance and reflectance measurements in combination with pressure-dependent x-ray powder diffraction experiments. For both compounds the infrared transmittance is suppressed above a critical pressure. The pressure-dependent reflectance and corresponding optical conductivity spectra reveal additional electronic excitations at high pressures extending down to the far-infrared range. These findings suggest the closure of the Mott-Hubbard gap under pressure. For TiOBr the pressure-induced suppression of the infrared transmittance also occurs at 23 K, where the compounds is in the spin-Peierls phase at ambient pressure. The orbital excitations in TiOBr shift linearly to higher frequency with increasing pressure. The shifts are not completely reversible upon pressure release, especially at low temperatures. The pressure-induced changes occur at somewhat lower pressure in the case of TiOBr compared to TiOCl. This difference cannot be attributed to the more two-dimensional character of TiOBr, since according to the ratio of the crystal parameters $a$ and $b$ the system becomes more one-dimensional under pressure, i.e., the high-pressure state seems to be rather of one-dimenisonal than of two-dimensional character. The closure of the Mott-Hubbard gap coincides with a structural phase transition. From the results of our pressure-dependent x-ray powder diffraction measurements on TiOBr and TiOCl we could furthermore extract the pressure-dependence of the lattice parameters and of the unit cell volume. The latter can be well described by the Murnaghan equation. The enhancement of the effective mass of the charge carriers around the critical pressure suggests the importance of electronic correlations for the mechanism driving the transition. However, the lattice degree of freedom seems to play in important role as well, since the crystal symmetry changes at the transition pressure. 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--- abstract: 'We have developed a terahertz radiation detector that measures both the amplitude and polarisation of the electric field as a function of time. The device is a three-contact photoconductive receiver designed so that two orthogonal electric field components of an arbitrary polarised electromagnetic wave may be detected simultaneously. The detector was fabricated on Fe$^{+}$ ion-implanted InP. Polarisation-sensitive detection is demonstrated with an extinction ratio better than 100:1. This type of device will have immediate application in studies of birefringent and optically active materials in the far-infrared region of the spectrum.' author: - 'E. Castro-Camus' - 'J. Lloyd-Hughes' - 'M.D. Fraser' - 'H.H. Tan' - 'C. Jagadish' - 'M.B. Johnston' title: ' Polarisation-sensitive terahertz detection by multicontact photoconductive receivers ' --- The far-infrared, or terahertz (THz), region of the electromagnetic spectrum encompasses the energy range of many collective processes in condensed matter physics and macromolecular chemistry. However, in the past this spectral region has been relatively unexplored owing to a lack of bright radiation sources and appropriate detectors. The technique of THz time domain spectroscopy (TDS),[@AustonC85; @SmithAN88] which has developed rapidly as a result of advances in ultra-short pulsed laser technology, now provides a very sensitive probe across the THz band. TDS is currently an invaluable tool in condensed matter physics [@HuberTBBAL01; @LeitenstorferHTBBA02; @KaindlHCLC03] and macromolecular chemistry.[@cpl03; @schmuttenmaer2004] To date THz-TDS techniques have relied on linearly polarised emitters and detectors. However, for spectroscopy of birefringent and optically active materials it is also important to measure the polarisation state of radiation before and after it has interacted with the material. Here we report on a detector that enables such a THz-TDS system to be realised. Vibrational circular dichroism (VCD) spectroscopy is a new technique which has substantial potential in the fields of macromolecular chemistry and structural biology.[@nafie96] Akin to the established technique of (ultraviolet) circular dichroism, VCD is used to analyse the structure of chiral molecules. It is predicted that VCD will be more powerful than conventional circular dichroism for stereo-chemical structure determination.[@nafie96] However the technique is currently limited by insensitive and narrow band spectrometers. Of particular interest to biochemists is the structure and function of proteins and nucleic acids. These chiral biomolecules have vibrational and librational modes in the THz region and the THz optical activity of these modes are starting to be studied experimentally.[@xu19; @XuRGSSBAP03] THz frequency VCD is already finding application in fields as distinct as biochemical research [@salzman:2175] and astrobiology.[@XuRGSSBAP03] In the future the ability to perform VCD using a polarisation sensitive THz-TDS technique should enhance the bandwidth and sensitivity of measurements, and allow dynamic time-resolved studies to be performed. In order to perform polarisation sensitive THz-TDS, it is necessary to be able to measure two (preferably orthogonal) electric field components of a terahertz transient. Theoretically it is possible to do this using a conventional (two contact) photoconductive receiver. That is, measure one electric field component and then rotate the receiver by $90^{\circ}$ and measure the other component. However in practice this procedure has two major disadvantages: Firstly, during the rotation of the photoconductive receiver, any slight misalignment will significantly shift the relative phase of the electric field components; secondly the data acquisition time is increased as both components are recorded separately. In order to avoid these disadvantages an integrated receiver capable of measuring both components simultaneously is needed. Such a detector may be realised by fabricating a three-contact photoconductive receiver. The three-contact receiver we developed is shown in Fig. \[f:Photo\]. When designing the three-contact receiver we considered two main constraints. Firstly, the unit vectors ($\mathbf{\hat{u}}_1$ and $\mathbf{\hat{u}}_2$) normal to the gaps formed between the earth contact and the other two contacts (1 and 2) need to be orthogonal. Secondly it is necessary that both gaps are within an area smaller than the laser beam waist and the focus spot size of the THz radiation (a circle of radius $\sim100\,\rm{\mu m}$). This last condition is necessary in order to have uniform laser and THz illumination across both gap regions. The performance of a photoconductive receiver depends strongly on the material from which the device is fabricated. Material dependent carrier trapping and recombination times play an essential role in photoconductive receiver device performance. Specifically, long carrier lifetimes will permit the reception of large amounts of noise and short carrier lifetimes will reduce the signal level and accuracy. Modified semiconductor materials such as low-temperature-grown or ion-implanted GaAs/InP are typically used, as carrier trapping times in these materials may be controlled.[@liu04; @ShenUBLDGBTE04] In order to fabricate the three-contact PCS device (shown in Fig. \[f:Photo\]) we have implanted semi-insulating InP (100) substrates using $2.0\,\rm{MeV}$ and $0.8\,\rm{MeV}$ $\rm{Fe^+}$ ions with doses of $1.0\times10^{13}\,\rm{cm^{-2}}$ and $2.5\times10^{12}\,\rm{cm^{-2}}$ respectively. These multi-energy implants gave an approximately uniform density of vacancies to a depth of 1$\mu$m, resulting in a carrier lifetime of $\sim130\,\rm{fs}$.[@carmody:1074] The samples were subsequently annealed at $500^{\circ}\rm{C}$ for 30 minutes under a PH$_3$ atmosphere. Finally the electrodes were defined using standard photolithography and lift-off techniques. The Cr/Au contacts were deposited to a thickness of 20/250nm using a thermal evaporator. ![ (Colour online) Diagram of experimental apparatus used for simultaneous detection of horizontal and vertical components of the electric field of a THz transient. A SI-GaAs photoconductive switch was used as emitter, parabolic mirrors were used to collect and focus the THz radiation onto the three-contact photoconductive receiver. One of the contacts was used as common (GND) and the other two were amplified independently to obtain the two orthogonal components. Inset: Photograph of a three-contact photoconductive receiver structure formed by two $16\mu \rm{m}$ gaps in orthogonal directions in order to measure the perpendicular components of the THz electric field. The unit vectors $\mathbf{\hat{u}}_1$ and $\mathbf{\hat{u}}_2$ represent the directions between the earth contact and contacts 1 and 2 respectively. The photograph was taken using an optical microscope. []{data-label="f:Photo"}](Fig1.eps){width="8.0cm"} In order to measure a THz electric field $\mathbf{E}_{\rm THz}$ using a photoconductive receiver it is necessary to gate the receiver with an ultra-short laser pulse. The laser pulse generates free charge carriers in the semiconductor substrate, which are accelerated by $\mathbf{E}_{\rm THz}$ thus generating a current $I$ between two contacts. Assuming a laser pulse of the form $\rm{sech}^2(1.76t/t_0)$ where $t_0$ is the full-width-at-half-maximum, the current measured through contact $i$ in the photoconductive receiver described here, is related to $\mathbf{E}_{\rm THz}$ by[@kono:898]: $${ I_{i}(t) \propto \int_{-\infty}^\infty \mathbf{E}_{\rm THz}(t') \cdot \mathbf{\hat{u}}_i\, e^{-(t'-t)/\tau} \times [1+\tanh(1.76(t'-t)/ t_0)] \rm{d}t' } \label{e:Iconventional}$$ where $\mathbf{E}_{\rm THz}(t')$ is the THz electric field, $\mathbf{\hat{u}}_1$ and $\mathbf{\hat{u}}_2$ are unit vectors in the direction of the two gaps between the contact ($i=$1,2) and the earth electrode. $\tau$ is the lifetime of free carriers. The three-contact photoconductive receiver was tested using the setup shown in Fig. \[f:Photo\]. A linearly polarised THz transient was generated by exciting a $400\,\rm{\mu m}$ gap SI-GaAs photoconductive switch emitter biased by a $\pm150\,\rm{V}$ square wave at a frequency of 25kHz. The emitted THz transients were collected in the back reflection geometry and then focused on to the receiver using off-axis parabolic mirrors. A Ti:Sapphire chirped mirror oscillator with a 75MHz repetition rate provided 10fs pulses of 4nJ and 800nm centre wavelength, which were used to excite the emitter. A 0.4nJ fraction split from the original pulse was used to gate the receiver. Two separate lock-in amplifiers were used to record the currents ($I_1$ and $I_2$) through the two contacts. The lock-in amplifiers and the common electrode of the receiver were connected to a common earth, and the references of both lock-in amplifiers were locked to a TTL signal provided by the 25kHz signal generator (used to drive the THz emitter). In all measurements the $I_{1}(t)$ signal from Lock-in 1 and $I_{2}(t)$ signal from Lock-in 2 were recorded simultaneously at each time step using a multichannel analogue to digital converter. The photoconductive switch emitter was mounted on a graduated rotation stage that allowed the gap, and hence the polarisation of the emitted THz transient, to be rotated. Both $I_1(t)$ and $I_2(t)$ where measured averaging over 90 scans at three different angles of the emitter ($0$, $45$ and $90^{\circ}$). The measurements were taken in an evacuated chamber at a pressure of 25mbar to avoid water vapor absorption. The THz electric field was calculated by differentiating numerically the two $I(t)$ traces measured by the lock-in amplifiers according to Eq. \[e:Iconventional\]. The horizontal $E_{\rm{H}}$ and vertical $E_{\rm{V}}$ electric field components are plotted against time in Figs. \[f:Lineal\] (a), (b) and (c) for the emitter at angles $0^\circ$, $45^\circ$ and $90^\circ$ respectively. The results demonstrate that the three-contact photoconductive receiver acts as a polarisation sensitive detector. The polarisation selectivity of the detector was assessed by measuring the cross polarised extinction ratio. This ratio was found to be 108:1 (128:1) for the horizontally (vertically) oriented emitter. It should be noted that the polarisation of the radiation arriving at the detector may not be perfectly linear, as photoconductive emitters do not produce purely dipolar radiation.[@rudd01] Therefore, the true extinction ratio of the detector may be higher. ![(Colour online) Horizontal (solid) and vertical (dashed) components of THz electric field (obtained form measured voltage) are plotted against time with the emitter at (a) $0$, (b) $45$ and (c) $90^\circ$ respectively.[]{data-label="f:Lineal"}](Fig2.eps){width="8.0cm"} ![ (Colour online) Parametric representation of horizontal and vertical components of THz electric field for three emitter orientations; $0$ (circles), $45$ (crosses) and $90^\circ$ (squares). The THz wavevector points out of the paper plane. In this plot the angle of polarisation for the three waves is easily observed. []{data-label="f:Parametric"}](Fig3.eps){width="8.0cm"} In Fig. \[f:Parametric\] a parametric plot of the data shown in Fig. \[f:Lineal\] is presented. For an ideal linearly polarised source the three sets of data should form straight lines at 0, 45 and 90$^{\circ}$ in the $E_{\rm{H}}-E_{\rm{V}}$ plane. However, the measured angles of polarisation (from the horizontal plane) are -5.5, 39, and 98$^{\circ}$ respectively and the polarisation appears to be slightly elliptical (especially in the 45$^{\circ}$ case). These discrepancies arise from a number of sources: It has been shown previously that photoconductive switch emitters produce a small quadrupole field leading to a cross polarised electric field component.[@cai97; @rudd01] Furthermore, low *f*-number collection systems (such as the *f*/1.5 system used in this experiment) inevitably lead to linearly polarised radiation becoming slightly elliptical.[@rudd01] The signal-to-noise (SNR) ratio in our experiments was measured to be 175:1 for the three contact receiver. We obtained a similar ratio for a conventional two-contact “bow-tie” receiver, which we fabricated on a piece of the same substrate material. This indicates that the SNR performance of this device is limited by the substrate material rather than the receiver design. Therefore the device sensitivity could be improved greatly by using optimised ion-implanted InP or GaAs substrates. Indeed optimised low temperature MBE-grown GaAs has been shown to have excellent SNR performance[@ShenUBLDGBTE04] which should be replicated in a three-contact device fabricated on that material. In conclusion, the design of a novel integrated detector capable of measuring both components of an arbitrarily polarised THz transient was presented as well as experimental evidence of its effectiveness. This integrated three-contact detector is expected to be very useful for further studies of time-domain circular dichroism spectroscopy and should have a wide range of applications in basic research and industry. The authors would like to thank the EPSRC (UK) and the Royal Society for financial support of this work, ECC wishes to thank CONACyT (México) for a scholarship. Australian authors would like to acknowledge the financial support of the Australian Research Council. [17]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , ****, (). , , , ****, (). , , , , , , ****, (). , , , , , , ****, (). , , , , , ****, (). , , , , , , ****, (). , ****, (). , ****, (). , , , , , , , (, ), vol. , pp. . , , , , , , , , ****, (). , ****, (). , , , , , , , ****, (). , , , , , , , , , ****, (). , , , , , ****, (). , , , ****, (). , , , ****, (). , , , , , , ****, ().
ZU-TH 22/01 [**Conférence du Club d’Information Scientifique**]{}\ [*Ecole Nationale Supérieure des Télécommunications*]{}\ Paris (France), le 11 Juin 2001 ------------------------------------------------------------------------ ------------------------------------------------------------------------ [**Denis Puy**]{}[^1] [ *Institute of Theoretical Physik, Zürich\ and Paul Scherrer Institut, Villigen (Switzerland)*]{} [**1- Introduction**]{}\ Titre étrange s’il en est. L’Univers sombre ou l’âge sombre de l’Univers relate ni une ère tragique, ni un antécédent ténébreux, ni une phase funeste de l’histoire de l’Univers ou un épisode de la saga des [*Star Wars* ]{}. Il s’agit d’une période peu connue et donc ... obscure d’un lointain passé de l’Univers.\ Martin Rees fut le premier à suggérer ce vocable pour désigner l’intermède de l’histoire de l’Univers qui a suivi le découplage de la matière et du rayonnement, et précédé la formation des premiers objets. Durant cette période, les conditions physiques de l’Univers permettent de penser que les premières molécules apparurent. Cette étape a été peu étudiée jusqu’à encore une décennie.\ La cosmologie s’appuie aujourd’hui essentiellement sur trois faits observationnels: - L’expansion de l’Univers, qui éloigne les galaxies les unes des autres. - La composition de l’Univers en éléments légers (environ 75 % d’atomes d’hydrogène, 25 % d’atomes d’helium). - L’existence d’un fond de rayonnement cosmologique de caractère thermique à environ 2.73 Kelvins. A partir de ces observations, un modèle d’évolution de l’Univers a pu être établi. C’est le modèle du [*Big Bang*]{}, qui fait aujourd’hui l’objet d’un assez large consensus. Selon ce modèle, l’Univers a connu une origine extrêmement dense et chaude, suivie d’une expansion et d’un refroidissement. C’est au cours de ce refroidissement que la matière a, peu à peu, pris la forme que nous lui connaissons aujourd’hui.\ Au cours de cet exposé j’analyserai, tout d’abord, les phénomènes qui ont prévalu avant la formation des premières molécules de l’Univers. Ainsi, je m’attacherai à une brève description de la phase de formation des noyaux, ou nucléosynthèse primordiale, suivie de celle de formation des premiers atomes ou recombinaison cosmologique. Cette dernière aura des conséquences particulièrement importantes sur l’évolution de l’Univers, car c’est à cette même époque que le rayonnement se découple de la matière, conduisant à libérer les photons dans un fond diffus. C’est précisément à cette période que les premières molécules vont apparaitre dans l’Histoire de l’Univers. Nous tenterons alors d’analyser les conséquences de l’existence de celles-ci sur la phase ultérieure de formation des premières structures gravitationnelles. [**2- La nucléosynthèse primordiale**]{}\ Le modèle standard du [*Big Bang*]{} est basé sur l’extrapolation de notre connaissance actuelle de l’Univers -expansion, existence du fond microonde- et sur celle de la physique des particules. Une description temporelle de l’Univers primordial, suivant son expansion, permet de décrire qualitativement les grandes étapes de son histoire à travers différentes transitions de phases successives: - Une première phase d’expansion extrêmement brutale, appelée [*inflation*]{}. - La transition de la [*théorie de grande unification*]{}, quand l’Univers avait un âge d’environ 10$^{-36}$ s et une température de 10$^{28}$ K, qui découple la force électrofaible et l’intéraction forte. - La transition [*électrofaible*]{} libérant la force électromagnétique et l’intéraction faible, quand il avait un âge d’environ 10$^{-10}$ s et une température d’environ 10$^{15}$ K. - La transition [*quark-hadron*]{}, ou confinement des quarks en hadrons, quand il avait un âge d’environ 10$^{-6}$ s et une température d’environ 10$^{13}$ K. Après cette transition quark-hadron, l’Univers est constitué essentiellement de neutrinos, d’anti-neutrinos, de positrons, d’électrons, de photons, de protons et de neutrons en équilibre thermodynamique (la soupe originelle).\ A ces températures, les neutrinos jouent un rôle stabilisateur dans cette agitation thermique. En effet absorbés et reémis sans cesse par les nucléons, ces neutrinos transforment continuellement les protons en neutrons et inversement. Ces réactions, gouvernées par l’intéraction faible, maintiennent en équilibre une population de neutrons tout à fait comparable à celle des protons. L’Univers est alors opaque aux neutrinos.\ Avec la décroissance de la température l’énergie des particules diminue progressivement, rapidement les neutrinos ne vont plus être en mesure d’interagir avec les nucléons. En dessous de la température de $10^{10}$ K, l’Univers devient transparent aux neutrinos. Ce passage à la transparence va avoir des incidences particulièrement importantes sur la population des neutrons. A ce stade l’équilibre entre les protons et neutrons est rompu. Certes le neutron peut continuer à se désintégrer en donnant naissance à un proton, un électron et un neutrino, mais en des temps de plus en plus longs. En revanche ce même neutron va réagir beaucoup plus vite avec un proton et former un noyau de deutérium. C’est le début de la phase de nucléosynthèse primordiale.\ Cette [*brisure*]{} de l’équilibre neutron-proton conduisant à la formation des premiers noyaux atomiques va s’intensifier à la température de 10$^9$ K. Peu à peu d’autres noyaux tels que l’helium 3, l’helium 4 vont apparaitre puis interagir entre eux pour constituer un réseau complexe de réactions nucléaires caractérisant le modèle standard de la nucléosynthèse.\ Le deutérium apparait donc comme la première [*brique*]{} de la synthèse des éléments légers. Une fois passée la [*chicane*]{} du deutérium, les noyaux plus lourds peuvent se constituer: le deutérium est un passage obligé dans la chaine des réactions qui conduisent à la synthèse de l’hélium 4. Peu à peu l’énergie du milieu diminue (par l’expansion), la barrière Coulombienne entre les noyaux va être de plus en plus efficace. La nucléosynthèse va stopper au noyau de beryllium $^9Be$, une description plus complète de la nucléosynthèse primordiale peut être trouvée dans Sarkar 1996 ou Puy & Signore 2001.\ Un calcul précis de quelques 150 réactions nucléaires connues conduit aux abondances primordiales suivantes (en unité de densité totale): $$\begin{aligned} & \bullet & H \sim 0.76 \nonumber \\ & \bullet & ^4He \sim 0.24 \nonumber \\ & \bullet & D \sim 4.3 \times 10^{-5} \nonumber \\ & \bullet & ^3He \sim 10^{-5} \nonumber \\ & \bullet & ^7Li \sim 2.2 \times 10^{-10} \nonumber \\ & \bullet & ^9Be \sim \, {\rm trace} \nonumber\end{aligned}$$ Les très récentes mesures du fond de rayonnement cosmologique effectuée par l’équipe conduite par Paolo de Bernardis (2000), à l’aide du ballon BOOMERANG lancé en 1998 en Antarctique, semblent confirmées ce modèle standard de la nucléosynthèse primordiale (Burles-Nollett-Turner 2001).\ D’autres alternatives théoriques furent tentées. L’une concerne une nucléosynthèse hétérogène, où lors du confinement quarks-hadrons certaines fluctuations de densité pourraient donner lieu à former des bulles riches en neutrons ou en protons, et faciliter ainsi la production d’éléments lourds (voir Jedamzik & Rehm 2001). L’autre tentative a été conduite par quelques théoriciens de physique des particules qui ont suggérer l’existence d’un quatrième type de neutrino: le neutrino stérile, caractérisé par le fait que sa force d’intéraction est beaucoup moins importante que la classique intéraction faible. Une première conséquence serait alors de modifier le rapport neutron-proton; paramètre important pour la nucléosynthèse primordial. Ce nouveau champ de recherche particulièrement actif au CERN de Genève pourrait conduire à de nouvelles valeurs d’abondance de noyaux primordiaux (voir Kirilova & Chizhov 2001). Dans le cadre de ces modèles, les noyaux plus lourds que les noyaux de nombre de masse supérieur à 11 pourraient être synthétisés. L’abondance primordiale des noyaux du $^7Li$ au $^{11}B$ serait également supérieure à celle du modèle standard. [**3- Chimie primordiale**]{}\ Après cette courte période de nucléosynthèse, l’Univers est encore dominé par le rayonnement qui est complétement couplé à la matière par diffusion élastique. Cette diffusion ou diffusion Thomson des photons par les électrons se prolonge jusqu’à ce que l’Univers ait une température comprise entre 5 000 K et 7 000 K. A ces températures, la photoionisation des atomes devient peu à peu négligeable. Les noyaux se [*recombinent*]{} alors avec les électrons libres pour former les atomes, c’est l’époque de la recombinaison.\ $He^{++}$, $He^+$ et $H^+$ se recombinent en suivant leurs potentiels d’ionisation décroissants. La matière change rapidement d’état, passant d’un plasma ionisé à un gaz neutre. Quand le nombre d’électrons libres a suffisamment décru, la diffusion Thomson devient négligeable: c’est l’époque de la dernière diffusion conduisant au découplage définitif entre la matière et le rayonnement. Des molécules sont alors capables de survivre.\ Puisque le modèle classique de nucléosynthèse primordiale favorise la production d’un nombre limité d’éléments légers, la chimie après la recombinaison est essentiellement une chimie gazeuse de l’hydrogène, du deutérium, de l’hélium et , à un degré moindre, du lithium. Bien que, chronologiquement, les premières molécules formées soient les ions moléculaires $He_2^+$ et $HeH^+$, et bien qu’on ne connaisse pas toutes les sections efficaces de toutes les réactions de création et de destruction de molécules, les molécules primordiales les plus abondantes sont l’hydrogène moléculaire $H_2$ et les hydrures $HD$ et $LiH$ (voir Puy et al. 1993, Puy & Signore 1999, Signore & Puy 1999).\ La formation de $H_2$ commence avec le processus radiatif: $$H^+ + H \, \to \, H_2 + h\nu$$ suivie d’un transfert de charge: $$H_2^+ + H \, \to \, H_2 + H^+.$$ Une autre voie pour la formation de $H_2$ commence par un attachement radiatif et la formation de l’ion $H^-$: $$H+ e^- \, \to \, H^- + h\nu$$ suivie par un détachement associatif avec H: $$H^- + H \, \to \, H_2 + e^-.$$ Quant à l’hydrure $HD$, forme isotopique de $H_2$, un transfert de charge détermine l’ionisation de $D$: $$D + H^+ \, \to \, D^+ + H$$ et la production d’une abondance significative de $H_2$ conduit à la formation rapide de $HD$: $$D^+ + H_2 \, \to \, HD + H^+.$$ Enfin l’hydrure de lithium $LiH$ outre par classique association radiative (réaction très lente): $$Li + H \, \to \, LiH + h\nu$$ cette molécule se forme principalement à partir de l’ion moléculaire $LiH^+$ par échange de charge: $$LiH^+ + H \, \to \, LiH + H^+ .$$ Le réseau complet comporte environ 120 réactions.\ L’évolution des abondances sera rapide après le découplage rayonnement- matière. Au décalage spectral $z=300$ l’essentiel des molécules primordiales et ions moléculaires sont formés, les abondances des molécules n’évoluent plus. l’expansion prive la chimie de réactions collisionnelles du fait de la décroissance de la densité. L’abondance est [*gelée*]{}. Les plus récents calculs concernant l’abondance [*finale*]{} des molécules et ions moléculaires donnent (voir Galli & Palla 1998 ou Stancil et al. 1998) donnent en unité de densité totale au décalage spectral $z = 5$: $$\begin{aligned} & \bullet & H_2 \sim 1.1 \times 10^{-6} \nonumber \\ & \bullet & HD \sim 1.2 \times 10^{-9} \nonumber \\ & \bullet & LiH \sim 7.1 \times 10^{-20} \nonumber \\ & \bullet & HeH^+ \sim 6.2 \times 10^{-13} \nonumber \\ & \bullet & LiH^+ \sim 9.4 \times 10^{-18} \nonumber \end{aligned}$$ [**-4 Nuages moléculaires primordiaux**]{}\ Tous les scénarii de formation de structures supposent la présence initiale de fluctuations de densité.\ Une étude détaillée de l’influence des molécules primordiales sur les différentes phases de la dynamique a été entreprise ces dernières années (Puy & Signore 1997, Abel et al. 2000 ou Fuller & Couchman 2000). Une approche simple est de considérer une fluctuation de densité sphérique homogène et isolée que l’on appelera [*nuage*]{} (Puy & Signore 1995). On peut montrer qu’il y a, dans un premier temps expansion, puis dans un second temps effondrement du nuage qui se découple alors de l’évolution cosmologique du milieu. La prérogative d’attraction de la gravité va se mettre en action au point de stopper la croissance de cette [*surdensité*]{}, et d’initier la phase d’effondrement. Pour un nuage de masse 10$^9$ fois la masse du soleil (caractéristique de la masse d’une galaxie), l’effondrement se produit quand la température du rayonnement atteint 150 K, l’Univers a alors un âge d’environ 5$\times 10^7$ ans. Les molécules peuvent rester présentes dans l’effondrement. Il est clair que celles-ci ne vont pas rester longtemps thermiquement inerte. On peut légitimement estimer que la matière en effondrement va devenir plus chaude que le fond microonde environnant. Les processus d’excitations collisionnelles suivis de dé-excitations radiatives (induite et spontanée) des molécules vont être alors prépondérants. Les molécules vont agir comme [*agent de liaison*]{} entre les photons du fond de rayonnement et la matière en effondrement, et conduire à refroidir cette dernière puisque le fond de radiation est plus froid que l’effondrement.\ On montra alors (Puy & Signore 1997) que dans ce cadre, la molécule $HD$ est le réfrigérant le plus efficace. Une première conséquence dynamique de l’existence de cette fonction de refroidissement est la possiblité de développer une instabilité thermique. La description de ce mécanisme peut s’effectuer très simplement. Si un système refroidit, il s’écarte donc de son état d’équilibre thermique. La fonction de refroidissement responsable de ce changement thermique peut, sous certaines conditions, engendrer un état d’instabilité permettant la croissance des fluctuations de densité, toujours présente dans un système réaliste. Néanmoins cette instabilité thermique favorise uniquement la croissance de surdensités à petites échelles à l’inverse de l’instabilité gravitationnelle qui ne privilégie pas d’échelles particulières. On montra alors que la molécule $HD$ peut provoquer une instabilité thermique autour de 200 K dans l’effondrement. Une telle instabilité thermique peut alors conduire à la formation de sous-unités dans le nuage en effondrement, puis à le fragmenter en nuages de plus faibles tailles et masses.\ Ce scénario conduit à penser que des structures du type stellaire pourraient être formées avant les galaxies. L’évolution de ces étoiles primordiales de grandes masses serait alors rapide puis conduirait, après leurs explosions en supernovas, à contaminer le milieu en éléments lourds. Chakrabarti (2000) montrèrent que dans ce cadre l’effondrement d’un nuage moléculaire carboné peut conduire à la formation significative d’adénine $H_5C_5N_5$, une des molécules constitutives de l’ADN. Cette molécule pourrait être apparu tôt dans l’histoire de l’Univers. De là à dire que la vie serait très primitive dans l’histoire de l’Univers, il y a un monde (de connaissance à franchir). A plus modeste échelle, une chimie plus complète est actuellement à l’étude.\ Néanmoins la connaissance de la masse des sous-structures produit par la fragmentation reste encore très imprécise. Les prochaines missions spatiales tels que HERSCHEL et le Télescope spatial de nouvelle génération NGST pourront donner une meilleure vision de l’Univers profond, et offrir des indications quant au processus de fragmentation. L’observation future des molécules primordiales peut être sérieusement envisagée avec les missions actuelles telles que FUSE ou ODIN, ainsi qu’avec le projet ALMA au Chili, d’interféromètre d’un système de 64 antennes de 12 m chacune, de l’organisation des observatoires sub-australs (ESO) qui permettra également d’obtenir une meilleure compréhension de la formation des premiers objets. L’avenir s’annonce passionnant. [*Il faut une infinie patience pour attendre toujours\ ce qui n’arrive pas...\ (Pierre Dac)* ]{} [**Remerciements:**]{} L’auteur remercie Guy Mizrahi et le comité directeur du CIS pour l’invitation à présenter cet exposé, l’ensemble des participants pour leur accueil chaleureux, ainsi que Patrick Koch et Monique Signore pour des remarques didactiques. [**Bibliographie**]{}\ [Abel T. et al. 2000 ApJ 540, 39\ Burles S. et al. 2001 ApJ 552, L1\ Chakrabarti S. 2000 Astr. & Astroph. 354, L6\ De Bernardis P. et al. 2000 Nature 404, 955\ Fuller T., Couchman H.2000 ApJ 544, 6\ Galli D., Palla F. 1998 Astr. & Astroph. 335, 403\ Jedamzik K., Rehm J. 2001 [`a`stro-ph/0101292]{}\ Kirilova D. Chizhov M. 2001 [`h`ep-ph/0102114]{}\ Puy D. et al. 1993 Astr. & Astroph. 267, 337\ Puy D. et al. 1995 Comptes Rendus Académie\ des Sciences 320, IIb, 619\ Puy D., Signore 1997 New Astr. 2, 299\ Puy D., Signore M. 1999 New Astr. Rev. 43, 223\ Puy D. Signore M. 2001 [`a`stro-ph/0101157]{}\ Sarkar S. 1996 Rep. Prog. Phys. 59, 1493\ Signore D., Puy D. 1999 New Astr. Rev. 43, 185\ Stancil P. et al. 1998 ApJ 509, 1]{} 5.7cm [**Site sur les futures missions et projets:**]{}\ [ALMA (projet ESO): [`h`ttp://www.mma.nrao.edu]{}\ \ FUSE (mission NASA en cours):\ [`h`ttp://fusewww.gsfc.nasa.gov/fuse]{}\ \ HERSCHEL (projet mission ESA):\ [`h`ttp://astro.estec.esa.nl/SA-general/Projects/First/first.html]{}\ \ ODIN (mission CNES-SNSB-CSA-TEKES en cours):\ [`h`ttp://www.snsb.se/Odin/Odin.html]{}\ \ NGST (projet NASA):\ [`h`ttp://ngst.gsfc.nasa.gov]{} ]{} [^1]: Email: puy@physik.unizh.ch
--- abstract: 'We comment upon a recent work of Zheng [*et al.*]{} concerning calculations of spectra of light nuclei with no core, where an effective interaction is constructed which spans over several shells. It is demonstrated that the omission of the particle-particle ladder diagrams in their calculations, explains the large differences between results obtained with various model spaces. We use this to infer that low-order perturbation theory works well in reproducing the binding energy of the system we consider.' address: 'Department of Physics, University of Oslo, N-0316 Oslo, Norway' author: - 'T. Engeland, M. Hjorth-Jensen, A. Holt and E. Osnes' title: 'Comment on “No core calculations” of the spectra of light nuclei' --- Introduction ============ Recently, Zheng [*et al*]{} [@prc48] (hereafter ZBJVM) have presented an approach meant to circumvent the notorious intruder state problem. Moreover, this approach was devised in order to avoid calculations of complicated Feynman-Goldstone diagrams which arise in perturbation theory. It is a well-known fact that the presence of so-called intruder states may lead to the divergence of the order-by-order pertubative expansion for the effective interaction $H_{\mathrm{eff}}$. The latter is understood to be evaluated from perturbative many-body techniques and is defined within a physically selected model space, which is given by a projection operator $P$. The remaining degrees of freedom are accounted for by the perturbative expansion. These degrees of freedom are represented by a projection operator $Q$, so that $P+Q=1$ and $PQ=0$. The idea behind the work of ZBJVM is, through the use of an enlarged model space, to avoid both the intruder state problem and that of calculating many perturbative contributions. In this comment we show that the effective interactions derived by Zheng [*et al.*]{} may not be consistent with the underlying theory for the effective interaction. Our points are discussed in the next three sections. In Section II we discuss the omission of particle-particle ladders diagrams in the calculations of ZBJVM. Section III critically discusses the use of the starting energy as a variable. In Section IV a brief discussion of the non-hermiticity of the effective interaction is also included, and our conclusions are drawn in Section V. No-core shell-model calculations with the $G$-matrix ==================================================== The first step in the calculations of Ref. [@prc48] is to evaluate the nuclear raction matrix $G$ given by $$G=V+V\frac{Q}{\omega - H_0}G,$$ where $\omega$ is the unperturbed energy of the interacting nucleons, and $H_0$ is the unperturbed hamiltonian. The operator $Q$, commonly referred to as the Pauli operator, is a projection operator which prevents the interacting nucleons from scattering into states occupied by other nucleons. There are many ways to handle the Pauli operator of Eq. (1). Two of these are demonstrated in Fig. \[fig:fig1\]. In (a) we show the Pauli operator obtained through the double-partitioned scheme of Ref. [@kkko76]. There one has to define a core, given by the boundary $n_1$, which represents the last single-hole state. $n_2$ is the last single-particle state of the model space. In the calculations of ZBJVM, the Pauli operator is defined as in (b) of Fig. \[fig:fig1\]. The model space is again limited by the boundary $n_2$, but we have no holes. This definition is the first step in the so-called “no-core” approach of Ref. [@prc48]. In this work we use the Pauli operator in (b) of Fig. \[fig:fig1\] and define the model space to consist of the $0s$-, $0p$-, $1s0d$- and $1p0f$-shells. We could then, in principle, use the corresponding $G$-matrix to obtain the eigenvalues within this model space. The authors of Ref. [@prc48] are also interested in studying how important various model spaces are. They therefore calculate the eigenvalues with smaller spaces first (see Table I of Ref. [@prc48]), say only the $0s$-shell. However, they use the $G$-matrix defined with a model space which includes also the $0p$-, the $1s0d$- and the $1p0f$-shells. In so doing, they have to include the ladder diagram of Fig. \[fig:fig2\], and higher-order ladder diagrams as well, with intermediate states from the $0p$-, $1s0d$- and $1p0f$-shells, since they use different model spaces in the calculations of spectra and the evaluation of the $G$-matrix, see e.g. the discussion in Ref. [@kkko76]. This is, to our knowledge, not done in Ref. [@prc48]. Actually, we will demonstrate that the omission of these ladder diagrams explains to a large extent why ZBJVM obtain rather different results when they compare results from diagonalizations with one, two and three oscillator shells, respectively. To demonstrate our point, we choose a fictitious system to consist of two particles only, and define various model spaces. The conclusions apply equally well to systems with more particles. Here we choose our hamiltonian $H$ to consist of $$H=H_0+G,$$ with the unperturbed single-particle energies which define $H_0$ given by the harmonic oscillator $$\varepsilon_{nl} =\left( 2n +l+\frac{3}{2}\right)\hbar\Omega +\Delta ,$$ where $\Omega$ is the oscillator energy. Here we set $\hbar\Omega =14$ MeV. We add a negative shift $\Delta=-71$ MeV in order to obtain negative starting energies only (see the discussion in Section III) and use, as in Ref. [@prc48], a fixed starting energy, chosen to be $-100$ MeV. This corresponds to twice the energy of a single particle state in the $0s$-shell, a choice we made in order to avoid poles in the calculation of ladder diagrams. As will be discussed in the next section, the use of a fixed starting energy implies that we have a degenerate model space, which is rather questionable if the model space spans over several shells. The choice of a fixed starting energy is also done in order to avoid the problems with the non-hermiticity of the effective interaction discussed in Section IV. The parameters of the Bonn B potential in Table A.1 of Ref. [@mac89] are used to define the nucleon-nucleon potential $V$. The Pauli operator is defined as in (b) of Fig. \[fig:fig1\], with $n_2$ given by the last single-particle state in the $1p0f$-shell. The resulting eigenvalues for the lowest lying $JT=10$ state is shown in Table I. The most important components in the wave functions of this state arise from single-particle states in the $0s$- and $0p$-shells. The results labelled $G$, include only the $G$-matrix, as done in the work of ZBJVM. As can be seen from Table I, there is clearly a large difference (of the order of $50\%$ or more) between results obtained with a model space defined by the $0s$-shell only and a model space which includes all shells up to the $1p0f$-shell. This qualitative pattern also agrees with Table I of ZBJVM. In their conclusions, Zheng [*et al.*]{} use this to infer that one needs to take into account large model spaces, since the binding energies do not stabilize as functions of the various model spaces[^1]. We show in Table I that this conclusion is misleading. The results obtained with the $G$-matrix plus the two-particle ladder (2P) diagram up to third order in $G$ (higher-order terms are negligible) for the $0s$ model space, show that these results are rather close to those obtained with $G$ for the model space which includes all four shells. This demonstrates clearly that the lack of stabilization in the calculation of the ground states in Ref. [@prc48], is simply due to the omission of the particle-particle ladder diagrams. Note that in our calculations with $G+2P$ for more than one oscillator shell, we use a degenerate model space, as done by ZBJVM. This means that if define the model space to include the $0s$-, $0p$- and $1s0d$-shells, all single-particle states have the same energy. This approximation explains also why the results of the $G+2P$ calculations differ slightly from model space to model space. In this sense, the result obtained with the $0s$-shell only, is the most rigorous one. This results shows also that low-order perturbation theory works well in reproducing the lowest $JT=10$ state. The results with $G$ should also have taken into account a non-degenerate model space, but here we have tried to follow ZBJVM as closely as possible. The problems with a non-degenerate model space are addressed in the next section. Role of the starting energy =========================== In principle, the effective interaction should not depend on the choice of starting energy $\omega$, though, since an approximation to the perturbation expansion is made, the effective interaction may depend on $\omega$. In Table II of Ref. [@prc48], it is shown that the excited spectra depend weakly on $\omega$, whereas the ground state of $^6$Li depends strongly on $\omega$. This state varies from $-23.044$ MeV to $-29.366$ MeV with starting energies between $\omega =20$ and $\omega =38$, respectively. The authors of Ref. [@prc48] give no physical arguments for why one should choose a given starting energy, except that certain starting energies give a better fit to the data. The fact that they get more attraction with the largest starting energy is rather simple. With a positive $\omega$ we are closer to the poles in the energy denominator of $G$, i.e.$$\frac{1}{\omega - H_0}.$$ However, the choice of a positive starting energy is not straightforward in the $G$-matrix calculation. With a negative starting energy (appropriate for the low-lying states of finite nuclei), there are no poles in the above energy denominator. Actually, a principle value integration should have been performed in the above calculation of $G$. This is however not our main objection against the use of the starting energy as a variable by ZBJVM. With a multi-shell model space, one can no longer use a fixed starting energy, rather, the starting energy should take into account the fact that the single-particle energies are no longer degenerate. As an example, consider the matrix element ${\left\langle (0d_{5/2})^2 \right|}G(\omega){\left| (0d_{5/2})^2 \right\rangle}$ coupled to $JT=10$. This matrix element would enter our multi-shell calculations in Table I. The correct starting energy should be the unperturbed energy of two particles in the $d_{5/2}$ orbit. This would correspond to $-44$ MeV in our example. This starting energy gives a matrix element of $0.20$ MeV. In the previous section we used a fixed starting energy of $-100$ MeV, which would give us a matrix element of $0.48$ MeV. Thus, if the other matrix elements behave in a similar way (and they do), the use of a degenerate model space as done by ZBJVM, becomes meaningsless. A scheme which takes the starting energy dependence into account, was recently proposed by Suzuki [*et al.*]{} [@npa94]. Thus, the starting energy is not a parameter which one can choose in order to obtain a good correspondence with the data. The dependence of $\omega$ must be taken into account in the calculations. However, this leads us to our last point, namely that of the non-hermiticity of the effective interaction. We have in our calculations used a fixed starting energy, in order to avoid this problem, which arises even at the level of the $G$-matrix. Non-hermiticity of the effective interaction ============================================ In the first point we stressed the need of including the ladder diagram of Fig. \[fig:fig2\]. However, if one does this, a more serious problem arises, namely that of the non-hermiticity of the effective interaction. Assume now that the intermediate states in the two-particle ladder diagram are those of the $1s0d$-shell only. Diagram (b) in Fig. \[fig:fig2\] is then proportional to $$-\frac{1}{4\hbar\Omega}{\left\langle (0p)^2 \right|}G{\left| (1s0d) \right\rangle}{\left\langle (1s0d) \right|}G {\left| (0s)^2 \right\rangle},$$ where the intermediate states must be those of the $1s0d$-shell if we use a model space for the effective interaction which consists of the $0s$- and the $0p$-shells. $\Omega$ is the oscillator frequency. The starting energy corresponds to the unperturbed energy of $0s$-shell. If we now evaluate diagram (c), we get $$-\frac{1}{2\hbar\Omega}{\left\langle (0s)^2 \right|}G{\left| (1s0d) \right\rangle}{\left\langle (1s0d) \right|}G {\left| (0p)^2 \right\rangle},$$ which yields a strongly non-hermitian effective interaction. The starting energy corresponds here to the unperturbed energy of two particles in the $0p$-shell. As done by ZBJVM, one could ignore ladder diagrams in the definition of the effective interaction, and thereby obtain a hermitian effective interaction in terms of $G$ only[^2]. However, as discussed in Section II, if one truncates the model space, one has to include the ladder diagram, yielding a non-hermitian interaction. It is important to note that this non-hermiticity arises only if we approximate RS perturbation theory to a given order. If all terms are taken into account, this problem does not occur. Viable approaches to obtain an order-by-order effective interaction which is hermitian, have recently been proposed by Lindgren [@lind91] and Kuo [*et al.*]{} [@kuo93]. This strong non-hermiticity is also present if one includes folded diagrams as well, as done in the recent work of Jaqua [*et al.*]{} [@jhbv94]. The same critical remarks in the above Sections apply to that work as well. Conclusion ========== In summary, we have shown that the differences between results for various model spaces obtained by the authors of Ref. [@prc48], is due to the omission of the particle-particle ladder diagrams in their calculations. Thus, the conclusion by ZBJVM, that the binding energies do not stabilize as functions of various model spaces, is not correct. Actually, we have demonstrated that low-order perturbation theory gives the same results within a small model space as the calculations in terms of the $G$-matrix in a large model space. Moreover, we argue that the use of a fixed starting energy by Zheng [*et al.*]{} may not be a viable approach, since the calculations involve several shells, and the starting energy should take this into account. However, if one wishes to properly evaluate the starting energy dependence and include the ladder diagrams, one has to face the problem of the non-hermiticity of the effective interaction, since one is dealing with an interaction defined for several shells. Finally, all such calculations, which involve many single-particle orbits from several shells, do become prohibitevely time-consuming for all nuclei but the lightest ones. Thus, even with the present increased computing power, the perspectives for computing properties of more interesting systems like $^{18}$O, are rather meagre. D. C. Zheng, B. R. Barrett, L. Jaqua, J. P.Vary and R. J. McCarthy, Phys. Rev. C [**48**]{}, 1083 (1993) E.M. Krenciglowa, C.L. Kung, T.T.S. Kuo and E. Osnes, Ann. of Phys. [**101**]{}, 154 (1976) R. Machleidt, Adv. Nucl. Phys. [**19**]{}, 189 (1989) K. Suzuki, R. Okamoto, T. T. S. Kuo and P. J. Ellis, Nucl. Phys. [**A567**]{}, 576 (1994) I. Lindgren, J. Phys. B: At. Mol. Opt. Phys., 1143 (1991) T. T. S. Kuo, P. J. Ellis, J. Hao, Z. Li, K. Suzuki and R. Okamoto, Nucl. Phys. [**A560**]{}, 621 (1993) L. Jaqua, B. R. Barrett, J. P.Vary and R. J. McCarthy, Nucl. Phys.  [**A571**]{}, 243 (1994) (50,50) (50,50) -- -------- -------- -------- -------- -7.83 -9.13 -11.03 -12.35 -12.37 -12.33 -12.30 – -- -------- -------- -------- -------- : Eigenvalues for a system with two particles for different model spaces. The second column lists the results for a model space consisting of the single-particle orbits of the $0s$-shell only. The third column includes the $0p$-shell while the fourth and fifth columns include the $1s0d$- and $1p0f$-shells, respectively. The row denoted by $G$ means that only the $G$-matrix defined in the text is used, while $G+2P$ includes the two-particle ladder diagram to third order in $G$. The results are scaled so that $\varepsilon_{0s_{1/2}}=0$. \[tab:tab1\] [^1]: Note that we omit any discussions on excited spectra, since these include in general more and more complicated configurations as one increases the model space. [^2]: In our $G+2P$ calculation in Section II we used a fixed starting energy, in order to avoid the non-hermiticity. Even with the $G$-matrix, the effective interaction will be non-hermitian if we do not use a fixed starting energy.
--- abstract: 'This is supplementary material for the paper *Degrees of freedom for nonlinear least squares estimation*. It contains the derivation of the divergence formulas, additional details related to the other proofs, technical details on the algorithms and implementations, and some additional simulation results.' address: 'Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark' author: - Niels Richard Hansen - Alexander Sokol bibliography: - '../texbib-1.bib' title: | Supplementary material:\ Divergence formulas and Algorithms --- Properties of the function $\rho$ ================================= In this section we give the central but well known result that the metric projection onto a closed set can be expressed as a subdifferential of a convex function. \[lem:1\] Assume that $K \subseteq \mathbb{R}^n$ is a nonempty and closed set. The function $$\rho(\mathbf{y}) = \sup_{\mathbf{x} \in K} \{ \mathbf{y}^T \mathbf{x} - ||\mathbf{x}||^2/2\}$$ is convex. With $\partial \rho$ denoting the subdifferential of $\rho$ then $\partial \rho(\mathbf{y})$ contains the set of points in $K$ closest to $\mathbf{y}$. If $\rho$ is differentiable in $\mathbf{\mathbf{y}}$ with gradient $\nabla \rho (\mathbf{y})$, then the metric projection of $\mathbf{y}$ onto $K$ is unique, and $\mathrm{pr}(\mathbf{y}) = \nabla \rho (\mathbf{y})$. Since $\rho$ is the pointwise supremum of the affine (thus convex) functions $$\mathbf{y} \mapsto \mathbf{y}^T \mathbf{x} - ||\mathbf{x}||^2/2 = ||\mathbf{y}||^2/2 - ||\mathbf{y} - \mathbf{x}||^2/2,$$ it is convex, and $$\rho(\mathbf{y}) = ||\mathbf{y}||^2/2 - \inf_{\mathbf{x} \in K} ||\mathbf{y} - \mathbf{x}||^2/2.$$ With $$\mathrm{Pr}(\mathbf{y}) = \operatorname*{arg\,min}_{\mathbf{x} \in K} ||\mathbf{y} - \mathbf{x}||^2$$ the nonempty set of points in $K$ closest to $\mathbf{y}$ it follows that $$\rho(\mathbf{y}) = \mathbf{y}^T \mathbf{x} - ||\mathbf{x}||^2/2$$ for all $\mathbf{x} \in \mathrm{Pr}(\mathbf{y})$. For $\mathbf{x} \in \mathrm{Pr}(\mathbf{y})$ $$\begin{aligned} \rho(\mathbf{y} + \mathbf{z}) & = & \sup_{\mathbf{x} \in K} \{ \mathbf{y}^T \mathbf{x} - ||\mathbf{x}||^2/2 + \mathbf{z}^T \mathbf{x}\} \geq \mathbf{y}^T \mathbf{x} - ||\mathbf{x}||^2/2 + \mathbf{z}^T\mathbf{x} \\ & = & \rho(\mathbf{y}) + \mathbf{z}^T \mathbf{x},\end{aligned}$$ which shows that $\mathrm{Pr}(\mathbf{y}) \subseteq \partial \rho(\mathbf{y})$ by definition of the subdifferential. If $\rho$ is differentiable, $$\partial \rho(\mathbf{y}) = \{ \mathrm{pr}(\mathbf{y}) \} = \mathrm{Pr}(\mathbf{y}),$$ and the last claim follows. Let $D \subseteq \mathbb{R}^n$ denote the domain of $\nabla \rho$ on which $\rho$ is differentiable. The following observation is useful. If $\mathbf{y} \in D$, if $\mathbf{y}_n \to \mathbf{y}$ and if $\mathbf{z}_n \in \mathrm{Pr}(\mathbf{y}_n)$ converges to $\mathbf{z}$ then $$\rho(\mathbf{y} + \mathbf{x}) = \lim_{n \to \infty} \rho(\mathbf{y}_n + \mathbf{x}) \geq \lim_{n \to \infty} \rho(\mathbf{y}_n) + \mathbf{x}^T \mathbf{z}_n \geq \rho(\mathbf{y}) + \mathbf{x}^T \mathbf{z},$$ which implies that $\mathbf{z} \in \partial \rho (\mathbf{y}) = \{\mathrm{pr}(\mathbf{y})\}$, whence $\mathbf{z} = \mathrm{pr}(\mathbf{y})$. This proves a continuity property of the metric projection: If $\mathbf{y} \in D$ and $U$ is a neighborhood of $\mathrm{pr}(\mathbf{y})$ then $\{ \mathbf{z} \in \mathbb{R}^n \mid \mathrm{Pr}(\mathbf{z}) \subseteq U \}$ contains a neighborhood of $\mathbf{y}$. We will need this continuity property when deriving the divergence formulas below. Proofs of the divergence formulas ================================= The formulas for computation of the divergence given in Section 3 of the paper will be proved using the implicit function theorem to compute the divergence of $\zeta(\hat{\beta})$. To connect such a local result expressed in the $\beta$ parametrization to the divergence of the globally defined metric projection we will first establish that there is a neighborhood of $\mathbf{y}$ where the (global) metric projection can be found by minimizing $||\mathbf{z} - \zeta(\beta)||^2_2$ in a neighborhood of $\hat{\beta}$. Note that $\mathrm{Pr}(\mathbf{z})$ denotes, as in the proof of Lemma \[lem:1\], the set of metric projections of $\mathbf{z}$. \[lem:local\] If the regularity assumptions on $\zeta$ as stated in Section 3 hold, then for all neighborhoods $V$ of $\hat{\beta}$ there exists a neighborhood $N$ of $\mathbf{y}$ such that $$\mathrm{Pr}(\mathbf{z}) = \zeta(\operatorname*{arg\,min}_{\beta \in V \cap \Theta} ||\mathbf{z} - \zeta(\beta)||^2_2)$$ for $\mathbf{z} \in N$. With $V$ a neighborhood of $\hat{\beta}$ there is, since $\zeta$ was assumed to be open at $\hat{\beta}$, a neighborhood $U$ of $\mathrm{pr}(\mathbf{y}) = \zeta(\hat{\beta})$ such that $$U \cap K \subseteq \zeta(V \cap \Theta).$$ By the continuity property of the metric projection there is a neighborhood $N$ of $\mathbf{y}$ such that $\mathrm{Pr}(\mathbf{z}) \subseteq U$ for $\mathbf{z} \in N$. By definition, $\mathrm{Pr}(\mathbf{z}) \subseteq K$, hence $$\mathrm{Pr}(\mathbf{z}) \subseteq \zeta(V \cap \Theta).$$ This proves first that $W = \operatorname*{arg\,min}_{\beta \in V \cap \Theta} ||\mathbf{z} - \zeta(\beta)||^2_2$ is not empty, and second that $\beta \in W$ if and only if $\zeta(\beta) \in \mathrm{Pr}(\mathbf{z})$. Below we use the implicit function theorem to show that for neighborhoods $N$ of $\mathbf{y}$ and $V$ of $\hat{\beta}$ there exists a $C^1$-map $\hat{\beta} : N \to V \cap \Theta$ such that $\zeta \circ \hat{\beta} : N \to K$ satisfies $$\{\zeta \circ \hat{\beta}(\mathbf{z})\} = \zeta(\operatorname*{arg\,min}_{x \in V \cap \Theta} ||\mathbf{z} - \zeta(\beta)||^2_2).$$ It follows from Lemma \[lem:local\] above that $$\mathrm{pr}(\mathbf{z}) = \zeta \circ \hat{\beta}(\mathbf{z})$$ for $\mathbf{z}$ in a neighborhood (contained in $N$) of $\mathbf{y}$. This ensures that $$\label{eq:divid} \nabla \cdot \mathrm{pr}(\mathbf{y}) = \nabla \cdot \zeta \circ \hat{\beta}(\mathbf{y}).$$ Now recall the definitions of the $G$ and $J$ matrices, $$\label{eq:Gdfn} G_{kl} = \sum_{i=1}^n \partial_k \zeta_i(\hat{\beta}) \partial_l \zeta_i(\hat{\beta})$$ and $$\label{eq:Jdfn} J_{kl} = G_{kl} - \sum_{i=1}^n (y_i - \zeta_i(\hat{\beta})) \partial_k \partial_l \zeta_i(\hat{\beta}).$$ The next lemma on differentiation of the quadratic loss is a straightforward computation, and its proof is left out. \[lem:help\] If $\zeta$ is $C^2$ in a neighborhood of $\beta$ then $f(\mathbf{z}, \beta) = \frac{1}{2} ||\mathbf{z} - \zeta(\beta)||_2^2$ is $C^2$ in a neighborhood of $(\mathbf{y}, \beta)$ with $$\partial_{z_i} \partial_k f(\mathbf{z}, \beta) = - \partial_k \zeta(\beta)$$ and $$\partial_k \partial_l f(\mathbf{z}, \beta) = J_{kl},$$ where $J_{kl}$ is given by (\[eq:Jdfn\]). Note that in the notation above, $\partial_k$ refers to differentiation w.r.t. to $\beta_k$ and $\partial_{z_i}$ refers to differentiation w.r.t. $z_i$. With $f$ as in Lemma \[lem:help\] the estimator $\hat{\beta}$ fulfills $$\nabla_{\beta} f(\mathbf{y}, \hat{\beta}) = 0,$$ with the Jacobian of the map $\beta \mapsto \nabla_{\beta} f(\mathbf{y}, \beta) $ being $J$ by Lemma \[lem:help\]. Since $J$ has full rank by assumption the implicit function theorem implies that there is a continuously differentiable solution map $\hat{\beta}(\mathbf{z})$, defined in a neighborhood of $\mathbf{y}$, such that $$\nabla_{\beta} f(\mathbf{z}, \hat{\beta}(\mathbf{z})) = 0.$$ Moreover, $D_\mathbf{z} \nabla_{\beta} f(\mathbf{y}, \hat{\beta}) = - D_{\beta} \zeta(\hat{\beta})^T$ by Lemma \[lem:help\], which gives by implicit differentiation that $$D_\mathbf{z} \hat{\beta}(\mathbf{y}) = J^{-1} D\zeta(\hat{\beta})^T.$$ Hence, $$D_\mathbf{z} (\zeta \circ \hat{\beta})(\mathbf{y}) = D\zeta(\hat{\beta}) J^{-1} D\zeta(\hat{\beta})^T.$$ It follows from (\[eq:divid\]) that $$\nabla \cdot \mathrm{pr}(\mathbf{y}) = \mathrm{tr}(D\zeta(\hat{\beta}) J^{-1} D\zeta(\hat{\beta})^T) = \mathrm{tr}(J^{-1} D\zeta(\hat{\beta})^T D\zeta(\hat{\beta})) = \mathrm{tr}(J^{-1} G),$$ since $G = D\zeta(\hat{\beta})^T D\zeta(\hat{\beta})$ as defined by (\[eq:Gdfn\]). With $f$ as in Lemma \[lem:help\] the estimator $\hat{\beta}$ fulfills, by assumption, $$\nabla_{\beta} f(\mathbf{y}, \hat{\beta}) = \hat{\lambda} \gamma$$ for $\hat{\lambda} > 0$, $\gamma \in \mathbb{R}^p$, $\gamma_k = \omega_k \mathrm{sign}(\hat{\beta}_k)$ if $\hat{\beta}_k \neq 0$ and $\gamma_k \in (-\omega_k,\omega_k)$ if $\hat{\beta}_k = 0$. Moreover, as $\hat{\lambda} > 0$ it holds that $\sum_{k=1}^p \gamma_k \beta_k = s.$ In the following we identify any $\mathbb{R}^{\mathcal{A}}$-vector denoted $\beta_{\mathcal{A}}$ with an $\mathbb{R}^p$ vector with 0’s in entries with indices not in $\mathcal{A}$. We introduce the map $$R(\mathbf{z}, \beta_{\mathcal{A}}, \lambda) = \left(\begin{array}{c} \nabla_{\beta_{\mathcal{A}}} f(\mathbf{z}, \beta_{\mathcal{A}}) - \lambda \gamma_{\mathcal{A}} \\ \sum_{i=1}^p \gamma_k \beta_{\mathcal{A},k} - s \end{array} \right),$$ and we observe that $R(\mathbf{y}, \hat{\beta}_{\mathcal{A}}, \hat{\lambda}) = 0$. The derivative of $R$ is found to be $$D_{\beta_{\mathcal{A}}, \lambda} R(\mathbf{y}, \hat{\beta}_{\mathcal{A}}, \hat{\lambda}) = \left(\begin{array}{cc} J_{\mathcal{A}, \mathcal{A}} & \gamma_{\mathcal{A}} \\ \gamma_{\mathcal{A}}^T & 0 \end{array} \right).$$ By the assumptions made on $J_{\mathcal{A}, \mathcal{A}}$ this matrix is invertible with $$\left(\begin{array}{cc} J_{\mathcal{A}, \mathcal{A}} & \gamma_{\mathcal{A}} \\ \gamma_{\mathcal{A}}^T & 0 \end{array} \right)^{-1} = \left(\begin{array}{cc} (J_{\mathcal{A}, \mathcal{A}})^{-1} - \frac{(J_{\mathcal{A}, \mathcal{A}})^{-1} \gamma_{\mathcal{A}} \gamma_{\mathcal{A}}^T (J_{\mathcal{A}, \mathcal{A}})^{-1}}{\gamma_{\mathcal{A}}^T (J_{\mathcal{A}, \mathcal{A}})^{-1} \gamma_{\mathcal{A}}} & *\\ * & * \end{array} \right).$$ It follows from the implicit function theorem that there is a neighborhood of $\mathbf{y}$ in which there is a continuously differentiable solution map $(\hat{\beta}_{\mathcal{A}}(\mathbf{z}), \hat{\lambda}(\mathbf{z}))$ that fulfills $R(\mathbf{z}, \hat{\beta}_{\mathcal{A}}(\mathbf{z}), \hat{\lambda}(\mathbf{z})) = 0.$ By the $C^2$-assumption the solution map fulfills the second order sufficient conditions in a neighborhood of $\mathbf{y}$, and $\hat{\beta}_{\mathcal{A}}(\mathbf{z})$ is a local solution to the constrained optimization problem. Since $D_\mathbf{z} \nabla_{\beta} f(\mathbf{y}, \hat{\beta}) = - D_{\beta} \zeta(\hat{\beta})^T$ by Lemma \[lem:help\], we get by implicit differentiation that $$D_\mathbf{z} \hat{\beta}_{\mathcal{A}} (\mathbf{y}) = \left((J_{\mathcal{A}, \mathcal{A}})^{-1} - \frac{(J_{\mathcal{A}, \mathcal{A}})^{-1} \gamma_{\mathcal{A}} \gamma_{\mathcal{A}}^T (J_{\mathcal{A}, \mathcal{A}})^{-1}}{\gamma_{\mathcal{A}}^T (J_{\mathcal{A}, \mathcal{A}})^{-1} \gamma_{\mathcal{A}}}\right) (D\zeta(\hat{\beta})_{\cdot, \mathcal{A}})^T.$$ Since $(D\zeta(\hat{\beta})_{\cdot, \mathcal{A}})^T D\zeta(\hat{\beta})_{\cdot,\mathcal{A}} = G_{\mathcal{A},\mathcal{A}}$ it follows as in the proof of Theorem 3 that $$\begin{aligned} \nabla \cdot \mathrm{pr}(\mathbf{y}) & = & \mathrm{tr}\left( (J_{\mathcal{A}, \mathcal{A}})^{-1} G_{\mathcal{A},\mathcal{A}} - \frac{(J_{\mathcal{A}, \mathcal{A}})^{-1} \gamma_{\mathcal{A}} \gamma_{\mathcal{A}}^T (J_{\mathcal{A}, \mathcal{A}})^{-1} G_{\mathcal{A},\mathcal{A}}}{\gamma_{\mathcal{A}}^T (J_{\mathcal{A}, \mathcal{A}})^{-1} \gamma_{\mathcal{A}}} \right) \\ & = & \mathrm{tr}\left( (J_{\mathcal{A}, \mathcal{A}})^{-1} G_{\mathcal{A},\mathcal{A}} \right)- \frac{\gamma_{\mathcal{A}}^T (J_{\mathcal{A}, \mathcal{A}})^{-1} G_{\mathcal{A},\mathcal{A}} (J_{\mathcal{A}, \mathcal{A}})^{-1} \gamma_{\mathcal{A}}}{\gamma_{\mathcal{A}}^T (J_{\mathcal{A}, \mathcal{A}})^{-1} \gamma_{\mathcal{A}}}.\end{aligned}$$ Summary of previous results in the mathematical literature ---------------------------------------------------------- There is an extensive mathematical literature on the uniqueness, and to some extent differentiability, of the metric projection – in particular in the infinite dimensional context. Some of these results are related to our derivations of the divergence formulas above. [@Haraux:1977] showed results on the directional differentiability of the metric projection onto a closed convex set in a Hilbert space. He showed, in particular, that in finite dimensions the projection onto a polytope is directionally differentiable in $\mathbf{y}$ for all $\mathbf{y}$ with the directional derivative being the projection onto $$(\mathbf{y} - \mathrm{pr}(\mathbf{y}))^{\perp} \cap T_{\mathrm{pr}(\mathbf{y})}$$ where $T_{\mathrm{pr}(\mathbf{y})}$ is the tangent cone, see [@Haraux:1977] for the details. This is a derivative if and only if it is linear, which happens if and only if $\mathrm{pr}(\mathbf{y})$ is in the relative interior of the face $(\mathbf{y} - \mathrm{pr}(\mathbf{y}))^{\perp} \cap K$. This is also the face of smallest dimension containing $\mathrm{pr}(\mathbf{y})$. If we consider an $\ell_1$-ball with radius $s$, and the solution is unique with $p(s)$ nonzero parameters, the corresponding face has dimension $p(s) - 1$. This result was also found in [@Kato:2009]. [@Haraux:1977] showed, in addition, in his Example 2 how to compute the derivative when the boundary of the set is $C^2$. The derivative is a form of regularized projection onto the tangent plane at $\mathrm{pr}(\mathbf{y})$ – the regularization being determined by the curvatures. Recently, [@Kato:2009] derived similar results in the context of shrinkage estimation. Abatzoglou derived results in [@Abatzoglou:1978], but without assuming convexity. These previous results are all closely related to our Theorem 3, but we chose to downplay the differential geometric content. Instead, we discussed in the paper its relation to TIC. More recent results on differentiability of the metric projection can be found in [@Rockafellar:1998]. Their Corollary 13.43 gives an abstract result for a specific point, $\mathbf{y}$, where $\mathrm{pr}(\mathbf{y})$ is prox-regular w.r.t. $\mathbf{y} - \mathrm{pr}(\mathbf{y})$, and the result applies, in particular, when $K$ is fully amenable (regular enough). The result by Haraux on projections onto polytopes follows from this general result – see Example 13.44 in [@Rockafellar:1998]. Algorithms and Implementations ============================== The general implementation that computes $\ell_1$-penalized nonlinear least squares estimates, as well as the implementation of computations specifically related to linear ODEs are available in the R package `smde`. See <http://www.math.ku.dk/~richard/smde/> for information on obtaining the R package and the R code used for the results reported in Section 4 in the paper. In the following sections we describe some of the technical results behind our implementation. In particular, the computation of derivatives related to the matrix exponential. Differentiation of the matrix exponential {#sec:diffexp} ----------------------------------------- The map $A \to e^A$ is well known to be $C^{\infty}$ as a map from $\mathbb{M}(d,d)$ to $\mathbb{M}(d,d)$. Moreover, its first and second partial derivatives can be efficiently computed. We summarize a few useful results from the literature. We denote by $L(A, F)$ the directional derivative of the matrix exponential in $A \in \mathbb{M}(d,d)$ in the general direction $F \in \mathbb{M}(d,d)$. It has the analytic integral representation $$\label{eq:frechet} L(A, F) = \int_0^1 e^{(1-u)A} F e^{uA} \, \mathrm{d} u.$$ See e.g. (10.15) in [@Higham:2008]. If we use $\partial_{kl}$ to denote the partial derivative w.r.t. the $(k,l)$’th entry, and if $E_{kl}$ denotes the $(k,l)$’th unit matrix, we have $\partial_{kl} e^A = L(A, E_{kl})$. This gives the identity $$\label{eq:directional} \mathrm{tr}(\partial_{kl} e^A M) = \mathrm{tr} \left( E_{kl} \int_0^1 e^{u A} M e^{(1-u) A} \, \mathrm{d} u \right) = L(A, M)_{l,k}.$$ for any $M \in \mathbb{M}(d,d)$. We will use this formula in the following section. Efficient algorithms exist for computing $L(A, F)$ for general matrices. It holds, for instance, that $$\exp\left( \left[\begin{array}{cc} A & F \\ 0 & A \end{array} \right] \right) = \left[ \begin{array}{cc} e^A & L(A, F) \\ 0 & e^A \end{array} \right],$$ see (10.43) in [@Higham:2008], so if we can efficiently compute matrix exponentials, we can compute the derivative. The `expmFrechet` function in the `expm` R package, [@expm:2012], implements a faster algorithm that avoids the dimension doubling. For the second partial derivatives it follows from (\[eq:frechet\]) that $$\partial_{hr} \partial_{kl} e^{A} = H(A, E_{hr}, E_{kl}) + H(A, E_{kl}, E_{hr}),$$ where $$H(A, F, G) = \int_0^1 \int_0^u e^{(1-u) A} F e^{(u - s) A} G e^{s A} \, \mathrm{d} s \mathrm{d} u.$$ The computation of these iterated integrals is based on Theorem 1 in [@loan:1978], which implies that [ $$\exp\left( \left[\begin{array}{cccc} A & F & 0 \\ 0 & A & G \\ 0 & 0 & A \\ \end{array} \right] \right) = \left[ \begin{array}{cccc} e^A & L(A, F) & H(A, F, G) \\ 0 & e^A & L(A, G) \\ 0 & 0 & e^A \\ \end{array} \right].$$ ]{} From the integral representation of $H(A, F, G)$ we find that for $M \in \mathbb{M}(d,d)$ $$\begin{aligned} \nonumber \mathrm{tr}(\partial_{hr} \partial_{kl} e^A M) & = & \mathrm{tr}(E_{hr} H(A, E_{kl}, M)) + \mathrm{tr}(E_{kl} H(A, E_{hr}, M)) \\ & = & H(A, E_{kl}, M)_{r,h} + H(A, E_{hr}, M)_{l,k}, \label{eq:seconddirectional}\end{aligned}$$ which was used for the computation of the $J$ matrix that enters in the formula in Theorem 4. Coordinate descent algorithm and sufficient transformations ----------------------------------------------------------- To solve the optimization problem $$\min_{\beta} ||y - \zeta(\beta)||_2^2 + \lambda \sum_{k=1}^p \omega_k |\beta_k|$$ for a decreasing sequence of $\lambda$’s we have implemented a plain coordinate wise descent algorithm based on a standard Gauss-Newton-type quadratic approximation of the loss function. That is, for given $\beta \in \Theta$ we approximate the loss in the $k$’th direction as $$\begin{aligned} ||y - \zeta(\beta + \delta e_k)||_2^2 & \simeq & ||r(\beta) - \partial_k \zeta(\beta) \delta||^2_2 \\ & = & ||r(\beta)||_2^2 - 2 \langle r(\beta), \partial_k \zeta(\beta) \rangle \delta + || \partial_k \zeta(\beta)||_2^2 \delta^2\end{aligned}$$ where $r(\beta) = y - \zeta(\beta)$. The coordinate wise penalized quadratic optimization problem can be solved explicitly, and we then iterate over the coordinates until convergence. We implemented two versions of the algorithm. Algorithm I is a generic algorithm that relies on two auxiliary functions for computing $\zeta(\beta)$ and $D\zeta(\beta)$. Algorithm II is specific to linear ODE models. With $m$ observations solving a $d$-dimensional linear ODE, the computation time for Algorithm I scales linearly with $m$, but the computation of $e^{tB} x$ and $D e^{tB} x$ can be implemented to take advantage of sparseness of $B$. Algorithm II relies, on the other hand, on the precomputation of three sufficient statistics, being $d \times d$ matrices, as outlined below. For dense matrices the current implementation of Algorithm II scales better with $d$, and after the precomputation of the sufficient statistics, all other computation times are independent of $m$. However, Algorithm II cannot take the same advantage of a sparse $B$. Since the loss is generally not convex, the steps may not be descent steps if the quadratic approximation is poor. We implemented Armijo backtracking as described in [@Tseng:2009] to ensure sufficient decrease and hence convergence. As mentioned above, Algorithm II for the linear ODE example relies on sufficient statistics for the computation of the loss as well as the quadratic approximation. We give here a brief derivation of the necessary formulas. On $\mathbb{M}(d,d)$ the inner product can be expressed in terms of the trace, $$\langle A, B \rangle = \mathrm{tr}(A^T B).$$ The corresponding norm, often referred to as the Frobenius norm, is the ordinary $2$-norm when matrices are identified with vectors in $\mathbb{R}^{d^2}$. For the linear ODE example, $\zeta(B) = e^{t B} x$, and $$||y - \zeta(B)||_2^2 = \mathrm{tr}(y y^T) - 2 \mathrm{tr}(e^{t B} x y^T) - \mathrm{tr} (e^{t B^T} e^{t B} x x^T),$$ which depends on the data through the three cross products $y y^T$, $x y^T$ and $x x^T$ only. These are $d \times d$ sufficient transformations. We also find that $$\begin{aligned} \langle r(B), \partial_{kl} \zeta(B) \rangle & = & \mathrm{tr}( \partial_{kl} e^{tB} x (y^T - x^T e^{t B^T})) \\ & = & \mathrm{tr}( \partial_{kl} e^{t B} (x y^T - x x^T e^{tB^T})) \\ & = & t L(tB, x y^T - x x^T e^{tB^T})_{l,k}\end{aligned}$$ by (\[eq:directional\]). Consequently, the entire gradient of the quadratic loss can be computed as $- 2 t L(tB, x y^T - x x^T e^{tB^T})^T$, which amounts to computing a single directional derivative of the exponential map. We also need to compute inner products of the derivatives, $ \partial_{kl} \zeta(B)$, of $\xi$, and to this end we observe that $$\begin{aligned} \langle \partial_{kl} \zeta(B), \partial_{hr} \zeta(B) \rangle & = & \mathrm{tr}(x^T \left(\partial_{kl} e^{tB}\right)^T \partial_{hr} e^{tB} x) \\ & = & \mathrm{tr}(\left(\partial_{kl} e^{tB} \right)^T \partial_{hr} e^{tB} xx^T) \\ & = & t^2 L(t B^T, L(tB, E_{hr}) xx^T)_{k,l}. \end{aligned}$$ That is, an entire column (or row) of the matrix of inner products can be computed by computing two directional derivatives of the exponential map. Penalized vs. constrained optimization ====================================== As mentioned above, our algorithms solve the penalized optimization problem for a given sequence of $\lambda$’s. A solution, $\hat{\beta}_{\lambda}$, for a given $\lambda$ is also a solution to the constrained optimization problem $$\min_{\beta \in \Theta_{s(\lambda)}} ||y - \zeta(\beta)||_2^2$$ where $s(\lambda) = \sum_{k=1}^p \omega_k |\hat{\beta}_{\lambda,k}|$ and $$\Theta_s = \left\{ \beta \;\middle|\; \sum_{k=1}^p \omega_k |\beta_k| \leq s \right\}.$$ The value of $s(\lambda)$ is decreasing in $\lambda$. Thus the algorithm provides a sequence of solutions to the constrained problems for increasing values of $s$. If the sequence of $\lambda$’s is fixed, the sequence of $s$’s will, however, be random. This is a small nuisance in the simulation study where we want to compute the degrees of freedom repeatedly for a fixed $s$. In practice we have solved this by linear interpolation to compute $\widehat{\mathrm{Risk}}(s)$ for a fixed set of constraints $s$. Further details and results from the simulation study ===================================================== In the simulation study on estimation of linear ODE models, data were generated using the following sparse $10 \times 10$ matrix: [$$B = \left( \begin{array}{rrrrrrrrrr} -1.0 & -1.0 & -0.9 & -0.8 & -0.7 & -0.6 & -0.4 & -0.3 & -0.2 & -0.1 \\ 1.0 & -1.0 & . & . & . & . & . & . & . & . \\ 0.9 & . & -1.0 & . & . & . & . & . & . & . \\ 0.8 & . & . & -1.0 & . & . & . & . & . & . \\ 0.7 & . & . & . & -1.0 & . & . & . & . & . \\ 0.6 & . & . & . & . & -1.0 & . & . & . & . \\ 0.4 & . & . & . & . & . & -1.0 & . & . & . \\ 0.3 & . & . & . & . & . & . & -1.0 & . & . \\ 0.2 & . & . & . & . & . & . & . & -1.0 & . \\ 0.1 & . & . & . & . & . & . & . & . & -1.0 \\ \end{array} \right)$$ ]{} The matrix exponential of $B$ is a dense matrix with most of the entries of comparable size. [$$e^B = \left( \begin{array}{rrrrrrrrrr} -0.11 & -0.19 & -0.17 & -0.15 & -0.12 & -0.10 & -0.08 & -0.06 & -0.04 & -0.02 \\ 0.19 & 0.23 & -0.12 & -0.11 & -0.09 & -0.08 & -0.06 & -0.04 & -0.03 & -0.01 \\ 0.17 & -0.12 & 0.26 & -0.09 & -0.08 & -0.07 & -0.05 & -0.04 & -0.03 & -0.01 \\ 0.15 & -0.11 & -0.09 & 0.29 & -0.07 & -0.06 & -0.05 & -0.03 & -0.02 & -0.01 \\ 0.12 & -0.09 & -0.08 & -0.07 & 0.31 & -0.05 & -0.04 & -0.03 & -0.02 & -0.01 \\ 0.10 & -0.08 & -0.07 & -0.06 & -0.05 & 0.33 & -0.03 & -0.02 & -0.02 & -0.01 \\ 0.08 & -0.06 & -0.05 & -0.05 & -0.04 & -0.03 & 0.34 & -0.02 & -0.01 & -0.01 \\ 0.06 & -0.04 & -0.04 & -0.03 & -0.03 & -0.02 & -0.02 & 0.35 & -0.01 & -0.00 \\ 0.04 & -0.03 & -0.03 & -0.02 & -0.02 & -0.02 & -0.01 & -0.01 & 0.36 & -0.00 \\ 0.02 & -0.01 & -0.01 & -0.01 & -0.01 & -0.01 & -0.01 & -0.00 & -0.00 & 0.37 \\ \end{array} \right)$$ ]{} ![Risks for the $\ell_1$-constrained estimator with adaptive weights as a function of the constraint $s$ compared to the risk of the MLE and hard thresholding of the MLE. In addition, expected values of risk estimates. The risk estimates underestimated the true risk when adaptive weights were used for the $\ell_1$-constrained estimator. \[fig:riskAdap\]](riskPlotAdaptive.pdf){width="60.00000%"} In addition to the results reported in the paper, Figure \[fig:riskAdap\] shows the results for the $\ell_1$-constrained estimator with adaptive weights. Using the divergence as an estimate of degrees of freedom resulted in this case in negatively biased risk estimates. This is because the divergence does not account for the data dependent weights. Despite of this, the data adaptive choice of the constraint was close to the optimal choice. It is notable that compared to using unit weights, the use of adaptive weights decreased the risk further. The adaptive weights also resulted in sparser estimates (41.7 nonzero entries on average) than when using unit weights (59.0 nonzero entries on average). Using $\hat{B}_{\hat{s}}$ to obtain a structural estimate of the nonzero entries the accuracy (fraction of correctly estimated zero and nonzero entries) was 0.81 with adaptive weights compared to 0.65 with unit weights. The forward stepwise model search gave 37.3 nonzero entries on average and an accuracy of 0.83. Thus for structural estimation, the model search was more accurate, though this comparison may not be entirely fair. The model search started from the diagonal matrix mainly for numerical reasons, which gave it 10 correct nonzero entries as a starting point. ![Computation time in seconds for Algorithm II as a function of the dimension $d$. The read line has slope 2.7. \[fig:scaling\]](scaling){width="90.00000%"} We finally carried out a small benchmark simulation to investigate how the current implementation of the coordinate descent algorithm for the nonlinear least squares problem scales with the dimension of the problem. It should be noted that there are many nobs to tweak to improve computation times. The simulation results presented in the paper with $d = 10$ were, for instance, carried out with a small relative tolerance of around $10^{-8}$ for the convergence criterion. In this benchmark study we used a relative tolerance of $10^{-4}$, which in our experience only occasionally will result in convergence problems. It is also possible to stop the algorithm when the model with the minimal estimated risk is reached to avoid the most expensive part of the optimization where many parameters are nonzero. We have not done that, but as in the simulation study in the paper we computed the optimal solution for 40 precomputed values of the penalty parameter. Then there is the specific choice of algorithm. In the benchmark we used Algorithm II described above. Figure \[fig:scaling\] shows the computation times for the optimization as a function of the dimension $d$ for $d \in \{2, 3, 4, 6, 8, 12, 16, 24, 32, 46, 64\}$. Note that the number of parameters is $p = d^2$, which for $d = 64$ gives $p = 4096$ parameters. For each value of $d$ we made 5 replications. We see from Figure \[fig:scaling\] that the computation time scales roughly like $d^3$. The bottleneck is the repeated computations of dense matrix exponentials.
--- abstract: 'We develop an algorithm that has the potential to relate the depth development of ultra high energy extensive air showers and the time delay for individual muons. The time distributions sampled at different positions at ground level by a large air shower array are converted into distributions of production distances using an approximate relation between production distance, transverse distance and time delay. The method is naturally restricted to inclined showers where muons dominate the signal at ground level but could be extended to vertical showers provided that the detectors used can separate the muon signal from electrons and photons. We explore the accuracy and practical uncertainties involved in the proposed method. For practical purposes only the muons that fall outside the central region of the shower can be used, and we establish cuts in transverse distance. The method is tested using simulated showers by comparing the production distance distributions obtained using the method with the actual distances in the simulated showers. It could be applied in the search for neutrinos to increase the acceptance to highly penetrating particles, as well as for unraveling the relative compositions of protons and heavy nuclei. We also illustrate that the obtained depth distributions have minimum width when both the arrival direction and the core position are well reconstructed.' author: - 'L. Cazón' - 'R.A. Vázquez' - 'E. Zas' title: Depth Development of Extensive Air Showers from Muon Time Distributions --- Introduction ============ When an Ultra High Energy Cosmic Ray (UHECR) particle enters the atmosphere it interacts producing an extensive air shower that propagates through it and reaches ground level. These showers are routinely detected by optical systems that collect fluorescence light emitted by nitrogen molecules excited as the front crosses the atmosphere, and by arrays of particle detectors that sample at ground level an enormous shower front which can exceed $10^{12}$ particles. In these arrays the relative times of the detected signals allow the reconstruction of the incoming particle arrival direction. The time distribution of the arriving signal has been known for long to be dependent on the depth distribution of the shower particles [@Lapikens; @Linsley; @Watson; @Antoni:2002vv] which is different for different primary particles. Exploring the highest energy particles is now considered to be a priority because the data are scarce, there are discrepancies between results obtained with the two techniques and because the origin of these particles is not at all understood [@NaganoWatson]. Their study is expected to provide both information on violent objects in the Universe where these particles originate and on their interactions (during propagation and in the atmosphere) at energies exceeding those achieved in accelerator experiments by many orders of magnitude. New experiments are being constructed and devised to improve the statistics, to increase the precision and to establish the mass of the primary particles. The Auger Observatory in Argentina is the first of a new generation of large aperture experiments. It combines the two techniques and for the ground array it uses water Čerenkov tanks with photomultiplier tubes and Flash Analog to Digital Converters (FADC) to record the time stamp of the signal in 25 ns intervals with unprecedented accuracy [@EA]. The perspective of improved detectors has triggered an increase of phenomenological activity in the study and characterization of extensive air showers. Evaluation of the time structure of showers is part of this effort motivated by both the practical need of controlling the uncertainties in the arrival direction reconstruction and also by the hope that its understanding might shed new light on the challenging problem of establishing their composition. The idea of relating the muon distributions to the shower development has been already quite successful [@Danilova; @Brancus:1997rr; @Pentchev1; @Ambrosio:1999nr; @model]. The arrival time distributions of muons has been characterized using simple geometrical and kinematical arguments and making a key simplifying assumption on the muon energy, transverse momentum, and distance of production distributions, namely that they are independent [@cazon3]. Here we have further developed the algorithm that relates the arrival time distribution of muons to the depth development of the shower following on these ideas. The scope of the method is limited because the shower front contains many other particles, mainly electrons and photons. In principle it requires muon identification but fortunately the muon signal dominates in two circumstances: when the shower is inclined and the electromagnetic part does not reach ground level [@model] but also for “more vertical” showers when the distance to shower axis is sufficiently large [@rhomu_rsignal]. The method complements alternative depth reconstruction methods which are always limited when only densities at ground level are taken into consideration. Moreover when the effects of arrival angle and impact point misreconstruction are taken into account it is seen that the induced distributions have a minimum in their spread for the correct angle and impact point. This effect opens the possibility of using this method for improving the confidence in the conventional angular and impact point reconstructions. While the precision obtained is possibly insufficient to be used for composition studies it will certainly have an important impact on improving the acceptance of air shower arrays to neutrinos through inclined showers. The accuracy in the depth development reconstruction is sufficient to exclude neutrino interactions at intermediate depths, when the electromagnetic shower would have been completely absorbed but the first interaction is sufficiently deep into the atmosphere to exclude both a cosmic ray hadron and a photon. The article is organized as follows: In Section II we summarize the factorization hypothesis for the muon distributions and the relations that follow, and motivate the inversion of the relation between the time and depth distribution from Ref. [@cazon3]. We also pay some attention to the relation between particle densities and detector geometric acceptance. In Section III we present the method to reconstruct the depth distribution. In Section IV we compare the depth distributions obtained with this reconstruction method to actual distributions from simulated showers to test it. In Section V discussing some practical limitations. In Section VI we explore the correlations between the reconstruction procedure and the assumed arrival direction and impact point as a check of its robustness; we summarize and conclude in Section VII. Technical details are presented in two appendices. Relation between depth development and muon time distributions ============================================================== The main features of the arrival time distributions of muons in extensive air showers can be accounted for by the different path lengths traveled by the muons from their production point. This has been recently shown using a simplified model to describe the muons in air showers which is based on the hypothesis that their energy, $E_i$, transverse momentum, $p_t$, production distance, $z$, and outgoing polar angle, $\zeta$ distributions factorize [@cazon3] $$\frac{d^4 N_0}{d z \;\! d\zeta \;\! d E_i \;\! d p_t } = \frac{1}{2 \pi} {\cal N}_0 \;\! h(z) \;\! f_1(E_i) \;\! f_2(p_t), \label{factorization}$$ In this expression $E_i$, $p_t$ and ${\cal N}_0$ refer to production, $z$ is measured along the shower axis from the muon production point to the ground. The transverse momentum is transverse to shower axis and has polar angle $\zeta$ in the transverse plane (perpendicular to shower axis). The functions $h(z)$, $f_1(E_i)$ and $f_2(p_t)$ are assumed independent and normalized to 1 and the factor $2 \pi$ accounts for a uniform polar angle distribution. Finally ${\cal N}_0$, the total number of produced muons, is the overall normalization. In this model the muons are assumed to travel in straight lines and these four variables are sufficient to determine the muon path uniquely. It is convenient to express the muon direction in terms of the angle $\alpha$ with respect to shower axis. For a muon produced with energy $E_i$ and transverse momentum $p_t$, the angle $\alpha$ is given by $$\sin\alpha=\frac{p_t c}{\sqrt{E_i^2-(m c^2)^2}}\simeq\frac{ p_t c}{E_i}, \label{geometry_relation}$$ We can approximate $p c = \sqrt{E_i^2-(m c^2)^2}\simeq E_i$, because the muon energy at ground level is typically greater than $mc^2$, this energy at production is even greater because of muon energy loss. We can now change the coordinates replacing $p_t$ in Eq. \[factorization\] using Eq. \[geometry\_relation\] to give $$\frac{d^4 N_0}{d z \;\! d\zeta \;\! d E_i \;\! d{\sin{\alpha}}} = {\cal N}_0 \;\! \frac{1}{2\pi} h(z) \;\! f_1(E_i) \;\! f_2\left(\frac{E_i}{c} \sin{\alpha} \right) \frac{E_i}{c}. \label{fact_alpha}$$ As the muons go through the atmosphere they lose energy and decay and even though we start with independent distributions at production, correlations between the relevant variables appear naturally when we consider the surviving muon distribution at ground level. This is explicitly shown in Appendix A using a simplified model for energy loss. The distribution of surviving muons can then be integrated in $E_i$ in order to obtain the depth distribution of the surviving muons, which is given in terms of two angles describing the muon direction, namely $ \alpha$ and $ \zeta$. It is convenient to relate them to the differential solid angle for the muon $d^2 \Omega=-d\zeta d\cos \alpha$. Then $$\frac{d^3 N}{d z \;\! d^2\Omega}= \frac{d^3 N}{d z \;\! d\zeta \;\! d\sin{\alpha}} \;\! \frac{\cos{\alpha}}{\sin{\alpha}}. \label{dNdOmega}$$ It is interesting to discuss in some detail the effect of a detector surface. From Eq. \[dNdOmega\] we can obtain the number of muons from a given production interval $dz$ that crosses an arbitrary surface $d^2A$ which subtends a solid angle $d^2\Omega$: $$d N_A = \frac{d^3 N}{ d z \;\! d^2 \Omega} d^2 \Omega ~d z = \frac{d^3 N}{ d z \;\! d^2 \Omega} \;\! \frac{\cos{\psi_{\mu A}} \;\! d^2 A}{l^2} \;\! d z. \label{eq:dN}$$ where $ \psi_{\mu A}$ is the angle between the normal to the surface and the muon direction. On the other hand the projection of $d^2A$ onto the shower transverse plane is $r d \zeta d r= d^2 A \cos{\psi_A} $, where $\psi_A$ is the angle between the normal to the surface and the shower direction. (See Fig. \[f:terra2\] in Appendix A.) Using this relation we can relate Eq. \[dNdOmega\] to the number of particles per unit area in the transverse plane: $$\frac{1}{r} \;\! \frac{d^3N_A}{ dz \;\! dr \;\! d \zeta} = \frac{d^3 N}{ d z \;\! d^2 \Omega} \;\! \frac{1}{l^2} \;\! \frac{\cos{\psi_{\mu A}}}{\cos{\psi_{A}}} = \frac{d^3 N}{ d z \;\! d^2 \Omega} \;\! \frac{1}{l^2} D_A(\Omega) = \frac{d^3 N}{ d z \;\! d \zeta \;\! d\sin{\alpha}} \;\! \frac{\cos{\alpha}}{\sin{\alpha}} \;\! \frac{1}{l^2} D_A(\Omega). \label{rho_mu}$$ Where $D_A(\Omega)$ denotes the geometrical factor involving these two angles. We stress that $D_A$ not only depends on the surface orientation with respect to shower axis but also on the direction of the incoming muons. Any detector can be regarded as a collection of such surfaces and as a result several such factors $D_{A_i}$ will have to be considered depending on the impact point to obtain the total effective collection area for a given arrival direction. We can divide Eq. \[rho\_mu\] by its integral in $z$ ($\hat N_{Ar \zeta}$) to normalize the function to 1. When this is done, using plausible functions for the distributions as described in Appendix A, and the relations between $r$, $l$, $z$ and $\alpha$ are used, a number of factors cancel out and we obtain the $z$-distribution of muons arriving at detector $A$ (normalized to 1) which can be related to a simple transform of the $z$ distribution: $$\frac{1}{{\hat N}_{Ar \zeta}} \;\! \frac{1}{r} \;\! \frac{d^3 N_A}{dr \;\! d z \;\! d\zeta} = \frac{h(z) \;\! l^{1-\gamma} \cos \alpha ~D_A(\Omega)}{ \int_0^{\infty} dz \;\! h(z) \;\! l^{1-\gamma} \cos \alpha ~D_A(\Omega)}. \label{mainfrac}$$ The proposed method relies on the above expression and the geometrical relation between $z$ and $t$ which is described in Ref. [@cazon3]. In that work it was shown that much of the time structure of the muons is due to geometrical effects which imply that to each $z$ there corresponds a given arrival time $t$. As a result we can relate the $t$ and $z$-distributions: $$\frac{1}{r}\;\! \frac{d^3 N_A}{dr \;\! d\zeta \;\! dt} = \frac{1}{r} \;\! \frac{d^3 N_A}{dr \;\! d\zeta \;\! dz } \;\! \frac{dz}{dt}. \label{dN_rdt}$$ We now define the normalized function $g(t)$ describing the shape of the time distribution through: $${\hat N}_{Ar \zeta} \;\! g(t) = \frac{1}{r}\;\! \frac{d^3 N_A}{dr \;\! d\zeta \;\! dz} \Big| \;\! \frac{dz}{dt} \Big|. \label{g(t)}$$ We can compare the $t$-distributions of the muons arriving at ground level given by Eq. \[g(t)\] to those obtained in simulations and agreement is found as will be shown in Section IV. The time distribution of the muons is related to the depth distribution of muon production. If we combine Eq. \[mainfrac\] and Eq. \[g(t)\] we obtain the following relation between $h(z)$ and the time distribution: $$g(t) \Big| \frac{dt}{dz} \Big| = \frac{1}{{\hat N}_{Ar \zeta}} \;\! \frac{1}{r} \;\! \frac{d^3 N_A}{dr \;\! d z \;\! d\zeta} = \frac{h(z) \;\! l^{1-\gamma} \cos \alpha ~D_A(\Omega)}{ \int_0^{\infty} dz \;\! h(z) \;\! l^{1-\gamma} \cos \alpha ~D_A(\Omega)}. \label{g(t)2}$$ This expression takes into account the fact that from different $r$ we effectively sample the $h(z)$ distribution with an extra $z$-dependence introduced as a overall factor through $l$ and the angles $\alpha$ and $\zeta$. Reconstruction of the depth distribution ======================================== In a typical air shower array a number of particle detectors sample the shower front at the Earth’s surface. We now consider a set of $M$ detectors labeled by a suffix $i$ (from $1$ up to $M$) each with a surface $A_i$ (which becomes $S_i$ when projected onto the shower plane) and located at position $(r,\zeta)_i$ in transverse plane coordinates. We calculate the time distribution of arriving muons to a detector $i$ by integrating Eq. \[dN\_rdt\] over the transverse surface $dS$, or, for all practical purposes, simply multiplying by the corresponding area $S_i$. Using Eq. \[g(t)\] the time distribution at detector $i$ becomes: $$\frac{dN_A}{dt}\Big|_{i} \equiv \int_{S_i} \frac{d^3N_{A_i}}{dt \;\! r \;\! dr \;\! d\zeta}(r,\zeta) dS \simeq S_i \hat{N}_{Ar\zeta} ~ g(t).$$ The number of muons falling in the detector $i$ can be considered as a finite sample of the continuous arrival time distribution probability $\frac{dN_A}{dt}\Big|_i$. Let us assume that we can fill a time histogram with $N_i$ entries corresponding to the $N_i$ muons detected by detector $i$. The entries of this histogram can be transformed into a $z$ histogram, using the correspondence $t\rightarrow z$ given by [@cazon3]: $$z=\frac{1}{2}\left( \frac{r^2}{ct}-ct\right)+ \Delta, \label{z_t_exact}$$ which can, in most cases, be approximated by: $$z \simeq \frac{1}{2}\frac{r^2}{ct} + \Delta. \label{z_t}$$ This mapping transforms each time entry (from $\frac{dN_A}{dt}\Big|_i \simeq S_i \hat{N}_{Ar\zeta} ~ g(t)$ ) into a $z$ entry (of $S_i \frac{1}{r}\;\! \frac{d^3N_A}{dr \;\! d\zeta \;\! dz }$) and finally into an entry of the $z$-distribution of the muons arriving at ground, $\frac{dN}{dz}$. Note that the delay $t$ is the time difference between the arrival time of a given particle and the arrival time of a reference plane perpendicular to the shower axis and traveling at the speed of light $c$, the [*time-reference plane*]{}. We have chosen the $0$-time origin corresponding to the arrival of the first particle at ground at $r=0$ (shower core). If the core hits ground at a universal time $ct_0'$, the relation between $t'$ and $t$ involves $\Delta$: $$ct=ct'-ct_0'+ \Delta. \label{times_relation}$$ Different detectors will give entries to a different time distribution, but they will be converted into samples of a [*unique*]{} $\frac{dN}{dz}$ distribution. As a result the converted entries of available detectors can be combined into a larger sample. These entries are naturally weighted by the number of muons detected at each detector. In Ref. [@cazon3] it was shown that there is an additional source of delay for muons because of their sub-luminal velocities $t_{\varepsilon}$. The total time delay is obtained adding it to the delay given by Eq. \[z\_t\]: $$t \simeq t_g+t_{\varepsilon}.$$ This delay is energy dependent and it is only dominant over the geometric delay for muons close to shower axis. In inclined showers the final muon energy is much larger than the muon mass, $m c^2 \ll E_i-\rho a l$, and therefore: $$t_\epsilon \simeq \frac{1}{2}\;\! \frac{(m c^2)^2}{c \rho a } \left[\frac{1}{E_i-\rho a l} -\frac{1}{E_i}\right]. \label{t:kinetic}$$ Since the muon energy is not measured in typical air shower measurements, we cannot account for these effects accurately. A solution is simply obtained by eliminating the measurements close to the axis to ensure that the kinematical delay can be neglected. This has an impact on statistics. In Fig. \[f:epsilon\_r\] we have plotted the factor $\epsilon (r,z)$, which can be taken as the relative value of the average kinematic delay with respect to the average geometric delay (see Appendix B), for $\zeta=90^{\circ}$, $\Delta=0$ and different $z$. We can see from this graph that the geometric delay dominates for distances above 600 m from shower core. We can include the kinematic delays on average. We obtain a simple parameterization for the average kinematic delay as a function of $z$ and $r$, (details are given in Appendix B): $$\epsilon (r,z) = p_0(z) \left(\frac{r}{\rm m}\right)^{p_1}.$$ If we now subtract the average kinematical delay from the measured delay, instead of Eq. \[z\_t\_exact\] we obtain: $$z \simeq \frac{1}{2} \left( \frac{r^2}{ct-c<t_{\varepsilon}>} -(ct-c< t_{\varepsilon} >)\right)+\Delta. \label{zcorr_t}$$ Since it is not possible to obtain $z(t)$ analytically from the previous expression we can use a simple numerical approach. We can for instance take zero kinematical delay as a first approximation to obtain $z$, we then get the average kinematical delay and substitute in Eq. \[zcorr\_t\]. Since the dependence of the $p$ coefficients on $z$ is mild (logarithmic) the procedure converges quickly and one iteration is sufficient. ![The factor $ \epsilon(r,z)$ which relates to the ratio of kinematic to geometric time delays (see text) versus $r$ for different $z$: 1.6, 3.2, 6.3, 13, 25, 50, 100, 200 km, from bottom to top.[]{data-label="f:epsilon_r"}](epsilon_r.eps){width="15"} Test ==== One approach to test the method is to simulate showers using a Monte Carlo generator that reproduces the time distribution of the signal in a collection of detectors and to apply the method to reconstruct the production distribution of the muons $h(z)$. Unfortunately major modifications have to be made in conventional simulation programs that are not designed to give the function $h(z)$ to compare with the reconstructed value. We have checked our reconstruction method by applying it to showers simulated with the Aires Monte Carlo package [@Aires]. It is straightforward to compare the $z$-distributions of the muons arriving at different locations in the ground. This has been done at different positions and agreement is found. This reflects the fact that the time distributions of the muons are well described by the model for muon time delays [@cazon3]. We have extended the test to combine detectors at different positions and to compare the results to the total distribution of the surviving muons $dN/dz$, which is straightforward to obtain in Aires and in most shower generators. In our model this distribution would be obtained by integrating Eq. \[mainfrac\] over $r$ and $\zeta$ to cover all the ground: $$\frac{dN}{dz}= \int \int \;\! \frac{1}{r}\;\! \frac{d^3N}{dr \;\! d\zeta \;\! dz} ~r \;\! dr \;\! d\zeta = \int \int \hat{N}_{r\zeta} \frac{h(z) \;\! l^{1-\gamma} \cos \alpha ~D_A(\alpha)}{ \int_0^{\infty} h(z) \;\! {l}^{1-\gamma} \cos \alpha ~D_A(\alpha) ~ dz} ~ r dr d\zeta. \label{dNdz_integral} $$ Using simulations we have studied how the total $dN/dz$ distribution at ground relates to the local distributions at different $r$, after integrating in $\zeta$ and $r$. We first note that to a first approximation the $\zeta$-integral is proportional to the integrand with $\Delta=0$. This is not surprising since $\Delta$ changes sign when integrating over $\zeta$. We have found that there is an effective value of $r$ ($r_*$) for local distribution that gives a very good approximation to the overall $dN/dz$ distribution. This value is slightly zenith angle dependent and ranges from about 400 m for showers at $0^\circ$, up to 1000 m for horizontal showers at $80^\circ$ or 1800 m at $86^{\circ}$ . This is reasonable because the bulk of the muons arrive to ground in a relatively constrained region: for instance, at $0^{\circ}$ this region is between $\sim 60$ m and $\sim 1000$ m. We can then substitute in Eq. \[dNdz\_integral\] $l$ for $l_{\star}=\sqrt{r_{\star}^2+z^2}$ and $\alpha$ for $\alpha_{\star}=\arcsin\frac{r_{\star}}{l_{\star}}$ and also consider that to compare with Aires that gives directly the muon position it is not necessary to include a geometric correction factor, i.e. $D_A=1$. We finally obtain the following approximation $$\frac{dN}{dz} \propto h(z) \;\! l_{\star}^{1-\gamma} \cos \alpha_{\star} . \label{dNdz_integral_aprox}$$ Since in a practical air shower array the detectors are going to be arranged on an unknown and arbitrary pattern around the shower axis it is convenient to correct the $z$-distribution obtained at each detector to a common observation distance $r_{\star}$ which approximately reproduces the overall $dN/dz$ as follows: $$\frac{dN}{dz} \propto g(t) \Big|\frac{dt}{dz}\Big| ~ \times ~ \frac{ l_{\star}^{1-\gamma} \cos \alpha_{\star} }{ l^{1-\gamma} \cos \alpha} \frac{1}{~D_A(\alpha)}. \label{geometric_correction}$$ In general, we must divide by $D_A(\alpha)$ to remove the dependence on the detector geometry if necessary. We note that the correction factors $\frac{ l_{\star}^{1-\gamma} \cos \alpha_{\star}}{l^{1-\gamma} \cos \alpha}$ approach 1 when $z$ increases (i.e. in horizontal showers). Taking this into account one can use a single $r_\star=400~$m for all zenith angles and still obtain relatively good approximations. To test the method we have used sets of 500 showers at different zenith angles. A given particle array will be limited to a sample of this distribution which is determined by the relative positions of the available detectors. We first take the muon output from the simulations and arrange the muons in a time histogram as can be done in an actual shower array (we use 25 ns bins). We then apply Eq. \[zcorr\_t\] to all the simulated muons to calculate $z$, where we have included the geometric corrections of Eq. \[geometric\_correction\] and the kinematic corrections as explained in the previous section. Finally an $r$ cut is applied. In Figs. \[f:00\] and \[f:70\] we illustrate the result for a $0^\circ$ and a $70^\circ$ showers. The shaded histogram is the distribution of all the muon production altitudes compared to that obtained from the reconstruction procedure using all the muons which reach the ground with $r>r_c$. This cut is necessary for the geometric inversion procedure to hold accurately. The result indicates that provided the muon time, the shower direction and impact point coordinates are known, the reconstruction procedure works well. Figs. \[f:00\] and \[f:70\] also show the same histogram without the $r$-cut which clearly fails to reproduce the $z$-distribution obtained in the simulation. This is mostly because of the time accuracy of the detectors assumed to be $\sim 12.5~$ns. ![Histograms of production distribution for 500 showers of $0^{\circ}$ zenith angle and $10^{19}$ eV energy: [**Light fill**]{}: original distribution. [**Unfilled Thick Line**]{}: Final reconstruction, after all corrections described in the text. [**Unfilled Thin Line**]{} Reconstruction with no r cut. []{data-label="f:00"}](p_00_noB.eps){width="13"} ![Histograms of production distribution for 500 showers of $70^{\circ}$ zenith angle and $10^{19}$ eV energy: [**Light fill**]{}: original distribution.[**Unfilled Thick Line**]{}: Final reconstruction, after all corrections described in the text. [**Unfilled Thin Line**]{} Reconstruction with no r cut. []{data-label="f:70"}](p_70_noB.eps){width="13"} We have verified that the reconstructed histogram is not sensitive to small changes in the $r_c$ but clearly these cuts in $r$ can have large impact on the statistics. In table \[t:shifts\] we compare the averages of the ratio of $z$ obtained with this method to that from the simulation $\Gamma=< \frac{z_{\rm rec}}{z_{\rm true}} >$. Clearly the method works best for moderately inclined showers between $60^\circ$ and $80^\circ$. At very low zenith angles, there is an overestimation of the production distance, which could be due to an slight overestimation of the energy loss. On the other hand, at very high zenith angles the magnetic field effects begin to be important and the time geometric relation underestimates the production distance. Nevertheless, in both cases, the precision obtained is quite good. [|c|c|c||c|c|]{}   $\theta \; \; ({\rm deg})$  &   $\phi$ (deg)   &  B   & cut r(m) & $\Gamma$\ 0 & - & no & 900 & 1.23\ 30 & - & no & 1000 & 1.16\ 60 & - & no & 1500 & 1.07\ 70 & - & no & 2000 & 1.03\ 80 & - & no & 2900 & 1.00\ 80 & 0 & yes & 2900 & 0.97\ 80 & 90 & yes & 2900 & 0.89\ 86 & 0 & yes & 4000 & 0.86\ 86 & 90 & yes & 4000 & 0.85\ \ \ 80 & 0 & yes & 1400 & 1.02\ \ 80 & 0 & yes & 900 & 1.14\ In Fig. \[f:p\_and\_nu500\] we have compared the results of the reconstruction procedure applied to protons and deeply injected protons arriving with $80^{\circ}$ zenith angle to illustrate how the method can be used to identify deeply interacting inclined showers at high zenith, which are natural neutrino candidates. A systematic study of the reconstruction procedure and the ability to identify neutrinos under realistic experimental conditions is left for future work. ![Histograms of production distribution for 500 showers of $80^{\circ}$ zenith angle and $10^{19}$ eV energy for normal protons and for protons injected at 500 ${\rm r/cm^2}$ of vertical depth to simulate a neutrino interaction (marked as $\nu$-like). The geomagnetic field is included. [**Light fill**]{}: original distribution. [**Unfilled Thick Line**]{}: Final reconstruction, after all corrections described in the text.[]{data-label="f:p_and_nu500"}](p_and_nu500.eps){width="13"} Limitations of the method ========================= The proposed method gives the reconstruction of the function $\frac{dN}{dz}$ i.e., the production altitude distribution for the surviving muons but clearly the method is only valid with restrictions. Since the method addresses the muon time distributions, it is essential to identify the muon contribution in the shower front. Clearly if the muon signal is separated at detector level, there are no limitations due to this issue but this is in general not possible for most of the detectors used in air shower arrays which typically have complex signals containing a mixture of electron, positron and photon signals (the electromagnetic component) and muons. Čerenkov detectors, such as the water tanks used in Haverah Park [@HP] and in the Auger observatory [@auger], have some advantages in this respect. Since shower muons are penetrating particles the signal they give in Čerenkov detectors is basically proportional to their track through the detector, while the electromagnetic component typically gives a signal which is simply proportional to the energy carried by photons, electrons and positrons, because it is all absorbed in it. As a result the volume of the detector determines the ratio between the muon and electromagnetic signals to a good extent. Large detectors give high muon contributions in spite of the muons being a small fraction of all the particles in the shower front. Moreover under some circumstances these large sharp pulses could in principle be isolated and there are efforts in this line to separate individual muon pulses from the time structure of the Auger tank signals [@auger]. In any case, since the muon lateral distribution is flatter than the electromagnetic contribution [@ldfs], the muons eventually dominate for sufficiently large distances to shower axis. As the zenith angle rises the muons dominate closer and closer to shower axis. In close to horizontal showers the muons dominate practically always [@HP]. Depending on the detector performance there are a number of limitations to the precision which are addressed in this subsection. It is firstly straightforward to see that the total number of entries of the $dN/dz$ histogram is the total number of muons detected in all the detectors, i.e $ N_{\mu}= \sum_{i=0}^M N_i$. If we assume that the $\frac{dN}{dz}$ distribution has a RMS width $\sigma$, this means that the position of the mean $<z>$ (which is related to $X_{max}$) can be obtained with no more precision than $\frac{\sigma}{\sqrt{N_\mu}}$ which is an intrinsic statistical limitation. A second limitation arises because of the intrinsic time resolution of the detectors used, $\delta t$, which will limit the precision of the muon arrival time. This will translate directly into an uncertainty in the production distance $z$, $\delta z$, through the map $t\rightarrow z$. According to Eq. \[z\_t\] $$\frac{\delta z}{z} \simeq -\frac{c \delta t }{ct} \left(1-\frac{\Delta}{z} \right)\simeq- \frac{\delta t}{t}.$$ This equation can be rewritten to relate $\frac{\delta z}{z}$ to $\delta t$ substituting $ct$ for the expression given by Eq. \[z\_t\]: $$\frac{\delta z}{z}=2 \frac{(z-\Delta)^2}{z}\frac{1}{r^2} c \delta t. \label{eq:Dz_z}$$ The time resolution of the detector affects the reconstruction precision depending on distance to shower core. As we look at the arrival time of muons closer to the shower axis the time delays become smaller and the relative error on the $z$ distribution reconstruction diverges. But again this problem can be solved by imposing the cut $r>r_c$. To have $\frac{\delta z}{z}$ less than a given value $e_z$, and provided that we can find an approximated upper limit for the production distance of the muons, $z<z_{u}$, we obtain the following condition for the cut in $r$: $$r > r_c(\zeta)=\frac{\sqrt{\frac{2 z_{u} c\delta t}{e_z}} } {1+ \sqrt{\frac{2c \delta t}{e_z z_u}}{\tan \theta \cos \zeta}}. \label{eq:r_c_a}$$ Here $r_c$ depends on $\zeta$ because of the asymmetry induced by the term $\Delta$. If we neglect this term, we obtain a simple expression that does not depend on the angle $\zeta$: $$r_c=\sqrt{\frac{2 z_{u} c \delta t}{e_z}}. \label{eq:r_c}$$ Notice that an $r$ cut can also avoid the regions near the shower axis where the kinematical delay dominates over the geometrical, and also the region where the muonic component signal is shadowed by the electromagnetic component. In necessary case, the most stringent of the restrictions must be applied. For example let us consider an air shower array with time resolution $\delta t=12.5$ ns (corresponding to half the sampling rate of the Auger detector), located at 1400 m altitude and detecting showers with a zenithal angle of $60^{\circ}$. We can easily identify an upper limit for production distance (for instance using simulation), for $60^{\circ}$ it is $z_{u}=31.6$ km. According to Eq. \[eq:r\_c\] we would require that $r>1500$ m in order that the resolution on the $z$-reconstruction, $\frac{\delta z}{z}$, was less that $10\%$ ($e_z=0.1$). Fig. \[Delta\_z\] illustrates the effect showing the contour lines of the precision as a function of $r$ and $z$ as given by Eq. \[eq:r\_c\]. The value of $r_c$ must increase as the zenith angle rises because the muons are produced at higher $z$. For high zenith the effect of the neglected $\zeta$-dependent term makes the $z$-reconstruction somewhat worse than the approximate expression of Eq. \[eq:r\_c\], but enough for our purposes (in necessary case the full expression (Eq. \[eq:r\_c\_a\]) could also be used). For a shower with $\theta=86^\circ$ if we use $e_z=0.1$ with the former expression to obtain $r_c$ we actually get a resolution $\frac{\delta z}{z}$ which rises up to $0.17$ for the worst case corresponding to $\zeta=180^{\circ}$. ![Contour lines for the function $\frac{\delta z}{z}=2 \frac{z}{r^2} c \delta t$, with $\delta t= 12.5$ ns. Different grey bands show the 1-$\sigma$ production distance of muons at different zenith angles.[]{data-label="Delta_z"}](Delta_z.eps){width="10"} Correlation with angular and core position uncertainties ======================================================== The method has a third intrinsic limitation because to convert the arrival time histogram into a histogram of production distances using Eq. \[h\_z\] the incoming direction and the position of shower axis must be known, so that the appropriate values of $r$ can be introduced. In a realistic case these will only be known to a given precision and further uncertainties will arise because of the correlations between both core position and direction uncertainties with the distance distribution obtained. The study of these effects with simulations shows that the reconstructed distributions have a minimum in width when the true shower direction and impact position are used to reconstruct the depth distribution. This adds a interest to the method since it can be used in principle as a further check of the reconstructed directions and impact points. To study these correlations we explore the stability of the reconstruction to shifts in the core positions and angular directions. Unfortunately the computing time necessary to test such stability can be very large if simulations are used in the same way they were used to test the method in the previous section. We will use instead the results of the muon time delay model of Ref. [@cazon3] to get distributions of the arrival time for the shower muons from simulated showers. In an attempt to be closer to experimental conditions we assume an array of particle detectors and calculate the number of muons that crosses each of them. We choose 10 $m^2~\times~1.2~m$ (area $\times$ height) detectors in a hexagonal grid, separated 1500 m corresponding to the Auger surface detector. This limits the statistics of the reconstructed distribution, $dN/dz$, in a realistic way. Fig. \[60\_19dNdlogz\_and\_original\] shows an example of the statistics that could be obtained for a $10^{19}$ eV shower with $\theta=60^\circ$ and $\phi=90^{\circ}$. ( The azimuth angle $\phi$ was measured counterclockwise respect to [*East*]{} direction.) ![Distribution for a reconstructed $10^{19}$ eV proton shower of $\theta=60^\circ$ and $\phi=90^\circ$ (histograms) compared with the original distribution when we have finite sampling produced by a finite number of detectors.[]{data-label="60_19dNdlogz_and_original"}](60_19dNdlogz_and_original.eps){width="13"} We first recalculate the depth distributions assuming that the incidence direction has been misreconstructed by $(\Delta \theta,\Delta \phi)$ with respect to the actual arrival direction chosen for the simulation. This procedure was repeated for different shifts in angular space within an interval of $4^\circ \times 4^\circ$. For each angular shift both the mean and RMS width of the $z$-distributions in log basis were calculated. In Figs. \[mean60\_19angle\_1500\] and \[RMS60\_19angle\_1500\] representative results showing the mean and RMS width for an example of a $10^{19}$ eV proton shower of $60^\circ$ zenith are shown as a function of $\Delta \theta$ ($\Delta \phi$) for fixed $\phi$ ($\theta$), Also the RMS width is shown in a two dimensional plot $(\Delta \theta, \Delta \phi)$ in Fig. \[RMS60\_19angle\_1500\]. ![Effects of reconstruction direction shifts for a $10^{19}$ eV proton shower of $\theta=60^\circ$ and $\phi=90^\circ$ with an $r$ cut $r>1500$ m. Mean of $\log_{10} (z/m)$ as a function of $\Delta \theta$ fixing $\Delta \phi=0$ (dashed line) and as a function of $\Delta \phi$ fixing $\Delta \theta=0$ (continuous line).[]{data-label="mean60_19angle_1500"}](mean60_19angle_1500.eps){width="7"} It is worth remarking that the mean value of $z$ is quite stable to shifts in azimuthal direction $\Delta \phi$, whereas there is a very slight rise of the reconstructed $z$ when increasing the zenith angle $\theta$. The behavior on this angle in not symmetrical since $\Delta$ introduces an asymmetry between early and late regions. On the other hand the RMS width of the distributions seems to have a local minimum when the correct arrival direction is used. In a typical air shower array the arrival direction is obtained using the relative arrival times of the signals at different locations. The observed minimum of the RMS width of the $z$-distribution suggests that this method could be also used to either reconstruct shower direction independently or, more likely, to check that the reconstruction obtained through conventional methods is consistent with the arrival time of the muons at large distances from shower core, on an individual shower basis. ![Effects of reconstruction direction shifts in the width of the $z$-distribution for a $10^{19}$ eV proton shower of $\theta=60^\circ$ and $\phi=90^\circ$ with an $r$ cut $r>1500$ m. [**Left**]{} RMS width of $\log_{10} (z/m)$ as a function of angular shifts $(\Delta \theta,\Delta \phi)$. [**Right**]{} RMS of $\log_{10} (z/m)$ as a function of $\Delta \theta$ fixing $\Delta \phi=0$ (dashed line) and as a function of $\Delta \phi$ fixing $\Delta \theta=0$ (continuous line).[]{data-label="RMS60_19angle_1500"}](RMS60_19angle_1500.eps){width="14" height="7cm"} A completely analogous method was followed to study the core position and $z$-reconstruction correlations. The reconstructed impact points were shifted by $(\Delta x,\Delta y)$ in the ground plane with respect to the core of the simulated shower over a grid covering a rectangle of 1000 $\times$ 1000 m. The means and RMS widths for shifts in both $x$ and $y$ positions are shown in Fig. \[mean\_and\_RMS\_60\_19core\_1500\]. The plots display important discontinuities. They are of statistical nature because the total number of muons in the detector is small and as the core position position is shifted individual detectors are rejected or accepted because of the $r$ cut. The detectors close to the cut are those that have most muons and including or not including them affects the $z$-distribution. These discontinuities are also present in some circumstances for angular shifts because the relative position of the detectors also change but clearly the changes of angular reconstruction modify the distances to shower axis at a second order level. These discontinuities can become smoother by increasing statistics, for instance relaxing the $r$-cut. ![Effects of impact point shifts on the mean (left panel) and RMS (right panel) of the $z$-distribution as a function of $\Delta x$ fixing $\Delta y=0$ (dashed line) and as a function of $\Delta y$ fixing $\Delta x=0$ (continuous line) for a $10^{19}$ eV proton shower at $\theta=60^{\circ}$ and $\phi=90^{\circ}$ with an $r$ cut $r>1500$ m. \[mean\_and\_RMS\_60\_19core\_1500\] ](mean_and_RMS_60_19core_1500.eps){width="15" height="7cm"} Notice that the mean value of $z$ is again quite stable to shifts in core position. This is not difficult to understand since each core location corresponds to a new [*time-reference plane*]{} which is only slightly shifted along the shower axis with respect to the planes obtained for other core positions. The effect is due to the curvature of the front and is therefore a second order effect. The RMS width of the distributions also displays a local minimum when the correct impact point is used. This is also not surprising given that approximately $z \propto r^2$. Differences in the $z$ reconstruction arise through the modification of the relative position of the tank with respect to shower core. It can be easily seen that when a tank gets a closer position ($r$ decreases) as a result of the shift, the tanks placed in the opposite $\zeta$ will get to a further one ($r$ increases). This suggests that the width of the reconstructed distribution should have a local minimum when the correct impact point is considered. An example is given relaxing the $r$-cut to 500 m in Figs. \[RMS60\_19core\_500\]. ![Effects of impact point shifts on the RMS width of the $z$-distribution for a $10^{19}$ eV proton at $\theta=60^{\circ}$ and $\phi=90^{\circ}$ with an relaxed $r$ cut $r>500$ m . [**Left**]{} RMS of $\log_{10} (z/m)$ as a function of a core shift $(\Delta x,\Delta y)$. [**Right**]{} RMS of $\log_{10} (z/m)$ as a function of $\Delta x$ (dashed line) fixing $\Delta y=0$, and as a function of $\Delta y$ fixing $\Delta x=0$ (continuous line).[]{data-label="RMS60_19core_500"}](RMS60_19core_500.eps){width="14" height="7cm"} In a typical air shower array the core position is obtained by the relative amplitude of the signals at different locations either through a fit or by some other means. As for shifts in angular direction the minimum of the RMS width of the $z$-distribution suggests that this method could be used either to reconstruct shower impact points independently or to check that the core position reconstruction obtained through conventional methods is consistent with the arrival time of the muons at large distances from shower core, on an individual shower basis. Summary ======= We have developed a method that has the potential of reconstructing the production altitude for the muons in inclined cosmic ray showers based on the time distribution of the muon signals in the detectors of an extensive air shower array. This method requires knowledge of the arrival times of muons in the detectors of the air shower array and it can be applied provided that a cut is made in distance to shower axis, $r>r_c$. Since the muon signal dominates at high $r$ it can be also used when the detectors cannot separate the muon signal provided that the $r$ cut is chosen so that the muon signal dominates. The method relies mainly on geometrical arguments and there are minor effects introduced through the kinematical delay of the muons which have little effect at large distances from shower axis. The model does not rely on any assumption about the interaction model for hadrons. Different models would give rise to different kinematic corrections, but the effect is small. Although we have assumed proton showers to explore the viability of this method the method can be also used for heavier nuclei, and similar results would be obtained in that case. The necessary cut introduces limitations because of statistics. We have checked that our method correctly reproduces the depth distribution of muon production using sets of simulated showers with AIRES to a degree of accuracy that is zenith angle dependent. The method works best in the $60^\circ-80^\circ$ region and it is fairly stable with respect to misreconstruction of the shower core and the incoming direction, in what concerns the mean of the distribution. The RMS width of the production distance distribution however is sensitive to both the reconstructed impact point and arrival direction. The width displays a minimum when the correct impact point and arrival direction are used in the reconstruction procedure. This work represents a new approach to studying extensive air showers. It will add information concerning the individual development of air showers and can be used to check the reconstructed impact point and arrival directions. The reconstruction of depth development in inclined showers can also have important implications in improving the potential of air shower arrays to detect neutrinos. Acknowledgments =============== We thank J Alvarez–Muñiz and A.A. Watson for many discussions on the time structure of shower, and many helpful comments after reading the manuscript. This work was partly supported by the Xunta de Galicia (PGIDIT02 PXIC 20611PN), by MCYT (FPA 2001-3837, FPA 2002-01161 and FPA 2004-01198). We thank CESGA, “Centro de Supercomputación de Galicia” for computer resources. [999]{} J. Linsley and L. Scarsi, Phys. Rev. 128 (1962) 2384. J. Lapikens, PhD Thesis, University of Leeds, 1974 A. A. Watson and J. G. Wilson J. Phys. A: Math. Nucl. Gen. 7 (1974) 1199; R. Walker and A. A. Watson, J. Phys. G 7 (1981) 1297; R.  Walker and A. A. Watson, J. Phys. G 8 (1982) 1131. T. Antoni [*et al.*]{}, Astropart. Phys.  18 (2003) 319. \[arXiv:astro-ph/0204266\]. Nagano and A.A. Watson, Rev. Mod. Phys. 72 (2000) 689. J. Abraham [*et al.*]{}, \[Auger Collaboration\], Nucl. Instr. Meth. A 523 (2004) 50. T.A. Danilova, D. Dumora, A.D. Erlykin, and J. Procureur J. Phys. G: Nucl. Part. Phys. 20 (1994) 961. I. M. Brancus, B. Vulpescu, H. Rebel, M. Duma and A. A. Chilingarian, Astropart. Phys.  7 (1997) 343. L. Pentchev, P. Doll, and H.O. Klages, J. Phys. G: Nucl. Part. Phys. 25 (1999) 1235. M. Ambrosio [*et al.*]{}, [*Proc. of the 26th Int. Cosmic Ray Conference, Salt Lake City, Utah, 17-25 Aug 1999*]{}, vol 5, p. 312. M. Ave, R.A. Vázquez, and E. Zas, Astropart. Phys. 14 (2000) 91. L. Cazón, R.A. Vázquez, A.A. Watson, and E. Zas, Astropart. Phys. 21 (2004) 71. J. Alvarez–Muñiz and I. Valiño, in preparation. S. J. Sciutto, AIRES: A System for Air Shower Simulation, [*Proc. of the 26th Int. Cosmic Ray Conference, Salt Lake City, Utah, 17-25 Aug 1999*]{}, vol. 1, p. 411; S. J. Sciutto astro-ph/9911331. M. Ave, J.A. Hinton, R.A. Vázquez, A.A. Watson, and E. Zas Phys. Rev. D 65 (2002) 063007. . By Auger Collaboration. FERMILAB-PUB-96-024, Jan 1996. ([*www.auger.org*]{}). E. J. Fenyves [*et al.*]{}, Phys. Rev. D 37 (1988) 649; C. Forti, [*et al.*]{}, Phys. Rev. D 42 (1990) 3668. Modelling the distribution of surviving muons ============================================= At ground level both muon energies and muon number are reduced because of energy loss and decay. As a first approximation, assuming a constant energy loss per unit depth $d E/dx=-{a}$ along an uniform atmospheric density $\rho$, both these effects can be easily accounted for. After traveling a distance $l$ a flux of muons $\phi_0$ of energy $E_i$ reduces by an energy dependent factor to: $$\phi(l) = \phi_0 \left[\frac{E_i-\rho {a}l} {E_i} \right]^{\kappa}. \label{N(l)}$$ The last factor takes into account both energy loss and decay in flight. The muon mean lifetime, $\tau$, enters through the exponent $\kappa={mc^2/(\rho {a}c \tau)} \sim 0.8 $. We can correct the energy of the produced muons given by Eq. \[factorization\] with such factor to obtain the distribution in $E_i$, $p_t$, $z$ and $\zeta$ after traveling a distance $l$: $$\frac{d^4 N(l)} {d z \;\! d\zeta \;\! d E_i \;\! d p_t} = {\cal N}_0 \;\! \frac{1}{2\pi}\;\! h(z) \;\! f_1(E_i) \;\! f_2(p_t) \left[\frac{E_i-\rho {a}l}{E_i} \right]^{\kappa}, \label{factorization_N}$$ Here $l$ is the distance traveled by the muon, which enters in Eq. \[N(l)\], given by $l^2=(z-\Delta)^2 + r^2$, where $r$ is the distance to shower axis at the end of the muon trajectory, and $z-\Delta$ is the distance travelled by the muon measured along the shower axis. The correction $\Delta=r \tan\theta\cos \zeta$ relates the distances measured along the axis between the intercepts of shower axis and the muon trajectory with ground level and depends on the zenith angle, $\theta$, as well as on the muon impact point coordinates in the transverse plane, $r$ and $\zeta$. For a muon to reach ground level there is a minimal production energy given by $E_i > m c^2+\rho {a}l$. Typical values of $\rho {a}l$ greatly exceed $m c^2$, particularly for inclined showers. After traveling a distance $l$ the transverse distance is simply $r=l \sin \alpha$. Performing the change the coordinates from $p_t$ to $\sin \alpha$ in Eq. \[factorization\_N\] we obtain: $$\frac{d^4 N(l)}{d z \;\! d\zeta \;\! d E_i \;\! d{\sin{\alpha}}} = {\cal N}_0 \;\! \frac{1}{2\pi} \;\! h(z) \;\! f_1(E_i)\;\! f_2\left(\frac{E_i}{c} \sin{\alpha} \right) \;\! \left[1-\frac{\rho {a}l}{E_i} \right]^{\kappa}\;\! \frac{E_i}{c}. \label{mu-dist}$$ Correlations between the ground variables appear naturally because of energy loss and decay. We can now introduce the following parameterizations for $f_1(E_i)$ and $f_2(p_t)$ which were shown in [@cazon3] to give good approximations to the muon time distributions: $$\begin{aligned} f_1(E_i)= \frac{\gamma -1}{ mc^2} \left(\frac{E_i}{mc^2}\right)^{-\gamma}\Theta(E_i-mc^2), \\ f_2(p_t)= \frac{p_t}{Q^2} \exp\left(-\frac{p_t}{Q}\right),\end{aligned}$$ where $\gamma\simeq 2.6$ and $Q \simeq 0.17$ GeV/c. Eq. \[mu-dist\] now becomes $$\frac{d^4 N}{d z \;\! d\zeta \;\! d E_i \;\! d{\sin{\alpha}}} = \frac{{\cal N}_0 (\gamma -1) }{2\pi} h(z) \;\! \left(\frac{E_i}{mc^2}\right)^{-\gamma+1} \;\! \frac{E_i \sin\alpha}{c^2 Q^2} \exp\left(-\frac{E_i \sin\alpha}{c Q}\right) \left[1-\frac{\rho {a}l}{E_i} \right]^{\kappa}.$$ We now integrate the distribution in $E_i$ for fixed $z$ and $\alpha$ to obtain: $$\frac{d^3 N}{d z \;\! d\zeta \;\! d{\sin{\alpha}}} = \int_{mc^2 + \rho{a}l} ^{\infty} \frac{d^4 N}{d z \;\! d\zeta d E_i \;\! d{\sin{\alpha}}} \;\! d E_i = { \frac{{\cal N}_0 (\gamma -1)}{ 2\pi}} \left( \frac{mc^2}{Q c}\right)^{\gamma -1} h(z) \;\! I(l,\sin \alpha) \sin^{\gamma -2} \alpha, \label{SinDistribution}$$ in which $I(l,\sin \alpha)$ is a dimensionless integral in the variable $x= \frac{E_i \sin \alpha}{c Q}$: $$I(l,\sin \alpha)= \int_{y + x_0} ^{\infty} x^{-\gamma+2} \left[1-\frac{x_0}{x}\right]^{\kappa} \exp \left(-x\right) dx,$$ with $x_0(l,\alpha)=\frac{\rho {a}l}{cQ} \sin \alpha$ and $y(\alpha)=\frac{mc^2}{cQ}\sin\alpha$. We note that $y\ll x_0$ when $l \gg mc^2/\rho {a}\simeq 500$ m and in that case the integral can be approximated replacing its lower limit by $x_0$. Since $z$, the distance of muon production, is relatively large, particularly for inclined showers, this approximation is adequate for most circumstances. Then it is easily seen that the integral depends only on the combination $l \sin \alpha=r$, i.e. on the transverse distance. We can then replace $I(l,\sin \alpha)$ by $I(r)$. In terms of the differential solid angle $d^2 \Omega=-d\zeta d \cos \alpha$ the number of muons arriving at ground level coming from production distance $z$ becomes: $$\frac{d^3 N(r)}{ d z \;\! d^2\Omega} = -\frac{d^3 N(r)}{ d z \;\! d \zeta \;\! d\cos \alpha} = \frac{{\cal N}_0 (\gamma -1)}{2\pi} \left(\frac{mc^2}{ Q c}\right)^{\gamma -1} h(z) ~ I(r) \left( \frac{ \cos \alpha }{\sin \alpha} \right) \sin^{\gamma-2} \alpha. \label{CosDistribution}$$ ![ Illustration of the relation between the ground surface, the transverse plane (shower plane) and the shower axis (big arrow). The small arrows represent a detector surface direction normal to the detector surface, the dashed lines are parallel to shower axis and the continuous lines are the muon trajectories. The expressions for number of muons $dN_A$ which go through the surface element $d^2A$ (labeled 1) are described in the text.[]{data-label="f:terra2"}](terra2.eps){width="16"} Using the relations between $z$, $l$ and $\alpha$, we can substitute Eq. \[CosDistribution\] into Eq. \[rho\_mu\] and integrate in $z$ to obtain: $$\hat{N}_{Ar \zeta} \equiv \frac{1}{r} \;\! \frac{d^2N_A}{dr \;\! d \zeta}= \frac{ {\cal N}_0 (\gamma -1) }{2\pi} \left(\frac{mc^2}{Q c}\right)^{\gamma -1} ~ \frac{I(r)}{r}~ \int h(z) \;\! \frac{ \cos \alpha ~ D_A(\Omega) \sin^{\gamma-2}\alpha}{l} d z. \label{N_r}$$ $\hat{N}_{Ar \zeta}$ is the number of muons going through $d^2 A$ (whose projection onto the transverse plane is $r dr d\zeta$) and has a non-trivial geometrical dependence through the $D_A(\Omega)$ factor. This factor can be both greater or less than 1 and becomes 1 as the angle $\alpha$ tends to zero, that is in the limit of small $r$. In that case $\hat{N}_{Ar \zeta}$ simply becomes the muon surface density in the transverse plane. $D_A(\Omega)$ can be regarded as a geometrical correction which accounts for the fact that the muons do not travel parallel to the shower axis and the fact that the detector planes are not perpendicular to the shower axis not to the muon fluxes (See Fig. \[f:terra2\]). It can be shown that when $z$ and $r$ are fixed $D_A$ depends on the angle $\zeta$ because the angles $\alpha$ and $\psi_{\mu A}$ differ. As a result the factor $D_A$ is responsible for part of the asymmetry in the muon signal. For instance a horizontal surface will have larger efficiency for collecting early arriving muons because $\psi_{\mu A}$ is smaller than for late arriving muons. Note that $D_A(\Omega)$ would be exactly 1 for an “ideal” detector of spherical shape. In that case these differences disappear. The expression obtained in Eq. \[g(t)2\] relates the time distribution to a transform of the depth distribution $h(z)$ which effectively accounts for muon decay in flight through $l$. If we are interested in obtaining the production distribution $h(z)$ formally we can rewrite Eq. \[g(t)2\] as: $$h(z) = g(t) \frac{dt}{dz} l^{-1+\gamma} \cos^{-1} \alpha ~ D_A^{-1}(\Omega) \int_0^{\infty} h(z) \;\! {l}^{1-\gamma} \cos \alpha ~D_A(\Omega) dz. \label{h_z}$$ This expression in principle allows us to obtain $h(z)$ from the time distribution of the arriving muons at a given point on the ground, $g(t)$. On its own it is not very useful because typically a single detector in an air shower array does not collect sufficient statistics to sample $h(z)$ reliably. in a practical situation one must combine the results of several detectors. Since $h(z)$ is normalized to 1, the unknown factor which is given by the integral on the RHS of the equation acts as an effective weight to be given to each detector. In a first attempt it is possible to ignore it. For inclined showers the weights to be applied for the relevant detectors are expected to be quite similar and the approximation works well. More sophisticated approaches could be deviced for instance using Eq. \[h\_z\] with a trial $h(z)$ function in an iterative process to sample $h(z)$. However since $h(z)$ is not directly available from the simulation program used we will not need to calculate $h(z)$ and we have instead compared our results to the $z$-distribution of the surviving muons. Parameterization of kinematical delays ====================================== The mean kinematical time delay can be obtained by applying the method developed in Ref. [@cazon3], summarized in Eq \[t:kinetic\] to the distributions discussed in this article: $$< t_{\varepsilon} > \ = \frac{\int t_{\varepsilon} \frac{d^4 N}{dz \;\! d\zeta \;\! dE_i \;\! dr } d E_i }{\int \frac{d^4 N}{dz \;\! d\zeta \;\! dE_i \;\! dr} dE_i }.$$ After some manipulation, using the results of the models in Appendix A a simple expression can be obtained for it: $$< t_{\varepsilon} > \ = \ \frac{1}{2c}\frac{r^2}{l} {\left(\frac{mc^2}{cQ}\right)^2} \frac{ \int_{y + x_0} ^{\infty} x^{-\gamma} \left[1-\frac{x_0}{x}\right]^{\kappa-1} \exp \left(-x\right) dx}{ \int_{y + x_0} ^{\infty} x^{-\gamma+2} \left[1-\frac{x_0}{x}\right]^{\kappa} \exp \left(-x\right) dx } = \frac{1}{2c}\frac{r^2}{l} \epsilon(r,z-\Delta), \label{eq:tepsilon_r}$$ with $x_0(l,\alpha)=\frac{\rho {a}l}{cQ} \sin \alpha$ and $y(\alpha)=\frac{mc^2}{cQ}\sin\alpha$. In the last equality of the above expressions we have introduced the dimensionless factor $\epsilon(r,z-\Delta)$. We note that for $z-\Delta \gg r$, for instance in inclined showers, it gives the ratio of the average kinematical delay to the geometrical delay at a given position $\frac{<t_\varepsilon>}{t_g}\simeq \epsilon(r,z-\Delta)$. This indicates the regions where the geometric delay dominates, which depend on production distance. For practical purposes we have parameterized $\epsilon(r,z)$ as follows: $$\epsilon (r,z) = p_0(z) \left(\frac{r}{\rm m}\right)^{p_1},$$ with $$\begin{aligned} \log_{10}p_0(z) &=& -0.6085 +1.955 \ \log_{10} (z/m) -0.3299 \ \log^2_{10} (z/m) +0.0186 \ \log^3_{10} (z/m), \\ \log_{10}p_1 &=& -1.176.\end{aligned}$$
--- abstract: 'CCD photometric observations in $VRI$ colors and spectroscopic observations of the newly discovered eclipsing binary GSC 2314-0530 (NSVS 6550671) with dMe components and very short period of $P=0.192636$ days are presented. The simultaneous light-curve solution and radial velocity solution allows to determine the global parameters of GSC 2314-0530: $T_{1}=3735$ K; $T_{2}=3106$ K; $M_{1}=0.51$ M$_{\sun}$; $M_{2}=0.26$ M$_{\sun}$; $R_{1}=0.55$ R$_{\sun}$; $R_{2}=0.29$ R$_{\sun}$; $L_{1}=0.053$ L$_{\sun}$; $L_{2}=0.007$ L$_{\sun}$; $i=72.5\degr$; $a=1.28$ R$_{\sun}$; $d=59$ pc. The chromospheric activity of its components is revealed by strong emission in the H$\alpha$ line (with mean $EW=5\ {\rm \AA}$) and observed several flares. Empirical relations mass–$M_{\rm {bol}}$, mass–radius and mass–temperature are derived on the basis of the parameters of known binaries with low-mass dM components.' author: - | Dinko P. Dimitrov$^{1}$[^1] and Diana P. Kjurkchieva$^{2}$\ $^{1}$Institute of Astronomy, Bulgarian Academy of Sciences, 72 Tsarigradsko shossee str., 1784 Sofia, Bulgaria\ $^{2}$Department of Physics, Shumen University, 115 Universitetska str., 9700 Shumen, Bulgaria date: 'Accepted –. Received –; in original form –' title: 'GSC 2314-0530: the shortest-period eclipsing system with dMe components [^2]' --- \[firstpage\] binaries: eclipsing – binaries: spectroscopic – stars: activity – stars: fundamental parameters – stars: late-type – stars: low-mass Introduction {#sec:intro} ============ Although the M dwarfs are the most numerous stars in our Galaxy, the mass, metalicity and age dependencies of their stellar luminosities and radii are poorly calibrated. The reason is the selection effect that plays against the detection of fainter and smaller stars. Less than 20 binaries with low-mass dM components have empirically-determined masses, radii, luminosities and temperatures (see Section \[sec:global\], Table \[tab:stars\]). As a result the mass-luminosity relation is determined by only a few low-mass stars. This deficiency hindered the development of the models for the cool dense atmospheres of the M dwarfs. It is established that all available models underestimate the radii (by around 10–15 per cent) and overestimate the temperatures (by 200–300 K) of short-period binaries with dM components [@ribas03; @maceroni04]. The Northern Sky Variability Survey (NSVS) contains a great number of photometric data [@wozniak04] that allows searching of variable stars and determination of their periods and types of variability. A multiparametric method for search for variable objects in large datasets was tested on the NSVS [@dimitrov09] and as a result many eclipsing stars were discovered. One of them was GSC 2314-0530 $\equiv$ NSVS 6550671 ($\alpha$=02$^{\rm h}20^{\rm m}50\fs9$, $\delta$=+$33\degr 20\arcmin 46\farcs6$). On the base of the NSVS photometry obtained in 1999–2000 we derived the ephemeris: $$\label{equ:nsvs} HJD({\rmn {MinI}})=2451352.062 + 0.192637 \times E$$ and built its light curve (Fig. \[fig:nsvs\]). ![NSVS photometry of GSC 2314-0530[]{data-label="fig:nsvs"}](fig1.eps){width="0.75\columnwidth"} We found that this star has been assigned also as SWASP J022050.85+332047.6 according to the SuperWASP photometric survey [@pollacco06]. @norton07 reported its coincidence with the [*ROSAT*]{} X-ray source 1RXS J022050.7+332049. Initially GSC 2314-0530 attracted our interest by its short orbital period because there were only several systems with non-degenerate components and periods below the short-period limit of 0.22 days [@rucinski07]: GSC 1387-0475 with $P=0.217811$ d [@rucinski07; @rucinski08], ASAS J071829-0336.7 with $P=0.211249$ d [@pribulla09], the star V34 in the globular cluster 47 Tuc with $P=0.2155$ d [@weldrake04] and BW3 V38 with orbital period $P=0.1984$ d [@maceroni97; @maceroni04]. When we established that the components of GSC 2314-0530 were dM stars our interest increased and we undertook intensive photometric and spectral observations in order to determine its global parameters and to add a new information for the dM stars as well as for the short-period binaries. ![Observed field around GSC 2314-0530[]{data-label="fig:chart"}](fig2.eps){width="0.75\columnwidth"} Observations and data reduction {#sec:observation} =============================== New photometry -------------- The CCD photometry of GSC 2314-0530 in $VRI$ bands was carried out at Rozhen National Astronomical Observatory with the 2-m RCC telescope equipped with VersArray CCD camera (1300 $\times$ 1340 pixels, 20 $\mu$m pixel, field of 5.25 $\times$ 5.35 arcmin) as well as with the 60-cm Cassegrain telescope using the FLI PL09000 CCD camera (3056 $\times$ 3056 pixels, 12 $\mu$m pixel, field of 17.1 $\times$ 17.1 arcmin). The average photometric precision per data point was 0.005 – 0.008 mag for the 60-cm telescope and 0.002 – 0.003 mag for the 2-m telescope. Table \[tab:log1\] presents the journal of our photometric observations. It should be noted that the observations on 2009 December 30 are synchronous in the $VRI$ colors. Date HJD(start) Phases Filter Exp. \[s\] N Telescope -------------- ---------------- --------------- -------- ------------ ----- ----------- 2009 July 25 2455038.482662 0.725 – 1.298 $R$ 120 126 60-cm 2009 July 26 2455039.484907 0.927 – 0.491 $R$ 120 54 60-cm 2009 July 27 2455040.468322 0.032 – 0.693 $R$ 120 83 60-cm 2009 July 28 2455041.501250 0.395 – 0.881 $R$ 120 62 60-cm 2009 Oct. 21 2455126.412740 0.210 – 1.294 $V$ 15 737 2-m 2009 Nov. 13 2455149.178102 0.393 – 1.389 $I$ 10 850 2-m 2009 Nov. 13 2455149.375822 0.419 – 1.391 $R$ 10 835 2-m 2009 Nov. 20 2455156.324421 0.489 – 0.521 $B$ 120 3 60-cm 2009 Nov. 20 2455156.325521 0.495 – 1.862 $V$ 60 183 60-cm 2009 Nov. 20 2455156.326088 0.498 – 0.529 $R$ 30 3 60-cm 2009 Nov. 20 2455156.326493 0.500 – 0.531 $I$ 30 3 60-cm 2009 Dec. 30 2455196.225416 0.610 – 1.785 $V$ 120 65 60-cm 2009 Dec. 30 2455196.226516 0.616 – 1.791 $R$ 60 65 60-cm 2009 Dec. 30 2455196.227256 0.619 – 1.810 $I$ 60 65 60-cm Standard stars of @landolt92 and standard fields of @stetson00 were used for transition from the instrumental system of each telescope to standard photometric system. The standard IDL procedures (adapted from DAOPHOT) were used for reduction of the photometric data. The standard stars were chosen on the basis of the method of @everett01 and Table \[tab:colors\] presents their colors. The values of $J-K$ are from the catalogue NOMAD [@zacharias05] while the values of other parameters are our estimations. The field of the variable and standard stars is shown in Fig. \[fig:chart\]. ------ -------------- --------- --------- --------- --------- --------- ------------------- ------------------- ID $V$ $B-V$ $V-R$ $V-I$ $J-K$ pmRA pmDE GSC/USNO-B1 \[mag\] \[mag\] \[mag\] \[mag\] \[mag\] \[mas yr$^{-1}$\] \[mas yr$^{-1}$\] Var 2314-0530 13.36 1.18 0.88 2.38 0.87 144.0 -112.0 St1 2314-1784 12.12 0.30 0.25 0.57 0.29 -000.8 -008.3 St2 2314-1378 12.24 0.29 0.24 0.58 0.34 -000.1 -001.6 St3 2314-1655 12.40 0.22 0.20 0.46 0.27 005.5 -004.0 Twin 1233-0046425 16.91 1.41 1.03 3.02 0.87 140.0 -112.0 ------ -------------- --------- --------- --------- --------- --------- ------------------- ------------------- Table \[tab:photometry\] presents a sample of our photometric data (the full table is available in the online version of the article, see Supporting Information). HJD Magnitude Filter ---------------- ----------- -------- 2455156.329669 14.8530 B 2455156.332679 14.8490 B 2455156.335689 14.8300 B 2455126.417320 13.3610 V 2455126.418512 13.3618 V 2455126.419890 13.3619 V 2455126.420167 13.3624 V 2455126.420700 13.3547 V 2455126.420978 13.3583 V 2455126.421128 13.3609 V : BVRI photometry of GSC 2314-0530[]{data-label="tab:photometry"} Some of our photometric runs covering well the orbital cycle are presented in Fig. \[fig:m/HJD\]. ![image](fig3.eps){width="70.00000%"} The Fourier analysis of all our photometric data performed by the software PERIOD-04 [@lenz05] leads to the ephemeris: $$\label{equ:rozhen} HJD({\rm MinI})=2451352.061633 + 0.1926359\times E .$$ The new-obtained period value is almost the same as that of the ephemeris (\[equ:nsvs\]) of the NSVS data that means that the orbital period of GSC 2314-0530 is stable. The color indices of our target (Table \[tab:colors\]) lead to M spectral type of the binary. Taking into account the almost equal eclipse depths of the light curve, i.e. the close temperatures of the components, as well as the short orbital period of the system, we may conclude that the two components of GSC 2314-0530 are dM stars. The value of the obtained period is below the short-period limit and reveals that our target is the shortest-period binary with dM components. Figure \[fig:folded\] shows the folded light curves from all our photometric data phased according to the ephemeris (\[equ:rozhen\]). ![The folded $V$, $R$, $I$ light curves of GSC 2314-0530 and their fits[]{data-label="fig:folded"}](fig4.eps){width="0.9\columnwidth"} Spectroscopy ------------ We obtained 26 spectra of GSC 2314-0530 with resolution $0.19\ {\rm{\AA}}$/pixel during November – December 2009 covering spectral range of $200\ {\rm{\AA}}$ around the H$\alpha$ line. We used a CCD Photometrics AT200 camera with the SITe SI003AB 1024 $\times$ 1024 pixels chip mounted on the Coude spectrograph (grating B$\&$L632/14.7$\degr$) on the 2-m RCC telescope at Rozhen. The exposure time was 15 min during 2009 November 26 and 20 min during 2009 December 31 and 2010 January 01. All stellar integrations were alternated with Th-Ar comparison source exposures for wavelength calibration. The bias frames and flat-field integrations were obtained at the beginning and at the end of the night. The mean S/N ratio for our observations was around 24, i.e. acceptable for radial velocity determination. Table \[tab:radvel\] presents the journal of our spectral observations. The reduction of the spectra was performed using IRAF packages by bias subtraction, flat fielding, cosmic ray removal, one-dimensional spectrum extraction and wavelength calibration. Figure \[fig:Ha-synoptic\] illustrates the orbital variability of the star spectra while Figure \[fig:Ha-phases\] presents the one-dimensional H$\alpha$ profiles at some orbital phases. ![The orbital variability of the spectra of GSC 2314-0530 from 2009 November 26[]{data-label="fig:Ha-synoptic"}](fig5.eps){width="\columnwidth"} ![The H$\alpha$ profiles at some phases[]{data-label="fig:Ha-phases"}](fig6.eps){width="0.75\columnwidth"} ![The two Gaussians (gray lines) reproducing the H$\alpha$ line (dots) of the two stellar components, and their sum (black line) fitting the total H$\alpha$ profile of GSC 2314-0530.[]{data-label="fig:Ha-fit"}](fig7.eps){width="0.75\columnwidth"} Analysis of the spectral data {#sec:analysis_sp} ============================= The obtained spectra of GSC 2314-0530 show wide emission H$\alpha$ lines implying high rotational velocities as well as absorption TiO bands at $6569\ {\rm{\AA}}$ and $6651\ {\rm{\AA}}$ (Fig. \[fig:Ha-synoptic\]). These spectral features suggest a dMe classification of GSC 2314-0530. The spectral contribution of the secondary component is visible only in the H$\alpha$ line (Fig. \[fig:Ha-synoptic\]). That is why we determined the radial velocities of the two stellar components by fitting the H$\alpha$ lines at each phase with Gaussians (Fig. \[fig:Ha-fit\]). Table \[tab:radvel\] and Figure \[fig:RV\] present the radial velocities of the stellar components of GSC 2314-0530. Their fit corresponds to values $K_{1}=V_{1}\sin i = 109.7\pm3.2$ km s$^{-1}$, $K_{2} = V_{2}\sin i = 211.3\pm5.8$ km s$^{-1}$ and $V_{0}\sin i =-1.2\pm5.7$ km s$^{-1}$. They lead to mass ratio $q=0.519\pm0.029$ and binary separation $a \sin i=1.22\pm0.04$ R$_{\sun}$. ![Radial velocities of the two components of GSC 2314-0530 (the sizes of the error bars correspond to 3$\sigma$) and their fits by the code PHOEBE [@prsa05].[]{data-label="fig:RV"}](fig8.eps){width="0.75\columnwidth"} ---- ---------------- ----- ------- -------- ---------- ------------------- ----------- ------ No HJD S/N phase $EW_{\rm{total}}$ \[$\rm{\AA}$\] 01 2455162.375293 23 0.88 89.2 $\pm3.3$ -157.5 $\pm6.2$ 4.97 02 2455162.385975 19 0.93 42.8 $\pm3.7$ 4.56 03 2455162.396657 22 0.99 -9.3 $\pm6.6$ 3.61 04 2455162.407335 21 0.04 -57.9 $\pm4.5$ 63.0 $\pm4.0$ 6.64 05 2455162.418012 27 0.10 -68.0 $\pm4.5$ 99.1 $\pm10.8$ 4.34 06 2455162.428689 28 0.15 -92.8 $\pm3.2$ 210.3 $\pm9.3$ 4.04 07 2455162.439367 28 0.21 -116.6 $\pm4.4$ 165.8 $\pm6.9$ 3.53 08 2455162.450048 29 0.27 -120.7 $\pm4.6$ 174.0 $\pm7.9$ 3.62 09 2455162.460726 28 0.32 -86.7 $\pm2.6$ 187.1 $\pm3.2$ 4.60 10 2455162.471405 30 0.38 -81.6 $\pm4.3$ 176.4 $\pm4.9$ 4.49 11 2455162.482903 25 0.44 -48.4 $\pm2.2$ 5.47 12 2455162.493582 26 0.49 -0.6 $\pm3.8$ 5.55 13 2455162.514939 25 0.60 73.4 $\pm5.6$ -106.5 $\pm12.9$ 5.34 14 2455162.526143 24 0.66 93.8 $\pm4.1$ -170.6 $\pm5.9$ 4.94 15 2455162.537036 25 0.72 96.2 $\pm3.4$ -222.2 $\pm8.6$ 6.18 16 2455197.222839 18 0.78 120.7 $\pm7.4$ 5.92 17 2455197.236667 29 0.85 93.9 $\pm3.0$ -160.6 $\pm9.1$ 5.07 18 2455197.250817 27 0.92 48.6 $\pm4.4$ -123.3 $\pm4.5$ 5.34 19 2455197.264965 28 0.99 -37.1 $\pm4.9$ 5.59 20 2455197.279111 27 0.07 -60.1 $\pm3.0$ 5.67 21 2455197.293257 29 0.14 -91.7 $\pm4.8$ 198.2 $\pm10.0$ 4.16 22 2455197.308131 29 0.22 -119.1 $\pm4.3$ 212.2 $\pm7.8$ 4.44 23 2455197.322278 30 0.29 -101.8 $\pm2.9$ 191.2 $\pm6.1$ 3.76 24 2455197.336431 28 0.37 -75.4 $\pm4.0$ 168.5 $\pm6.1$ 3.65 25 2455198.274197 25 0.23 -91.5 $\pm3.4$ 205.3 $\pm4.3$ 6.04 26 2455198.288351 25 0.31 -72.2 $\pm4.0$ 199.8 $\pm4.0$ 6.29 ---- ---------------- ----- ------- -------- ---------- ------------------- ----------- ------ Analysis of the photometric data {#sec:analysis_ph} ================================ The qualitative analysis of the new photometric data (Fig. \[fig:folded\]) leads to several conclusions. (1) : The Min I is deeper than Min II. This means that the secondary’s temperature is lower than the primary’s temperature. (2) : The light maxima are not equal. This O’Connell effect implies presence of surface temperature spot(s). (3) : The Max I appears at the expected phase 0.25 while the phase of Max II is around 0.78. As a result the second half of the light curves is quite distorted. Similar asymmetry is visible also on the NSVS light curve (Fig. \[fig:nsvs\]) of the star almost 10 years earlier, i.e. this distortion is possibly permanent. We noted that the shape of the light curve of GSC 2314-0530 at phase range 0.5–0.8 resembles at some degree that of the cataclysmic stars with their peculiar standstills causing delay of the light increasing after the light minimum. (4) : The $V-I$ light curve of GSC 2314-0530 (Fig. \[fig:V-I\]) clearly reveals that the system becomes redder after the two eclipses and bluer after the two quadratures. The phases of the extrema of the $V-I$ light curve have around 0.05 phase delays in respect to those of the light curves **except for** the second maximum of $V-I$ which delay is more than 0.10. (5) : We observed several flares of GSC 2314-0530 (Fig. \[fig:m/HJD\]) resembling those of UV Ceti stars (see more in Section \[sec:activity\]). ![$V-I$ light curve of GSC 2314-0530 from synchronous $VRI$ observation with 60-cm telescope[]{data-label="fig:V-I"}](fig9.eps){width="0.75\columnwidth"} In order to determine the global parameters of GSC 2314-0530 we modeled our $VRI$ folded curves simultaneously using the software PHOEBE [@prsa05] by the following procedure. (a) : We fixed the mass ratio $q=0.519$ from our radial velocity solution. (b) : The obtained components of the heliocentric space velocity $U=-23$ km/s, $V=-44$ km/s and $W=-12$ km/s allow us to assume solar metalicity for the emission of GSC 2314-0530 [@leggett00]. (c) : We adopted coefficients of gravity brightening $g_1=g_2=0.32$ and reflection $A_1=A_2=0.5$ (appropriate for late stars) while the limb-darkening coefficients for each star and each color were taken from the tables of @vanhamme93. (d) : Taking into account that $E(V-I)=0.03$ mag in the GSC 2314-0530’ direction [@schlegel98] we obtained its de-reddened color index $(V-I)_{0}=2.35$ mag. According to table 2 of @vandenberg03 this out-of-eclipse color index corresponds to mean temperature of the binary $T_{\rm m}=3560$ K. It should be noted that the index $B-V=1.18$ mag of GSC 2314-0530 corresponds to mean temperature around 4400 K, i.e. 840 K higher than that obtained by the $V-I$ index. This is a new confirmation of the conclusion that the majority of the dMe stars have $B-V$ colors too blue for their $V-I$ colors [@stauffer86]. Our result also shows that the temperature difference obtained by the two color indices ($V-I$ and $B-V$) is higher than 200–300 K [@ribas03; @maceroni04] and can reach 800 K. (e) : At the first stage we fixed $T_{1}=3700$ K (taking into account that the temperature of the primary component $T_{1}$ is higher than $T_{\rm m}$) and varied the secondary’ temperature $T_{2}$, the orbital inclination $i$ and the potentials $\Omega_{1,2}$. In order to reproduce the O’Connell effect and light curve distortions we had to add two cool spots on the primary’s surface and to vary their parameters: longitude $\lambda$, latitude $\beta$, angular size $\alpha$ and temperature $T_{\rm {sp}}$. Moreover, in order to get a good simultaneous fit for the three colors $VRI$ by the same stellar and spot parameters we added a third light $L_3$ which contributes differently to the different colors. We consider the last supposition as artificial step to compensate the peculiar energy distribution of the dM stars that appear especially faint in the $V$ band probably to the big TiO absorption as well as to the big contribution of the spots. (f) : After getting a good fit of our $VRI$ photometric data we began to vary also the primary’s temperature. As a result we obtained the best light curve solution which parameters are given in Table \[tab:lightsolution\]. The respective synthetic $VRI$ light curves are shown in Fig. \[fig:folded\] as gray lines. They coincide very well with the observational data at all phases except for the flares. ----------------------- ------- ------- ------------ $ i $ \[\] 72.5 $\pm0.1$ $ T_1 $ \[K\] 3735 $\pm10$ $ T_2 $ \[K\] 3106 $\pm10$ $\Omega_1$ 2.944 $\pm0.002$ $\Omega_2$ 3.545 $\pm0.009$ $\lambda_{\rm {Sp1}}$ \[\] 147 $\pm5$ $\beta_{\rm {Sp1}}$ \[\] 70 $\pm10$ $\alpha_{\rm {Sp1}}$ \[\] 20 $\pm1$ $T_{\rm {Sp1}}$ \[K\] 3175 $\pm50$ $\lambda_{\rm {Sp2}}$ \[\] 195 $\pm5$ $\beta_{\rm {Sp2}}$ \[\] 75 $\pm10$ $\alpha_{\rm {Sp2}}$ \[\] 8 $\pm1$ $T_{\rm {Sp2}}$ \[K\] 3175 $\pm50$ $L_3$(V) 0.171 $\pm0.003$ $L_3$(R) 0.222 $\pm0.002$ $L_3$(I) 0.298 $\pm0.002$ ----------------------- ------- ------- ------------ : Best light curve solution from [phoebe]{}[]{data-label="tab:lightsolution"} ![3D model of GSC 2314-0530 at phase 0.75[]{data-label="fig:3D-config"}](fig10.eps){width="0.95\columnwidth"} The obtained potentials correspond to relative mean stellar radii $r_{1}=0.431$ and $r_{2}=0.228$ revealing that the primary component almost fills-in its Roche lobe (Fig. \[fig:3D-config\]). Global parameters of GSC 2314-0530 {#sec:global} ================================== Using the photometric value of the orbital inclination $i=72.5\degr$ we determined consecutively the following global parameters of GSC 2314-0530: (a) : orbital velocities of the two components $V_{1}=115.1\pm3.4$ km s$^{-1}$, $V_{2}=221.6\pm6.1$ km s$^{-1}$; (b) : orbital separation $a=1.28\pm0.04$ R$_{\sun}$; (c) : masses of the components $M_{1}=0.51\pm0.02$ M$_{\sun}$ and $M_{2}=0.26\pm0.02$ M$_{\sun}$; (d) : absolute (mean) radii of the components $R_{1}=0.55\pm0.01$ R$_{\sun}$ and $R_{2}=0.29\pm0.01$ R$_{\sun}$; (e) : surface gravity $\log g_1 = 4.68 $ and $\log g_2 = 4.95 $; (f) : stellar luminosities $L_{1}=0.053\pm0.002$ L$_{\sun}$ and $L_{2}=0.0070\pm0.0006$ L$_{\sun}$; (g) : bolometric absolute magnitudes of the components (using $M^{\sun}_{\rm {bol}}=4.72$) $M_{\rm {bol1}}=7.91\pm0.04$ mag and $M_{\rm {bol2}}=10.11\pm0.09$ mag as well as bolometric absolute magnitude of the binary $M_{\rm {bol}}(\rm total)=7.77\pm0.05$ mag; (h) : absolute $V$ magnitude of the binary $M_{V}(\rm total)=9.5\pm0.05$ mag (using $BC_{V}=-1.73$ corresponding to $T_{\rm m}$ from table 2 of @vandenberg03); (i) : distance to the binary $d=59\pm2$ pc. It should be noted that while the masses and radii of the components were directly determined, their temperatures and absolute magnitudes required external calibrations which are poorly known for the late stars. We calculated the equatorial velocities of the components by measuring the rotation broadening of their H$\alpha$ lines (using $i=72.5\degr$). The obtained values $V_{\rm {rot1}}=145\pm15$ km s$^{-1}$ and $V_{\rm {rot2}}=69\pm15$ km s$^{-1}$ reveal that the components of GSC 2314-0530 are quite fast rotators (see Table \[tab:activity\]). Thus our target confirms the conclusion of @stauffer86 that the stars with larger velocities have centrally peaked H$\alpha$ emission while the slower rotators have centrally reversed profiles as well as the conclusion of @worden81 that stars with centrally peaked H$\alpha$ emission profiles belong to short-period binaries. ------------------------- ------- ------- ---------------- ---------------- ---------------- ------ ------ --------- --------------------- ---------------- -------- ------ ------ Name $P$ $T$ $M$ $R$ $L$ $i$ $q$ $V-I$ $M_{\mathrm {bol}}$ $a$ $d$ Type Ref. \[d\] \[K\] \[M$_{\sun}$\] \[R$_{\sun}$\] \[L$_{\sun}$\] \[\] \[mag\] \[mag\] \[R$_{\sun}$\] \[pc\] CU Cnc=GJ 2069A 2.77 3160 0.43 0.43 0.016 86 0.92 2.80 9.19 0.92 12.8 D (1) 3125 0.40 0.39 0.013 9.45 2MASS J01542930+0053266 2.64 3700 0.66 0.64 0.069 86 0.95 7.62 8.70 623 D (2) 3300 0.62 0.61 0.039 8.24 NSVS 06507557 0.51 3960 0.65 0.60 0.079 83 0.42 2.13 7.48 2.65 111 D (3) 3360 0.28 0.44 0.022 8.86 NSVS 07394765 2.26 3170 0.56 0.58 0.030 84 1.16 8.52 2.60 D (4) 3860 0.65 0.69 0.095 7.27 NSVS 07453183 0.37 3340 0.68 0.72 0.060 89 1.07 1.40 7.77 7.75 D (4) 3570 0.73 0.79 0.090 7.33 UNSW-TR-2 2.11 3870 0.53 0.64 0.082 83 0.95 7.43 7.05 169 D (5) 3845 0.51 0.61 0.073 7.56 CM Dra 1.27 3150 0.23 0.25 0.005 90 0.93 10.47 3.75 D (6) 3125 0.21 0.23 0.004 10.71 TrES HerO-07621 1.12 3500 0.49 0.45 0.027 83 0.95 8.64 2.25 118 D (7) 3400 0.49 0.45 0.024 8.77 YY Gem 0.81 3820 0.60 0.62 0.070 86 1.00 1.92 7.57 3.87 D (8) 3820 0.60 0.62 0.070 7.57 GJ 3226 0.77 3313 0.38 0.37 0.016 83 0.75 2.73 9.20 3.08 42 D (9) 3247 0.28 0.32 0.009 9.83 2MASS 04463285+1901432 0.62 3320 0.47 0.56 0.034 81 0.41 2.59 8.39 2.66 540 D (10) 2910 0.19 0.21 0.003 11.03 V405 And 0.496 4050 0.49 0.78 0.147 66 0.98 6.80 2.25 D (11) 3000 0.21 0.23 0.004 10.71 GU Boo 0.49 3920 0.61 0.62 0.082 88 0.98 1.90 7.43 2.79 100 D (12) 3810 0.60 0.62 0.073 7.60 SDSS MEB-1 0.41 3320 0.27 0.27 0.008 85 0.98 9.96 1.85 D (13) 3300 0.24 0.25 0.007 10.11 NSVS 01031772 0.37 3615 0.54 0.53 0.043 86 0.92 8.08 2.20 40 D (14) 3513 0.50 0.51 0.036 8.27 OGLE BW3 V38 0.198 3500 0.44 0.51 0.035 86 0.95 2.45 8.39 1.35 400 SD (15) 3450 0.41 0.44 0.025 8.78 GSC 2314-0530 0.192 3735 0.51 0.55 0.053 72 0.52 2.34 7.91 1.28 59 SD (16) 3106 0.26 0.29 0.007 10.11 ------------------------- ------- ------- ---------------- ---------------- ---------------- ------ ------ --------- --------------------- ---------------- -------- ------ ------ References: (1) Ribas 2003, Delfosse et al. 1999; (2) Becker et al. 2008; (3) Cakirli $\&$ Ibanoglu 2009; (4) Coughlin $\&$ Shaw 2007; (5) Young et al. 2006; (6) Metcalfe et al. 1996; (7) Creevey et al. 2005; (8) Bopp 1974, Torres $\&$ Ribas 2002; (9) Irwin et al. 2009; (10) Hebb et al. 2006; (11) Vida et al. 2008; (12) Lopez-Morales $\&$ Ribas 2005; (13) Blake et al. 2008; (14) Lopez-Morales et al. (2006); (15) Maceroni $\&$ Montalban (2004); (16) this paper Some of the determined global parameters of GSC 2314-0530 together with those of the other known binaries with low-mass dM components are given in Table \[tab:stars\] which columns are: star name; period $P$ in days; temperatures $T$ of the components; masses $M$, radii $R$ and luminosities $L$ of the components in solar units; orbital inclination $i$ in degrees; mass ratio $q$; color index $V-I$ of the binary; bolometric absolute magnitudes $M_{\rm {bol}}$ of the components; orbital separation $a$ in solar radii; distance $d$ in pc; type of binary configuration (D – detached, SD – semidetached); references. Figure \[fig:relations\] shows the empirical diagrams mass-$M_{\rm {bol}}$, mass-radius and mass-temperature for the low-mass stars from Table \[tab:stars\] (total number 34). They occupy relative narrow bands on these diagrams. This means that the luminosities, radii and temperatures of these stars depend on their masses. These statistical relations can be described by the following formulas: ![Empirical relations mass-$M_{\rm {bol}}$, mass-radius and mass-temperature for the low-mass stars from Table \[tab:stars\] and their fits. The locus of the primary stars are signed by diamonds, those of the secondaries – by triangles, while those of our star – by large filled symbols.[]{data-label="fig:relations"}](fig11.eps){width="0.85\columnwidth"} $$\begin{aligned} M_{\rm {bol}} & = & 13.0 - 13.4 \times M + 7.7 \times M^2 \\ \nonumber R & = & 0.019 + 1.002 \times M \\ \nonumber T & = & 2983 + 396 \times M + 1333 \times M^2\end{aligned}$$ We assume that the bigger scatter of the mass-temperature diagram is due mainly to the weakly established calibration $T$/$(V-I)$ for the late low-mass stars. Moreover, some star temperatures probably have been determined without taking into account the reddening. Activity of GSC 2314-0530 {#sec:activity} ========================= The manifestations of stellar activity as H$\alpha$ emission, spots, flares, etc., are consequences of magnetic fields. It is assumed that the fully-convective late stars have strong, long-lasting, magnetic field. According to @mullan01 the larger radii and lower temperatures of dM stars can be explained by the presence of strong magnetic fields and their activity is at the saturation limit. Perhaps the significant spot coverage decreases the photospheric temperature which the star compensates by increasing its radius to conserve the total radiative flux. Surface spots ------------- The photospheric activity of the late stars is demonstrated mainly by O’Connell effect and distorted light curves. They can be reproduced by surface temperature inhomogeneities (spots). It is reasonable to assume existence of cool spots by analogy with our Sun. Usually they are put on the primary star although the same effect can be reached by spots on the secondary but then the spots should be larger and/or cooler. There are also fits of the light curves of late binaries with bright spots [@torres02; @maceroni94]. These are interpreted by uniform distribution of dark spots covering however most of stellar surface except for a spot-free area, i.e. “bright spots” represent the true photosphere. The light curves of all binaries with low-mass dM components from Table \[tab:stars\] are distorted and they have been reproduced by large cool spots which angular radii reach up to 80$\degr$. The distorted light curves of GSC 2314-0530 were reproduced by two cool spots on the primary component (see their parameters in Table \[tab:lightsolution\]) covering 3.5 per cent of its surface. The fact that the shape of the light curve distortions of GSC 2314-0530 remains the same almost 10 years means that the main (larger) spot visible at phase 0.6 presents long-lived active region on the primary surface. H$\alpha$ emission ------------------ The $EW$ of the H$\alpha$ line is an useful indicator of chromospheric activity for M dwarfs because those stars are much brighter at 6500 ${\rm{\AA}}$ than at 3900 ${\rm{\AA}}$. @stauffer86 divided dM into 4 subsets ordered by chromospheric activity. The least chromospheric active dM have weak H$\alpha$ absorption line. As the chromosphere increases the $EW$ of the H$\alpha$ absorption first increases, then decreases and finally H$\alpha$ goes into the emission. Table \[tab:radvel\] presents the orbital variations of the $EW$ of the total H$\alpha$ emission of GSC 2314-0530. Although it seemed to change irregularly in the range 3.6-6.6 ${\rm{\AA}}$ during the cycle we noted a trend of the $EW$ to be smaller around the first quadrature than around the second quadrature. The exceptions from this trend are the big $EW$ values of the only two spectra from 2010 January 01 at phases 0.23 and 0.31. They may due to flare event. Such a supposition is reasonable because two of the observed flares are around the first quadrature (see Table \[tab:flares\]). The foregoing trend of the H$\alpha$ emission is opposite to that of total light of GSC 2314-0530 that is bigger at the first quadrature than at the second one. Such an anti-correlation is typical for the chromospherically active stars of types RS CVn and BY Dra. --------------- ------------------ ------------------ ------------------ -------- Star $V_{\rm {rot1}}$ $V_{\rm {rot2}}$ $EW$ Flares \[km s$^{-1}$\] \[km s$^{-1}$\] \[${\rm{\AA}}$\] CM Dra em Y CU Cnc 4 V405 And Y GU Boo 64 64 1.7 YY Gem 37 37 2 Y NSVS 06507557 59 43 \[-3,+2\] NSVS 01031772 72 70 BW3 V38 131 113 5.4 GJ3236 25 19 GSC 2314-0530 145 69 $\leq$6.6 Y --------------- ------------------ ------------------ ------------------ -------- : Activity of low-mass dM stars[]{data-label="tab:activity"} Table \[tab:activity\] presents the $EW$ of the H$\alpha$ emission of some binaries with low-mass dM components from Table \[tab:stars\] at normal state (out of flare). The comparison reveals the strong H$\alpha$ emission of GSC 2314-0530. This result is not surprising taking into account the low temperature and fast rotation of its components. The mean value $EW=5\ {\rm{\AA}}$ of the H$\alpha$ emission of GSC 2314-0530 is considerably smaller than that of the accreting pre-main-sequence dMe stars which H$\alpha$ emission has $EW > 10\ {\rm{\AA}}$. Flares ------ Flare activity is typical for the late stars. The last column of Table \[tab:activity\] shows those stars from our Table \[tab:stars\] in which some flares have been registered (signed by “Y”). During our observational runs we were witnesses of six flares of GSC 2314-0530 that revealed its high flare activity. The amplitudes $A$ and durations $\tau$ of the observed flares are given in Table \[tab:flares\]. It should be noted that 3 of the observed 6 flares occurred around the phase of maximum visibility 0.6 of the large, stable spot (Sp1). This implies correlation between the two signs of stellar activity: spots and flares. Both of them are appearances of the long-lived active area on the primary star. Besides the optical flares there is information about X-flares of GSC 2314-0530 [@fuhrmeister03]. -------------- ------------------- ------- -------- --------- --------- Date HJD$_{\rm {max}}$ Phase Filter $A$ $\tau$ $2455000 +$ \[mag\] \[min\] 2009 Oct. 26 126.49373 0.61 $V$ 0.022 4 2009 Nov. 13 149.23146 0.64 $I$ 0.085 22 2009 Nov. 13 149.26995 0.84 $I$ 0.027 13 2009 Nov. 13 149.41788 0.61 $R$ 0.085 19 2009 Nov. 13 149.55281 0.31 $R$ 0.015 9 2009 Nov. 20 156.48180 0.31 $V$ 0.092 25 -------------- ------------------- ------- -------- --------- --------- : Observed flares of GSC 2314-0530[]{data-label="tab:flares"} Angular momentum ---------------- The small orbital angular momentum is characteristic feature of all short-period systems ranging from CVs to CB that seem to be old, being at later stages of the angular momentum loss evolution as a result of the period decrease. We calculated the orbital angular momentum of the target by the expression [@popper77] $$\label{equ:am} J_{\rm {rel}}=M_{1}M_{2} \left( \frac {P}{M_{1}+M_{2}} \right)^{1/3}$$ where $P$ is in days and $M_{i}$ are in solar units. The obtained value $\log J_{\rm {rel}}=-1.01$ of GSC 2314-0530 is considerably smaller than those of the RS CVn binaries and detached systems which have $\log J_{\rm {rel}}\geq +0.08$. The orbital angular momentum of GSC 2314-0530 is smaller even than those of the contact systems which have $\log J_{\rm {rel}} \geq -0.5$. It is bigger only than those of the short-period CVs of SU UMa type. The small orbital angular momentum of GSC 2314-0530 implies existence of past episode of angular momentum loss during the binary evolution. It means also that GSC 2314-0530 is not pre-MS object. This conclusion is supported by the values of $\log g$ of its components. X-ray emission -------------- The X-ray emission of the stellar coronae are directly related to the presence of magnetic fields and consequently gives information about the efficiency of the stellar dynamo. @rucinski84 established that the X-ray luminosity decreased for later M stars while the ratio $L_{X}/L_{bol}$ did not change significantly from M0 to M6. As a result he proposed the ratio $L_{X}/L_{bol}$ as most relevant measure of activity of M dwarfs. @vilhu87 found that the upper boundary of $L_{X}/L_{bol}$ for late M stars is $\sim 10^{-3}$ . Besides all indicators of stellar activity in the optical (surface inhomogeneities, emission lines, flares) the star GSC 2314-0530 shows also X-ray emission (it is identified as [*ROSAT*]{} X-ray source 1RXS J022050.7+332049) and X-ray flares. On the basis of the measured X-ray flux $F_{X}=4.266 \times 10^{-13}$ ergs cm$^{-2}$ $s^{-1}$ of GSC 2314-0530 at quiescence [@voges99; @schmitt95] and derived distance 59 pc we calculated its X-ray luminosity $L_{X}=1.68 \times 10^{29}$ ergs s$^{-1}$. This value is at the upper boundary $\log L_{X}\approx $ 29 for dM stars [@rosner81; @caillault86]. The value $f_{X}/f_{bol}= L_{X}/L_{bol}=0.7 \times 10^{-3}$ of GSC 2314-0530 is almost at the upper boundary of this ratio and considerably bigger than those of the M dwarfs studied by @rucinski84 and @caillault86. It is known that the activity and angular momentum loss tend to be saturated at high-rotation rates [@vilhu87]. Due to its short period and high activity GSC 2314-0530 is perhaps an example of such saturation. Is GSC 2314-0530 alone? {#sec:twin} ======================= Our observed field (Fig. \[fig:chart\]) contains the weak star USNO-B1 1233-0046425. We called it Twin due to the same tangential shift as our target star GSC 2314-0530. Table \[tab:colors\] presents the proper motion and the colors of Twin according to the catalogue NOMAD. USNO-B1 1233-0046425 has $V-I=3.02$ corresponding to temperature less than 3200 K. We suspect that our “twins” may form visual binary. The angular distance between them of 61 arcsec corresponds to linear separation around 3500 au for distance of 59 pc. Such a supposition is reasonable because it is known that the short-period close binaries often are triple systems [@pribulla06]. Particularly, the object TrES Her0-07621 from our Table \[tab:stars\] has a red stellar neighbor at a distance 8 arcsec with close proper motion [@creevey05]. The check of the supposition if Twin is physical companion of GSC 2314-0530 needs astrometric observations of the “twins”. Conclusions {#sec:conclusion} =========== The analysis of our photometric and spectral observations of the newly discovered eclipsing binary GSC 2314-0530 allows us to derive the following conclusions: \(1) This star is the shortest-period binary with dM components which period is below the short-period limit. \(2) By simultaneous radial velocity solution and light curve solution we determined the global parameters of GSC 2314-0530: inclination $i=72.5\degr$; orbital separation $a=1.28$ R$_{\sun}$; masses $M_{1}=0.51$ M$_{\sun}$ and $M_{2}=0.26$ M$_{\sun}$; radii $R_{1}=0.55$ R$_{\sun}$ and $R_{2}=0.29$ R$_{\sun}$; temperatures $T_{1}=3735$ K and $T_{2}=3106$ K; luminosities $L_{1}=0.053$ L$_{\sun}$ and $L_{2}=0.007$ L$_{\sun}$; distance $d=59$ pc. \(3) We derived empirical relations mass–$M_{\rm {bol}}$, mass–radius and mass–temperature on the basis of the parameters of known binaries with low-mass dM components. \(4) The distorted light curve of GSC 2314-0530 were reproduced by two cool spots on the primary component. The next sign of the activity of GSC 2314-0530 is the strong H$\alpha$ emission of its components. Moreover we registered 6 flares of GSC 2314-0530. Half of them occurred at the phases of maximum visibility of the larger stable cool spot on the primary. The analysis of all appearances of magnetic activity revealed existence of long-lived active area on the primary of GSC 2314-0530. The high activity of the target is natural consequence of the fast rotation and low temperatures of its components. Our study of the newly discovered short-period eclipsing binary GSC 2314-0530 presents a next small step toward understanding dMe stars and adds a new information to the poor statistic of the low-mass dM stars. Recently they became especially interesting as appropriate targets for planet searches due to the relative larger transit depths. Acknowledgments {#acknowledgments .unnumbered} =============== The research was supported partly by funds of projects DO 02-362 of the Bulgarian Scientific Foundation. This research make use of the SIMBAD and Vizier databases, operated at CDS, Strasbourg, France, and NASA’s Astrophysics Data System Abstract Service. The authors are very grateful to the anonymous referee for the valuable notes and advices. 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--- abstract: 'Structural fluctuations in the thermal equilibrium of the kinesin motor domain are studied using a lattice protein model with Gō interactions. By means of the multi-self-overlap ensemble (MSOE) Monte Carlo method and the principal component analysis (PCA), the free-energy landscape is obtained. It is shown that kinesins have two subdomains that exhibit partial folding/unfolding at functionally important regions: one is located around the nucleotide binding site and the other includes the main microtubule binding site. These subdomains are consistent with structural variability that was reported recently based on experimentally-obtained structures. On the other hand, such large structural fluctuations have not been captured by B-factor or normal mode analyses. Thus, they are beyond the elastic regime, and it is essential to take into account chain connectivity for studying the function of kinesins.' author: - | Hiroo Kenzaki$^{1 2 3}$, and Macoto Kikuchi$^{2 1 4}$\ $^1$ Department of Physics, Osaka University, Toyonaka 560-0043, Japan\ $^2$ Cybermedia Center, Osaka University, Toyonaka 560-0043, Japan\ $^3$ Core Research for Evolutional Science and Technology,\ Japan Science and Technology Agency, Nagoya, Japan\ $^4$ Core Research for Evolutional Science and Technology,\ Japan Science and Technology Agency, Suita, Japan title: 'Free-Energy Landscape of Kinesin by a Realistic Lattice Model' --- [**Present address**]{}: Department of Computational Science and Engineering, Nagoya University, Nagoya 464-8603, Japan [**E-mail address**]{}: kenzaki@tbp.cse.nagoya-u.ac.jp INTRODUCTION {#introduction .unnumbered} ============ Biomolecular motors are proteins that exhibit biolocomotion by converting chemical energy into mechanical energy [@mehta99; @vale00; @schliwa03]. A detailed mechanism of this energy conversion has been a long-standing problem. Unlike macroscopic motors, biomolecular motors operate under a strong influence of thermal fluctuations, since the chemical energy provided by ATP hydrolysis is of the same order as that of the thermal fluctuations. Kinesin, which is responsible for intracellular transport and cell division [@hirokawa98], moves stepwise in one direction along the microtubule with one ATP hydrolysis per step [@svoboda93; @schnitzer00; @taniguchi05]. Kinesin has attracted considerable attention and was studied intensively in recent years. Conventional kinesin functions as a dimer; it is widely accepted that kinesin walks hand-over-hand on a microtubule [@yildiz04]. The gliding movement of kinesin is believed to be realized by controlling the binding with and dissociation from the microtubule [@cross04]. Kinesin binds tightly to the microtubule in the nucleotide-free and ATP states, whereas it detaches from the microtubule in the ADP state [@romberg93]. The structures of kinesin and myosin share highly conserved regions known as switch I and switch II, named after their counterparts in G proteins [@vale96]; G proteins are signaling proteins that bind to GTP instead of ATP. The switch regions of the G proteins move during nucleotide hydrolysis and thereby control the binding with and detaching from a target protein [@vetter01]. As for kinesin, both the switch I and switch II regions have different conformations among different X-ray structures [@sack99; @kull02; @nitta04]; further, they show microtubule-dependent structural changes [@rice99]. This strongly suggests that kinesin also controls its affinity with the rail protein by structural transformations of the switch regions. It is commonly considered that functionally important fluctuations of a protein can be read from the B-factor of the X-ray crystal structure. For kinesin, however, it was reported that a large B-factor does not correspond to a (supposedly) functionally important fluctuation [@sack99]. In particular, it was pointed out that the fluctuations of the switch regions are largely underevaluated. Moreover, a normal mode analysis based on the elastic-network model has revealed that small fluctuations in the elastic regime are insufficient to capture the conformational change of kinesin in contrast to the other molecular motors [@zheng03]. These results indicate that larger-scale structural fluctuations beyond the elastic regime are relevant to the function of kinesins. In order to investigate such large structural fluctuations, we follow a methodology of folding study in this work. The energy landscape theory of protein folding has been accepted widely in the last decade; this theory states that proteins have a funnel-like energy landscape toward the native structure [@bryngelson95; @onuchic97]. This new view of the protein folding has resulted in a revival of the Gō model [@taketomi75; @go83], which was originally introduced for explaining the mechanism of two-state folding transitions of small globular proteins. Recent studies of protein folding have shown that Gō-like models (namely, models constructed with the same design as the Gō model) are applicable far beyond the original scope; they can describe the folding of some proteins with intermediate states [@zhou97; @clementi00; @koga01; @karanicolas03; @levy04; @das05; @kenzaki06; @kenzaki06b]. In this work, we examine the large structural fluctuations of kinesins by applying a Gō-like model for calculating a free-energy landscape. We use a realistic lattice protein model [@kenzaki06; @kenzaki06b], because their relatively small conformation spaces are fairly favorable for numerical simulations. MODEL AND METHODS {#model-and-methods .unnumbered} ================= In the lattice model of its simplest form, each amino acid residue is expressed as a bead that is allowed to move on a simple cubic lattice, and the polypeptide chain is represented as a string of connected beads with a unit bond length [@sali94; @dill95]. If a pair of unconnected beads occupies neighboring sites, the pair is called to form a contact and the interaction is applied to it. Although such a lattice model can reproduce various universal properties of the protein folding, it is too simple to represent the chain geometry of real proteins; thus, we need to introduce a lattice model of a higher resolution for the present purpose. Some realistic lattice models that can represent flexible protein structures have been proposed [@kolinski96; @kolinski04; @hao96]. In the present work, we use a 210-211 hybrid lattice model in which C$^{\alpha}$ atoms of amino acids are again located on a simple cubic lattice; however, consecutive atoms are connected by vectors of the type (2,1,0) or (2,1,1) and all their possible permutations [@kenzaki06; @kenzaki06b]. The length of (1,0,0) vector is set to $1.62$ Å, and the length of a C$^{\alpha}-$C$^{\alpha}$ bond is then $3.63$ Å or $3.97$ Å. In order to express the excluded volume effect, we consider that each amino acid occupies seven lattice sites, that is, a center site at the bead position and its nearest neighbors. We introduce Gō interactions between amino acids, which act only on pairs of amino acids that form native contacts. We call such pair a “native pair” from now on. The X-ray crystal structure of the kinesin motor domain of [*Homo sapiens*]{} (PDB ID code 1bg2) is used as a reference of the native structure (Figure 1(a)) [@kull96]. This crystal structure is of the ADP-binding state (note that the ADP molecule is not included explicitly in the following simulations). The native structure of the lattice protein model is defined by fitting to the C$^{\alpha}$ trace of this crystal structure. The accuracy of the fitting in terms of the root mean square deviation (rmsd) is $0.92$ Å(Figure 1(b)). The Gō potential between the $i$ th and $j$ th amino acids is defined as follows: $$\begin{aligned} V_{g\bar{o}} & = & - \sum_{j-i>2} \varepsilon_{i,j} C_{i,j} \Delta (r_{i,j}, r_{i,j}^{nat}),\\ \Delta(x,y) & = & \left\{ \begin{array}{lc} 1 & |x^2 - y^2| \le W\\ 0, & \textrm{otherwise} \end{array} \right.\end{aligned}$$ where $r_{i,j}$ is the distance between C$^{\alpha}$ atoms, $r_{i,j}^{nat}$ is their native distance, $\varepsilon_{i,j}$ is the interaction strength between residue $i$ and $j$, and $C_{i,j}$ is the constant that equals $1$ or $0$ depending on whether the pair is a native pair or not. $W$ is the width of the Gō potential. We add a local potential to maintain a protein-like local structure. Here, we introduce a harmonic potential between the $i$ th and ($i+2$) th residues to bias the bond angles toward the native structure. The local potential is $$\begin{aligned} V_{local} & = & \frac{K_{b}}{2} \sum_{i=1} (r_{i,i+2} - r_{i,i+2}^{nat})^2\end{aligned}$$ where $K_{b}$ is the strength of the local interaction. We use $W = 2$ and $K_b = 1$. The interaction strength $\varepsilon_{i,j}$ takes either the uniform value $\varepsilon_{i,j} = 1$ or normalized Miyazawa-Jernigan (MJ) contact-energy parameters [@miyazawa96]. We compute the density of states and several physical quantities in thermal equilibrium by using a Monte Carlo scheme called multi-self-overlap ensemble (MSOE) Monte Carlo method [@iba98; @chikenji99]. In this method, the self-avoiding condition is systematically weakened, and a flat histogram both in energy and degree of the excluded volume is obtained by a bivariate extension of the multicanonical ensemble Monte Carlo method. By calculating the physical quantities only for the self-avoiding conformations, correct canonical averages are obtained. The MSOE method gives a high performance, especially for long proteins, in comparison with the standard multicanonical ensemble for this realistic lattice model. It should be noted that both lattice model and multicanonical methods are not suitable for studying dynamics of protein. In the present work we focus on equilibrium properties, the free-energy landscape in particular, which can be treated by the lattice model as long as the conformations are realistically expressed. RESULTS {#results .unnumbered} ======= Simulations are prepared for the following three models: (i) C$^{\alpha}$($7.5$): A pair of amino acids is considered to be a native pair if the distance between their C$^{\alpha}$ atoms is less than $7.5$ Å in the X-ray crystal structure. (ii) All($4.5$): A pair of amino acids is considered to be a native pair if the minimum distance between their heavy atoms is less than $4.5$ Å in the X-ray crystal structure. (iii) All($4.5$)/MJ: Definition of the native pair is same as All($4.5$), but heterogeneous interactions based on normalized MJ contact-energy parameters [@miyazawa96] is used as the Gō interaction. The density of state of kinesin across a wide energy range including both the native state and unfolded state are calculated by MSOE. Figure 2(a) shows the entropy of the kinesin motor domain. The figure represents the folding funnel if rotated by $90$ degrees. Free-Energy Landscape and Fluctuation of C$^{\alpha}$($7.5$) Model {#free-energy-landscape-and-fluctuation-of-calpha7.5-model .unnumbered} ------------------------------------------------------------------ We first focus on the C$^{\alpha}$($7.5$) model. The energy shown in Figure 2(b) (solid line) jumps at two temperatures. Both the jumps correspond to cooperative transitions. The gyration radius significantly changes at the higher transition point, and slightly at the lower transition point (Figure 2(c)). As shown below, the hydrophobic core is broken and kinesin unfolds completely above the higher transition point. Thus, the higher transition is the main folding/unfolding transition and is considered to be irrelevant to the stepping motion. On the other hand, folding/unfolding of local structures takes place at the lower transition point with the hydrophobic core maintained folded. In order to examine each transition in detail, we carry out the principal component analysis (PCA) for contact formation of the native pairs. The variable $x_k$ is introduced, where the index $k$ runs all the native pairs; $x_k = 1$ if $k$-th native pair is in contact and 0 otherwise. The variance-covariance matrix $$M_{kl} = \langle x_kx_l \rangle - \langle x_k \rangle \langle x_l\rangle$$ is calculated, where $\langle \rangle$ indicates thermal average. Then $M_{kl}$ is diagonalized to yield the eigenvalues and eigenvectors. The eigenvectors are called the principal components (PC); PC expresses which contacts are formed/unformed cooperatively, and PC with a large eigenvalue represents an important structural fluctuation. We calculate the average value of amplitudes in PC of all the contacts belonging to each residue. Residues are regarded to constitute a fluctuation unit, if they have relatively large averages in the same PC. At the higher transition point, only one eigenvalue is large and the fluctuation unit is located at the hydrophobic core \[central anti-parallel $\beta$-sheet (Figure 1(a))\]. The eigenvalue distribution at the lower transition point is shown in Figure 3(a). There are two large eigenvalues. The fluctuation unit of the first eigenvector (PC1) is composed of helix 4, loop 12, helix 5, helix 6, and strands 5a/b (insertion of strand 5), and that of the second eigenvector (PC2) is composed of helix 3 and helix 2 \[Figure 3(b)\] We found that structural fluctuations of these fluctuation units actually are partial folding and unfolding, because the components in the principal eigenvectors have the same sign. The former region overlaps with the switch II region and the latter with the switch I region. Hence, we call these regions SWII subdomain and SWI subdomain, respectively. Figure 3(c) shows the free-energy landscape in two-dimensional space spanned by the first two principal components PC1 and PC2 at the transition point. Four states are distinguishable as four minima in the free-energy landscape. They are classified according to the formation of SWI and SWII: both SWI and SWII are unfolded (UU state); only SWI is formed (FU state); only SWII is formed (UF state); both the subdomains are formed (FF state). The two subdomains behave as mutually independent folding units because they belong to different eigenvectors. In other words, partial folding/unfolding transitions of FF $\leftrightarrow$ FU $\leftrightarrow$ UU and FF $\leftrightarrow$ UF $\leftrightarrow$ UU occur independently. It should be noted that the FU $\leftrightarrow$ UF transition is not realized because of the free-energy barrier. If these fluctuation units just reflect intrinsic properties of the lower transition point and do not have any relevance for other temperature, they may not be of interest from the viewpoint of function of kinesin. In order to demonstrate that this is not the case and the fluctuation units are rather robust, we show the energy dependence of the mean contact probability of the native contacts for each residue in Figure 3(d). Five states are distinguishable, each of which corresponds to a different energy range. The highest energy range corresponds to the completely unfolded state, so that it is not relevant to the lower transition. The $\beta$-sheet hydrophobic core is formed in the other four states. They actually correspond to UU, FU, UF and FF states described above and appear in Figure 3(d) in this order from a higher energy to a lower energy. Thus, we can safely say that SWI and SWII subdomains behave as two independent fluctuation units as long as the hydrophobic core is formed. Units of Fluctuation Are Robust {#units-of-fluctuation-are-robust .unnumbered} ------------------------------- Let us now turn to the All($4.5$) and All($4.5$)/MJ models. As is seen in Figure 2(b), only a single jump is observed in the energy. The protein is completely unfolded at temperatures above this jump. Hence, in these cases, the folding/unfolding of local structures are not separated from that of the hydrophobic core with respect to the temperature. Thus, PCA is not suitable to describe partial folding/unfolding. Nevertheless, we can discuss the folding unit from the energy dependence of the mean contact probability of the native contacts. As is seen in Figures 4(a) and 4(b), five states - unfold, UU, UF, FU, and FF - are again distinguishable. In other words, they are separated in energy similar to the C$^{\alpha}$($7.5$) models. Thus, two subdomains - SWI and SWII - are also included as the folding units in both the All($4.5$) and All($4.5$)/MJ models. The structure of the boundary of the subdomains is slightly different; strand 1 is included in the hydrophobic core in the C$^{\alpha}$($7.5$) model, but in the SWII subdomain in the All($4.5$ ) and All($4.5$)/MJ models. Strand 5a/b is included in the SWII subdomain in the C$^{\alpha}$($7.5$) model, but not in the All($4.5$) and All($4.5$)/MJ models. In brief, although the number of transitions is different for different models, the overall structure of folding units (subdomains) is robust. DISCUSSION {#discussion .unnumbered} ========== The simulation result indicate that the kinesin motor domain has two subdomains other than the hydrophobic core: SWI and SWII, which overlap with the switch I and switch II regions, respectively. These two subdomains exhibit structural fluctuations, which actually are partial folding/unfolding, and nearly independent when the nucleotide is absent. The largest fluctuation is localized at the SWII subdomain. A question may occur whether these subdomains are relevant to the real kinesin protein or not. Quite recently, Grant et al. studied structural variability of kinesin using 37 available structures in the protein data bank (PDB) [@grant07]. The PCA revealed that there are two significant principal components. The first principal component describes the concerted displacement located at helix 4, loop 12, helix 5, and loop 13. This subdomain corresponds to the central part of the SWII subdomain we found. They suggested that the second principal component is localized at helix 3 and the vicinity of loop 6 and loop 10. Seeing their result, we found that the component has a large value at helix 2, which actually is located in the vicinity of loop 6. Although they did not call this region as subdomain explicitly, the region and the SWI subdomain overlap in large part with each other. Thus, our simulation result is consistent to their analysis. This indicates that the SWI and SWII subdomains successfully captured the functionally important fluctuations. These two subdomains are considered to play important roles in the stepping motion, based on the following experimental facts: (i) Helix 4, loop 12, and helix 5 belonging to the SWII subdomain are known to undergo a large conformational change during the ATP hydrolysis cycle [@nitta04]; this region also includes the main microtubule binding site [@woehlke97; @hoenger00]. Moreover, the SWII subdomain interacts directly with the neck linker, which connects the motor domain and neck region (the neck linker is not included in our simulation), because the neck linker is adjacent in sequence to helix 6 (included in the SWII subdomain). (ii) Helix 3 in the SWI subdomain also undergoes a large conformational change during the ATP hydrolysis cycle [@naber03]. In addition, judging from the number of included residues, we may safely assume that a time scale of the conformational changes of these two subdomains of real kinesin is approximately the same order as that of the stepping motion ($\sim$ msec). In contrast, it is unlikely that the hydrophobic core unfolds on the stepping motion, because the time scale of the core unfolding should be much longer ($\sim$ sec or longer). The present results suggest that conformational changes of these functionally important regions are realized by a partial unfolding-refolding. We showed that SWI and SWII behaves as independent folding unit in the absence of nucleotide. Binding to the nucleotide may induce coupling between these two folding units, which will be left for future study. The subdomain structure is largely determined by the topology of the native structure because three variations of coarse-grained Gō-like models show similar subdomain structures. It has been pointed out that the cooperativity of folding transition depends on the detail of the interaction within the framework of the Gō-like model, such as inclusion of many-body interaction [@chan04], interaction range [@kenzaki06], and so on. However, as we found in previous study subdomain structure is rather insensitive to such detail [@kenzaki06]. We note that the large fluctuations of these subdomains are totally different from the small fluctuations observed by, for example, the B-factor in the X-ray crystal structure and elastic network model. In fact, the B-factor of the subdomains is not particularly larger than that of other parts of the motor domain. Thus, the large fluctuations described by the present model are beyond the elastic regime and could not be captured by the B-factor or elastic network model. It is essential to take into account the chain connectivity explicitly for studying the function of kinesins. Myosin motor domain is more than twice the size of kinesins, but their nucleotide binding regions and switch I/II regions have similar topology to those of kinesins. This indicates that the basic properties of the free-energy landscape of myosin may also be similar. Before closing the discussion, some comments should be made on related simulations. All-atom molecular dynamics simulations of the kinesin motor domain have been performed [@wriggers98; @minehardt01]. Wriggers et al. captured the nucleotide-dependent small fluctuations of the switch regions [@wriggers98], and Minehardt et al. simulated conformational change of the switch I region [@minehardt01]. Although, these results are in part consistent with the present result, these simulations are extremely short in time (order of nanosecond) as compared to the time scale of the stepping motion (order of millisecond), since the all atom simulation of a large protein as kinesin is still a hard task. Recently, Hyeon and Onuchic performed the simulation of the kinesin dimer bounded on the microtubule by the Gō-like model, and investigated the allosteric transition regulated by the inter-monomer strain through the neck-linker [@hyeon07]. In contrast, we focused on the structural fluctuations of a kinesin monomer in thermal equilibrium. We found that kinesin has two subdomains, which give rise to the conformational fluctuation of the functionally important regions. This result is supported by the recent analysis on experimental data [@grant07]. The coarse-grained Gō-like model made it possible to describe the large structural fluctuation that is relevant to the function. ACKNOWLEDGMENTS {#acknowledgments .unnumbered} =============== We thank G. Chikenji, M. Sasai and F. Takagi for fruitful discussions. 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Phys Rev Lett 1999;83:1886-1889. Grant BJ, McCammon JA, Caves LSD, Cross RA. Multivariate analysis of conserved sequence-structure relationships in kinesins: coupling of the active site and a tubulin-binding sub-domain. J Mol Biol 2007;368:1231-1248. Woehlke G, Ruby AK, Hart CL, Ly B, Hom-Booher N, Vale RD. Microtubule interaction site of the kinesin motor. Cell 1997;90:207-216. Hoenger A, Thormählen M, Diaz-Avalos R, Doerhoefer M, Goldie KN, Müller J, Mandelkow E. A new look at the microtubule binding patterns of dimeric kinesins. J Mol Biol 2000;297:1087-1103. Naber N, Minehardt TJ, Rice S, Chen X, Grammer J, Matuska, M Vale, RD, Kollman PA, Car R, Yount RG, Cooke R, Edward Pate E. Closing of the nucleotide pocket of kinesin-family motors upon binding to microtubules. Science 2003;300:798-801. Chan HS, Shimizu S, Kaya H. Cooperativity principles in protein folding. Methods Enzymol 2004;380:350-379. Wriggers W, Schulten K. Nucleotide-dependent movements of the kinesin motor domain predicted by simulated annealing. Biophys J. 1998;75:646-661. Minehardt TJ, Cooke R, Pate E, Kollman PA. Molecular dynamics study of the energetic, mechanistic, and structural implications of a closed phosphate tube in ncd. Biophys J. 2001;80:1151-1168. Hyeon C, Onuchic JN. Internal strain regulates the nucleotide binding site of the kinesin leading head. Proc Natl Acad Sci USA. 2007;104:2175-2180. ![ (a) The X-ray structure of kinesin motor domain (PDB ID code 1bg2). It consists of a central antiparallel $\beta$-sheet of eight strands, sandwiched between three $\alpha$-helices on either side. Orange: ADP, blue: helix 2 (91-122), strand 5 (141-144, 171-173), helix 3 (176-189) and switch I (198-203), red: switch II (231-236), loop 11 (238-254), helix 4 (257-269), loop 12 (272-280), and helix 5 (281-292), green: strand 1 (9-15), and helix 6 (306-320). This figure was prepared using the programs MOLSCRIPT [@kraulis91] and Raster3D [@merritt97]. (b) The superposition of native structure of the lattice protein model of kinesin (blue) and the C$^{\alpha}$ trace of the X-ray crystal structure (red). Accuracy of the fitting in terms of the root mean square deviation (rmsd) is $0.92$ Å.](fig1.eps) ![ (a) Relative entropy $S(E)$ of kinesin as a function of energy $E$. (b) The average energy $E$ as a function of temperature $T$. (c) The radius of gyration $R$ as a function of temperature $T$.](fig2.eps) ![ Free-energy landscape and structural fluctuation of C$^{\alpha}$(7.5) model. (a) Eigenvalue distribution of PCA at the lower transition point. (b) The first two principal components PC1 and PC2 at the lower transition point are shown. The residues belonging to the fluctuation unit of PC1 are shown in red and those of PC2 are blue. Black indicates unstable residues with low contact formation probabilities. ADP (not included in the simulation) is orange. (c) Free-energy landscape at the lower transition point in two dimensional space spanned by the first two principal components PC1 and PC2. (d) Energy dependence of the mean contact probability of the native contact for each residue.](fig3.eps) ![ Energy dependence of the mean contact probability of the native contact for each residue for (a) All($4.5$) and (b) All($4.5$)/MJ.](fig4.eps)
--- abstract: | We consider diffusion of vibrations in $3d$ harmonic lattices with strong force-constant disorder. Above some frequency $\omega_{\rm IR}$, corresponding to the Ioffe-Regel crossover, notion of phonons becomes ill defined. They cannot propagate through the system and transfer energy. Nevertheless most of the vibrations in this range are not localized. We show that they are similar to [*diffusons*]{} introduced by Allen, Feldman et al., Phil. Mag. B [**79**]{}, 1715 (1999) to describe heat transport in glasses. The crossover frequency $\omega_{\rm IR}$ is close to the position of the boson peak. Changing strength of disorder we can vary $\omega_{\rm IR}$ from zero value (when rigidity is zero and there are no phonons in the lattice) up to a typical frequency in the system. Above $\omega_{\rm IR}$ the energy in the lattice is transferred by means of diffusion of vibrational excitations. We calculated the diffusivity of the modes $D(\omega)$ using both the direct numerical solution of Newton equations and the formula of Edwards and Thouless. It is nearly a constant above $\omega_{\rm IR}$ and goes to zero at the localization threshold. We show that apart from the diffusion of energy, the diffusion of particle displacements in the lattice takes place as well. Above $\omega_{\rm IR}$ a displacement structure factor $S({\bf q}, \omega)$ coincides well with a structure factor of random walk on the lattice. As a result the vibrational line width $\Gamma(q)=D_u q^2$ where $D_u$ is a diffusion coefficient of particle displacements. Our findings may have important consequence for the interpretation of experimental data on inelastic x-ray scattering and mechanisms of heat transfer in glasses. author: - 'Y. M. Beltukov' - 'V. I. Kozub' - 'D. A. Parshin' title: 'The Ioffe-Regel criterion and diffusion of vibrations in random lattices' --- Introduction ============ Propagation of vibrational excitations in disordered systems is one of the advanced problems in condensed matter physics. In particular, transport mediated by these excitations is responsible for the thermal conductivity of amorphous dielectrics (glasses). However mechanisms of heat transfer in glasses above the plateau region are still poorly understood. At low temperatures below 1K the low frequency plane long wave acoustical phonons are well defined excitations which transfer the heat in glasses. At these temperatures the thermal conductivity $\varkappa(T)\propto T^2$ and is controlled by a resonant scattering of phonons on two-level systems (TLS) [@Hunklinger; @Phillips]. Between 4K and 20K the thermal conductivity $\varkappa(T)$ saturates and displays a well known plateau [@Zeller]. As was shown in [@Buchenau1] it can be explained by resonant scattering of phonons by quasilocal vibrations (QLV). The QLV, together with TLS and phonons are vibrational excitations responsible for many universal properties of glasses [@Parshin1]. However, above approximately 20K the thermal conductivity rises again and finally saturates on the level of one order of magnitude higher, at temperatures about several hundreds Kelvin [@CahillPohl]. As generally believed, the origin of this second rise of the thermal conductivity (above the plateau) is not related to phonons. It was established long ago [@BirchClark; @Kittel; @Graebner], that in this temperature (frequency) range the mean free path of phonons $l$ becomes of the order of their wave length $\lambda$ (or even smaller, of the order of interatomic distance). Correspondingly, the Ioffe-Regel criterion for phonons [@Ioffe] becomes violated. The existence of such crossover was confirmed by molecular dynamics calculations for some real and model glasses [@taraskin; @schober] and disordered lattices [@schirm; @taras]. In the regime of such strong scattering a standard concept of plane waves (phonons) with a well defined wave vector $\bf q$ becomes inapplicable. The question then arises: what physical mechanism is responsible for the heat transfer in glasses in this temperature range? The numerical simulations show that majority of the vibrational modes in the corresponding frequency range are not localized [@jin; @oligschleger; @taraskin2]. As was shown in [@Cahill1; @Cahill2; @Cahill3], a lower limit of the thermal conductivity of amorphous solids above 30 K can be correctly estimated within the framework of the Einstein’s model [@Einstein]. It was assumed that the mechanism of heat transport above the plateau is a random walk of thermal energy between clusters of neighboring atoms vibrating with random phases. In fact, a diffusion mechanism for the heat transfer in this temperature range was proposed. At the same time, delocalized vibrations in glasses of a new type, different from phonons, were introduced. They were called [*diffusons*]{} [@Nature1; @Nature2; @Nature3; @Nature4; @Nature5]. These are vibrations spreading through the system not ballistically, as phonons (on distances of the order of mean free path) but by means of diffusion. It is an important class of excitations which occupy in glasses the dominant part of the spectrum [@Nature5]. In these papers it was put forward the hypothesis that the boundary between phonons and diffusons is determined by the Ioffe-Regel criterion for phonons. Since diffusons are delocalized excitations, they may be responsible for the thermal conductivity of glasses above the plateau. The similar conclusion was made by the authors of [@strong1; @strong2]. They considered the case of strong scattering of phonons in disordered lattices with a significant fraction of randomly located missing sites, but which is still far from the percolation threshold. It was shown that, in contrast to the electronic case, the Ioffe-Regel criterion is inaccurate in the prediction of phonon localization. Instead of localization, the vibrational transport above the Ioffe-Regel threshold becomes diffusive with approximately constant energy diffusivity $D(\omega)$. The diffusivity was calculated by numerical solution of the Newton equations for particle displacements. Similar calculations but for real glasses were done in the papers [@sim; @similar] using molecular dynamics methods. The diffusons above the Ioffe-Regel crossover were identified also in granular jammed systems with repulsive forces between the particles [@jammed1; @jammed2]. They also have diffusivity which is independent of frequency $\omega$. It was calculated making use of the Kubo-Greenwood formula for the thermal conductivity derived in [@Nature2]. In jammed systems the Ioffe-Regel crossover frequency $\omega_{\rm IR}$ can vary. It is shifted to zero when the system approaches the jamming transition point and rigidity goes to zero. Therefore, as we believe, it is important to study properties of diffusons systematically in systems where they exist. They bring a new physics to our understanding of vibrational properties in strongly disordered systems and energy/heat transfer in glasses. To study these properties, we should have a model being sufficiently simple but still allowing to describe all of them. Since we consider harmonic models, the simplest but still rather general, are (scalar or vector) models where particles, placed in equilibrium positions, are connected by random elastic springs. The equilibrium positions can be taken on a lattice [@schirm; @taras; @bunde; @taraskin3], or randomly distributed in space [@parisi1; @grigera]. With some exceptions, there is no principal difference between these two cases because equilibrium positions do not enter to the dynamical matrix. The only important features are the type of disorder in elastic spring constants and the topology of the bonds. If all spring constants are positive, one can study different situations taking different distributions of random springs to explain existing experimental data [@bunde]. However a problem appears when some spring constants take negative values [@schirm; @taraskin3; @grigera]. Even if the number of negative springs and their absolute values are relatively small, in the absence of any correlation with positive springs they produce an inevitable mechanical instability in all infinite systems. Therefore, a situation of strong force-constant disorder when appreciable concentration of springs can take negative values comparable with positive ones, is not possible in these models. On the other hand, an inclusion of negative spring constants can considerably improve an agreement between theory and existing experimental data [@schirm; @taraskin3; @grigera]. One can solve this problem mathematically, using a stable random matrix approach [@PositiveDefine; @Chalker; @our1; @our2]. In this approach positive and negative springs are tangly correlated with each other. These correlations automatically guaranty the mechanical stability of the system independently of character and strength of the force constant disorder. In the present paper we are going to use this approach to investigate diffusion of vibrational excitations in disordered lattices with strong force-constant disorder. Some of our preliminary results have been presented in short form elsewhere [@our2]. The paper is organized as follows. In Section \[rma\] for the sake of clarity of further consideration, we outline the main properties of the model. We consider disordered lattices with strong force-constant disorder, described by stable positive definite random dynamical matrix $AA^T$ having positive eigenvalues only. Matrix $A$ is a random matrix (not necessary symmetric) built on simple cubic lattice, with statistically independent matrix elements between the nearest neighbors $A_{i\ne j}$, having zero mean $\left<A_{i\ne j}\right>=0$ and equal variance $\left<A_{i\ne j}^2\right>=V^2$. We show that the density of states $g(\omega)$ is not zero at $\omega=0$ and phonons cannot propagate through the lattice. Similarly to systems at jamming transition point, the rigidity of the lattice is also zero. However the physical reason is different. In our case it is due to high concentration of negative springs (about 45%) in the system what makes it extremely soft. The participation ratio $P(\omega)$ indicates that all modes with exception of high frequency part are delocalized. As it is shown by further investigation, all of them are diffusons. In Section \[phonons\] we consider slightly additively deformed dynamical matrix $AA^T+\mu M_0$ which has phonon-like excitations at small frequencies. Here positive definite matrix $M_0$ (random or non-random) is independent of $A$. $\mu$ is a parameter of the model which can vary in the interval $0\leqslant\mu < \infty$. Analyzing properties of this matrix, we calculate the Young modulus $E$, sound velocity, the density of states and participation ratio, the dynamical structure factor $S({\bf q}, \omega)$, the phonon dispersion law $\omega_{\bf q}$, and also their mean free path $l(\omega)$. Comparison of the later with phonon wave length $\lambda$ determines the Ioffe-Regel crossover frequency $\omega_{\rm IR}$. It goes to zero when $\mu\to 0$. We show that above $\omega_{\rm IR}$, phonons cease to exist. They are transformed to diffusons. In Section \[diff\] we consider properties of diffusons. The number of diffusing physical quantities coincides with the number of integrals of motion. In a closed free mechanical system there are two integrals of motion, momentum and energy. In Section \[diffmomentum\] we investigate diffusion of momentum. We show that for all masses equal, it is equivalent to the diffusion of particle displacements since center of inertia is conserved. The displacement structure factor $S({\bf q}, \omega)$ coincides well with the structure factor $S_{\rm rw}({\bf q}, \omega)$ of the random walk on a lattice. We introduce new additional diffusion coefficient $D_u$ which is [*diffusivity of particle displacements*]{}. It is different from the energy diffusivity $D(\omega)$ investigated in [@Nature1; @Nature2; @Nature3; @Nature4; @Nature5]. We calculate the correlation function of particle displacements $C({\bf r}, \omega)$ and radius of diffuson. In Section \[ub55p\], we investigate the diffusion of energy $D(\omega)$ using two different approaches. The first approach is a direct numerical solution of Newton’s equations. In the second approach the diffusivity is calculated by means of Edwards and Thouless formula which relates the energy diffusivity with an infinitesimal change of boundary conditions. Both approaches give similar results. We show that diffusivity $D(\omega)$ is independent of frequency in the diffuson range. In Section \[scaling\] we discuss scaling properties of the model (their dependence on parameters $V$ and $\mu$). We show that they are similar with systems near jamming transition point. In Section \[disc\] we discuss the obtained results and compare them with experiment. A random matrix approach {#rma} ======================== In harmonic approximation vibrational properties of a mechanical system of $N$ particles are determined by the dynamical matrix $M_{ij}=\Phi_{ij}/\sqrt{m_im_j}$, where $\Phi_{ij}$ is the force constant matrix and $m_i$ are the particle masses. The matrices $M$ and $\Phi$ are real, symmetric and [*positively definite*]{} matrices $N\times N$ ( for simplicity we will consider a scalar model). The condition of positive definiteness is important. It ensures mechanical stability of the system. One can always present every real, symmetric and positive definite matrix $M$ in the following form [@PositiveDefine; @Chalker] $$M = AA^T, \quad \mbox{or}\quad M_{ij}=\sum_k A_{ik} A_{jk}. \label{we45}$$ Here $A$ is some real matrix of a general form (not necessarily symmetric). And, vice versa, for every real matrix $A$ the product $AA^T$ is always a positively definite symmetric matrix. Matrices $AA^T$ belong to Wishart ensemble [@Wishart]. The eigenvalue distribution for such kind of large random matrices was firstly investigated in [@Marchenko]. For a free mechanical system it is necessary to satisfy also conditions [@maradudin] $$\sum_iM_{ij}=\sum_jM_{ij}=0 \label{s6cv1}$$ (for simplicity we consider below all masses $m_i=1$). It ensures that the potential energy of the system $$U=\frac{1}{2}\sum\limits_{ij}M_{ij}u_iu_j=-\frac{1}{2}\sum\limits_{i,j<i}M_{ij}(u_i-u_j)^2$$ and forces between the particles depend only on the differences of particle displacements $u_i-u_j$. As a result, the potential energy and forces are not changed under any translation of the system as a whole. These conditions are necessary (but not sufficient) for existence of low frequency acoustic phonon-like modes in the system. If conditions (\[s6cv1\]) are violated, we have spatially pinned system where propagation of Goldstone modes (phonons) is not possible. In structural glasses in many cases (as, for example, in vitreous silica or amorphous silicon) a mass disorder is not important and we usually deal with the force constant disorder. It is related to fluctuations of valence bond lengths and valence bond angles because of an absence of crystalline ordering. Since valence forces depend exponentially on the distances between the atoms, they can experience strong fluctuations. Due to positional disorder there are also fluctuations of long distance Coulomb forces in non covalent materials. Thus the force-constant disorder plays an essential role in glassy dynamics. Therefore, one may expect that some important properties of glasses can be reproduced if we take matrix $A$ as a random one. As soon as the random elastic spring constants $-M_{ij}$ connecting the particles are fixed, the exact equilibrium particle positions are no longer important for particle dynamic on a long length scales much bigger than the interatomic distances. They do not enter to the dynamical matrix $M$. Therefore, it is reasonable to consider harmonic [*lattice models*]{} involving only force constant disorder. If disorder is sufficiently strong, then it automatically includes the coordination number disorder since any weak bonding is equivalent to a negligibly small interaction between the two neighbors. Also, if the average coordination number is sufficiently big, its exact value and, therefore, the type of the lattice is of no importance as well. The similar considerations concern polarization of the modes. A vectorial character of vibrations in real glasses makes the issue to be more complicated. Many universal properties of glasses (as, for example, the thermal conductivity) are not related to polarization of the modes. Therefore, for better understanding of the physics involved it is often instructive to exploit scalar models. In these models the problem of zero frequency modes existing in vectorial isostatic lattices [@mao] does not exist [@schirm]. For these reasons, different scalar models were successfully used in glassy physics in the past [@schirm; @parisi1; @grigera; @bunde]. Due to the reasons mentioned above we, as in [@our2], consider the case of a simple cubic lattice with $N$ particles and lattice constant $a_0=1$. Each particle has its unique integer index $i$ which takes values from $1$ to $N$. We construct the random matrix $A$ as follows. The non-diagonal elements $A_{ij}$ (for $i\ne j$) we take as independent random numbers from Gaussian distribution with zero mean $\left<A_{ij}\right>=0$ and unit variance $\left<A^2_{ij}\right>=V^2=1$ if $i$-th and $j$-th particles are nearest neighbors. For each particle in a simple cubic lattice there are six nearest neighbors. As a result, for given $i$ we have 6 non zero non-diagonal elements $A_{ij}$ for matrix $A$. Non-diagonal elements $A_{ij}$ and $A_{ji}$ are statistically independent from each other (matrix $A$ is non-symmetric). All other non-diagonal elements (for non-nearest neighbors) $A_{ij}=0$. To ensure the property (\[s6cv1\]) the diagonal elements $A_{ii}$ are calculated as follows $$A_{ii}= -\sum\limits_{j\ne i} A_{ji} . \label{67vb}$$ Then, according to Eq. (\[we45\]), the Eq. (\[s6cv1\]) will be also met. ![Structure of the dynamical matrix $M$ in $2d$ case. Particles 1-12 interact with the central black particle.[]{data-label="fig:dyn_matrix"}](Fig_1) The matrix $M$ is then constructed according to Eq. (\[we45\]). As was shown in [@our1], in a $3d$ simple cubic lattice each particle is connected by elastic springs with $24$ neighbors. The elastic spring constants are random and can be either positive or negative. A negative spring by definition is a spring which expands after initial stretching and shrinks after initial contraction. The effect of negative spring constants on atomic vibrations was discussed in different papers [@schirm; @resmod; @surface; @frenkel; @electron; @parisi1; @taraskin3]. To elucidate where this coordination number 24 in $3d$ case comes from, let us first consider as example a $2d$ simple square lattice. In this case each particle interacts with 12 neighbors shown on Fig. \[fig:dyn\_matrix\]. In accordance to Eq. (\[we45\]), matrix elements connecting central black particle with its 4 nearest neighbors are of the type $$M_{01}=\sum\limits_k A_{0k}A_{1k}=A_{00}A_{10}+A_{01}A_{11}. \label{z8bn}$$ From Eq. (\[67vb\]) it follows that diagonal elements meet the following relations $$A_{00}=-(A_{10}+A_{20}+A_{30}+A_{40}), \label{sc45}$$ $$A_{11}=-(A_{91}+A_{51}+A_{61}+A_{01}). \label{sc34}$$ One has to insert them in Eq. (\[z8bn\]) $$\begin{aligned} M_{01}= &-& A_{10}^2 - A_{01}^2 - A_{10}(A_{20}+A_{30}+A_{40})-\nonumber \\ &-& A_{01}(A_{91}+A_{51}+A_{61}) . \label{h7cv}\end{aligned}$$ Since averaged values $\left<A_{i\ne j}\right>=0$ and different non-diagonal matrix elements $A_{ij}$ are statistically independent from each other, the average value $\left<M_{01} \right>$ is determined by the first two quadratic terms in Eq. (\[h7cv\]). As a result, it is non-zero and negative. It corresponds to positive average elastic spring $k_{01}=-M_{01}$ between particles 0 and 1 $$\left<k_{01}\right>=-\left<M_{01}\right>=\left<A_{10}^2\right>+\left<A_{01}^2\right>=2. \label{c6x4}$$ Though, according to Gaussian distribution of $A_{i\ne k}$, the spring constant $k_{01}$ can take negative values as well. All the aforesaid is valid for other nearest neighbor matrix elements $M_{02}$, $M_{03}$ and $M_{04}$. The next nearest neighbor matrix elements are given by $$M_{05}=\sum\limits_k A_{0k}A_{5k}=A_{01}A_{51}+A_{04}A_{54} , \label{5xwe}$$ $$M_{09}=\sum\limits_k A_{0k}A_{9k}=A_{01}A_{91} . \label{3vcb}$$ It is easy to see that the average values of $\left<M_{05}\right>$ and $\left<M_{09}\right>$ and corresponding average elastic springs $\left<k_{05}\right>$ and $\left<k_{09}\right>$ are zero. So the next nearest neighbor springs can be either positive or negative with equal probability. The same is valid for 6 other next nearest neighbor matrix elements $M_{06}$, $M_{07}$, $M_{08}$ and $M_{0,10}$ $M_{0,11}$, $M_{0,12}$. ![Distributions of random elastic spring constants in $3d$ simple cubic lattice.[]{data-label="fig:kDistr"}](Fig_2) In 3d case for simple cubic lattice there are 6 springs of the type $M_{01}$, 12 springs of the type $M_{05}$ and 6 springs of the type $M_{09}$. As a result all together we have 24 particles interacting with the central black particle. All these 24 spring constants can be either positive or negative but to ensure the mechanical stability of the whole system they are correlated with each other in a rather tangly way. Distributions of different spring constants are shown on Fig. \[fig:kDistr\]. The distribution of $k_{01}$ is asymmetric with positive mean value. The distributions of $k_{05}$ and $k_{09}$ are even (with zero average value) and for $k_{09}$ are given by zeroth-order Macdonald function [@our1] which logarithmically diverges at $k=0$. The resulting distribution of all spring constants was calculated numerically in [@our2]. The number of negative springs was found to be about 45%. One can find a similarity between our spring constant distributions and dynamical matrix element distributions obtained in [@taras] for IC-glass, in [@huang] for simple fluid with short-ranged interactions (see Fig. 1 in these papers), and in [@christie0] for realistic model of amorphous silicon (see Figs 2.12, 2.13). Though it is difficult to compare our scalar model with vector models analyzed in [@taras; @huang; @christie0]. Concluding this part, we can easily include into consideration the next neighbor shell for matrix $A$. Then, in addition to the previous case, the matrix elements of the type $A_{05}$ should be taken into account. As a result the coordination number for matrix $M$ in simple cubic lattice increases up to 90. Just opposite, applying some additional constraints, we can reduce the coordination number from 24 to smaller numbers or make it fluctuating quantity, etc. We have checked that all these modifications can lead to quantitative changes but do not change qualitatively the main results of the paper. Therefore, we will restrict our consideration by the simplest case outlined before. ![The normalized DOS $g(\omega)$ for random matrix $M=AA^T$ built on a simple cubic lattice with $N=20\times 20\times 20$ particles and averaged over $1000$ realizations. Inset: The participation ratio $P(\omega)$ for $N=10^3$ (a) and $N=27^3$ (b) for one realization.[]{data-label="fig:Dos1"}](Fig_3) Fig. \[fig:Dos1\] shows the normalized density of vibrational states (DOS) $g(\omega)$ of matrix $M=AA^T$ in $3d$ simple cubic lattice. The periodic boundary conditions were used. As follows from the figure, the spectrum is gapless i.e. $g(\omega)$ is nonzero at $\omega\to 0$. In spite of the fact that the conditions (\[s6cv1\]) are fulfilled, we do not see the expected phonon modes with their DOS $g_{\rm ph}(\omega)\propto\omega^2$ for $\omega\to 0$. It means that phonons as plane wave excitations cannot propagate through the lattice. This result is not changed qualitatively  [@our2] neither by including next neighbor shells to build matrix $A$ nor by switch to vector model. As was shown in [@our2], such a behavior of DOS at $\omega\to 0$ is related to the fact that the affine assumptions are violated and the macroscopic elasticity theory becomes inapplicable in this case. The average value of the static Young modulus of the lattice $E\propto 1/N$. Therefore, in the thermodynamic limit ($N\to\infty$) $E\to 0$. As a result, the rigidity of the lattice and sound velocity are also tend to zero. This unusual behavior is due to a presence of high concentration of negative springs (45%) in the lattice which makes it to be extremely soft. To determine whether vibrational modes are localized or delocalized, we have calculated the participation ratio $$P(\omega)=\left[N\sum\limits_{i=1}^Ne_i^4(\omega) \right]^{-1}. \label{r6gh}$$ Here $e_i(\omega)$ is $i$-th particle projection of the normalized eigenvector with frequency $\omega$. As one can see from the inset of Fig. \[fig:Dos1\], all modes with exception of small high frequency part are [*delocalized*]{}. They have $P(\omega)\approx 0.2$ which is independent of the system size. This value is close to the theoretical value $1/3$ for Porter-Thomas distribution of $e_i^2(\omega)$ [@porter; @our1]. We have verified also that the level spacing distribution obeys the Wigner-Dyson statistics [@our1]. It also indicates the mode delocalization. As we will show in Section \[diff\], all these delocalized gapless vibrational modes can be identified as diffusons. They spread in the lattice by means of diffusion. To elucidate a spacial structure of the eigenmodes for matrix $M=AA^T$ we considered as an example a two dimensional square lattice with $N=400\times 400$ particles and calculated eigenvector $e_i(\omega_{\rm min})$ ($i=1, 2, ... , N$) for the lowest frequency $\omega_{\rm min}$ in the system. The result is shown on Fig. \[fig:fractal\]. Particles with positive and negative displacements are shown by white and black dots correspondingly. As one can see from the figure, the mode is delocalized. Its spatial structure is random (fractal) and has nothing to do with a plane wave. Similar picture takes place in a 3d case. Phonons ======= To introduce phonons into the picture we should have finite rigidity of the lattice. The rigidity can be introduced by different means. Since a sum of positive definite matrices is a positive definite matrix, then simplest possibility is to add to the random matrix $AA^T$ a “crystalline part” [@our2] $$M=AA^T+\mu M_0. \label{rt56e}$$ Here $A$ is the same random matrix built on a $3d$ simple cubic lattice with $a_0=1$ as in the previous Section. Matrix $M_0$ is a positively definite crystal dynamical matrix for the same lattice with unit masses, and all spring constants (between the nearest neighbors) equal to unity. As was shown in [@our2] the tune parameter $\mu\geqslant 0$ controls the rigidity of the lattice. Adding the regular part $\mu M_0$, changes the distribution of spring constants $k_{01}$ between the nearest neighbors, as shown on Fig. \[fig:kDistr\]. The average value is equal to $\left<k_{01}\right>=2+\mu$. At small values of $\mu\ll 1$ the change is negligible. The distribution mainly consists from strongly fluctuating part $AA^T$ (compare the distributions of $k_{01}$ for $\mu=0$ and $\mu=0.1$). Therefore, it is not obvious at all that such small perturbation is able to introduce a finite rigidity and phonons into the system. A strong scattering of phonons by diffusons may leave the diffuson spectrum unchanged. Also we can consider elastic springs in matrix $\mu M_0$ to be fluctuating quantities, somehow distributed in the closed interval $[0, \mu]$. Otherwise we can cut out a big amount of springs $\mu$ from the lattice, so the phonons cease to exist in the term $\mu M_0$ at all (see Section \[cutlat\]). Another nontrivial possibility is shown in Section \[superposition\]. In the paper we limit ourselves by the most simple case described by Eq. (\[rt56e\]). ![Young modulus $E$ as a function of $\mu$ for dynamical matrix $M=AA^T+\mu M_0$ built on a cubic lattice with $N=100\times 100\times 100$ particles (one realization). Black dots are calculated values, the line is the best least-square fit.[]{data-label="fig:Emu"}](Fig_5) To find the rigidity (as a function of $\mu$), we calculated numerically the Young modulus $E$ of the lattice with dynamical matrix given by Eq. (\[rt56e\]) for $\mu\ne 0$. In modeling of amorphous solids, the standard method to do that is to use Irving-Kirkwood stress tensor formula [@irving]. However, it is difficult to implement this procedure in our case of strong local fluctuations of elastic springs when microscopic displacement field $u({\bf r})$ is not a differentiable function of atomic positions [@our2]. Therefore, to avoid these difficulties we, as in [@our2], used a direct numerical method. We took a very big cubic sample with $N=L\times L\times L=10^6$ particles and side $L-1$ to reduce fluctuations and possible non-affine response. According to the standard textbook formula of the macroscopic elasticity theory (see Eq. 5.2 in [@Landau]), Young modulus is given by $E=\sigma_{zz}/u_{zz}$. Here $\sigma_{zz}$ is the stress, and $u_{zz}$ is the strain. The component $u_{zz}$ gives the relative lengthening of the sample $\Delta L/(L-1)$. Then we fixed the strain and calculated the stress $\sigma_{zz}$. For that we fixed particles on the left hand side of our cubic sample and displaced all particles on the opposite (right hand) side by the unit distance $\Delta L=1$. Since Newton equations are linear, the final result is independent of the value of the step strain used. In other two directions we used the periodic boundary conditions. Then, solving the system of linear Newton equations, we found the new equilibrium positions of all other particles in the sample and calculated restoring forces $f_i$ acting on the displaced particles on the right boundary. Due to randomness of the elastic bonds, the restoring forces are also random. Let $\bar f$ be the average restoring force. Then, by definition, the stress $\sigma_{zz}=\sum_if_i/L^2=\bar{f}$ and the Young modulus $E$ can be calculated as follows $$E=\frac{\bar{f}(L-1)}{\Delta L} . \label{d6vg}$$ To avoid confusion, we remind that we are using here a scalar version of the elasticity theory. Therefore, all forces in the lattice are parallel (or antiparallel) to the particle displacements. The results of these calculations are shown on Fig. \[fig:Emu\] for cubic sample with $N=10^6$ particles [@fluct]. As we can see from the fit, the Young modulus has a following dependence on $\mu$: $$\begin{aligned} E &=& \mu, \quad \mu\gg 1, \label{a6df}\\ E &=& 1.5\sqrt{\mu}, \quad \mu \ll 1 . \label{x6sd}\end{aligned}$$ As a result, for $\mu\gg 1$ we have a usual crystal, where disorder is relatively small and relation (\[a6df\]) is obvious. For $\mu\ll 1$ the force constant disorder is strong. The fluctuations of the nondiagonal matrix elements $M_{i\ne j}$ are much bigger than the averaged values [@our1; @our2]. In this case Young modulus $E\propto\sqrt{\mu}$. It is much bigger than the crystal result (\[a6df\]). Strong fluctuations of the positive and negative elastic springs which in average almost compensate each other make the lattice much more rigid than in the case of crystal. Therefore for $\mu\ll 1$ one can not consider our lattice as a simple superposition of two systems $AA^T$ and $\mu M_0$. The origin of this behavior $E\propto\sqrt{\mu}$ is unclear and it should be elucidated in further work (see also Section \[scaling\]). But below we will support our numerical findings by calculation of the sound velocity and of the phonon density of states (for small $\omega$) and by a comparison of the latter with total DOS calculated numerically for matrix (\[rt56e\]). Below in this paper we will consider the case of strong and moderate force constant disorder when $0\leqslant\mu\leqslant 1$. ![The normalized DOS $g(\omega)$ for dynamical matrix $M=AA^T+\mu M_0$ and four different $\mu$ (0, 0.001, 0.01, 0.1, 1) calculated with precise numerical KPM solution for cubic lattice with $N=200^3$ (full lines). Straight lines correspond to Eq. (\[st56\]) with sound velocity $v=\sqrt{E}$. Filled and open diamonds correspond to phonon contribution to the DOS below and above the Ioffe-Regel crossover frequency $\omega_{\rm IR}$ correspondingly (see further text for details). Inset: dependence $\omega_{\rm max}(\mu)\propto\sqrt{\mu}$.[]{data-label="fig:gwmu"}](Fig_6) To calculate the phonon contribution to the DOS at small $\omega$, we need to know the sound velocity $v$ at zero frequency. It is related to the Young modulus in a standard way: $$v=\sqrt{E} \label{s7cvbf}$$ (since all particle masses $m_i=1$ and lattice constant $a_0=1$). Then for the phonon DOS (in the scalar model) we have $$g_{\rm ph}(\omega)=\frac{1}{2\pi^2}\frac{\omega^2}{v^3}. \label{st56}$$ The total DOS $g(\omega)$, normalized to unity and calculated numerically by the kernel polynomial method (KPM) [@kpm; @kpmreview] for dynamical matrix (\[rt56e\]) and different values of $\mu$, is shown on Fig. \[fig:gwmu\]. We see from the figure that for $\mu\ne 0$ the DOS at low enough frequencies is proportional to $\omega^2$ which corresponds to acoustical phonon excitations. Thus, introducing finite values of $\mu$, we open up a soft [*phonon gap*]{} in the gapless diffuson spectrum, existing at $\mu=0$. The DOS in the gap, as we will show in the paper, is built by acoustic phonon-like modes and at low frequencies goes to zero as $g(\omega)\propto\omega^2$. The term [*phonon gap*]{} is motivated since, if conditions (\[s6cv1\]) are violated, then addition $\mu M_0$ to random matrix $AA^T$ opens a [*hard gap*]{} in the gapless vibrational spectrum (see Fig. \[fig:hardgap\] below). Just above this gap the DOS has a sharp maximum at frequency $\omega_{\rm max}$ which we will identify with the width of the gap. As follows from the figure, the maximum frequency for $\mu\ll 1$ increases as $\omega_{\rm max}\propto\sqrt{\mu}$. Above the maximum the vibrational excitations remain to be diffusons (see Section \[diff\]). One can try to explain the dependence $\omega_{\rm max}\propto\sqrt{\mu}$ as follows. In the absence of random part $AA^T$ the dynamical matrix $M$ is determined by the crystalline part $\mu M_0$ only. Then (for simple cubic lattice) we have well defined phonon modes with dispersion law $$\omega^2_{\rm cryst}=4\mu\left(\sin^2\frac{q_x}{2} + \sin^2\frac{q_y}{2}+ \sin^2\frac{q_z}{2}\right).$$ The maximum frequency in this case is equal to $\omega_{\rm max,\, cryst}=2\sqrt{3\mu}\propto\sqrt{\mu}$ which qualitatively (but not quantitatively) explains aforesaid dependence $\propto\sqrt{\mu}$. However the sound velocity in this pure crystallyne lattice case $v_{\rm cryst}=\sqrt{\mu}$. Though according to Eqs. (\[s7cvbf\], \[x6sd\]) $v\propto\mu^{1/4}$ for $M=AA^T+\mu M_0$ what is much bigger then $\sqrt{\mu}$ for small values of $\mu\ll 1$. It means that simple superposition approach does not work in this case and physical picture is more complicated. As we will show in Section \[scaling\] the Young modulus $E$ depends also on the amplitude of the random part $AA^T$. Since the DOS $g(\omega)$ is normalized to unity for all values of $\mu$, we conclude from Fig. \[fig:gwmu\] (comparing the DOS for $\mu\ne 0$ with DOS for $\mu=0$) that vibrations corresponding to the maximum for $\mu\ne 0$ were pushed out from the region of small frequencies $\omega<\omega_{\rm max}$ for $\mu=0$. We see also from the figure that, after initial $\omega^2$ dependence, the DOS for $\mu\ne 0$ increases much faster than $\omega^2$. It is a clear signature of the presence of the boson peak in our disordered lattice. As we will show further (see Table I), the frequency $\omega_{\rm max}$ is correlated with position of the boson peak $\omega_b$ (the maximum in the reduced DOS $g(\omega)/\omega^2$). Therefore appearance of the boson peak in disordered systems is not necessarily related to the acoustic van Hove singularity in crystals as was proposed recently [@schirm; @taraskin3; @vanHove2]. The straight lines on the Fig. \[fig:gwmu\] correspond to the phonon DOS $g_{\rm ph}(\omega)$ determined by Eq. (\[st56\]) with the sound velocity $v=\sqrt{E}$ and $E$ calculated from Fig. \[fig:Emu\]. One can see a good agreement of the total $g(\omega)$ at low frequencies with the phonon contribution $g_{\rm ph}(\omega)$. From that we can conclude that at least the low frequency excitations in the phonon gap are the usual long-wave acoustical phonons. However, actually, as we will show further, nearly all excitations in the gap up to the frequencies close to $\omega_{\rm max}$ correspond to phonons, but with a nonlinear dispersion law. ![Participation ratio for different $\mu$ as a function of $\omega$ for $N=27^3$ (one realization). The arrows indicate positions of $\omega_{\rm max}$ in $g(\omega)$ for corresponding values of $\mu$ (see Fig. \[fig:gwmu\]).[]{data-label="fig:PR"}](Fig_7) This conclusion is supported by calculations of the participation ratio $P(\omega)$. It is shown in Fig. \[fig:PR\] for various values of $\mu$. For $\mu\ne 0$, one can clearly distinguish in the function $P(\omega)$ a presence of the two different frequency regions. As follows from Fig. \[fig:gwmu\], the low frequency part (below $\omega_{\rm max}$) corresponds to the phonons. In this range the participation ratio increases with decreasing frequency. It is related to increase of the phonon mean free path $l(\omega)$ as $\omega\to 0$ (see Fig. \[fig:IR\]). In the high frequency part (above $\omega_{\rm max}$) $P(\omega)$ is approximately independent of the frequency and coincides with participation ratio for $\mu=0$. As we will show in Section \[diff\] this range corresponds to diffusons. A similar rise of the participation ratio with decreasing frequency was found recently in 2d Lennard-Jones glasses [@tanguy] (see Fig. 1b of this paper). ![The normalized DOS $g(\omega)$ for dynamical matrix $M=AA^T+\mu M_0$ and different $\mu$ (0, 0.001, 0.01, 0.1, 1) calculated with precise numerical KPM solution for cubic lattice with $N=200^3$ (full lines). The conditions (\[s6cv1\]) are violated. Inset: dependence $\omega_{\rm max}(\mu)\propto\sqrt{\mu}$.[]{data-label="fig:hardgap"}](Fig_8) It is important to emphasize that for existence of the acoustical phonon excitations the conditions (\[s6cv1\]) are crucial. If they are not obeyed, then, instead of soft phonon gap in the vibrational spectrum shown on Fig. \[fig:gwmu\], we have a hard gap shown on Fig. \[fig:hardgap\]. Inside the hard gap there are no vibrations at all. The dynamical matrix $M$ in this case was taken in the same form (\[rt56e\]). But diagonal elements $A_{ii}$ of the matrix $A$ were taken as independent Gaussian random variables with average $\left<A_{ii}\right>=0$ and unite variance $\left<A^2_{ii}\right>=1$. As a result the condition (\[67vb\]) (and therefore (\[s6cv1\])) was violated and we have got a spatially pinned lattice where low frequency acoustical phonon modes cannot exist. However, the width of the hard gap in this case has the same $\mu$ dependence as the width of the phonon gap, $\omega_{\rm max}\propto\sqrt{\mu}$. To find the phonon dispersion curve (dependence of the phonon frequency $\omega$ on the wave vector $\bf q$) and phonon mean free path $l(\omega)$ we should calculate space and time Fourier transform of the particle displacement field $u({\bf r}, t)$. For that we ascribed to all the particles at the initial moment $t=0$ random displacements $u({\bf r}, 0)$ (from Gaussian distribution with zero mean and unit variance) and zero velocities. Then, numerically solving Newton equations (with all masses $m_i=1$) we analyzed the particle dynamics at $t\ne 0$. In calculations we used Runge-Kutta-4 method with sufficiently small time step $\Delta t=0.01$. We have checked that in this case the total energy of the system is conserved over the whole investigated time interval $T$ with relative precision $10^{-7}$ without use of any damping. The calculated values of particle displacements also have relative precision higher then $10^{-7}$ (we compared the results with time step $\Delta t=0.01$ with results obtained with two times smaller time step $\Delta t=0.005$). Let $u({\bf r}_i, t)$ be the $i$-th particle displacement as a function of particle coordinate ${\bf r}_i$ and time t. We define the displacement structure factor (DSF) of the displacement field as follows $$S({\bf q}, \omega) = \frac{2}{NT}\left|\sum\limits_{i=1}^Ne^{-i{\bf q}{\bf r}_i}\int\limits_0^{T} u({\bf r}_i, t) e^{i\omega t}dt \right|^2 . \label{45gto}$$ For better frequency resolution, the upper time limit $T$ was taken sufficiently large ($T=3000$), while the integration time step was chosen as $\Delta t=0.01$. Since vectors ${\bf r}_i$ in a cubic lattice are discrete, the wave vectors ${\bf q}\equiv {\bf q}_n$ are also discrete and are defined on the corresponding reciprocal lattice. For example, for cubic sample $L\times L\times L$ and ${\bf q}\parallel \left<100\right>$ direction we have $q_n=2\pi n/L$ where integer numbers $n$ are $-L/2 \leqslant n \leqslant L/2$. One can show (see Section \[DSF\]) that definition (\[45gto\]) is equivalent to the usual expression $$S({\bf q}, \omega) = \frac{\pi}{N}\sum\limits_{j=1}^N\Big|\sum\limits_{i=1}^Ne_i(\omega_j)e^{-i{\bf q}{\bf r}_i}\Big|^2\delta(\omega-\omega_j) . \label{eq:sqw_def2}$$ Here $e_i(\omega_j)$ — is $i$-th component of the eigenvector of the dynamical matrix $M$ corresponding to $i$-th particle and eigenfrequency $\omega_j$ [@maradudin]. The normalized density of states is related to the structure factor by the sum rule $$g(\omega)=\frac{1}{\pi}\sum\limits_{\bf q}S({\bf q}, \omega) . \label{2we}$$ According to definition (\[45gto\]) $S(0,\omega)=0$ since the position of center of inertia is conserved and $\sum_i u({\bf r}_i, t)=0$. ![The Lorentz dispersion curves for different wave vectors ${\bf q}\parallel \left<100\right>$ direction and $\mu=0.1$. Closed diamonds correspond to the calculated values of $S({\bf q}, \omega)$ and lines are fitting curves according to Eq. (\[s556v\]). The number of particles $N=50^3$ (one realization). Insets: the Lorentian dispersion curves for $q=0.5$ and $q=0.75$.[]{data-label="fig:Lorentz"}](Fig_9) ![image](Fig_10) To analyze phonon excitations, we have found the maximum of $S({\bf q}, \omega)$ as a function of $\omega$ for each discrete value of ${\bf q}_n$, for several values of $\mu$. As an example, the results for $\mu=0.1$ and one $\bf q$ direction are shown on Fig. \[fig:Lorentz\]. For the fitting curves we used the Lorentz distribution $$S({\bf q}, \omega)\propto \frac{1}{\left(\omega - \omega_{\bf q} \right)^2 + \left(\Delta \omega \right)^2}. \label{s556v}$$ From this fit we can find both the phonon frequency $\omega_{\bf q}$ and the phonon line width $\Delta\omega$. The results for $\omega_{\bf q}$ are shown on Fig. \[fig:sqw\_ph\] for three values of $\mu$ and ${\bf q}\parallel \left<100\right>$. For sufficiently small values of wave vector $q$ we see a nice linear dispersion curve $\omega_q=vq$, with the sound velocity $v$ given by Eq. (\[s7cvbf\]). It is independent of the $\bf q$ direction (i.e. the sound velocity is isotropic). With increase of $q$, the frequency $\omega_q$ shows a pronounced negative dispersion of the group velocity $v_g=d\omega_q/dq$ and approaches the maximum frequency $\omega_{\rm max}$ where the dependence $\omega_q$ saturates. In this $\bf q$ region we observed a weak anisotropy of the dispersion curves for $\mu=1$. At smaller values of $\mu$ the dependence $\omega_{\bf q}$ is isotropic. Since $\omega_{\rm max}\propto \sqrt{\mu}$, the vertical axis on Fig. \[fig:sqw\_ph\] scales approximately as $\sqrt{\mu}$ and the horizontal axis scales as $\mu^{1/4}$ (sound velocity $v\propto\sqrt{E}\propto \mu^{1/4}$, and $q_{\rm max}\approx \omega_{\rm max}/v \propto \mu^{1/4}$ as well). The strong negative dispersion of the group velocity $v_g$ for big $q$ values can be explained by [*avoided crossing principle*]{} (or level repulsion effect) due to the coupling of phonons to quasilocal vibrations near frequency $\omega_{\rm max}$, corresponding to sharp maximum in DOS $g(\omega)$ (see Fig. \[fig:gwmu\]). Similar phenomenon exists in polariton physics [@polariton]. The dip in the participation ratio $P(\omega)$ for $\mu=0.001$, $\mu=0.01$ and $\mu=0.1$ at $\omega\approx\omega_{\rm max}$ (see Fig. \[fig:PR\]) evidences in favor of this idea. The vibrations inside the dip correspond to frequencies near $\omega_{\rm max}$ and have smaller participation ratio than the others. Therefore they can be referred to as quasilocal vibrations. In the following we will see that this strong scattering is also responsible for the deep minimum in the diffusivity $D(\omega)$ at $\omega\approx\omega_{\rm max}$ (see Fig. \[fig:Dw\_mu\]). The negative dispersion of the group velosity $v_g$ is responsible also for the pronounced rise of the phonon DOS above the $\omega^2$ dependence, given by Eq. (\[st56\]). It is clearly seen on the Fig. \[fig:gwmu\]. Indeed, taking the dispersion into account and disregarding weak anisotropy (taking place only for $\mu=1$) we can write instead of Eq. (\[st56\]) $$g_{\rm ph}(\omega)= \frac{1}{2\pi^2}\frac{q^2(\omega)}{v_g(\omega)}. \label{d56v}$$ Here $v_g(\omega)=d\omega/dq$ is the group velocity shown in Insets on Fig. \[fig:sqw\_ph\]. Taking for $q(\omega)$ and $v_g(\omega)$ the data from Fig. \[fig:sqw\_ph\] we obtain the points (filled and open diamonds) shown on Fig. \[fig:gwmu\]. Since they perfectly coincide with numerical data for $g(\omega)$ below $\omega_{\rm max}$, we conclude that [*all*]{} the excitations in the phonon gap belong to phonons (with nonlinear dispersion at higher values of $q$). ![The phonon line width $\Delta\omega$ as a function of $\omega$ for different $\mu$ in cubic sample with $N=50^3$ (one realization). Different symbols correspond to different $\bf q$ directions. ${\text{\protect\rotatebox[origin=c]{45}{$\square$}}}$ for ${\bf q}\parallel \left<100\right>$, $\triangle$ for ${\bf q}\parallel \left<110\right>$, $\square$ for ${\bf q}\parallel \left<111\right>$. Filled and open symbols refer to excitations below and above the Ioffe-Regel crossover frequency $\omega_{\rm IR}$ correspondingly (see text for details).[]{data-label="fig:Dw(w)"}](Fig_11) The phonon line width $\Delta\omega$ can be also found from fits similar to those shown on Fig. \[fig:Lorentz\]. It is related to the phonon life time $\tau=1/2\Delta\omega$. The factor 2 takes into account that $\Delta\omega$ corresponds to decay of the amplitude of the vibration. The results are shown on Fig. \[fig:Dw(w)\]. As follows from this figure, $\Delta\omega\propto \omega^4$ and does not depend on the direction of $\bf q$. We think that this frequency dependence is not due to Rayleigh scattering of phonons on a static disorder. In such a case $\Delta\omega$ would be proportional to $q^4$. Due to nonlinear dispersion in $\omega_{\bf q}$, these dependencies do not correspond to each other. More likely, the phonon line width is due to strong resonant scattering of phonons by quasilocal vibrations responsible for the sharp peak in the DOS, similar to those introduced in [@Buchenau1]. The deep minimum in the diffusivity $D(\omega)$ around frequency $\omega_{\rm max}$ also supports this idea (see Fig. \[fig:Dw\_mu\]). We hope to investigate this important question in future work. With known value of $\Delta\omega$, the phonon mean free path $l(\omega)$ can be calculated as follows $$l(\omega) = v_g\tau=\frac{v_g}{2\Delta\omega}. \label{we35}$$ The phonons are well defined excitations if their mean free path $l(\omega)$ exceeds the phonon wave length $\lambda=2\pi/q$ (Ioffe-Regel criterium for phonons). As we will see in the next Section, phonons transform to diffusons when $l(\omega)\approx \lambda/2$. We will call the corresponding crossover frequency as $\omega_{\rm IR}$. Fig. \[fig:IR\] shows the ratio $l(\omega)/\lambda$ as a function of $\omega$ for several values of $\mu$ and different directions of the wave vector $\bf q$. The boundary between filled and open symbols (the full horizontal line) corresponds to frequency $\omega_{\rm IR}$. Thus filled and open symbols on Figs. \[fig:gwmu\], \[fig:sqw\_ph\], \[fig:Dw(w)\], \[fig:IR\] belong to phonons with frequencies below and above the Ioffe-Regel crossover frequency correspondingly. ![The ratio $l(\omega)/\lambda$ as a function of $\omega$ for different $\mu$. Different symbols correspond to different $\bf q$ directions as explained on Fig. \[fig:Dw(w)\]. The full horizontal line (separating filled and open symbols) corresponds to Ioffe-Regel crossover $l(\omega)=\lambda/2$.[]{data-label="fig:IR"}](Fig_12) ![image](Fig_13) Usually in glasses the Ioffe-Regel crossover frequency $\omega_{\rm IR}$ is correlated with position of the boson peak $\omega_b$, see [@laermans1; @laermans2; @laermans3; @laermans4; @tanaka] and references therein. It is the frequency where the reduced DOS $g(\omega)/\omega^2$ has a maximum. We also have a rather sharp boson peak in our disordered lattices [@our2]. As follows from Fig. \[fig:gwmu\] the left side of the boson peak is built from phonons having negative dispersion of the group velocity $d\omega_{\bf q}/d{\bf q}$. Similar conclusion was made recently for $2d$ and $3d$ Lennard-Jones glasses [@tanguy; @tan2005; @monaco]. The right side of the boson peak consists from diffuson modes shifted from the region of small frequencies $0<\omega<\omega_{\rm max}$ by additional term $\mu M_0$ and further modified by interaction with phonons. But more work is necessary to elucidate the precise structure of these modes. The frequencies $\omega_{\rm max}$, $\omega_{\rm IR}$, and $\omega_b$ are collected in Table \[tab1\] for different $\mu$. As we can see from the table, $\omega_{\rm IR}$ is close to the frequency $\omega_{\rm max}$ and to the position of the boson peak $\omega_b$. Above $\omega_{\rm IR}$ phonons cease to exist as well defined excitations. They are smoothly transformed to diffusons which we will consider in the next Section. The relative number of phonons in the lattice can be estimated as follows $$N_{\rm ph} = \int\limits_{0}^{\omega_{\rm IR}}g(\omega) d\omega . \label{pnononsnumber}$$ These values are also given in the Table \[tab1\]. We see that for all investigated values of $\mu$ the relative number of phonons in the lattice is small. It is in agreement with similar estimates for amorphous silicon [@Nature5]. [|\*[5]{}[@c@|]{}]{} $\mu$ & $\omega_{\rm max}$ & $\omega_b$ & $\omega_{\rm IR}$ & $N_{\rm ph}$\ 1 & 2.5 & 2.4 & 2.2$^*$ & 0.12\ 0.1 & 0.78 & 0.74 & 0.62 & 0.027\ 0.01 & 0.23 & 0.23 & 0.19 & 0.0066\ 0.001 & 0.072 & 0.07 & &\ Diffusons {#diff} ========= In this section we are going to consider properties of diffusons. As is well known, the diffusion phenomenon usually takes place for physical quantities which are conserved. In a free closed mechanical system we have two integrals of motion, momentum and energy. Therefore one should discriminate between diffusion of momentum and energy. Diffusion of momentum {#diffmomentum} --------------------- First let us consider diffusion of momentum. Usually the diffusion of momentum is related to viscosity in the system. When all particle masses being equal ($m_i=1$), the diffusion of momentum is equivalent to the diffusion of particle displacements. It is because in our system the position of the center of inertia is conserved and we can put it at the origin of the coordinate system. Then the sum of all particle displacements vanishes $$\sum\limits_i u_i(t)=0, \label{xty5}$$ i.e. it is an integral of motion. The diffusion of displacements in this case looks like a diffusion of ”particles” in a lattice where the total number of particles is conserved. By analogy with diffusion of ”particles” the information about diffusivity of displacements is absorbed in the displacement structure factor $S({\bf q}, \omega)$ (\[45gto\]). We remind that to calculate this structure factor we ascribed at the initial moment $t=0$ the random displacements to all the particles with Gaussian distribution (with zero mean and unit variance) and velocities equal to zero. So the condition (\[xty5\]) at $t=0$ was satisfied. Therefore let us analyze now this structure factor in the diffuson frequency range. Consider first the case of $\mu=0$ when phonons are absent and only diffusons are present in the lattice. Fig. \[fig:Sqw\_RW\] shows the structure factor $S({\bf q}, \omega)$ as a function of wave vector $q$ for three different directions in ${\bf q}$ space (symbols) and for three different frequencies $\omega$. Let us compare this displacement structure factor with structure factor of the random walk $S_{\rm rw}({\bf q}, \omega)$ on the lattice. As was shown in [@difcondmat] for the case of the random walk on a lattice, $S_{\rm rw}({\bf q}, \omega)$ is given by expression $$S_{\rm rw}({\bf q}, \omega)= \frac{2\Gamma({\bf q})}{\omega^2 + \Gamma^2({\bf q})}. \label{s6as}$$ It is a Lorentzian, with a width $\Gamma({\bf q})$ given by $$\Gamma({\bf q})=D_{\rm rw}Q^2({\bf q}) , \label{s7gh}$$ where $D_{\rm rw}$ is a diffusion constant of the random walk. In a simple cubic lattice (with lattice constant $a_0=1$) the function $Q({\bf q})$ reads $$Q({\bf q})=2\sqrt{\sin^2\frac{q_x}{2} + \sin^2\frac{q_y}{2} + \sin^2\frac{q_z}{2}}. \label{tu67}$$ For small values of $q\ll 1$, $Q({\bf q})=q$ and in the continuum limit we have the well known result for the diffusion structure factor $$S_{\rm rw}({\bf q}, \omega)= \frac{2D_{\rm rw}q^2}{D^2_{\rm rw}q^4+\omega^2}. \label{uhg7}$$ Let us note that the structure factor (\[s6as\]) has a maximum at ${\bf q}$ values obeying the condition $$\omega=\Gamma({\bf q})=D_{\rm rw}Q^2({\bf q}). \label{2wdr}$$ We can specify it as a [*dispersion law for diffusons*]{}. The width of the maximum is $\Gamma({\bf q})$. For $q\ll 1$, $\Gamma({\bf q})= D_{\rm rw} q^2$. ![The correlation function $C({\bf r},\omega)$ for $\mu=0$ and six frequencies $\omega$ (0.14, 0.31, 0.49, 0.66, 0.84, 1.01) for sample with $N=50^3$ particles averaged over 300 realizations. The full lines are our numerical results obtained from Eq. (\[45gto\]). Each line starts from $r=r_{\rm min}$ which is about 2.5 interatomic distances (marked by arrows). The dashed line corresponds to Eq. (\[eq:Corr\]) with $D_{\rm rw} = 0.7$.[]{data-label="fig:sqwCorr"}](Fig_14) ![image](Fig_15) A comparison of the displacement structure factor $S({\bf q}, \omega)$, (\[45gto\]), and the structure factor of the random walk $S_{\rm rw}({\bf q}, \omega)$, (\[s6as\]), is shown on Fig. \[fig:Sqw\_RW\]. One fitting parameter was the diffusion coefficient $D_{\rm rw}$ in Eq. (\[s7gh\]). From comparison of these data we obtained $D_{\rm rw}\approx 0.7$. It means that the diffusion coefficient of particle displacements $D_u\approx 0.7$ (see Section \[disc\]). Another fitting parameter was a height $h(\omega)$ of the random walk structure factor in the maximum. According to Eq. (\[s6as\]), in the maximum $\Gamma({\bf q})=\omega$ and $h(\omega)=1/\omega$, but to fit the data points on Fig. \[fig:Sqw\_RW\] we used slightly higher values of $h(\omega)$. The small difference between $h(\omega)$ and $1/\omega$ can be explained by different frequency dependencies of the density of states $g(\omega)$ for vibrations and for the random walk (following from the sum rule similar to Eq. (\[2we\])). As we can see from the figure, for the investigated frequencies the fit is perfect. With increasing frequency above $\omega\approx 2-3$, the fitting becomes more and more poor since we approach the localization threshold at $\omega_{\rm loc}\approx 5.5\pm 0.5$ (see below) which is not described well by a simple model of Markovian random walk. Now let us consider a behavior of a correlation function. The correlation function of particle displacements at some frequency $\omega$, expressed through eigenvectors $e_{\bf r}(\omega)$ of the dynamical matrix $M$, reads $$C({\bf r},\omega) = \sum\limits_{{\bf r}'}e_{{\bf r}'+{\bf r}}(\omega)e_{{\bf r}'}(\omega).$$ It is a Fourier transform of the displacement structure factor (\[45gto\]) $$C({\bf r},\omega) = \frac{1}{8\pi^4}\int S({\bf q},\omega) e^{i {\bf qr}} d{\bf q}. \label{s55fg}$$ Let us compare this correlation function with correlation function of the random walk. For distances bigger than the period of the lattice ($a_0=1$) we can make use of the limit of small $q\ll 1$ and integrate Eq. (\[uhg7\]) for the random walk structure factor taken in approximation of continuous media. As a result, we derive $$C_{\rm rw}({\bf r},\omega) = \frac{\exp\left(-r\sqrt{ \frac{\displaystyle\omega}{\displaystyle2D_{\rm rw}}}\,\right)\cos\left(r\sqrt{\frac{\displaystyle\omega}{\displaystyle2D_{\rm rw}}}\,\right)}{2\pi^2 r D_{\rm rw}}. \label{eq:Corr}$$ Fig. \[fig:sqwCorr\] shows a good agreement of our correlation function (\[s55fg\]) with the correlation function of the random walk (\[eq:Corr\]). For all investigated frequencies the numerical data collapse together and become indistinguishable from the theoretical prediction (\[eq:Corr\]). We can see also on this figure the anticorrelation phenomenon (the region of negative values of the correlation function). As follows from Eq. (\[eq:Corr\]), the correlation function of the random walk changes its sign for the first time at $$r\sqrt{\frac{\displaystyle\omega}{\displaystyle2D_{\rm rw}}}=\frac{\pi}{2} . \label{s5tt}$$ It is also in a good agreement with our numerical results. Therefore we can call a corresponding value of $r$ found from Eq. (\[s5tt\]) as a [*radius of diffuson*]{}. It is a typical size of the regions vibrating with frequency $\omega$ and having the same sign of all particle displacements. According to (\[s5tt\]), the radius of diffuson is given by $$r_{\rm d}(\omega)=\frac{\pi}{\sqrt{2}}\sqrt{\frac{D_{\rm rw}}{\omega}}\propto \omega^{-1/2} . \label{q8bn}$$ At $\omega=0$ the correlation function (\[eq:Corr\]) decays slowly as $1/r$. In disordered systems at critical point the correlation function decays as $C(r)\propto 1/r^{d-D_2}$ where $d$ is the space dimension and $D_2$ is a correlation dimension. From this we conclude that in our case $D_2=2$ what corresponds to diffusion. Now let us analyze the displacement structure factor $S({\bf q}, \omega)$ for $\mu\ne 0$. For better visual effect we will show a map of the function $S({\bf q}, \omega)$ on the plane ($\omega$, $q$) for different directions in $\bf q$ space. To do that, for each frequency $\omega$ we have found the maximum $S({\bf q}, \omega)$ as a function of $q$ along some directions in $\bf q$ space. Then we normalized function $S({\bf q}, \omega)$ along this line $\omega$=const to the magnitude of this maximum. The results are shown on Fig. \[fig:sqw\] for four different values of $\mu$ and two directions in $\bf q$ space. The white color corresponds to the maximum when normalized structure factor $S_n({\bf q}, \omega)=1$ while the black color to the case where $S_n({\bf q}, \omega)=0$. For $\mu\ne 0$ we can see clearly two types of excitations in the lattice. At low enough frequencies, below $\omega_{\rm IR}$, we see phonons with well defined dispersion law $\omega_{\bf q}$, the same as in the previous Section. At the Ioffe-Regel crossover frequency $\omega_{\rm IR}$, the structure factor strongly broadens and phonon dispersion line disappears. Above $\omega_{\rm IR}$ the displacement structure factor coincides well with the structure factor for $\mu=0$ case shown on Fig. \[fig:sqw\]a, which corresponds to diffusons. The maximum of the normalized structure factor $S_n({\bf q},\omega)$ (white regions) agrees well with Eq. (\[2wdr\]) (with the same diffusion coefficient $D_{\rm rw}$) giving the maximum of the random walk structure factor $S_{\rm rw}({\bf q}, \omega)$ (black line). It means that diffusion coefficient of particle displacements is independent of $\mu$. Deviations from $S_{\rm rw}({\bf q}, \omega)$ take place at high frequencies near the localization threshold. For $\mu\ne 0$ the radius of diffuson (\[q8bn\]) takes a maximum value at $\omega\approx\omega_{\rm IR}$. At smaller frequencies we have well defined phonons. Since $\omega_{\rm IR}\propto\sqrt{\mu}$ and $D_{\rm rw}\approx 1$ we can write for $0<\mu\lesssim 1 $ $$r_{\rm d}(\omega_{\rm IR})\equiv r_c \simeq \sqrt{D_{\rm rw}/\omega_{\rm IR}}\simeq \mu^{-1/4}. \label{3b6g}$$ The value $r_c$ plays a role of correlation length in our lattice. It diverges when $\mu\to 0$. The physical meaning of this length is that it by the order of the value coincides with the Ioffe-Regel wave length $\lambda_{\rm IR}=2\pi/q_{\rm IR}$ corresponding to frequency $\omega_{\rm IR}$ (see Section \[scaling\]). Samples with size smaller than $r_c$ have no phonon-like modes at al. ![The same normalized structure factor $S_n({\bf q},\omega)$ as on Fig. \[fig:sqw\] but in $\bf q$ space in plane $q_xq_y$ ($q_z=0$) for $\omega=0.5$. The left picture corresponds to $\mu=0$ (a) and the right to $\mu=0.1$ (b).[]{data-label="fig:qxqy"}](Fig_16) To compare phonon and diffuson structure factors, a cross section of the structure factor $S_n({\bf q},\omega)$ in $\bf q$ space for $q_z=0$ and frequency $\omega=0.5$ is shown on Fig. \[fig:qxqy\] for $\mu=0$ and $\mu=0.1$. At the left side (a) of this figure we see the structure factor of diffuson. On the right side we see the structure factor of phonon (b). As compared with phonon structure factor, the diffuson structure factor is much more broadened. Diffusion of energy {#ub55p} ------------------- Now let us consider the diffusion of energy. The diffusion of energy is different from diffusion of particle displacements (see Section \[disc\]). The first approach to calculate the diffusivity of energy $D(\omega)$ for vibrations with frequency $\omega$ is a direct numerical solution of Newton’s equations. For that we have used the Runge-Kutta-4 method with time step $\Delta t=0.01$ applied to a cubic sample with $N=L\times L\times L$ particles (lattice constant $a_0=1$) and with free boundary conditions along the $x$ direction. Along other two directions we take the periodic boundary conditions. ![The dependence of $R^2(t)$ in the case of $\mu = 0$ for one sample with $N=100\times 100\times 100$ particles and $14$ different frequencies $\omega=0.5, 1, 1.5, \ldots, 7$ (from top to bottom). The numbers indicate integer frequencies. The slope of each line corresponds to each black dot in Fig. \[fig:Dw\]. Two points at $\omega=2$ and $\omega=6$ correspond to two distributions of energy $E(x,t)$ over the sample for delocalized and localized modes correspondingly. They are shown on Fig. \[fig:Ex\] (see below).[]{data-label="fig:fan"}](Fig_17) Assuming zero initial conditions for displacements and velocities of all the particles, let us apply external forces with frequency $\omega$ and random phases $\varphi_i$ to all the particles in the central layer $x=0$ of our sample [@central] $$f^{\rm ext}_i(t)=\sin(\omega t+\varphi_i)\exp\left(-\frac{t^2}{2T^2}\right) \label{eq:Nf}$$ where $\omega T\gg 1$. The right and the left sides of the sample have coordinates $x_{\rm r,l}=\pm L/2$. In such a way we excite vibrations with frequencies near frequency $\omega$ distributed in a small frequency interval $(\omega-1/T,\, \omega+1/T)$. In calculations we used $T=5$ for all frequencies $\omega$. We started our calculations at time $t_{0}=-5T$ when the external force is still negligible. After applying the force to the central layer $x=0$, vibrations will spread to the left and to the right ends of the sample. The average squared distance to the energy diffusion front we define as usual $$R^2(t)=\frac{1}{E_{\rm tot}}\sum\limits_{i=1}^N x_i^2 E_i(t) = \frac{1}{E_{\rm tot}} \int\limits_{-L/2}^{L/2}x^2 E(x,t)dx . \label{eq:NR2}$$ Here $x_i$ is the $x$ coordinate of the $i$-th particle, $E_i(t)$ is the energy of $i$-th particle and sum is taken over all particles in the sample. $E_{\rm tot}=\sum_i E_i(t)$ is the total energy of the system. It is independent of time after the external force $f^{\rm ext}_i(t)$ becomes negligibly small (i.e. for $t>5T$). The energy of $i$-th particle $E_i(t)$ we define as a sum of the kinetic energy and a half of the potential energy of connected bonds ($m_i=1$) $$E_i(t) = \frac{v_i(t)^2}{2}-\frac{1}{4}\sum\limits_j M_{ij}\big(u_i(t)-u_j(t)\big)^2.$$ Here $v_i(t)={\dot u}_i(t)$ is a particle velocity. Summation over all particles in Eq. (\[eq:NR2\]) we can divide in two steps. First we sum over all particles in the layer $x$ and then we sum over all layers. Let $E(x,t)$ be a total energy confined to the layer $x$ at time $t$. Having in mind that in our case we have lattice constant $a_0=1$ and sample size $L\gg 1$, we can change summation over different layers to integration over coordinate $x$ for times where $R(t)\gg 1$. ![The dependence of diffusivity $D(\omega)$ on $\omega$ for $\mu = 0$. Black dots are calculated by the direct solution of Newton’s equations from Eqs. (\[eq:NR2\], \[eq:ND\]) and Fig. \[fig:fan\] for $N=100^3$ particles (one realization). Full lines for $N=10^3, 14^3, 20^3$ are calculated using formula of Edwards and Thouless (\[eq:ETp\]) with $c=1$ (see below). Averaging for lines is performed over frequencies in the small interval $(\omega-\delta\omega,\omega+\delta\omega)$ with $\delta\omega=0.25$ and over several thousands realizations.[]{data-label="fig:Dw"}](Fig_18) We will apply this method to the case of $\mu=0$ (i.e. for the lattice without phonons). The results are shown on Fig. \[fig:fan\]. As we can see from the figure for small and middle frequencies, $R^2(t)\propto t$. Therefore for these frequencies vibrations indeed spread along the $x$ axis by means of diffusion. The slope of the lines decreases with frequency $\omega$. For calculating the slope, we take the time interval $\Delta t$ where, on the one hand $t > 5T$, and on the other hand, $R \ll L/2$. From the slope of $R^2(t)$ we can calculate the diffusivity of modes $D(\omega)$ using one dimensional formula $$R^2(t) = 2D(\omega)t. \label{eq:ND}$$ This diffusivity is shown by black dots on Fig. \[fig:Dw\]. At small frequencies it is approximately constant, then it decreases with frequency approaching zero at the localization threshold, $\omega_{\rm loc}\approx 5.5\pm 0.5$. At higher frequencies above $\omega_{\rm loc}$ the dependence $R^2(t)$ saturates with increasing $t$. This indicates localization of the vibrational modes. ![Black points (diamonds and triangles) show the distribution of energy $E(x,t)$ contained in the layer $x$ as a function of $x$ for two different frequencies $\omega = 2$ and $\omega=6$ at times $t=234$ and $t=900$, respectively, calculated numerically with Newton method. Full lines are theoretical predictions for delocalized (diffusive) and localized modes given by Eqs. (\[eq:Exdiff\], \[eq:Exloc\]) with $R^2\approx 166$ and $R^2\approx 22$ correspondingly.[]{data-label="fig:Ex"}](Fig_19) The difference between delocalized and localized modes is clearly seen if we examine the dependence $E(x,t)$ as a function of coordinate $x$ at some moment $t$ for two different frequencies below and above the localization threshold. These two points for investigation are shown on Fig. \[fig:fan\]. Black diamond corresponds to delocalized mode with frequency $\omega=2$ and has coordinates $t=234$ and $R^2=166$. The distribution of energy $E(x,t)$ over the sample calculated numerically at this moment is shown by black diamonds on Fig. \[fig:Ex\]. The data are perfectly fitted by solid line drawn according to the solution of diffusion equation in $1d$ case $$E(x,t) = \frac{E_{\rm tot}}{\sqrt{2\pi R^2}} \exp\left(-\frac{x^2}{2R^2}\right), \label{eq:Exdiff}$$ with value of $R^2=166$. Black triangle on Fig. \[fig:fan\] corresponds to localized mode with frequency $\omega=6$ and has coordinates $t=900$ and $R^2=22$. The distribution of energy $E(x,t)$ over the sample calculated numerically at this moment is shown by black triangles on Fig. \[fig:Ex\]. This distribution is drastically different from the previous case. For localized modes we expect the usual exponential decay $$E(x,t) = \frac{E_{\rm tot}}{\sqrt{2}R}\exp\left(-\frac{\sqrt{2}|x|}{R}\right). \label{eq:Exloc}$$ The fit of the numerical data with this function and $R^2=22$ is shown on Fig. \[fig:Ex\]. The fit is perfect except for the central point at $x=0$ which lies noticeably above prediction of Eq. (\[eq:Exloc\]). The coefficients in Eqs. (\[eq:Exdiff\], \[eq:Exloc\]) were taken to satisfy the obvious rules $$\int\limits_{-\infty}^{\infty}E(x,t)dx=E_{\rm tot}, \quad \frac{1}{E_{\rm tot}}\int\limits_{-\infty}^{\infty}x^2 E(x,t) dx = R^2. \label{cf45}$$ To find the diffusivity $D(\omega)$ for $\mu\ne 0$, the method of numerical solution of Newton’s equations is not appropriate, because in this case we have phonons in the lattice with long mean free paths. Correspondingly samples with much bigger sizes are necessary to use this approach. Therefore for $\mu\ne 0$ we used a second approach. In this approach, the diffusivity $D(\omega_i)$ at eigenfrequency $\omega_i$ was calculated by means of the formula of Edwards and Thouless [@thouless] $$D(\omega_i) \simeq L^2 |\Delta\omega_i| \label{eq:ET}$$ where $L$ is the length of the sample and $\Delta\omega_i$ is sensitivity of the eigenfrequency $\omega_i$ to a twist of boundary conditions. More precisely, we defined the diffusivity as follows: $$D(\omega) = c\lim_{\varphi \to 0}\frac{L^2}{\varphi^2}\langle|\Delta\omega(\omega)|\rangle \label{eq:ETp}$$ where $\varphi$ is the angle of twisting, and $c$ is some constant of the order of unity. It will be determined from comparison with the Newton method. The averaging in Eq. \[eq:ETp\] is performed over frequencies $\omega$ in the small interval $(\omega-\delta\omega,\omega+\delta\omega)$ with $\delta\omega=0.25$ and/or over several thousands realizations. ![The diffusivity $D(\omega)$ for various $\mu$ (0, 0.01, 0.1, 1) for sample with $N=14^3$ (crosses). The diffusivity was calculated using formula of Edwards and Thouless (\[eq:ETp\]) with $c=1$ and averaged over two thousand realizations. The arrows indicate frequencies $\omega_{\rm max}$ in the DOS $g(\omega)$ for corresponding values of $\mu$. Open symbols correspond to phonon diffusivity (\[9ngh\]) below the Ioffe-Regel crossover frequency $\omega_{\rm IR}$.[]{data-label="fig:Dw_mu"}](Fig_20) The symmetric real matrix $M$ was defined as usual (\[rt56e\]) with periodic boundary conditions. The twisting of the matrix $M$ by angle $\varphi$ gives a new Hermitian matrix $M'$ obtained as follows. For bonds between the left ($l$) and the right ($r$) boundaries of our cubic sample $$M'_{lr}=M_{lr}\exp(i\varphi),\quad M'_{rl}=M_{rl}\exp(-i\varphi) .$$ For all other bonds $M'_{jk}=M_{jk}$. So $\Delta\omega_i$ is the difference between $i$-th eigenfrequencies of matrices $M$ and $M'$ $$\Delta\omega_i = \omega_i - \omega'_i.$$ Twisting of boundary conditions was performed for $x$ direction only. For others two directions the periodic boundary conditions were used. For $\mu=0$ the results for $D(\omega)$ are shown on Fig. \[fig:Dw\] for three different cubic samples (full lines). We compared these results with numerical solution of Newton equations for $\mu=0$ (black dots) and get for the constant $c \approx 1$. Then we used this $c$ value for $\mu\ne 0$. The results are shown on Fig. \[fig:Dw\_mu\]. For $\mu\ne 0$ we see clearly two different frequency regions in the function $D(\omega)$. At low frequencies, diffusivity increases with decreasing of $\omega$. This range corresponds to the phonons. Indeed, the diffusivity of phonons $D(\omega)$ can be calculated as follows $$D(\omega) = \frac{1}{3}l(\omega)v_g(\omega) . \label{9ngh}$$ Open symbols on Fig. \[fig:Dw\_mu\] show contribution calculated from this equation (just below Ioffe-Regel threshold). We see a good agreement with Edwards and Thouless formula. After a deep minimum at frequency $\omega\approx \omega_{\rm max}$ the diffusivity $D(\omega)$ saturates at a constant level (independent of $\mu$) coinciding with $D(\omega)$ for $\mu=0$. The diffusivity in this range corresponds to diffusons. Similar behavior of $D(\omega)$ was found recently in jammed systems [@jammed1; @jammed2]. The deep minimum in the diffusivity at $\omega\approx\omega_{\rm max}$ corresponds to strong scattering of phonons by the quasilocal vibrations near the sharp peaks in the DOS $g(\omega)$ (see Fig. \[fig:gwmu\]). Scaling relations {#scaling} ================= Finally, the concept of diffusons allows us to establish useful scaling relations between observable values and important parameters of our model. One parameter is $\mu$. It has a dimensionality of frequency squared. The second important parameter of the model is the variance of non-diagonal elements $A_{ij}$ of the random matrix $A$ which we hitherto considered to be equal to unity $$\left<A_{ij}^2\right>=V^2. \label{s6cv0}$$ The parameter $V$ has dimension of frequency and assigns the scale of typical frequencies in the system. In particular, the normalized density of states $g(\omega)$ for $\mu=0$ shown on Fig \[fig:Dos1\] has the following scaling relation $$g(\omega)\simeq 1/V.$$ Since for $\omega$ below $\omega_{\rm IR}$ in our disordered lattice we have phonons with $\omega=vq$ (here $v$ is sound velocity) and above $\omega_{\rm IR}$ we have diffusons with $\omega=Dq^2$ (here $D=D_{\rm rw}$) we can write at the Ioffe-Regel threshold the order of the magnitude estimates $$\omega_{\rm IR}\simeq vq_{\rm IR}, \quad \omega_{\rm IR}\simeq Dq^2_{\rm IR}. \label{28bn}$$ From these equations it follows that $$v^2\simeq D\omega_{\rm IR}, \quad q_{\rm IR}^{-1} \simeq D/v. \label{2ghb}$$ Since, according to Eq. (\[s7cvbf\]) $v=\sqrt{E}$ (the units of mass and length we put equal to unity, $m=a_0=1$), we find for the Young modulus a useful relation $$E\simeq D\omega_{\rm IR}\simeq D\sqrt{\mu} . \label{8c5g}$$ Because, as we have shown in Section \[diff\], the diffusion coefficient $D$ is independent of $\mu$, the Young modulus has the same $\mu$ dependence as $\omega_{\rm IR}\simeq\sqrt{\mu}$ (see inset on Fig. \[fig:gwmu\]). It is in a full agreement with Fig. \[fig:Emu\] for $\mu\ll 1$. From the dimensionality considerations (since $V$ has a dimension of frequency) we have for the diffusivity $D$ the following estimate $$D\simeq V .$$ It is quite natural since the typical diffusion jump length is of the order of lattice constant $a_0=1$ and typical jump frequency is of the order of typical frequency in the system $V$. Therefore, the diffusivity $D\simeq Va_0^2$. Taking this into account, for the Young modulus and sound velocity at small frequencies we have for $V\gg \sqrt{\mu}$ $$E\simeq V\sqrt{\mu} ,\quad v=\sqrt{E}\simeq (\mu V^2)^{1/4}$$ i.e. the Young modulus is proportional to the characteristic frequency in the system $V$. The correlation length (\[3b6g\]) $$\lambda_{\rm IR} \simeq l(\omega_{\rm IR}) \simeq q^{-1}_{\rm IR} \simeq \sqrt{D/\omega_{\rm IR}} \simeq D/v \simeq \left(V^2/\mu\right)^{1/4} . \label{qkl9}$$ Though our paper is not aimed at jamming transition and we consider completely different model, it is interesting to note that these scaling relations are identical to those found in jamming transition [@jammed2]. Authors [@jammed2] study a model of amorphous packing of frictionless spheres interacting via the repulsive pair potential $$U(r_{ij})\propto(1-r_{ij}/\sigma_{ij})^\alpha \quad \mbox{if} \quad r_{ij}<\sigma_{ij} , \nonumber \label{99ff}$$ $$U(r_{ij})=0 \quad \mbox{if} \quad r_{ij}>\sigma_{ij}, \label{a8cv}$$ where the distance between the centers of particles $i$ and $j$ is denoted by $r_{ij}$ and the sum of their radii by $\sigma_{ij}$. This model system, irrespective of the value of $\alpha$, exhibits a jamming/unjamming transition at $T=0$ at a packing fraction $\phi=\phi_c$ at which the particles are just touching each other and there is no overlap [@hern]. At densities lower than $\phi_c$ particles are free to rearrange while above $\phi_c$ at $\Delta\phi \equiv \phi - \phi_c$, the system behaves as a weakly connected amorphous solid with an average coordination number that scales as a power law with an exponent $$\Delta z\equiv z-z_c\sim \Delta\phi^{1/2}$$ where $z_c=2d$, with $d$ being the space dimension. It was found that different quantities exhibit scaling behavior near the jamming point. According to [@jammed2] the Ioffe-Regel crossover frequency $\omega^*$ and the shear modulus $G$ behave as (we use below the notation of the paper [@jammed2]) $$\omega^* \sim \Delta\phi^{(\alpha-1)/2}, \quad G\sim \Delta\phi^{(2\alpha-3)/2}. \label{a6mn}$$ The transverse sound velocity $v_t$ and the diffusivity in the plateau region $d_0$ scale $$v_t\sim\Delta\phi^{(2\alpha-3)/4} , \quad d_0\sim \Delta\phi^{(\alpha-2)/2}. \label{abc5}$$ The applied pressure $p$ and the plateau in the density of states $D_0$ depend on the packing fraction as follows [@hern] $$p\sim \Delta\phi^{\alpha-1}, \quad D_0\sim \Delta\phi^{(2-\alpha)/2}. \label{a7xc}$$ Thus if we put $$\mu\sim\Delta\phi^{\alpha-1}, \quad V\sim\Delta\phi^{(\alpha-2)/2} , \label{s7vb}$$ then the crossover frequency $\omega_{\rm IR}$, the Young modulus $E$, sound velocity $v$, the diffusivity at the plateau $D$, and the density of states $g(\omega)$ in our model have the same scaling as the crossover frequency $\omega^*$, the shear modulus $G$, transverse sound velocity $v_t$, the diffusivity in the plateau $d_0$, and the density of states $D_0$ in the jamming transition model respectively. In particular, the parameters $\mu$ and $V$ in our model are equivalent to pressure $p$ and inverse density of states $1/D_0$ in the jamming transition model correspondingly. In the paper we mainly considered a case of strong disorder, $\mu \ll V^2$. Taking into account Eq. (\[s7vb\]) we find that the small parameter of our model $$\mu/V^2 \sim \Delta\phi \label{6x7}$$ coincides with the small parameter $\Delta\phi$ in the jamming transition model. The mean free path at the crossover as follows from (\[qkl9\]) and (\[s7vb\]) is given by $$l(\omega_{\rm IR})\sim \Delta\phi^{-1/4} ,$$ what also coincides with [@jammed2]. It would be very interesting to investigate physical reasons for this striking “mapping” of two models to each other in more details in a future work. Discussion {#disc} ========== We have developed a stable random matrix approach to describe vibrations in strongly disordered systems, which have properties similar to what one observes in granular matter at the jamming transition point, in jammed systems and, finally, in real glasses. This approach has one important advantage in comparison to other models. It describes mechanical systems which are always stable independently of the degree of disorder. Previous random matrix models [@schirm; @taraskin3; @grigera] suffer from an inherent mechanical instability that occurs at some critical amount of disorder. As a result they are limited by consideration of “relatively weak” or “moderate” disorder. We use scalar model and take the dynamical matrix in the form $M=AA^T+\mu M_0$. Here $A$ is a random matrix $N\times N$ built on a simple cubic lattice with $N$ particles and interaction between nearest neighbors only. The only non zero non-diagonal matrix elements $A_{ij}$ between the nearest neighbors are taken as independent random numbers from Gaussian distribution with zero mean $\left<A_{ij}\right>=0$ and unit variance $\left<A^2_{ij}\right>=V^2=1$. The variance controls the degree of disorder in the lattice. To ensure the important property (\[s6cv1\]) the diagonal elements are calculated as a minus sum of non-diagonal elements $A_{ii}= -\sum_{j\ne i} A_{ji}$. $M_0$ is a crystalline dynamical matrix with unit springs between the nearest neighbors. As a result each particle in this lattice is connected by random elastic springs with 24 surrounding particles. Since matrix $AA^T$ is always positive definite, such form of the dynamical matrix guarantees the mechanical stability of the system for any positive value of $\mu$. If the first term $AA^T$ is responsible for the disorder in the system, the second term $\mu M_0$ describes the ordered part of the Hamiltonian. The parameter $\mu$ controls the relative amplitude of this part and the rigidity of the lattice. It can vary in the interval $0\leqslant \mu < \infty$, changing the rigidity and relative amount of disorder. In this paper we have mainly considered the case of strong and moderate disorder when $0\leqslant \mu\lesssim V^2$ and fluctuating part of the dynamical matrix is bigger then the ordered part. In this case the Young modulus of the lattice $E\propto V\sqrt{\mu}$. The parameter $\mu$ plays the same role as pressure in jammed systems. We have found that the delocalized vibrational excitations in this disordered lattice are of two types. At low frequencies below the Ioffe-Regel crossover, $\omega < \omega_{\rm IR}$, they are the usual phonons (plane waves) which can be characterized by frequency $\omega$ and wave vector $\bf q$. However, with increasing of $\omega$, due to the disorder-induced scattering, the phonon line width $\Delta\omega$ increases rapidly as $\Delta\omega\propto\omega^{4}$ and at some frequency $\omega\approx\omega_{\rm IR}$ the phonon mean free path $l$ becomes of the order of the wave length $\lambda$. Though this crossover is not sharp and has no critical behavior at $\omega=\omega_{\rm IR}$, the structure of the eigenmodes at higher frequencies quite soon become very different from the plane waves. As a result, at higher frequencies the original notion of phonons is lost and delocalized vibrational modes have a diffusive nature. They are similar to [*diffusons*]{} introduced by Allen and Feldman, et al. [@Nature5]. The diffusons again can be characterized by frequency $\omega$, but have no well defined wave vector $\bf q$. Above $\omega\approx\omega_{\rm IR}$ the structure factor of particle displacements $S({\bf q}, \omega)$ becomes very similar to the structure factor $S_{\rm rw}({\bf q}, \omega)$ of a random walk on the lattice. The former has a broad maximum as a function of $q$ at $q=\sqrt{\omega/D_u}$, where $D_u\simeq V$ is a diffusion coefficient of the particle displacements. The displacement structure factor $S(q,t)$ in the diffuson range, for small $q\ll 1/a_0$, decays as following, $S(q,t)\propto\exp(-D_uq^2t)$. As a result the vibrational line width $\Gamma(q)=D_u q^2$. Such quadratic dependence of $\Gamma(q)$ was found in many glasses in the experiments on inelastic x-ray scattering, see for example [@sette; @ruoccosette] and references therein. It was also found in molecular dynamic simulation of amorphous silicon [@christie]. However in these and other papers this line width was attributed to phonons without discussion of its physical origin. We guess that the observed $q^2$ dependence of $\Gamma(q)$ has nothing to do with phonons and is in fact related to diffusons. However, a more detailed investigation is necessary for a definite conclusion. The crossover between phonons and diffusons takes place at the Ioffe-Regel crossover frequency $\omega_{\rm IR}$ which is close to the position of the boson peak. Since for phonons $\Delta\omega\propto\omega^4$ and for diffusons $\Gamma(q)=D_uq^2$, there should exist a crossover from $\omega^4$ to $q^2$ dependence of the line width. Such a crossover was indeed found recently in inelastic x-ray scattering in lithium diborate glass [@laermans3], densified vitreous silica [@ruffle2003], vitreous silica [@valentina1; @valentina2; @valentina3], glassy sorbitol [@valentina4] and glycerol glass [@valentina5]. The crossover frequency was found to be close to the BP position. As a result, if our guess is true, we can calculate the diffusion coefficient of particle displacements, $D_u=\Gamma(q)/q^2$, from the experimental line width $\Gamma(q)$ in the range, where it is proportional to $q^2$. Taking into account that $D_u\approx a_0^2/\tau$ where $a_0$ is the lattice constant and $\tau$ is an average time for a jump, we come to the order of the value estimate $D_u\approx 1$mm$^2$/sec for $a_0\approx 2$Å and $\tau\approx 0.4\times 10^{-13}$sec. Let us compare this value with experimental data. In the paper [@valentina2] it was found that in vitreous silica $\hbar\Gamma/(\hbar\omega)^2=0.07$meV$^{-1}$ for $q\ge 2$ nm$^{-1}$. Taking the sound velocity $v_L=5250$ msec$^{-1}$ for $q=2$ nm$^{-1}$ we get for diffusion coefficient $D_u=1.3$mm$^2$/sec. Let us compare this value with the diffusivity of energy $D(\omega)$ for small $\omega$ in the same glass. We expect that both coefficients should be of the same order of magnitude. The diffusivity of energy $D(\omega)$ in vitreous silica was calculated in the paper [@sim]. It was obtained that $D(0)=1.4$mm$^2$/sec. A close estimate $D(0)=1.1$mm$^2$/sec was given in [@similar]. As one can see the agreement between $D_u$ and $D(0)$ is unexpectedly good. In glycerol glass [@ruocco] we found the diffusivity about factor of two smaller, $D_u=0.46$mm$^2$/sec. For amorphous silicon from molecular dynamic calculations [@christie] we get $D_{ul}=3.2$mm$^2$/sec for longitudinal vibrations, and $D_{ut}=1.2$mm$^2$/sec for transverse vibrations. For the diffusivity of energy we have in this glass the estimate [@Nature5] $D(0)=0.6$mm$^2$/sec. Since $\omega_{\rm IR}\propto\sqrt{\mu}$ (and independent of the strength of disorder $V$), we can vary the Ioffe-Regel crossover frequency and, therefore, the relative number of phonons $N_{\rm ph}$ in the system, changing the parameter $\mu$. It is zero when $\mu=0$ and there are no phonons in the lattice. In this case all delocalized vibrations are diffusons. If $0 <\mu\ll 1$ we have phonons, but their relative number is small. One can show that in this case $N_{\rm ph}\propto\mu^{3/4}$. In the opposite case, $\mu\gg 1$, the disorder is relatively small and nearly all vibrations in the lattice are well defined plane waves, i.e. phonons. In amorphous silicon the relative number of phonons (plane waves) was estimated to be only 4% from all of the vibrational modes in the system [@Nature5]. The estimates show that in our model we have such a small amount of propagating modes, as in a-Si, for $\mu\approx 0.1$. In the silica glass we can estimate the relative number of phonons from the data [@taraskin]. Taking into account that Ioffe-Regel crossover frequency in amorphous silica was estimated to be [@taraskin] $\nu_{\rm IR}=1$THz, and integrating density of states [@taraskin] up to this frequency we come to the relative number $N_{\rm ph}=0.002\pm 0.0005$. As a result in the typical glass such as amorphous silica only $0.2$% of all modes are phonons. As follows from Table \[tab1\] it corresponds to very small values of $\mu <0.01$. It means that small amount of phonons in disordered systems is a signature of strong disorder. Usually the phenomenon of diffusion takes place for conserved quantities. In our system we have two integrals of motion. They are the momentum and the energy of the lattice. Therefore, first of all, one has to discriminate the diffusion of particle momentums (or particle displacements) from the diffusion of energy. Conservation of displacement is related to conservation of the center of inertia in the system. As a result, the diffusion of particle displacements has the same diffusion coefficient as the diffusion of particle momentums. The diffusion coefficient of displacements/momentums $D_{u/v}$ is hidden in the displacement structure factor $S({\bf q}, \omega)$ (\[45gto\]). Comparing this structure factor with the structure factor of the random walk on the lattice, we found that for the case of $\mu=0$ the diffusion coefficient $D_{u/v}=D_{\rm rw}=0.7$. We can check that it is indeed the diffusion coefficient of particle displacements/momentums in a similar way we used for finding the diffusivity of energy $D(\omega)$ in Section \[ub55p\]. Let us consider a cubic random lattice $L\times L\times L$ with $\mu=0$ and unit masses $m_i=1$ with periodic boundary conditions. At initial moment $t=0$ let us displace all particles in a thin layer around the central layer (with coordinate $x=0$) according to Gaussian distribution $$u(x,0)=u_0 e^{-x^2/2x_0^2}. \label{ts45n}$$ Here the thickness of the layer $x_0$ should be small enough in comparison to the sample size $L$, i.e. $x_0\ll L/2$. Initial velocities ${\dot u}(0)$ of all the particles are equal to zero. After initial displacements in the thin central layer, the particle displacements will diffuse to the left and to the right ends of the sample. Solving numerically the Newton equations, we find the average squared distance to the displacement diffusion front, similar to Eq. (\[eq:NR2\]) $$R^2_u(t)=\frac{1}{u_{\rm tot}} \sum\limits_i x_i^2 u_i(t), \quad u_{\rm tot}=\sum\limits_i u_i(t). \label{et55}$$ Since the center of inertia does not move, the total displacement of all particles $u_{\rm tot}$ is independent of time and equal to the total displacement at $t=0$. From the slope of $R^2_u(t)$ we can calculate the diffusion coefficient of the displacements $D_u$ as follows $$R_u^2(t)=2D_u t \label{x5as}$$ similar to Eq. (\[eq:ND\]). In the same way we can calculate the diffusion of momentum. For that at the moment $t=0$ initial displacements of all the particles we put equal to zero. However initial velocities $v={\dot u}(0)$ in the thin central layer we take distributed similar to Eq. (\[ts45n\]) $$v(x)=v_0 e^{-x^2/2x_0^2}. \label{ts46n}$$ Then, as in the previous case, solving numerically the Newton equations we find $$R^2_v(t)=\frac{1}{v_{\rm tot}} \sum\limits_i x_i^2 v_i(t), \quad v_{\rm tot}=\sum\limits_i v_i(t). \label{et56}$$ Since the total momentum is conserved, $v_{\rm tot}$ is also independent of time and equal to its initial value at $t=0$. From the slope of $R^2_v(t)$ we can calculate the diffusion coefficient of the momentum $D_v$ using one dimensional equation $$R_v^2(t)=2D_v t \label{x5asd}$$ similar to Eq. (\[x5as\]). In both cases we have obtained for diffusion coefficients $D_u$ and $D_v$ the same value as was derived from the structure factor, $D_u\approx D_v\approx D_{\rm rw}=0.7$. It confirms our statement that the displacement structure factor $S({\bf q}, \omega)$ gives us the information about diffusion of particle displacements (or momentums). The diffusion of momentum is usually related to viscosity $\eta$ of the medium. Therefore in the case of $\mu=0$ our lattice has no rigidity but has a finite value of viscosity. In disordered lattices the diffusion of energy is different from the diffusion of particle displacements (momentums). In the harmonic approximation the eigenmodes with different frequencies do not interact with each other. Therefore the energy cannot be transferred from one eigenmode to other eigenmodes. It means that energy of every eigenmode $E(\omega_i)$ is conserved (with time). The total energy $E_{\rm tot}$ is just a sum of these eigenmode contributions $$E_{\rm tot}=\sum\limits_iE(\omega_i). \label{x5a9}$$ As a result, instead of one integral of motion (the total energy $E_{\rm tot}$), in a scalar harmonic system with $N$ particles we have $N$ integrals of motion $E(\omega_i)$. And for each frequency $\omega_i$ we have its own unique energy diffusivity $D(\omega_i)$. At this point our model decidedly confirms the physical picture suggested in papers [@Nature1; @Nature2; @Nature3; @Nature4; @Nature5] for amorphous silicon. We believe that it can be applied to some other glasses as well. Usually this diffusivity is hidden in a displacement/momentum structure factor of the $4$-th order. However, we calculated the diffusivity of energy $D(\omega)$ in a different way using two different approaches as it was discussed in Section \[ub55p\]. The first approach is based on the direct solution of Newton equations. In the second approach we calculated the diffusivity using Edwards and Thouless formula [@thouless]. Both approaches give the same result. In the first approach we used a short external force pulse $\Delta t$ exciting vibrations in a small space region of the lattice and in a small frequency interval $\Delta\omega\approx 1/\Delta t$ near frequency $\omega$. Then on a time scale $t\gg \Delta t$ the energy diffused through the lattice. Using Newton equations of motion we calculated this diffusion directly. It was supposed that the interval $\Delta\omega$ is much bigger than the interlevel spacing $\delta\omega$ and therefore the former consists of many eigenmodes. In the thermodynamic limit $\delta\omega\propto 1/N\to 0$ if $N\to\infty$. Therefore in an infinite system we can take the interval $\Delta\omega$ arbitrary small. The energy diffusion coefficient $D(\omega)$ in this case is a function of frequency $\omega$. Approaching the localization threshold $\omega_{\rm loc}$ the diffusivity $D(\omega)$ should go to zero. We applied this method for $\mu=0$, when there are no phonons in the lattice. In this case we obtained for diffusivity at zero frequency $D(0)\approx 0.4$, i.e. the value about factor of two smaller than for diffusivity of displacements, $D_u$. However this approach is rather difficult to implement for computer simulations in the case when $\mu\ne 0$. In this case we have phonons in the lattice with long mean free paths. And samples with much bigger sizes are necessary. Therefore, to calculate the diffusivity $D(\omega)$ for arbitrary value of $\mu$ (including the case of $\mu=0$), we used another approach. In this approach Edwards and Thouless formula [@thouless], $D(\omega_i) = c L^2 |\Delta\omega_i|$, was used. It relates the diffusivity $D(\omega_i)$ with shift of the eigenfrequencies $\Delta\omega_i$ due to change of the boundary conditions in one direction. The proportionality coefficient $c$ we found from the comparison with the Newton method for $\mu=0$. In this case both methods result in the same frequency dependence of $D(\omega)$. The diffusivity of vibrational modes $D(\omega)$ in disordered lattices is a very important quantity. It determines the thermal conductivity [@Nature2; @jammed1; @jammed2] $$\varkappa(T) \propto \int\limits_0^\infty d\omega g(\omega)D(\omega)C(\omega, T). \label{thcond1}$$ Here $g(\omega)$ is density of states and $C(\omega,T)$ is specific heat of harmonic oscillator $$C(\omega,T)=\left(\frac{\hbar\omega}{T} \right)^2\frac{e^{\hbar\omega/T}}{\left(e^{\hbar\omega/T}-1 \right)^2}. \label{specificheat1}$$ Localized modes have $D(\omega_i)=0$ and make no contribution to $\varkappa(T)$. If functions $g(\omega)$ and $D(\omega)$ are approximately constant in some frequency interval (the case that we have, for example, in our picture for $\omega>\omega_{\rm IR}$), then we find from Eq. \[thcond1\] that approximately $\varkappa(T)\propto T$ in the corresponding temperature range [@jammed1]. It explains a quasi-linear temperature dependence of the thermal conductivity above the plateau observed in glasses [@CahillPohl]. With increasing frequency the functions $g(\omega)$ and $D(\omega)$ finally drop to zero and thermal conductivity saturates at some constant level independent of temperature. Thus the conception of diffusons gives clear explanation for the temperature dependence of the thermal conductivity of glasses and other disordered systems. Summarizing, using a stable random matrix approach we have presented a consequent theory of vibrational properties in strongly disordered systems. In these systems a relative amount of phonons is small and almost all delocalized vibrations are diffusons. The diffusons play an important role and are responsible for the transport properties of glasses at higher temperatures. Presumably they are also accounted for the mysterious $q^2$ dependence of the vibrational line width $\Gamma(q)$ observed in many experiments on inelastic x-ray scattering in glasses. Therefore we think that it is necessary to take them into account in interpretation of experimental data. Acknowledgments =============== We are very grateful to V. L. Gurevich and Anne Tanguy for many stimulating discussions and gratefully acknowledge interesting discussions with B. Rufflé and E. Courtens as well. One of the authors (DAP) thanks the University Lyon 1 for hospitality. This work was supported by St. Petersburg Government (diploma project no. 2.4/29-06/143C), Dynasty Foundation, RF President Grant “Leading Scientific Schools” NSh-5442.2012.2 and Russian Ministry of Education and Science (contract N 14.740.11.0892). Appendices ========== Lattices with cut out bonds {#cutlat} --------------------------- ![The normalized DOS $g(\omega)$ for dynamical matrix $M=AA^T+\mu M_0$ with $\mu=1$ and different percentage $100\%-p$ of cut out springs calculated with precise numerical KPM solution for cubic lattice with $N=200^3$ (full lines). Straight lines are calculated according to Eq. (\[st56\]) with sound velocity $v=\sqrt{E}$. The Young modulus $E$ is calculated in the same way as in the Section \[phonons\].[]{data-label="fig:cut"}](Fig_21) Consider here the case when some part of springs $\mu$ are cut out from the matrix $\mu M_0$ in dynamical matrix (\[rt56e\]). The value of parameter $\mu=1$ we will keep fixed. Let parameter $p$ gives the percentage of remaining springs. The percolation threshold in the simple cubic lattice for bond percolation problem is at $p_c\approx 25\%$ [@percolation]. If $p<p_c$, then there is no infinite cluster of connected springs and therefore matrix $\mu M_0$ with cut out springs itself has no acoustical phonon-like modes at all. Nevertheless, the full dynamical matrix (\[rt56e\]) still has well defined phonon modes with density of states $\propto\omega^2$ for all positive values of $p$ even below the percolation threshold. The normalized density of states $g(\omega)$ for $\mu=1$ and different values of $p$ is shown on Fig. \[fig:cut\]. The straight lines show the phonon contribution to the DOS calculated from Eq. (\[st56\]) with sound velocity given by Eq. (\[s7cvbf\]). The Young modulus $E$ was calculated numerically using Eq. (\[d6vg\]) for the lattice with $N=10^6$ particles (one realisation) in the same way as it was done in Section \[phonons\]. The details of these calculations will be published elsewhere. Superposition of two random matrices {#superposition} ------------------------------------ Another (less obvious) possibility to get phonons is to add to the random dynamical matrix $AA^T$ a random matrix $\beta BB^T$. Here $\beta$ is a parameter and the random matrix $B$ is build in the same way as random matrix $A$ but they are statistically independent from each other. Though both terms $AA^T$ and $\beta BB^T$ taken separately have zero rigidity (and do not have phonons) their superposition introduces a finite rigidity $E$ to the system. The rigidity changes when we vary parameter $\beta$ as $E\propto\sqrt{\beta}$ and goes to zero when $\beta\to 0$. So the scaling relations ![The normalized DOS $g(\omega)$ for dynamical matrix $M=AA^T+\beta BB^T$ with different $\beta$ calculated with precise numerical KPM solution for simple cubic lattice with $N=100^3$ (full lines). Straight lines are calculated according to Eq. (\[st56\]) with sound velocity $v=\sqrt{E}$. The Young modulus $E$ is calculated in the same way as in the Section \[phonons\].[]{data-label="fig:beta"}](Fig_22) in this case for $\beta\ll V^2$ are the same as in Section \[scaling\] with replacement of $\mu$ by $\beta$. The preliminary results obtained within this approach are shown on Fig. \[fig:beta\]. Further details will be published elsewhere. These two examples show clearly, that appearance of phonons in the system is not related to the crystalline order in the term $\mu M_0$. The issue is more complicated. We are going to discuss this problem in more details elsewhere. Displacement structure factor {#DSF} ----------------------------- Let us consider the displacement structure factor given by Eq. (\[45gto\]) $$S({\bf q}, \omega) = \frac{2}{NT}\left|\sum\limits_{i=1}^Ne^{-i{\bf q}{\bf r}_i}\int\limits_0^{T} u({\bf r}_i, t) e^{i\omega t}dt \right|^2 . \label{45gto1}$$ We will assume that initial velocities of all particles at $t=0$ are zero. Then the displacement of $i$-th particle $u({\bf r}_i, t)$ as a function of time can be written in the form $$u({\bf r}_i, t) = \sum\limits_{j=1}^N a_j e_i(\omega_j) \cos(\omega_j t). \label{eq:ut}$$ Here $e_i(\omega_j)$ — is eigenvector of the dynamical matrix $M$ corresponding to $i$-th particle and eigenfrequency $\omega_j$. The eigenvectors satisfy equations $$\sum\limits_{j=1}^N M_{ij}e_j(\omega_k)=\omega_k^2e_i(\omega_k). \label{s66f}$$ They form an orthogonal set [@maradudin], so that $$\sum\limits_{j=1}^N e_{i}(\omega_j)e_{k}(\omega_j)= \sum\limits_{j=1}^N e_j(\omega_i)e_j(\omega_k) = \delta_{ik}. \label{z7bv}$$ Using (\[z7bv\]), one can write the coefficients $a_j$ in (\[eq:ut\]) in terms of the particle displacements for $t=0$ $$a_j=\sum\limits_{i=1}^N u({\bf r}_i, 0) e_i(\omega_j). \label{s55cv}$$ The initial displacements $u({\bf r}_i, 0)$ are independent Gaussian random variables with zero mean and unit variance $$\left<u({\bf r}_i, 0)\right>=0, \quad \left<u({\bf r}_i, 0)u({\bf r}_j, 0)\right>=\delta_{ij} . \label{a6xc}$$ Basing on this equation and making use of (\[s55cv\]) and of (\[z7bv\]) one can prove that the coefficients $a_j$ are also independent random Gaussian variables $$\left<a_j\right>=0, \quad \left<a_i a_j\right>=\delta_{ij}. \label{q1b6}$$ Using this property, one can evaluate the average (\[45gto1\]) as $$\left<S({\bf q}, \omega)\right>=\frac{2}{NT}\sum\limits_{j=1}^N \left|\sum\limits_{i=1}^N e_i(\omega_j) e^{-i{\bf q}{\bf r}_i} \right|^2 \left|\int\limits_0^T \cos(\omega_j t)e^{i\omega t}dt\right|^2 . \label{88vb}$$ Having in mind that $$\lim_{T\to \infty}\frac{2}{T}\left|\int\limits_0^T\cos(\omega_jt) e^{i\omega t} dt \right|^2 = \pi \left(\delta(\omega-\omega_j)+\delta(\omega+\omega_j)\right)$$ and taking only positive frequencies, we arrive to $$\left<S({\bf q}, \omega)\right> = \frac{\pi}{N}\sum\limits_{j=1}^N\left|\sum\limits_{i=1}^N e_i(\omega_j)e^{-i{\bf q}{\bf r}_i}\right|^2\delta(\omega-\omega_j). \label{eq:VSqw2}$$ [99]{} S. 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--- abstract: | Solar-like oscillations are stochastically excited by turbulent convection. In this work we investigate changes in the acoustic oscillation power spectrum of solar-type stars by varying the treatment of convection in the equilibrium structure and the properties of the stochastic excitation model. We consider different stellar models computed with the standard mixing-length description by [-@Bohm58] and with a generalized formulation of the mixing-length approach by Gough (1976, 1977). We calculate the acoustic power generated by the turbulent convection which is injected stochastically into the acoustic pulsation modes. Large differences in the oscillation powers are obtained depending on the choice of the assumed convection formulation. We show that the high-quality data Eddington will provide, will allow us to distinguish between theoretical predictions of acoustic power spectra obtained with different convection models. author: - 'R. Samadi' - 'G. Houdek' - 'M.-J. Goupil' - 'Y. Lebreton' - 'A. Baglin' nocite: - '[@Gough76; @Gough77]' - '[@Samadi00I]' - '[@Gough76; @Gough77]' - '[@Bohm58]' - '[@Gough76]' - '[@Samadi00III; @Samadi00b]' title: 'Oscillation power across the HR diagram : sensitivity to the convection treatment' --- Introduction ============ The amplitudes of solar-like oscillations are defined by the balance between excitation and damping. The oscillations are excited stochastically by the acoustic noise generated by the turbulent motion of the convective elements. In Samadi & Goupil (2001, PaperI hereafter) a theoretical formulation for the acoustic power injected into solar-like oscillations is proposed and which supplements previous theories. We refer the reader to [-@Samadi01] for a detailed summary and discussion on some recent unsolved problems. The excitation process depends on the assumed turbulence spectrum, as discussed, for example, by [-@Samadi00b] and [-@Samadi01]. It also depends crucially on the convection model to compute the stratification of the convectively unstable layers in the equilibrium model. The amount of energy injected into the oscillations depends strongly on the velocity of the convective elements. The main goal of this work is to asses changes in the oscillation power spectrum due to modifications of the convection treatment in the equilibrium model. We consider two different formulations: the classical description of mixing-length theory by [-@Bohm58] and a nonlocal generalization of the mixing-length formulation by Gough (1976, 1977). Additionally we study the dependence of the oscillation power on the assumed turbulence spectrum in the excitation model for both convection formulations. The theoretical formulation in PaperI involves two free parameters. These parameters are calibrated such as to reproduce for a solar model the observed acoustic power spectrum (Section 2). In Section 3 we compute oscillation power spectra for several stellar models using the two convection formulations. We conclude that Eddington’s performance will allow us to distinguish between the two treatments of convection and consequently that Eddington will provide further constraints on convection models. Power injected into solar-like oscillations =========================================== Theory of the stochastic excitation ----------------------------------- The acoustic power $P$ injected into the oscillations is defined ($e.g.$, [@GMK94]) in terms of the mode damping rate $\eta$, the oscillation mean-square amplitude $\langle A^2 \rangle$, the mode inertia $I$ and oscillation frequency $\omega$ as : [ $$\begin{aligned} P(\omega) = \eta\; {\langle A^2 \rangle}\;I\;\omega^2\,. \label{eq:P_omega} \end{aligned}$$]{} The mean-square amplitude is defined by both the excitation by turbulent convection and by the damping process. It can be written as [ $$\begin{aligned} \left < A^2 \right > & \propto & \eta^{-1} \int_{0}^{M}{\rm d}m \, \rho \, w^3 \, \Lambda^4 \,\left (\frac{{\rm d}\xi_{\rm r}} {{\rm d}r} \right )^2 \nonumber \\ & & \times \, \left \{ \mathcal{S}_{\rm R} (\omega,m) + \mathcal{R}^2(m) \, \mathcal{F}^2( \xi_{\rm r},m) \, \mathcal{S}_{\rm S} (\omega,m)\right \}\,, \label{eqn:A2} \end{aligned}$$]{} where $\displaystyle{\xi_{\rm r}}$ is the radial displacement eigenfunction, $\rho$ the density, $\Lambda$ is the mixing length, $w$ the vertical rms velocity of the convective elements; $\mathcal{F}^2(\xi_{\rm r},m)$ is a function which includes the second derivative of $\xi_{\rm r}$, and $\mathcal{R}^2(m)$ describes the ratio of the excitation by the entropy fluctuations to that by the Reynolds fluctuations. The source functions $\mathcal{S}_{\rm R}(\omega,m)$ and $\mathcal{S}_{\rm S}(\omega,m)$ describe the contributions from the Reynolds and entropy fluctuations, respectively, arising from the smaller scales of the turbulent cascade. Detailed expressions for $\left < A^2 \right >$, $\mathcal{S}_{\rm R}(\omega,m)$, $\mathcal{S}_{\rm S}(\omega,m)$, $\mathcal{R}^2$ and $\mathcal{F}^2$ are given in PaperI. The source functions include the turbulent kinetic energy spectrum $E(k)$, and the turbulent spectrum of the entropy fluctuations $E_s(k)$ which can be related to $E(k)$ by a simple expression ($e.g.,$ [@Samadi00II], PaperII hereafter). Both source functions $\mathcal{S}_{\rm R}(\omega,m)$ and $\mathcal{S}_{\rm S}(\omega,m)$ are integrated first over all eddy wavenumbers $k$, followed by an integration over the stellar mass $M$ to obtain the acoustic power $P$ (see Eq.\[eqn:A2\]). In the present work, $\rho$ and $w$ are obtained from the two equilibrium models, computed with the aforementioned two convection formulations. The eigenfunctions $\xi_{\rm r}$ and eigenfrequencies $\omega$ are obtained from two different oscillation codes and the $k$-dependence of $E(k)$ is inferred from different observations of the solar granulation and from theory. The equilibrium models ---------------------- We consider two sets of stellar models: the first set consists of complete models obtained with the CESAM evolutionary code in which the convective heat flux is computed according to the classical mixing-length theory by [Böhm-Vitense]{} (1958, C-MLT hereafter). The momentum flux (sometimes referred to as turbulent pressure) is neglected in this set of models. The eigenfunctions are obtained from the adiabatic FILOU pulsation code by [-@Tran95]. The detailed input physics used in this set of models is described in Paper II. The second set consists of envelope models computed in the manner of [-@Balmforth92a] and [-@Houdek99]. In these models convection is treated with the nonlocal mixing-length formulation by Gough (1976, G-MLT hereafter), which consistently includes the momentum flux (i.e., the $rr$-component of the Reynolds stress tensor). The eigenfunctions are obtained from the nonadiabatic pulsation code by [-@Balmforth92a], which includes both the Lagrangian perturbations of the heat and momentum fluxes in the manner of [-@Gough77]. For both model sets the mixing length $\Lambda = \alpha \, H_p$, where $H_p$ is the local pressure scale height and $\alpha$ the mixing-length parameter, is calibrated first to a solar model to obtain the helioseismically determined depth of the convection zone of $0.287$ solar radii ([@C-DGT91]). Models for stellar turbulence ----------------------------- Several turbulent spectra $E(k)$ were discussed in Paper II: the “Nesis Kolmogorov Spectrum” (NKS hereafter) and the “Raised Kolmogorov Spectrum” (RKS hereafter) were suggested from observations of the solar granulation by [-@Espagnet93] and [-@Nesis93]. Here we also consider the “Broad Kolmogorov Spectrum” (BKS hereafter) by [-@Musielak94]. These spectra are depicted in Figure \[fig:spc\_cinetique\]. For wavenumbers $k>k_0$, where $k_0$ is the smallest wavenumber of the classical Kolmogorov spectrum, all spectra follow the classical Kolmogorov scaling law, $k^{-5/3}$. Only in the low-wavenumber range, $k < k_0$, where kinetic energy is injected into the turbulent cascade, the spectra exhibit different scaling laws. These spectra were considered by [-@Samadi00b] to compute acoustic power spectra for various solar-type stars and are also considered in this paper. From observations of the solar granulation [-@Espagnet93] and [-@Nesis93] suggest various values for $k_0$. Because of this ambiguity in the choice of $k_0$, we relate $k_0$ to the mixing length $\Lambda$ by $k_0=2\pi \, / \, (\beta \Lambda)$, where $\beta$ is a free parameter of order unity (PaperI). Another uncertainty in the formulation of the excitation model it the eddy correlation time scale. Because of our still poor understanding of turbulent convection the eddy correlation time scale is not well defined and therefore needs to be scaled, leading to an additional free parameter $\lambda$ of order unity ($e.g.$, [@Balmforth92b]). As it was demonstrated in Paper II the oscillation power computed for the Sun depends crucially on the values of the free parameters $\lambda$ and $\beta$. Calibration of the free parameters ---------------------------------- The mean square surface velocity $v_{\rm s}$ is related to the acoustic power $P(\omega)$ and the mode damping rate $\eta$ by the expression : [ $$v_{\rm s}^2 = \xi_{\rm r}^2(r_{\rm s}) \; P(\omega) \, / \, 2 \eta I\,, \label{eqn:vs}$$]{} where $r_{\rm s}$ is the radius at which the oscillations are observed. The acoustic power $P$ is computed for the two calibrated solar models using the aforementioned convection formulations. For the velocity estimates $v_{\rm s}$, Eq.\[eqn:vs\], we assume the observed linewidths (damping rates) by [-@Libbrecht88]. The free parameters $\beta$ and $\lambda$ are calibrated for all turbulent spectra (see Section 2.3) and for both solar models to fit the estimated velocities $v_{\rm s}$ to the observations by [-@Libbrecht88]. Figure \[fig:VRSEP\_sungh\] displays the results for the solar model computed with G-MLT. The corresponding calibrated values of $\beta$ and $\lambda$ are listed in Table \[tab:adjusted\_parameters\_G-MLT\] (see Paper II for the calibration results of the models computed with the C-MLT). At low frequencies the velocity $v_{\rm s}$ computed with G-MLT is in better agreement with the data than the results published in Paper II obtained with the C-MLT. At high frequencies the differences in $v_{\rm s}$ obtained with the various turbulent spectra are of similar small magnitude than the results of Paper II which assumed the C-MLT. Best agreement between theory and measurements is obtained with the NKS for both convection formulations. [llll]{} spectrum&$\beta\lambda$&$\beta$&$\lambda$RKS & 0.6 & 1.96 & 0.31 BKS & 1.6 & 4.06 & 0.39 NKS & 2.6 & 3.11 & 0.83 Scanning the HR diagram ======================= Stellar models -------------- We consider various solar-type stars with masses between $1\,M_\odot$ and $2\,M_\odot$ in the vicinity of the main sequence. The model parameters are listed in Table \[tab:models\_param\], which correspond to the models considered previously by [-@Samadi00b]. As for the solar models in Section 2 we consider two sets of stellar models: the first set, computed with the C-MLT, is adopted from [-@Samadi00b]. The second set, computed with G-MLT, assumes the calibrated mixing length of the solar model discussed in Section 2. The acoustical cut-off frequencies are very similar between the two sets of models and are displayed in Table \[tab:models\_param\] for the first set. -------- --------------- ------------------------ --------------- --------- ---------- -- Models $L$ $T_{\mathrm{\rm eff}}$ $M$ Age $ \nu_c$ \[$L_\odot$\] \[K\] \[$M_\odot$\] \[Gyr\] \[mHz\] $A$ 12.1 6350 1.68 1.79 1.0 $B$ 9.0 6050 1.44 3.05 1.0 $C$ 6.6 6400 1.46 2.40 1.5 $D$ 3.7 5740 1.08 7.33 1.5 $E$ 3.5 6120 1.25 4.10 2.3 $F$ 2.6 6420 1.25 1.76 3.6 -------- --------------- ------------------------ --------------- --------- ---------- -- : Model parameters of solar-type stars. The model age and acoustical cut-off frequency $\nu_c$ are listed for the models obtained with the CESAM evolutionary code which assumes the C-MLT for computing the convective heat flux. []{data-label="tab:models_param"} Dependence on convection models ------------------------------- For all stellar models in both sets we assume the NKS and the calibrated values of $\beta$ and $\lambda$ quoted in [-@Samadi00b] and in Table \[tab:adjusted\_parameters\_G-MLT\]. The position in the HR diagram of all stellar models and a qualitative overview of their acoustic power spectra $P$ are depicted in Fig.  \[fig:pRSEPnkcs\_stars2stars\_HR\]. Detailed results of $P$ are shown in Fig. \[fig:pRSEP\_CMLTvsGMLT\]. At high frequencies the differences in $P$ between models computed with the C-MLT and G-MLT increase with increasing effective temperature $T_{\rm eff}$ and luminosity $L$. As discussed in [-@Houdek96], the nonlocal formulation (G-MLT) predicts smaller temperature gradients in the upper superadiabatic region relative to the C-MLT. This means that convection is more efficient in models computed with G-MLT which leads to a different profile (depth-dependence) of the superadiabatic temperature gradient between the C-MLT and the G-MLT. The differences in the superadiabatic temperature gradient between the two model sets increase with $L$ and $T_{\rm eff}$. These results are illustrated in Fig. \[fig:cmp\_gradT\] which shows the superadiabatic temperature gradient $\nabla-\nabla_{ad}$ versus $R_* -r$, with $R_*$ being the radius of the star. With increasing $L$ or $T_{\rm eff}$, the maximum of $\nabla - \nabla_{ad}$ is shifted more rapidly to deeper layers for the G-MLT models than for the C-MLT models. This leads to progressively larger differences in the convective velocities $w$ and in the shape of the eigenfunctions between the two sets of models. These differences in $w$ (note that $P$ depends crucially on $w$, see Eq.\[eqn:A2\]) and in the shape of the eigenfunctions are particular large in the superficial layers and result in a larger amount of acoustic power injected into high frequency modes for models computed with the C-MLT. There is an additional age effect: it can be seen from comparing the results of the models E and F (see Fig.\[fig:pRSEP\_CMLTvsGMLT\]). Model E has the same mass than model F but is older. The increase of the maximum acoustic power with age is found to be larger for models computed with the C-MLT than for the G-MLT models. Dependence on turbulence models ------------------------------- We compute $P$ for all the kinetic turbulent spectra discussed in Section 2.3. The results are shown in Fig.\[fig:pRSEP\_stars2stars\_spc\_CE\] for the models C and E. For the set of models computed with the C-MLT (top panels), the dependence of $P$ on the assumed turbulence spectra is more pronounced for the hotter model C than for model E. This is not observed for the model set computed with G-MLT (bottom panels). Models computed with the C-MLT exhibit a more pronounced decrease of the depth of the surface convection zone with increasing $T_{\rm eff}$. This leads to a smaller extend of the excitation region for hotter models compared to the models computed with G-MLT. As discussed in more detail by Samadi et al. (2001b,c) a shallower excitation region results in a more pronounced frequency-dependence of $P$ on the assumed turbulent spectra. Model C, obtained with G-MLT, exhibit a deeper excitation region and consequently the dependence of $P$ on the assumed turbulent spectra is smaller. This property can be explained by the larger efficacy with which G-MLT transports the convective heat flux. Discussion ---------- One of Eddington’s tasks will be the continuous observation of stellar luminosity oscillations over a time period of 30 days with a frequency accuracy of $0.3\,\mu$Hz ([@Favata00]). Furthermore the large telescope of Eddington (approximately ten times larger than that of the COROT mission) will lead to a noise level of $\sim 0.2$ ppm for stars with a magnitude $m_v=6$ and for an observing period of 30 days. For comparison, COROT will reach a noise level of $0.7$ ppm ([@Auvergne00]) for stars with a magnitude $m_v\simeq6$ and for a continuous observing period of 5 days. Will the high-quality observations of the Eddington mission be accurate enough to allow us to distinguish between the changes in the predicted oscillation powers $P$ of Section 3.2 and Section 3.3 ? In order to answer this question we estimate the expected accuracy of Eddington’s measurements of $P$. [-@Kjeldsen95] suggested a very simplified scaling law between oscillation amplitude ratios $(\delta L/L)/v_{\rm s}$ and $T_{\rm eff}$: [ $$\delta L/ L \propto v_{\rm s} \, T_{\rm eff}^{-1/2}\,, \label{eqn:dl_vs}$$]{} assuming adiabatic oscillations in a purely radiative star. It suggests that the amplitude ratios scale inverse proportionally to the effective temperature (but see also Houdek et al., 1999, Fig.14, who found that the amplitude ratios scale directly with $T_{\rm eff}$). According to Eq.(\[eqn:vs\]) and Eq.(\[eqn:dl\_vs\]) the relative error for $P$ can be expressed as [ $$\frac{\Delta P}{P} = 2 \, \frac{\Delta \delta L}{\delta L} + \frac{\Delta \eta}{\eta}\,, \label{eqn:DeltaP_P}$$]{} where $\delta L$ is obtained from Eqs.(\[eqn:vs\], \[eqn:dl\_vs\]). The damping rates $\eta$ are supplied from the nonadiabatic pulsation code of [-@Balmforth92a] assuming equilibrium models computed with G-MLT. In Figure \[fig:pRSEP\_stars2stars\_spc\_CE\] and \[fig:pRSEP\_CMLTvsGMLT\] the error bars $\Delta P$ are plotted according to Eq.(\[eqn:DeltaP\_P\]) assuming a noise level of $\Delta(\delta L)\sim 0.2$ ppm and $\Delta(\eta / 2 \pi)\sim 0.3 \, \mu$Hz. We conclude that for hotter stars with a magnitude $m_v\leq6$, Eddington will provide data of sufficient accuracy which will allow us to distinguish between the results obtained with the two considered convection formulations and consequently will provide further details on how to improve stellar convection models. GH acknowledges the support by the UK Particle Physics and Astronomy Research Council. [19]{} natexlab\#1[\#1]{}url \#1[[\#1]{}]{}urlprefix M., [the COROT Team]{}, 2000, In: The Third MONS Workshop : Science Preparation and Target Selection, Aarhus University, 135–138 N.J., 1992a, [MNRAS]{}, 255, 603 N.J., 1992b, [MNRAS]{}, 255, 639 E., 1958, Zeitschr. 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--- author: - 'S. Ansoldi' --- The study of the dynamics of an (infinitesimally)[^1] thin surface layer separating two domains of spacetime is an interesting problem in General Relativity. The system can be described in a very concise and geometrically flavored way using Israel’s junction conditions [@bib:NuCim1966.B44.....1I; @bib:NuCim1967.B48...463I; @bib:PhReD1991..43..1129I]. Starting from these toeholds many authors have then tackled the problem of finding some hints about the properties of the *still undiscovered* quantum theory of gravitational phenomena using shells as convenient models. In this context, just as examples of what can be found in the literature, we quote the seminal works of Berezin [@bib:PhLeB1990.241...402B] and Visser [@bib:PhReD1991..43...402V], that date back to the early nineties, or the more recent [@bib:JMPhA2002..17...979B; @bib:PhReD2002..65064006R] and references therein. What we are going to shortly discuss in the present contribution is set in this last perspective and suggests a semiclassical approach to define WKB quantum states for spherically symmetric shells. This method has already been used in [@bib:ClQuG2002..19..6321A]. Let us then consider a spherical shell (we refer the reader to [@bib:PhReD1991..43..1129I] for very concise/clear background material and for definitions). For our purpose the relevant result is equation (4) in [@bib:PhReD1991..43..1129I], i.e. the junction conditions[^2] $ K ^{-} _{ij} - K ^{+} _{ij} \propto S _{ij} - g _{ij} S / 2 . $ $K _{ij}$ is the extrinsic curvature of the shell and can have different values on the two sides ($+$ and $-$ spacetime regions) of it. $S _{ij}$ is the stress energy tensor describing the energy/matter content of the shell ($S$ is its trace). For a spherical shell these equations reduce to the single condition $$\epsilon _{-} ( \dot{R} ^{2} + f _{-} (R) ) ^{1/2} - \epsilon _{+} ( \dot{R} ^{2} + f _{+} (R) ) ^{1/2} = M (R) / R , \label{eq:sphjuncon}$$ where $f _{\pm} (r)$ are the metric functions in the static coordinate systems adapted to the spherical symmetry of the $4$-dimensional spacetime regions joined across the shell; $\epsilon _{\pm} = +1, -1$ are signs and $R$ and $M (R)$ are the the radius (a function of the proper time $\tau$ of a co-moving observer[^3]) and the matter content (what remains of $S _{ij}$) of the spherical shell, respectively. As shown for example in [@bib:NuPhB1990.339...417G; @bib:ClQuG1997..14..2727S; @bib:Thes.1994....TriestA] the above equation can be obtained from an effective Lagrangian[^4] $L _{\mathrm{EFF}} (R , \dot{R})$, as a first integral of the second order Euler-Lagrange equation. From $L _{\mathrm{EFF}} (R , \dot{R})$ the momentum conjugated to the single degree of freedom $R$ can be obtained as usual, $P (R, \dot{R}) = \partial L _{\mathrm{EFF}} (R , \dot{R}) / (\partial \dot{R})$. We are not interested in the explicit form of $P$ here, we just point out that it is a function of $R$ and $\dot{R}$ highly *non-linear* in $\dot{R}$. This non-linearity spoils the natural and simple interpretation of the momentum than we know from classical analytical mechanics. Nevertheless we can still solve for the classically allowed trajectories of the shell, using a *standard* analogy with the motion of a *unitary mass particle with zero energy in an effective potential* [@bib:PhReD1987..36..2919T; @bib:PhReD1987..35..2961G; @bib:PhReD1989..40..2511S]. This gives $\dot{R}$ as a function of $R$ and, substituting this expression for $\dot{R}$ in $P (R, \dot{R})$, we obtain the *conjugated momentum on a solution of the classical equations of motion*. In what follows we are going to indicate the momentum *evaluated on a classical trajectory* as $P (R ; {\mathcal{S}})$: this emphasizes that it is a function of $R$, that it depends on the set ${\mathcal{S}}$ of the other parameters of the problem, but, of course, it is not a function of $\dot{R}$. ![\[fig:actlev\][Plot of some WKB levels for the example discussed in the text. The curves $a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$, $i$, $j$ correspond to the quantum numbers $1,2,3,4,5,10,20,30,40,50$, respectively.]{}](ansoldifig.eps){width="8cm"} By integrating the expression for $P (R ; {\mathcal{S}})$ on a classically allowed trajectory with turning points $R _{\mathrm{MIN}}$ and $R _{\mathrm{MAX}}$, we can compute the value of the classical action for that trajectory. It is a function of the set of parameters characterizing the matter content and of the geometry, ${\mathcal{S}}$, and WKB quantum states of the system can now be defined as states for which the above action is a multiple of the quantum $$S ( {\mathcal{S}} ) = 2 \int _{R _{\mathrm{MIN}}} ^{R _{\mathrm{MAX}}} P (R ; {\mathcal{S}}) d R \sim n, \quad n = 0 , 1 , 2 , 3 , \dots{}. \label{eq:BohSom}$$ In quantum gravity we expect to have a theory that selects some geometries from the set of all possible ones consistently with quantum dynamics. In our discussion we have limited the treatment to a *minisuperspace approximation*, but, indeed, we see that the quantization condition does select only some of the possible geometries, those in which the parameters of the model are related by the quantum number $n$ as in . We can see this more explicitly in the following simple model: a dust shell ($M (R) = m$) separating two domains of anti-de Sitter spacetime with the same cosmological constant ($f _{-} (r) = f _{+} (r) = f (r) = 1 + r ^{2} / H ^{2}$): it is possible to prove that a non-trivial junction of the two spacetimes can be performed, although we are not going to discuss this aspect here nor we are going to describe the resulting global spacetime structure. In this case the set of parameters is ${\mathcal{S}} = \{ m , H \}$ and the quantization condition becomes $S ( m , H ) \sim n$, $n = 0 , 1 , 2 , 3 , \dots{}$. Some levels are plotted in figure \[fig:actlev\] and clearly show that given one of the parameters (say $m$) only a discrete set of values for the other $H$ is allowed: thus the quantization condition restricts the possible values that can be given to the parameters that characterize the spacetime geometry and/or the matter content of the shell. This is consistent with the general picture of a quantum gravitational scenario. Other applications of this general approach are presently under study. In particular, generalizations to higher dimensions [@bib:none.when........AGI] could be relevant, for example, in the context of brane cosmological models. [19]{} natexlab\#1[\#1]{}\[1\][“\#1”]{} url \#1[`#1`]{}urlprefix , W., *Nuovo Cimento B*, **44**, 1 (1966). , W., *Nuovo Cimento B*, **48**, 463 (1967). , C., and [Israel]{}, W., *Phys. Rev. D*, **43**, 1129–1142 (1991). , S., *Class. Quantum Grav.*, **19**, 6321–6344 (2002). , V. A., *Phys. Lett. B*, **241**, 194–200 (1990). , M., *Phys. Rev. D*, **43**, 402–409 (1991). , V., *J. Mod. Phys. A*, **17**, 979–988 (2002). , A., [Cruz-Pacheco]{}, G., [Minzoni]{}, A., [Padilla]{}, P., [Rosenbaum]{}, M., [Ryan Jr.]{}, M. P., [Smyth]{}, N. F., and [Vukasinac]{}, T., *Phys. Rev. D*, **65**, 064006(1–13) (2002). , C. W., [Thorne]{}, K. S., and [Wheeler]{}, J. A., *“Gravitation”*, W. H. 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[^1]: Far from being only idealizations of more realistic situations, shells have been extensively used to build concrete models of many astrophysical and cosmological scenarios (for a detailed bibliography, please, see the references in [@bib:ClQuG2002..19..6321A]). [^2]: Conventions are as in [@bib:Freem1970...1..1279W] and the definition of the (quantities relevant to the concept of) extrinsic curvature can be found in [@bib:PhReD1991..43..1129I; @bib:Freem1970...1..1279W]. [^3]: As usual an overdot, “$\dot{\quad}$”,denotes a total derivative with respect to $\tau$. [^4]: For a more general and deeper discussion see [@bib:PhReD2000..62044025K] and references therein or also, in addition, [@bib:JMaPh2001..42..2590G; @bib:Thes.2004....TriestS].
--- abstract: 'The application of artificial neural networks (ANNs) for the estimation of HI gas mass fraction () is investigated, based on a sample of 13,674 galaxies in the Sloan Digital Sky Survey (SDSS) with HI detections or upper limits from the Arecibo Legacy Fast Arecibo L-band Feed Array (ALFALFA). We show that, for an example set of fixed input parameters ($g-r$ colour and $i$-band surface brightness), a multidimensional quadratic model yields  scaling relations with a smaller scatter (0.22 dex) than traditional linear fits (0.32 dex), demonstrating that non-linear methods can lead to an improved performance over traditional approaches. A more extensive ANN analysis is performed using 15 galaxy parameters that capture variation in stellar mass, internal structure, environment and star formation. Of the 15 parameters investigated, we find that $g-r$ colour, followed by stellar mass surface density, bulge fraction and specific star formation rate have the best connection with . By combining two control parameters, that indicate how well a given galaxy in SDSS is represented by the ALFALFA training set () and the scatter in the training procedure (), we develop a strategy for quantifying which SDSS galaxies our ANN can be adequately applied to, and the associated errors in the  estimation. In contrast to previous works, our  estimation has no systematic trend with galactic parameters such as M$_{\star}$, $g-r$ and SFR. We present a catalog of  estimates for more than half a million galaxies in the SDSS, of which $\sim$ 150,000 galaxies have a secure selection parameter with average scatter in the  estimation of 0.22 dex.' author: - | Hossen Teimoorinia$^1$, Sara L. Ellison$^1$ & David R. Patton$^2$\ $^1$ Department of Physics & Astronomy, University of Victoria, Finnerty Road, Victoria, British Columbia, V8P 1A1, Canada.\ $^{2}$Department of Physics and Astronomy, Trent University, 1600 West Bank Drive, Peterborough, ON K9L 0G2, Canada.\ title: 'Pattern recognition in the ALFALFA.70 and Sloan Digital Sky Surveys: A catalog of $\sim$ 500,000 HI gas fraction estimates based on artificial neural networks.' --- \[firstpage\] galaxies: fundamental parameters -galaxies: evolution - methods: data analysis- methods: statistical- Astronomical data bases: surveys introduction ============ Neutral hydrogen plays an important role in the fuelling pipeline for star formation activity in galaxies. The study of HI can therefore provide insights into some of the main physical processes that drive galaxy evolution. To this end, numerous surveys (see Giovanelli & Haynes 2015 for a recent review) have been conducted with the aim of determining the HI mass () of large numbers of galaxies, such as the Arecibo Dual Beam Survey (ADBS, Rosenberg & Schneider 2000), the HI Parkes All-Sky Survey (HIPASS, Barnes et al. 2001) and the HI Jodrell All-Sky Survey (HIJASS, Lang et al. 2003). Most recently, the Arecibo Legacy Fast Arecibo L-band Feed Array (ALFALFA) survey has provided a wide area, moderate depth (log /M$_{\odot} >$8.5), blind survey of HI in the local ($z<0.06$) universe (Giovanelli et al. 2005). Due to the broad inverse correlation between gas fraction and stellar mass, relatively few massive galaxies are included in the ALFALFA detections. To complement the strategy of ALFALFA, the GALEX Arecibo SDSS Survey (GASS) has therefore performed deep 21 cm observations of $\sim$ 1000 massive (log M$_{\star}$/M$_{\odot} > 10$) galaxies down to a fixed gas fraction limit (Catinella et al. 2010, 2013). Coupled with multi-wavelength data from other large surveys, these large HI surveys have provided unprecedented insight into the connections between global gas properties and environment, nuclear activity, star formation and chemical enrichment (Cortese et al. 2011; Fabello et al. 2011; Bothwell et al. 2013; Lara-Lopez et al. 2013; Hughes et al. 2013). Despite the tremendous efforts of HI surveys, the measurement of HI masses continues to lag behind optical surveys. For example, in contrast to the $\sim$ 5000 detections available for the 40 per cent ALFALFA data release (Haynes et al. 2011), the main galaxy sample of the Sloan Digital Sky Survey (SDSS) contains more than two orders of magnitude more galaxies out to $z \sim 0.2$ (e.g. Strauss et al. 2002). For this reason, there is a long history of efforts that attempt to infer  from more readily available optical properties (Roberts 1969; Bothun 1984; Roberts & Haynes 1994). In the last decade, such efforts have been able to capitalize on large, homogeneous datasets, often in combination with multi-wavelength data in the UV and IR regimes (Kannappan 2004; Zhang et al. 2009; Catinella et al 2010, 2012; Toribio et al. 2011; Wang et al. 2011; Li et al. 2012; Denes, Kilborn, & Koribalski 2014; Eckert et al. 2015). These works have used different linear combinations of physical parameters to reduce scatter between the estimated and observed HI gas mass fraction (defined as  throughout this paper). However, despite the availability of extensive physical parameters, and the identification of the ‘best’ parameters for  estimation, the improvement in the accuracy of scaling laws has been small. For example, Zhang et al. (2009) used a linear combination of $i$-band surface brightness and $g-r$ colour that was motivated by the Kennicutt-Schmidt law to achieve a scatter in their  estimator of $\sim$ 0.31 dex. Catinella et al. (2010) found that the near ultraviolet (NUV) NUV$-r$ colour was the single best estimator of HI mass, and determined a ‘gas fraction plane’ by combining the stellar mass surface density ($\mu_{\star}$), NUV$-r$ and HI gas mass fraction. However, the scatter in this relation remained at 0.3 dex (see also Catinella et al. 2012). Motivated by the finding that gas fractions are linked to outer disk colours (Wang et al. 2011), Li et al. (2012) add the colour gradient to their linear estimator, applied to ALFALFA and GASS samples, but again the scatter in the HI gas fraction was not improved beyond $\sim0.3$ dex. Moreover, many of these linear scaling relationships still have significant systematics relative to stellar mass, age, colour or concentration indices (e.g., Zhang et al. 2009). Despite their current limitations, linear scaling relations and the gas fraction plane are useful as indicators of ‘HI normalcy’, which can be used to identify outliers (within a given sample) that are either particularly gas-rich or gas-poor (e.g. Catinella et al. 2012). Similarly, Cortese et al. (2011) demonstrate that the distance from the gas fraction plane is a good alternative to the classic ‘HI deficiency’ parameter (Haynes & Giovanelli 1984). However, extreme caution must be applied in using scaling relations to *estimate* , due to the inherently biased nature of the samples from which they are (currently) constructed. For example, the application of scaling relations derived from the blind, but shallow, ALFALFA data tends to over-estimate  in deeper, or targeted surveys (e.g., Huang et al. 2012). Problems may persist even when precautions are taken to identify galaxies with broadly similar properties as the calibration sample. For example, Zhang et al. (2009) identify a sample of star-forming galaxies from which they determine their scaling relation. It might therefore be expected that the distinction between star-forming and quiescent galaxies may help to identify samples of galaxies appropriate for the application of the Zhang et al. (2009) parametrization. However, Huang et al. (2012) show that the Zhang et al. (2009) relation is not a good predictor of  in ALFALFA, despite the domination of star-forming galaxies in the latter. Therefore, whilst linear scalings reveal the typical relationship between  and other optical/NUV/infrared (IR) properties *for a given sample*, they can not be used as a universal predictor of HI gas mass fraction, particularly in circumstances when galaxies are expected to deviate from the ‘normal’ relation (e.g. Cortese et al. 2011). In this paper, we explore the application of non-linear methods, such as artificial neural networks (ANNs), to the challenge of gas fraction estimation. We have previously demonstrated the application of ANN to various astronomical problems, including pattern recognition for the spectral classification of galaxies (Teimoorinia 2012), fitting applications in order to estimate line fluxes which can be used for Active Galactic Nucleus (AGN) classifications, metal abundances etc. (Teimoorinia & Ellison 2014) and a large public catalog of estimated IR luminosities (Ellison et al. 2016a) that can be used to determined star formation rates for galaxies dominated by AGN (Ellison et al. 2016b). Most recently, we have presented a method for ranking the parameters involved in galaxy quenching, which effectively disentangles the complex multi-variate nature of this physical process (Teimoorinia et al. 2016). We refer the interested reader to those works for details of the general ANN method. The non-linear ANN techniques used in these previous works are readily applied to the estimation of the gas fraction from large input datasets. In this work, we will train and validate the ANN using galaxies drawn from the ALFALFA.70 data release, matched to galaxies in the SDSS DR7. The trained network can then be applied to all galaxies in the DR7 (not covered by ALFALFA.70) in order to predict the gas fraction based on their optical properties. The layout of the paper is as follows. In Section \[sec-sample\] we describe the samples that are used for training the ANN and the method used for rejecting galaxies that may have their 21 cm fluxes contaminated by the presence of a close companion within the Arecibo beam. The results in the following sections are then presented with respect to three main objectives. First, we will explore whether non-linear methods are able to reduce the scatter in  scaling relations below what has been previously achieved for linear calibrations for a given set of input parameters (Section \[sec-example\]). Specifically, we use the same optical input parameters as Zhang et al. (2009) in order to compare the scatter of their linear calibration with that of the non-linear methods. Although this example uses a limited number of variables for training, it serves as a direct comparison of the performance of linear and non-linear techniques. In Section \[sec-example\] we demonstrate how differences in HI surveys can influence the application of scaling relations. In Section \[sec-performance\] the input data for the ANN are described and also a performance function, R, is introduced for ranking the weights of input parameters in the fitting procedure. A second performance metric is then used to determined which galaxy variables show the most important physical connection with . The third objective is then to develop the non-linear relationships between  and galactic properties into a procedure that can be robustly used to estimate  for galaxies in the SDSS, the results of which are presented in Section \[sec-fitting\]. In Section \[sec-pr\] we describe a pattern recognition procedure that determines how similar a given galaxy is to those that have been used in the training set (i.e. detections in ALFALFA). We hence are able to assign an ALFALFA detection probability, , to any SDSS galaxy; a robust gas fraction can be best determined for galaxies with high detection probabilities (i.e. the galaxy is similar to the ALFALFA detected galaxies on which the ANN was trained). The final strategy for defining a robust sample for applying the ANN is presented in Section \[sec-comb\]. In section \[sec-sum\] a short summary is presented. A public catalog of the HI gas mass fraction, the control parameters and errors accompanies this paper in electronic format. Samples {#sec-sample} ======= Training Set ------------ At the heart of our ANN approach is a training set based on matches between the SDSS DR7 galaxy catalog and the ALFALFA.70 data release which contains 6837 galaxies with 21 cm detections at a S/N$>$ 6.5. Following Haynes et al. (2011), we allocate a code = 1 to these objects (hereafter AC1). Haynes et al. (2011) also identify lower S/N detections (code = 2, hereafter, AC2). Although AC2 galaxies have been matched to galaxies in the SDSS and are therefore likely detections, we take the conservative measure of not including them as detections in our training sample. Our analysis also makes use of galaxies not detected by ALFALFA. Non-detections of SDSS galaxies are not explicitly reported in the ALFALFA data releases, so we make use of the non-detections for the ALFALFA.40 footprint as computed by Ellison et al. (2015). The 23,652 non-detections are allocated a code = 3 (AC3). ) It should be noted that the training sample is sufficiently low redshift (and also validation sets which are discussed in this paper) that it is safe to assume no evolution with redshift in the quantities of interest. While each of the galaxies in AC1 consists of an HI source with a single optical counterpart, the relatively large diameter of the Arecibo beam (3.3$\times$3.8 arcminutes) means that more than one galaxy may contribute to the 21 cm flux of a given source (Haynes et al. 2011; Jones et al. 2015), leading to overestimates of the HI fluxes of some galaxies. We therefore undertake a three-step process to remove from our AC1 sample galaxies whose HI flux may be contaminated by neighbouring galaxies within the beam. First, for every galaxy in our initial training set, we search for all known spectroscopic companions (from Simard et al. 2011) which may contribute significant additional flux to a given galaxy’s HI measurement. Given the paucity of gas in red sequence galaxies, we consider only companions which are blue. Following Patton et al. (2011), we classify galaxies as blue if they have Simard et al. (2011) global rest-frame colours of $g-r < -0.01*(M_r+21) + 0.65$. If a galaxy has one or more blue companions which lie within one beam radius (1.9 arcminutes) and within a relative velocity of 250 km s$^{-1}$, we conclude that there is a risk of significant HI contamination, and remove the galaxy from our training set. A total of 431 galaxies were removed from our training set for this reason. Second, we remove galaxies which lie within one beam radius and 250 km s$^{-1}$ of a known ALFALFA source. While approximately half of these sources have already been removed in the previous step, a comparable number remain due to spectroscopic incompleteness in SDSS. In particular, fibre collisions lead to a high rate of spectroscopic incompleteness at angular separations less than 55 arcsec (Blanton et al. 2003, Patton & Atfield 2008). A total of 27 additional galaxies are removed from our training set for this reason. Finally, we address the possibility of contamination by companions whose centres lie outside the ALFALFA beam and yet whose HI flux is likely to overlap the beam. We use ALFALFA HI gas fractions and the Mendel et al. (2014) stellar masses to compute the HI mass of all companions which lie outside the beam but within 250 km s$^{-1}$. We then estimate the HI radius of each companion, using the relationship between HI mass and diameter reported in Wang et al. (2016)[^1] If a companion’s estimated HI radius overlaps the beam, we remove the given host galaxy from our training set. An additional 50 galaxies are removed from our training set by this criterion, leaving a training set of 6329 galaxies whose HI fluxes are unlikely to be contaminated by neighbouring galaxies. Visual inspection of the SDSS images of a random subset of these galaxies confirms this interpretation, as no obvious blue galaxies are seen within one beam radius (with the exception of those with relative velocities greater than 250 km s$^{-1}$). Additional Samples and Ancillary Data ------------------------------------- In order to test our  estimator on independent datasets, we will also make use of several other samples that have been matched with the SDSS. The largest of these is the GASS sample[^2], for which we use the 342 detections from the data release 3 (Catinella et al. 2013). Additionally, we use the Cornell catalog ($\sim 30$ galaxies, Giovanelli et al 2007) as incorporated into the GASS representative sample and a sample of $\sim40$ post-mergers (PM) presented by Ellison et al. (2015).We have also used 279 galaxies (of 2839, matched with our ANN’s input parameter space) from the Nancay Interstellar Baryons Legacy Extragalactic Survey (NIBLES) sample, presented by van Driel et al. (2016). They use a sample in the redshift range z$<$0.04 selected on z-band magnitude ($-24 < \rm{M_z} < -13.5$) as a proxy for stellar mass. Galaxies in the NIBLES sample are at the bright end of the SDSS distribution. All of the above samples have large amounts of ancillary information available, which may be used as input variables for the  estimation, from the following sources: - Photometry is taken from SDSS DR7. Structural parameters, such as bulge fractions and galaxy sizes are taken from the re-processed SDSS images by Simard et al. (2011) and Mendel et al. (2014). The fluxes in different bands ($u, g, r, i, z$) are all corrected for Galactic extinction. - Stellar masses are taken from Mendel et al. (2014) based on their re-assessment of SDSS photometry. - Total star formation rates were taken from the MPA/JHU catalogs, which applied a colour dependent aperture correction to account for the light outside of the SDSS fibre (Brinchmann et al. 2004; Salim et al. 2007). Star formation rates are only used for those galaxies classified as ‘star-forming’ by the definition of Kauffmann et al. (2003). Specific SFRs are determined by combining the SFRs described above with the stellar masses from Mendel et al. (2014). - The halo masses come from the group catalogue of Yang et al. (2007, 2009). - Local environmental densities are computed as $\Sigma_n = \frac{n}{\pi d_n^2}$, where $d_n$ is the projected distance in Mpc to the $n^{th}$ nearest neighbour within $\pm$1000 . Normalized densities, $\delta_n$, are computed relative to the median $\Sigma_n$ within a redshift slice $\pm$ 0.01. In this study we adopt $n=5$. - Stellar mass density is defined as log($\mu_{*i})= \log(M_*) -\log(2 \pi R_{50i}^2$) in which M$_*$ is the stellar mass and R$_{50i}$ is the radius (in kpc) enclosing 50 per cent of the total Petrosian $i$-band flux. A simple example of HI mass estimation from non-linear methods {#sec-example} ============================================================== Before launching into a complex many-parameter ANN application for  estimation, we show a simple example of comparing the linear estimation of Zhang et al (2009) to a non-linear method. Zhang et al (2009) used a sample of $\sim$ 800 galaxies with HI masses in the Hyperleda database that are matched to galaxies in the SDSS to determine a calibration between and the $g-$ and $r$-band Petrosian apparent magnitudes and $i$-band surface brightness, defined as $\mu_i = m_i +2.5 \log(2 \pi R_{50i}^2$), where R$_{50i}$ is the radius (in units of arcsecond) enclosing 50 per cent of the total Petrosian $i$-band flux. The calibration presented by Zhang et al. (2009) is: $$\rm{log~M_{HI}/M_{*}}=-1.73\it{(g-r)}+\rm{0.22\mu_{i}-4.08} \label{eq-zhnag}$$ Using Eq. \[eq-zhnag\] we estimate  for the samples described in Section \[sec-sample\]: AC1, GASS, PM and Cornell, where we group the latter two samples together for plotting purposes due to their small size. In Fig. \[fig-zhang-valid\] we compare the estimated by Zhang et al. (2009)’s calibration (Eq. \[eq-zhnag\]) and the observed values. In each panel, the values in the lower right corner give the mean difference between the estimated and observed , and the scatter ($\sigma$) in these differences. There are systematic offsets between the observed and estimated  values, that vary between the samples. For example, the top panel of Fig. \[fig-zhang-valid\] shows that  in AC1 (ALFALFA detections) is underestimated, on average, by 0.22 dex by Zhang’s formula. The same seems to be true in the NIBLES data, with the offset occurring in a similar regime (i.e. log $>0$). The offset between AC1 and the Zhang et al. formulation has been previously reported by Huang et al. (2012) who suggest that differences in the methods of calculating the stellar masses may account for a systematic deviation of $\sim0.2$ dex. The top panel of Fig. \[fig-zhang-valid\] and explanation by Huang et al. (2012) demonstrate an important caveat for the application of any calibration method – if parameters are not uniformly derived, even ‘perfect’ calibrations will perform poorly. It is therefore vital to apply calibrations to datasets whose parameters have been derived as closely as possible to the original data. The second plot from the top of Fig. \[fig-zhang-valid\] shows that, in contrast to what was seen in AC1, the  estimated for the GASS sample using Eq. \[eq-zhnag\] has a tendency to be mildly over-estimated. Taken together, the top and middle panels therefore imply a total difference between AC1 and GASS estimations $\sim$ 0.45 dex. As described above, Huang et al. (2012) have suggested that this may be due, at least in part, to differences in stellar mass calibrations. In order to test this suggestion, we can re-derive the best fit linear coefficients of ($g-r$) and $\mu_i$ in Eq.\[eq-zhnag\] using AC1 and test this calibration on the GASS data. Since both samples have consistent sources of input parameters and stellar mass measurements, any systematic trend caused by inconsistencies therein should be removed. The AC1-calibrated version of Eq.\[eq-zhnag\] is then $$\rm{log~M_{HI}/M_{*}}=-2.332\it{(g-r)}+\rm{0.168\mu_{i}-2.528} \label{eq-zhnag_new}$$ In Fig. \[fig-alfa-gass-two-par\] we now compare the observed and estimated  for the newly derived coefficients in Eq.\[eq-zhnag\_new\] in the AC1 and GASS samples (blue and red points respectively). The first important result demonstrated by Fig. \[fig-alfa-gass-two-par\], is that Eq.\[eq-zhnag\_new\] does not perform well at estimating the data on which it is calibrated (AC1), indicating that AC1 can not be well fit with the parametrization in Eqs. \[eq-zhnag\] and \[eq-zhnag\_new\]. Application to the GASS sample also results in a strong systematic offset. Importantly, the mean offset between the estimated (using Eq. \[eq-zhnag\_new\]) and the observed GASS data is still $\sim0.45$ dex, the same as was inferred between AC1 and GASS using the original coefficients in Eq. \[eq-zhnag\]. This difference can no longer be attributed to differences in inhomogeneous stellar mass estimates or photometry, which were derived identically for the two samples. As we will demonstrate below, the offset is due to the fundamentally different nature of the galaxies in the two samples. ![Comparison of observed and estimated  based on the  calibration of Zhang et al. (2009), given by Eq.\[eq-zhnag\]. The mean offset between the estimated and observed  and the scatter ($\sigma$) therein are shown in each plot. The gray dashed line shows the one-to-one relation of  ($_{est}) =$($_{obs})$.[]{data-label="fig-zhang-valid"}](fig-01-1.pdf "fig:"){width="6.7cm" height="5.2cm"} ![Comparison of observed and estimated  based on the  calibration of Zhang et al. (2009), given by Eq.\[eq-zhnag\]. The mean offset between the estimated and observed  and the scatter ($\sigma$) therein are shown in each plot. The gray dashed line shows the one-to-one relation of  ($_{est}) =$($_{obs})$.[]{data-label="fig-zhang-valid"}](fig-01-2.pdf "fig:"){width="6.7cm" height="5.2cm"} ![Comparison of observed and estimated  based on the  calibration of Zhang et al. (2009), given by Eq.\[eq-zhnag\]. The mean offset between the estimated and observed  and the scatter ($\sigma$) therein are shown in each plot. The gray dashed line shows the one-to-one relation of  ($_{est}) =$($_{obs})$.[]{data-label="fig-zhang-valid"}](fig-01-3.pdf "fig:"){width="6.7cm" height="5.2cm"} ![Comparison of observed and estimated  based on the  calibration of Zhang et al. (2009), given by Eq.\[eq-zhnag\]. The mean offset between the estimated and observed  and the scatter ($\sigma$) therein are shown in each plot. The gray dashed line shows the one-to-one relation of  ($_{est}) =$($_{obs})$.[]{data-label="fig-zhang-valid"}](fig-01-4.pdf "fig:"){width="6.7cm" height="5.2cm"} ![A re-calibration of Eq.\[eq-zhnag\] based on AC1 data yield the new coefficients given in Eq.\[eq-zhnag\_new\]. This new linear relationship is applied to AC1 (blue points) and GASS (red points). The gray dashed line shows the one-to-one relation of  ($_{est}) =$ ($_{obs}$). The deviations of the estimated from the observed  in AC1 indicate that a linear combination of $g-r$ and $\mu_i$ is insufficient for estimating the HI gas fraction. A systematic error of $\sim0.45$ dex is found when the AC1 linear calibration in Eq.\[eq-zhnag\_new\] is applied to the GASS sample. This offset can no longer be due to inconsistencies in the input data. []{data-label="fig-alfa-gass-two-par"}](fig-02.pdf){width="8cm" height="6.5cm"} To compare the performance of the linear (Zhang et al. 2009) and non-linear approaches, we follow the methods outlined in Teimoorinia & Ellison (2014), in which non-linear methods were used to determine Balmer decrements and emission line fluxes. Here, we use the AC1 sample of HI detections from ALFALFA.70, combined with the same parameters used by Zhang et al. (2009) and tested in the above linear fit tests ($g-r$ and $\mu_i$). Therefore, the only change we are making is in the methodology of the fitting, not in the parameters used. We use the Levenberg-Marquardt algorithm (Marquardt 1963) to find the coefficients in the following equation: $$f(C,X)=\sum\limits_{i,j=1}^N C_{ij}X_iX_j+\sum\limits_{i=1}^N C^{'}_iX_i+C^{''}. \label{eq-basis-function}$$ In the above equation, $f(x)$= log   and X is: $$X= \left( \begin{array}{c} m_g\\ m_r\\ m_i\\ R_{50i}\\ \end{array} \right)$$ Here, N=4 and X$_i$ (i=1 to 4) in which $m_{g}$ , $m_{r}$, $m_i$ are apparent magnitudes. These are the individual variables that comprise $g-r$ and $\mu_i$ in the parametrization of Zhang et al. (2009) in Eqs. \[eq-zhnag\] and \[eq-zhnag\_new\]. We use AC1 as the training set and determine the 15 coefficients required for the above parametrization. Fig. \[fig-alfa-gass-4-par\] shows the estimated vs. observed gas fractions derived from Eqn. \[eq-basis-function\], which is a representation of the matrix-based solution. The residual tilt in Fig. \[fig-alfa-gass-two-par\] in the AC1 data (blue points) is now completely removed and scatter is reduced from 0.31 dex in the linear calibration (Eq. \[eq-zhnag\_new\]) to 0.22 dex. This demonstrates that although $g-r$ and $\mu_i$ *can* be used for  calibration, a more complex representation, such as the matrix form, can provide a better match (due to more complex connections between the parameters). Moreover, this matrix form can also remove the deviation and skew seen in Fig. \[fig-alfa-gass-two-par\] for AC1, which is obtained using only three (different) coefficients. Despite the significant improvement in AC1, Fig. \[fig-alfa-gass-4-par\] shows that no improvement in scatter or systematic offset is seen in the GASS dataset. The persistent systematic offset shown in Fig. \[fig-alfa-gass-4-par\] for the GASS data could be due to the two samples (AC1 and GASS) representing galaxies of different physical nature, so that even the matrix form of the estimator cannot extrapolate it in a suitable manner. This is demonstrated in Fig. \[fig-gr-mu\] where we plot $\mu_i$ vs. $g-r$ for the two samples; whilst there is some overlap, the GASS galaxies are preferentially located at the reddest $g-r$ colours and lowest values of $\mu_i$, that is poorly represented in AC1. It is possible to select a subset of the GASS data that are relatively well represented in the AC1 data, for which we might expect the calibration to perform better. In the lower panel of Fig. \[fig-gr-mu\] we again plot the comparison of estimated and observed  shown in Fig. \[fig-alfa-gass-4-par\], but now limiting the GASS data to the range $\mu_i>20$ and $g-r<0.6$, which is more populated by the AC1 sample. The estimated   does not remove all the offsets; however, this estimate is now closer to the observed one for this limited GASS sample. These tests demonstrate that whilst a single ANN model could not be found that is a good representation of all of the galaxies in the combined GASS and ALFALFA samples, it is nonetheless possible to post-facto exclude those galaxies for which the ANN is not suitable. For this reason, our approach for the remainder of the paper is to train our networks only using the ALFALFA sample, and then to develop a set of criteria from which we can assess the robustness of the gas fraction estimate. The cuts used in Fig. \[fig-gr-mu\] represent a rather crude approach, and practical here only because we are dealing with a two-dimensional input parameter space. With greater dimensionality (higher number of input parameters for the calibration), such simple cuts are cumbersome and complex, and ultimately subjective with no quantitative assessment of suitability. Therefore, whilst the general approach of limiting the data suitable for the application of a given calibration is desirable, a more sophisticated method is required. We return to this issue in Section \[sec-pr\]. ![The same as Figure \[fig-alfa-gass-two-par\], using the same input variables, but now using the matrix representation in Eq. \[eq-basis-function\] with 15 independent coefficients rather than a linear fit. The systematic trend in AC1 (blue points) is removed and the scatter is reduced from 0.32 to 0.22 dex. This demonstrates the improvement that is possible with the application of non-linear methods. However, a systematic offset remains in the GASS sample (red points) and there is no improvement in the relative systematic error.[]{data-label="fig-alfa-gass-4-par"}](fig-03.pdf){width="8cm" height="6.5cm"} ![Top panel: $\mu_i$ vs. $g-r$ for AC1 (blue) and GASS (red); the two samples are preferentially located in different regions of parameter space. Lower panel: As for Fig. \[fig-alfa-gass-4-par\], but restricting GASS data points to the region $\mu_i>20$ and $g-r<0.6$, where the AC1 sample is mostly located.[]{data-label="fig-gr-mu"}](fig-04-1.pdf "fig:"){width="8.cm" height="6.5cm"} ![Top panel: $\mu_i$ vs. $g-r$ for AC1 (blue) and GASS (red); the two samples are preferentially located in different regions of parameter space. Lower panel: As for Fig. \[fig-alfa-gass-4-par\], but restricting GASS data points to the region $\mu_i>20$ and $g-r<0.6$, where the AC1 sample is mostly located.[]{data-label="fig-gr-mu"}](fig-04-2.pdf "fig:"){width="8cm" height="6.5cm"} In this section, we have demonstrated that for a given set of input parameters a non-linear approach can yield a significant improvement in the estimation of  over a linear fit to the data. The matrix representation provides a very good option for determining the gas fraction for galaxies with only photometric data. However, using a larger number of input parameters, the accuracy of the estimation may be further improved. As we move to larger numbers of input parameters, the complexity of the matrices increases, and ultimately becomes very cumbersome, such that ANN can provide a better framework for the extension into many variable space. Another important result from this section is the potential pitfalls associated with combining different datasets for training. Homogeneity is key in the training process. If the training sample is heterogeneous (either in its intrinsic properties or in the methods used to obtain measured or derived properties) the network will be compromised. Moreover, the mixed quality or depth of a combined dataset (e.g. ALFALFA+GASS) makes it very challenging to determine a unique set of control and quality parameters, which is a critical part of our analysis. It is therefore extremely rare that different datasets are (or can be) combined for training. Instead, we use a single sample for our training set (ALFALFA) and then define the regime for which it can be robustly applied to other samples. This might be done in linear examples using a simple criterion such as a cut on a physical parameter. Since the parameters can be correlated with each other in different and complex ways, if we want to use a higher dimensional space as a training set then some simple cuts on only some input data might not be a suitable approach. In Section \[sec-pr\] we will return to this point and describe a pattern recognition method using an ANN model to achieve this objective. Input parameters, performance function and ranking {#sec-performance} ================================================== An important pre-analysis step is the selection of the input parameters on which the ANN will be trained. Galaxy evolution involves a complex interplay between many parameters and the manual exploration of parameter space normally restricts investigation to a handful of variables at a time. However, with an ANN approach, the parameter space can be expanded straightforwardly to enable a multi-variate analysis. Selection of galaxy variables for training ------------------------------------------ In selecting the parameters used for our training set, we have attempted to incorporate the primary physical variables that may affect the HI gas fraction. Some of the parameters are not independent; for example, we use both g- and r-band magnitudes as well as g-r colour. This repetition is not detrimental to the network’s performance, but contributes stability. Photometry and galaxy colour represent some of the raw observables that have been shown to correlate with gas fraction (e.g. Kannappan 2004; Denes et al. 2014; Eckert et al. 2015). These are in turn related to physical parameters such as galactic stellar mass, which has been shown to exhibit a strong anti-correlation with  (e.g. Zhang et al. 2009; Catinella et al. 2010; Huang et al. 2012). However, internal galaxy structure, size and the distribution of stellar mass appear to be even more tightly correlated with gas fraction than stellar mass itself (Zhang et al. 2009; Catinella et al. 2010; Toribio et al. 2011; Wang et al. 2016). Furthermore, Brown et al. (2015) have argued that specific star formation rate also modulates  at fixed M$_{\star}$. Environment also appears to play a role in determining gas fraction with gas fractions suppressed in both the cluster (e.g. Chung et al. 2007; Cortese et al. 2011; Denes et al. 2014) and group (e.g. Verdes-Montenegro et al. 2001; Rasmussen et al. 2008; Kilborn et al. 2009; Catinella et al. 2013; Hess & Wilcots 2013; Denes et al. 2016; Odekon et al. 2016) environments. We have included two environmental metrics in our list of training variables, halo mass and $\delta_5$, although Brown et al. (2016) have recently shown that it is the former of these that dominates the environmental dependence of . The full list of 15 parameters used in our work, based on the SDSS imaging and spectroscopic data, is presented in Table \[tab1\], which includes photometry, metrics of internal size and structure, star formation and environment. We note that not all the 15 parameters are available for all galaxies. This requirement reduces the number of galaxies in both the GASS and NIBLES samples from their complete data release. In the analysis that follows, the full AC1 and AC3 samples are therefore sometimes restricted further by the lack of input parameter data. If a certain input variable (such as a magnitude in a band) is required for a given training run, galaxies without a robust measurement of that input variable are excluded. Amongst the parameters in Table \[tab1\], there are two notable omissions of parameters that may significantly dictate . The first is angular momentum, which has recently be proposed by Obreschkow et al. (2016) to dictate HI gas fractions based on arguments of gas instability in the interstellar medium (see also Huang et al. 2012; Maddox et al. 2015). Unfortunately, we do not have metrics of angular momentum available for our sample. We have also not included a NUV colour, proposed by several works (e.g. Cortese et al. 2011; Catinella et al. 2013, Brown et al. 2015) to be the single most important variable in their samples. Requiring NUV photometry reduces our sample size by a factor of more than 4, to only $\sim$ 1400 galaxies, which was found to be inadequate for ANN training. Input data Description ------------------ ----------------------------------------- M$_*$ stellar mass M$_u$ $u$ band absolute magnitude M$_g$ $g$ band absolute magnitude M$_r$ $r$ band absolute magnitude M$_i$ $i$ band absolute magnitude M$_z$ $z$ band absolute magnitude $g-r$ observed colour $\mu_{*i}$ $i$-band stellar mass density SFR star formation rate sSFR specific star formation rate M$_{\rm{Halo}}$ halo mass $\delta_{5}$ local galaxy density $\rm{rhalf_r}$ half light radius (kpc) in the $r$ band B/T bulge-to-total fraction in the $r$ band $\rm{rd_{disk}}$ disk radius (kpc) in the $r$ band : Galaxy variables used in training sets. \[tab1\] The ANN performance metric, R ----------------------------- In this section, we introduce the use of a performance function, as a metric of the quality of the fit between gas fraction and a given variable. This performance metric can be a simple linear regression of the estimated and the observed values or a Spearman’s rank correlation number (see Huang et al. 2012 for more details), but it may also be a more complicated figure of merit (e.g. Teimoorina et al. 2016). We use the coefficient of determination, $R^2$, which is a measure of goodness of fit and is defined as: $$R\rm{^2=1-\frac{\sum\limits_{i=1}^n (target-y_{fit})^2}{(n-1)\times Var(target)}}. \label{eq-R}$$ In this equation $y_{\rm{fit}}$ is a linear fit to the target (observed data) and the estimated values (obtained by ANN). Var is the variance and $n$ is the number of objects in the sample. $R$ ranges from 0 to 1 in which $R\sim0$ indicates that the fit is not significantly better than a model in which $y_{\rm{fit}}$ = constant. A value of $R$=1 indicates that the linear equation ($y_{\rm{fit}}=aX+b$), where X is the observed , predicts 100% of the variance in the target ($y$, in this case the predicted gas fraction). Each parameter listed in Table \[tab1\] has a certain contribution to estimating  which can be considered as a weight in the fitting procedure, such that parameters with higher values of R contribute more significantly to the combined estimate of . In Figure \[fig-perform\] we show four different parameters taken from Table \[tab1\] as an example of the performance metric functionality. The value of R for each variable is given in the lower right of each panel. Amongst these four examples, it can be seen that stellar mass has the highest value of R=0.85, and indeed the scatter between the observed and predicted  is relatively small. Although parameters with low R, such as B/T (R=0.29) provide little improvement in our predictions of their inclusion with the ensemble of parameters can still provide stability to the network. We also note that in some regimes, some of our parameters may become unreliable, such as M$_{halo}$ in the low mass regime. In these cases, variables act simply as random numbers, and in the limit of a large training sample such as ours, such random variables do not decrease the performance of the network. For these reasons, all 15 parameters are used in our final ANN. ![Comparison of predicted and observed  for four individual parameters taken from our complete list of 15 (Table 1). These examples are selected because they span a wide range of performance, as parametrized by the R value given in the lower right corner of each panel. []{data-label="fig-perform"}](fig-05.pdf){width="9cm" height="7.5cm"} In Figure \[fig-perform15\] we show the performance numbers for all the 15 input parameters. It should be noted that these performance numbers are only the weights that show the contribution of each single parameter in the fitting procedure (i.e., when we use only a parameter from 15 for fitting) and should not be considered as a ranking in physical importance. In order to extract physical determism, different statistical methods such as receiver operating characteristics (ROC) (Teimoorinia et al. 2016 and reference therein) can be applied. The physical link between galaxy variables and gas fraction {#sec-rank} ----------------------------------------------------------- In Teimoorinia et al. (2016) we described a different performance number, namely the area under the curve (AUC) of ROC plots which can be used to link the physical significance of one variable to the value of another. This method was applied by Teimoorinia et al. (2016) to determine which were the most important variables for determining whether or not a galaxy’s SFR was quenched. A full description and worked examples of the AUC parameter can be found in that paper. In brief, the AUC ranges from 0.5 (a random number with no physical import) to 1 (outstanding performance governing the physical link between two variables). In Figure \[fig-AUC\] we show the ordered values of AUC for the 15 parameters used in this work. The numerical values of AUC are traditionally associated with qualitative ranks, ranging from ‘outstanding’ (AUC$>$0.9) to ‘random’ (no physical importance, AUC$<$0.6), e.g. Hosmer & Lameshow (2000). Figure \[fig-AUC\] shows that $g-r$ colour and stellar mass surface density are the two best performing indicators of , and indeed these parameters have featured widely in the literature (e.g. Zhang et al. 2009; Catinella et al. 2010, 2012; Cortese et al. 2011). Whereas B/T has a low value of R $\sim$ 0.3, it has an AUC = 0.71, placing it as the 3rd most important parameter for determining . Specific SFR also plays a marginally acceptable role in governing , but the 11 other parameters in our list have a formally low impact on the gas fraction. This includes parameters that are linked to environment: halo mass and $\delta_5$, indicating that such parameters are not the *prime* drivers of , although they may still contribute at a lower level once higher performing variables have been accounted for. Moreover, although the value of R for stellar mass is high, we can see from Figure \[fig-AUC\] that the AUC value associated with M$_{\star}$ is 0.63, and this parameter therefore performs little better than a random variable. Although none of the individual parameters has a particularly high AUC, when all 15 are considered together, the AUC = 0.86, characterized as an ‘excellent’ (but not ‘outstanding’) indicator. ![The performance number R for all 15 parameters listed in Table \[tab1\]. The R parameter reflects the amount of scatter present between the observed and predicted gas fraction and hence the relative weight of a given parameter in a fit. []{data-label="fig-perform15"}](fig-06.pdf){width="9cm" height="4.5cm"} ![The performance number AUC for all 15 parameters listed in Table \[tab1\]. In contrast to R, the AUC indicates the relative importance of a given parameter for determining .[]{data-label="fig-AUC"}](fig-auc.pdf){width="9cm" height="4.5cm"} Fitting results {#sec-fitting} =============== Having defined our samples (Section 2.1), removed galaxies with contamination (Section 2.1), chosen 15 relevant variables for fitting the target data (Section 4.1 and Table 1), defined their respective weights (Section 4.2) and ranked their relative physical importance (Section 4.3), we now proceed with a Bayesian neural network model (e.g., Ellison et al. 2016a) to fit the 6329 uncontaminated galaxies from AC1. In practice, we train 25 different networks with different initialization conditions and select the 20 best performing networks. Hence, each galaxy has 20 estimations of , from which we adopt the mean value. We can also quantify the scatter () in the estimated  (for a given galaxy, and for a given variable) for these best 20 performing networks, which can be used to quantify an uncertainty in the network estimation. That is, if a given galaxy’s  is robustly estimated by the 20 networks (for a given variable)  will be small. If the  estimation is unstable,  will be large. We have previously used  as a way of identifying which subset of galaxies have robust ANN estimations (see Ellison et al. 2016a, for a more technical discussion). In Figure \[fig-fit-ac1\] we show the fitting results for the 15 parameters for the training sample AC1, by comparing the estimated and observed  in the top panel. The scatter is low: $\sigma=0.184$ dex and there is no systematic offset at any value of . In the lower panels, we demonstrate that there is no systematic error in the estimated  as a function of four of the 15 variables. These four variables are selected as representative examples; indeed, we find no systematic offset for any of the 15 parameters in our input list. ![The top panel shows the estimated vs observed gas fraction for AC1. The lower panels show the difference in estimated and observed gas fraction vs. four different physical parameters, to demonstrate that there are no systematic trends or residuals.[]{data-label="fig-fit-ac1"}](fig-12-1.pdf "fig:"){width="8.2cm" height="6.3cm"} ![The top panel shows the estimated vs observed gas fraction for AC1. The lower panels show the difference in estimated and observed gas fraction vs. four different physical parameters, to demonstrate that there are no systematic trends or residuals.[]{data-label="fig-fit-ac1"}](fig-12-2.pdf "fig:"){width="8.2cm" height="5.5cm"} In Figure \[fig-fit-ac1-unclean\] we show the predicted  for the 508 galaxies that were removed from AC1 due to the possibility of contamination from a near neighbour galaxy in the Arecibo beam (Section \[sec-sample\]). There is now a systematic difference between the predicted  and the observed value, with the latter being on average 0.14 dex higher than the prediction. This result confirms that a significant fraction of the 508 galaxies that were excluded from the training set due to suspected contamination do indeed have additional 21 cm flux from a companion. ![Comparison between the observed and predicted  for the 508 galaxies in the A70 sample suspected of being contaminated by near neighbour within the Arecibo beam. On average, the observed  is 0.14 dex higher than predicted by the ANN, indicating that these galaxies do indeed have contaminated 21 cm fluxes.[]{data-label="fig-fit-ac1-unclean"}](fig-125.pdf){width="8.2cm" height="6.3cm"} Pattern recognition and detection probabilities {#sec-pr} =============================================== Before we can apply the trained networks to the SDSS, we must develop a technique for identifying which of its galaxies are suitable targets for the ANN to be applied to. As we have shown in Fig. \[fig-gr-mu\], applying a solution to galaxies that are not well represented in the training set can lead to large errors in the predictions. In the case of ALFALFA, which is a shallow, blind survey, approximately 4/5 of the SDSS galaxies are undetected at 21 cm. Galaxies detected by ALFALFA are therefore those with the highest gas masses for their stellar mass. This well known selection effect contributes to the observed anti-correlation between M$_{\star}$ and HI gas fraction and motivates surveys such as GASS that are designed to detect galaxies down to a fixed gas fraction. The selection of gas rich galaxies in ALFALFA has an important impact on our approach to   estimation, since we are only training our network with this biased population. That is, if the calibration is determined for the relatively gas-rich galaxies in AC1 and then applied to all SDSS galaxies, we are essentially forcing all galaxies to follow gas-rich behaviour. It is a critical step of our analysis to determine which galaxies can be legitimately calibrated using our estimator. The approach adopted here is to attempt to distinguish which galaxies are represented in AC1, as opposed to those that appear as non-detections in AC3. Artificial neural networks are powerful tools for such pattern recognition problems and they can also provide us with statistical information about a data set in order to categorize the data in a quantitative way (Teimoorinia 2012). We recall that sample AC1 contains 6837 galaxies detected at 21 cm, all of which have measurements of the 15 parameters listed in Table \[tab1\]. In order to create a statistical balance within the network, we randomly select 6837 galaxies out of the 23,652 non-detections in AC3. These 13674 galaxies are used as the main training sample for the pattern recognition step of our analysis. The objective of this step is to train a network that can distinguish the 21 cm detections from the non-detections, based solely on the 15 input SDSS parameters. Note that we use all 6837 of AC1 regardless of the potential contamination described in Section \[sec-sample\], since what is important here is whether or not a galaxy is detected, not its exact 21 cm flux. We use a binary classification for our training set such that a value of 0/1 is initially assigned to all galaxies in AC3 (non-detections) and AC1 (detections), respectively. During the training procedure, the 13674 galaxies of the training set are randomly separated into two sub-samples of training (70%) and validation sets (30%), to avoid any over-fitting problems. Using these input and target values, we train 40 networks and use the average output of the best 20. The network output will be the estimated probability that the input pattern (of SDSS parameters) belongs to one of the two categories. We refer to this probability of detection and non-detection galaxies as the pattern recognition detection metric, , which has a value between 0 and 1. In the ideal case, for the training set of 13674 galaxies, we would expect to see two completely separated groups with  values of 0 and 1, i.e. HI detections and non-detections that are completely separable in terms of their SDSS properties. However, in practice, the detection pattern exhibits a more continuous behaviour, due to the smoothly varying properties within the galaxy populations (see also Teimeoorinia et al. 2016). In Figure \[fig-dp\] we show the actual (normalized) distribution of the detection pattern for the galaxies in the training set, plotted separately for AC1 (blue line) and AC3 (red line). The figure shows that the detection pattern peaks near 1 and 0 for AC1 and AC3 respectively, demonstrating that the network is largely successful in discriminating the two samples based on their SDSS properties. However, both samples show long tails in their  distributions, indicating that not all galaxies are correctly classified with the 15 parameters in our training set. In other words, this is not a perfect classification so that some detections in AC1 have low probabilities estimated by the ANN, and some non-detections in AC3 have high estimated values of . For a decision boundary of =0.5 such that AC1 ($<$0.5) and AC3 $>$0.5) represent misclassifications, we find more than 80 per cent of galaxies of AC1 sample are correctly classified. The exact choice of  threshold will depend on the specific application of the data and the requisite combination of purity and accuracy. According to binary classification methods, the level of separation shown in Figure \[fig-dp\] can be considered as ‘successful’. In this method, the AUC = 0.86 and is therefore indicative of a very good classification. ![Normalized distribution of detection pattern () for 13674 galaxies in the pattern recognition training set. Probabilities peak at high values for 21 cm detections (AC1, blue line) and low values for non-detections (AC3, red line). Although both samples have long tails, more than 80 per cent of galaxies are correctly classified in sample AC1 for a threshold of $>$0.5.[]{data-label="fig-dp"}](fig-13.pdf){width="9cm" height="4.5cm"} ![image](fig-14-1.pdf){width="8.5cm" height="4.5cm"} ![image](fig-14-2.pdf){width="8.5cm" height="4.5cm"} ![image](fig-14-3.pdf){width="8.5cm" height="4.5cm"} ![image](fig-14-4.pdf){width="8.5cm" height="4.5cm"} ![image](fig-14-5.pdf){width="8.5cm" height="4.5cm"} ![image](fig-14-6.pdf){width="8.5cm" height="4.5cm"} In Figure \[fig-class-pr\] we compare the physical parameter space of galaxies in AC1 (left panels) and AC3 (right panels), where points are colour coded by . Figure \[fig-class-pr\] shows that the distribution of $g-r$ colours, stellar masses, B/T and SFRs are quite different between the two samples, as expected based on the ALFALFA survey design. For example, most of the detections are located on the star forming main sequence and have blue colours, whereas non-detections additionally populate the region of quiescent galaxies with red colours and high bulge fractions. However, the distribution of  between AC1 and AC3 is qualitatively similar. For example, the highest values of in AC3 are seen for galaxies on the main sequence (bottom right panel) and blue colours (top and middle right panels). The values are typically lower than seen for AC1, as expected given that, in reality, the AC3 galaxies are indeed not detected. It should be noted that in the pattern recognition procedure we do not use any direct connection between the observed  values and the 15 parameters.  is better interpreted as a connection between the best parameters and the nature of the survey. The closer the resemblance of a given galaxy with those in the original survey, the smaller the error in the estimation. We now apply the pattern recognition networks to all 561,585 galaxies in SDSS for which all 15 of our parameters are measured. The resultant distribution of  values is shown in Figure \[fig-pr-det-sdss\]. In the lower panel of Figure \[fig-pr-det-sdss\] we show the number of galaxies remaining in the sample as a function of progressively more stringent  cut. ![Top panel: The distribution of  for 561,585 galaxies in the SDSS for which all 15 parameters are available. Bottom panel: Number of galaxies remaining in the sample as a function of  threshold.[]{data-label="fig-pr-det-sdss"}](fig-15-1.pdf "fig:"){width="8.9cm" height="5.cm"} ![Top panel: The distribution of  for 561,585 galaxies in the SDSS for which all 15 parameters are available. Bottom panel: Number of galaxies remaining in the sample as a function of  threshold.[]{data-label="fig-pr-det-sdss"}](fig-15-2.pdf "fig:"){width="8.9cm" height="3.5cm"} In Section \[sec-fitting\] we described how the scatter amongst the 20 best trained networks yields an uncertainty on the ANN estimation, ; we now compute for the networks trained with our 15 parameters. For reference, in the top panel of Figure \[fig-sig-fit\] we show the distribution of  obtained for AC1. The lower panel shows  for the 561,585 galaxies with the 15 available parameters from SDSS. As can be seen from the top panel, log  spans the interval $-2.5$ to $-1$ for AC1. On the other hand, when we apply the networks to the SDSS, the distribution (in the lower panel) exhibits a more extended distribution, notably with a tail to higher values. This indicates (not surprisingly) that some galaxies in the SDSS have an estimation of  that is not as stable as in the training set. From a pattern recognition point of view the two distributions can be considered as two distinguishable groups with considerable overlap. A cut at log = $-1$ removes $\sim135,000$ galaxies from the SDSS sample, indicating that $\sim$1/3 of galaxies in SDSS have a higher uncertainty in their  estimation than in the training set. ![Top panel: Distribution of log  for the AC1 training sample. Lower panel:Distribution of log  for SDSS. []{data-label="fig-sig-fit"}](fig-16.pdf){width="8.8cm" height="6.5cm"} ![The left panels, from top to bottom, show GASS, (PM & Cornell) and NIBLES, respectively. They include all  and . The right panels show the same plots when  and  are limited, as indicated in the panel legends.[]{data-label="fig-fit-gass-pm-2p"}](fig-17-1.pdf "fig:"){width="4cm" height="3.8cm"} ![The left panels, from top to bottom, show GASS, (PM & Cornell) and NIBLES, respectively. They include all  and . The right panels show the same plots when  and  are limited, as indicated in the panel legends.[]{data-label="fig-fit-gass-pm-2p"}](fig-17-3.pdf "fig:"){width="4cm" height="3.8cm"} ![The left panels, from top to bottom, show GASS, (PM & Cornell) and NIBLES, respectively. They include all  and . The right panels show the same plots when  and  are limited, as indicated in the panel legends.[]{data-label="fig-fit-gass-pm-2p"}](fig-17-2.pdf "fig:"){width="4cm" height="3.8cm"} ![The left panels, from top to bottom, show GASS, (PM & Cornell) and NIBLES, respectively. They include all  and . The right panels show the same plots when  and  are limited, as indicated in the panel legends.[]{data-label="fig-fit-gass-pm-2p"}](fig-17-4.pdf "fig:"){width="4cm" height="3.8cm"} ![The left panels, from top to bottom, show GASS, (PM & Cornell) and NIBLES, respectively. They include all  and . The right panels show the same plots when  and  are limited, as indicated in the panel legends.[]{data-label="fig-fit-gass-pm-2p"}](fig-17-5.pdf "fig:"){width="4cm" height="3.8cm"} ![The left panels, from top to bottom, show GASS, (PM & Cornell) and NIBLES, respectively. They include all  and . The right panels show the same plots when  and  are limited, as indicated in the panel legends.[]{data-label="fig-fit-gass-pm-2p"}](fig-17-6.pdf "fig:"){width="4cm" height="3.8cm"} We now have at our disposal two parameters that can be used to select a sample of SDSS galaxies for which robust  estimations can be made, based on their similarity to the training set () and the uncertainty in the networks (). Both of these parameters are included in our public catalog. As a demonstration of how judicious cuts in these parameters can reduce the scatter between the estimated and observed , in Fig. \[fig-fit-gass-pm-2p\] we show results for the GASS (left panels) and PM & Cornell samples (right). The top panels show all galaxies in the sample, and the lower panels show the galaxies remaining after we restrict  $>$ 0.5 and  $<0.1$. These cuts can be viewed as nominal threshold values for the SDSS catalog, given the natural decision boundary at PR=0.5 and the distribution of  in the training set (as shown in the top panel of Figure \[fig-sig-fit\]). However, different cuts may be required for surveys with different properties. For example, for the GASS sample, imposing these default cuts does reduce the scatter from 0.33 to 0.24 dex, although a small offset still persists, indicating that even more stringent thresholds may be required. The PM & Cornell sample is small and shows some outliers; excluding the single most significant outlier reduces the scatter of the full sample from 0.35 to 0.29 dex. Furthermore, placing a  $>$ 0.5 and  $<0.1$ cut on the PM & Cornell sample further reduction in scatter to 0.24 dex (lower right panel of Figure \[fig-sig-fit\]). One approach for a practical application of  and  is to make cuts on each of these parameters, as we have done above, with choices that are best suited to the science application in hand. However, an alternative approach is to combine  and  into a single parameter, which we explore in the next section. A combination of  and  and a determination of final error {#sec-comb} ========================================================= In Fig. \[fig-dp\], we find that $\sim$25% of non-detections (i.e., AC3) are predicted to be detections (i.e., false positive); so a cut only on PR is not the best way to distinguish and separate more secure estimations. On the other hand, we have introduced a control parameter taken only from the detections in the fitting step: . We want to go a step further to introduce a more secure control parameter: a combination of PR and . To do this we notice that in Figure \[fig-sig-fit\], log  is extended in the interval $-2.5$ – 0, a range broader than AC1. The SDSS distribution can be normalized to AC1 via a simple ‘inverse’ mapping process, by setting the minimum and maximum values of the two distributions to 0 and 1, where 1 is the ’best’ value, representing the minimum scatter in the distribution. We call this normalized distribution . Figure \[fig-sig-fit-map\] shows  for both SDSS (lower panel) and AC1 (upper panel). ![The distribution of  after the application of an inverse mapping process in which the distributions of  are normalized the minimum and maximum values of the distribution of Figure \[fig-sig-fit\], with values of 0 and 1 respectively. Top panel: AC1. Bottom panel: SDSS.[]{data-label="fig-sig-fit-map"}](fig-18.pdf){width="8.8cm" height="6.5cm"} We now have two ‘quality control’ parameters that are distinct indicators of the estimation:  and , where both have values between 0 and 1. The best estimations will be when both  and  are large (i.e. tend towards 1). Moreover, these two parameters can now be trivially combined ($\times$), and then normalized to define a single variable, , that represents the confidence of the  estimation. This confidence metric also has a value between 0 and 1, where the higher the number, the more secure the  estimation will be. This is demonstrated in Figure \[fig-pr-total\] where we plot  vs  colour coded by  for the SDSS sample. The area above the black solid line shows $\sim150000$ galaxies with $>$0.5. ![ vs  colour coded by , defined as $\times$. The most reliable  estimations can be made for galaxies in which both  and  are large. The area above the black solid line shows $\sim150000$ galaxies with $>$0.5.[]{data-label="fig-pr-total"}](fig-19.pdf){width="8.2cm" height="7cm"} In Fig. \[fig-gass1-color\] we show how the  metric performs on one of our validation samples, namely the GASS sample. In the top panel, we show the estimated vs. observed  colour coded by . The familiar over-estimate of  by the ANN at low gas fractions is seen. However, the colour coding by  shows that the greatest offsets correspond to the lowest values of . This trend is emphasized in the lower panel by plotting the difference between the estimated and observed  as a function of the observed value. In the inset panel, we show the GASS points with $>$ 0.5; the scatter is reduced to 0.22 dex and with a small systematic error, which can be eliminated by increasing the cut in  to $\sim$ 0.7. For other surveys in our combined sample, a threshold of $>$ 0.5 provides a good nominal selection of robust  determinations, and is our recommended default value. ![Top panel: the estimated vs observed gas fraction colour coded by  for the GASS sample. The inset show points for $>0.5$. Bottom panel: The difference between estimated and observed  vs the observed value, again colour coded by .[]{data-label="fig-gass1-color"}](fig-20-1.pdf "fig:"){width="8.5cm" height="5.8cm"} ![Top panel: the estimated vs observed gas fraction colour coded by  for the GASS sample. The inset show points for $>0.5$. Bottom panel: The difference between estimated and observed  vs the observed value, again colour coded by .[]{data-label="fig-gass1-color"}](fig-20-2.pdf "fig:"){width="8.5cm" height="3.2cm"} In addition to using  for imposing robustness thresholds on the  estimations, we can also use the scatter in the observed vs. estimated gas fractions as an indicator of the likely uncertainty therein. To do this, first we construct a combined sample of all the validation sets (633 galaxies) shown in the left panels of Fig. \[fig-fit-gass-pm-2p\]. Then, in order to make a compromise and also to avoid any bias, we add 633 galaxies randomly selected from AC1 to our combined validation set. In the left panel of Fig. \[fig-combine\], we show the overall distribution of  in the combined sample (N=1266) and in the right panel the  comparison. Again, it can be seen that points that have deviant estimations of  have low values of . Now, we can obtain a relationship between the scatter of the new sample and . Since,  estimations are more reliable for higher , we use data only in region $0.33 <\rm{C_{fgas}} <1$. In this way, we avoid using small  and also, at the same time, we have adequate data points to fit a function to them and extrapolate it to smaller . The upper plot of Fig. \[fig-pr-err-color\] shows these data points (in red circles) and also a polynomial function fitted to them. Here, we show the scatter in the  estimation for bins in  ($\pm0.1$, the value on the x-axis). We can see that, for example, the fitted function shows a scatter of $\sim$0.8 dex for $\rm{C_{fgas}}=0$, which is obtained by an extrapolation. In the lower plot of this figure we use the data points (in blue rectangles) in the same regions to obtain mean offset and again fit the similar function to the points. The average estimated offset (by the function) for $<$0.5 is 0.23 (median 0.22) and the scatter in this region is 0.38. This is well matched with the data from sample GASS, which is the dominant sample in this region. There is no significant average offset for $>$0.5 and the mean scatter in this region is 0.22 dex. The recommended values to use are $0.5\leq\rm{C_{fgas}}\leq1$ or $0.09\leq\rm{\sigma_{fgas}}\leq0.29$. The resulting distribution of uncertainties for the SDSS sample is shown in the top panel of Fig. \[fig-err-sdss\], and both  and  are provided in our online catalog. The distribution of uncertainties has a narrow peak at small values, corresponding to galaxies with a high  value, and for which  is expected to be well estimated. In the lower panel, we plot the star forming main sequence, colour coded by . The bimodal distribution of  seen in the top panel of Fig. \[fig-err-sdss\] is clearly present in the distribution of SFR as a function of stellar mass. The ANN can make robust estimations for most galaxies on the main sequence, but performs poorly for passive galaxies, as expected from the distribution of  shown in the lower panels of Fig. \[fig-class-pr\]. As we have seen repeatedly in this paper, this is due to the nature of the AC1 training sample, whose galaxies are mostly star-forming. In Table \[table-samp\], we show the first 10 entries of the ANN estimated  along with all the control parameters. The full catalog is available in the online material that accompanies this paper. ![Left panel: The distribution of  for the combined sample. Right panel: Estimated values of   for the sample, colour coded by , compared with the observed values.[]{data-label="fig-combine"}](fig-21-1.pdf "fig:"){width="4.1cm" height="3.9cm"} ![Left panel: The distribution of  for the combined sample. Right panel: Estimated values of   for the sample, colour coded by , compared with the observed values.[]{data-label="fig-combine"}](fig-21-2.pdf "fig:"){width="4.1cm" height="3.9cm"} ![The scatter between the estimated and observed  as a function of . The top panel shows the error in 7 different bins of  ( $0.33<\rm{C_{fgas}}<1$). The lower panel shows the mean offset between observed and predicted . in the same area. The dashed lines are polynomial functions fitted to the data.[]{data-label="fig-pr-err-color"}](fig-22.pdf){width="9cm" height="5.8cm"} ![The top panel shows the distribution of errors; . In the bottom panel we plot the SFR vs. stellar mass for the SDSS sample (for presentation purposes, 60 000 galaxies have been selected at random). []{data-label="fig-err-sdss"}](fig-23-1.pdf "fig:"){width="8.8cm" height="5cm"} ![The top panel shows the distribution of errors; . In the bottom panel we plot the SFR vs. stellar mass for the SDSS sample (for presentation purposes, 60 000 galaxies have been selected at random). []{data-label="fig-err-sdss"}](fig-23-2.pdf "fig:"){width="8.8cm" height="5cm"} A final caveat on the derivation of  from {#sec_mhi_caveat} ------------------------------------------ In this paper we have concentrated on the estimation of , rather than , although the latter can be derived from the former since we have stellar masses available. In Figure \[fig-mhi\] we show the estimated value of  vs. the observed one, colour coded by the number of galaxies in each point. A mild skewness is seen in this plot, whereby low  galaxies have their gas masses slightly over-estimated and vice versa. This effect persists even after the application of  thresholds. Moreover, we have found that this effect persists if the network is re-trained with  as the target. This check includes a complete re-assessment of the most relevant parameters for estimation (indeed, the parameter rankings for the prediction of  are quite different to those for ). As was previously shown in the top panel of Figure \[fig-fit-ac1\], no such skewness exists in the estimation of . The skewness in the  estimates is again due to the nature of the training sample, the large observed scatter in  at a given stellar mass and the shallow nature of the survey. We therefore caution that whilst we have demonstrated the robustness of our  estimates, there may be errors of several tenths of a dex in attempts to convert these to gas masses. It is worth noting that previous efforts to calibrate the gas content of galaxies have similarly used  rather than . ![ The estimated value of  vs. the observed one, colour coded by number of galaxies in each point.[]{data-label="fig-mhi"}](fig-24.pdf){width="8.8cm" height="7cm"} ![ Distribution of galaxy gas fractions as a function of stellar mass for ALFALFA.70 (grey points), GASS (green points), HiGHz (red points) and the ANN predictions for the SDSS (blue contours and points). The ANN predicts a significant population of high mass (log M$_{\star} >11$) galaxies with gas fractions log  $> -1$ that are rare in current samples. For clarity, only detections (no upper limits) are shown.[]{data-label="fig-25"}](fig-25.pdf){width="8.5cm" height="6.8cm"} ID (SDSS) RA       Dec $z$ log M$_*$ log PR -------------------- ------------- -------------- -------- ----------- --------- -------- -------- -------- -------- -------- -- -- -- -- -- -- -- -- 587736781457719496 237.4554943 34.90548896 0.1670 11.4110 -0.9456 0.1178 0.1972 0.5249 0.1006 0.6169 588018091616174223 237.5470295 34.70075261 0.0701 9.9120 -0.2466 0.3310 0.4111 0.7075 0.0269 0.3913 587736781457850591 237.6627645 34.78976388 0.1511 10.7487 -0.9023 0.3493 0.4945 0.6207 0.0503 0.3778 587736781457850627 237.691242 34.70155096 0.0751 10.5140 -0.5599 0.4970 0.6302 0.6929 0.0298 0.2922 587739130812694794 225.2513585 29.37813267 0.0793 10.6881 -0.8689 0.3917 0.4506 0.7637 0.0179 0.3491 587742773490155647 192.5694372 16.66269477 0.0252 9.2378 -0.0193 0.6708 0.7546 0.7809 0.0158 0.2259 587739630095040681 223.1457785 26.26263514 0.0759 10.0936 -0.1939 0.3101 0.3877 0.7027 0.0278 0.4078 587736781457916109 237.8366547 34.56556243 0.0488 10.5449 -0.5856 0.6167 0.6857 0.7901 0.0148 0.2443 587730816291569798 344.8364624 -10.13584859 0.0600 9.9198 -0.0108 0.5532 0.5998 0.8103 0.0128 0.2680 587742773490221138 192.5891741 16.65422642 0.0680 10.1461 -0.0377 0.7943 0.9399 0.7424 0.0209 0.1842 \[table-samp\] Summary {#sec-sum} ======= In this paper we have presented a novel method to estimate HI gas mass fraction and the associated uncertainties based on the patterns found in our data sets, using machine learning methods. The ALFALFA survey is used as our main training sample, and we check our model estimations with a range of validation sets, comprised of the GASS and Cornell surveys and a small sample of post-merger galaxies. We have shown that, for a given set of input parameters, non-linear methods can significantly reduce the scatter in the estimation of , compared with traditional linear fits. Specifically, we demonstrate that using only the $g-r$ colour and $i$-band surface brightness, a matrix-based model can reduce the scatter in a linear model from 0.32 dex to 0.22 dex. In order to extend our models to include more galactic parameters, we assess the correlations between  and 15 parameters derived from the SDSS imaging and spectroscopic datasets. Two performance parameters are presented: R, which represents the relative weights of the fit in minimizing scatter and AUC, which quantifies the physical relevance of a given galaxy parameter to its gas fraction. The AUC ranking shows that $g-r$ colour and stellar mass surface density both strongly govern a galaxy’s gas fraction, with specific SFR and B/T having a further marginal relevance. We introduce several parameters that permit the assessment of how accurately is estimated for a given galaxy in the SDSS. The scatter in the estimation of is determined from 20 individually trained networks () and provides an indication of variation in the fitting process. The inverse normalized version of  () has a value from 0 to 1, where higher values indicate a smaller scatter. We also use a pattern recognition technique to identify the similarity of a given galaxy to the ALFALFA detections used in our ANN training set, . Again, this parameter has a value between 0 and 1, where higher values indicate a greater similarity to ALFALFA detections, and hence, a higher likelihood that the network can provide an appropriate estimation of .  and  can by multiplied together to yield a single parameter, , whose value again ranges from 0 to 1, where higher values indicate more robust estimations.  can also be used to determine an uncertainty associated with a given  estimation. We demonstrate how the application of these various parameters to the validation sets effectively removes outliers from the ANN estimations. A  cut of 0.5 yields $\sim$ 150 000 galaxies with  estimations from our ANN approach, a factor of $\sim$ 20 greater than the number of firm 21 cm detections in the 70 per cent data release of ALFALFA. All of the quality control parameters accompany the  estimations in the online table. Our catalog of predicted gas fractions offer several advantages over previous attempts to calibrate gas fractions. For example, we have shown that there is no systematic error in our  estimates with various galaxy properties, and the scatter in our estimates is lower than has been previously achieved. Perhaps most importantly, we have provided a quantitative prescription for assessing the robustness of our estimates, on a galaxy-by-galaxy basis. Our ANN  estimates also potentially offer advantages over even direct Arecibo observations. The large size of the Arecibo beam (3.3$\times$3.8 arcminutes) is not able to distinguish contributions from multiple galaxies with close separations. Indeed, as shown in Figs. \[fig-fit-ac1\] and \[fig-fit-ac1-unclean\], whilst our predictions reproduce well the gas fractions in the ‘clean’ training sample, galaxies with close companions have observed gas fractions that are typically 0.14 dex above the ANN predictions. Finally, our ANN-based gas fractions extend the number of galaxies in the nearby universe with robust  estimates by more than an order of magnitude. The growth in sample size yields predictions for large numbers of galaxies in parameter space beyond current samples. For example, most HI surveys are currently limited to $z < 0.06$, whereas our  predictions extend to $z \sim 0.2$. Indeed, our catalog contains $\sim$ 61,000 robust ( $>$ 0.5)  predictions at $z>0.1$. Figure \[fig-25\] shows the distribution of gas fractions for the GASS, HiGHz (Catinella & Cortese 2015) and ALFALFA.70 samples (green, red and grey points respectively), compared with the ANN predictions (blue contours and points, where the latter show individual galaxies with counts below the lowest contour). It can be seen that the ANN predicts a significant population of high stellar mass (log M$_{\star} > 11$) galaxies with gas fractions log  $> -1$. Such galaxies are largely absent from current samples and are predicted to contain some of the highest HI masses in the local universe. Follow-up observations of these high stellar mass, high HI mass galaxies to confirm our ANN predictions would be of great interest. Acknowledgments {#acknowledgments .unnumbered} =============== We are grateful to Barbara Catinella and Luca Cortese for comments on an earlier draft of this paper and for help and advice with the ALFALFA.70 sample. We thank Wim van Driel for providing us with his table of HI measurements in electronic format. We also thank the anonymous referee, whose comments led to an improved paper. SLE and DRP gratefully acknowledge the receipt of NSERC Discovery Grants. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/. 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--- abstract: 'Regularization in the optimization of deep neural networks is often critical to avoid undesirable over-fitting leading to better generalization of model. One of the most popular regularization algorithms is to impose $L_2$ penalty on the model parameters resulting in the decay of parameters, called weight-decay, and the decay rate is generally constant to all the model parameters in the course of optimization. In contrast to the previous approach based on the constant rate of weight-decay, we propose to consider the residual that measures dissimilarity between the current state of model and observations in the determination of the weight-decay for each parameter in an adaptive way, called adaptive weight-decay (AdaDecay) where the gradient norms are normalized within each layer and the degree of regularization for each parameter is determined in proportional to the magnitude of its gradient using the sigmoid function. We empirically demonstrate the effectiveness of AdaDecay in comparison to the state-of-the-art optimization algorithms using popular benchmark datasets: MNIST, Fashion-MNIST, and CIFAR-10 with conventional neural network models ranging from shallow to deep. The quantitative evaluation of our proposed algorithm indicates that AdaDecay improves generalization leading to better accuracy across all the datasets and models.' author: - 'Kensuke Nakamura, and Byung-Woo Hong\* [^1] [^2]' bibliography: - 'image.bib' title: Adaptive Weight Decay for Deep Neural Networks --- [Shell : Adaptive Weight Decay for Deep Neural Networks]{} Adaptive Regularization, Deep Learning, Neural Networks, Stochastic Gradient Descent, Weight-decay Introduction {#sec:intro} ============ The deep neural network model consists of a nested architecture of layers where the number of parameters is in millions[@bottou2010large; @simonyan2014very; @szegedy2015going; @he2016deep; @he2016identity; @huang2017densely; @bottou2018optimization]. Due to its high degrees of freedom, the deep model can approximate linear and nonlinear functions; however it is always at risk of over-fitting to training data. Thus the deep neural network requires regularization techniques in training process in order to achieve generalization, resulting in a good prediction for unknown data. Since the number of example data is also huge, the deep model is trained using the stochastic gradient descent (SGD) [@robbins1951stochastic; @rumelhart1988learning; @zhang2004solving; @bottou2010large; @bottou2018optimization] that updates parameters using a small subset of data (mini-batch) in combination with regularization techniques. A simple yet major regularization technique is, so-called, early stopping [@prechelt1998early; @zhang2016understanding] where the training process is terminated manually at a certain epoch before the validation loss increases. The noise injection techniques, e.g., the mini-batch procedure that induces noise to gradients [@zhu2019anisotropic] and the dropout that randomly zeros the activation of nodes [@srivastava2014dropout], give implicit regularization effects for the training process. The weight-decay is an explicit regularization such that an additional penalty term is defined in the energy function and the regularization effect can be tuned by its coefficient. In contrast to the recent development of adaptive methods on deep optimization, a constant weight-decay has been employed while it played an important role in both classic and modern deep neural networks [@krogh1992simple; @simonyan2014very; @chollet2017xception; @zhang2016understanding]. Layer-wise weight-decay was considered in [@ishii2017layer; @bengio2012practical] but it is limited to classical neural networks without the skip-connection. We propose an adaptive regularization method of weight-decay, called Adaptive Weight-Decay (AdaDecay), that varies in spatio-temporal domains during the training process and is beneficial to both shallow and deep neural network models. The proposed AdaDecay determines the weight-decay rate in parameter-wise at each optimization iteration using the magnitude of the loss gradient that measures the difference between the current state of the model with the mini-batch data. We normalize the gradient norm in each layer in order to make the algorithm independent to the model architecture and robust to hyper-parameter selection. We also present experimental results of the presented AdaDecay in comparison to the state-of-the-art optimization algorithms using major benchmark datasets on the image classification with shallow and deep networks, where our AdaDecay has overtakes the others in validation accuracy. In the remaining of this paper we summarize the related studies with our contributions in Section \[sec:works\] and then describe notations in Section \[sec:prelimminary\]. We present AdaDecay in Section \[sec:adadecay\], followed by experimental results in Section \[sec:experimental\] and conclusion in Section \[sec:conclusion\] Related Work {#sec:works} ============ [**Learning Rate Annealing:**]{} The stochastic gradient, calculated by a subset of data, gives a noise to gradient and provides an implicit regularization effect [@zhu2019anisotropic]. In SGD, parameters are updated by subtracting the gradient with the stochastic noise multiplied by the learning rate. The learning rate should shrink in order to reduce the noise and converge the algorithm. To this aim, a variety of learning rate annealing, e.g. exponential [@george2006adaptive] and staircase [@smith2017don], and the adaptive learning rates, e.g., AdaGrad [@duchi2011adaptive], have been proposed, The sophisticated adaptive techniques, e.g., RMSprop [@tieleman2012lecture] and Adam [@kingma2014adam], enable parameter-wise control of the learning rates. The drawback of learning rate techniques on the regularization is that it reduces or increases both the step-size and the noise. [**Dropout**]{} is another regularization technique that is in particular used with classical shallow networks. The dropout zeros the activation of randomly selected nodes with a certain probability during the training process [@srivastava2014dropout]. The dropping rate is generally set to be constant but its variants have been considered with adaptive rates depending on parameter value [@ba2013adaptive], estimated gradient variance [@kingma2015variational], biased gradient estimator [@srinivas2016generalized], layer depth [@huang2016deep], or marginal likelihood over noises [@noh2017regularizing]. However, in fact, recent deep models do not support the dropout and its variants. The reason may be that the number of parameters in a layer is relatively smaller than the the classic neural networks, and random masking to nodes can be erroneous to the model. [**Energy Landscape:**]{} The geometrical property of energy surface is helpful in optimization of highly complex non-convex problems associated with deep network architecture. It is preferred to drive a solution toward local minima on a flat energy surface that is considered to yield better generalization [@hochreiter1997flat; @Chaudhari2017EntropySGD; @dinh2017sharp] where flatness is defined around the minimum by its connected region, its curvature of the second order structure, and the width of its basin, respectively. A geometry-driven optimization based on SGD has been developed in deep learning problems such as Entropy-SGD [@Chaudhari2017EntropySGD]. In our approach, we do not attempt to measure geometric property of loss landscape such as flatness with extra computational cost, but instead consider explicit regularization to model parameters. [**Variance Reduction:**]{} The variance of stochastic gradients is detrimental to SGD, motivating variance reduction techniques [@Roux2012; @johnson2013accelerating; @Chatterji2018OnTT; @zhong2014fast; @shen2016adaptive; @Difan2018svrHMM; @Zhou2019ASim] that aim to reduce the variance incurred due to their stochastic process of estimation, and improve the convergence rate mainly for convex optimization while some are extended to non-convex problems [@allen2016variance; @huo2017asynchronous; @liu2018zeroth]. One of the most practical algorithms for better convergence rates includes momentum [@sutton1986two], modified momentum for accelerated gradient [@nesterov1983method], and stochastic estimation of accelerated gradient (Accelerated-SGD) [@Kidambi2018Acc]. These algorithms are more focused on the efficiency in convergence than the generalization of model for accuracy. [**Weight-Decay:**]{} is an explicit way of regularization such that a regularization term is added into the energy function. Specifically $L^2$-norm is used as the regularization term in order to penalize large weight values. Different with the other implicit methods, e.g., stochastic update and dropout, one can directly control the regularization effect by the weight-decay coefficient. The weight-decay coefficient is tuned by hand [@simonyan2014very; @chollet2017xception], or learned by Bayesian optimization [@snoek2015scalable; @shahriari2016unbounded]. However, in contrast to recent development of adaptive methods of dropout [@ba2013adaptive; @kingma2015variational; @srinivas2016generalized; @huang2016deep; @noh2017regularizing] and learning-rate [@duchi2011adaptive; @tieleman2012lecture; @kingma2014adam] in deep optimization, a constant weight-decay coefficient has been employed in usual. Layer-wise weight-decay has been considered in [@ishii2017layer; @bengio2012practical] where different weight-decay coefficients are given for different layers of network model using the variance of gradients in layer. The drawback of the layer-wise method [@ishii2017layer; @bengio2012practical] is that it assumes that layers are aligned in a single sequence. The skip-connection [@he2016deep; @he2016identity; @Balduzzi2017shattered], that is one of the key architectures in the recent deep networks, makes it non-trivial. The main contributions of this work are three folds: First, we propose an adaptive regularization method (AdaDecay) in which parameter-wise weight-decay varies in spatio-temporal domains reflecting the currant state of model and the mini-batch data. Second, the proposed AdaDecay determines the weight-decay rate of each parameter based on the norm of gradient normalized in layer. This makes the algorithm independent to model architecture and beneficial to both shallow and deep neural network models. Third, we empirically demonstrate the robustness and effectiveness of the presented AdaDecay in comparison to the state-of-the-art optimization algorithm using both shallow neural network models and modern deep models with three of the major benchmark datasets on image classification. Preliminary {#sec:prelimminary} =========== We consider an energy optimization problem based on a given set of training data in a supervised machine learning framework. Let $\chi = \{ (x_i, y_i) \}_{i = 1}^n$ be a set of training data where $x_i \in \X \subset \mathbb{R}^{N}$ is the $i$-th input and $y_i \in \Y \subset \mathbb{R}^{M}$ is its desired output. Let $h_w \colon \X \to \Y$ be a prediction function that is associated with its model parameters $w = ( w_1, w_2, \cdots, w_m ) \in \R^m$ where $m$ denotes the dimension of the feature space. The objective of the supervised learning problem under consideration is to find an optimal set of parameters $w^*$ by minimizing the empirical loss $\L(w)$ that typically consists of a data fidelity term $\rho(w)$ and a regularization term $\gamma(w)$ as follows: $$\begin{aligned} w^* = \arg\min_w \mathcal{L}(w), \quad \mathcal{L}(w) = \rho( w ) + \lambda \, \gamma( w ), \label{eq:loss} \end{aligned}$$ where $\lambda > 0$ is a control parameter, called weight-decay coefficient, that determines the trade-off between the data fidelity term $\rho(w)$ and the regularization $\gamma(w)$. The data fidelity term $\rho(w)$ is of the additive form over a set of training data $\{ (x_i, y_i) \}_{i=1}^n$ as follows: $$\begin{aligned} \rho(w) &= \frac{1}{n} \sum_{i=1}^n f_i( w ), \label{eq:fidelity} \end{aligned}$$ where $f_i( w )$ denotes a data fidelity incurred by a set of model parameters $w$ for a sample pair $(x_i, y_i)$. The data fidelity $f_i( w )$ is designed to measure the discrepancy between the prediction $h_w( x_i )$ with an input $x_i$ and its desired output $y_i$ for a given sample pair $(x_i, y_i)$. The regularization $\gamma( w )$ is designed to impose a smoothness constraint to the solution space, thus avoid undesirable over-fitting to the model. The weight-decay coefficient $\lambda \in \R$ is determined based on the relation between the underlying distribution of the data and the prior distribution of the model. In the optimization of the objective function $\mathcal{L}(w)$ defined by: $$\begin{aligned} \mathcal{L}(w) &= \frac{1}{n} \sum_{i=1}^n f_i( w ) + \lambda \, \gamma( w ), \label{eq:objective} \end{aligned}$$ where $f_i(w)$ and $\gamma(w)$ are assumed to be differentiable, we consider a first-order optimization algorithm leading to the following gradient descent step at each iteration $t$: $$\begin{aligned} w^{t+1} \coloneqq w^{t} - \eta^{t} \left( \frac{1}{n} \sum_{i=1}^n \nabla f_{i}(w^{t}) + \lambda \nabla \gamma(w^t) \right), \label{eq:update:vanilla} \end{aligned}$$ where we denote by $\nabla f_{i}(w^{t})$ gradient of $f_i$ with respect to $w$ at iteration $t$, and by $\eta^{t}$ the learning rate at iteration $t$. The computation of the above full gradient over the entire training data is often intractable due to a large number of data, which leads to the use of stochastic gradient that is computed using a subset uniformly selected at random from the training data. The iterative step of the stochastic gradient descent algorithm at iteration $t$ reads: $$\begin{aligned} w^{t+1} \coloneqq w^{t} - \eta^{t} \left( \frac{1}{B} \sum_{i \in \beta^t} \nabla f_{i}(w^{t}) + \lambda \nabla \gamma(w^t) \right), \label{eq:update:std} \end{aligned}$$ where $\beta^t$ denotes a mini-batch that is the index set of a subset uniformly selected at random from the training data. The mini-batch size $B = |\beta^t|$ is known to be related to the variance of the gradient norms, and thus to the regularization of the model. We assume that the mini-batch size is fixed in the optimization procedure to simplify the problem and emphasize the role of regularization parameter $\lambda$. Regularization via Adaptive Weight-Decay (AdaDecay) {#sec:adadecay} =================================================== We present a regularization algorithm that is designed to determine the degree of regularization for each model parameter considering its current state of solution in the course of optimization procedure in an adaptive way. The optimization of interest aims to minimize the objective function that consists of a data fidelity term, a regularization term, and a control parameter for their relative weight. The control parameter that determines the relative significance between the data fidelity and the regularization is generally chosen to be constant based on the assumption that the underlying distributions of the residual and the prior smoothness follow uni-modal distributions. However, it is often ineffective to model the trade-off between the data fidelity and the regularization distributions using a static control parameter based on the ratio between the variances of their distributions. Thus, we propose an adaptive regularization scheme that considers residual in the determination of regularity for both the spatial domain of model parameters and the temporal domain of optimization. Weight-Decay for Individual Model Parameter ------------------------------------------- The computation of empirical stochastic gradient involves the noise process following a certain distribution with zero mean, and its variance is related to the degree of regularization that is desired to be imposed. We consider the regularization $\gamma(w)$ in Eq.  by the squared Euclidean norm leading to the following objective function: $$\begin{aligned} \L(w) &= \rho( w ) + \frac{\lambda}{2} \, \| w \|_2^2, \label{eq:objective:l2} \end{aligned}$$ where $\lambda \in \R$ denotes the coefficient for the regularization term. Then, the gradient descent step at each iteration $t$ by a first-order optimization algorithm reads: $$\begin{aligned} w^{t+1} & \coloneqq w^{t} - \eta^{t} \left( \nabla \rho( w^t ) + \lambda w^t \right) \nonumber \\ & = (1 - \eta^{t} \lambda) \, w^{t} - \eta^{t} \, \nabla \rho(w^{t}), \label{eq:update:weightdecay} \end{aligned}$$ where $\eta^t \lambda$ is constrained to be $[0, 1)$ leading to the shrinkage of the unknown model parameters $w$ in iteration $t$ and this regularization scheme based on the $L_2^2$ norm is called weight-decay. In contrast to the static coefficient $\lambda \in \R$ in the conventional weight-decay regularization, we propose a regularization scheme that is designed to impose adaptive regularity to each model parameter $w_j$ with an additional term $\theta_j$ as follows: $$\begin{aligned} \L(w) &= \rho( w ) + \frac{\lambda}{2} \, \| \theta \odot w \|_2^2, \label{eq:objective:l2:parameterwise} \end{aligned}$$ where $\theta = (\theta_1, \theta_2, \cdots, \theta_m) \in \R^{m}, w = (w_1, w_2, \cdots, w_m) \in \R^{m}$ and the symbol $\odot$ denotes the Hadamard product defined by: $\theta \odot w = (\theta_1 w_1, \theta_2 w_2, \cdots, \theta_m w_m)$. The degree of regularization for each model parameter $w_j^t$ at iteration $t$ is determined by the adaptive term $\theta_j^t$, leading to the following modified iterative update step: $$\begin{aligned} \label{eq:update:weightdecay:adaptive} w_j^{t+1} & \coloneqq (1 - \eta^{t} \lambda \theta_j^{t}) \, w_j^{t} - \eta^{t} \, g_j^t, \end{aligned}$$ where $g_j^t = \frac{\partial \rho(w^t)}{\partial w_j}$ is the gradient of the data fidelity $\rho$ with respect to the parameter $w_j$ at iteration $t$. The weight-decay coefficient $\lambda \ge 0$ determines the constant degree of regularity for all the model parameters whereas the adaptive term $\theta$ determines the relative significant of each model parameter at each step in the course of optimization, i.e., the decay rate for each model parameter $w_j^t$ at iteration $t$ is determined by the global regularization parameter $\lambda$ multiplied by the adaptive term $\theta_j^t$. Note that Eq.  with $\theta_j^t = 1$ for all $j$ and $t$ becomes the same as Eq.  with a constant weight-decay. Adaptive Weight-Decay based on Residual --------------------------------------- We now consider the parameter-wise adaptive term $\theta_j^{t}$ in Eq. . Our proposed regularization scheme is designed to impose an adaptive regularity to each model parameter based on its associated residual at each iteration of optimization leading to an adaptive regularization in both the spatial domain of the model parameter and the temporal domain of the optimization. The degree of regularity for each model parameter is determined in consideration of residual, or norm of gradient, that determines a discrepancy between the current state of model and the observation. The gradient norm $|g_j^t|$, known as Gauss-Southwell rule, has been used in importance sampling of parameters, e.g.,  [@glasmachers2013accelerated; @nutini2017let; @namkoong2017adaptive]. This is, however, not directly applicable to our case since the magnitude of gradients varies exponentially over layers in deep model. We thus normalize the gradient norm $|g_j^t|$ to have mean $0$ and standard deviation (std) $1$ within each layer at each iteration in order to consider the relative significance of the local parameters within the layer. The normalized gradient-norm $\tilde{g}_j^t$ is given by: $$\begin{aligned} \tilde{g}_j^t &= \frac{|g_j^t| - \mu_l^t}{\sigma_l^t}, \label{eq:grad:normalized} \end{aligned}$$ where $l$ denotes the index of the layer that includes the parameter $w_j$, and $\mu_l^t$ and $\sigma_l^t$ denotes the mean and standard deviation of all the gradient norms for the parameters within the layer $l$ at iteration $t$, respectively. We assume that the degree of regularity $\theta_j^t$ for each parameter $w_j$ at iteration $t$ follows a distribution of the residual leading to the following data-driven regularity: $$\begin{aligned} \theta_j^t \propto \tilde{g}_j^t, \label{eq:weight:distribution} \end{aligned}$$ where the degree of regularization for each parameter is proportional to the norm of its gradient. In the determination of our adaptive regularization, we use the scaled sigmoid function defined by: $S(x; \alpha) = 2 / (1 + \exp(- \alpha x))$, where $\alpha \in \R$ is a control parameter for the steepness of function value transition. Then, the relative degree of regularization $\theta_j^t$ for each parameter $w_j$ at iteration $t$ is determined by the scaled sigmoid function $S$ of the normalized gradient norm $\tilde{g}_j^t$ as follows: $$\begin{aligned} \theta_j^t = S(\tilde{g}_j^t; \alpha) = \frac{2}{1 + \exp(- \alpha \tilde{g}_j^t)}, \label{eq:theta} \end{aligned}$$ where $\alpha$ determines the slope of the decay rate transition according to the gradient norm, and $\theta_j^t$ ranges from $0$ to $2$ and its average is $1$ since $\tilde{g}_j^t$ is normalized to have mean $0$ and standard deviation $1$. The pseudo code of the proposed algorithm for the adaptive weight-decay is described in Algorithm \[alg:adadecay\] where the degree of regularization for each parameter is determined based on the scaled sigmoid function of the norm of its gradient leading to the adaptive regularization in both the spatial domain of model parameters and the temporal domain of optimization. The complexity of AdaDecay shown in Algorithm \[alg:adadecay\] remains $O(1)$ to the number of training examples as SGD with constant weight-decay. $\{g_j^t\}_j$ : gradients of the parameters $j$ computed by back-propagation at iteration $t$ $\eta^t$ : learning rate at iteration $t$ $\lambda$ : global weight-decay coefficient $\alpha$ : hyper-parameter for the adaptation to the gradient norm $\mu_l^t$ : compute the mean of gradient norms $|g_j^t|$ in the layer $l$ $\sigma_l^t$ : compute the std of gradient norms $|g_j^t|$ in the layer $l$ $\tilde{g}_j^t = (|g_j^t| - \mu_l^t) / \sigma_l^t$ $\theta_j^t = S(\tilde{g}_j^t ; \alpha)$ using Eq.  $w_j^{t+1} \coloneqq (1 - \eta^{t} \lambda \theta_j^{t}) \, w_j^{t} - \eta^{t} \, g_j^t$ Experimental Results {#sec:experimental} ==================== We provide quantitative evaluation of the presented AdaDecay in comparison to the state-of-the-art optimization algorithms. For experiments, we use three of major benchmark datasets on image recognition: MNIST, Fashon-MNIST, and CIFAR-10. MNIST [@lecun1998gradient] is a simple yet fundamental dataset that consists of 60K training and 10K test gray images of hand-written 10 digits. Fashion-MNIST [@xiao2017online] is a modern dataset that consists of 60K training and 10K test gray images with 10 categories of clothes and fashion items. CIFAR-10 [@krizhevsky2009learning] is a more challenging task that consists of 50K training and 10K test object images with 10 categories. Regarding the network architecture, we employ four of shallow networks and four of deep networks: The shallow networks include: fully-connected neural networks with two hidden layers (NN-2) and with three hidden layers (NN-3) [@blum1991approximation], LeNet-4 [@lecun1998gradient] with two convolution layers followed by two of fully-connected layers, and VGG-9 [@simonyan2014very]. The deep networks used in our experiments are: ResNet-18 [@he2016deep; @he2016identity], ResNet-50 [@Balduzzi2017shattered], GoogLeNet [@szegedy2015going], and the densely connected convolutional networks (DenseConv) [@huang2017densely]. The batch normalization [@ioffe2015batch] is used in VGG-9, ResNet-18, ResNet-50, GoogLeNet and DenseConv. Our comparative analysis involves the following optimization algorithms: the stochastic gradient descent with the constant weight-decay (SGD), SGD with RMSprop [@tieleman2012lecture] (RMS), SGD with Adam [@kingma2014adam] (Adam), Entropy-SGD [@Chaudhari2017EntropySGD] (eSGD), Accelerated-SGD [@Kidambi2018Acc] (aSGD), and SGD with the presented adaptive weight-decay (AdaDecay). Regarding hyper-parameters in our experiments, we use a practical condition including the mini-batch size of $B=128$ with the momentum of 0.9. The weight-decay coefficient is inspected in Section \[sec:wd-tuning\] and fixed at $\lambda=5\times10^{-4}$ for all the algorithms. The hyper-parameter $\alpha$ of AdaDecay that determines the adaptation to the gradient norm is inspected in Section \[sec:alpha\] and fixed at $\alpha=4$. The learning-rate annealing with a sigmoid function that starts from $\eta=0.1$ and ends at $\eta=0.001$ is applied for SGD, eSGD, aSGD, and AdaDecay based on a pre-experiment result in which SGD with the sigmoid learning-rate annealing has achieved better accuracy than those with the fixed learning rate, exponential function, and staircase. We use grid search and set 0.95 as the weighting factor in RMSprop, 0.9 and 0.999 for the first and second momentum factors in Adam. The learning-rate scale in RMS and Adam is set as 0.001 for the shallow networks, and 0.0001 for the deep networks. The hyper-parameters for eSGD and aSGD are also set as the recommended in the original papers [@Chaudhari2017EntropySGD; @Kidambi2018Acc] including the Langevin loop number of 5 for eSGD. We perform the training process for 100 epochs, and use the maximum and the last $10\%$-epoch mean of the validation accuracy for the test data as the evaluation measures of each trial. For quantitative comparison, we repeat the training process of the shallow networks with MNIST and Fashion-MNIST datasets for 50 independent trials, and the deep networks with CIFAR-10 dataset for 32 trials. We consider both the maximum of the validation accuracy across epochs and trials, and the $10\%$-trimmed average of the last $10\%$-epoch accuracy over the trials. Selection of Weight-Decay Coefficient {#sec:wd-tuning} ------------------------------------- The weight-decay coefficient $\lambda$ determines the balance between the stochastic loss with the regularization and thus plays a critical role in both the standard weight-decay and the presented AdaDecay. In Figure \[fig:wd-tuning\], we test the coefficient ranging in $\lambda = 1\times10^{-3}, 7\times10^{-4}, 5\times10^{-4}, ... , 1\times10^{-4}$, using MNIST with NN-2, Fashion-MNIST with NN-2, and CIFAR-10 with ResNet-18 as instances. We compare SGD using the constant weight-decay (constant) with SGD using our AdaDecay with fixed $\alpha=4$ (ours) where the two algorithms share the same weight-decay coefficient. Figure \[fig:wd-tuning\] successfully demonstrates that the presented AdaDecay overtakes the constant weight-decay irrespective of the weight-decay coefficient $\lambda$ across both datasets and the model architecture. Based on the these results and the related works [@simonyan2014very; @chollet2017xception], we employ $\lambda=5\times10^{-4}$ in our experiments. \[htb\] ------------------------------------------------- -------------------------------------------------------- ------------------------------------------------ ![image]({{wd.MNIST.FFNet_MNIST}}){width="\fw"} ![image]({{wd.FashionMNIST.FFNet_MNIST}}){width="\fw"} ![image]({{wd.cifar10.ResNet18}}){width="\fw"} \(1) MNIST \(2) Fashion-MNIST \(3) CIFAR-10 ------------------------------------------------- -------------------------------------------------------- ------------------------------------------------ [p[6mm]{} | PPPPP | PPPPP | PPPPP]{} & & &\ $\alpha$ & -1 & 1 & 2 & 4 & 8 & -1 & 1 & 2 & 4 & 8 & -1 & 1 & 2 & 4 & 8\ ave & 98.48 & 98.55 & 98.55 & 98.56 & 98.55 & 89.01 & 89.27 & 89.31 & 89.49 & 89.58 & 93.92 & 94.79 & 94.78 & 94.80 & 94.74\ max & 98.57 & 98.67 & 98.70 & 98.72 & 98.69 & 89.35 & 89.56 & 89.68 & 89.84 & 89.92 & 94.24 & 95.05 & 94.98 & 95.04 & 94.94\ Adaptation to Gradient Norm {#sec:alpha} --------------------------- We empirically demonstrates the effect of hyper-parameter $\alpha$ in Eq.  that determines the adaptation to the gradient norm normalized in layers. We present the $10\%$-trimmed average and the maximum accuracy over the trials in Table \[tab.alpha.tuning\] where $\alpha$ is set as -1, 1, 2, 4, 8 and we trained NN-2 using MNIST and Fashion-MNIST datasets and ResNet-18 using CIFAR-10 for instances. As shown in Table \[tab.alpha.tuning\], $\alpha=-1$ is inferior to $\alpha=1$ in accuracy. This allows our algorithm to impose higher weight-decay rate due to larger gradient-norms. Table \[tab.alpha.tuning\] also demonstrates the presented AdaDecay is robust to the choice of the hyper-parameter $\alpha>0$. We use $\alpha=4$ throughout the following experiment, that has achieved the best result for MNIST and CIFAR10 and second best for Fashion-MNIST in Table \[tab.alpha.tuning\] Comparison to Randomized Weight-Decay {#sec:comparison.to.randomized.wd} ------------------------------------- Since we normalize the gradient norm in layer with mean of 0 and std of 1, one may argue that AdaDecay involves a randomization effect to weight-decay. We thus compare AdaDecay with a noise injection to the weight-decay, namely randomized weight-decay, that follows Algorithm \[alg:adadecay\] but replaces Eq.  by $$\theta_j^t = S(\mathcal{N}_{(0,1)}; \alpha) = \frac{2}{1 + \exp(- \alpha \mathcal{N}_{(0,1)})}, \label{eq:RND}$$ where $\mathcal{N}_{(0,1)}$ is a random variable following the Normal distribution with mean of 0 and std of 1. Table \[tab:randomized-wd\] presents the validation accuracy for MNIST, Fashion-MNIST, and CIFAR-10 by fundamental models of NN-2, NN-2, and ResNet-18 respectively, trained by SGD with constant weight-decay, the randomized weight-decay, and the proposed AdaDecay with $\lambda=5\times10^{-4}$ and $\alpha=4$. It is successfully demonstrated that the benefit of our AdaDecay is not due to the randomization effect to the weight-decay but the use of adaptive weight-decay based on the gradient norm. [P[4mm]{} | PPP | PPP | PPP]{} & & &\ & const & rnd & ours & const & rnd & our & cons & rnd & ours\ ave & 98.53 & 98.53 & **98.56** & 89.23 & 89.25 & **89.49** & 94.70 & 94.70 & **94.80**\ max & 98.63 & 98.65 & **98.72** & 89.50 & 89.59 & **89.84** & 94.98 & 95.00 & **95.04**\ \[htb\] [PP]{} ![Validation accuracy over drop-out rate $P = 0, 2^{-4}, 2^{-3}, 2^{-2}$ by SGD with the constant weight-decay (black) and the proposed AdaDecay with fixed $\alpha=4$ (red) for MNIST (left) and Fashion-MNIST (right) using LeNet-4. The weight-decay coefficient is fixed at $\lambda=5\times10^{-4}$. The $10\%$-trimmed average of the last $10\%$-epoch mean accuracy over 50 trials for MNIST and Fashion-MNIST is shown. []{data-label="fig:dropout"}]({{dropout.mnist.ConvNet_MNIST}} "fig:"){width="\fw"} & ![Validation accuracy over drop-out rate $P = 0, 2^{-4}, 2^{-3}, 2^{-2}$ by SGD with the constant weight-decay (black) and the proposed AdaDecay with fixed $\alpha=4$ (red) for MNIST (left) and Fashion-MNIST (right) using LeNet-4. The weight-decay coefficient is fixed at $\lambda=5\times10^{-4}$. The $10\%$-trimmed average of the last $10\%$-epoch mean accuracy over 50 trials for MNIST and Fashion-MNIST is shown. []{data-label="fig:dropout"}]({{dropout.FashionMNIST.ConvNet_MNIST}} "fig:"){width="\fw"}\ (1) MNIST & (2) Fashion-MNIST\ Effect of Dropout {#sec:effect.of.dropout} ----------------- We demonstrate that our AdaDecay can be combined with dropout that gives an implicit regularization effect. We present accuracy curve over the dropping rate ($P$) within 50 trials in Figure \[fig:dropout\] where we trained LeNet-4 that supports the dropout for MNIST and Fashion-MNIST using SGD with the constant weight-decay of $\lambda=5\times10^{-4}$ (SGD) and with our AdaDecay (Ours) with $\lambda=5\times10^{-4}$ and $\alpha=4$. Figure \[fig:dropout\] demonstrates that our AdaDecay overtakes the constant weight-decay irrespective of the use of dropout. We employ the dropping rate of $P=0$ for LeNet-4 in the other experiment for a fair comparison with other network models. Comparison to State-of-the-Arts {#sec:compariosn.to.soa} ------------------------------- We now compare SGD with the presented adaptive weight-decay (AdaDecay) to SGD with the constant weight-decay (SGD), SGD with RMSprop [@tieleman2012lecture] (RMS), SGD with Adam [@kingma2014adam] (Adam), Entropy-SGD [@Chaudhari2017EntropySGD] (eSGD), and Accelerated-SGD [@Kidambi2018Acc] (aSGD). We fix the weight-decay coefficient at $\lambda=5\times10^{-4}$ for all the algorithms and $\alpha=4$ for ours. In Table \[tab:accuracy\], we present the $10\%$-trimmed average (upper) and the maximum (lower) validation accuracy with the shallow networks: NN-2, NN-3, LeNet-4, and VGG-9 for MNIST (1) and Fashion-MNIST (2) over 50 trials, and the deep models: ResNet-18, ResNet-50, GoogLeNet, and DesnseConv for CIFAR-10 (3) over 32 trials, respectively. It is shown that SGD powered by our AdaDecay outperforms all the others in the average accuracy consistently regardless of model and dataset. The visualization of the average accuracy curve over epochs in Figure \[fig:accuracy\_curve\] indicating that SGD with our AdaDecay (red) achieves better accuracy than SGD with the constant weight-decay (black), RMSprop (yellow), Adam (blue), Engropy-SGD (magenta), and Accelerated-SGD (green) across both the shallow networks with MNIST (1) and Fashion-MNIST (2), and the deep networks with CIFAR-10 (3). \(1) Validation accuracy for MNIST\ [l | PPPPPP | PPPPPP]{} & &\ & SGD & RMS & Adam & eSGD & aSGD & Ours & SGD & RMS & Adam & eSGD & aSGD & Ours\ ave & 98.53 & 98.22 & 98.19 & 98.16 & 98.15 & **98.56** & 98.66 & 98.31 & 98.26 & 98.29 & 98.23 & **98.69**\ max & 98.63 & 98.36 & 98.31 & 98.31 & 98.35 & **98.72** & 98.80 & 98.49 & 98.47 & 98.41 & 98.45 & **98.82**\ & &\ & SGD & RMS & Adam & eSGD & aSGD & Ours & SGD & RMS & Adam & eSGD & aSGD & Ours\ ave &99.31 & 99.30 & 99.27 & 99.23 & 99.17 & **99.32** & 99.62 & 99.37 & 99.37 & 99.58 & 99.52 & **99.63**\ max &**99.48** & 99.39 & 99.37 & 99.38 & 99.28 & 99.45 & **99.71** & 99.50 & 99.43 & 99.63 & 99.59 & 99.70\ \ (2) Validation accuracy for Fashion-MNIST\ [l | PPPPPP | PPPPPP]{} & &\ & SGD & RMS & Adam & eSGD & aSGD & Ours & SGD & RMS & Adam & eSGD & aSGD & Ours\ ave & 89.23 & 88.89 & 88.98 & 87.63 & 89.12 & **89.49** & 89.71 & 89.17 & 89.26 & 88.13 & 89.24 & **89.95**\ max & 89.50 & 89.15 & 89.28 & 87.83 & 89.44 & **89.84** & 89.87 & 89.54 & 89.49 & 88.32 & 89.56 & **90.23**\ & &\ & SGD & RMS & Adam & eSGD & aSGD & Ours & SGD & RMS & Adam & eSGD & aSGD & Ours\ ave & 90.65 & 90.81 & 90.78 & 89.76 & 90.12 & **90.87** & 93.45 & 91.97 & 92.08 & 93.33 & 93.05 & **93.51**\ max & 91.36 & 91.30 & 91.23 & 90.54 & 90.48 & **91.51** & 93.73 & 92.36 & 92.46 & 93.72 & 93.37 & **93.83**\ \ (3) Validation accuracy for CIFAR-10\ [l | PPPPPP | PPPPPP]{} & &\ & SGD & RMS & Adam & eSGD & aSGD & Ours & SGD & RMS & Adam & eSGD & aSGD & Ours\ ave & 94.70 & 90.72 & 90.92 & 91.46 & 93.07 & **94.80** & 94.61 & 91.22 & 91.26 & 90.76 & 92.82 & **94.71**\ max & 94.98 & 91.25 & 91.36 & 91.99 & 93.38 & **95.04** & 95.16 & 91.82 & 91.73 & 91.38 & 93.36 & **95.22**\ & &\ & SGD & RMS & Adam & eSGD & aSGD & Ours & SGD & RMS & Adam & eSGD & aSGD & Ours\ ave & 94.91 & 90.48 & 90.62 & 93.00 & 93.39 & **95.17** & 94.72 & 83.17 & 83.80 & 88.05 & 90.27 & **94.91**\ max & 95.43 & 90.94 & 90.97 & 93.45 & 93.79 & **95.50** & 95.08 & 83.60 & 84.40 & 88.60 & 90.56 & **95.22**\ \[htb\] ![image]({{legend.adadecay}}){width="260pt"}\ [PPPP]{} ![image]({{cmpAccuracy,MNIST,FFNet_MNIST,e,100,B,128,mo,0.9}}){width="\fw"} & ![image]({{cmpAccuracy,MNIST,FFFNet_MNIST,e,100,B,128,mo,0.9}}){width="\fw"} & ![image]({{cmpAccuracy,MNIST,ConvNet_MNIST,e,100,B,128,mo,0.9}}){width="\fw"} & ![image]({{cmpAccuracy,MNIST,VGG9_1d,e,100,B,128,mo,0.9}}){width="\fw"}\ NN-2 & NN-3 & LeNet-4 & VGG-9\ \ ![image]({{cmpAccuracy,FashionMNIST,FFNet_MNIST,e,100,B,128,mo,0.9}}){width="\fw"} & ![image]({{cmpAccuracy,FashionMNIST,FFFNet_MNIST,e,100,B,128,mo,0.9}}){width="\fw"} & ![image]({{cmpAccuracy,FashionMNIST,ConvNet_MNIST,e,100,B,128,mo,0.9}}){width="\fw"} & ![image]({{cmpAccuracy,FashionMNIST,VGG9_1d,e,100,B,128,mo,0.9}}){width="\fw"}\ NN-2 & NN-3 & LeNet-4 & VGG-9\ \ ![image]({{cmpAccuracy,cifar10,ResNet18,e,100,B,128,mo,0.9}}){width="\fw"} & ![image]({{cmpAccuracy,cifar10,ResNet50,e,100,B,128,mo,0.9}}){width="\fw"} & ![image]({{cmpAccuracy,cifar10,GoogLeNet,e,100,B,128,mo,0.9}}){width="\fw"} & ![image]({{cmpAccuracy,cifar10,DenseConv,e,100,B,128,mo,0.9}}){width="\fw"}\ ResNet-18 & ResNet-50 & GoogLeNet & DenseConv\ \ Conclusion {#sec:conclusion} ========== We have presented an adaptive regularization method for deep neural networks driven by spatio-temporal weight-decay. The proposed algorithm is designed to consider parameter-wise weight-decay and determine it based on the norm of gradient that reflects the current model and the given data at each optimization iteration. The proposed AdaDecay penalizes large gradient norm and leads to better generalization of the model independent to network architectures and is performed without any additional cost of back-propagation or inner loop. The robustness and effectiveness of the AdaDecay has been empirically supported by experimental results in which SGD using our AdaDcay outperforms a number of other optimization methods for the image classification task with the shallow and deep networks using the major datasets. We have focused on the image classification yet the presented adaptive regularization would have a potential impact to other machine-learning tasks using neural networks in essential. Acknowledgment {#acknowledgment .unnumbered} ============== This work was partially supported by the National Research Foundation of Korea: NRF-2017R1A2B4006023 and NRF-2018R1A4A1059731. [^1]: \* corresponding author: B.-W. Hong e-mail: hong@cau.ac.kr [^2]: K Nakamura and B.-W. Hong are with the Computer Science Department, Chung-Ang University, Korea (see https://www.image.cau.ac.kr/).
--- abstract: 'For an $R$-module $M$, projective in $\sigma[M]$ and satisfying ascending chain condition (ACC) on left annihilators, we introduce the concept of Goldie module. We also use the concept of semiprime module defined by Raggi et. al. in [@S] to give necessary and sufficient conditions for an $R$-module $M$, to be a semiprime Goldie module. This theorem is a generalization of Goldie’s theorem for semiprime left Goldie rings. Moreover, we prove that $M$ is a semiprime (prime) Goldie module if and only if the ring $S=End_R(M)$ is a semiprime (prime) right Goldie ring. Also, we study the case when $M$ is a duo module.' author: - | Jaime Castro Pérez[^1]\ *Escuela de Ingeniería y Ciencias, Instituto Tecnolólogico y de*\ *Estudios Superiores de Monterrey*\ *Calle del Puente 222, Tlalpan, 14380, México D.F., México.*\ Mauricio Medina Bárcenas[^2] José Ríos Montes[^3]\ Angel Zaldívar[^4]\ *Instituto de Matemáticas, Universidad Nacional*\ *Autónoma de México*\ *Area de la Investigación Científica, Circuito Exterior, C.U.,*\ *04510, México D.F., México.* title: On Semiprime Goldie Modules --- *Keywords*: Prime module, Semiprime module, Goldie module, Essentially compressible module, Duo module. *2010 Mathematics Subject Classification*: 16D50, 16D80, 16P50, 16P70. Introduction {#introduction .unnumbered} ============ Goldie’s Theorem states that a ring $R$ has a semisimple artinian classical left quotient ring if and only if $R$ is a semiprime ring with finite uniform dimension and satisfies ACC on left annihilators. Wisbauer proves in ([@W], Theorem 11.6) a version of Goldie’s Theorem in terms of modules. For a retractable $R$-module $M$ with $S=End_R(M)$ the following conditions are equivalent: $\textit{1}$. $M$ is non $M$-singular with finite uniform dimension and $S$ is semiprime, $\textit{2}$. $M$ is non $M$-singular with finite uniform dimension and for every $N\leq_e{M}$ there exists a monomorphism $M\rightarrow{N}$, $\textit{3}$. $End_R(\widehat{M})$ is semisimple left artinian and it is the classical left quotient ring of $S$, here $\widehat{M}$ denotes the $M$-injective hull of $M$. Also, in [@H] the authors study when the endomorphism ring of a semiprojective module is a semiprime Goldie ring. In this paper we give another generalization of Goldie’s Theorem. For this, we use the product of submodules of a module $M$ defined in [@Bic] to say when a module is a semiprime module. This product extends the product of left ideals of a ring $R$, so $R$ is a semiprime module (over itself) if and only if $R$ is a semiprime ring in the usual sense. In order to have a definition of Goldie Module such that it extends the classical definition of left Goldie ring, we introduce what ascending chain condition on left annihilators means on a module. A left annihilator in $M$ is a submodule of the form $\mathcal{A}_X=\bigcap_{f\in{X}}{Ker(f)}$ for some $X\subseteq{End_R(M)}$. This definition with $R=M$ is the usual concept of left annihilator. The main concept of this work is that an $R$-module $M$ is a Goldie module if $M$ satisfies ACC on left annihilators and has finite uniform dimension. We prove some characterizations of semiprime Goldie modules (Theorem \[110\], Theorem \[144\] and Corollary \[145\]) which generalize the Goldie’s Theorem and extends the Theorem 11.6 of [@W] and corollary 2.7 of [@H]. We organize this paper in three sections. Section 1 proves several results for semiprime modules. We also generalize Theorem 10.24 of [@LN] to semiprime artinian modules. In section 2 we introduce the concept of Goldie modules. We prove the main Theorem of this paper and a characterization of semiprime Goldie modules. We also obtain some examples of Goldie modules. We also prove that if $M$ has finitely many minimal prime submodules $P_1$,...,$P_t$ in $M$ such that $M/P_i$ $(1\leq{i}\leq{t})$ has finite uniform dimension, then $M$ is Goldie module if and only if each $M/P_i$ is Goldie module for $(1\leq{i}\leq{t})$. We also give a description of the submodule $\mathcal{Z}(N)$ with $N\in\sigma[M]$. In the last section we apply the previous results to duo modules which extend results for commutative rings. In [@DM] the authors say that they do not know a duo module with a quotient not duo, in this section we show an example. Throughout this paper $R$ will be an associative ring with unit and $R$-Mod will denote the category of unitary left $R$-modules. A submodule $N$ of an $R$-module $M$ is denoted by $N\leq{M}$. If $N$ is a proper submodule we write $N<M$. We use $N\leq_e{M}$ for an essential submodule. Let $M$ and $X$ be $R$-modules. $X$ is said to be $M$-generated if there exists an epimorphism from a direct sum of copies of $M$ onto $X$. Every $R$-module $X$ has a largest $M$-generated submodule called the trace of $M$ in $X$, defined by $tr^M(X)=\sum\{f(M)|f:M\to X\}$. The category $\sigma[M]$ is defined as the smallest full subcategory of $R$-Mod containing all $R$-modules $X$ which are isomorphic to a submodule of an $M$-generated module. A module $N\in\sigma[M]$ is called singular in $\sigma[M]$ or $M$-singular, if there is an exact sequence in $\sigma[M]$, $0\rightarrow{K}\rightarrow{L}\rightarrow{N}\rightarrow{0}$ with $K\leq_e{L}$. The class $\mathcal{S}$ of all $M$-singular modules in $\sigma[M]$ is closed under submodules, quotients and direct sums. Therefore, any $L\in\sigma[M]$ has a largest $M$-singular submodule $$\mathcal{Z}(L)=\sum\{f(N)|N\in\mathcal{S}\;\rm{and}\;f\in{Hom_R(N,L)}\}$$ $L$ is called non $M$-singular if $\mathcal{Z}(L)=0$. Let $M$ be an $R$-module. In [@B] the annihilator in $M$ of a class $\mathcal{C}$ of modules is defined as $Ann_M(\mathcal{C})=\bigcap_{K\in\Omega}{K}$, where $$\Omega=\{K\leq{M}|\rm{there}\;\rm{exists}\;W\in\mathcal{C}\;and\;f\in{Hom_R(M,W)}\;\rm{with}\;K=Ker(f)\}$$ Also in [@B], the author defines a product in the following way: Let $N\leq{M}$. For each module $X$, $N\cdot{X}=Ann_M(\mathcal{C})$ where $\mathcal{C} $ is the class of modules $W$ such that $f(N)=0$ for all $f\in{Hom_R(M,W)}$. For an $R$-module $M$ and $K,L$ submodules of $M$, in [@Bic] the product $K_ML$ is defined by $K_ML=\sum\{f(K)|f\in{Hom_R(M,L)}\}$. Moreover, in [@B] it is showed that if $M$ is projective in $\sigma[M]$, and $N\leq{M}$, then $N\cdot{X}=N_MX$ for every module $X$. A nonzero $R$-module $M$ is called monoform if for each submodule $N$ of $M$ and each morphism $f:N\rightarrow{M}$, $f$ is either zero or a monomorphism. $M$ has enough monoforms if each nonzero submodule of $M$ contains a monoform submodule. Let $M$-tors be the frame of all hereditary torsion theories on $\sigma[M]$. For a family $\{M_\alpha\}$ of modules in $\sigma[M]$, let $\chi(\{M_\alpha\})$ the greatest element of $M$-tors for which all $M_\alpha$ are torsion free. Let $\xi(\{M_\alpha\})$ be the least element of $M$-tors for which all $M_\alpha$ are torsion. $\xi(\{M_\alpha\})$ and $\chi(\{M_\alpha\})$ are called the hereditary torsion theory generated by the family $\{M_\alpha\}$ and the hereditary torsion theory cogenerated by the same family. In particular, the greatest and least elements in $M$-tors are denoted by $\chi$ and $\xi$ respectively. If $\tau\in{M-tors}$, let $\mathbb{T}_\tau$, $\mathbb{F}_\tau$ and $t_\tau$ denote the torsion class, the torsion free class and the preradical associated to $\tau$, respectively. For details about concepts and terminology concerning torsion theories in $\sigma[M]$, see [@Foun] and [@W]. Semiprime Modules ================= \[section\] \[I11\] Let $M\in{R-Mod}$ and $K$, $L$ submodules of $M$. Put $K_ML=\sum\{f(K)|f\in{Hom_R(M,L)}\}$. For the properties of this product see [@P] Proposition 1.3. \[11\][Definition]{} Let $M\in{R-Mod}$. We say a fully invariant submodule $N\leq{M}$ is a *prime submodule* in $M$ if for any fully invariant submodules $K,L\leq{M}$ such that $K_ML\leq{N}$, then $K\leq{N}$ or $L\leq{N}$. We say $M$ is a *prime module* if $0$ is a prime submodule. \[11\][Proposition]{} \[121\] Let $M$ be projective in $\sigma[M]$ and $P$ a fully invariant submodule of $M$. The following conditions are equivalent: 1. $P$ is prime in $M$. 2. For any submodules $K$, $L$ of $M$ containing $P$ and such that $K_ML\leq{P}$, then $K=P$ or $L=P$. $\textit{1}\Rightarrow\textit{2}:$ By Proposition 1.11 of [@P]. $\textit{2}\Rightarrow\textit{1}:$ Suppose that $K$, $L$ are submodules of $M$ such that $K_ML\leq{P}$. We claim that $K_M(L+P)\leq{P}$. Since $K_ML\leq{L\cap{P}}$, by Proposition 5.5 of [@B] $K_M({L}/{L\cap{P}})=0$ so $K_M({L+P}/{P})=0$. Thus $K_M(L+P)\leq{P}$. On the other hand, $$(K+P)_M(L+P)=K_M(L+P)+P_M(L+P)\leq{P}$$ because $P$ is fully invariant in $M$. Then, by hypothesis $K+P={P}$ or $L+P={P}$, hence $K\leq{P}$ or $L\leq{P}$. \[11\][Definition]{} We say a fully invariant submodule $N\leq{M}$ is a *semiprime submodule* in $M$ if for any fully invariant submodule $K\leq{M}$ such that $K_MK\leq{N}$, then $K\leq{N}$. We said $M$ is a *semiprime module* if $0$ is a semiprime submodule. \[11\][Lemma]{} \[143\] Let $M$ be projective in $\sigma[M]$ and $N$ a fully invariant submodule of $M$. The following conditions are equivalent: 1. $N$ is semiprime in $M$. 2. For any submodule $K$ of $M$, $K_MK\leq{N}$ implies $K\leq{M}$. 3. For any submodule $K\leq{M}$ containing $N$ such that $K_MK\leq{N}$, then $K=N$. $\textit{1}\Rightarrow\textit{2}:$ Let $K\leq{M}$ such that $K_MK\leq{N}$. Consider the submodule $K_MM$ of $M$. This is the minimal fully invariant submodule of $M$ which contains $K$ and $K_MX=(K_MM)_MX$ for every module $X$. Hence by Proposition 1.3 of [@P] we have that $$K_MK=(K_MM)_MK\leq{((K_MM)_MK)_MM)}\leq{N_MM}$$ Since $N$ is fully invariant submodule of $M$ then $N_MM=N$ and by Proposition 5.5 of [@B] $(K_MM)_M(K_MM)=((K_MM)_MK)_MM)\leq{N}$. Since $N$ is semiprime in $M$, $K_MM\leq{N}$. Hence $K\leq{N}$. $\textit{2}\Rightarrow\textit{1}:$ By definition. $\textit{1}\Leftrightarrow\textit{3}:$ Similar to the proof of Proposition \[121\]. In Remark \[aa\] below, we give an example where the associativity of the product $(\cdot)_M(\cdot)$ is not true in general. \[11\][Definition]{} Let $M\in{R-Mod}$ and $N$ a fully invariant submodule of $M$. We define the *powers* of $N$ as: 1. $N^0=0$ 2. $N^1=N$ 3. $N^m=N_MN^{m-1}$ \[11\][Lemma]{} \[124\] Let $M$ be projective in $\sigma[M]$ and $N$ semiprime in $M$. Let $J$ be a fully invariant submodule of $M$ such that $J^n\leq{N}$ then $J\leq{N}$. By induction on $n$. If $n=1$ the result is clear. Suppose $n>1$ and the Proposition is valid for $n-1$. We have that $2n-2\geq{n}$ then $$J^{2n-2}\leq{N}$$ so $$(J^{n-1})^2={J^{n-1}}_M{J^{n-1}}\leq{N}$$ since $N$ is semiprime $J^{n-1}\leq{N}$ then $J\leq{N}$. \[11\][Proposition]{} \[108a\] Let $S:=End_R(M)$ and assume $M$ generates all its submodules. If $N$ is a fully invariant submodule of $M$ such that $Hom_R(M,N)$ is a prime (semiprime) ideal of $S$, then $N$ is prime (semiprime) in $M$. Let $K$ and $L$ be fully invariant submodules of $M$ such that $K_ML\leq{N}$. Put $I=Hom_R(M,L)$ and $J=Hom_R(M,K)$. Let $m\in{M}$ and $\sum{f_ig_i}\in{IJ}$. Since $g_i\in{J}$ and $g_i(m)\in{K}$ then $\sum{f_i(g_i(m))}\in{K_ML}\leq{N}$. Hence $IJ\leq{Hom_R(M,N)}$. Since $Hom_R(M,N)$ is prime (semiprime) in $S$, then $I\leq{Hom_R(M,N)}$ or $J\leq{Hom_R(M,N)}$. Hence $tr^M(L):=Hom(M,L)M\leq{N}$ or $tr^M(K)\leq{N}$ and since $M$ generates all its submodules then $L\leq{N}$ or $K\leq{N}$. Thus $N$ is a prime (semiprime) submodule. Next definition aper in [@Kh] \[11\][Definition]{} A module $M$ is *retractable* if $Hom_R(M,N)\neq{0}$ for all $0\neq{N}\leq{M}$ \[11\][Corollary]{} \[108\] Let $S:=End_R(M)$ with $M$ retractable. If $S$ is a prime (semiprime) ring then $M$ is prime (semiprime). Let $K$ and $L$ be fully invariant submodules of $M$ such that $K_ML=0$. Since $Hom_R(M,0)$ is a prime (semiprime) ideal of $S$ then by the proof of \[108a\], $tr^M(K)=0$ o $tr^M(L)=0$. Since $M$ is retractable, $K=0$ or $L=0$. Hence $0$ is prime (semiprime) in $M$. Thus $M$ is prime (semiprime). \[11\][Proposition]{} \[123\] Let $M$ be projective in $\sigma[M]$ and $N$ a proper fully invariant submodule of $M$. The following conditions are equivalent: 1. $N$ is semiprime in $M$. 2. If $m\in{M}$ is such that ${Rm}_M{Rm}\leq{N}$, then $m\in{N}$. 3. $N$ is an intersection of prime submodules. $\textit{1}\Rightarrow\textit{2}:$ By Lemma \[143\]. $\textit{2}\Rightarrow\textit{3}:$ Since $N$ is proper in $M$, let $0\neq{m_0}\in{M\setminus{N}}$. Then ${Rm_0}_M{Rm_0}\nleq{N}$. Now, let $0\neq{m_1}\in{{Rm_0}_M{Rm_0}}$ but $m_1\notin{N}$ Then ${Rm_1}_M{Rm_1}\nleq{N}$ and ${Rm_1}_M{Rm_1}\leq{{Rm_0}_M{Rm_0}}$. We obtain a sequence of non-zero elements of $M$, $\{m_0,m_1,...\}$ such that $m_i\notin{N}$ for all $i$ and ${Rm_{i+1}}_M{Rm_{i+1}}\leq{{Rm_i}_M{Rm_i}}$. By Zorn’s Lemma there exists a fully invariant submodule $P$ of $M$ with $N\leq{P}$, maximal with the property that $m_i\notin{P}$ for all $i$ . We claim $P$ is a prime submodule. Let $K$ and $L$ submodules of $M$ containing $P$. Since $P\leq{K}$ and $P\leq{L}$, then there exists $m_i$ and $m_j$ such that $m_i\in{K}$ and $m_j\in{L}$. Suppose $i\leq{j}$, then ${Rm_i}_M{Rm_i}\leq{K}$ and by construction $m_j\in{Rm_i}_M{Rm_i}$ and thus $m_j\in{K}$. If we put $k=max\{i,j\}$, then $m_k\in{K}$ and $m_k\in{L}$. Hence, ${Rm_k}_M{Rm_k}\leq{K_ML}$, and so $K_ML\nleq{P}$. By Proposition \[121\], $P$ is prime in $M$. $\textit{3}\Rightarrow\textit{1}:$ It is clear. \[11\][Proposition]{} \[116c\] Let $0\neq{M}$ be a semiprime module and projective in $\sigma[M]$. Then $M$ has minimal prime submodules in $M$. By the proof Proposition of \[123\], $M$ has prime submodules. Let $P\leq{M}$ be a prime submodule. Consider $\Gamma=\{Q\leq{P}|Q\;is\;prime\}$. This family is not empty because $P\in\Gamma$. Let $\mathcal{C}=\{Q_i\}$ be a descending chain in $\Gamma$. Let $N,K\leq{M}$ be fully invariant submodules of $M$ such that $N_MK\leq\bigcap\mathcal{C}$. Suppose that $N\nleq\bigcap\mathcal{C}$. Then there exists $Q_j$ such that $N\nleq{Q_j}$ and $N\nleq{Q_l}$ for all $Q_l\leq{Q_j}$. Therefore $K\leq{Q_l}$ for all $Q_l\leq{Q_j}$, and since $\mathcal{C}$ is a chain then $K\leq\bigcap\mathcal{C}$. Therefore $\bigcap\mathcal{C}\in\Gamma$. By Zorn’s Lemma $\Gamma$ has minimal elements. \[11\][Remark]{} Notice that if $M$ is projective in $\sigma[M]$ and $M$ has prime submodules in $M$, then $M$ has minimal prime submodules. \[11\][Corollary]{} \[116a\] Let $0\neq{M}$ be a semiprime module and projective in $\sigma[M]$. Then $$0=\bigcap\{P\leq{M}|P\;is\;a\;minimal\;prime\;in\;M\}.$$ Let $x\in\bigcap\{P\leq{M}|P\;is\;a\;minimal\;prime\;in\;M\}$ and $Q\leq{M}$ be a prime submodule in $M$. By Proposition \[116c\] there exists a minimal prime submodule $P$ such that $P\leq{Q}$ then $x\in{Q}$ and $x$ is in the intersection of all primes in $M$. By Proposition \[123\], $x=0$. \[11\][Lemma]{} \[126\] Let $M\in{R-Mod}$ and $N$ a minimal submodule of $M$. Then $N^2=0$ or $N$ is a direct summand of $M$. Suppose that $N_MN\neq{0}$. Then there exists $f:M\rightarrow{N}$ such that $f(N)\neq{0}$. Since $0\neq{f(M)}\leq{N}$ and $N$ is a minimal submodule, $f(M)=N$. On the other hand, $Ker(f)\cap{N}\leq{N}$, since $f(N)\neq{0}$ then $Ker(f)\cap{N}=0$. We have that $({M}/{Ker(f)})\cong{N}$ and since $N$ is a minimal submodule $Ker(f)$, then is a maximal submodule of $M$. Thus $Ker(f)\oplus{N}=M$. \[11\][Corollary]{} \[127\] Let $M$ be a retractable module. If $N$ is a minimal submodule in a semiprime module $M$, then $N$ is a direct summand. Since $M$ is semiprime, $N_MN\neq{0}$. \[11\][Theorem]{} \[128\] The following conditions are equivalent for a retractable $R$-module $M$: 1. $M$ is semisimple and left artinian. 2. $M$ is semiprime and left artinian. 3. $M$ is semiprime and satisfies DCC on cyclic submodules and direct summands. $\textit{1}\Rightarrow\textit{2}:$ If $M$ is semisimple then it is semiprime. $\textit{2}\Rightarrow\textit{3}:$ Since $M$ is left artinian, then it satisfies DCC on cyclic submodules and direct summands. $\textit{3}\Rightarrow\textit{1}:$ Since $M$ satisfies DCC on cyclic submodules, there exists $K_1$ a minimal submodule of $M$. By Corollary \[127\], $M=K_1\oplus{L_1}$. Now there exists $K_2$ a minimal submodule of $L_1$ and $L_1=K_2\oplus{L_2}$. With this process we obtain a descending chain of direct summands, which by hypothesis it is finite $L_1\supseteq{L_2}\supseteq{L_3}\supseteq...\supseteq{L_m}$. Since $L_m$ is simple and $M=K_1\oplus{K_2}\oplus...\oplus{K_m}\oplus{L_m}$, then $M$ is semisimple. Now, if $M$ is semisimple and satisfies DCC on direct summands then $M$ is artinian. \[11\][Definition]{} \[11\] Let $M\in{R-Mod}$ and $N\leq{M}$. We say $N$ is an *annihilator submodule* if $N=Ann_M(K)$ for some $0\neq{K}\leq{M}$. \[11\][Lemma]{} \[112a\] Let $M$ be semiprime and projective in $\sigma[M]$. Let $N,L\leq{M}$. If $L_MN=0$, then $N_ML=0$ and $L\cap{N}=0$. Since $L_MN=0$, then $$0=N_M(L_MN)_ML=(N_ML)_M(N_ML).$$ Hence $N_ML=0$ . Now, since $L\cap{N}\leq{L}$ and $L\cap{N}\leq{N}$, then $$(L\cap{N})_M(L\cap{N})\leq{L_MN}=0.$$ Thus $L\cap{N}=0$ \[11\][Corollary]{} \[112\] Let $M$ be semiprime and projective in $\sigma[M]$. If $N\leq{M}$, then $N_MAnn_M(N)=0$. \[11\][Proposition]{} \[113\] Let $M$ be semiprime and projective in $\sigma[M]$ and $N\leq{M}$. Then $N$ is an annihilator submodule if and only if $N=Ann_M(Ann_M(N))$ $\Rightarrow:$ By Lemma \[112\] $N\leq{Ann_M(Ann_M(N))}$. There is $K\leq{M}$ such that $N=Ann_M(K)$, hence $$K_MN=K_MAnn_M(K)=0$$ and thus $K\leq{Ann_M(N)}$. Therefore, $$Ann_M(Ann_M(N))\leq{Ann_M(K)}=N$$ It follows that $N=Ann_M(Ann_M(N))$. $\Leftarrow:$ By definition of annihilator submodule. \[11\][Proposition]{} \[114\] Let $M$ be semiprime and $N\leq{M}$. Then, $Ann_M(N)$ is the unique pseudocomplement fully invariant of $N$. Moreover, $N\bigoplus{Ann_M(N)}$ intersects all fully invariant submodules of $M$. Let $L\leq{M}$ be a fully invariant pseudocomplement of $N$ in $M$. Then $$L_MN\leq{L\cap{N}}=0$$ Thus $L\leq{Ann_M(N)}$. Observe that $$(Ann_M(N)\cap{N})_M(Ann_M(N)\cap{N})\leq(Ann_M(N)\cap{N})_MN=0$$ Since $M$ is semiprime, $Ann_M(N)\cap{N}=0$. Thus $L=Ann_M(N)$. \[11\][Lemma]{} \[116b\] Let $M$ be a semiprime module and $N\leq{M}$. Let $S$ be the set of all minimal prime submodules of $M$ which do not contain $N$. Then $Ann_M(N)=\bigcap\{P|P\in{S}\}$. Put $K=\bigcap\{P|P\in{S}\}$. Any element in $K\cap{N}$ is in the intersection of all minimal prime submodules of $M$ which is zero. Then $K\cap{N}=0$. Since $K$ is fully invariant in $M$, $K_MN\leq{K\cap{N}}=0$. Thus, $K\leq{Ann_M(N)}$. Now, let $P\in{S}$. Since $Ann_M(N)_MN=0\leq{P}$ and $N\nleq{P}$, then $Ann_M(N)\leq{K}$. \[11\][Lemma]{} \[148\] Let $M$ be projective in $\sigma[M]$. If $M$ is semiprime then $M$ is retractable. Let $N\leq{M}$ and suppose $Hom_R(M,N)=0$. Then $Ann_M(N)=M$. So $M_MN=0$ but $N_MN\subseteq{M_MN}=0$. Since $M$ is semprime then $N=0$ by Lemma \[143\]. \[11\][Proposition]{} \[115\] Let $M$ be projective in $\sigma[M]$ and semiprime. The following conditions are equivalent for $N\leq M$: 1. $N$ is a maximal annihilator submodule. 2. $N$ is an annihilator submodule and is a minimal prime submodule. 3. $N$ is prime in $M$ and $N$ is an annihilator submodule. $\textit{1}\Rightarrow\textit{2}:$ Let $K\leq{M}$ such that $N=Ann_M(K)$. Let $L,H\leq{M}$ be fully invariant submodules of $M$ such that $L_MH\leq{N}$. Assume $H\nleq{N}$. Then $0\neq{H_MK}$. Hence $Ann_M(K)\leq{Ann_M(H_MK)}$, but since $Ann_M(K)$ is a maximal annihilator submodule, then $Ann_M(K)=Ann_M(H_MK)$. As $M$ is projective in $\sigma[M]$, by Proposition 5.5 of [@B], we have that $$L_M(H_M(H_MK))=(L_MH)_M(H_MK)\leq{N_M(H_MK)}=0$$ Now, since $H_M(H_MK)\leq{H_MK}$, then $$Ann_M(K)=Ann_M(H_MK)\leq{Ann_M(H_M(H_MK))}$$ Therefore $Ann_M(H_MK)=Ann_M(H_M(H_MK))$. Thus $L\leq{Ann_M(K)}=N$. Now, let $P\leq{M}$ be a prime submodule of $M$ such that $P<{N}$. We have that $N_MK=0\leq{P}$. So $K\leq{P}<{N}$. Hence $K_MK=0$. Thus $K=0$, but $M$ is semiprime, a contradiction. It follows that $N$ is a minimal prime submodule of $M$. $\textit{2}\Rightarrow\textit{3}:$ By hypothesis. $\textit{3}\Rightarrow\textit{1}:$ Suppose $N<{K}$ with $K$ an annihilator submodule. Then $$Ann_M(K)_MK=0\leq{N}$$ Since $N$ is prime in $M$, then $Ann_M(K)\leq{N}<{K}$. By Proposition \[114\] $Ann_M(K)\cap{K}=0$, hence $Ann_M(K)=0$. Since $K$ is an annihilator submodule, by Proposition \[113\], $K=Ann_M(Ann_M(K))=Ann_M(0)=M$. \[11\][Remark]{} \[aa\] Following the notation of Example 1.12 of [@P] Let $R=\mathbb{Z}_2\rtimes(\mathbb{Z}_2\oplus\mathbb{Z}_2)$. This ring has only one maximal ideal $I$ and it has three simple ideals: $J_1$, $J_2$, $J_3$, which are isomorphic. Then, the lattice of ideals of $R$ has the form $$\xymatrix{ & \stackrel{R}{\bullet}\ar@{-}[d] & \\ & \stackrel{I}{\bullet}\ar@{-}[d]\ar@{-}[dl]\ar@{-}[dr] & \\ \stackrel{J_1}{\bullet}\ar@{-}[dr] & \stackrel{J_2}{\bullet}\ar@{-}[d] & \stackrel{J_3}{\bullet}\ar@{-}[dl] \\ & \stackrel{0}{\bullet} &}$$ Moreover, $R$ is artinian and $R$-Mod has only one simple module up to isomorphism. Let $S$ be a simple module. By Theorem 2.13 of [@BP], the lattice of fully invariant submodules of $E(S)$ has tree maximal submodules $N$, $L$ and $K$, and it has the form $$\xymatrix{ & \stackrel{E(S)}{\bullet}\ar@{-}[dl]\ar@{-}[d]\ar@{-}[dr] & \\ \stackrel{K}{\bullet}\ar@{-}[dr] & \stackrel{L}{\bullet}\ar@{-}[d] & \stackrel{N}{\bullet}\ar@{-}[dl] \\ & \stackrel{S}{\bullet}\ar@{-}[d] & \\ & \stackrel{0}{\bullet} & }$$ Put $M=E(S)$. Since $K\cap{L}=S$ and $K_ML\leq{K\cap{L}}$, then $K_ML\leq{S}$. On the other hand consider the composition $$\xymatrix{M\ar[r]^-{\pi} & M/N\ar[r]^-{\cong} & S\ar[r]^-{i} & L}$$ $f=i\circ\pi$ where $\pi$ is the natural projection and $i$ is the inclusion. Then, $f(K)=S$ and $S\leq{K_ML}$. Thus, $K_ML=S$. Notice that $K_ML\leq{N}$ but $K\nleq{N}$ and $L\nleq{N}$. Hence $N$ is not prime in $M$. Analogously, we prove that neither $K$ nor $L$ are prime in $M$. We also note that $K_MK=S$. Moreover, $\pi(K)=S$, so $K_MS=S$. In the same way $L_MS=S$ and $N_MS=S$ Let $g:M\rightarrow{K}$ be a non zero morphism. If $Ker(g)\cap{S}=0$ then $g$ is a monomorphism, a contradiction. So $Ker(g)\cap{S}=S$. Thus $S_MK=0$ and $Ann_M(K)=S$. Analogously $Ann_M(L)=S=Ann_M(N)=Ann_M(S)$. Since $S_MS\leq{S_MK}$, $S_MS=0$. Thus $M$ is not semiprime. Hence, $S$ is a maximal annihilator submodule of $M$ which is not prime because $K_MK=S$. With this we can see that associativity is not true in general, because $L_M(K_MS)=L_MS=S$ and $(L_MK)_MS=S_MS=0$. Notice that, in this example $Hom_R(M,H)\neq{0}$ for all $H\in\sigma[M]$ in particular $M$ is retractable, but $M$ is not projective in $\sigma[M]$. \[11\][Proposition]{} \[115b\] Let $M$ be projective in $\sigma[M]$ and semiprime. For $N\leq M$, if $N=Ann_M(U)$ with $U\leq{M}$ a uniform submodule, then $N$ is a maximal annihilator in $M$. Suppose that $N<{K}$ with $K$ an annihilator submodule in $M$. Since $N=Ann_M(U)$ by Proposition \[114\], $K\cap{U}\neq{0}$. By hypothesis $U$ is uniform and thus $K\cap{U}\leq_{e}U$. Then $$(K\cap{U})\oplus{Ann_M(U)}\leq_{e}U\oplus{Ann_M(U)}$$ Now, notice that if $L\leq_{FI}M$, by Proposition \[114\] $(U\oplus{Ann_M(U)})\cap{L}\neq{0}$. So $((K\cap{U})\oplus{Ann_M(U)})\cap{L}\neq{0}$. Therefore, $K\cap{L}\neq{0}$ and $K$ intersects all fully invariant submodules of $M$ . Since $K\cap{Ann_M(K)}=0$ and $Ann_M(K)\leq_{FI}M$, then $Ann_M(K)=0$. Thus, $K=Ann_M(Ann_M(K))=Ann_M(0)=M$. \[11\][Proposition]{} \[117\] Let $M$ be projective in $\sigma[M]$ and semiprime with finite uniform dimension. Then: 1. $M$ has finitely many minimal prime submodules. 2. The number of annihilators submodules is finite. 3. $M$ satisfies ACC on annihilators submodules. $\textit{1}:$ Let $U_1,..,U_n$ be uniform submodules of $M$ such that $U_1\oplus...\oplus{U_n}\leq_eM$. By Proposition s \[115\] and \[115b\], $P_i:=Ann_M(U_i)$ is a minimal prime submodule of $M$ for each $i$. By Proposition \[114\], $(U_1\oplus...\oplus{U_n})\cap{Ann_M(U_1\oplus...\oplus{U_n})}=0$ and $P_1\cap...\cap{P_n}\leq{Ann_M(U_1\oplus...\oplus{U_n})}=0$. Now, if $P$ is a minimal prime submodule of $M$, then $${P_1}_M{P_2}_M...{_M}P_n\leq{P_1\cap...\cap{P_n}}=0\leq{P}$$ Hence, there exists $j$ such that $P_j\leq{P}$, a contradiction. $\textit{2}:$ By Lemma \[116b\]. $\textit{3}:$ It is clear by $\textit{2}$. Goldie Modules ============== The following definition was taken from [@E] \[section\] Let $M\in{R-Mod}$. $M$ is *essentially compressible* if for every essential submodule $N\leq_e{M}$ there exists a monomorphism $M\rightarrow{N}$. \[112\][Definition]{} Let $M\in{R-Mod}$. We call a left annihilator in $M$ a submodule $$\mathcal{A}_X=\bigcap\{Ker(f)|f\in{X}\}$$ for some $X\subseteq{End_R(M)}$. \[112\][Definition]{} We say $M$ is a *Goldie module* if it satisfies ACC on left annihilators and has finite uniform dimension. \[112\][Lemma]{} \[107\] Suppose $M$ is projective in $\sigma[M]$. If $N\in\sigma[M]$ is essentially compressible, then $Ann_M(N)$ is a semiprime submodule of $M$. Let $L\leq{M}$ be a fully invariant submodule of $M$ such that $L_ML\leq{Ann_M(N)}$. Put $$\Gamma=\{K\leq{N}|L_MK=0\}$$ Then $\Gamma\neq\emptyset$ and by Zorn’s Lemma there exists a maximal independent family $\{K_i\}_I$ in $\Gamma$. Notice that $\bigoplus_I{K_i}\in\Gamma$ because $$L_M\bigoplus_I{K_i}=\bigoplus_I{L_MK_i}=0$$ Let $0\neq{A}\leq{N}$ be a submodule. Since $(L_ML)_MA=0$ then $L_MA\in\Gamma$. If $L_MA=0$ then $A\in\Gamma$ and $A\cap\bigoplus_I{K_i}\neq{0}$ because $\{K_i\}$ is a maximal independent family in $\Gamma$. Now, if $L_MA\neq{0}$ we also have $(L_MA)\cap\bigoplus_I{K_i}\neq{0}$ and $(L_MA)\cap\bigoplus_I{K_i}\leq{A\cap\bigoplus_I{K_i}}$. Thus $\bigoplus_I{K_i}\leq_eN$. By hypothesis there exists a monomorphism $\theta:N\rightarrow\bigoplus_I{K_i}$. Then $$\theta(L_MN)\leq{L_M\bigoplus_I{K_i}}=0$$ and hence $L_MN=0$. Thus $L\leq{Ann_M(N)}$. \[112\][Proposition]{} \[142\] Let $M$ be projective in $\sigma[M]$. If $N\in\sigma[M]$ is an $M$-singular module, then $Ker(f)\leq_e{M}$ for all $f\in{Hom_R(M,N)}$. Let $f\in{Hom_R(M,N)}$. Since $N$ is $M$-singular, there exists an exact sequence $$\xymatrix{0\ar[r] & K\ar[r]^i & L\ar[r]^\pi & N\ar[r] & 0}$$ in $\sigma[M]$ with $K\leq_e{L}$. Since $M$ is projective in $\sigma[M]$, there exists $\hat{f}:M\rightarrow{L}$ such that $\pi\hat{f}=f$: $$\xymatrix{ & M\ar[d]^f\ar[dl]_{\hat{f}} & \\ L\ar[r]^\pi & N\ar[r] & 0}$$ As $K\leq_e{L}$, then $\hat{f}^{-1}(K)\leq_e{M}$. Then $$f(\hat{f}^{-1}(K))=\pi(\hat{f}(\hat{f}^{-1}(K)))\leq\pi(K)=0.$$ Therefore, $\hat{f}^{-1}(K)\leq{Ker(f)}$ and hence $Ker(f)\leq_e{M}$. \[112\][Proposition]{} \[125\] Let $M$ be projective in $\sigma[M]$. If $M$ is essentially compressible then $M$ is non $M$-singular. Suppose $\mathcal{Z}(M)\neq{0}$. If $\mathcal{Z}(M)\leq_eM$, then there exists a monomorphism $\theta:M\rightarrow{\mathcal{Z}(M)}$, by Proposition \[142\] $Ker\theta\leq_eM$, a contradiction. Therefore $\mathcal{Z}(M)$ has a pseudocomplement $K$ in $M$ and thus $\mathcal{Z}(M)\oplus{K}\leq_eM$. Hence, there exists a monomorphism $\theta:M\rightarrow{\mathcal{Z}(M)\oplus{K}}$. Let $\pi:\mathcal{Z}(M)\oplus{K}\rightarrow{\mathcal{Z}(M)}$ be the canonical projection, then $Ker(\pi\theta)\leq_eM$ and so $Ker(\pi\theta)=\theta^{-1}(Ker\pi)=\theta^{-1}(K)\leq_eM$. But $\mathcal{Z}(M)\cap\theta^{-1}(K)=0$,contradiction. Thus $\mathcal{Z}(M)=0$. \[112\][Lemma]{} \[132\] Let $M\in{R-Mod}$ with finite uniform dimension. Then, for every monomorphism $f:M\rightarrow{M}$, $Im(f)\leq_e{M}$. Let $f:M\rightarrow{M}$ be a monomrfism. If the uniform dimension of $M$ is $n$, $(Udim(M)=n)$ and there exists $K\leq{M}$ such that $f(M)\cap{K}=0$, then $Udim(f(M)\oplus{K})=n+1$, a contradiction. \[112\][Theorem]{} \[110\] Let $M$ be projective in $\sigma[M]$ with finite uniform dimension. The following conditions are equivalent: 1. $M$ is semiprime and non $M$-singular 2. $M$ is semiprime and satisfies ACC on annihilators 3. Let $N\leq{M}$, then $N\leq_e{M}$ if and only if there exists a monomorphism $f:M\rightarrow{N}$. $\textit{1}\Rightarrow\textit{2}:$ Since $M$ is non $M$-singular and has finite uniform dimension then, by Proposition 3.6 of [@K] $M$ satisfies ACC on annihilators. This proves $\textit{2}$. $\textit{2}\Rightarrow\textit{3}:$ Let $N\leq{M}$. Suppose that $N\leq_e{M}$. Since $M$ is semiprime with uniform dimension and satisfies ACC on annihilators, then $M$ is essentially compressible by Proposition 3.13 of [@K]. Now, if $f:M\rightarrow{N}$ is a monomorphism then $N\leq_e{M}$ by lemma \[132\]. $\textit{3}\Rightarrow\textit{1}:$ It follows from Lemma \[107\] and Proposition \[125\]. \[112\][Remark]{} Notice that Theorem \[110\] is a generalization of Goldie’s Theorem. See [@L] Theorem 11.13. In Proposition 3.13 of [@K], $M$ is a generator of $\sigma[M]$, but by Lemma \[148\] this hypothesis is not necessary. \[112\][Corollary]{} \[129\] Let $M$ be projective in $\sigma[M]$ and semiprime. Then, $M$ has finite uniform dimension and enough monoforms if and only if $M$ is a Goldie module. $\Rightarrow:$ Since $M$ is semiprime with finite uniform dimension and enough monoforms, then $M$ is non $M$-singular by Proposition 3.8 of [@K]. By Theorem \[110\], $M$ is a Goldie module. $\Leftarrow:$ If $M$ is a Goldie module, $M$ has finite uniform dimension and by Theorem \[110\] $M$ is non $M$-singular. Hence the uniform submodules of $M$ are monoform. Since $M$ has finite uniform dimension every submodule of $M$ contains a uniform, hence every submodule contains a monoform. For the definition of $M$-Gabriel dimension see [@P] section 4. \[112\][Corollary]{} \[135\] Let $M$ be projective in $\sigma[M]$ with finite uniform dimension. If $M$ is a semiprime module and has $M$-Gabriel dimension, then $M$ is a Goldie module. Let $N\leq{M}$. Since $M$ has $M$-Gabriel dimension, by Lemma 4.2 of [@P], $N$ contains a cocritical submodule $L$. Then $L$ is monoform. By Proposition \[129\] $M$ is a Goldie module. \[112\][Corollary]{} \[130\] Let $M$ be projective in $\sigma[M]$ and semiprime with Krull dimension. Then $M$ is a semiprime Goldie module. Since $M$ has Krull dimension, $M$ has finite uniform dimension and enough monoforms. By Proposition \[129\] $M$ is a Goldie module. \[112\][Proposition]{} \[140\] Suppose that $M$ is progenerator of $\sigma[M]$. Let $N\in\sigma[M]$, then $$\mathcal{Z}(N)=\sum\{f(M)|f:M\rightarrow{N}\;ker(f)\leq_e{M}\}.$$ By definition of $M$-singular module, it is clear that $\sum\{f(M)|f:M\rightarrow{N}\;ker(f)\leq_e{M}\}\leq{\mathcal{Z}(N)}$. Now, let $n\in{\mathcal{Z}(N)}$ and consider $Rn\leq{\mathcal{Z}(N)}$. Since $Rn\in\sigma[M]$ there exists a natural number $t$ and an epimorphism $\rho:M^t\rightarrow{Rn}$. Suppose that $(m_1,..,m_t)$ is such that $\rho(m_1,...,m_t)=n$. If $j_i:M\rightarrow{M^t}$ are the inclusions $(i=1,...,t)$, then by Proposition \[142\] $Ker(\rho\circ{j_i})\leq_e{M}$. Thus, $n=\sum_{i=1}^{t}{\rho\circ{j_i}(m_i)}\in\sum\{f(M)|f:M\rightarrow{N}\;ker(f)\leq_e{M}\}$. \[112\][Remark]{} \[140k\] Let $M\in{R-Mod}$ and consider $\tau_g\in{M-tors}$, where $\tau_g=\xi(\{S\in\sigma[M]|S\;is\;M-singular\})$. If $M\in\mathbb{F}_{\tau_g}$, by [@W] Proposition. 10.2, we have that $\chi(M)=\tau_g$. Let $t_{\tau_g}$ be the preradical associated to $\tau_g$. Then $$t_{\tau_g}(N)=\sum\{S\leq{N}|S\in\mathbb{T}_{\tau_g}\}=\sum\{S\leq{N}|S\;is\;M-singular\}=\mathcal{Z}(N).$$ \[112\][Proposition]{} \[141\] Suppose $M$ is progenerator of $\sigma[M]$. If $M$ is semiprime Goldie, then $$\mathcal{Z}(N)=\sum{f(M)}$$ where the sum is over the $f:M\rightarrow{N}$ such that there exists $\alpha\in{End_R(M)}$ monomorphism with $\alpha(M)\leq_eM$ and $f\alpha=0$. Let $N\in\sigma[M]$. By Proposition \[140\] $$\mathcal{Z}(N)=\sum\{f(M)|f:M\rightarrow{N}\;ker(f)\leq_e{M}\}.$$ If $f:M\rightarrow{N}$ with $Ker(f)\leq_e{M}$, by Theorem \[110\] there exists a monomorphism $\alpha:M\rightarrow{Ker(f)}$. We have that $f\alpha=0$ and by Lemma \[132\] $\alpha(M)\leq_e(M)$. Let $f:M\rightarrow{N}$ such that there exists $\alpha:M\rightarrow{M}$ $f\alpha=0$ and $\alpha(M)\leq_e(M)$. Then $\alpha(M)\leq{Ker(f)}$. Therefore $Ker(f)\leq_e(M)$. \[112\][Remark]{} Let $R$ be a ring such that $R$-Mod has an infinite set of non-isomorphic simples modules. Consider $M=\bigoplus_I{S_i}$, $I$ an infinite set, such that $S_i$ is a simple module for all $i\in{I}$ and with $Si\ncong{S_j}$ if $i\neq{j}$. This module does not have finite uniform dimension and, in $M$-tors, $\tau_g=\chi$. Then, if $N\in\sigma[M]$ $$t_{\tau_g}(N)=\mathcal{Z}(N)=\sum{f(M)}$$ where the sum is over the $f:M\rightarrow{N}$ such that there exists $\alpha\in{End_R(M)}$ monomorphism with $\alpha(M)\leq_eM$ and $f\alpha=0$. This example shows that the converse of the last Proposition is not true in general. Following [@A] \[112\][Definition]{} A module $M$ is *weakly compressible* if for any nonzero submodule $N$ of $M$, there exists $f:M\rightarrow{N}$ such that $f\circ{f}\neq{0}$. \[112\][Remark]{} \[146\] Notice that if $M$ is weakly compressible then $M$ is a semiprime module. The converse hold if $M$ is projective in $\sigma[M]$ Next definition was taken from [@H] \[112\][Definition]{} A module $M$ is a *semiprojective* module if $I=Hom(M,IM)$ for any cyclic right ideal $I$ of $End_R(M)$ For other characterizations see [@Foun]. \[112\][Proposition]{} \[109\] Let $M$ be projective in $\sigma[M]$ and retractable. Then, $S:=End_R(M)$ is semiprime if and only if $M$ is semiprime. $\Rightarrow:$ Corollary \[108\]. $\Leftarrow:$ If $M$ is semiprime, since $M$ is projective in $\sigma[M]$ then $M$ is weakly compressible and semiprojective. Then, by \[[@H]. Theorem 2.6 (b)\] $S$ is semiprime. \[112\][Lemma]{} \[147\] Let $M$ be projective in $\sigma[M]$ and retractable. $M$ is non $M$-singular if and only if $Hom_R(M/N,M)=0$ for all $N\leq_e{M}$. $\Rightarrow$: If $N\leq_e{M}$ then $M/N$ is $M$-singular, then $Hom_R(M/N,M)=0$. $\Leftarrow$: Suppose $\mathcal{Z}(M)\neq{0}$. Since $M$ is retractable there exists $0\neq f:M\to \mathcal{Z}(M)$. By Proposition \[142\] $Ker(f)\leq_eM$, so there exists a non zero morphism form $M/Ker(f)\to M$. For a retractable $R$-module $M$, Theorem 11.6 of [@W] gives necessary and sufficient conditions in order to $T:=End_R(\widehat{M})$ being semisimple, left artinian, and being the classical left quotient ring of $S=End_R(M)$. Also, in ([@H], Corollary 2.7) the authors give necessary and sufficient conditions for a semiprojective module $M$ to $S$ being a semiprime right Goldie ring. We give an extension of these results. \[112\][Theorem]{} \[144\] Let $M$ be projective in $\sigma[M]$, $S=End_R(M)$ and $T=End_R(\widehat{M})$. The following conditions are equivalent: 1. $M$ is a semiprime Goldie module. 2. $T$ is semisimple right artinian and is the classical right quotient ring of $S$. 3. $S$ is a semiprime right Goldie ring. 4. $M$ is weakly compressible with finite uniform dimension, and for all $N\leq_e{M}$, $Hom_R(M/N,M)=0$ . $\textit{1}\Rightarrow\textit{2}:$ By Proposition \[109\], $S$ is a semiprime ring. Since $M$ is a Goldie module, then $M$ is non $M$-singular with finite uniform dimension, hence by [@W] Proposition 11.6, $T$ is right semisimple and is the classical right quotient ring of $S$. $\textit{2}\Rightarrow\textit{3}:$ By [@L] Theorem 11.13, $S$ is a semiprime right Goldie ring . $\textit{3}\Rightarrow\textit{4}:$ By [@H] Corollary 2.7. $\textit{4}\Rightarrow\textit{1}:$ Since $M$ is weakly compressible then $M$ is semiprime. By Lemma \[147\] $M$ is non $M$-singular. Thus, by Theorem \[110\] $M$ is a Goldie module. \[112\][Corollary]{} \[145\] Let $M$ be projective in $\sigma[M]$, $S=End_R(M)$ and $T=End_R(\widehat{M})$. The following conditions are equivalent: 1. $M$ is a prime Goldie module. 2. $T$ is simple right artinian and is the classical right quotient ring of $S$. 3. $S$ is a prime right Goldie ring. 4. Given nonzero submodules $N$, $K$ of $M$ there exists a morphism $f:M\rightarrow{N}$ such that $K\nsubseteq{Ker(f)}$. $M$ has finite uniform dimension and for all $N\leq_e{M}$, $Hom(M/N,M)=0$. $\textit{1}\Rightarrow\textit{2}:$ By Proposition \[144\], $S$ is a semiprime ring and $T$ is right semisimple and the classical right quotient ring of $S$. Let $0\neq{I}\leq{T}$ be an ideal. Since $T$ is semisimple, there exits an ideal $J\leq{T}$ such that $T=I\oplus{J}$. Put $M_1=I\widehat{M}$ and $M_2=J\widehat{M}$. Then $M_1$ and $M_2$ are fully invariant submodules of $\widehat{M}$ and $M_1\cap{M_2}=0$ because $I\cap{J}=0$. Consider $M_1\cap{M}$ and $M_2\cap{M}$. If $f\in{S}$, then there exists $\hat{f}\in{T}$ such that $f=\hat{f}|_M$. Let $x\in{M_1\cap{M}}$. Then $f(x)=\hat{f}(x)\in{M_1\cap{M}}$ since $M_1$ is a fully invariant submodule of $\widehat{M}$. Thus $M_1\cap{M}$ is a fully invariant submodule of $M$. In the same way, $M_2\cap{M}$ is fully invariant in $M$. Since $(M_1\cap{M})\cap{(M_2\cap{M})}=0$, then $(M_1\cap{M})_M{(M_2\cap{M})}=0$. Hence $M_1\cap{M}=0$ or $M_2\cap{M}=0$ because $M$ is prime. On the other hand, $M\leq_e\widehat{M}$ and so $M_1=0$ or $M_2=0$. Since $0\neq{I}$, then $M_2=0$. Thus $J=0$, and it follows that $T$ is a simple ring. $\textit{2}\Rightarrow\textit{3}:$ By [@L] Corollary 11.16, $S$ is a prime right Goldie ring. $\textit{3}\Rightarrow\textit{4}:$ Let $N$, $K$ be nonzero submodules of $M$, if $K\subseteq{Ker(f)}$ for all $f:M\rightarrow{N}$ then $0=Hom_R(M,N)Hom(M,K)\leq{S}$. Then $Hom_R(M,N)=0$ or $Hom_R(M,K)=0$. By retractability, $N=0$ or $K=0$, a contradiction. $\textit{4}\Rightarrow\textit{1}$ It is clear. \[112\][Remark]{} \[sing\] Suppose that $M$ and $N$ are $R$-modules such that $\sigma[N]\subseteq\sigma[M]$. If $N$ is non $M$-singular, then $N$ is non $N$-singular. This is because if there exists an exact sequence $0\rightarrow{L}\rightarrow{K}\rightarrow{N}\rightarrow{0}$ in $\sigma[N]$ such that $L\leq_e{N}$, then this sequence is in $\sigma[M]$ which implies that $N$ is $M$-singular, a contradiction. \[112\][Proposition]{} \[139\] Let $M$ be projective in $\sigma[M]$ and semiprime with finitely many minimal prime submodules $P_1,...,P_t$. Suppose every quotient $M/P_{i}$ $(1\leq{i}\leq{t})$ has finite uniform dimension. Then $M$ is a Goldie module if and only if each $M/P_i$ is a Goldie module. $\Rightarrow:$ Suppose $M$ is a Goldie module and $P_i$ is a minimal prime submodule of $M$. By hypothesis, each $M/P_i$ has finite uniform dimension. Notice that by proposition \[116a\] $$P_i\subseteq{Ann_M(P_1\cap...\cap{P_{i-1}}\cap{P_{i+1}}\cap...\cap{P_n})}$$ Since $M$ has finite uniform dimension there exist a uniform submodule $U_i$ of $P_1\cap...\cap{P_{i-1}}\cap{P_{i+1}}\cap...\cap{P_n}$. So $P_i\subseteq{Ann_M(U_i)}$. By Propositions \[115\] and \[115b\], $P_i=Ann_M(U_i)$. Then, there exists a monomorphism $M/P_i\rightarrow{{U_i}^X}$ and since $U_i$ is non $M$-singular, then $M/P_i$ is non $M$-singular. Thus $M/P_i$ is non $({M}/{P_i})$-singular by Remark \[sing\]. Since $M/P_i$ is a prime module, by Theorem \[110\] $M/P_i$ is a Goldie module. $\Leftarrow:$ By Corollary \[116a\] there exists a monomorphism $M\rightarrow\bigoplus_{i=1}^{t}{M/P_i}$. Since each $M/P_i$ has finite uniform dimension then $M$ has finite uniform dimension. Let $0\neq{N}$ be a submodule of $M$. Since there exists a monomorphism $M\rightarrow\bigoplus{M/P_i}$ then there exists $1\leq{i}\leq{t}$ and submodules $0\neq{K}\leq{M/P_i}$ and $0\neq{N'}\leq{N}$ such that $K\cong{N'}$. We have that $M/P_i$ is a Goldie module, thus it has enough monoforms. Hence $N'$ has a monoform submodule, that is $M$ has enough monoforms, and so by Corollary \[129\] $M$ is Goldie module. \[112\][Remark]{} \[132a\] Notice that if $M$ is a semiprime Goldie module then $M$ has finitely many minimal prime submodules by Proposition \[117\]. So in the proof $\Rightarrow:$ of Proposition \[139\] this hypothesis is not used. \[112\][Definition]{} \[131\] Let $M\in{R-Mod}$ and $N\leq{M}$. We say $N$ is a *regular submodule* if there exists a monomorphism $M\rightarrow{N}$. Denote $$\textit{Reg}(M):=\{N\leq{M}|N\;regular\;submodule\}$$ \[112\][Remark]{} \[bb\] There exists modules with regular submodules which are nonessential. For example, a pure infinite module, see [@MM]. \[112\][Proposition]{} \[133\] Let $M$ be projective in $\sigma[M]$ and a semiprime Goldie module. Then, $N$ is a regular submodule of $M$ if and only if $N$ is essential in $M$. Since $M$ is Goldie, every regular submodule is essential by Lemma \[132\]. Now, let $N\leq_eM$. By Theorem \[110\], $N$ is a regular submodule. If $K\in\sigma[M]$, we say that $K$ is $\textit{Reg}(M)$-injective if any morphism $f:N\to K$ with $N\in\textit{Reg}(M)$ can be extended to a endomorphism of $M$. \[112\][Corollary]{} \[134\] Let $M$ be projective in $\sigma[M]$ and a semiprime Goldie module. Let $K\in\sigma[M]$. If $K$ is $\textit{Reg}(M)$-injective, then $K$ is $M$-injective. Duo Modules =========== Following [@DM] \[section\] \[118\] Let $M\in{R-Mod}$. $M$ is a *duo module* if every submodule of $M$ is fully invariant in $M$. Examples: 1. If $_RS$ is a simple module then, $S$ is a duo module. 2. If $_RM=\bigoplus_I{S_i}$ with $S_i$ simple and $S_i$ not isomorphic to $S_j$ $i\neq{j}$ then $M$ is a duo module. 3. An $R$-module $M$ is called a multiplication module if every $N\leq{M}$ is of the form $IM=N$ for some ideal $I$ of $R$. These modules are examples of duo modules. See [@T] 4. Consider the example in Remark \[aa\] that was taken from [@P]. In that paper it is proved that $M/K\cong{S}\cong{M/L}\cong{M/N}$, hence $L$, $K$ and $N$ are maximal submodules of $M$. It follows that $K/S$, $L/S$ and $N/S$ are maximal submodules of $M/S$. Moreover, since $K\cap{L}=S=K\cap{N}=N\cap{L}$, then $M/S=K/S\oplus{L/S}$. Thus $$K/S\cong\frac{M/S}{L/S}\cong{M/L}\cong{S}$$ This implies that $K/S$ is simple, and analogously $L/S$ and $N/S$ are simple. Let $0\neq{T}<{M}$. Since $S\leq_e{M}$, then $S\leq{T}$. If $T=S$, then $T$ is fully invariant. Suppose that $T\neq{S}$ and $T\notin\{K,L,N\}$. We have that $S\leq{T\cap{K}}\leq{K}$. Moreover, since $K/S$ is simple, then $T\cap{K}=S$ or $T\cap{K}=K$. If $T\cap{K}=K$ then $K\leq{T}<M$; but $K$ is maximal, then $K=T$, a contradiction. Thus, $T\cap{K}=S$. Analogously $T\cap{L}=S=T\cap{N}$. Let $0\neq{x}\in{M}$. If $ann_R(x)=0$, there exists a monomorphism $R\rightarrow{M}$ and thus $E(R)=M$, a contradiction, because $E(R)\cong{M}\oplus{M}$ (see [@P], Example 1.12) and $M$ is a indecomposable injective module. Thus, $ann_R(x)\neq{0}$ for all $0\neq{x}\in{M}$. Let $0\neq{x}\in{T}$. Since $ann_R(x)\neq{0}$, then $ann_R(x)\in\{I,J_1,J_2,J_3\}$. By Theorem 2.13 of [@BP] we have that: - If $ann_R(x)=I$ then $x\in{S}$ - If $ann_R(x)=J_1$ then $x\in{K\cap{T}}=S$ - If $ann_R(x)=J_2$ then $x\in{L\cap{T}}=S$ - If $ann_R(x)=J_3$ then $x\in{N\cap{T}}=S$ Therefore $T\leq{S}$, a contradiction. Thus, all submodules of $M$ are fully invariant. \[114\][Remark]{} In [@DM] the authors state that they did not know an example of a duo module $M$ and a submodule $N$ such that $M/N$ is not a duo module. In this example, $M$ is a duo module, but $M/S\cong{S}\oplus{S}$ is not a duo module. \[114\][Proposition]{} \[136\] $M$ is a duo module as $R$-module and it generates all its submodules if and only if $M$ is a multiplication module as $End_R(M)$-module. $\Rightarrow:$ Let $S=End_R(M)$ and let $N$ be a submodule of $M$. Since $M$ is a duo module, $N$ is fully invariant, thus $Hom_R(M,N)$ is an ideal of $S$. Since $M$ generates all its submodules, then $N=tr^M(N)=Hom_R(M,N)M$. Thus, $M$ is a multiplication module as $End_R(M)$-module. $\Leftarrow:$ It is clear. \[114\][Proposition]{} \[119\] Let $M$ be projective in $\sigma[M]$. Suppose that $M$ is a semiprime and non $M$-singular duo module. Then, for every subset $X\subseteq{End_R(M)}$ we have that: $$Ann_M(Ann_M(\bigcap_X{Ker(f)}))=\bigcap_X{Ker(f)}$$ Since $M$ is a duo module, by Proposition \[114\], $Ann_M(\bigcap_X{Ker(f)})$ is the unique pseudocomplement of $\bigcap_X{Ker(f)}$. Then $$\bigcap_X{Ker(f)}\leq_eAnn_M(Ann_M(\bigcap_X{Ker(f)})).$$ Since $M$ is non $M$-singular, $\bigcap_X{Ker(f)}$ has no essential extensions in $M$ by Lemma 3.5 of [@K]. Thus, we have the equality. \[114\][Proposition]{} \[120\] Let $M$ projective in $\sigma[M]$. Suppose that $M$ is a semiprime and non $M$-singular duo module. The following conditions are equivalent: 1. $M$ has finite uniform dimension. 2. $M$ has a finite number of minimal prime submodules. 3. The number of annihilators in $M$ is finite. 4. $M$ satisfies the ACC on annihilators. 5. $M$ satisfies the ACC on pseudocomplements. $\textit{1}\Rightarrow\textit{2}\Rightarrow\textit{3}:$ Are true by Proposition \[117\]. $\textit{3}\Rightarrow\textit{4}:$ By Proposition \[119\]. $\textit{4}\Rightarrow\textit{5}:$ By Proposition \[114\]. $\textit{5}\Rightarrow\textit{1}:$ By [@L] Proposition 6.30. \[114\][Proposition]{} \[137\] Let $M$ be projective in $\sigma[M]$. Suppose $M$ is a prime duo module with finite uniform dimension. Then, $Udim(M)=1$ Since $M$ is prime, $0$ is the unique minimal prime submodule of $M$. By Proposition \[117\], there exists a uniform submodule $U$ of $M$ such that $0=Ann_M(U)$. By Proposition \[114\], $U\leq_eM$. Thus, $Udim(M)=1$. \[114\][Theorem]{} \[138\] Let $M$ be projective in $\sigma[M]$. If $M$ is a semiprime duo module, then the following conditions are equivalent: 1. $M$ is a prime Goldie module. 2. $\widehat{M}$ is indecomposable and $M$ is non $M$-singular. 3. $M$ is uniform and non $M$-singular. $\textit{1}\Rightarrow\textit{2}:$ Since $M$ is a prime module, by Proposition \[137\], $Udim(M)=1$ and then $\widehat{M}$ is indecomposable. Since $M$ is a Goldie module, by Theorem \[110\] $M$ is non $M$-singular. $\textit{2}\Rightarrow\textit{3}:$ Let $0\neq{K}\leq{M}$. Then, there exists $L\leq{M}$ such that $K\oplus{L}\leq_eM$. Hence, $\widehat{K}\oplus\widehat{L}=\hat{M}$, but since $\widehat{M}$ is indecomposable, then $L=0$. Thus $K\leq_e{M}$. $\textit{3}\Rightarrow\textit{1}:$ Let $K$ and $0\neq{L}$ be submodules of $M$ such that $K_ML=0$. Then, $K\leq{Ann_M(L)}$, and thus $K\cap{L}=0$ by Proposition \[114\]. Since $M$ is uniform, $K=0$. Thus, $M$ is prime and by Theorem \[110\] $M$ is Goldie. [15]{} O.D. Avraamova, A generalized density theorem, *Abelian Group and Modules*, no. 8, Tomsk. Gos. Univ., Tomsk, 3-16 (1989). J. Beachy, M-Injective Modules and Prime $M$-Ideals, *Communications in Algebra*, 30:10, 4649-4676 (2002) L. Bican, P. Jambor, T. Kepka, P. Nemec. Prime and coprime modules, *Fundamenta matematicae*, 107:33-44 (1980). J. Castro, J. Ríos Prime Submodules and Local Gabriel Correspondence in $\sigma[M]$ , *Communications in Algebra*, 40:1, 213-232 (2012) J. Castro, J. Ríos, FBN Modules, *Communications in Algebra*, 40:12, 4604-4616 (2012) J. Castro, J. 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(2005) F. Raggi, J. Ríos, H. Rincón, R. Fernández-Alonso, Basic Preradicals and Main Injective Modules , *J. Algebra Appl.* 8(1):1-16 (2009) F. Raggi, J. Ríos, H. Rincón, R. Fernández-Alonso, Semiprime preradicals , *Communications in Algebra*, **37**, no. 8, 2811-2822 (2009). P.F. Smith, M.R. Vedadi, Essentially Compressible Modules and Rings , *J. of Algebra*, 304, 812-831 (2006). A. A. Tuganbaev, Multiplication modules over noncommutative rings , *Sb. Math*, 194:1837-1864 (2003). R. Wisbauer, *Foundations of Module and Ring Theory.*, Reading: Gordon and Breach (1991). R. Wisbauer, *Modules and Algebras: Bimodule Structure and Group Actions on Algebras*, England: Addison Wesley Longman Limited (1996). [^1]: jcastrop@itesm.mx,correspondingauthor [^2]: mmedina@matem.unam.mx [^3]: jrios@matem.unam.mx [^4]: zaldivar@matem.unam.mx
--- abstract: 'We compute the geometric part of algebraic cobordism over Dedekind domains of mixed characteristic after inverting the positive residue characteristics and prove cases of a Conjecture of Voevodsky relating this geometric part to the Lazard ring for regular local bases. The method is by analyzing the slice tower of algebraic cobordism, relying on the Hopkins-Morel isomorphism from the quotient of the algebraic cobordism spectrum by the generators of the Lazard ring to the motivic Eilenberg-MacLane spectrum, again after inverting the positive residue characteristics.' author: - Markus Spitzweck bibliography: - 'ma.bib' title: Algebraic Cobordism in mixed characteristic --- Introduction ============ Algebraic cobordism is a theory for smooth schemes over a base scheme $S$ defined by a motivic ring spectrum ${\mathsf{MGL}}_S$ in the stable motivic homotopy category ${\mathsf{SH}}(S)$. It is the motivic counterpart of complex cobordism ${\mathsf{MU}}$. A famous Theorem of Quillen states that the natural map from the Lazard ring $L_*$ classifying formal group laws to the coefficients of ${\mathsf{MU}}$ is an isomorphism, moreover $L_* \cong {{\mathbb Z}}[x_1,x_2,x_3,\ldots]$ with $\deg(x_i)=i$ (here we divide the usual topological grading by $2$). For an oriented motivic ring spectrum $E$ the geometric part $E_{(2,1)*}$ of the coefficients also carries a formal group law constructed in the exact same way as in topology by evaluating the theory on ${{\mathbb P}}^\infty$ and using that ${{\mathbb P}}^\infty$ is naturally endowed with a multiplication. Thus there is a classifying map $L_* \to E_{(2,1)*}$. It is known that for $E={\mathsf{MGL}}_k$ for a field $k$ of characteristic $0$ this map is an isomorphism using the Hopkins-Morel isomorphism, see [@hoyois.hopkins-morel Proposition 8.2]. More generally in [@levine.comparison] it is shown that over such fields the Levine-Morel algebraic cobordism $\Omega^*(-)$ is isomorphic to ${\mathsf{MGL}}_k^{(2,1)*}(-)$ on smooth schemes over $k$. If the base field $k$ has positive characteristic the map $L_* \to {\mathsf{MGL}}_{(2,1)*}$ becomes at least an isomorphism after inverting the characteristic, see again [@hoyois.hopkins-morel Proposition 8.2]. The main ingredient in the proof is that the Hopkins-Morel isomorphism yields a computation of the slices of ${\mathsf{MGL}}_S$ with respect to Voevodsky’s slice filtration, that ${\mathsf{MGL}}_S$ is complete with respect to this filtration and that the slices have a simple form, namely they are shifted twists of the motivic Eilenberg-MacLane spectrum. The facts about the slices of ${\mathsf{MGL}}_S$ hold more generally true over spectra $S$ of Dedekind domains of mixed characteristic (after inverting the positive residue characteristics), using the motivic Eilenberg-MacLane spectrum introduced in [@spitzweck.em]. The main new input of this note is that in this case ${\mathsf{MGL}}_S$ is also complete with respect to the slice filtration (Corollary \[gfrerz\]), a consequence of the fact that ${\mathsf{MGL}}_S$ is connective with respect to the homotopy sheaves, see Proposition \[gdrttt\]. This yields a computation of the geometric part of the homotopy groups of ${\mathsf{MGL}}_S$ (Theorem \[grfe5z4\]), again after inverting the residue characteristics. In our formulation we always assume a Hopkins-Morel isomorphism for the given coefficients, hoping that the Hopkins-Morel isomorphism will be settled completely in the future. We prove cases of a Conjecture of Voevodsky ([@voevodsky.icm Conjecture 1]), see Theorem \[hte4234t\], comparing the Lazard ring to $({\mathsf{MGL}}_S)_{(2,1)*}$ for $S$ the spectrum of a regular local ring. We also give applications to some homotopy groups or sheaves of ${\mathsf{MGL}}_S$ outside the geometric diagonal, see section \[dzu634e\], and discuss generalizations of our results to motivic Landweber spectra. .4cm We note that the observation that the Hopkins-Morel isomorphism yields the computation of the zero-slice of the sphere spectrum (after inverting suitable primes), see Theorem \[dhu5t3\], was independently made by Oliver Röndigs. .7cm [**Acknowledgements:**]{} I would like to thank Peter Arndt, Christian Häsemeyer, Marc Hoyois, Moritz Kerz, Marc Levine, Niko Naumann, Oliver Röndigs, Manfred Stelzer, Florian Strunk, Jörg Wildeshaus and Paul Arne [Ø]{}stv[æ]{}r for very helpful discussions and suggestions on the subject. Preliminaries ============= By a base scheme we always mean a separated Noetherian scheme of finite Krull dimension. For a base scheme $S$ we let ${\mathsf{SH}}(S)$ be the stable motivic homotopy category. We let ${\mathsf{M}\mathbb{Z}}_S \in {\mathsf{SH}}(S)$ be the motivic Eilenberg-MacLane spectrum over $S$ constructed in [@spitzweck.em]. Also we let ${{\mathcal M}}(r) \in {{\mathsf D}}({\mathrm{Sh}}({\mathrm{Sm}}_{S,{\mathit{Zar}}},{{\mathbb Z}}))$ (for notation see [@spitzweck.em]) be the motivic complexes of weight $r \in {{\mathbb Z}}$, so as a ${\mathbb{G}_{m,S}}$-spectrum ${\mathsf{M}\mathbb{Z}}_S$ has ${{\mathcal M}}(r)[r]$ in level $r$. If $S$ is the spectrum of a Dedekind domain of mixed characteristic we note that ${{\mathcal M}}(0)= S^0 \underline{{{\mathbb Z}}}$, thus for $X \in {\mathrm{Sm}}_S$ we have $H^{0,0}(X,{{\mathbb Z}})={{\mathbb Z}}^{\pi_0(X)}$. Also ${{\mathcal M}}(1) \cong {{\mathcal O}}^*[-1]$, so $H^{1,1}(X,{{\mathbb Z}}) \cong {{\mathcal O}}^*(X)$ and $H^{2,1}(X,{{\mathbb Z}}) \cong {\mathrm{Pic}}(X)$. We have ${{\mathcal M}}(r) \cong 0$ for $r < 0$. For general $S$ we denote by ${\mathsf{MGL}}_S \in {\mathsf{SH}}(S)$ the algebraic cobordism spectrum. There is a natural map $L_* \to ({\mathsf{MGL}}_S)_{2*,*}$, where $L_*$ denotes the Lazard ring. Fixing generators $x_i \in L_i$ there is a map $$\Phi_S \colon {\mathsf{MGL}}_S/(x_1,x_2,\ldots){\mathsf{MGL}}_S \to {\mathsf{M}\mathbb{Z}}_S,$$ see [@spitzweck.em §11.1], which is an isomorphism after inverting all positive residue characteristics of $S$, see [@spitzweck.em Theorem 11.3]. For any ring or abelian group $R$ we let $M_R \in {\mathsf{SH}}(S)$ be the Moore spectrum on $R$ and ${\mathsf{M}R}_S$ the version of ${\mathsf{M}\mathbb{Z}}_S$ with $R$-coeffcients. Slices ====== For $i \in {{\mathbb Z}}$ denote by $f_i$ resp. $l_i$ the $i$-th colocalization resp. localization functor for Voevodsky’s motivic slice filtration on ${\mathsf{SH}}(S)$. For any $E \in {\mathsf{SH}}(S)$ and $k \ge n$ we set $E\left<n,k \right>:=l_{k+1}(f_n(E))$. Thus we have exact triangles $$f_{k+1}(E) \to f_n(E) \to E\left<n,k \right> \to f_{k+1}(E)[1]$$ and $s_n(E) = E\left<n,n \right>$. We note that all these functors commute with homotopy colimits. \[dhu5t3\] Let $X$ be an essentially smooth scheme over a Dedekind domain of mixed characteristic and $R$ a localization of ${{\mathbb Z}}$ such that $\Phi_X \wedge M_R$ is an isomorphism (e.g. if every positive residue characteristic of $X$ is invertible in $R$). Then $$s_0 M_R \cong s_0 ({\mathsf{MGL}}_X \wedge M_R) \cong {\mathsf{M}R}_X.$$ More generally $$s_n ({\mathsf{MGL}}_X \wedge M_R) \cong \Sigma^{2n,n} {\mathsf{M}R}_X \otimes L_n.$$ The first isomorphism of the first line follows from [@spitzweck.rel Corollary 3.3]. From the assumption that $\Phi_X \wedge M_R$ is an isomorphism it follows that the map ${\mathsf{MGL}}_X \wedge M_R \to {\mathsf{M}R}_X$ induces an isomorphism on zero-slices and that ${\mathsf{M}R}_X$ is effective. Moreover $l_1 {\mathsf{M}\mathbb{Z}}_X \cong {\mathsf{M}\mathbb{Z}}_X$, since negative weight motivic cohomology vanishes in our situation. Thus the second isomorphism of the first line follows. The second line is a version of [@spitzweck.rel Theorem 4.7] with $R$-coefficients. It is then also possible to determine the slices of motivic Landweber spectra with $R$-coeffcients, see [@spitzweck.slices], for example of ${\mathsf{KGL}}_X \wedge M_R$. Subcategories of the stable motivic homotopy category ===================================================== Fix a base scheme $S$. We let ${\mathsf{SH}}(S)_{\ge n}$ be the $\ge n$ part (in the homological sense) of ${\mathsf{SH}}(S)$ with respect to the homotopy $t$-structure, see e.g. [@hoyois.hopkins-morel §2.1]. Thus ${\mathsf{SH}}(S)_{\ge n}$ is generated by homotopy colimits and extensions by the objects $\Sigma^{p,q} \Sigma^\infty_+ X$ for $X \in {\mathrm{Sm}}_S$ and $p-q \ge n$. For each $E \in {\mathsf{SH}}(S)$ we let ${\underline{\pi}}^{\mathrm{pre}}_{p,q}(E)$ be the presheaf $$X \mapsto {\mathrm{Hom}}_{{\mathsf{SH}}(S)}(\Sigma^{p,q} \Sigma^\infty_+ X,E)$$ on ${\mathrm{Sm}}_S$. Let ${\underline{\pi}}_{p,q}(E)$ be the sheafification of ${\underline{\pi}}^{\mathrm{pre}}_{p,q}(E)$ with respect to the Nisnevich topology. We also set $\pi_{p,q}(E):={\underline{\pi}}^{\mathrm{pre}}_{p,q}(E)(S)=E_{p,q}$. We let ${\mathsf{SH}}(S)_{h \ge n}$ be the full subcategory of ${\mathsf{SH}}(S)$ of objects $E$ such that ${\underline{\pi}}_{p,q}(E)=0$ for $p-q< n$. The categories ${\mathsf{SH}}(S)_{h \ge n}$ are closed under homotopy colimits and extensions in ${\mathsf{SH}}(S)$. Th functors ${\underline{\pi}}_{p,q}$ respect sums. Moreover the long exact sequences of homotopy sheaves associated to an exact triangle in ${\mathsf{SH}}(S)$ show that ${\mathsf{SH}}(S)_{h \ge n}$ is closed under cofibers and extensions. This shows the claim. \[gegrsy\] Let $i \colon Z \hookrightarrow S$ be a closed inclusion of base schemes. Then $i_*({\mathsf{SH}}(Z)_{h \ge n}) \subset {\mathsf{SH}}(S)_{h \ge n}$. Let $E \in {\mathsf{SH}}(Z)_{h \ge n}$. Let $Y$ be the spectrum of the henselization of a local ring of a scheme from ${\mathrm{Sm}}_S$. Then $Y_Z:=Y \times_S Z$ is also the spectrum of a henselian local ring, and ${\underline{\pi}}^{\mathrm{pre}}_{p,q}(i_*E)(Y) \cong {\underline{\pi}}^{\mathrm{pre}}_{p,q}(E)(Y_Z)=0$ for $p-q < n$ (the first isomorphism holds since $i_*$ commutes with homotopy colimits). We let ${\mathsf{SH}}^{{S^1}}_s(S)$ be the homotopy category of presheaves of ${{S^1}}$-spectra on ${\mathrm{Sm}}_S$ localized with respect to the Nisnevich topology, and ${\mathsf{SH}}^{{S^1}}(S)$ the further ${{\mathbb A}}^1$-localization of that category. We let ${\mathsf{SH}}^{{S^1}}_s(S)_{\ge n}$ be the $\ge n$ part (in the homological sense) of ${\mathsf{SH}}^{{S^1}}_s(S)$ with respect to the standard $t$-structure, and for $E \in {\mathsf{SH}}^{{S^1}}_s(S)$ we let $E_{\ge n}$ and $E_{\le n}$ be the corresponding truncations. We let $E_{=n} := (E_{\ge n})_{\le n}$. As above for $E \in {\mathsf{SH}}^{{S^1}}_s(S)$ we have the presheaves ${\underline{\pi}}^{\mathrm{pre}}_p(E)$ and the sheaves ${\underline{\pi}}_p(E)$. For $E \in {\mathsf{SH}}^{{S^1}}_s(S)$ we have $E \in {\mathsf{SH}}^{{S^1}}_s(S)_{\ge n}$ if and only if ${\underline{\pi}}_k(E)=0$ for $k < n$. Note that ${\mathsf{SH}}^{{S^1}}_s(S)_{\ge n}$ is generated by homotopy colimits and extensions by the objects $\Sigma^n \Sigma^\infty_+ X$, $X \in {\mathrm{Sm}}_S$, thus the canonical functor $\sigma \colon {\mathsf{SH}}^{{S^1}}_s(S) \to {\mathsf{SH}}(S)$ sends ${\mathsf{SH}}^{{S^1}}_s(S)_{\ge n}$ to ${\mathsf{SH}}(S)_{\ge n}$. We have ${\mathsf{SH}}(S)_{h \ge n} \subset {\mathsf{SH}}(S)_{\ge n}$. If $S$ is the spectrum of a field then the inclusion is an equality. Let $E \in {\mathsf{SH}}(S)_{h \ge n}$. For any $i \in {{\mathbb N}}$ let $E_i$ be the image of $\Sigma^{i,i} E$ in ${\mathsf{SH}}^{{S^1}}_s(S)$. By assumption we have $E_i \in {\mathsf{SH}}^{{S^1}}_s(S)_{\ge n}$. Thus $\Sigma^{-i,-i} \sigma(E_i) \in {\mathsf{SH}}(S)_{\ge n}$. The proof of the first statement concludes by noting that $E \cong {\mathrm{hocolim}}_{i \to \infty} \Sigma^{-i,-i} \sigma(E_i)$. The second statement is [@hoyois.hopkins-morel Theorem 2.3]. \[hterge\] Let $E \in {\mathsf{SH}}^{{S^1}}_s(S)$. Then $E \to {\mathrm{holim}}_{n \to \infty} E_{\le n}$ is an isomorphism. We show that for all $n \in {{\mathbb Z}}$ we have ${\underline{\pi}}_n(E) \cong {\underline{\pi}}_n({\mathrm{holim}}_{k \to \infty} E_{\le k})$. Fix $n \in {{\mathbb Z}}$ and let $X \in {\mathrm{Sm}}_S$ be of dimension $d$. We are ready if we show $${\underline{\pi}}_n(E)|_{X_{\mathit{Nis}}} \cong {\underline{\pi}}_n({\mathrm{holim}}_{k \to \infty} E_{\le k})|_{X_{\mathit{Nis}}} \; \; (*).$$ For $m > n+d$ we have ${\underline{\pi}}^{\mathrm{pre}}_n(E_{=m}[j])(Y)=0$ for $Y \in X_{\mathit{Nis}}$ and $j \ge 0$, so homing out of $\Sigma^\infty_+ Y$ into the exact triangle $$E_{=m} \to E_{\le m} \to E_{\le (m-1)} \to E_{=m}[1]$$ shows that ${\underline{\pi}}^{\mathrm{pre}}_n(E_{\le m})(Y) \cong {\underline{\pi}}^{\mathrm{pre}}_n(E_{\le (m-1)})(Y)$. Using the Milnor exact sequence this shows that $${\underline{\pi}}^{\mathrm{pre}}_n({\mathrm{holim}}_{k \to \infty} E_{\le k})|_{X_{\mathit{Nis}}} \cong {\underline{\pi}}^{\mathrm{pre}}_n(E_{\le m})|_{X_{\mathit{Nis}}}$$ for $m \ge n+d$. Sheafifiying proves $(*)$. \[dwrzjj\] Let $$\cdots \to E_{i+1} \to E_i \to E_{i-1} \to \cdots \to E_1 \to E_0$$ be an inverse system of objects in ${\mathsf{SH}}(S)$. Suppose for each $n \in {{\mathbb N}}$ there is an $N \in {{\mathbb N}}$ such that $E_i \in {\mathsf{SH}}(S)_{h \ge n}$ for $i \ge N$. Then ${\mathrm{holim}}_{i \to \infty} E_i \cong 0$. Fix $q \in {{\mathbb Z}}$ and let $F_i$ be the image of $\Sigma^{q,q} E_i$ in ${\mathsf{SH}}^{{S^1}}_s(S)$. We are ready if we show ${\mathrm{holim}}_{i \to \infty} F_i \cong 0$. By assumption for every $n \in {{\mathbb N}}$ there is a $N \in {{\mathbb N}}$ such that $F_i \in {\mathsf{SH}}^{{S^1}}_s(S)_{\ge n}$ for each $i \ge N$. By Lemma \[hterge\] we have $F_i \cong {\mathrm{holim}}_{k \to \infty} (F_i)_{\le k}$. Thus $${\mathrm{holim}}_{i \to \infty} F_i \cong {\mathrm{holim}}_i {\mathrm{holim}}_k (F_i)_{\le k} \cong {\mathrm{holim}}_k {\mathrm{holim}}_i (F_i)_{\le k} \cong {\mathrm{holim}}_k 0 \cong 0.$$ We also have the Let $E \in {\mathsf{SH}}(S)_{h \ge n}$ and $X \in {\mathrm{Sm}}_S$ of dimension $d$. Then $${\underline{\pi}}^{\mathrm{pre}}_{p,q}(E)(X)=0$$ for $p-q<n-d$. \[gr455\] Let $$\cdots \to E_{i+1} \to E_i \to E_{i-1} \to \cdots \to E_1 \to E_0$$ be an inverse system of objects in ${\mathsf{SH}}(S)_{h \ge n}$. Suppose for each $p,q \in {{\mathbb Z}}$ and $d \in {{\mathbb N}}$ there is an $N \in {{\mathbb N}}$ such that for $X \in {\mathrm{Sm}}_S$ of dimension $d$ the map $${\underline{\pi}}^{\mathrm{pre}}_{p,q}(E_{i+1})(X) \to {\underline{\pi}}^{\mathrm{pre}}_{p,q}(E_i)(X)$$ is an isomorphism for all $i \ge N$. Then ${\mathrm{holim}}_{i \to \infty} E_i \in {\mathsf{SH}}(S)_{h \ge n}$. (Here the latter homotopy limit is computed in ${\mathsf{SH}}(S)$.) Let $p,q \in {{\mathbb Z}}$, $d \in {{\mathbb N}}$ and $X \in {\mathrm{Sm}}_S$ of dimension $d$. Choose $N \in {{\mathbb N}}$ such that for any $Y \in {\mathrm{Sm}}_S$ of dimension $\le d$ the map $${\underline{\pi}}^{\mathrm{pre}}_{p,q}(E_{i+1})(Y) \to {\underline{\pi}}^{\mathrm{pre}}_{p,q}(E_i)(Y)$$ is an isomorphism for all $i \ge N$. We claim that $${\underline{\pi}}_{p,q}({\mathrm{holim}}_k E_k)|_{X_{\mathit{Nis}}} \cong {\underline{\pi}}_{p,q}(E_i)|_{X_{\mathit{Nis}}}$$ for all $i \ge N$. For every $Y \in X_{\mathit{Nis}}$ we have the Milnor short exact sequence $$0 \to \text{$\lim_i$}^1 {\underline{\pi}}^{\mathrm{pre}}_{p+1,q}(E_i)(Y) \to {\underline{\pi}}^{\mathrm{pre}}_{p,q}({\mathrm{holim}}_i E_i)(Y) \to \lim_i {\underline{\pi}}^{\mathrm{pre}}_{p,q}(E_i)(Y) \to 0.$$ The $\lim^1$-term vanishes because the inverse system of abelian groups stabilizes by assumption. Sheafifying we see that ${\underline{\pi}}_{p,q}({\mathrm{holim}}_k E_k)|_{X_{\mathit{Nis}}} \cong {\underline{\pi}}_{p,q}(E_i)|_{X_{\mathit{Nis}}}$ for $i \ge N$, in particular ${\underline{\pi}}_{p,q}({\mathrm{holim}}_k E_k)|_{X_{\mathit{Nis}}} = 0$ in case $p-q<n$. Since this is true for all $X \in {\mathrm{Sm}}_S$ we conclude ${\underline{\pi}}_{p,q}({\mathrm{holim}}_k E_k) =0$ for $p-q < n$. Connectivity of algebraic cobordism =================================== \[vftjt\] Let $X$ be a smooth scheme over a Dedekind domain of mixed characteristic or over a field. Then for any abelian group $A$ we have ${\mathsf{M}A}_X \in {\mathsf{SH}}(X)_{h \ge 0}$. This follows from the fact that the motivic complexes ${{\mathcal M}}(r)$ have vanishing $i$-th cohomology sheaf for $i>r$, see [@geisser.dede Corollary 4.4]. \[fhtrth\] Let $S$ be the spetrum of a discrete valuation ring of mixed characteristic, $j \colon \eta \to S$ the inclusion of the generic point. Then for any abelian group $A$ we have $j_* {\mathsf{M}A}_\eta \in {\mathsf{SH}}(S)_{h \ge 0}$. Let $i \colon s \to S$ be the inclusion of the closed point. We have an exact triangle $$i_!i^! {\mathsf{M}A}_S \to {\mathsf{M}A}_S \to j_* {\mathsf{M}A}_\eta \to i_!i^! {\mathsf{M}A}_S[1]$$ and an isomorphism $i^! {\mathsf{M}A}_S \cong {\mathsf{M}A}_s(-1)[-2] \in {\mathsf{SH}}(s)_{h \ge -1}$, see [@spitzweck.em Theorem 7.4]. We conclude with Proposition \[gegrsy\] and Lemma \[vftjt\]. \[gfewet\] Let the situation be as in Proposition \[fhtrth\]. Then $$j_*{\mathsf{MGL}}_\eta\left<0,n\right> \wedge M_A \in {\mathsf{SH}}(S)_{h \ge 0}$$ for all $n \ge 0$. We can assume $A={{\mathbb Z}}$. Since $\eta$ is of characteristic $0$ we have $s_n {\mathsf{MGL}}_\eta \cong \Sigma^{2n,n} {\mathsf{M}\mathbb{Z}}\otimes L_n$. Moreover we have exact triangles $$s_n {\mathsf{MGL}}_\eta \to {\mathsf{MGL}}_\eta\left<0,n\right> \to {\mathsf{MGL}}_\eta \left<0,n-1 \right> \to s_n {\mathsf{MGL}}_\eta [1].$$ Applying $j_*$ to these triangles and using Proposition \[fhtrth\] one concludes by induction on $n$. \[rergr\] Let the situation be as in Proposition \[fhtrth\]. Let $p,q \in {{\mathbb Z}}$ and $X \in {\mathrm{Sm}}_S$ of dimension $d$. Then $${\underline{\pi}}^{\mathrm{pre}}_{p,q}(j_* {\mathsf{MGL}}_\eta\left< 0,n+1 \right>)(X) \to {\underline{\pi}}^{\mathrm{pre}}_{p,q}(j_* {\mathsf{MGL}}_\eta\left< 0,n \right>)(X)$$ is an isomorphism for $n \ge p-q+d$. Consider the exact triangle $$j_* s_{n+1} {\mathsf{MGL}}_\eta \to j_* {\mathsf{MGL}}_\eta \left< 0,n+1 \right> \to j_* {\mathsf{MGL}}_\eta \left< 0,n \right> \to s_{n+1} {\mathsf{MGL}}_\eta [1].$$ We have $${\underline{\pi}}^{\mathrm{pre}}_{p,q}(j_* s_{n+1} {\mathsf{MGL}}_\eta)(X) = H_{\mathrm{mot}}^{2(n+1)-p,n+1-q}(X_\eta, L_{n+1}).$$ The latter group vanishes for $2(n+1)-p> n+1 -q+d$, showing the claim. \[eweggg\] Let the situation be as in Proposition \[fhtrth\]. Then $j_* {\mathsf{MGL}}_\eta \in {\mathsf{SH}}(S)_{{\mathsf{h}}\ge 0}$. Consider the inverse system $$\cdots \to j_* {\mathsf{MGL}}_\eta \left<0,n+1 \right> \to j_* {\mathsf{MGL}}_\eta \left< 0,n \right> \to \cdots \to j_* s_0 {\mathsf{MGL}}_\eta$$ in ${\mathsf{SH}}(S)$. Since $j_*$ preserves homotopy limits the homotopy limit over this system is $j_* {\mathsf{MGL}}_\eta$, using [@hoyois.hopkins-morel Corollary 2.4 and Lemma 8.10 or Theorem 8.12]. By Lemma \[gfewet\] every object of this system is in ${\mathsf{SH}}(S)_{h \ge 0}$. Moreover by Lemma \[rergr\] the assumptions of Proposition \[gr455\] are satisfied. Thus this Proposition implies the claim. \[greerg\] Let the situation be as in Proposition \[fhtrth\] and let $i \colon s \to S$ be the inclusion of the closed point. Then $i^! {\mathsf{MGL}}_S \in {\mathsf{SH}}(s)_{\ge -1}$. Note first that $i^*$ sends ${\mathsf{SH}}(S)_{\ge 0}$ to ${\mathsf{SH}}(s)_{\ge 0}$. We have ${\mathsf{MGL}}\in {\mathsf{SH}}(S)_{\ge 0}$ and by Lemma \[eweggg\] also $j_* {\mathsf{MGL}}_\eta \in {\mathsf{SH}}(S)_{h \ge 0} \subset {\mathsf{SH}}(S)_{\ge 0}$. Applying $i^*$ to the exact triangle $$i_!i^! {\mathsf{MGL}}_S \to {\mathsf{MGL}}_S \to j_* {\mathsf{MGL}}_\eta \to i_!i^! {\mathsf{MGL}}_S [1]$$ shows the claim. \[gferghd\] Let $S$ be the spectrum of a discrete valuation ring of mixed characteristic. Then ${\mathsf{MGL}}_S \in {\mathsf{SH}}(S)_{h \ge -1}$. Let the notation be as above. The claim follows from the exact triangle $$i_!i^! {\mathsf{MGL}}_S \to {\mathsf{MGL}}_S \to j_* {\mathsf{MGL}}_\eta \to i_!i^! {\mathsf{MGL}}_S [1],$$ Lemma \[eweggg\], Proposition \[greerg\] and Proposition \[gegrsy\]. \[gdrttt\] Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Then ${\mathsf{MGL}}_S \in {\mathsf{SH}}(S)_{h \ge -1}$. The henselization of a local ring of a scheme in ${\mathrm{Sm}}_S$ lies over a local ring of $S$, thus the claim follows from Lemma \[gferghd\]. Compare the following result to [@voevodsky.open Conjecture 15]. \[gfrerz\] Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $R$-module $A$ we have $$f_n {\mathsf{MGL}}_S \wedge M_A \cong {\mathrm{holim}}_{k \to \infty} {\mathsf{MGL}}_S \left< n,k \right> \wedge M_A.$$ Under the assumption we have ${\mathsf{f}}_k {\mathsf{MGL}}_S \wedge M_A \in {\mathsf{SH}}(S)_{h \ge k-1}$, since this is a homotopy colimit of objects of the form $\Sigma^{2i,i} {\mathsf{MGL}}_S \wedge M_A$ with $i \ge k$, see the proof of [@spitzweck.rel Theorem 4.7], using Proposition \[gdrttt\]. Thus by Corollary \[dwrzjj\] we have ${\mathrm{holim}}_{k \to \infty} f_k {\mathsf{MGL}}_S \wedge M_A \cong 0$ implying the claim. A similar result holds for motivic Landweber spectra using the same argument as in the proof of [@hoyois.hopkins-morel Lemma 8.11]. For example ${\mathsf{KGL}}_S \wedge M_A$ is complete with respect to the slice filtration. The geometric part of algebraic cobordism ========================================= \[jewet45\] Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Let $p,q \in {{\mathbb Z}}$ and $X \in {\mathrm{Sm}}_S$. Then for any $R$-module $A$ the inverse system of abelian groups $({\underline{\pi}}^{\mathrm{pre}}_{p,q}({\mathsf{MGL}}_S \left< 0,k \right> \wedge M_A)(X))_k$ eventually becomes constant for $k \to \infty$. This follows from the exact triangle $$s_k {\mathsf{MGL}}_S \wedge M_A \to {\mathsf{MGL}}_S \left<0,k \right> \wedge M_A \to {\mathsf{MGL}}_S \left< 0,k-1 \right> \wedge M_A \to s_k {\mathsf{MGL}}_S \wedge M_A [1]$$ and $s_k {\mathsf{MGL}}_S \wedge M_A \cong \Sigma^{2k,k} {\mathsf{M}A}\otimes L_k$ since ${\underline{\pi}}^{\mathrm{pre}}_{p,q}(\Sigma^{2k+j,k} {\mathsf{M}A})(X)=0$, $j \ge 0$, for $k$ big enough. \[dsgtjt\] Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Let $p,q \in {{\mathbb Z}}$ and $X \in {\mathrm{Sm}}_S$. Then for any $R$-module $A$ the canonical map $${\underline{\pi}}^{\mathrm{pre}}_{p,q}({\mathsf{MGL}}_S \wedge M_A)(X) \to \lim_k {\underline{\pi}}^{\mathrm{pre}}_{p,q}({\mathsf{MGL}}_S \left<0,k \right> \wedge M_A)(X)$$ is an isomorphism. This follows from Corollary \[gfrerz\], the Milnor short exact sequence and Lemma \[jewet45\]. \[xfhtjt\] Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Let $n \in {{\mathbb Z}}$. Then for $k \ge n+1$ and any $R$-module $A$ the natural map $$\pi_{2n,n} {\mathsf{MGL}}_S \left<n,k+1 \right> \wedge M_A \to \pi_{2n,n} {\mathsf{MGL}}_S \left<n,k \right> \wedge M_A$$ is an isomorphism. This follows from the exact sequence $$\pi_{2n,n} s_{k+1} {\mathsf{MGL}}_S \wedge M_A \to \pi_{2n,n} {\mathsf{MGL}}_S \left<n,k+1 \right> \wedge M_A \to$$ $$\pi_{2n,n} {\mathsf{MGL}}_S \left<n,k \right> \wedge M_A \to \pi_{2n-1,n} s_k {\mathsf{MGL}}_S \wedge M_A$$ and the fact the the two outer terms are $0$ for $k \ge n+1$. \[sgtjtr\] Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $R$-module $A$ the canonical map $$\pi_{2n,n} {\mathsf{MGL}}_S \wedge M_A \to \pi_{2n,n} {\mathsf{MGL}}_S \left< n,n+1 \right> \wedge M_A$$ is an isomorphism. This follows from Corollary \[dsgtjt\] and Lemma \[xfhtjt\]. \[grfe5z4\] Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for every $n \in {{\mathbb Z}}$ and $R$-module $A$ there is a canonical isomorphism $$\pi_{2n,n} {\mathsf{MGL}}_S \wedge M_A \cong L_n \otimes A \oplus L_{n+1} \otimes \mathrm{Pic}(S) \otimes A.$$ We have the exact sequence $$\pi_{2n+1,n} s_n {\mathsf{MGL}}_S \wedge M_A \to \pi_{2n,n} s_{n+1} {\mathsf{MGL}}_S \wedge M_A \to \pi_{2n,n} {\mathsf{MGL}}_S \left<n,n+1 \right> \wedge M_A$$ $$\to \pi_{2n,n} s_n {\mathsf{MGL}}_S \wedge M_A \to \pi_{2n-1,n} s_{n+1} {\mathsf{MGL}}_S \wedge M_A.$$ The two outer terms are $0$. Also $\pi_{0,0} \Sigma^{2,1} {\mathsf{M}A}_S \cong \mathrm{Pic}(S) \otimes A$. Moreover there is a canonical map $L_n \otimes A \to \pi_{2n,n} {\mathsf{MGL}}_S \wedge M_A$ splitting the resulting short exact sequence, whence the claim follows form Corollary \[sgtjtr\]. Let $S$ be the spectrum of a Dedekind domain of mixed characteristic and $R$ the localization of ${{\mathbb Z}}$ obtained by inverting all positive residue characteristics of $S$. Then $$(\pi_{2n,n} {\mathsf{MGL}}_S) \otimes R \cong (L_n \oplus L_{n+1} \otimes \mathrm{Pic}(S)) \otimes R.$$ We have the following case of a Conjecture of Voevodsky (see [@voevodsky.icm Conjecture 1]): \[hte4234t\] Let $S={\mathrm{Spec}}(R)$, where $R$ is a (regular) Noetherian local ring which is regular over some discrete valuation ring of mixed characteristic. Then the natural map $$L_* \to ({\mathsf{MGL}}_S)_{2*,*}$$ becomes an isomorphism after inverting the residue characteristic of the closed point of $S$. By Popescu’s Theorem $R$ is a filtered colimit of smooth algebras over a discrete valuation ring $V$ of mixed characteristic. Thus we are reduced to the case where $R$ is the local ring of a scheme $X \in {\mathrm{Sm}}_{{\mathrm{Spec}}(V)}$ by a colimit argument. Let $p$ be the residue characteristic of the closed point of ${\mathrm{Spec}}(V)$. By the same type of argument as above and the vanishing of $(p,q)$-motivic cohomology of such local rings for $p>q$ we have $$(MGL_S)_{2n,n} [1/p] \cong (s_n {\mathsf{MGL}}_S)_{2n,n} [1/p] \cong L_n [1/p],$$ using that for a fixed dimension only a fixed finite number of slices of ${\mathsf{MGL}}_S[1/p]$ contribute to the value of $\pi^{\mathrm{pre}}_{2n,n}({\mathsf{MGL}}_S [1/p])$ on schemes of that dimension. More generally we have Let $S$ be as in the previous Theorem and $E \in {\mathsf{SH}}(S)$ a motivic Landweber spectrum modelled on $E^{\mathrm{top}}_{2*}$. Then the natural map $$E^{\mathrm{top}}_{2*} \to E_{2*,*}$$ is an isomorphism after inverting the residue characteristic of the closed point of $S$. This follows from the definition of motivic Landweber spectrum. Some other parts of algebraic cobordism {#dzu634e} ======================================= We have the following vanishing result: Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $p,q \in {{\mathbb Z}}$ and $R$-module $A$ we have ${\underline{\pi}}_{p,q} {\mathsf{MGL}}_S \wedge M_A \cong 0$ for $p< 2q$ or $p<q$. In particular we have ${\mathsf{MGL}}_S \wedge M_R \in {\mathsf{SH}}(S)_{h \ge 0}$. Let $p,q \in {{\mathbb Z}}$ satisfying the condition of the statement. Let $d \in {{\mathbb N}}$. Then there is a $N \ge q$ such that for any scheme of dimension $\le d$ and $k \ge N$ the map $${\underline{\pi}}^{\mathrm{pre}}_{p,q}({\mathsf{MGL}}_S \wedge M_A)(X) \to {\underline{\pi}}^{\mathrm{pre}}_{p,q}({\mathsf{MGL}}_S\left< 0,k \right> \wedge M_A)(X)$$ is an isomorphism. The assertion then follows by an induction argument on $i$ showing that ${\underline{\pi}}_{p,q}({\mathsf{MGL}}_S \left< q,q+i \right> \wedge M_A)=0$. Generalizing the argument given in the last proof we get \[dh5t43\] Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $p,q \in {{\mathbb Z}}$ and $R$-module $A$ we have $${\underline{\pi}}_{p,q}({\mathsf{MGL}}_S \wedge M_A) \cong \lim_k {\underline{\pi}}_{p,q}({\mathsf{MGL}}_S \left<0,k \right> \wedge M_A) \cong {\underline{\pi}}_{p,q}({\mathsf{MGL}}_S \left<\max(0,q),n \right> \wedge M_A)$$ for $n \ge p-q$ or $n \ge p-2q$. Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $R$-module $A$ and $n \in {{\mathbb Z}}$ we have ${\underline{\pi}}_{n,n}({\mathsf{MGL}}_S \wedge M_A) \cong \underline{K}^M_{-n} \otimes A$, where $\underline{K}^M_{-n}$ is the $(-n)$-th Milnor-$K$-theory sheaf defined via the degree $(-n,-n)$-motivic cohomology. Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $R$-module $A$ and $n \in {{\mathbb Z}}$ we have ${\underline{\pi}}_{2n,n}({\mathsf{MGL}}_S \wedge M_A) \cong \underline{L}_n \otimes A$, where the latter sheaf is the constant sheaf on $L_n \otimes A$. Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $R$-module $A$ and $n \in {{\mathbb Z}}$ we have ${\underline{\pi}}_{2n+1,n}({\mathsf{MGL}}_S \wedge M_A) \cong {{\mathcal O}}^* \otimes L_{n+1} \otimes A$. By Lemma \[dh5t43\] we have $${\underline{\pi}}_{2n+1,n}({\mathsf{MGL}}_S \wedge M_A) \cong {\underline{\pi}}_{2n+1,n}({\mathsf{MGL}}_S \left< n,n+1 \right> \wedge M_A).$$ The long exact sequence of sheaves associated to the exact triangle $$s_{n+1} {\mathsf{MGL}}_S \wedge M_A \to {\mathsf{MGL}}_S \left< n,n+1 \right> \wedge M_A \to s_n {\mathsf{MGL}}_S \wedge M_A \to s_{n+1} {\mathsf{MGL}}_S \wedge M_A [1]$$ together with $${\underline{\pi}}_{2n+1,n}(s_n {\mathsf{MGL}}_S \wedge M_A [-1])= {\underline{\pi}}_{2n+1,n}(s_n {\mathsf{MGL}}_S \wedge M_A)=0$$ and $${\underline{\pi}}_{0,0}(\Sigma^{1,1} {\mathsf{M}A}_S) \cong {{\mathcal O}}^* \otimes A$$ gives the result. Similarly we get Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $R$-module $A$ and $n \in {{\mathbb Z}}$ there is an exact sequence $$\underline{K}^M_{1-n} \otimes A \to {\underline{\pi}}_{n+1,n}({\mathsf{MGL}}_S \wedge M_A) \to {{\mathcal H}}_{\mathrm{mot}}^{-n-1,-n}(-,A) \to 0,$$ where the latter group denotes the motivic cohomology sheaf in degees $(-n-1,-n)$ and $A$-coefficients. We also have Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $R$-module $A$ and $n \in {{\mathbb Z}}$ there is an exact sequences $$H^{3,2}(S) \otimes A \otimes L_{n+2} \to \pi_{2n+1,n} {\mathsf{MGL}}_S \to H^{1,1}(S,A) \otimes L_{n+1} \to 0.$$ If $A$ is torsionfree the first map is also injective. Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Let $X$ be an essentially smooth scheme over $S$. Then for any $R$-module $A$, $n \in {{\mathbb Z}}$ and $i \ge 2$ we have ${\mathsf{MGL}}_S^{2n+i,n}(X,A)=0$. This follows from the above considerations and the fact that for $X \in {\mathrm{Sm}}_S$ we have $H^{p,q}(X)=0$ for $p \ge 2q+2$, since motivic cohomology is computed as hypercohomology over $S$ of the Bloch-Levine cycle complexes. We leave assertions about the groups $\pi_{2n-1,n} {\mathsf{MGL}}_S \wedge M_A$, $\pi_{n,n} {\mathsf{MGL}}_S \wedge M_A$ and $\pi_{n-1,n} {\mathsf{MGL}}_S \wedge M_A$ to the interested reader. Fakult[ä]{}t f[ü]{}r Mathematik, Universit[ä]{}t Osnabrück, Germany.\ e-mail: markus.spitzweck@uni-osnabrueck.de
--- abstract: 'Ferrario & Wickramasinghe (2006) explored the hypothesis that the magnetic fields of neutron stars are of fossil origin. In this context, they predicted the field distribution of the progenitor OB stars, finding that 5 per cent of main sequence massive stars should have fields in excess of 1kG. We have carried out sensitive ESPaDOnS spectropolarimetric observations to search for direct evidence of such fields in all massive B- and O-type stars in the Orion Nebula Cluster star-forming region. We have detected unambiguous Stokes V Zeeman signatures in spectra of three out of the eight stars observed (38%). Using a new state-of-the-art Bayesian analysis, we infer the presence of strong (kG), organised magnetic fields in their photospheres. For the remaining five stars, we constrain any dipolar fields in the photosphere to be weaker than about 200G. Statistically, the chance of finding three $\sim1\rm\,kG$ fields in a sample of eight OB stars is quite low (less than 1%) if the predictions of Ferrario & Wickramasinghe are correct. This implies that either the magnetic fields of neutron stars are not of fossil origin, that the flux-evolution model of Ferrario & Wickramasinghe is incomplete, or that the ONC has unusual magnetic properties. We are undertaking a study of other young star clusters, in order to better explore these possibilities.' author: - Véronique Petit - 'Gregg A. Wade' - Laurent Drissen - Thierry Montmerle bibliography: - 'onc\_pulsar.bib' title: 'Exploring the origin of neutron star magnetic field: magnetic properties of the progenitor OB stars' --- [address=[Dépt. de Physique, Université Laval, Québec (QC), Canada, G1K 7P4]{},altaddress=[Observatoire du Mont-Mégantic, Québec, Canada]{} ]{} [address=[Dept. of Physics, Royal Military College of Canada, PO Box 17000, Stn Forces, Kingston, Canada, K7K 4B4]{}]{} [address=[Dépt. de Physique, Université Laval, Québec (QC), Canada, G1K 7P4]{},altaddress=[Observatoire du Mont-Mégantic, Québec, Canada]{}]{} [address=[Laboratoire d’Astrophysique de Grenoble, Université Joseph-Fourier, 38041 Grenoble Cedex 9, France]{}]{} Introduction ============ The origin of the magnetic fields of massive stars, present at all evolutionary stages (PMS, MS, post-MS [@Alecian2007MNRAS; @Borra1980ApJS42p421; @Schmidt2003ApJ595p1101]), remains an open question. Two possible models may explain the presence of magnetic fields: \(i) In the dynamo model, the field is generated by a dynamo mechanism, occurring classically in convective regions or induced by strong shear during differential rotation. \(ii) In the fossil model, the field is a remnant from a dynamo active during a previous evolutionary phase, or swept up from the interstellar medium (ISM) during star formation. This scenario implies that the field must somehow survive the various structural changes encountered during stellar evolution. The magnetic flux is usually assumed to be conserved to some extent. The dynamo model reproduces well the characteristics of late-type main sequence stars and giants. However, it fails to explain the fields of magnetic Ap stars, as their envelopes are entirely radiative. Some models of convection in the small convective core of those stars have been put forward, but they still have fundamental difficuties reproducing the observed fields [@Charbonneau2001ApJ559p1094]. Their simple magnetic geometries, lack of significant mass-field strength or period-field strength relation, and the fact that the observed characteristics of Herbig star magnetic fields [e.g. @Wade2007MNRAS376p1145; @Wade2005AA442pL31; @Catala2007AA462p293; @Alecian2006ArXiv0612186; @Folsom2007MNRAS376p361] are qualitatively identical to those of Ap stars, point toward a fossil origin. Furthermore, the incidence, geometries and strengths of white dwarf magnetic fields are at least qualitatively compatible with evolution from Ap-Bp stars, suggesting that the fields of white dwarfs may also be of fossil origin [@Wickramasinghe2005MNRAS356p1576]. Similar questions can be asked about neutron stars. Neutron star field strengths, inferred from spin down rates of radio pulsars, are in the range of $10^{11}$-$10^{14}\rm\,G$. A particular class of pulsars called the magnetars, which include the anomalous X-ray pulsars and soft gamma repeaters, have fields that range up to $10^{14}$-$10^{15}\rm\,G$. The two possible scenarios that have been put forward to explain the existence of these fields are again the fossil and dynamo mechanisms. The fossil hypothesis implies that neutron star fields come from the progenitor OB star fields which have survived the post main sequence and the core collapse phases. There is some observational evidence that neutron stars may evolve from stars as massive as $45\rm\,M_\odot$ [@Gaensler2005ApJ620pL95; @Muno2006ApJ636pL41]. The only two known O stars with directly measured magnetic fields are $\theta^1\rm\,Ori\,C$ and HD191612 ($\sim40\rm\,M_\odot$), with dipolar field strengths ranging between 1000-1500G [@Donati2002MNRAS333p55; @Donati2006MNRAS365pL6]. The magnetic flux of these stars ($\sim10^{27}\rm\,G\,cm^2$) is of the same scale as the magnetic flux of the highest field magnetars. Provided that OB star fields are remnants from the ISM, the fossil hypothesis could provide a powerful explanation of the wide range of magnetic fields present in neutron stars, and may also explain the super strong fields seen in magnetars. In this fossil model, the properties of neutron star magnetic fields are a function of the field properties of the progenitor OB stars, plus any physics that occurs during post main sequence evolution to alter those fields. Depending on the relative importance of these two ingredients, the fields of neutron stars might be nearly identical to those of OB stars (but stronger, of course), or rather different. On the other hand, it has been suggested that neutron star magnetic fields could instead be generated by a dynamo mechanism taking place during the core collapse itself, and induced by differential rotation. Present studies [@Heger2005ApJ626p350] assume that any primordial fields present in the progenitor star are weak enough to be expelled by the dynamo process. However, if the initial field is strong enough, the evolution will be different, as this field is likely to interfere with the differential rotation and therefore with the dynamo process itself [@Spruit1999AA349p189]. Hence, in both cases, and the likely combination of them, a primordial field present during the formation of a neutron star will play a fundamental role. Knowing the magnetic properties of the progenitor OB stars would therefore be an important asset for constraining models of stellar evolution leading to neutron star birth. Observations ============ Magnetic fields can be directly detected in stellar atmospheres by the means of the Zeeman effect. If the field is strong enough, and the spectral lines narrow enough, one can directly see the Zeeman splitting of the lines in the intensity spectrum. However, if the field is weaker, and the lines broadened either intrinsically or by fast stellar rotation, the splitting is much more difficult to detect, even at high spectral resolution. In that case, the most effective way to detect the Zeeman effect is by looking at circular polarisation signatures across photospheric spectral lines. Recently, a technique called “Least Squares Deconvolution” (LSD) has been developed by @Donati1997MNRAS291p658 for extracting a mean Stokes V profile, using all the lines present in a spectrum simultaneously. This allows for better detection limits as it substantially increases the signal-to-noise ratio. This technique has been proven to be very useful for detecting magnetic fields in Ap-Bp stars [@Auriere2007] and late-type stars [@Wade2001ASPC248p403]. However, magnetic fields in hotter OB stars remain difficult to detect. The few photospheric lines present in the optical spectrum and the large intrinsic width of the lines, worsened by the usual fast rotation of those stars, require large-bandwidth and high signal-to-noise ratio observations to start with, even using LSD. For those reasons, the magnetic characteristics of the population of neutron star progenitors are currently poorly known, and mostly only by indirect ways, such as non-thermal radio and X-ray emission and variability, peculiar abundances or cyclical variations of photospheric and wind spectral lines (see [@Fullerton2003ASPC305p333; @Wade2001ASPC248p403] for a review). Nevertheless, those indicators are often interpreted to show that magnetism may be widespread among massive OB stars. The advent of a new generation of spectropolarimeters such as ESPaDOnS at the Canada-France-Hawaii Telescope (CFHT) and its twin NARVAL at the Télescope Bernard-Lyot (TBL) now allows new investigations of magnetic fields in massive OB stars. ESPaDOnS consists of a polarimetric module located at the Cassegrin focus of the CFHT, linked by optical fibres to the high-resolution echelle spectrometer. A resolution of 65,000 for a spectral range covering from $360\rm\,nm$ to $1\rm\,\mu m$ can be achieved in a single observation. A complete circular polarisation observation consists of series of 4 sub-exposures between which the polarimeter quarter-wave plate is rotated back and forth between position angles. This procedure results in exchanging orthogonally polarised beams throughout the entire instrument, which makes it possible to reduce systematic errors. The Orion Nebular Cluster ------------------------- ![ \[x\] X-ray efficiency as a function of effective temperature. The detected stars are circled. Filled symbols are for stars with indirect indications of the presence of a magnetic field and gray symbols are for confirmed or suspected binaries. Plotting symbols indicate the following properties: circles are for T-Tauri type emission, triangles are chemically peculiar (CP) stars, and the diamond star was not observed. The dotted line indicates the typical efficiency for massive stars. ](figure_x){width="45.00000%"} ![ \[lsd\] Least Squares Deconvolved profiles for Par1772 (B2V) and $\theta^1\rm\,Ori\,D$ (B0.5V). The curves are the mean Stokes I profiles (bottom), the mean Stokes V profiles (top) and the N diagnostic null profiles (middle), in solid line for January 2006 and dashed line for March 2007.](figure_lsd){width="95.00000%"} ![ \[bayes\] Marginalized posterior probability densities for Par1772 (one of the 3 detected OB stars) and $\theta^1\rm\,Ori\,D$ (one of the 5 undetected stars). The magnetic field strength 90% credible region (filled) is \[900, 2900\]G for Par1772 and \[0, 160\]G for $\theta^1\rm\,Ori\,D$.](figure_bayes){width="95.00000%"} The first goal of this project was to explore the connection between magnetic fields and X-ray production in massive OB stars. Stellar magnetic fields are well known to produce X-rays in late-type convection stars like the Sun. However, X-ray emission coming from OB stars is often explained by radiative instabilities resulting in a multitude of shocks in their winds [@Lucy1980ApJ241p300; @Cohen1999ApJ520p833]. The [*Chandra*]{} Orion Ultradeep Project (COUP) was dedicated to observe the Orion Nebula Cluster (ONC) in X-rays. The OBA sample (20 stars) was studied with the goal of disentangling the respective roles of winds and magnetic fields in producing X-rays [@Stelzer2005ApJS160p557]. The production of X-rays by radiative shocks should be the dominant mechanism for the subsample of 9 OB stars with strong winds. However, aside from two of those stars, all targets showed X-ray flux intensity and/or variability which were inconsistent with the small shock model predictions (Figure \[x\]). For these reasons, we started our investigation with the 9 OB stars of this young star cluster. They range from B3V ($\sim8\rm\,M_\odot$) to O7V ($\sim40\rm\,M_\odot$), approximately the mass range from which neutron stars are thought to be formed. We conducted spectropolarimetric observations of 8 of those massive stars in January 2006 and March 2007 at CFHT. Using the LSD technique, we found clear Stokes V signatures for 3 stars: the previously-detected $\theta^1\rm\,Ori\,C$, as well as Par1772 (shown in Figure \[lsd\], along with the non-detection case $\theta^1\rm\,Ori\,D$), and NUOri. Magnetic analysis ----------------- In order to extract the surface field characteristics from the observed Stokes V profiles, we compared them with theoretical profiles for a large grid of dipolar magnetic configurations, calculated with the polarised LTE radiative transfer code Zeeman2 [@Landstreet1988ApJ326p967; @Wade2001AA374p265]. We sampled the 4-dimentional parameter space ($i$, $\beta$, $\phi$, B) which describes a centered dipolar magnetic configuration. In such a model, $i$ is the projected angle of the rotation axis to the line of sight, $\beta$ is the angle between the magnetic axis and the rotation axis, $\phi$ is the rotational phase and B is the polar field strength. For each configuration, we calculated the reduced $\chi^2$ of the model fit to the observed mean Stokes V profiles. Assuming that only the phase may change between two observations of a given star, the goodness-of-fit of a given ($i$, $\beta$, B)-configuration is expressed in terms of Bayesian probability density. This ensures that a good magnetic ($i$, $\beta$, B)-configuration possesses phases that fit both observations, as the rotational period is not known with enough accuracy to determine [*a priori*]{} the phase difference. We can determine the probability density of the field strength, by marginalising over inclination ($i$) and obliquity ($\beta$). Then, we extract a 90% credible region for the field strength of each star (Figure \[bayes\]) with the technique described by @Gregory2005book. Our first results show that any dipolar fields present in the 5 undetected stars are weaker than $\sim200\rm\,G$ with a 90% confidence. The field strengths of the 3 detected stars are approximately 1-3kG. Discussion and conclusion ========================= ![ \[prob\] Predicted magnetic field repartition of 8 randomly-drawn stars according to the predicted distribution of @Ferrario2006MNRAS367p1323 along with the observed distribution of field strengths in the ONC.](figure_prob){width="45.00000%"} As an illustrative example, we can compare our new observational results with the predictions made by @Ferrario2006MNRAS367p1323 about the magnetic field distribution of massive stars (8-$45\rm\,M_\odot$) on the main sequence. They parametrized the magnetic flux distribution on the main sequence $\chi(\Phi)$ as the sum of two Gaussians, along with the birth spin period of neutron stars. Assuming a complete conservation of magnetic flux, they have calculated the expected properties of isolated radio pulsars. They used the 1374-MHz Parkes Multi-Beam Survey of isolated radio pulsars in order to constrain the model parameters. They obtained a continuous magnetic field distribution in the progenitor OB stars peaking at $\sim46\rm\,G$ with 5 per cent[^1] of the stars having a field in excess of 1kG. Of course, our sample contains only 8 stars, but we can still make some rough comparaisons. Taking the predicted field strength distribution, we assume that it is the true parent distribution from which we draw a random sample of 8 stars. We define three possible outcomes: \[0-200\]G, \[200-1000\]G and over 1000G, with respective probabilities derived from the parent theoretical distribution (Figure \[prob\]). According to the multinomial distribution, the probability of observing the distribution of magnetic field strengths observed in the ONC is below 1%. This result might be interpreted, at first glance, to suggest that the fields of neutron stars are not of fossil origin. However, some points are important to consider: 1. The sample of stars may not be representative of a general parent distribution, as the stars all come from the same cluster. This region could be unusually magnetic, especially if the fields of the OB stars themselves are also of fossil origin from the ISM. 2. There may be a fossil component to the magnetic field origin, but the assumed parent distribution is not in fact the true parent distribution because some assumptions are incorrect, or some elements are missing from the model. Examples of such missing physics might be partial flux conservation or the influence of dynamo processes during core collapse. In order to better explore these possibilities, a larger sample of OB stars, from clusters and from the field, must be studied in order to increase the population of neutron star progenitors with known magnetic properties. Our team has undertaken an extensive spectropolarimetric study of massive stars in other young star clusters to provide these important data. VP acknowledges support from Fonds québécois de la recherche sur la nature et les technologies. LD acknowledges support from the Canada Research Chair program and the Discovery Grants programme of the Natural Science and Engineering Research Council of Canada. GAW acknowledges support from the Discovery Grants programme of the Natural Science and Engineering Research Council of Canada. Based on observations obtained at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii. The ESPaDOnS data were reduced using the data reduction software Libre-ESpRIT, written by J.-F. Donati from the Observatoire Midi-Pyrénées and made available to observers at the CFHT. [^1]: Although the paper states 8%, recalculation based on the detailed model distribution provided by L. Ferrario gives 5%.
--- abstract: 'The [[Teichmüller curve]{}]{} is the fiber space over [[Teichmüller space]{}]{} $T_g$ of closed [[Riemann surface]{}]{}s, where the fiber over a point $(\Sigma,\sigma) \in T_g$ is the underlying surface $\Sigma$. We derive formulas for [[sectional curvature]{}]{}s on the [[Teichmüller curve]{}]{}. In particular, our method can be applied to investigate the geometry of the [[Weil-Petersson geodesic]{}]{} as a three-manifold, and the degeneration of the curvatures near the infinity of the augmented [[Teichmüller space]{}]{} along a [[Weil-Petersson geodesic]{}]{}, as well as the minimality of hyperbolic surfaces in this [[3-manifold]{}]{}.' address: - 'School of Mathematics, University of Minnesota, Minneappolis, MN 55455, USA' - 'Department of Mathematics, Yale University, New Haven, CT 06520, USA' - 'Department of Mathematics, The City University of New York, Staten Island, NY 10314, USA.' author: - Ren Guo - Subhojoy Gupta - Zheng Huang date: 'January 16, 2010' title: Curvatures on the Teichmüller Curve --- Introduction ============ [[Teichmüller space]{}]{} $T_{g}$ is the space of [[hyperbolic metric]{}]{}s on $\Sigma$, modulo an equivalent relationship, where two [[conformal structure]{}]{}s $\sigma$ and $\rho$ are considered equivalent if there is a biholomorphic map between $(\Sigma,\sigma)$ and $(\Sigma,\rho)$, in the homotopy class of the identity. Here and throughout this paper, we always assume $\Sigma$ is a smooth, oriented, closed [[Riemann surface]{}]{} of genus $g > 1$. [[Teichmüller space]{}]{} is a complex manifold of [[complex dimension]{}]{} $3g - 3$ ([@Ahl61]), and the co[[tangent space]{}]{} at a base point $(\Sigma, \sigma)$ is identified with $Q(\sigma)$, the space of [[holomorphic quadratic differential]{}]{}s on $(\Sigma, \sigma)$. Let $\{\phi_1,\cdots,\phi_{3g-3}\}$ be a basis of $Q(\sigma)$. The local coordinates of the [[Teichmüller space]{}]{} $T_g$ in the neighborhood of $(\Sigma, \sigma)$ are given by $(t^1,t^2,...,t^{3g-3}) \in {\mathbb{C}}^{3g-3}$. A generic [[holomorphic quadratic differential]{}]{} $\phi dz^2$ on $(\Sigma, \sigma)$ is written as $\phi dz^2=\sum_{k=1}^{3g-3}t^k\phi_k dz^2$, where $z$ is the conformal coordinate on $(\Sigma,\sigma)$. Throughout the paper, $\sigma$ and $\rho$ are [[conformal structure]{}]{}s, with conformal coordinates $z$ and $w$, respectively. We also denote $g_{\sigma}dzd\bar{z}$ and $g_{\rho}dwd\overline{w}$ as the [[hyperbolic metric]{}]{}s on $(\Sigma,\sigma)$ and $(\Sigma, \rho)$, respectively. Corresponding to each $\phi dz^2$, there is a [[hyperbolic metric]{}]{} $g_{\rho}dwd\overline{w}$ on $\Sigma$, from the work of Samson-Wolf [@Sam78; @Wol89]: given a pair of points (two [[conformal structure]{}]{}s) $(\sigma, \rho)$ in [[Teichmüller space]{}]{}, there is a unique [[harmonic map]{}]{} $w: (\Sigma, \sigma) \to (\Sigma, \rho)$ in the homotopy class of identity, and $\phi dz^2$ is the [[Hopf differential]{}]{} of the map $u$. This induces a homeomorphism $\phi: T_g \to Q(\sigma)$ which sends $\rho$ to $\phi dz^2$. Above correspondence is well-defined via the inverse of this homeomorphism. The [[Teichmüller curve]{}]{} ${\mathcal{T}}_g$ is the fiber space over the [[Teichmüller space]{}]{} $T_g$ of closed [[Riemann surface]{}]{}s, where the fiber over a point $(\Sigma,\sigma) \in T_g$ is the underlying surface $\Sigma$. This is a manifold of real dimension $6g-4$. In this paper, we obtain curvature formulas for a general Riemannian metric on the [[Teichmüller curve]{}]{} ${\mathcal{T}}_g$, and use this to study the geometry of a Riemannian [[3-manifold]{}]{} formed by a [[Weil-Petersson geodesic]{}]{}, particularly, the degeneration of its curvatures as the [[Weil-Petersson geodesic]{}]{} heads towards the boundary of the augmented [[Teichmüller space]{}]{}. A point in ${\mathcal{T}}_g$ is represented as $(\sigma,z_0)$, where $\sigma \in T_g$ and $z_0$ is a point on the marked surface $(\Sigma,\sigma)$. Let $\pi: T{\mathcal{T}}_g \to TT_g$ be the fiberation and the kernel of the differential map $d\pi:T{\mathcal{T}}_g \to TT_g$ defines a line bundle $\nu$ over the [[Teichmüller curve]{}]{} ${\mathcal{T}}_g$. Wolpert [@Wol86] calculated the Chern form $c_{1}(v)$ of such a line bundle and showed that it is a negative differential $2$-form. He suggested to define a [[Kähler metric]{}]{} $G$ on the [[Teichmüller curve]{}]{} ${\mathcal{T}}_g$ such that its [[Kähler form]{}]{} is $-c_{1}(v)$. The [[Kähler potential]{}]{} of $G$ is given by $\log\|{{\frac{\partial{}}{\partial{w}}}}\|$, where $\|\cdot\|$ is the length of $\cdot$ with respect to the [[hyperbolic metric]{}]{} on a fiber $(\Sigma, \rho(w))$. Here $\rho(w)$ is the [[conformal structure]{}]{} $\rho$ with the conformal coordinate $w$. When restricted to each fiber $(\Sigma, \rho(w))$, the metric $G$ is ${\frac{1}{2}}g_{\rho(w)}dwd\overline{w}$. When evaluated at $\sigma \in T_g$, the metric $G = \frac12\sigma dzd\overline{z}+ \sum_{k,\ell=1}^{3g-3}\delta_{k\ell}dt^k d\overline{t^{\ell}}$. Under this metric $G$ on the [[Teichmüller curve]{}]{} ${\mathcal{T}}_g$, Jost calculated the holomorphic [[sectional curvature]{}]{} in the fiber direction: [@Jos91a] Under metric $G$, at the base point $(\sigma,z_0)$, the [[sectional curvature]{}]{} of the tangent plane expanded by $\frac{\partial}{\partial z}$ and $\sqrt{-1}\frac{\partial}{\partial z}$ is $$\begin{aligned} \label{J} K(\frac{\partial}{\partial z}, \sqrt{-1}\frac{\partial}{\partial z}) = -1+\sum_{\ell=1}^{3g-3}|\mu_{\ell}|^2(z_0).\end{aligned}$$ where $\{\mu_{\ell}(z)\}$ is a basis for the [[tangent space]{}]{}, denoted by $B_{h}(\sigma)$, of [[Teichmüller space]{}]{} at $(\Sigma,\sigma)$, normalized according to $D(\mu_{\alpha}\overline{\mu}_{\beta})(z_0) =\delta_{\alpha\beta}$, for the operator $D = -2(\Delta_{g_{\sigma}}-2)^{-1}$ on the hyperbolic surface $(\Sigma, g_{\sigma}dzd\bar{z})$. In the present work, using a general Riemannian metric on ${\mathcal{T}}_g$, we find the same curvature formula holds in the fiber directions. Moreover, we determine the [[sectional curvature]{}]{}s of the directions spanned by one fiber direction and the other by a tangent vector in [[Teichmüller space]{}]{}. Denote $z=x+\sqrt{-1}y$ and $t^{\ell}=x^{\ell}+\sqrt{-1}y^{\ell}.$ We consider a Riemannian metric $G'$ on the [[Teichmüller curve]{}]{} ${\mathcal{T}}_g$: $$\label{metric} G'= g_{\rho(w)}dwd\overline{w}+ \sum h_{\alpha\beta}(w)d\nu^\alpha d\nu^\beta$$ where $\nu^\alpha, \nu^\beta\in \{x^1, y^1,...,x^{3g-3}, y^{3g-3}\}$. The metric $G'$ restricted on the fiber $(\Sigma,\rho)$ is identified with the [[hyperbolic metric]{}]{} $g_{\rho(w)}dwd\overline{w}$. \[thm:curve\] With respect to the metric $G'$ on the [[Teichmüller curve]{}]{} ${\mathcal{T}}_g$, we assume functions $\{h_{\alpha\beta}\}$ satisfy the following: $$\label{h1} \sum h_{\alpha\beta}(z)d\nu^\alpha d\nu^\beta=2\sum_{\ell=1}^{3g-3}((dx^\ell) ^2+(dy^\ell) ^2), \forall z \in (\Sigma, \sigma),$$ i.e., it is Euclidean when restricted on $(\Sigma, \sigma)$ in $T_g$. Then at the base point $(\sigma, z_0)$, the [[sectional curvature]{}]{} of the [[tangent plane]{}]{} expanded by $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$ is $$\begin{aligned} \label{k-fiber} K(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}) = -1+\sum_{\ell=1}^{3g-3}|\mu_\ell|^2(z_0),\end{aligned}$$ where $\mu_\ell=\frac{\overline{\phi_{\ell}}}{g_{\sigma}}$ and the basis $\{\phi_{\ell}dz^2\}$ on $Q(\sigma)$ is chosen such that holds. If we require further that functions $\{h_{\alpha\beta}\}$ satisfy the following: $$\label{h2} \sum h_{\alpha\beta}(w)d\nu^\alpha d\nu^\beta=2\sum_{\ell=1}^{3g-3}((dx^\ell)^2+(dy^\ell)^2), \forall \rho(w) \in T_g,$$ i.e., a global Euclidean structure rather than a local one as in , then we have: \[thm:ft\] With respect to the metric $G'$ on the [[Teichmüller curve]{}]{} ${\mathcal{T}}_g$, we choose the basis $\{\phi_{\ell}dz^2\}$ on $Q(\sigma)$ such that holds for functions $\{h_{\alpha\beta}\}$ in $G'$. Then at the base point $(\sigma, z_0)$, the [[sectional curvature]{}]{}s of the [[tangent plane]{}]{}s expanded by $\frac{\partial}{\partial x}$ and ${\frac{\partial}{\partial x^\ell}}$, $\ell = 1, 2, \cdots, 3g-3$, are: $$\begin{aligned} \label{fib-tan1} K(\frac{\partial}{\partial x}, {\frac{\partial}{\partial x^\ell}}) = -{\frac{1}{2}}D(|\mu_{\ell}|^2)(z_0), \end{aligned}$$ where $\mu_{\ell} =\frac{\bar{\phi}_{\ell}}{g_{\sigma}}$. Similarly, $$\begin{aligned} \label{fib-tan2} K(\frac{\partial}{\partial y}, {\frac{\partial}{\partial x^\ell}}) = K(\frac{\partial}{\partial x}, {\frac{\partial}{\partial y^\ell}}) = K(\frac{\partial}{\partial y}, {\frac{\partial}{\partial y^\ell}}) = -{\frac{1}{2}}D(|\mu_{\ell}|^2)(z_0).\end{aligned}$$ They are all non-positive. One of the motivations of this paper is to study the [[Weil-Petersson geodesic]{}]{}s in [[Teichmüller space]{}]{} from its intrinsic geometry. As [[conformal structure]{}]{}s of a topological surface travel along a curve in [[Teichmüller space]{}]{}, they form a three-space which is homeomorphic to $\Sigma \times {\mathbb{R}}$. We would like to set up as follows to study the shape of a [[Weil-Petersson geodesic]{}]{}: we take one tangent vector $\mu_0 {\frac{d\bar{z}}{dz}}$ at the point $\sigma \in T_g$. Consider the [[Weil-Petersson geodesic]{}]{} $\gamma = \gamma(s)$ in [[Teichmüller space]{}]{} $T_g$ through the point $\sigma$ and in the direction $\mu_0 {\frac{d\bar{z}}{dz}}$. We consider the germ $N_{\sigma}$ at $\sigma$, and a natural local metric denoted by $H$, near $t=0$, takes a very simple form: $$\label{H} H = g_{\rho(w)}dwd\overline{w} + dt^2,$$ where $g_{\rho(w)}dwd\overline{w}$ is the pull-back metric will be made clear in $(3.1)$. \[thm:g\] At any point $z_0 \in (\Sigma, \sigma)$, we determine the [[sectional curvature]{}]{}s of the [[tangent plane]{}]{}s spanned by two of the three vectors $\frac{\partial}{\partial x}$, $\frac{\partial}{\partial y}$, and $\frac{\partial}{\partial t}$, with respect to the metric $H$ on a [[Weil-Petersson geodesic]{}]{} $\gamma$ through $\sigma$ in the direction of $\mu_0 {\frac{d\bar{z}}{dz}} \in B_h(\sigma)$, where there is a $\phi_0dz^2 \in Q(\sigma)$ such that $\mu_0 {\frac{d\bar{z}}{dz}}=\frac{\bar{\phi}_0d\bar{z}^2}{g_\sigma |dz|^2}$: 1. $K(\frac{\partial}{\partial x}, \frac{\partial}{\partial y})(\sigma,z_0) = -1+|\mu_0(z_0)|^2$; 2. $K(\frac{\partial}{\partial x}, \frac{\partial}{\partial t})(\sigma,z_0) = K(\frac{\partial}{\partial y}, \frac{\partial}{\partial t}) (\sigma,z_0)= -D(|\mu_0|^2)(z_0)$. Moreover, we define a Riemannian structure on the surface bundle $N = \bigcup_{\sigma \in \gamma}N_\sigma$ over the [[Weil-Petersson geodesic]{}]{} $\gamma$, and find Let $\gamma$ be a [[Weil-Petersson geodesic]{}]{} in [[Teichmüller space]{}]{}. Then there exists a choice of a metric in each conformal class along $\gamma$, such that the surface bundle $N$ over $\gamma$ acquires a Riemannian metric whose germ at each fiber is $N_\sigma$ as in . In particular, 1. each fiber over $\sigma$ is a hyperbolic surface; 2. each fiber over $\sigma$ is also a minimal surface; 3. the sectional curvatures of the metric are given in the Theorem 1.3. Generally, it is difficult to carry out calculations on the [[Weil-Petersson]{}]{} metric on [[Teichmüller space]{}]{}. Part of the reason is that the metric itself, as well as the curvature tensor formula, are given as an integral (or a sum of integrals) over the underlying [[conformal structure]{}]{} $(\Sigma, \sigma)$, rather than in terms of any global coordinates of [[Teichmüller space]{}]{}. In this work, the technical tool we intensively rely on is the [[Teichmüller theory]{}]{} of [[harmonic map]{}]{}s, begun with the pioneer work of Eells-Sampson ([@ES64]). This theory has many applications in many different areas in geometry and analysis. In the case of compact hyperbolic surfaces, the analytical theory does not involve any regularity issue: the degree one [[harmonic map]{}]{} between hyperbolic surfaces is a diffeomorphism ([@SY78]). It has become an important computational tool in [[Teichmüller theory]{}]{}, as well as hyperbolic geometry (see, for example, [@Wol89], [@Jos91a; @Jos91b; @JY09], [@Min92a; @Min92b], [@Hua05; @Hua07]). Plan of the paper {#plan-of-the-paper .unnumbered} ----------------- We start with a brief collection of preliminary facts in §2, where we introduce [[harmonic map]{}]{}s between hyperbolic surface in §2.1, [[Teichmüller space]{}]{} and the [[Teichmüller curve]{}]{} in §2.2, and the [[Weil-Petersson metric]{}]{} in §2.3. Theorems \[thm:curve\] and \[thm:ft\] are proved in §3, where we obtain formulas for the curvatures of the [[Teichmüller curve]{}]{} for the Riemannian metric $G'$ (). We will focus on the geometry of the [[Weil-Petersson geodesic]{}]{} in the last section §4, prove the Theorems \[thm:g\] and 1.4. Acknowledgment {#acknowledgment .unnumbered} -------------- We wish to express our gratitude to Jürgen Jost, Yair Minsky for their generous help and interest, and especially Michael Wolf and Scott Wolpert for their invaluable suggestions. The research of Huang is partially supported by a PSC-CUNY research grant. Background ========== Harmonic maps between surfaces ------------------------------ We review our basic computational scheme, which is based on [[harmonic map]{}]{}s between compact hyperbolic surfaces. Let $w:(\Sigma, g_{\sigma}|dz|^2) \rightarrow (\Sigma, g_{\rho}|dw|^2)$ be a Lipschitz map, where $g_{\sigma}|dz|^2$ and $g_{\rho}|dw|^2$ are [[hyperbolic metric]{}]{}s on the surface $\Sigma$, associated to the [[conformal structure]{}]{}s $\sigma$ and $\rho$, respectively. And $z$ and $w$ are conformal coordinates on $(\Sigma, \sigma)$ and $(\Sigma,\rho)$. We (following [@Sam78]) define important density functions: the [*holomorphic energy density*]{} $${\mathcal{H}}(z) = {\frac{g_{\rho}}{g_{\sigma}}}|w_z|^2,$$ and the [*anti-holomorphic energy density*]{} $${\mathcal{L}}(z) = {\frac{g_{\rho}}{g_{\sigma}}}|w_{\bar{z}}|^2.$$ The energy density function of the map $w$ is now simply $$\label{e} e(w(z))= {\mathcal{H}}(z) + {\mathcal{L}}(z),$$ and the [*total energy*]{} and the [*Jacobian determinant*]{} of the map are given by $$\label{E} E(w,\sigma,\rho) = \int_{\Sigma}e(z)g_{\sigma}|dz|^2 = \int_{\Sigma}e(z)dA$$ and $${\mathcal{J}}(z) = {\mathcal{H}}(z) - {\mathcal{L}}(z),$$ respectively. Here $dA$ in is the area element for $(\Sigma, \sigma)$. The map $w$ is [*harmonic*]{} if it is a critical point of this total energy functional . The $(2,0)$ part of the pullback $w^{*}\rho$ is particularly important, and it is called [*[Hopf differential]{}*]{} of $w$: $$\phi(z)dz^2 = (w^{*}\rho)^{(2,0)} = g_{\rho}w_z {\bar{w}}_zdz^2.$$ It is well-known that there is a unique [[harmonic map]{}]{} $w:(\Sigma, \sigma) \rightarrow (\Sigma, \rho)$ in each homotopy class, and $w$ is a diffeomorphism with positive Jacobian determinant. [[Teichmüller space]{}]{} and the [[Teichmüller curve]{}]{} ----------------------------------------------------------- For a closed surface $\Sigma$ of genus $g > 1$, it is a consequence of the Uniformization Theorem that the notion of [[conformal structure]{}]{}s, complex structures, and [[hyperbolic metric]{}]{}s on $\Sigma$ are equivalent. [[Teichmüller space]{}]{} $T_g$ is the space of [[conformal structure]{}]{}s on $\Sigma$, modulo the group of orientation preserving diffeomorphisms isotopic to the identity. For a [[conformal structure]{}]{} $\sigma$ on $\Sigma$, it represents a point in $T_g$, and we denote by $z$ its conformal coordinate. We routinely use $(\Sigma, \sigma)$ to indicate the marking by a [[conformal structure]{}]{}. The co[[tangent space]{}]{} of $T_g$ at $\sigma$ is identified as $Q(\sigma) = \{\phi(z)dz^2: \overline{\partial} \phi =0\}$, the space of [[holomorphic quadratic differential]{}]{}s on $(\Sigma, \sigma)$. The [[Hopf differential]{}]{} of a [[harmonic map]{}]{} is holomorphic, hence belongs to $Q(\sigma)$, and the map $w$ is conformal if and only if its [[Hopf differential]{}]{} is $0$. This is essentially the entrance of [[harmonic map]{}]{} theory to [[Teichmüller theory]{}]{}. Note that $Q(\sigma)$ is a Banach space of complex dimension $3g-3$, while the space of [[Beltrami differential]{}]{}s is of infinite dimension. We denote $g_{\sigma}|dz|^2$ the [[hyperbolic metric]{}]{} on $(\Sigma, \sigma)$, and its Laplacian is $$\Delta_{\sigma} = {\frac{4}{g_{\sigma}}}{\frac{\partial^2}{\partial z \partial \bar{z}}},$$ with nonpositive eigenvalues. We also define an operator $D = -2(\Delta_{\sigma} -2)^{-1}$. It is $L^{2}$-self-adjoint with respect to $(\Sigma, g_{\sigma}|dz|^2)$. This operator plays an essential role in understanding the [[Weil-Petersson]{}]{} geometry of [[Teichmüller space]{}]{}. The [[tangent space]{}]{} of $T_g$ at $\sigma$ can be identified with the space of [[harmonic Beltrami differential]{}]{}s $B_h(\sigma)$. A [[Beltrami differential]{}]{} $\mu(z){\frac{d\bar{z}}{dz}}$ on $(\Sigma, \sigma)$ is harmonic if $\mu(z){\frac{d\bar{z}}{dz}} = {\frac{\bar{\phi}d\bar{z}^2}{g_{\sigma}dzd\bar{z}}}$ for some $\phi dz^2 \in Q(\sigma)$. The [[Teichmüller curve]{}]{} ${\mathcal{T}}_g$ is a fiber bundle over [[Teichmüller space]{}]{} $T_g$, and the fiber over $\sigma \in T_g$ is the marked surface $(\Sigma, \sigma)$. As a manifold of real dimension $6g -4$, every point in ${\mathcal{T}}_g$ can be represented as $(\sigma, z_0)$, where $\sigma \in T_g$ and $z_0 \in (\Sigma, \sigma)$. The first Chern class $c_1(v) = \frac{\sqrt{-1}}{2\pi} \Theta$ of the line bundle $\nu$ over ${\mathcal{T}}_g$ is computed by Wolpert ([@Wol86]), where the curvature 2-form $\Theta$ is found to satisfy $\Theta({\frac{\partial{}}{\partial{z}}}, {\frac{\partial{}}{\partial{\bar{z}}}}) = \frac{-2}{(z-\bar{z})^2}$, and $\Theta({\frac{\partial{}}{\partial{\bar{z}}}}, \tau_{\mu}) = 0$, and $\Theta(\bar{\tau}_{\nu}, \tau_{\mu}) = D(\mu\bar{\nu})$. It is negative, therefore one can define a [[Kähler metric]{}]{} $-c_1(v)$ on ${\mathcal{T}}_g$. The [[Weil-Petersson metric]{}]{} and [[3-manifold]{}]{}s --------------------------------------------------------- The [[Weil-Petersson]{}]{} co-metric is defined on the cotangent space $Q(\sigma)$ by the natural $L^2$-norm: $$\begin{aligned} ||\phi||_{WP}^2 = \int_{\Sigma} \frac {|\phi|^2}{g_{\sigma}^2}dA_{\sigma}, \forall \phi \in Q(\sigma),\end{aligned}$$ where $dA_{\sigma}$ is the hyperbolic area element of $(\Sigma,\sigma)$. By duality, we obtain the [[Weil-Petersson metric]{}]{} on $T_g$. The [[Weil-Petersson geodesic]{}]{}s are intimately related to the geometry of three manifolds. Given $X, Y \in {T}_g(\Sigma)$, they uniquely determine a [[quasi-Fuchsian]{}]{} hyperbolic [[3-manifold]{}]{}, $Q(X,Y)$, with $X$ and $Y$ as conformal boundaries, by the means of Bers’ simultaneous uniformization ([@Ber72]). Brock ([@Bro03]) showed that the hyperbolic volume of the convex core of $Q(X,Y)$ is quasi-isometric to the length of the [[Weil-Petersson geodesic]{}]{} joining $X$ and $Y$ in [[Teichmüller space]{}]{}. We obtained a [[Weil-Petersson]{}]{} potential from varying [[quasi-Fuchsian]{}]{} manifolds in [[quasi-Fuchsian]{}]{} space near the Fuchsian locus ([@GHW09]). Many mysterious properties of the [[Weil-Petersson]{}]{} geometry of [[Teichmüller space]{}]{} are largely due to the incompleteness of the metric ([@Chu76; @Wol75]). When a [[Weil-Petersson]{}]{} geodesic can not be extended, a short simple closed curve on the surface is pinched to a single point ([@Mas76]), while curvatures on the surface remain hyperbolic. A basic property is that the [[Weil-Petersson metric]{}]{} is geodescially convex ([@Wol87]), hence any two points can be joined by a unique [[Weil-Petersson]{}]{} geodesic. The [[Weil-Petersson]{}]{} geometry of [[Teichmüller space]{}]{} is quite satisfying: it is a space of negative curvature ([@Tro86], [@Wol86]). However, there is neither negative upper bound ([@Hua05]) nor lower bound ([@Hua07], [@Sch86]) of the sectional curvatures. We refer more detailed discussions on the [[Weil-Petersson]{}]{} geometry of [[Teichmüller space]{}]{} to articles ([@Wol03; @Wol06; @Wol09]). We want to understand the infinitesimal geometry of the [[Weil-Petersson geodesic]{}]{}. To this end, we consider a simple [[Weil-Petersson geodesic]{}]{} $\gamma$ determined by the point $\sigma \in T_g$ and the direction $\mu_{\ell}$, where $\mu_{\ell} \in B_h(\sigma)$ is a tangent vector at $\sigma$. We denote $N_\sigma$ the germ of a hyperbolic surface associated to the [[conformal structure]{}]{} at the point $\sigma \in \gamma$. The set of these germs over the [[Weil-Petersson geodesic]{}]{} $\gamma$ form a three-dimensional space. Curvature Formulas of the [[Teichmüller curve]{}]{} =================================================== We prove theorems 1.1 and 1.2 in this section. Curvatures in the fiber directions are obtained in §3.1, and fiber-tangential directions follow in §3.2. Fiber directions ---------------- In this subsection, we determine the [[sectional curvature]{}]{} of the [[Teichmüller curve]{}]{} ${\mathcal{T}}_g$ in the fiber directions. The metric $G'$ is given as in : $$G'= g_{\rho(w)}dwd\overline{w}+ \sum h_{\alpha\beta}(w)d\nu^\alpha d\nu^\beta,$$ where the functions $\{h_{\alpha\beta}\}$ satisfy : $$\sum h_{\alpha\beta}(z)d\nu^\alpha d\nu^\beta=2\sum_{\ell=1}^{3g-3}((dx^\ell)^2+(dy^\ell)^2), \forall z \in (\Sigma, \sigma).$$ In other words, we only require it to be Euclidean at $w=z$ in $T_g$. We now proceed to calculate the [[sectional curvature]{}]{} in the fiber directions, spanned by vectors $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$, where $z =x+\sqrt{-1}y$. \[Proof of Theorem \[thm:curve\]\] Fixing $\sigma \in T_g$, for any [[hyperbolic metric]{}]{} $g_{\rho}$ on $\Sigma$, denote the unique [[harmonic map]{}]{} $w: (\Sigma,\sigma) \rightarrow (\Sigma,g_{\rho}dwd\bar{w})$ in the homotopy class of the identity, one obtains the [[Hopf differential]{}]{} $\phi(z)dz^2$ of $w$, which is holomorphic. Therefore we have a map $\phi$ from [[Teichmüller space]{}]{} to $Q(\sigma)$. This map is a homeomorphism ([@Sam78; @Wol89]). Therefore, via the inverse map, $Q(\sigma)$ provides global coordinates for $T_g$. The space of [[holomorphic quadratic differential]{}]{}s $Q(\sigma)$ is a Banach space, so for $\phi_{0}dz^{2} \in Q(\sigma)\backslash \{0\}$, and $\phi(t) = t\phi_{0}$ is a ray in $Q(\sigma)$. We denote the [[hyperbolic metric]{}]{}s $g_{\rho(t)}|dw(t)|^{2}$ as the points in $T_g$ determined by the ray $\phi(t)$ in $Q(\sigma)$, via the Sampson-Wolf Theorem. Here $w(t)$ is the family of [[harmonic map]{}]{}s, in the homotopy class of the identity, whose associated [[Hopf differential]{}]{}s are given by $\phi(t) = \rho(w(t)) w_z(t) {\bar{w}}_z(t)dz^2$. Clearly, at $t = 0$, $\rho(0) = \sigma$. For this family of [[harmonic map]{}]{}s $w(t): (\Sigma, g_{\sigma}|dz|^2) \rightarrow (\Sigma, g_{\rho(t)}|dw(t)|^{2})$, we pull back the metric to find $$\begin{aligned} \label{pb} w^{*} g_{\rho(t)}|dw(t)|^{2} &=& 2Re(g_{\rho(w(t))}w_z(t){\bar{w}}_z(t)dz^2) + g_{\rho(t)}(|w_{z}(t)|^{2}+ |{\bar{w}}_{z}(t)|^{2}) |dz|^{2} \nonumber \\ &=& \phi(t) dz^{2} + g_{\sigma}e(t)|dz|^{2} + {\bar{\phi}}(t)d{\bar{z}}^{2}, \end{aligned}$$ where $e=e(t)$ is the energy density of $w(t)$, as in . For $\phi(t)dz^{2}$, we have $\mu(t)\frac{d\overline{z}}{d z} = {\frac{\bar{\phi}(t)d\bar{z}^{2}}{g_\sigma dzd{\bar{z}}}} \in B_h(\sigma)$, a family of [[harmonic Beltrami differential]{}]{}s. They represent [[tangent vector]{}]{}s at $\sigma \in T_g$. Let $\{\phi_1,\cdots,\phi_{3g-3}\}$ be a basis of $Q(\sigma)$ such that holds for the functions $\{h_{\alpha\beta}\}$ in the definition of the metric $G'$ of ${\mathcal{T}}_g$. We write $t =(t^1, \cdots, t^{3g-3})$ such that $\phi(t) dz^2=\sum_{\ell=1}^{3g-3}t^{\ell}\phi_{\ell}dz^2$. And we use $|_0$ to indicate evaluation at $t^{\alpha} = 0$ for any $\alpha =1, \cdots, 3g-3$. The variations of the energy density $e(t)$ are calculated by Wolf ([@Wol89]) as follows: $$\begin{aligned} e(t)|_0&=1,\\ \frac{\partial e(t)}{\partial t^\alpha}|_0 &=\frac{\partial e(t)}{\partial \overline{t^{\alpha}}} |_0 =0,\\ \frac{\partial^2 e(t)}{\partial t^\alpha \partial t^\beta}|_0 & =\frac{\partial^2 e(t)}{\partial \overline{t^{\alpha}} \partial \overline{t^{\beta}}}|_0 =0,\\ \frac{\partial^2e(t)}{\partial t^{\alpha} \partial \overline{t^{\beta}}}|_0 & =(D+1)\frac{\phi_\alpha \overline{\phi}_{\beta}}{g_{\sigma}^2}.\end{aligned}$$ Using real coordinates, we can rewrite the above results as: $$\begin{aligned} e(t)|_0&=1,\\ \frac{\partial e(t)}{\partial x^\alpha}|_0 &=\frac{\partial e(t)}{\partial y^\alpha } |_0 =0,\\ \frac{\partial^2 e(t)}{\partial x^\alpha \partial x^\beta}|_0 &=\frac{\partial^2 e(t)}{\partial y^\alpha \partial y^\beta}|_0 =(D+1)\frac{2Re(\phi_\alpha \overline{\phi}_{\beta}) }{g_{\sigma}^2},\end{aligned}$$ Using the pullback , the metric $G'$ on ${\mathcal{T}}_g$ is written as $$\begin{aligned} \label{pb-r} G'&=\phi(t)dz^2+g_{\sigma}e(t)dzd\overline{z} + \overline{\phi}(t) d\overline{z}^2+ \sum h_{\alpha\beta}(w)d\nu^\alpha d\nu^\beta \nonumber\\ &=(g_{\sigma}e+ 2Re \phi)dx^2-4(Im\phi)dxdy+(g_{\sigma}e- 2Re \phi)dy^2 + \sum h_{\alpha\beta}(w)d\nu^\alpha d\nu^\beta.\end{aligned}$$ To simply our notation, we denote $R_{1221} = R_{xyyx}$, and utilize Einstein notation to compute this curvature tensor as follows: $$\begin{aligned} \label{r} R_{1221}|_0 &\ =g_{11}(\Gamma_{22,1}^1-\Gamma_{12,1}^1+ \Gamma_{22}^\beta\Gamma_{1\beta}^1-\Gamma_{12}^\beta\Gamma_{2\beta}^1)|_0\nonumber \\ &\ =g_{11}(\Gamma_{22,1}^1-\Gamma_{12,1}^1\nonumber\\ &\ \ \ \ \ \ \ \ \ \ +\Gamma_{22}^1\Gamma_{11}^1+\Gamma_{22}^2\Gamma_{12}^1 +\sum_{\ell=1}^{3g-3}(\Gamma_{22}^{x^\ell}\Gamma_{1x^\ell}^1+\Gamma_{22}^{y^\ell}\Gamma_{1y^\ell}^1)\nonumber\\ &\ \ \ \ \ \ \ \ \ \ -\Gamma_{12}^1\Gamma_{21}^1-\Gamma_{12}^2\Gamma_{22}^1 -\sum_{\ell=1}^{3g-3}(\Gamma_{12}^{x^\ell}\Gamma_{2x^\ell}^1+\Gamma_{12}^{y^\ell}\Gamma_{2y^\ell}^1))|_0.\end{aligned}$$ Here we recall from that, at $t=0$, the functions $h_{\alpha\beta}$ satisfy the condition that $\sum h_{\alpha\beta}(z)d\nu^\alpha d\nu^\beta=2\sum_{\ell=1}^{3g-3}((dx^\ell)^2+(dy^\ell)^2)$. We calculate the values of Christoffel symbols evaluated at $t^\alpha=0$ for any $\alpha$: $$\left\{ \begin{matrix} \Gamma_{11}^{1}=\displaystyle \frac{(g_{\sigma})_1}{2g_{\sigma}}, & \Gamma_{11}^{2}= \displaystyle -\frac{(g_{\sigma})_2}{2g_\sigma}, & \Gamma_{11}^{x^\ell}= \displaystyle -\frac{Re\phi_\ell}2, & \Gamma_{11}^{y^\ell}= \displaystyle \frac{Im\phi_\ell}2,\\ \\ \Gamma_{12}^{1}=\displaystyle \frac{(g_{\sigma})_2}{2\sigma}, & \Gamma_{12}^{2} =\displaystyle \frac{(g_{\sigma})_1}{2\sigma}, & \Gamma_{12}^{x^\ell} = \displaystyle \frac{Im\phi_\ell}2, & \Gamma_{12}^{y^\ell}= \displaystyle \frac{Re\phi_\ell}2, \\ \\ \Gamma_{22}^{1}=\displaystyle -\frac{(g_{\sigma})_1}{2g_\sigma}, & \Gamma_{22}^{2}=\displaystyle \frac{(g_{\sigma})_2}{2g_\sigma}, & \Gamma_{22}^{x^\ell}=\displaystyle \frac{Re\phi_\ell}2, & \Gamma_{22}^{y^\ell}=\displaystyle -\frac{Im\phi_\ell}2. \end{matrix} \right.$$ Some mixed terms are calculated as follows: $$\left\{ \begin{matrix} \Gamma_{1x^\ell}^{1}=\displaystyle \frac{Re\phi_\ell}{g_{\sigma}}, & \Gamma_{1x^\ell}^{2}= \displaystyle -\frac{Im\phi_\ell}{g_{\sigma}}, & \Gamma_{1x^\ell}^{x^\ell}= \displaystyle 0, & \Gamma_{1x^\ell}^{y^\ell}= \displaystyle 0, \\ \\ \Gamma_{1y^\ell}^{1}=\displaystyle -\frac{Im\phi_\ell}{g_{\sigma}}, & \Gamma_{1y^\ell}^{2} =\displaystyle -\frac{Re\phi_\ell}{g_{\sigma}}, & \Gamma_{1y^\ell}^{x^\ell} =\displaystyle 0, & \Gamma_{1y^\ell}^{y^\ell} =\displaystyle 0,\\ \\ \Gamma_{2x^\ell}^{1}=\displaystyle -\frac{Im\phi_\ell}{g_{\sigma}}, & \Gamma_{2x^\ell}^{2}= \displaystyle -\frac{Re\phi_\ell}{g_{\sigma}}, & \Gamma_{2x^\ell}^{x^\ell}= \displaystyle 0, & \Gamma_{2x^\ell}^{y^\ell}= \displaystyle 0 ,\\ \\ \Gamma_{2y^\ell}^{1}=\displaystyle -\frac{Re\phi_\ell}{g_{\sigma}}, & \Gamma_{2y^\ell}^{2}=\displaystyle \frac{Im\phi_\ell}{g_{\sigma}}, & \Gamma_{2y^\ell}^{x^\ell}=\displaystyle 0, & \Gamma_{2y^\ell}^{y^\ell}=\displaystyle 0, \end{matrix} \right.$$ and $$\left\{ \begin{matrix} \Gamma_{x^\ell x^\ell}^{1}=\displaystyle 0, & \Gamma_{x^\ell x^\ell}^{2}= \displaystyle 0,\\ \\ \Gamma_{x^\ell y^\ell}^{1}=\displaystyle 0 , & \Gamma_{x^\ell y^\ell}^{2} = \displaystyle 0,\\ \\ \Gamma_{y^\ell y^\ell}^{1}=\displaystyle 0 , & \Gamma_{y^\ell y^\ell}^{2}= \displaystyle 0.\\ \end{matrix} \right.$$ Using that the curvature of the [[hyperbolic metric]{}]{} $g_{\sigma}(dx^2+dy^2)$ is $-1$, and $g_{11}|_0 = g_{\sigma}$ from , we have: $$\begin{aligned} R_{1221}|_0 &\ =g_{\sigma}(-(\frac{(g_\sigma)_1}{2g_{\sigma}})_1-(\frac{(g_\sigma)_2}{2g_{\sigma}})_2\\ &\ \ \ \ \ \ \ \ \ \ -\frac{((g_\sigma)_1)^2}{4g_{\sigma}^2}+\frac{((g_\sigma)_2)^2}{4g_{\sigma}^2} +\frac12\sum_{\ell=1}^{3g-3}(\frac{(Re\phi_\ell)^2}{g_\sigma}+\frac{(Im\phi_\ell)^2}{g_{\sigma}})\\ &\ \ \ \ \ \ \ \ \ \ -\frac{((g_\sigma)_2)^2}{4g_\sigma^2}+\frac{((g_\sigma)_1)^2}{4g_\sigma^2} +\frac12\sum_{\ell=1}^{3g-3}(\frac{(Re\phi_\ell)^2}{g_\sigma}+\frac{(Im\phi_\ell)^2}{g_\sigma}))\\ &=g_{\sigma}(-(\frac{(g_\sigma)_1}{2g_\sigma})_1-(\frac{(g_\sigma)_2}{2g_\sigma})_2+ \frac{1}{g_\sigma}\sum_{\ell=1}^{3g-3}|\phi_\ell|^2 )\\ &\ =-g_{\sigma}^2+\sum_{\ell=1}^{3g-3}|\phi_\ell|^2.\end{aligned}$$ Therefore the curvature in directions $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$, at $t=0$, is $$\begin{aligned} K(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}) = {\frac{R_{1221}}{g_\sigma^2}} &=& -1+\sum_{\ell=1}^{3g-3}\frac{|\phi_{\ell}|^2}{g_{\sigma}^2}(z_0) \\ &=& -1+ \sum_{\ell=1}^{3g-3}|\mu_\ell|^{2}(z_0).\end{aligned}$$ The normalization for the metric $G = -c_{1}(v)$ on ${\mathcal{T}}_g$ provides the following estimates of the curvatures in the fiber directions, in particular, it is impossible for the curvatures to be non-positive everywhere in ${\mathcal{T}}_g$: With respect to the metric $G = -c_{1}(v)$ on ${\mathcal{T}}_g$, we have 1. [@Jos91a]: $Sup_{(\sigma,z_0) \in {\mathcal{T}}_g}K(\frac{\partial}{\partial z}, \sqrt{-1}\frac{\partial}{\partial z}) > 0$; 2. $Sup_{(\sigma,z_0) \in {\mathcal{T}}_g}K(\frac{\partial}{\partial z}, \sqrt{-1}\frac{\partial}{\partial z}) \le 9g-10$. 0.1in 1. This is proved in ([@Jos91a]). The idea is that the explicit [[Kähler potential]{}]{} of the metric $G$ forces $D(|\mu_\ell|^2)(z_0) =1$ for all $\ell = 1, \cdots, 3g-3$, then the lower bound $0$ is a consequence of the [[maximum principle]{}]{}. 2. The upper bound is a consequence of the following point-wise estimate from next Lemma. \[thm:w\][@Wol08] For any $\mu(z){\frac{d\bar{z}}{dz}} \in B_h(\sigma)$, and $\forall z \in (\Sigma, \sigma)$, we have the following: $$3D(|\mu|^2)(z) \ge |\mu|^2(z).$$ Fiber-tangential directions --------------------------- To calculate the [[sectional curvature]{}]{}s spanned by one fiber direction and one tangential direction, we have to complete the mixed terms in Christoffel symbols which were not required in the last subsection. For this reason, we require functions $\{h_{\alpha\beta}\}$ satisfy : $$\sum h_{\alpha\beta}(w)d\nu^\alpha d\nu^\beta=2\sum_{\ell=1}^{3g-3}((dx^\ell)^2+(dy^\ell)^2), \forall \rho(w) \in T_g.$$ Therefore the metric $G'$ takes the form: $$\label{metric2} G'= g_{\rho(w)}dwd\overline{w}+ 2\sum_{\ell=1}^{3g-3}(dx_\ell^2+dy_\ell^2).$$ And we are now going to prove the Theorem \[thm:ft\]. \[Proof of Theorem \[thm:ft\]\] We will continue to use notation in the proof of Theorem \[thm:curve\]. We will only calculate the curvature tensor $R_{1x^{\ell}x^{\ell}1} = R_{xx^{\ell}x^{\ell}x}$ at $t=0$, which is given by: $$\label{r2} R_{1x^{\ell}x^{\ell}1}|_0 = g_{\sigma}(\Gamma_{x^{\ell}x^{\ell},1}^1-\Gamma_{1x^{\ell},x^\ell}^1+ \Gamma_{x^{\ell}x^{\ell}}^\beta\Gamma_{1\beta}^1-\Gamma_{1x^{\ell}}^\beta\Gamma_{x^{\ell}\beta}^1)|_0.$$ We have $$\begin{aligned} & \Gamma_{1x^{\ell}}^1=\frac12\frac{g_{\sigma}e- 2 Re \phi}{g_{\sigma}^2 e^2-4 |\phi|^2} (g_{\sigma} \frac{\partial e}{\partial x^{\ell}}+ 2 Re \phi_{\ell})+ \frac12\frac{2Im\phi }{g_{\sigma}^2 e^2-4|\phi|^2} (-2Im\phi_{\ell}),\\ & \Gamma_{1y^{\ell}}^1=\frac12\frac{g_{\sigma}e- 2 Re \phi}{g_{\sigma}^2 e^2-4 |\phi|^2} (g_{\sigma}\frac{\partial e}{\partial y^{\ell}}- 2 Im \phi_{\ell})+ \frac12\frac{2Im\phi }{g_{\sigma}^2 e^2-4|\phi|^2} (2Re\phi_{\ell}),\\ &\Gamma_{2x^{\ell}}^2=\frac12\frac{2Im\phi }{g_{\sigma}^2 e^2-4|\phi|^2} (-2Im\phi_{\ell})+ \frac12\frac{g_{\sigma}e+ 2 Re \phi}{g_{\sigma}^2 e^2-4|\phi|^2} (g_{\sigma}\frac{\partial e}{\partial x^{\ell}}- 2 Re \phi_{\ell}),\\ &\Gamma_{2y^{\ell}}^2=\frac12\frac{2Im\phi }{g_{\sigma}^2 e^2-4|\phi|^2} (2Re\phi_{\ell})+ \frac12\frac{g_{\sigma}e+ 2 Re \phi}{g_{\sigma}^2 e^2-4|\phi|^2} (\sigma \frac{\partial e}{\partial y^{\ell}}+ 2 Im \phi_{\ell}).\end{aligned}$$ And their derivatives at $t=0$ are as follows: $$\begin{aligned} & \Gamma_{1x^{\ell},x^{\ell}}^1|_0=\Gamma_{2x^{\ell},x^{\ell}}^1|_0=-2\frac{|\phi_{\ell}|^2}{g_\sigma^2}+ \frac12 \frac{\partial^2 e(w)}{\partial x^{\ell} \partial x^{\ell}} =(D-1)\frac{|\phi_{\ell}|^2}{g_\sigma^2},\\ & \Gamma_{1y^{\ell},y^{\ell}}^1|_0=\Gamma_{2y^{\ell},y^{\ell}}^1|_0=-2\frac{|\phi_{\ell}|^2}{g_\sigma^2}+ \frac12 \frac{\partial^2 e(w)}{\partial y^{\ell} \partial y^{\ell}} =(D-1)\frac{|\phi_{\ell}|^2}{g_\sigma^2}. \end{aligned}$$ Moreover, because of condition , we can complete the table of Christoffel symbols by: $$\left\{ \begin{matrix} & \Gamma_{x^{\ell}x^{\ell}}^{x^{\ell}}= \displaystyle 0 , & \Gamma_{x^{\ell}x^{\ell}}^{y^{\ell}}= \displaystyle 0 , \\ \\ & \Gamma_{x^{\ell}y^{\ell}}^{x^{\ell}} =\displaystyle 0, & \Gamma_{x^{\ell}y^{\ell}}^{y^{\ell}} =\displaystyle 0,\\ \\ & \Gamma_{y^{\ell}y^{\ell}}^{x^{\ell}}= \displaystyle 0 , & \Gamma_{y^{\ell}y^{\ell}}^{y^{\ell}}= \displaystyle 0 .\\ \end{matrix} \right.$$ Substituting these terms into , we find $$\begin{aligned} R_{1x^{\ell}x^{\ell}1}|_0 &= g_{\sigma}(\Gamma_{x^{\ell}x^{\ell},1}^1-\Gamma_{1x^{\ell},x^\ell}^1+ \Gamma_{x^{\ell}x^{\ell}}^\beta\Gamma_{1\beta}^1-\Gamma_{1x^{\ell}}^\beta\Gamma_{x^{\ell}\beta}^1)|_0\\ &= g_{\sigma}(0-(D-1)\frac{|\phi_{\ell}|^2}{g_\sigma^2}+0-\frac{(Re\phi_{\ell})^2}{g_\sigma^2}-\frac{(Im\phi_{\ell})^2}{g_\sigma^2})\\ &= -g_\sigma D(\frac{|\phi_{\ell}|^2}{g_\sigma^2})(z_0).\end{aligned}$$ Then the curvature $$K(\frac{\partial}{\partial x}, {\frac{\partial}{\partial x^\ell}}) = \frac{R_{1x^{\ell}x^{\ell}1}}{2g_\sigma}= -{\frac{1}{2}}D(|\mu_{\ell}|^2)(z_0).$$ Similarly, one can work out other curvatures: $$\frac{R_{1y^{\ell}y^{\ell}1}}{2g_\sigma}=\frac{R_{2x^{\ell}x^{\ell}2}}{2g_\sigma}=\frac{R_{2y^{\ell}y^{\ell}2}}{2g_\sigma}= -{\frac{1}{2}}D(|\mu_{\ell}|^2)(z_0).$$ They are all bounded from above by $-\frac16 |\mu_{\ell}|^2(z_0)$, hence non-positive. Application: The geometry of the [[Weil-Petersson geodesic]{}]{} ================================================================ An application of our method is to study the geometry of the [[Weil-Petersson metric]{}]{} in [[Teichmüller space]{}]{}, in particular, three-manifold formed by a surface bundle over a [[Weil-Petersson geodesic]{}]{} $\gamma$. This manifold is the union of germs $N_\sigma$ over $\gamma$, where $\sigma \in \gamma$, and the fiber at $\sigma$ is a hyperbolic surface in the conformal class determined by $\sigma$. We always assume the curve $\gamma$ is parametrized by its arc length. Recall the [[Hopf differential]{}]{}s $\phi(t)dz^2$ of the family of [[harmonic map]{}]{}s from $(\Sigma,\sigma)$ to $(\Sigma, g_{\rho(t)}dwd\bar{w})$ determine a curve $\rho(t) \in T_g$. It is crucial to us that, when $\phi(t)dz^2$ is a ray in $Q(\sigma)$, i.e., for some $\phi_0 dz^2 \in Q(\sigma)\backslash 0$, $\phi(t) = t\phi_0$, then the slice $\rho(t)$ is a [[Weil-Petersson]{}]{} geodesic at $t=0$ ([@Ahl61]). This permits calculation of curvatures and the second fundamental form of the fibers of $N=\bigcup_\sigma N_\sigma$ by local computation on the germs. This section is organized in subsections. §4.1 contains local calculations where we determine the [[sectional curvature]{}]{}s of the germ $N_\sigma$. The asymptotic behavior of these curvatures near the infinity is also investigated in the subsection; in §4.2, we study the [[second fundamental form]{}]{} of the fiber hyperbolic surface of $N_\sigma$ at $\sigma$, and show that this fiber is minimal; in §4.3, we equip the [[surface bundle]{}]{} $N=\bigcup_\sigma N_\sigma$ a Riemannian structure and prove the Theorem 1.4. Curvatures of the germ $N_\sigma$ --------------------------------- Let $\gamma$ be a [[Weil-Petersson geodesic]{}]{} arc in [[Teichmüller space]{}]{} that passes through $(\Sigma, \sigma)$ and in the direction of the harmonic [[Beltrami differential]{}]{} $\mu_0(z){\frac{d\bar{z}}{dz}} =\frac{\bar{\phi}_0d\overline{z}^2}{g_\sigma dz d\overline{z}} \in B_h(\sigma)$, and $z = x+\sqrt{-1}y$. The [[Weil-Petersson geodesic]{}]{} arc $\gamma$ is parametrized by its arc length, so we have $$\|\mu_0\|_{WP}=(\int_{(S,\sigma)}|\mu_0(z)|^2 dA(z))^{\frac12} =1.$$ In this subsection, we focus on local geometry near $\sigma$, i.e., we consider the germ $N_\sigma$ over the point $\sigma \in \gamma$, with the metric $H$ as in : $$H = g_{\rho(w)}dwd\overline{w} + dt^2.$$ We obtain [[sectional curvature]{}]{}s of $N_\sigma$ with the metric $H$ for tangent vectors $\frac{\partial}{\partial x}$, $\frac{\partial}{\partial y}$ and $\frac{\partial}{\partial t}$, at $t=0$, and evaluating at $z_0 \in (\Sigma,\sigma)$, that is: \[Proof of Theorem \[thm:g\]\] We set up similarly as in the proof of the Theorem \[thm:curve\]. For a family of [[hyperbolic metric]{}]{}s $g_{\rho(w)}dwd\bar{w}$, we have a family of [[holomorphic quadratic differential]{}]{}s $\phi(t)dz^2 = t\phi_0 dz^2 \in Q(\sigma)$ associated to the [[harmonic map]{}]{}s $w(t)$ from $(\Sigma,\sigma)$ to $(\Sigma, g_{\rho(w)}dwd\bar{w})$. The pullback metric on $(\Sigma,\sigma)$ is given by : $$w^{*} g_{\rho(t)}|dw(t)|^{2} = \phi(t) dz^{2} + g_{\sigma}e(t)|dz|^{2} + {\bar{\phi}}(t)d{\bar{z}}^{2},$$ where $e=e(t)$ is again the energy density of $w(t)$, and $\phi(t) =t\phi_0$. The metric $H$ on the germ $N_\sigma$ is, in real coordinates, the following: $$\begin{aligned} \label{H-r} &t\phi_0 dz^2+\sigma e(t)dz d\overline{z}+t\overline{\phi}_0d\overline{z}^2+dt^2 \\ =&\ t\phi_0 (dx^2-dy^2+2idxdy)+ g_\sigma e(t)(dx^2+dy^2)+ t\overline{\phi}_0(dx^2-dy^2-2idxdy)+dt^2 \\ =&\ (g_\sigma e+ 2t Re \phi_0)dx^2-4t Im\phi_0 dxdy+(g_\sigma e- 2t Re \phi_0)dy^2 +dt^2\end{aligned}$$ The pullback metric on the family of surfaces can be represented by its matrix form as follows: $$\label{fff} (g_{ij}(t))=\left( \begin{array}{ccc} g_\sigma e(t)+ 2t Re \phi_0 & -2t Im\phi_0 \\ -2t Im\phi_0 &g_\sigma e(t)- 2t Re \phi_0 \end{array} \right).$$ As before, we use indices $1$, $2$ for variables $x$ and $y$, respectively, to simplify the notation in the [[Christoffel symbol]{}]{}s. We now use index $3$ for the variable $t$ for the same purpose. The values of relevant [[Christoffel symbol]{}]{}s, at $t=0$, can be computed as follows: $$\left\{ \begin{matrix} \Gamma_{11}^{1}=\displaystyle \frac{(g_{\sigma})_1}{2g_{\sigma}}, & \Gamma_{11}^{2}= \displaystyle -\frac{(g_{\sigma})_2}{2g_\sigma}, & \Gamma_{11}^3= \displaystyle -Re\phi_0, \\ \\ \Gamma_{12}^{1}=\displaystyle \frac{(g_{\sigma})_2}{2\sigma}, & \Gamma_{12}^{2} =\displaystyle \frac{(g_{\sigma})_1}{2\sigma}, & \Gamma_{12}^3 = \displaystyle Im\phi_0,\\ \\ \Gamma_{22}^{1}=\displaystyle -\frac{(g_{\sigma})_1}{2g_\sigma}, & \Gamma_{22}^{2}=\displaystyle \frac{(g_{\sigma})_2}{2g_\sigma}, & \Gamma_{22}^3=\displaystyle Re\phi_0, \end{matrix} \right.$$ and $$\left\{ \begin{matrix} \Gamma_{13}^{1}=\displaystyle \frac{Re\phi_0}{g_\sigma}, & \Gamma_{13}^{2}=\displaystyle -\frac{Im\phi_0}{g_\sigma}, & \Gamma_{13}^3= \displaystyle 0, \\ \\ \Gamma_{23}^{1}=\displaystyle -\frac{Im\phi_0}{g_\sigma}, & \Gamma_{23}^{2} =\displaystyle -\frac{Re\phi_0}{g_\sigma}, & \Gamma_{23}^3 = \displaystyle 0,\\ \\ \Gamma_{33}^{1}=\displaystyle 0, & \Gamma_{33}^{2}=\displaystyle 0, & \Gamma_{33}^3=\displaystyle 0. \end{matrix} \right.$$ The curvature tensor $R_{1221}$, at $t=0$, is $$\begin{aligned} R_{1221}|_0 &=& g_\sigma ( \Gamma_{22,1}^{1} - \Gamma_{21,2}^{1} + \Gamma_{22}^{\beta}\Gamma_{\beta 1}^{1}-\Gamma_{21}^{\beta}\Gamma_{\beta2}^{1})|_0 \nonumber.\end{aligned}$$ Applying the curvature on $(\Sigma,\sigma)$ is $-1$, and above values for [[Christoffel symbol]{}]{}s, we have: $$\begin{aligned} R_{1221}|_0 &=& g_\sigma (-g_\sigma + \frac{(Re\phi_0)^2}{g_\sigma} + \frac{(Im\phi_0)^2}{g_\sigma}) \\ &=& -g_\sigma^2+ |\phi_0|^2.\end{aligned}$$ So we find that, at $(\sigma, z_0)$, the curvature in the fiber directions is $$K(\frac{\partial}{\partial x}, \frac{\partial}{\partial y})(\sigma, z_0) = \frac{R_{1221}}{g_\sigma^2}= -1+|\mu_0(z_0)|^2.$$ We are left to determine curvatures $K(\frac{\partial}{\partial x}, \frac{\partial}{\partial t})$ and $K(\frac{\partial}{\partial y}, \frac{\partial}{\partial t})$. The [[curvature tensor]{}]{} $R_{1331}$ can be computed as follows: $$R_{1331}|_0 = g_\sigma ( \Gamma_{33,1}^{1} - \Gamma_{31,3}^{1} + \Gamma_{33}^{\beta}\Gamma_{\beta1}^{1}-\Gamma_{31}^{\beta}\Gamma_{\beta3}^{1})|_0,$$ where Einstein notation is employed for $\beta = 1, 2, 3$. It is easy to verify from the values of the [[Christoffel symbol]{}]{}s at $t=0$, that $$\begin{aligned} \sum_{\beta=1}^3 (\Gamma_{33}^{\beta}\Gamma_{\beta1}^{1}-\Gamma_{31}^{\beta}\Gamma_{\beta3}^{1})|_0 &=& -(\Gamma_{31}^{1})^2|_0 - (\Gamma_{31}^2\Gamma_{23}^1)|_0\\ &=& -|\mu_0|^2,\end{aligned}$$ and we apply the second variation of $e(t)$ to find $$\begin{aligned} \Gamma_{13,3}^{1}|_0 &=& \frac12 (-\frac{4(Re\phi_0)^2}{g_\sigma^2} + 2D(|\mu_0|^2) + 2|\mu_0|^2 -\frac{4(Im\phi_0)^2}{g_\sigma^2}) \\ &=& D(|\mu_0|^2) - |\mu_0|^2.\end{aligned}$$ Therefore we have $$\begin{aligned} R_{1331}|_0 &=& g_\sigma ( 0 - D(|\mu_0|^2) + |\mu_0|^2 -|\mu_0|^2) \\ &= & g_\sigma (-D(|\mu_0|^2)),\end{aligned}$$ and at $z_0$ $$K(\frac{\partial}{\partial x}, \frac{\partial}{\partial t})(\sigma, z_0) = \frac{R_{1331}}{g_\sigma}= -D(|\mu_0|^2)(z_0) \le 0.$$ The last curvature is calculated in the similar fashion and this completes the proof of the Theorem \[thm:g\]. We are particularly interested in the case when $\sigma \in T_g$ travels along a [[Weil-Petersson geodesic]{}]{} $\gamma$ near the infinity of the augmented [[Teichmüller space]{}]{}, in which case, at least one simple closed curve on the surface $(\Sigma,\sigma)$ is being pinched, as the result, the norm $|\mu|$ is unbounded. Therefore, we find: \[thm:de\] $Sup_{\sigma \in T_g} K(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}) = +\infty$, and $Inf_{\sigma \in T_g}K(\frac{\partial}{\partial x}, \frac{\partial}{\partial t}) = -\infty$. When $\sigma \in T_g$ is near the infinity of the augmented [[Teichmüller space]{}]{}, a short essential curve on $(\Sigma,\sigma)$ is being pinched, and therefore $Sup_{z \in (\Sigma,\sigma)}|\mu(z)| = +\infty$. Corollary \[thm:de\] is then the consequence of this and the Lemma \[thm:w\]. Naturally one hopes that $N_\sigma$ is negatively curved. Theorem 1.3 gives a negative answer to this. Therefore, as often seen in Riemannian geometry, it is natural to modify a given metric for better property. On the germ $N_\sigma$, based on the metric $H$ in , we consider the modified metric $H_f$ as follows: $$H_f = g_{\rho(w)}dwd\overline{w} + f(t)dt^2,$$ where $f(t) > 0$ and $f(0) =1$. Proceeding as in the proof of Theorem \[thm:g\], it is easy to show the following: The curvature $K(\frac{\partial}{\partial x}, \frac{\partial}{\partial y})$ of the [[Weil-Petersson geodesic]{}]{} $\gamma$ in the fiber directions, with respect to the metric $H_f$, at $t=0$, and $z=z_0$ is equal to $-1+|\mu_0|^2(z_0)$. Therefore $(N_\sigma, H_f)$ is not negatively curved. Minimality: the germ $N_\sigma$ ------------------------------- We now consider how the fiber hyperbolic surface $(\Sigma, g_\sigma |dz|^2)$ interacts with the germ $N_\sigma$. Naturally we study the [[second fundamental form]{}]{}. Since the $t$-direction is perpendicular to each surface, the [[second fundamental form]{}]{} $(h_{ij}(t))$ can be calculated as $$(h_{ij}(t))=\left( \begin{array}{ccc} g_\sigma e'(t)+ 2 Re \phi_0 & -2 Im\phi_0 \\ -2 Im\phi_0 &g_\sigma e'(t)- 2 Re \phi_0 \end{array} \right).$$ Here $e'(t)$ is the $t-$derivative of the energy density $e(t)$ of the family of [[harmonic map]{}]{}s $w(t)$. Since $e(0)=1$ and $e'(0)=0$, when evaluated at $t=0$, we have $$(h_{ij}(0))=\left( \begin{array}{ccc} 2 Re \phi_0 & -2 Im\phi_0 \\ -2 Im\phi_0 &- 2 Re \phi_0 \end{array} \right).$$ Now the principal curvatures of surface at time $t=0$ are the eigenvalues of the following matrix: $$(g_{\sigma})^{-1}(h_{ij}(0))= \frac{1}{g_\sigma} \left( \begin{array}{ccc} 2 Re \phi_0 & -2 Im\phi_0 \\ -2 Im\phi_0 &- 2 Re \phi_0 \end{array} \right).$$ Its two eigenvalues are now: $$\lambda = \pm2\frac{|\phi_0|}{g_\sigma} =\pm2 |\mu_0|.$$ Therefore we have proved: \[thm:minimal\] Each fiber hyperbolic surface $(S, g_\sigma dzd\bar{z})$ is a minimal surface of the germ $N_\sigma$. We note that the [[principal curvature]{}]{}s are unbounded if $\sigma$ is near the infinity of the augmented [[Teichmüller space]{}]{}. Minimality: surface bundle $N$ over $\gamma$ -------------------------------------------- Let $\gamma(t)$ be a [[Weil-Petersson geodesic]{}]{} arc, for $0 \leq t \leq T$, parametrized by its arc length. We denote the collection of germs $N_\sigma$ for $\sigma \in \gamma(t)$ by $N$, and it is clear that the three-manifold $N = \Sigma \times [0,T]$ is a surface bundle over $\gamma(t)$. In this subsection, we prove the Theorem 1.4. \[proof of Theorem 1.4\] We now equip $N$ with a Riemannian structure. We wish to equip $N$ with a metric of the form $$\label{M} g_{\gamma(t)}dw(t)d\bar{w}(t) + dt^2$$ where $g_{\gamma(t)}$ is the [[hyperbolic metric]{}]{} in the conformal class of $\gamma(t)$. Let ${\mathcal{M}}_{-1}$ be the space of [[hyperbolic metric]{}]{}s on a topological surface $S$, and recall that [[Teichmüller space]{}]{} is the space of [[hyperbolic metric]{}]{}s on $S$, up to orientation-preserving diffeomorphisms in the homotopy class of the identity: $T_g = {\mathcal{M}}_{-1}/D_0$, where $D_0$ is the identity component of the diffeomorphism group. Let $[g_{\gamma(t)}]$ denote the fiber over $g_{\gamma(t)} \in T_g$. At any $g_1 \in [g_{\gamma(t)}]$, the [[tangent space]{}]{} $T_{g_1}{\mathcal{M}}_{-1}$ is identified as $Sym(0,2)$, the space of symmetric $(0,2)$-tensors on $S$. Let $h_1 \in Sym(0,2)$ be the symmetric $(0,2)$-tensor which induces a deformation of $g_1$ preserving the scalar curvature. It is divergence-free and traceless, and moreover the deformation is smoothly dependent in $t$ by Theorem A of [@FM75], or Theorems 2.4.2 of [@Tro92]. In our case, $g_1$ is the hyperbolic metric $\sigma\in [g_{\gamma(t)}]$ and $h_1$ is the holomorphic quadratic differential in the tangent direction $\phi_0 dz^2$, and the smooth dependence results in a $C^\infty$ Riemannian metric on $N$ such that the germs at each fiber is identical to $N_\sigma$. Note that the metric agrees with metric associated to $\rho(t)$ up to second order ([@Ahl61]), and now parts (1) and (2) follow from Theorem 4.3, and part (3) follows from the Theorem 1.3. By a theorem of Sullivan ([@Sul79]), any compact Riemannian manifold with a taut foliation admits a Riemannian metric such that each leaf of the foliation is a minimal surface. Theorem 1.4 shows that the $3$-manifold associated to a closed [[Weil-Petersson geodesic]{}]{} is such an example: If $\gamma$ is a closed [[Weil-Petersson geodesic]{}]{} loop in moduli space $\mathcal{M}(S)$, then the metric on the associated [[3-manifold]{}]{} $N$ is a Sullivan metric. The [[Weil-Petersson geodesic]{}]{} loop lifts to a path of hyperbolic metrics $\gamma(t)$ in ${\mathcal{M}}_{-1}$, as in the proof of Theorem 1.4, such that the hyperbolic surfaces $\Sigma_0=\Sigma\times\{0\}$ and $\Sigma_1=\Sigma\times \{T\}$ are isometric by an orientation-preserving mapping class $\psi:\Sigma_0\to \Sigma_1$. We glue the boundary of $\Sigma \times [0,T]$ equipped with the metric by $\psi$ to obtain a surface bundle $N$. Since the normal direction is $\partial/ \partial t$, the normal vectors $\partial/\partial t$ on $\Sigma_0$ match up with the normal vectors on $\Sigma_1$, and the resulting metric on $N$ is smooth. The property of all the fibers being minimal follows from Theorem 1.4. 1. The minimality of the germs does not require $\gamma \subset T_g$ to be a geodesic. It is essentially due to the property that deformations $h_1$ can be chosen as traceless. 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--- abstract: 'The purpose of this paper is to determine quantum master and filter equations for systems coupled to fields in certain non-classical continuous-mode states. Specifically, we consider two types of field states (i) single photon states, and (ii) superpositions of coherent states. The system and field are described using a quantum stochastic unitary model. Master equations are derived from this model and are given in terms of systems of coupled equations. The output field carries information about the system, and is continuously monitored. The quantum filters are determined with the aid of an embedding of the system into a larger non-Markovian system, and are given by a system of coupled stochastic differential equations.' author: - 'John E. Gough[^1]' - 'Matthew R. James[^2]' - 'Hendra I. Nurdin[^3]' title: 'Quantum Filtering for Systems Driven by Fields in Single Photon States and Superposition of Coherent States using Non-Markovian Embeddings[^4]' --- Keywords: quantum filtering, continuous-mode single photon states, continuous-mode superpositions of coherent states, quantum stochastic processes Introduction ============ In recent years single photon states of light and superpositions of coherent states have become increasingly important due to applications in quantum technology, in particular, quantum computing and quantum information systems, [@MHNWGBRH97], [@GM08], [@KLM01], [@GRTZ02], [@VWSRVSKW06]. For instance, the light may interact with a system, say an atom, quantum dot, or cavity, and this system may be used as a quantum memory, [@MHNWGBRH97], or to control the pulse shape of the single photon state [@GM08]. When light interacts with a quantum system, information about the system is contained in the scattered light. This information may be useful for monitoring the behavior of the system, or for controlling it. The topic of this paper concerns the extraction of information from the scattered light when the incoming light is placed in a single photon state $\vert \Psi \rangle = \vert 1_\xi \rangle$, or a superposition of coherent states $\vert \Psi \rangle = \sum_j \alpha_j \vert f_j \rangle$, as illustrated in Figure \[fig:filter-one-1\]. ![A system initialized in a state $\vert \eta \rangle$ coupled to a field in a state $\vert \Psi \rangle$ (single photon or superposition of coherent states). The output field is continuously monitored by homodyne detection (assumed perfect) to produce a classical measurement signal $Y(t)$. The output $Y(t)$ is filtered to produce estimates $\hat X(t)=\pi_t(X)$ of system operators $X$ at time $t$. []{data-label="fig:filter-one-1"}](images/system-filter-qip) The problem of extracting information from continuous measurement of the scattered light is a problem of *quantum filtering*, [@BB91], [@VPB92], [@VPB92a], [@HC93], [@WM93], [@AB03], [@BHJ07], [@WM10]. The current state of the art for quantum filtering considers incoming light in a vacuum or other Gaussian state, with quadrature or counting measurements. Both single photon states of light, and superpositions of coherent states of light, are highly non-classical, and are fundamentally different from Gaussian states. In view of the increasing importance of these non-Gaussian states of light, the purpose of this paper is to solve a quantum filtering problem for systems driven by fields in single photon states and superpositions of coherent states. In the case of single photon fields, the master equation describing unconditional dynamics was shown to be a system of coupled equations in [@GEPZ98], a feature of non-Markovian character. Markovian embeddings were used in [@HPB04] to derive quantum trajectory equations (quantum filtering equations) for a class of non-Markovian master equations. In recent work, the authors have shown how to construct ancilla systems to combine with the system of interest to form a Markovian extended system driven by vacuum from which quantum filtering results may be obtained for single photon states and superpositions of coherent states from the standard filter for the extended system, [@GJN11a], [@GJNC11]. However, depending on the complexity of the non-classical state, it may possibly be difficult to determine suitable ancilla systems, and indeed the superposition case was not straightforward. In this paper we present an alternative approach to the embedding that also allows for the derivation of the quantum filter. The extended system forms a non-Markovian system, with the ancilla, system and field initialized in a superposition state. While standard filtering results do not apply, the quantum stochastic methods can nevertheless be applied to determine the quantum filters. In this way, we expand the range of methods that may be applied to derive quantum filters for non-classical states. The paper is organized as follows. In section \[sec:problem\] the idealized filtering problem to be solved in this paper is formulated. The continuous mode single photon states are defined in Section \[sec:photon-state\]. The master equation for the single photon field state is derived in Section \[sec:master\] using the model presented in Section \[sec:problem\]. This leads naturally to Section \[sec:the-matrix\], where the system is embedded in a larger model, inspired by the approach used in [@HPB04] but differing in the details. This extended system provides a compact and transparent description of the problem, and may readily be generalized to $n$-photon states, and indeed, multiple channels of $n$-photon states. The quantum filter for the extended system is presented in Section \[sec:matrix-filter\], with a derivation extending the reference method appearing in Section \[sec:extended-filter\]. The filtering results for the extended system are used to find the filtering equations for the original problem involving a single photon field in Section \[sec:single-photon-filter\]. The superposition of coherent field states is defined in Section \[sec:cat-super\], and a suitable embedded system for this case is described in Section \[sec:cat-embed\]. The corresponding master and filtering equations are presented in Sections \[sec:cat-master\] and \[sec:cat-super-filter\], respectively. Some concluding remarks are made in Section \[sec:conclusion\]. In this paper we are not concerned with technical issues concerning domains of unbounded operators and related matters, and indeed, we assume that the system operators are bounded, and that all quantum stochastic integrals are well-defined in the sense of Hudson-Parthasarathy, [@HP84]. *Notation:* We use the standard Dirac notation $\vert \psi \rangle$ to denote state vectors (vectors in a Hilbert space) [@EM98], [@AFP09]. The superscript $^\ast$ indicates Hilbert space adjoint or complex conjugate. The inner product of state vectors $\vert \psi_1 \rangle $ and $\vert \psi_2 \rangle$ is denoted $\langle \psi_1 \vert \psi_2 \rangle$. The expected value of an operator $X$ when the system is in state $\vert \psi \rangle$ is denoted $\mathbb{E}_\psi[ X ] = \langle \psi \vert X \vert \psi \rangle$. For operators $A$ and $B$ we write $ \langle A, B \rangle = \mathrm{tr}[ A^\ast B ]. $ Problem Formulation {#sec:problem} =================== We consider a quantum system $S$ coupled to a quantum field $B$, as shown in Figure \[fig:filter-one-1\]. The field $B$ has two components, the input field $B_{in}$ and the output field, $B_{out}$. In this paper we consider two non-classical cases for the state $\vert \Psi \rangle$ of the input field (i) a single photon state $\vert \Psi \rangle = \vert 1_\xi \rangle$, where $\xi$ is a complex valued function such that $\int_0^\infty \vert \xi(s) \vert^2 ds =1$ (representing the wave packet shape), or (ii) a superposition of coherent states $\vert \Psi \rangle = \sum_j \alpha_j \vert f_j \rangle$, where $\vert f_j \rangle$ are coherent states and the complex numbers $\alpha_j$ ($j=1,\ldots,n$) are normalized weights. As illustrated in Figure \[fig:filter-one-1\], the field interacts with the quantum system $S$, and the results of this interaction provide information about the system that may be obtained through continuous measurement of an observable $Y(t)$ of the output field $B_{out}(t)$. The filtering problem of interest in this paper is to determine the conditional state from which estimates $\hat X(t)$ of system operators $X$ may be determined at time $t$ based on knowledge of the observables $\{ Y(s)$, $0 \leq s \leq t \}$. In what follows the system $S$ is assumed to be defined on a Hilbert space $\mathfrak{H}_S$, with an initial state denoted $\vert \eta \rangle \in \mathfrak{H}_S$. The input field $B_{in}$ is described in terms of annihilation $B(\xi)$ and creation $B^\ast(\xi)$ operators defined on a Fock space $\mathfrak{F}$, [@KRP92 Chapter II], [@BHJ07 Section 4]. Quantum expectation will be denoted by the symbol $\mathbb{E}$, and when we wish to display the underlying state, we employ subscripts; for example, $\mathbb{E}_{\eta\Psi}$ denotes quantum expectation with respect to the state $\vert \eta \rangle \otimes \vert \Psi \rangle$. The dynamics of the system will be described using the quantum stochastic calculus, [@HP84], [@GC85], [@KRP92], [@GZ00], [@BHJ07]. Quantum stochastic integrals are defined in terms of fundamental field operators $B(t)$, $B^\ast(t)$ and $\Lambda(t)$, [@KRP92 Chapter II], [@BHJ07 Section 4].[^5] The non-zero Ito products for the field operators are $$dB(t) dB^\ast(t) = dt, \ \ dB(t) d\Lambda(t) = dB(t), \ \ d\Lambda(t) d\Lambda(t) = d\Lambda(t), \ \ d\Lambda(t) dB^\ast(t)=dB^\ast(t). \label{eq:Ito-sp}$$ The dynamics of the composite system is described by a unitary $U(t)$ solving the Schrödinger equation, or quantum stochastic differential equation (QSDE), $$dU(t) = \{ (S-I)d\Lambda(t) + L dB^\ast (t)- L^\ast S dB(t) - (\frac{1}{2} L^\ast L +iH )dt \} U(t), \label{eq:unitary}$$ with initial condition $U(0)=I$. Here, $H$ is a fixed self-adjoint operator representing the free Hamiltonian of the system, and $L$ and $S$ are system operators determining the coupling of the system to the field, with $S$ unitary. In this paper, for simplicity we assume that the parameters $S,L,H$ are bounded operators on the system Hilbert space $\mathfrak{H}_S$. However, we remark under some suitable additional conditions the results and equations obtained in this paper should also be extendable to some special classes of QSDEs with unbounded parameters, exploiting the results in [@Fagno90; @FW03]. A system operator $X$ at time $t$ is given in the Heisenberg picture by $X(t)=j_{t}( X) =U( t) ^{\ast } ( X\otimes I ) U( t) $ and it follows from the quantum Ito calculus that $$\begin{aligned} dj_{t}( X) &=&j_{t}(S^{\ast }XS-X) d\Lambda ( t) +j_{t}(S^{\ast }[X,L]) dB(t) ^{\ast } \notag \\ && +j_{t}([L^{\ast },X]S) dB( t) +j_{t}(\mathcal{L}(X)) dt , \label{eq:X-dyn}\end{aligned}$$ where $$\mathcal{L}(X)=\frac{1}{2}L^{\ast }[X,L]+\frac{1}{2}[L^{\ast },X]L-i[ X,H] .$$ The map $X\mapsto \mathcal{L} (X)$ is known as the *Lindblad generator*, while the quartet of maps $X \mapsto \mathcal{L} (X), S^\ast XS -X, \, S^\ast [X,L ], \, [L^\ast , X] S$ are known as *Evans-Hudson maps*. The output field is defined by $B_{out}(t) = U(t)^\ast B(t) U(t)$.[^6] In this paper we consider the output field observable $Y(t)$ defined by $$Y(t) = U(t)^\ast Z(t) U(t) , \label{eq:Y-out}$$ where $$Z(t)= B(t)+ B^\ast(t), \label{eq:Z-def}$$ is a quadrature observable of the input field (the counting case $Z(t)=\Lambda(t)$ is discussed briefly in Section \[sec:conclusion\]). Note that both $Z(t)$ and $Y(t)$ are self-adjoint and self-commutative: $[Z(t), Z(s)]=0$ and $[Y(t), Y(s)]=0$. We write $\mathscr{Z}_t$ and $\mathscr{Y}_t$ for the subspaces of commuting operators generated by the observables $Z(s)$, $Y(s)$, $0\leq s \leq t$, respectively.[^7] They are related by the unitary rotation $\mathscr{Y}_t = U(t)^\ast \mathscr{Z}_t U(t)$. Physically, $Y(t)$ may represent the integrated photocurrent arising in an idealized (perfect) homodyne photodetection scheme, as in Figure \[fig:filter-one-1\]. For further information on homodyne detection, we refer the reader to the literature; for example, [@BR04], [@AB03], [@WM10]. The primary goal of this paper is to determine the *quantum filter* for the quantum conditional expectation (see, e.g. [@BHJ07 Definition 3.13]) $$\hat X(t) = \mathbb{E}_{\eta\Psi}[ X(t) \, \vert \, \mathscr{Y}_t ] . \label{eq:cond-exp}$$ This conditional expectation is well defined, since $X(t)$ commutes with the subspace $\mathscr{Y}_t $ (non-demolition condition). The conditional estimate $\hat X(t)$ is affiliated to $\mathscr{Y}_t$ (written in abbreviated fashion as $\hat X(t) \in \mathscr{Y}_t$) and is characterized by the requirement that $$\mathbb{E}_{\eta\Psi} [ \hat X(t) K ] = \mathbb{E}_{\eta\Psi} [ X(t) K ] \label{eq:c-exp-def}$$ for all $K \in \mathscr{Y}_t$. Single Photon Input Fields {#sec:photon} ========================== Single Photon Fields States {#sec:photon-state} --------------------------- In this section we consider the continuous-mode single photon state $\vert \Psi \rangle = \vert 1_\xi \rangle$ defined by [@RL00 sec. 6.3], [@GM08 eq. (9)] $$\vert 1_\xi \rangle = B^\ast(\xi) \vert 0 \rangle, \label{eq:xi-create}$$ where $\xi$ is a complex valued function such that $\int_0^\infty \vert \xi(s) \vert^2 ds =1$, and $\vert 0 \rangle$ is the vacuum state of the field. Expression (\[eq:xi-create\]) says that the single photon wavepacket with temporal shape $\xi$ is created from the vacuum using the field operator $B^\ast(\xi)$. The Hilbert space for the composite system is $$\mathfrak{H} = \mathfrak{H}_S \otimes \mathfrak{F}= \mathfrak{H}_S \otimes \mathfrak{F}_{t]} \otimes \mathfrak{F}_{(t},$$ where here we have exhibited the continuous temporal tensor product decomposition of the Fock space $\mathfrak{F}=\mathfrak{F}_{t]} \otimes \mathfrak{F}_{(t}$ into past and future components, which is of basic importance in what follows. Write $$\mathbb{E}_{11} [ X \otimes F ] = \langle \eta 1_\xi \vert (X \otimes F) \vert \eta 1_\xi \rangle = \langle \eta \vert X \vert \eta \rangle \langle 1_\xi \vert F \vert 1_\xi \rangle$$ for the expectation with respect to the product state $\vert \eta 1_\xi \rangle$, where the field is in the single photon state. Here and in what follows $X$ is a bounded system operator acting on $\mathfrak{H}_S$, and $F$ is a field operator acting on the Fock space $\mathfrak{F}$. Similarly, we may define the expectation when the field is in the vacuum state, $$\mathbb{E}_{00} [ X \otimes F ] = \langle \eta 0 \vert (X \otimes F) \vert \eta 0 \rangle = \langle \eta \vert X \vert \eta \rangle \langle 0 \vert F \vert 0 \rangle .$$ We will also have need for the cross-expectations $$\begin{aligned} \mathbb{E}_{10}[ X \otimes F ] = \langle \eta 1_\xi \vert (X \otimes F) \vert \eta 0 \rangle, \ \text{and} \ \mathbb{E}_{01}[ X \otimes F ] = \langle \eta 0 \vert (X \otimes F) \vert \eta 1_\xi \rangle.\end{aligned}$$ A crucial difference between the single photon state and the vacuum state is that the later state factorizes $\vert 0 \rangle =\vert 0_{t]} \rangle \otimes \vert 0_{(t} \rangle $ with respect to the temporal factorization $\mathfrak{F}=\mathfrak{F}_{t]} \otimes \mathfrak{F}_{(t}$ of the Fock space, with $\vert 0_{t]} \rangle \in \mathfrak{F}_{t]} $ and $\vert 0_{(t} \rangle \in \mathfrak{F}_{(t}$, while the former does not. Rather, we have $$\vert 1_\xi \rangle = B^\ast(\xi) \vert 0 \rangle = \vert {1_\xi}_{t]} \rangle \otimes \vert 0_{(t} \rangle + \vert 0_{t]} \rangle \otimes \vert {1_\xi}_{(t} \rangle , \label{eq:factor-additive-1}$$ where $$\vert {1_\xi}_{t]} \rangle = B^{-\ast}_t(\xi) \vert 0_{t]} \rangle, \ \text{and} \ \vert {1_\xi}_{(t} \rangle = B^{+\ast}_t(\xi) \vert 0_{(t} \rangle ,$$ and $$B^-_t(\xi) = B(\xi \chi_{[0,t]}), \ \ B^+_t(\xi) = B(\xi \chi_{(t,\infty]}), \ \ B(\xi) = B^-_t(\xi) + B^+_t(\xi) . \label{eq:B-xi-decomp-1}$$ Here, $\chi_{[0,t]}$ is the indicator function for the time interval $[0,t]$. Note that while $\vert 1_\xi \rangle$ has unit norm, we have $$\parallel \vert {1_\xi}_{t]} \rangle \parallel^2 = \int_0^t \vert \xi(s) \vert^2 ds, \ \text{and} \ \parallel \vert {1_\xi}_{(t} \rangle \parallel^2 = \int_t^\infty \vert \xi(s) \vert^2 ds .$$ A consequence of the additive decomposition (\[eq:factor-additive-1\]) and the definitions (\[eq:B-xi-decomp-1\]) is the following. Let $K(t)$ be a bounded operator acting on the full Hilbert space $\mathfrak{H}$ that is adapted, i.e. $K(t)$ acts trivially on $\mathfrak{F}_{(t}$, the field in the future. Then the expectation with respect to the single photon field may be expressed in terms of the vacuum state as follows: $$\begin{aligned} \mathbb{E}_{11} [ K(t) ] &=& \mathbb{E}_{00} [ B^-_t(\xi) K(t) B^{-\ast}_t(\xi) + r(t) K(t) ] \label{eq:xi-phi}\end{aligned}$$ where $r(t) = \int_t^\infty \vert \xi(s) \vert^2 ds$. Master Equation {#sec:master} --------------- Before deriving the quantum filter, we work out dynamical equations for the unconditioned single photon expectation, [@GEPZ98]. To assist us in evaluating this expectation, we make use of the following lemma. \[lemma:expectation-basic\] Let $K(t)$ be a bounded quantum stochastic process defined by $$K(t) = \int_0^t M_0(s) ds + \int_0^t M_- (s) dB(s) + \int_0^t M_+(s) dB^\ast(s) + \int_0^t M_1(s) d\Lambda(s), \label{eq:Kt-def}$$ where $M_0$, $M_\pm$ and $M_1$ are bounded and adapted. Then we have $$\begin{aligned} \mathbb{E}_{11}[ K(t) ] &=& \mathbb{E}_{11}[ \int_0^t M_0(s) ds ] + \mathbb{E}_{10}[ \int_0^t M_- (s) \xi(s) ds ] \notag \\ && + \mathbb{E}_{01}[ \int_0^t M_+(s) \xi^\ast(s) ds ] + \mathbb{E}_{00}[ \int_0^t M_1 (s) \vert \xi(s) \vert^2 ds ] , \label{eq:Kt-11} \\ \mathbb{E}_{10}[ K(t) ] &=& \mathbb{E}_{10}[ \int_0^t M_0(s) ds ] + \mathbb{E}_{00}[ \int_0^t M_+ (s) \xi^\ast(s) ds ] , \label{eq:Kt-10} \\ \mathbb{E}_{01}[ K(t) ] &=& \mathbb{E}_{01}[ \int_0^t M_0(s) ds ] + \mathbb{E}_{00}[ \int_0^t M_- (s) \xi(s) ds ] , \label{eq:Kt-01} \\ \mathbb{E}_{00}[ K(t) ] &=& \mathbb{E}_{00}[ \int_0^t M_0(s) ds ] . \label{eq:Kt-00}\end{aligned}$$ Using (\[eq:xi-phi\]), the expressions $B^-_t(\xi) = \int_0^t \xi^\ast(s) dB(s)$, $B^{-\ast}_t(\xi) = \int_0^t \xi(s) dB^\ast(s)$, and the Ito rule we have $$\begin{aligned} \mathbb{E}_{11}[ d K(t) ] &=& \mathbb{E}_{00} [ d ( B^-_t (\xi) K(t) B^{-\ast}_t (\xi) + r(t) K(t) ) ] \notag \\ &=& \mathbb{E}_{00} [ B^-_t (\xi) M_0(t) B^{-\ast}_t (\xi) + r(t) M_0(t) \notag \\ && + M_+(t) B^{-\ast}_t(\xi) \xi^\ast(t) + B^-_t (\xi) M_-(t) \xi(t) +M_1(t) |\xi(t)|^2] dt \notag \\ &=& \mathbb{E}_{11} [ M_0(t) ] dt + \mathbb{E}_{00} [ M_+(t) B^{\ast} (\xi) \xi^\ast(t) + B(\xi) M_-(t) \xi(t) + M_1 |\xi(t)|^2] dt .\end{aligned}$$ This last line is justified since $M_\pm$ are adapted and $\mathbb{E}_{00}[ B^+_t(\xi) ] = 0$. That is, $$\mathbb{E}_{11}[ dK(t) ] = \mathbb{E}_{11} [ M_0(t) ] dt + \mathbb{E}_{01}[ M_+(t) ] \xi^\ast(t) dt + \mathbb{E}_{10} [ M_-(t) ] \xi(t) dt + \mathbb{E}_{00}[ \int_0^t M_1 (s) \vert \xi(s) \vert^2 ds ] .$$ This proves (\[eq:Kt-11\]). The remaining expressions are proven in a similar manner. [$\Box$]{} We will first express the master equation in Heisenberg form using the expectations $$\mu^{jk}_t(X) = \mathbb{E}_{jk}[ X(t) ].$$ Note that for all $t \geq 0$ we have $$\mu_t^{00}(I)= 1 = \mu^{11}_t(I) , \ \ \ \mu^{01}_t(I)=0=\mu^{10}_t(I).$$ \[thm:master\] The master equation in Heisenberg form for the system when the field is in the single photon state $\vert 1_\xi \rangle$ is given by the system of equations $$\begin{aligned} \dot{\mu}^{11}_t (X) &=& \mu^{11}_t(\mathcal{L}(X)) + \mu^{01}_t( S^\ast [X,L] ) \xi^\ast(t) + \mu^{10}_t( [L^\ast, X] S ) \xi(t) \notag \\ && + \mu^{00}_t( S^\ast X S - S) \vert \xi(t) \vert^2, \label{eq:rho-dyn-a-11} \\ \dot{\mu}^{10}_t (X) &=& \mu^{10}_t(\mathcal{L}(X)) + \mu^{00}_t( S^\ast [X, L] ) \xi^\ast(t) , \label{eq:rho-dyn-a-10} \\ \dot{\mu}^{01}_t (X) &=& \mu^{01}_t(\mathcal{L}(X)) + \mu^{00}_t( [L^\ast, X] S ) \xi(t) , \label{eq:rho-dyn-a-01} \\ \dot{\mu}^{00}_t (X) &=& \mu^{00}_t(\mathcal{L}(X)) . \label{eq:rho-dyn-a-00}\end{aligned}$$ The initial conditions are $$\mu^{11}_0(X)= \mu^{00}_0(X)= \langle \eta, X \eta \rangle, \ \ \mu^{10}_0(X)= \mu^{01}_0(X)=0.$$ Equations (\[eq:rho-dyn-a-11\])- (\[eq:rho-dyn-a-00\]) are obtained by applying Lemma \[lemma:expectation-basic\] to the Heisenberg equation (\[eq:X-dyn\]). [$\Box$]{} It is apparent from Theorem \[thm:master\] that the single photon expectation $\mu^{11}_t(X) = \mathbb{E}_{11}[ X(t)]$ cannot be determined by a single differential equation, and that instead a system of coupled equations is required, equation (\[eq:rho-dyn-a-11\])-(\[eq:rho-dyn-a-00\]). Note that the unitary matrix $S$ appearing in the Schrödinger equation (\[eq:unitary\]) does appear in the single photon master equations (\[eq:rho-dyn-a-11\])-(\[eq:rho-dyn-a-00\]), in contrast to the vacuum case (which corresponds to (\[eq:rho-dyn-a-00\])). Embedding {#sec:the-matrix} --------- In this section we construct a suitable embedding for the system and single photon field, and show how the system of master equations from Section \[sec:master\] can be compactly represented as a single equation for a larger system. This embedding will be used in subsequent sections to derive the quantum filter. We should emphasize, however, that our embedding is not the same as that used in [@HPB04], [@GJN11a], [@GJNC11]. The embedding is illustrated in Figure \[fig:extend-1\]. ![System embedded in the extended system. While the analysis does not employ any coupling between the system and ancilla two-level system, the ancilla, system and field are assumed to be initialized in a superposition state $\vert \Sigma \rangle$ defined in equation (\[eq:super\]).[]{data-label="fig:extend-1"}](images/extend-2) Recall that the system and field are defined on a Hilbert space $\mathfrak{H} = \mathfrak{H}_S \otimes \mathfrak{F}$. We define an extended space $$\tilde {\mathfrak{H}}= \mathbb{C}^2 \otimes \mathfrak{H} = \mathfrak{H} \oplus \mathfrak{H}$$ which includes the system, field and an ancilla two-level system. Let $\vert e_0 \rangle$ and $\vert e_1 \rangle$ be an orthonormal basis for $\mathbb{C}^2$, $$\vert e_0 \rangle = \left[ \begin{array}{c} 0 \\ 1 \end{array} \right], \ \ \vert e_1 \rangle = \left[ \begin{array}{c} 1 \\ 0 \end{array} \right],$$ and let $A$ be an operator acting on $\mathbb{C}^2$, i.e. a complex $2 \times 2$ matrix $$A = \left[ \begin{array}{cc} a_{11} & a_{10} \\ a_{01} & a_{00} \end{array} \right].$$ It may be helpful to think of operators $A \otimes X \otimes F$ on the extended space $\tilde {\mathfrak{H}}$ represented in the Kronecker product form $$A \otimes (X \otimes F) = \left[ \begin{array}{cc} a_{11} (X \otimes F) & a_{10} (X \otimes F) \\ a_{01} (X \otimes F) & a_{00} (X \otimes F) \end{array} \right].$$ We allow the extended system to evolve unitarily according to $I \otimes U(t) $, where $U(t)$ is the unitary operator for the system and field, given by the Schrödinger equation (\[eq:unitary\]). Note in particular that the system is not coupled to the ancilla $\mathbb{C}^2$, and observables of this two-level system are static. We initialize the extended system in the superposition state $$\vert \Sigma \rangle = \alpha_1 \vert e_1 \eta 1_\xi \rangle + \alpha_0 \vert e_0 \eta 0 \rangle, \label{eq:super}$$ where $\vert \alpha_0 \vert^2 + \vert \alpha_1 \vert^2 =1$. This state evolves according to $$\vert \Sigma(t) \rangle = (I \otimes U(t)) \vert \Sigma \rangle.$$ For notational convenience we write $$w_{jk} = \alpha_j^\ast \alpha_k \label{w_jk}$$ and note that $w = \sum_{jk} w_{jk} \vert e_j \rangle \langle e_k \vert$ is a density matrix for $\mathbb{C}^2 $. The expectation with respect to the superposition state $\vert \Sigma \rangle$ is given by $$\tilde \mu_t( A \otimes X) = \mathbb{E}_{\psi} [ A \otimes X(t) ] = \langle \Sigma \vert (A \otimes X(t)) \vert \Sigma \rangle = \sum_{jk} w_{jk} a_{jk} \mu^{jk}_t(X). \label{eq:master-mu-rep1}$$ This expectation is correctly normalized, $\mu_t(I \otimes I)=1$, and the expectations $\mu^{jk}_t(X)$ defined in Section \[sec:master\] are scaled components of $\tilde \mu_t(A\otimes X)$: $$\mu^{jk}_t(X) = \frac{ \tilde \mu_t( \vert e_j \rangle \langle e_k \vert \otimes X)}{ w_{jk} }, \label{eq:master-mu-rep2}$$ for $w_{jk} \neq 0$, otherwise it can be set to, say, 0. We also have $$\mu^{jk}_t(X) = \frac{ w_{11} \tilde\mu_t( \vert e_j \rangle \langle e_k \vert \otimes X) }{ w_{jk} \tilde \mu_t( \vert e_1 \rangle \langle e_1 \vert \otimes I) } . \label{eqmui-jk-bayes}$$ Note that in the extended space the Schrödinger and Heisenberg pictures are related by $$\mathbb{E}_{\Sigma(t) } [ A \otimes X \otimes F ] = \mathbb{E}_{\Sigma} [ A \otimes U^\ast(t) (X \otimes F) U(t) ] .$$ In order to derive the equation for expectations in the extended system, we need the following lemma, which follows from Lemma \[lemma:expectation-basic\], and makes use of the matrices $$\sigma_+ = \vert e_1 \rangle \langle e_0 \vert = \left[ \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right], \ \ \sigma_- = \vert e_0 \rangle \langle e_1 \vert = \left[ \begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array} \right].$$ \[lemma:expectation-basic-extended\] Assume $\alpha_0\neq 0$, and let $M(t)$ be bounded and adapted. Then $$\begin{aligned} \mathbb{E}_\Sigma [ \int_0^t A \otimes M(s) dB(s) ] &=& \nu \mathbb{E}_\Sigma [ \int_0^t ( A \sigma_+)\otimes M(s) \xi(s) ds ], \\ \mathbb{E}_\Sigma [ \int_0^t A \otimes M (s) dB^\ast(s) ] &=& \nu^\ast \mathbb{E}_\Sigma [ \int_0^t (\sigma_-A )\otimes M(s) \xi^\ast(s) ds], \\ \mathbb{E}_\Sigma [ \int_0^t A \otimes M(s) d\Lambda(s) ] &=& \vert \nu \vert^2 \mathbb{E}_\Sigma [ \int_0^t ( \sigma_- A \sigma_+ )\otimes M(s) \vert \xi(s) \vert^2 ds ],\end{aligned}$$ where $$\nu = \frac{\alpha_1}{\alpha_0}.$$ In Lemma \[lemma:expectation-basic-extended\], expectations of stochastic integrals with respect to the superposition state $\vert \Sigma \rangle$ are expressed in terms of expectations of non-stochastic integrals again with respect to $\vert \Sigma \rangle$ with the aid of the matrices $\sigma_\pm$ acting on the ancilla system $\mathbb{C}^2$. The action of the field annihilation, creation and gauge operators is therefore captured algebraically and all expectations in these relations are with respect to the same state. We now have \[thm:master-matrix\] Assume $\alpha_0\neq 0$. Then the expectation $\tilde \mu_t(A \otimes X)$ (defined by (\[eq:master-mu-rep1\])) evolves according to $$\dot {\tilde \mu}_t(A \otimes X) = \tilde \mu_t (\mathcal{G}_t(A \otimes X) ), \label{eq:matrix-master}$$ where $$\begin{aligned} \mathcal{G}_t(A \otimes X) &=& A \otimes \mathcal{L}(X) + ( A\sigma_+) \otimes [L^\ast, X] S \nu \xi(t) + (\sigma_- A ) \otimes S^\ast [X,L] \nu^\ast \xi^\ast(t) \notag \\ && + ( \sigma_- A \sigma_+ ) \otimes (S^\ast X S - X)) \vert \nu \xi(t) \vert^2.\end{aligned}$$ The reader may easily verify that the system of master equations (\[eq:rho-dyn-a-11\])-(\[eq:rho-dyn-a-00\]) for $\mu^{jk}_t(X)$, $j,k=1,0$, follows from equation (\[eq:matrix-master\]) by setting $A=\vert e_j \rangle \langle e_k \vert$. Quantum Filter for the Extended System {#sec:matrix-filter} -------------------------------------- The extended system provides a convenient framework for quantum filtering, since all expectations can be expressed in terms of the superposition state $\vert \Sigma \rangle$. Our immediate goal in this section is to determine the equation for the quantum conditional expectation $$\tilde \pi_t( A \otimes X) = \mathbb{E}_{\Sigma}[ A \otimes X(t) \, \vert \, I \otimes \mathscr{Y}_t] , \label{eq:matrix-c-exp-def}$$ and in Section \[sec:single-photon-filter\] we will explain how the quantum filter for the single photon field may be obtained from this equation. The continuously monitored field observable that corresponds to the conditional expectation (\[eq:matrix-c-exp-def\]) is $I \otimes Y(t)$, and from (\[eq:Y-out\]) we have the corresponding output equation for the extended system: $$d(I \otimes Y(t)) = I \otimes (L(t)+L^\ast(t)) dt + I \otimes (S(t) dB(t) + S^\ast(t) dB^\ast(t) ) .$$ In what follows we will make use of the following lemma concerning expectations of the process $$V(t) = \int_0^t ( S(s) dB(s) + S^\ast(s) dB^\ast(s) )$$ with respect to the single photon state. \[lemma:Z-compensated-mtg\] For any $K \in \mathscr{Y}_s$, we have $$\begin{aligned} \mathbb{E}_{11}[ (V(t) - V(s)) K ] &=& \mathbb{E}_{10}[ \int_s^t S(r) \xi(r)dr K] + \mathbb{E}_{01}[ \int_s^t S^\ast(r) \xi^\ast(r)dr K] . \label{eq:V-compensated} \end{aligned}$$ Equation (\[eq:V-compensated\]) is obtained using the additive decomposition (\[eq:factor-additive-1\]) and Lemma \[lemma:expectation-basic\]. [$\Box$]{} Note that an operator $K$ in the unital commutative algebra $I \otimes \mathscr{Y}_t$ has the form $K=I \otimes \tilde K$, where $\tilde K \in \mathscr{Y}_t$. By the spectral theorem, [@BHJ07 Theorem 3.3], we may identify $K$ and $\tilde K$, both of which are equivalent to a classical stochastic process $K_t(s)$, $0 \leq s \leq t$. In the remainder of this paper, we use these identifications without further comment. The quantum conditional expectation $\tilde \pi_t( A \otimes X) \in I \otimes \mathscr{Y}_t$ is well defined because $A \otimes X(t)$ is in the commutant $I \otimes \mathscr{Y}_t^{\prime}$ of the algebra $I \otimes \mathscr{Y}_t$, and is characterized by the requirement that $$\mathbb{E}_{\Sigma} [ \tilde \pi_t(A \otimes X) I \otimes K ] = \mathbb{E}_{\Sigma} [ (A \otimes X(t))( I \otimes K) ] \label{eq:c-exp-def-matrix}$$ for all $K \in \mathscr{Y}_t$, see, e.g. [@BHJ07 Definition 3.13]. \[thm:matrix-filter\] Assume $\alpha_0 \neq 0$. The conditional expectation $\tilde \pi_t(A \otimes X)$ defined by (\[eq:matrix-c-exp-def\]) for the extended system satisfies $$\begin{aligned} d\tilde \pi_t(A\otimes X) &=&\tilde \pi_t( \mathcal{G}_t(A \otimes X) ) dt + \mathcal{H}_t(A\otimes X) dW(t), \label{pi_filter}\end{aligned}$$ where $$\begin{aligned} \mathcal{H}_t(A \otimes X) &=& \tilde \pi_t( A \otimes (XL+L^\ast X) ) - \tilde \pi_t( A \otimes X) \pi_t( I \otimes (L+L^\ast) ) \notag \\ && + \tilde \pi_t( ( A \sigma_+) \otimes XS) \nu \xi(t) + \tilde \pi_t( ( \sigma_-A ) \otimes S^\ast X) \nu^\ast \xi^\ast(t) \notag \\ && - \tilde \pi_t( A \otimes X) \tilde \pi_t( ( \sigma_+ \otimes S) \nu \xi(t) +( \sigma_- \otimes S^\ast ) \nu^\ast \xi^\ast(t) ) \label{eq:matrix-filter}\end{aligned}$$ and $$dW(t) = dY(t) - \tilde \pi_t( I \otimes (L+L^\ast) + ( \sigma_+ \otimes S) \nu \xi(t) + (\sigma_- \otimes S^\ast) \nu^\ast \xi^\ast(t) ) dt . \label{eq:innovation-matrix}$$ The process $W(t)$ defined by (\[eq:innovation-matrix\]) is a $I \otimes \mathscr{Y}_t$ Wiener process with respect to $\vert \Sigma \rangle$ and is called the *innovations process*. We follow the characteristic function method [@HSM05], [@VPB93], [@VPB92], whereby we postulate that the filter has the form $$d \tilde \pi_t( A \otimes X) = \mathcal{F}_t(A \otimes X) dt + \mathcal{H}_t(A \otimes X) I\otimes dY(t) ,$$ where $\mathcal{F}_t$ and $\mathcal{H}_t$ are to be determined. Let $f$ be square integrable, and define a process $c_f$ by $$dc_f(t) = f(t) c_f(t) dY(t), \ \ c_f(0)=1.$$ Then $I \otimes c_f(t)$ is adapted to $I \otimes \mathscr{Y}_t$, and the defining relation (\[eq:c-exp-def-matrix\]) implies that $$\mathbb{E}_\Sigma [ A \otimes (X(t) c_f(t) ) ] = \mathbb{E}_\Sigma [ \tilde \pi_t( A \otimes X) I \otimes c_f(t) ) ]$$ holds for all $f$. By calculating the differentials of both sides, taking expectations and conditioning we obtain $$\begin{aligned} \mathbb{E}_\Sigma [ A \otimes (dX(t) c_f(t)) ] &=& \mathbb{E}_\Sigma [ (I \otimes c_f(t) ) \tilde \pi_t( \mathcal{G}(A \otimes X) ) \\ && + (I \otimes f(t) c_f(t) ) \{\tilde \pi_t( A \otimes (XL+L^\ast X) ) \notag \\ && + \tilde \pi_t( A \sigma_+ \otimes XS ) \nu \xi(t) + \tilde \pi_t( \sigma_-A \otimes S^\ast X) \nu^\ast \xi^\ast(t) \} ]dt \notag\end{aligned}$$ and $$\begin{aligned} && \mathbb{E}_\Sigma [ A \otimes (d\tilde \pi_t(A\otimes X) c_f(t)) ] \\ &=& \mathbb{E}_\Sigma [ (I \otimes c_f(t) \{ \mathcal{F}_t(A\otimes X) + \mathcal{H}_t(A\otimes X) \tilde \pi_t( I \otimes (L+L^\ast))\nonumber \\ && +\mathcal{H}_t(A\otimes X) \pi_t( ( \sigma_+ \otimes S) \nu \xi(t) + (\sigma_- \otimes S^\ast ) \nu^\ast \xi^\ast(t) ) \} \notag \\ && + (I \otimes f(t) c_f(t) ) \{ \tilde \pi_t( A \otimes X) \tilde \pi_t( I \otimes (L+L^\ast) ) + \mathcal{H}_t \notag \\ && + \tilde \pi_t( A \otimes X)\tilde \pi_t( \sigma_+ \otimes S \nu \xi(t) + \sigma_- \otimes S^\ast \nu^\ast \xi^\ast(t) ) \} ] dt. \notag\end{aligned}$$ Now equating coefficients of $c_f(t)$ and $f(t) c_f(t)$ we solve for $\mathcal{F}_t(A\otimes X)$ and $\mathcal{H}_t(A\otimes X)$ to obtain the filter equation. We now prove the martingale property $\mathbb{E}_{\Sigma} [ I \otimes ( W(t) - W(s) ) \, \vert \, I \otimes \mathscr{Y}_s ] = 0$, that is, $ \mathbb{E}_{\Sigma} [ I \otimes ( W(t) - W(s) ) (I \otimes K) ] =0 $ for all $K \in \mathscr{Y}_t$. Now $$\begin{aligned} && \mathbb{E}_{\Sigma} [ I \otimes ( W(t) - W(s) ) (I \otimes K) ] \notag \\ &=& \mathbb{E}_{\Sigma} [ \{ I \otimes (Y(t) - Y(s) ) \notag \\ && - \int_s^t \pi_r( I \otimes (L+L^\ast) + \sigma_+ \otimes S \xi(r) + \sigma_- \otimes S^\ast \xi^\ast(r) ) dr \} I \otimes K ] \notag \\ &=& \mathbb{E}_{\Sigma} [ \{ I \otimes ( Y(t) - Y(s) ) \notag \\ && - \int_s^t ( I \otimes (L(r)+L^\ast(r) ) + \sigma_+ \otimes S(r) \xi(r) + \sigma_- \otimes S^\ast(r) \xi^\ast(r) ) dr \} I \otimes K ] \notag \\ &=& \mathbb{E}_{\Sigma} [ \{ I \otimes (V(t) - V (s)) - \int_s^t ( \sigma_+ \otimes S(r) \nu \xi(r) + \sigma_- \otimes S^\ast(r) \nu^\ast \xi^\ast(r) ) dr \} I \otimes K ] =0. \notag\end{aligned}$$ To see that this last expression is zero, we make use of Lemma \[lemma:Z-compensated-mtg\], the multiplicative factorization of the vacuum state, the fact that $V(t)$ has zero expectation in the vacuum state, to find that $$\begin{aligned} \mathbb{E}_{\Sigma} [ I \otimes (V(t) - V(s)) I \otimes K ] &=& w_{11} \mathbb{E}_{11}[ (V(t) - V(s)) K ] + w_{00} \mathbb{E}_{00}[ (V(t) - V(s)) K ] \notag \\ &=& w_{11} ( \int_s^t \mathbb{E}_{10}[ S(r)K] \xi(r)dr + \int_s^t \mathbb{E}_{01}[ S^\ast(r) K] \xi^\ast(r)dr ) , \notag\end{aligned}$$ and $$\begin{aligned} && \mathbb{E}_{\Sigma} [ \int_s^t ( \sigma_+ \otimes S(r)K) \xi(t) + \sigma_- \otimes S^\ast (r) K \xi^\ast(t) ) dr ] \notag \\ &=&w_{11} ( \int_s^t \mathbb{E}_{10}[ S(r) K] \xi(r) dr + \int_s^t \mathbb{E}_{01}[ S^\ast (r) K] \xi^\ast(r) dr ) . \notag\end{aligned}$$ Finally, since $dW(t) dW(t)=dt$, Levy’s Theorem implies that $W(t)$ is a $\mathscr{Y}_t$ Wiener process. This completes the proof. [$\Box$]{} Notice the terms involving $\sigma_\pm$ in the filter (equation (\[eq:matrix-filter\])) and in the innovations process (equation (\[eq:innovation-matrix\])). These terms arise from expectations involving the single photon state. Note that due to the martingale property of the innovations process $W(t)$ we see that if we take the expected value of equation (\[eq:matrix-filter\]) we recover equation (\[eq:matrix-master\]), consistent with $\mathbb{E}_\Sigma [ \tilde \pi_t(A\otimes X) ] = \tilde \mu_t(A\otimes X) $ and the definition of conditional expectation. Single Photon Quantum Filter {#sec:single-photon-filter} ---------------------------- We return now to the main goal of the paper, namely the determination of the quantum filter for the conditional state when the field is in the single photon state, as stated in equation (\[eq:cond-exp\]). As discussed earlier, our strategy is to make use of the filtering results obtained in Section \[sec:matrix-filter\] for the extended system. Assume $\alpha_0 \neq 0$. Define the conditional quantities $\pi _{t}^{jk}(X)$ by $$\pi^{jk}_t(X) = \frac{ w_{11} \tilde \pi_t( \vert e_j \rangle \langle e_k \vert \otimes X) }{ w_{jk} \tilde \pi_t( \vert e_1 \rangle \langle e_1 \vert \otimes I) } . \label{eq:pi-jk-def}$$ where $\tilde \pi_t(A \otimes X)$ is the conditional state for the extended system defined by (\[eq:matrix-c-exp-def\]). Then for all $K \in \mathscr{Y}_{t}$ we have $$\mathbb{E}_{11}[\pi _{t}^{jk}(X)\,K]=\mathbb{E}_{jk}[j_{t}(X)K]. \label{eq:pi-jk-c-exp-0}$$ We have $$\begin{aligned} \mathbb{E}_{11}[ \pi^{jk}_t(X) K ] &=& \frac{1}{w_{11}} \mathbb{E}_{\Sigma} [ \vert e_1 \rangle \langle e_1 \vert \otimes (\pi^{jk}_t(X)K) ] \\ &=& \frac{1}{w_{11}} \mathbb{E}_{\Sigma} [ \pi_t( \vert e_1 \rangle \langle e_1 \vert \otimes I) (I \otimes \pi^{jk}_t(X)K) ] \\ &=& \frac{1}{w_{jk}} \mathbb{E}_{\Sigma} [ \pi_t( \vert e_j \rangle \langle e_k \vert \otimes X) (I \otimes K) ] \\ &=& \frac{1}{w_{jk}} \mathbb{E}_{\Sigma} [ ( \vert e_j \rangle \langle e_k \vert \otimes j_t(X)) (I \otimes K) ] \\ &=& \mathbb{E}_{jk}[ j_t(X) K ]\end{aligned}$$ as required. [$\Box$]{} We can now present our main theorem for the quantum filter for the single photon field state. \[thm:main\] The quantum filter for the conditional expectation with respect to the single photon field is given in the Heisenberg picture by $$\hat X(t)= \mathbb{E}_{11}[ X(t) \, \vert \, \mathscr{Y}_t ] = \pi^{11}_t(X),$$ where $\pi^{11}_t(X)$ is defined by (\[eq:pi-jk-def\]) (for $j=k=1$), and is given by the system of equations $$\begin{aligned} d\pi^{11}_t (X) &=& (\pi^{11}_t(\mathcal{L}(X)) + \pi^{01}_t( S^\ast [X,L] ) \xi^\ast(t) + \pi^{10}_t( [L^\ast, X] S ) \xi(t) \notag \\ && + \pi^{00}_t( S^\ast X S - X) \vert \xi(t) \vert^2)dt \notag \\ && + ( \pi^{11}_t( XL + L^\ast X) + \pi^{01}_t(S^\ast X) \xi^\ast(t) + \pi^{10}_t( XS) \xi(t) \notag \\ && - \pi^{11}_t(X) ( \pi^{11}_t(L+L^\ast) + \pi^{01}_t(S) \xi(t) + \pi^{10}_t(S^\ast) \xi^\ast(t) ) )dW(t), \label{eq:pi-dyn-a-11} \\ d\pi^{10}_t (X) &=& ( \pi^{10}_t(\mathcal{L}(X)) + \pi^{00}_t( S^\ast [X, L] ) \xi^\ast(t) )dt \notag \\ && + ( \pi^{10}_t( XL + L^\ast X) + \pi^{00}_t(S^\ast X) \xi^\ast(t) \notag \\ && - \pi^{10}_t(X) ( \pi^{11}_t(L+L^\ast) + \pi^{01}_t(S) \xi(t) + \pi^{10}_t(S^\ast) \xi^\ast(t) ) )dW(t), \label{eq:pi-dyn-a-10} \\ d\pi^{01}_t (X) &=& ( \pi^{01}_t(\mathcal{L}(X)) + \pi^{00}_t( [L^\ast, X] S ) \xi(t) )dt \notag \\ && + ( \pi^{01}_t( XL + L^\ast X) + \pi^{00}_t( XS) \xi(t) \notag \\ && - \pi^{01}_t(X) ( \pi^{11}_t(L+L^\ast) + \pi^{01}_t(S) \xi(t) + \pi^{10}_t(S^\ast) \xi^\ast(t) ) )dW(t), \label{eq:pi-dyn-a-01} \\ d\pi^{00}_t (X) &=& \pi^{00}_t(\mathcal{L}(X)) dt + ( \pi^{00}_t( XL + L^\ast X) \notag \\ && - \pi^{00}_t(X) ( \pi^{11}_t(L+L^\ast) + \pi^{01}_t(S) \xi(t) + \pi^{10}_t(S^\ast) \xi^\ast(t) ) )dW(t) . \label{eq:pi-dyn-a-00}\end{aligned}$$ Here, the innovations process $W(t)$ is a $\mathscr{Y}_t$ Wiener process with respect to the single photon state and is defined by $$dW(t) = dY(t) - ( \pi^{11}_t( L+L^\ast) + \pi^{10}_t(S) \xi(t) + \pi^{01}_t(S^\ast) \xi^\ast(t) ) dt . \label{eq:innovation-11}$$ The initial conditions are $$\pi^{11}_0(X)= \pi^{00}_0(X)= \langle \eta, X \eta \rangle, \ \ \pi^{10}_0(X)= \pi^{01}_0(X)=0. \label{eq:pi-jk--initial}$$ Suppose first that $\alpha_0 \neq 0$. Setting $j=k=1$ in equation (\[eq:pi-jk-c-exp-0\]) above, and noting that $K\in \mathscr{Y}_{s}$ was otherwise arbitrary, we deduce that $\pi _{t}^{11}(X)$ is the desired conditional expectation for the single photon field state, as characterized by equation (\[eq:c-exp-def\]). The differential equations (\[eq:pi-dyn-a-11\])-(\[eq:pi-dyn-a-00\]) follow from the definition (\[eq:pi-jk-def\]), the filter (\[pi\_filter\]) for the extended system, and the Ito rule. Next, we note that the coefficients of the QSDEs (\[eq:pi-dyn-a-11\])-(\[eq:pi-dyn-a-00\]), the initial conditions, and $Y_t$ do not depend on $\alpha_0$ and $\alpha_1$. Hence, the solutions $\pi_t^{jk}(X)$ of this system of equations are independent of $\alpha_0$ and $\alpha_1$. Therefore, $\pi_t^{jk}(X)$ can be defined for $\alpha_j \in \{0,1\}$, $j=0,1$, and is in fact identical for all $0 \leq |\alpha_0|,|\alpha_1| \leq 1$. We now prove that $W(t)$ is a $\mathscr{Y}_t$-martingale, that is, $\mathbb{E}_{11}[W(t)-W(s)\,|\mathscr{Y}_{s}]=0$. To this end, let $K \in \mathscr{Y}_{s}$. Then $$\begin{aligned} \mathbb{E}_{11}[\{W(t)-W(s)\}K] &=&\mathbb{E}_{11}[\{Y(t)-Y(s) \notag \\ &&-\int_{s}^{t}(\pi _{r}^{11}((L+L^{\ast })+\pi _{r}^{10}(S)\xi (t)+\pi _{r}^{01}(S^{\ast })\xi ^{\ast }(t))dr\}K] \notag \\ &=&\mathbb{E}_{11}[\{\int_{s}^{t}(L(r)+L^{\ast }(r))dr+V(t)-V(s) \notag \\ &&-\int_{s}^{t}(\pi _{r}^{11}((L+L^{\ast })+\pi _{r}^{10}(1)\xi (t)+\pi _{r}^{01}(1)\xi ^{\ast }(t))dr\}K] \notag \\ &=&\mathbb{E}_{11}[\{\int_{s}^{t}(L(r)+L^{\ast }(r)-\pi _{r}^{11}((L+L^{\ast }))dr\}K] \notag \\ &&+\mathbb{E}_{11}[\{V(t)-V(s)-\int_{s}^{t}(\pi _{r}^{10}(S)\xi (t)+\pi _{r}^{01}(S^{\ast })\xi ^{\ast }(t))dr\}K]\end{aligned}$$ however, this vanishes from (\[eq:pi-jk-c-exp-0\]), Lemma \[lemma:Z-compensated-mtg\], and $$\begin{aligned} && \mathbb{E}_{11}[\{\int_{s}^{t}(\pi _{r}^{10}(S)\xi (t)+\pi _{r}^{01}(S^{\ast })\xi ^{\ast }(t))dr\}K] \\ && =\mathbb{E}_{10}[\int_{s}^{t}S(r)\xi (r)dr K]+\mathbb{E}_{01}[\int_{s}^{t}S^{\ast }(r)\xi ^{\ast }(r)dr K ].\end{aligned}$$ Finally, since $dW(t) dW(t)=dt$, Levy’s Theorem implies that $W(t)$ is a $\mathscr{Y}_t$ Wiener process. [$\Box$]{} Reference Method for Filtering in the Extended System {#sec:extended-filter} ----------------------------------------------------- The reference method is one of the standard approaches to filtering theory, with its origins in the work of Duncan, Mortensen, Zakai, Holevo and Belavkin, see [@RE82], [@AH91], [@VPB92], [@BH06], [@BHJ07]. In this section we apply this approach, as described in [@BHJ07 sec. 6], to the filtering problem in the extended system, giving an independent derivation of the fundamental filtering equation. Our first step is the following. \[lemma:matrix-F-rep\] Assume $\alpha_0 \neq 0$. Then we have $$\mathbb{E}_{\Sigma}[ (A \otimes X(t)) ] = \mathbb{E}_{\Sigma}[ F^\ast(t) (A \otimes X) F(t) ]$$ where $F(t) \in (I\otimes \mathscr{Z}_t)^{\prime}$ is given by $$\begin{aligned} dF(t) &=& ( G_0(t) dt +G_1(t)dZ(t)) F(t) , \label{dF}\end{aligned}$$ $F(0)=I$, and $$\begin{aligned} G_0(t) &=& -I \otimes ( \frac{1}{2} L^\ast L + iH) - \sigma_+ \otimes (L+L^\ast S) \nu \xi(t) , \\ G_1(t) &=& I \otimes L + \sigma_+ \otimes (S-I) \nu \xi(t) .\end{aligned}$$ Let us suppose that $F\left( t\right) $ satisfies (\[dF\]) with coefficients $G_{0}\left( t\right) $, $G_{1}\left( t\right) $ which both commute with $\sigma _{+}\otimes I$, that is, $$G_{i}\left( t\right) =I\otimes g_{i0}\left( t\right) +\sigma _{+}\otimes g_{i1}\left( t\right) .$$ Then $$\mathbb{E}_{\Sigma }\left[ d\left\{ F\left( t\right) ^{\ast }A \otimes XF\left( t\right) \right\} \right] = \mathbb{E}_{\Sigma }\left[ F\left( t\right) ^{\ast }\left\{ T_t (A \otimes X) dt +Q_t (A \otimes X)dZ(t) \right\} F\left( t\right) \right] ,$$ where $$\begin{aligned} T_{t}\left( A \otimes X\right) &=&G_1 (t)^{\ast }A \otimes XG_1 (t)+A \otimes XG_0 (t)+G_0 (t)^{\ast }A \otimes X, \\ Q_{t}\left( A \otimes X\right) &=&A \otimes XG_1 (t)+G_1 (t)^{\ast }A \otimes X.\end{aligned}$$ Using lemma \[lemma:expectation-basic-extended\] we see that this equals $$\mathbb{E}_{\Sigma}[F\left( t\right) ^{\ast }\{ T_t (A \otimes X) + Q_t (A \otimes X) \nu \xi \left( t\right) \left( \sigma _{+}\otimes I\right) +\nu ^{\ast }\xi \left( t\right) ^{\ast }\left( \sigma _{-}\otimes I\right) Q_t (A \otimes X) \}F\left( t\right) ]dt.$$ We now require that this equals $\mathbb{E}_{\Sigma }\left[ \mathcal{G}_t\left( A \otimes X\right) \right] dt$. This implies the four identities $$\begin{aligned} g_{11}{}^{\ast }Xg_{11}+\nu \xi g_{11}^{\ast }X+\nu ^{\ast }\xi ^{\ast }Xg_{11} &=&|\nu \xi |^{2}(S^{\ast }XS-X) \label{id1} \\ g_{10}^{\ast }Xg_{11}+Xg_{01}+\nu \xi g_{10}^{\ast }X+\nu \xi Xg_{10} &=&\nu \xi \lbrack L^{\ast },X]S \label{id2} \\ g_{11}^{\ast }Xg_{10}+g_{01}{}^{\ast }X+\nu ^{\ast }\xi ^{\ast }g_{10}{}^{\ast }X+\nu ^{\ast }\xi ^{\ast }Xg_{10} &=&\nu ^{\ast }\xi ^{\ast }S^{\ast }[X,L] \label{id3} \\ g_{10}^{\ast }Xg_{10}+g_{00}^{\ast }X+Xg_{00} &=&\mathcal{L}\left( X\right) . \label{id4}\end{aligned}$$ The first identity (\[id1\]) is satisfied if $g_{11}=\nu \xi (S-1)$. Substituting $X=I$ into (\[id2\]) we deduce that $g_{01}=-\nu \xi g_{10}^{\ast }S-\nu \xi g_{10}$ and thus $$g_{10}^{\ast }Xg_{11}+Xg_{01}+\nu \xi g_{10}^{\ast }X+\nu \xi Xg_{10}=\nu \xi \lbrack g_{10}^{\ast },X]S.$$ Therefore (\[id2\]) is satisfied if $g_{10}=L$, and consequently $g_{01}=-\nu \xi \left( L+L^{\ast }S\right) $. It then follows that (\[id3\]) will be automatically satisfied, while (\[id4\]) then only requires that $g_{00}=-(\frac{1}{2}L^{\ast }L+iH)$ in order to obtain the Lindblad generator $\mathcal{L}\left( X\right) $. This leads us precisely to the coefficients $G_i (t)$ stated in the lemma, and the identity $$T_t (A \otimes X) + Q_t (A \otimes X) \nu \xi \left( t\right) \left( \sigma _{+}\otimes I\right) +\nu ^{\ast }\xi \left( t\right) ^{\ast }\left( \sigma _{-}\otimes I\right) Q_t (A \otimes X) = \mathcal{G}_t\left( A \otimes X\right)$$ The following Bayes’ relation is proven along similar lines to [@BHJ07 Theorem 6.2]. \[lemma:matrix-bayes\] For $\alpha_0 \neq 0$, define $$\begin{aligned} \varsigma_t( A \otimes X) = (I \otimes U^\ast(t)) \mathbb{E}_{\Sigma} [ F^\ast(t) (A \otimes X) F(t) \, \vert \, I \otimes \mathscr{Z}_t ] (I \otimes U(t)). \label{eq:varsigma-def}\end{aligned}$$ Then $$\pi_t(A \otimes X) = \frac{\varsigma_t(A \otimes X) }{\varsigma_t( I \otimes I)} . \label{eq:matrix-bayes}$$ In order to determine the differential equation for $\varsigma_t(A\otimes X)$, we first define, for $ A\otimes X\in \mathcal{B}(\mathbb{C}^{2}\otimes \mathfrak{h}_{S})$ the process $$\gamma _{t}\left( A\otimes X \right) =\mathbb{E}_{\Sigma }[F(t)^{\ast }A\otimes X \, F(t)|I\otimes \mathscr{Z}_t],$$ so that $\varsigma _{t}(A\otimes X)\equiv (I\otimes U(t))^{\ast }\gamma _{t}(A\otimes X)(I\otimes U(t))$. We then have \[lemma:matrix-ref-c-exp\] Let $\alpha_0 \neq 0$. The process $\gamma _{t}\left( A\otimes X \right) $ satisfies the QSDE $$d\gamma _{t}\left( A\otimes X \right) =\tau _{t}\left( A\otimes X \right) dt+\beta _{t}\left( A\otimes X \right) dZ\left( t\right) \label{eq:qsde-gamma}$$ where $\tau _{t}\left( A \otimes X\right) ,\beta _{t}\left( A \otimes X\right) \in \mathscr{Z}_t $, are given by $$\begin{aligned} \beta _{t}\left( A \otimes X\right) &=&\gamma _{t}\left( Q_{t}\left( A \otimes X\right) \right) \nonumber \\ && +\nu \xi (t)\gamma _{t}\left( A \otimes X\left( \sigma _{+}\otimes I\right) \right) +\nu ^{\ast }\xi (t)^{\ast }\gamma _{t}\left( \left( \sigma _{-}\otimes I\right) A \otimes X\right) -\gamma _{t}\left( A \otimes X\right) \theta _{t}, \\ \tau _{t}\left( A \otimes X\right) &=&\gamma _{t}\left( T_{t}\left( A \otimes X\right) \right) \nonumber \\ && +\nu \xi (t)\gamma _{t}\left( Q_{t}\left( A \otimes X\right) \sigma _{+}\right) +\nu ^{\ast }\xi (t)^{\ast }\gamma _{t}\left( \sigma _{-}Q_{t}\left( A \otimes X\right) \right) -\beta _{t}\left( A \otimes X\right) \theta _{t},\end{aligned}$$ with $$\theta _{t}=\mathbb{E}_{\psi }[\left( \nu \xi (t)\sigma _{+}+\nu ^{\ast }\xi ^{\ast }(t)\sigma _{-}\right) \otimes I|I\otimes \mathscr{Z}_{t}].$$ Setting $R_{t}=F(t)^{\ast }(A\otimes X)F(t)$, we have that $$dR_{t}=F(t)^{\ast }T_{t}\left( A\otimes X \right) F(t)dt+F(t)^{\ast }Q_{t}(A\otimes X)F(t)dZ\left( t\right)$$ and our aim is compute $\gamma _{t}\left( A\otimes X \right) =\mathbb{E}_{\Sigma }[R_{t}|I\otimes \mathscr{Z}_{t}]$. In particular, $$\mathbb{E}_{\Sigma }\left[ \left( R_{t}-\gamma _{t}\left( A\otimes X \right) \right) D_{t}\right] =0$$ for every $D_{t}\in I\otimes \mathscr{Z}_{t}$ and we now apply a technique similar to the characteristic function method, this time using the input process $Z$ and taking the process $D_{t}$ to satisfy the QSDE $dD_{t}=f\left( t\right) D_{t}dZ\left( t\right) $ with $D_{0}=I$, for given integrable $f$. From the Ito product rule we then have $$0=\mathbb{E}_{\Sigma}\left[ \left( dR_{t}-d\gamma _{t}\left( A\otimes X \right) \right) D_{t}+\left( R_{t}-\gamma _{t}\left( A\otimes X \right) \right) dD_{t}+\left( dR_{t}-d\gamma _{t}\left( A\otimes X \right) \right) dD_{t}\right]$$ and making the ansatz that $d\gamma _{t}\left( A\otimes X \right) =\tau _{t}\left( A\otimes X \right) dt+\beta _{t}\left( A\otimes X \right) dZ\left( t\right) $ for unknown coefficients $\tau _{t}\left( A\otimes X \right) $ and $\beta _{t}\left( A\otimes X \right) $ we see that $$\begin{aligned} 0 &=&\mathbb{E}_{\Sigma } [ ( F(t)^{\ast }T_{t}\left( A\otimes X \right) F(t)-\tau _{t}\left( A\otimes X \right) ) D_{t}dt \nonumber \\ && \ \ \ \ \ + ( F(t)^{\ast }Q_{t}\left( A\otimes X \right) F(t)-\beta _{t}\left( A\otimes X \right) ) D_{t}dZ\left( t\right) ] \\ &&+\mathbb{E}_{\psi }\left[ \left( R_{t}-\gamma _{t}\left( A\otimes X \right) \right) D_{t}f\left( t\right) dZ(t)\right] \\ &&+\mathbb{E}_{\psi }\left[ \left[ F(t)^{\ast }Q_{t}\left( A\otimes X \right) F(t)-\beta _{t}\left( A\otimes X \right) \right] D_{t}f\left( t\right) dt\left( t\right) \right] .\end{aligned}$$ We now make use of Lemma \[lemma:expectation-basic-extended\] again and apply the commutation relations $F(t)\sigma _{+}=\sigma _{+}F(t)$, $\sigma _{-}F(t)^{\ast }=F^{\ast }\left( t\right) \sigma _{-}$. (Note that $\sigma _{+}$ will not commute with $F^{\ast }\left( t\right) $.) Inserting $\mathbb{E}_{\Sigma}\left[ \cdot |I\otimes \mathscr{Z}_t \right] $ under the expectation sign, then separating coefficients of $D_t$ and $D_tf(t)$, we obtain the equations $$\begin{aligned} 0 &=&\gamma _{t}\left( T_{t}\left( A\otimes X \right) \right) -\tau _{t}\left( A\otimes X \right) \\ && +\nu _{t}\gamma _{t}\left( Q_t\left( A\otimes X \right) \left( \sigma _{+}\otimes I\right) \right) +\nu _{t}\gamma _{t}\left( \left( \sigma _{-}\otimes I\right) Q_t\left( A\otimes X \right) \right) \\ && -\beta _{t}\left( A\otimes X \right) \mathbb{E}_{\Sigma }\left[ \nu \xi (t)\sigma _{+}\otimes I+\nu ^{\ast }\xi (t)^{\ast }\sigma _{-}\otimes I|I\otimes \mathscr{Z}_t \right] , \\ 0 &=&\gamma _{t}\left( Q_t\left( A\otimes X \right) \right) -\beta _{t}\left( A\otimes X \right) \\ && +\nu \xi (t)\gamma _{t}\left( A\otimes X \left( \sigma _{+}\otimes I\right) \right) +\nu ^{\ast }\xi (t)^{\ast }\gamma _{t}\left( \left( \sigma _{-}\otimes I\right) A\otimes X \right) \\ && -\gamma _{t}\left( A\otimes X \right) \mathbb{E}_{\Sigma }\left[ \nu \xi (t)\sigma_{+}\otimes I+\nu ^{\ast }\xi (t)^{\ast } \sigma _{-}\otimes I|I\otimes \mathscr{Z}_t \right] .\end{aligned}$$ Rearranging these expressions yields the relations in the statement of the lemma. Let $\alpha_0 \neq 0$. The unnormalized conditional expectation $\varsigma_t(A \otimes X)$ defined by (\[eq:varsigma-def\]) satisfies the equation $$d\varsigma _{t}\left( A \otimes X \right) =\varsigma _{t}\left( \mathcal{G}_{t}(A \otimes X)\right) dt+\lambda _{t}\left( A \otimes X \right) d\tilde{Y}(t), \label{eq:matrix-sigma-dyn}$$ where $$\begin{aligned} \lambda _{t}\left( A \otimes X \right) &=&\varsigma _{t}(A \otimes X \tilde{L}_{t}+\tilde{L}_{t}^{\ast }A \otimes X)-\varsigma _{t}\left( A \otimes X \right) \kappa _{t}, \label{eq:def_lambda_t}\\ \tilde{L}_{t} &=&I\otimes L+\nu _{t}\xi (t)\sigma _{+}\otimes S, \label{Ltilde} \\ d\tilde{Y}(t) &=&dY\left( t\right) -\kappa _{t}dt,\quad \tilde{Y}(0)=0, \label{Ytilde} \\ \kappa _{t} &=&\varsigma _{t}\left( \left( \nu \xi (t)\sigma _{+}+\nu ^{\ast }\xi (t)^{\ast }\sigma _{-}\right) \otimes I\right) . \label{eq:kappa-def}\end{aligned}$$ We remark that by inspection the coefficients in the QSDE for $\gamma _{t}\left( A \otimes X \right) $ simplify to $$\begin{aligned} \beta _{t}\left( A \otimes X \right) &=&\gamma _{t}\left( Q_{t}\left(A \otimes X \right) \right) \\ && +\nu \xi (t)\gamma _{t}\left( A \otimes X \left( \sigma _{+}\otimes I\right) \right) +\nu ^{\ast }\xi (t)^{\ast }\gamma _{t}\left( \left( \sigma _{-}\otimes I\right) A \otimes X \right) -\gamma _{t}\left( A \otimes X \right) \theta _{t} \\ &=&\gamma _{t} ( A \otimes X \left( I\otimes L+\nu _{t}\sigma _{+}\otimes S\right) +\left( I\otimes L^{\ast }+\nu ^{\ast }\xi (t)^{\ast }\sigma _{-}\otimes S^*\right) A \otimes X ) \\ && -\gamma _{t}\left( A \otimes X \right) \theta _{t}, \\ \tau _{t}\left( A \otimes X \right) &=&\gamma _{t}\left( T_{t}\left( A \otimes X \right) \right) +\nu \xi (t)\gamma _{t}\left( Q_{t}\left( A \otimes X \right) \sigma _{+}\right) \\ && +\nu ^{\ast }\xi (t)^{\ast }\gamma _{t}\left( \sigma _{-}Q_{t}\left( A \otimes X \right) \right) -\beta _{t}\left( A \otimes X \right) \theta _{t} \\ &\equiv &\gamma _{t}\left( \mathcal{G}\left( A \otimes X \right) \right) -\beta _{t}\left( A \otimes X \right) \theta _{t},\end{aligned}$$ and therefore $$d\gamma _{t}\left( A \otimes X \right) =\gamma _{t}\left( \mathcal{G}(A \otimes X )\right) dt+\beta _{t}\left( A \otimes X \right) \left[ dZ\left( t\right) -\theta _{t}dt\right] .$$ The QSDE for $\varsigma _{t}(A \otimes X)\equiv (I\otimes U(t))^{\ast }\gamma _{t}(A \otimes X)(I\otimes U(t))$ is then readily deduced from the unitary rotation noting that $\kappa _{t}\equiv (I\otimes U(t))^{\ast }\theta _{t}(I\otimes U(t))$ and $\lambda _{t}\left( A \otimes X \right) \equiv (I\otimes U(t))^{\ast }\beta _{t}\left( A \otimes X \right) (I\otimes U(t))$. For $\alpha_0 \neq 0$, the conditional expectation $\pi_t(A \otimes X)$ defined by (\[eq:matrix-c-exp-def\]) and given by (\[eq:matrix-bayes\]) satisfies equation (\[pi\_filter\]) derived in Theorem \[thm:matrix-filter\]. We see that $d\varsigma _{t}\left( I\otimes I\right) =\lambda _{t}\left( I\otimes I\right) \,d\tilde{Y}\left( t\right) $ and so $$d\frac{1}{\varsigma _{t}\left( I\otimes I\right) }=-\frac{\lambda _{t}\left( I\otimes I\right) }{\varsigma _{t}\left( I\otimes I\right) ^{2}}\,d\tilde{Y}\left( t\right) +\frac{1}{\varsigma _{t}\left( I\otimes I\right) ^{3}}\lambda _{t}\left( I\otimes I\right) ^{2}dt.$$ However, we note from (\[eq:def\_lambda\_t\]) that $$\frac{\lambda _{t}\left( I\otimes I\right) }{\varsigma _{t}\left( I\otimes I\right) }\equiv \pi _{t}\left( \tilde{L}_{t}+\tilde{L}_{t}^{\ast }\right) -\kappa _{t}.$$ By an application of the Ito product rule, the normalized filter therefore satisfies $$\begin{aligned} && d\pi _{t}(A \otimes X) \\ &=&\pi _{t}(\mathcal{G}_{t}(A \otimes X))dt \\ &&+\left\{ \frac{\lambda _{t}\left( A \otimes X \right) -\pi _{t}\left( A \otimes X \right) \lambda _{t}\left( I\otimes I\right) }{\varsigma _{t}\left( I\otimes I\right) }\right\} \left[ dY(t) -\kappa _{t}dt-\frac{\lambda _{t}\left( I\otimes I\right) }{\varsigma _{t}\left( I\otimes I\right) }dt\right] \\ &=&\pi _{t}(\mathcal{G}_{t}(A \otimes X))dt+ \{ \pi _{t}\left( A \otimes X \tilde{L}_{t}+\tilde{L}_{t}^{\ast }A \otimes X \right) \\ && -\pi _{t}\left( A \otimes X \right) \pi _{t}\left( \tilde{L}_{t}+\tilde{L}_{t}^{\ast }\right) \} \left[ dY(t)-\pi _{t}\left( \tilde{L}_{t}+\tilde{L}_{t}^{\ast }\right) dt\right] \\ &\equiv &\pi _{t}(\mathcal{G}_{t}(A \otimes X))dt+\mathcal{H}_{t}(A \otimes X)dW\left( t\right),\end{aligned}$$ since we have from (\[Ltilde\]), (\[eq:matrix-filter\]) and(\[eq:innovation-matrix\]) $$\begin{aligned} \mathcal{H}_{t}(A \otimes X) &\equiv &\pi _{t}\left( A \otimes X \tilde{L}_{t}+\tilde{L}_{t}^{\ast }A \otimes X \right) -\pi _{t}\left( A \otimes X \right) \pi _{t}\left( \tilde{L}_{t}+\tilde{L}_{t}^{\ast }\right) , \\ dW(t) &\equiv &dY-\pi _{t}\left( \tilde{L}_{t}+\tilde{L}_{t}^{\ast }\right) dt.\end{aligned}$$ Therefore we recover equation (\[pi\_filter\]). Fields in a Superposition of Coherent States {#sec:cat} ============================================ Superposition of Coherent States {#sec:cat-super} -------------------------------- In this section we take the field to be in a superposition state $$\vert \Psi \rangle = \sum_j \alpha_j \vert f_j \rangle , \label{eq:super-state}$$ where $\vert f_j \rangle$ are coherent states and the complex numbers $\alpha_j$ ($j=1,\ldots,n$) are non-zero normalized weights (described further below). Coherent vectors $\vert f \rangle$ may be expressed in terms of the vacuum vector using the Weyl (or displacement) operator [@KRP92] $W(f)$ which serves as a : $$\vert f \rangle = W(f) \vert 0 \rangle.$$ While the collection of all coherent vectors is dense in the Fock space, they are not orthogonal, and indeed the inner product (in the Fock space) is given by $$\langle f \vert g \rangle = \exp( -\frac{1}{2} \parallel f \parallel_2^2 -\frac{1}{2} \parallel g \parallel_2^2 + \langle f, g \rangle_2 ).$$ Here, $\parallel f \parallel_2^2 = \langle f,f \rangle_2$ and $\langle f, g \rangle_2$ are the $L^2([0,\infty), \mathbf{C})$ norm and inner product respectively. The superposition state $\vert \psi \rangle$ given by (\[eq:super-state\]) is specified by a choice of coherent vectors $\vert f_j \rangle$, with weights $\alpha_j$ ensuring normalization: $\langle \psi \vert \psi \rangle = \sum_{jk} \alpha_j^\ast \alpha_k g_{jk}=1$, where $g_{jk}= \langle f_j \vert f_k \rangle$. For a system operator $X$ acting on $\mathfrak{H}_S$, and $F$ is a field operator acting on the Fock space $\mathfrak{F}$, the expectation with respect to the state $ \vert \eta \rangle \otimes \vert \Psi \rangle$ is defined by $$\begin{aligned} \mathbb{E}_{\eta\Psi}[ X \otimes F] &=& \langle \eta \Psi \vert (X \otimes F)\vert \eta\Psi \rangle = \langle \eta \vert X \vert \eta \rangle \langle \Psi \vert F \vert \Psi \rangle \nonumber \\ &=& \langle \eta \vert X \vert \eta \rangle \sum_{jk} \alpha_j^\ast \alpha_k \langle f_j \vert F \vert f_k \rangle \nonumber \\ &=& \sum_{jk} \alpha_j^\ast \alpha_k \mathbb{E}_{jk}[ X \otimes F ], \label{eq:expect-def}\end{aligned}$$ where $$\mathbb{E}_{jk}[ X \otimes F ] = \langle \eta \vert X \vert \eta \rangle \langle f_j \vert F \vert f_k \rangle$$ for $j,k=1,\ldots,n$. We write $\mathbb{E}_{00}[ X \otimes F ] = \langle \eta \vert X \vert \eta \rangle \langle 0 \vert F \vert 0 \rangle $ for the vacuum case. Consider now the expectation of an adapted operator $K(t)$ on the composite system $\mathfrak{H} = (\mathfrak{H}_S \otimes \mathfrak{F}_{t]}) \otimes \mathfrak{F}_{(t}$; this means that $K(t)$ acts trivially on the future component $\mathfrak{F}_{(t}$. Let $\chi_{[0,t]}$ is the indicator function for the time interval $[0,t]$. Now coherent vectors and Weyl operators factorize as $\vert f \rangle = \vert f \chi_{[0,t]} \rangle \otimes \vert f \chi_{(t,\infty)} \rangle$ and $W(f)= W(f \chi_{[0,t]} ) \otimes W(f \chi_{(t,\infty)} ) $, respectively. Write $$W^-_t(f)= W(f \chi_{[0,t]} ), \ \ W^+_t(f)=W(f \chi_{(t,\infty)} ) .$$ Then we can express the coherent expectations of adapted processes $K(t)$ in terms of the vacuum: $$\begin{aligned} \mathbb{E}_{jk}[ K(t)] &=& \mathbb{E}_{00} [ W^{-\ast}_t(f_j) K(t) W^-_t(f_k) ] r^{jk}(t) \label{eq:Kt-00-0}\end{aligned}$$ where $ r^{jk}(t) = \langle 0 \vert W^{+\ast}_t(f_j) W^+_t(f_k) \vert 0 \rangle $ satisfies $$\dot r^{jk}(t) = -( f_j^\ast (t)f_k(t) - \frac{1}{2} \vert f_j(t) \vert^2 - \frac{1}{2} \vert f_k (t) \vert^2 )r^{jk}(t), \ \ r^{jk}(0)=1.$$ Note that $j=k$ is the standard coherent expectation, in which case $r^{jj}(t)=1$. The following lemma shows how expectations of stochastic integrals with respect to the superposition state can be evaluated. \[lemma:expectation-basic-cat\] Let $K(t)$ be a bounded quantum stochastic process defined by (\[eq:Kt-def\]), where $M_0$, $M_\pm$ and $M_1$ are bounded and adapted. Then we have $$\begin{aligned} \mathbb{E}_{jk}[ K(t) ] &=& \mathbb{E}_{jk}[ \int_0^t M_0(s) ds ] + \int_0^t M_- (s) f_k(s) ds \nonumber \\ && + \int_0^t M_+ (s) f^\ast_j(s) ds + \int_0^t M_1 (s) f_j^\ast(s) f_k(s) ds ] . \label{eq:Kt-00-cat}\end{aligned}$$ Equation (\[eq:Kt-00-cat\]) follows from the following eigenstate property of coherent vectors: $$\begin{aligned} dB(t) \vert f \rangle &=& f(t) \vert f \rangle dt, \notag \\ d\Lambda(t) \vert f \rangle &=& dB^\ast(t) f(t) \vert f \rangle .\end{aligned}$$ [$\Box$]{} Embedding {#sec:cat-embed} --------- For the superposition of $n$ coherent states, we use an $n$-level ancilla system, leading to the extended space $$\tilde {\mathfrak{H}}= \mathbb{C}^n \otimes \mathfrak{H} = \mathfrak{H} \oplus \mathfrak{H} \oplus \cdots \oplus \mathfrak{H} \ \ (n \ \mathrm{times}).$$ As in the single photon case, we allow the extended system to evolve unitarily according to $I \otimes U(t) $, where $U(t)$ is the unitary operator for the system and field, given by the Schrödinger equation (\[eq:unitary\]). Let $\vert e_j \rangle$, $j=1,\ldots,n$, be an orthonormal basis for $\mathbf{C}^n$. We initialize the extended system in the state $$\vert \Sigma \rangle = \frac{1}{\vert \alpha \vert} \sum_{j} \alpha_j \vert e_j \rangle \otimes \vert \eta \rangle \otimes \vert f_j \rangle, \label{eq:super-cat}$$ where $\alpha_j \neq 0$ for all $j$ and $\vert \alpha \vert^2= \sum_j \alpha_j^\ast \alpha_j$ (so that $\langle \Sigma \vert \Sigma \rangle =1$). This state evolves according to $\vert \Sigma(t) \rangle = (I \otimes U(t)) \vert \Sigma \rangle$. Let $A$ be an operator acting on $\mathbb{C}^n$, i.e. a complex $n \times n$ matrix, $A=( a_{jk} )$, $j,k=1,\ldots,n$. Then expectation in the extended system is defined by $$\mathbb{E}_{\Sigma}[ A \otimes X \otimes F] = \langle \Sigma \vert (A \otimes X \otimes F) \vert \Sigma \rangle = \frac{1}{\vert \alpha \vert^2} \sum_{jk} a_{jk} \alpha_j^\ast \alpha_k \mathbb{E}_{jk}[ X \otimes F]. \label{eq:expect-extended-def}$$ Expectations of quantum stochastic integrals can be compactly expressed in the extended system, as the following lemma shows. \[lemma:expectation-basic-extended-cat\] Let $M(t)$ be adapted. Then $$\begin{aligned} \mathbb{E}_{\Sigma} [ \int_0^t A \otimes M(s) dB(s) ] &=& \mathbb{E}_{\Sigma} [\int_0^t ( A C(s) )\otimes M(s) ds ], \label{eq:extended-expect-1} \\ \mathbb{E}_{\Sigma} [ \int_0^t A \otimes M (s) dB^\ast(s) ] &=& \mathbb{E}_{\Sigma} [ \int_0^t ( C^\dagger (s) A )\otimes M(s) ds], \label{eq:extended-expect-2} \\ \mathbb{E}_{\Sigma} [ \int_0^t A \otimes M(s) d\Lambda(s) ] &=& \mathbb{E}_{\Sigma} [ \int_0^t ( C^\dagger(s) A C(s) )\otimes M(s) ds ], \label{eq:extended-expect-3}\end{aligned}$$ where $$C(t) = \mathrm{diag}[ f_1(t), \ldots, f_n(t) ].$$ Notice that the expectations of the stochastic integrals are expressed in terms of the action of the matrix $C(t)$ on the ancilla factor $A$. Master Equation {#sec:cat-master} --------------- In this section we show how the the unconditional expectation $$\mu_t(X) = \mathbb{E}_{\eta\Psi}[ X(t) ] \label{eq:mu-def}$$ may be computed from a collection of differential equations. We do this through a differential equation for the unconditional expectation $$\tilde\mu_t(A\otimes X) = \mathbb{E}_{\Sigma} [ A \otimes X(t) ] \label{eq:mu-tilde-def}$$ for the extended system. Let $R$ be an $n \times n$ matrix defined by $R_{jk}=1$ for all $j,k = 1, \ldots, n$, and define $$\begin{aligned} \mathcal{G}_t(A \otimes X) &=& A \otimes \mathcal{L}(X) + ( AC(t)) \otimes [L^\ast, X] S + (C^\dagger(t) A ) \otimes S^\ast [X,L] \notag \\ && + ( C^\dagger(t) A C(t) ) \otimes (S^\ast X S - X)) .\end{aligned}$$ \[lemma:master-1-cat\] The unconditional expectation (\[eq:mu-def\]) with respect to the superposition state $\vert \Psi \rangle$ (defined by (\[eq:super-state\])) is given by $$\mu_t(X)= \frac{ \tilde\mu_t(R \otimes X) }{ \tilde\mu_t( R \otimes I) }, \label{eq:mu-relation}$$ and the master equation for the expectation (\[eq:mu-tilde-def\]) in the extended system is $$\frac{d}{dt} \tilde \mu_t(A \otimes X) = \tilde \mu_t( \mathcal{G}_t(A \otimes X)) , \label{eq:mu-master}$$ with initial condition $\tilde \mu_0(A\otimes X)= \frac{1}{\vert \alpha \vert^2} \langle \eta \vert X \vert \eta \rangle \sum_{jk} a_{jk} \alpha_j^\ast \alpha_k$. By definitions (\[eq:expect-extended-def\]) and (\[eq:expect-def\]) we have $$\begin{aligned} \mathbb{E}_{\Sigma} [ R\otimes X(t) ] &=& \frac{1}{\vert \alpha \vert^2} \sum_{jk} \alpha_j^\ast \alpha_k \mathbb{E}_{jk}[ X(t) ] \\ &=& \frac{1}{\vert \alpha \vert^2} \mathbb{E}_{\eta\Psi} [ X(t) ] ,\end{aligned}$$ and in particular $$\mathbb{E}_{\Sigma} [ R \otimes I ] = \frac{1}{\vert \alpha \vert^2} .$$ From these expressions, we see that $$\mathbb{E}_{\eta\Psi} [ X(t) ] = \vert \alpha \vert^2 \mathbb{E}_{\Sigma} [ R\otimes X(t) ] = \frac{\mathbb{E}_{\Sigma} [ R\otimes X(t) ] }{\mathbb{E}_{\Sigma} [ R\otimes I] },$$ which proves (\[eq:mu-relation\]). The differential equation (\[eq:mu-master\]) follows from the QSDE (\[eq:X-dyn\]) for $X(t)=j_t(X)$ and relations (\[eq:extended-expect-1\])-(\[eq:extended-expect-3\]) upon evaluating the differential $d \mathbb{E}_{\Sigma}[ A \otimes X(t) ]$. [$\Box$]{} \[thm:master-cat\] The unconditional expectation $\mu_t(X)$ when the field is in the superposition state $\vert \Psi \rangle$ (defined by (\[eq:super-state\])) is given by $$\mu_t(X) = \frac{ \sum_{jk} \alpha_j^\ast \alpha_k \mu^{jk}_t(X) }{\sum_{jk} \alpha_j^\ast \alpha_k \mu^{jk}_t(I) }, \label{eq:mu-cat-rep}$$ where $\mu^{jk}_t(X)$ is given by the system of equations $$\frac{d}{dt} \mu^{jk}_t(X) = \mu^{jk}_t( \mathcal{G}^{jk}_t(X) ), \label{eq:dot-mu-jk-c}$$ and where $$\mathcal{G}^{jk}_t(X) = \mathcal{L}(X) + S^\ast [X, L ] f_j^\ast(t) + [ L^\ast, X] S f_k(t) + (S^\ast X S - X) f_j^\ast(t) f_k(t) .$$ The initial conditions are $$\mu^{jk}_0(X)= \langle \eta \vert X \vert \eta \rangle g_{jk}.$$ Define $$\mu^{jk}_t(X) = \mathbb{E}_{jk}[ X(t) ].$$ Then as in the proof of Lemma \[lemma:master-1-cat\] we may show that $$\mu^{jk}_t(X) = \frac{\vert \alpha \vert^2}{\alpha_j^\ast \alpha_k} \tilde\mu_t( \vert e_j \rangle \langle e_k \vert \otimes X) .$$ The the relation (\[eq:mu-cat-rep\]) follows from (\[eq:mu-relation\]). The differential equation (\[eq:dot-mu-jk-c\]) follows from equation (\[eq:mu-master\]) with $A=\vert e_j \rangle \langle e_k \vert$. [$\Box$]{} Superposition State Filter {#sec:cat-super-filter} -------------------------- In this section we show how the conditional expectation $\hat X(t)=\pi_t(X)$ defined by (\[eq:cond-exp\]) can be evaluated using a system of conditional equations. This will make use of the conditional expectation $$\tilde \pi_t( A \otimes X) = \mathbb{E}_{\Sigma} [ A \otimes X(t) \, \vert \, I \otimes \mathscr{Y}_t].$$ for the extended system. \[lemma:filter-extended-cat\] The conditional expectation $\hat X(t)=\pi_t(X)$ defined by (\[eq:cond-exp\]) with respect to the superposition state $\vert \Psi\rangle $ is given by $$\pi_t(X)= \frac{ \tilde\pi_t(R \otimes X) }{ \tilde\pi_t( R \otimes I) }. \label{eq:pi-relation}$$ The quantum filter for the conditional expectation $\tilde\pi_t(A\otimes X)$ is $$\begin{aligned} d \tilde \pi_t( A \otimes X) & = & \tilde \pi_t( \mathcal{G}_t(A \otimes X)) dt + \mathcal{H}_t( A \otimes X) dW(t) \label{eq:super-filter}\end{aligned}$$ with initial condition $\tilde\pi_0(A\otimes X)= \frac{1}{\vert \alpha \vert^2} \langle \eta, X \eta \rangle \sum_{jk} a_{jk} \alpha_j^\ast \alpha_k$, where $$\begin{aligned} \mathcal{H}_t( A \otimes X) &=& \tilde \pi_t( A \otimes X (I \otimes L + C(t) \otimes S) + (I \otimes L^\ast + C^\dagger(t) \otimes S^\ast) A \otimes X ) \nonumber \\ && - \tilde \pi_t( A \otimes X) \tilde \pi_t( I \otimes L + C(t) \otimes S + I \otimes L^\ast + C^\dagger(t) \otimes S^\ast )\end{aligned}$$ and $W(t)$ is a $\mathscr{Y}_t$-Wiener process given by $$\begin{aligned} dW(t) &=& dY(t) - \tilde \pi_t( I \otimes L + C(t) \otimes S + I \otimes L^\ast + C^\dagger(t) \otimes S^\ast ) dt, \ W(0)=0.\end{aligned}$$ The initial condition is $\tilde \pi_0(A\otimes X)= \frac{1}{\vert \alpha \vert^2} \langle \eta \vert X \vert \eta \rangle \sum_{jk} a_{jk} \alpha_j^\ast \alpha_k$. Let $K \in \mathscr{Y}_t$. Then we have $$\begin{aligned} \mathbb{E}_{\Sigma} [ \tilde\pi_t( R\otimes X) (I\otimes K) ] &=& \mathbb{E}_{\Sigma} [ (R \otimes X(t)) (I \otimes K) ] \notag \\ &=& \frac{1}{\vert \alpha \vert^2} \sum_{jk} \alpha_j^\ast \alpha_k \mathbb{E}_{jk}[ X(t) K] \notag \\ &=& \frac{1}{\vert \alpha \vert^2} \mathbb{E}_{\eta\Psi}[ X(t)K ] \notag \\ &=& \frac{1}{\vert \alpha \vert^2} \mathbb{E}_{\eta\Psi }[ \pi_t(X) K ] \notag \\ &=& \mathbb{E}_{\Sigma}[ R \otimes \pi_t(X) K ] \notag \\ &=& \mathbb{E}_{\Sigma}[ \mathbb{E}_{\Sigma} [R \otimes \pi_t(X) K \vert I \otimes \mathscr{Y}_t ]] \notag \\ &=& \mathbb{E}_{\Sigma}[ \tilde\pi_t(R\otimes I) (I \otimes \pi_t(X)) (I \otimes K) ] .\end{aligned}$$ This proves (\[eq:pi-relation\]). The filtering equation (\[eq:super-filter\]) is derived using the characteristic function method [@HSM05], [@VPB93], [@VPB92]. We postulate that the filter has the form $$d \tilde\pi_t( A \otimes X) = \mathcal{F}_t(A \otimes X) dt + \mathcal{H}_t(A \otimes X) I\otimes dY(t) ,$$ where $\mathcal{F}_t$ and $\mathcal{H}_t$ are to be determined. Let $f$ be square integrable, and define a process $c_f(t)$ by $$dc_f(t) = f(t) c_f(t) dY(t), \ \ c_f(0)=1.$$ Then $I \otimes c_f(t)$ is adapted to $I \otimes \mathscr{Y}_t$, and the definition of quantum conditional expectation [@BHJ07 sec. 3.3] implies that $$\mathbb{E}_\Sigma [ A \otimes (X(t) c_f(t) ) ] = \mathbb{E}_\Sigma [ \tilde \pi_t( A \otimes X(t)) (I \otimes c_f(t) ) ]$$ holds for all $f$. By calculating the differentials of both sides, taking expectations and conditioning we obtain $$\begin{aligned} \mathbb{E}_\Sigma [ A \otimes (dX(t) c_f(t)) ] &=& \mathbb{E}_\Sigma [ (I \otimes c_f(t) ) \tilde\pi_t( \mathcal{G}(A \otimes X) ) \\ && + (I \otimes f(t) c_f(t) ) \tilde \pi_t( (A \otimes X)(I \otimes L + C(t) \otimes S) \notag \\ && + (I \otimes L + C^\dagger(t) \otimes S^\ast) (A \otimes X) )]dt \ \ \ \notag\end{aligned}$$ and $$\begin{aligned} && \mathbb{E}_\Sigma [ A \otimes (d\tilde \pi_t(A\otimes X) c_f(t)) ] \\ &=& \mathbb{E}_\Sigma [ (I \otimes c_f(t) \{ \mathcal{F}_t(A\otimes X) + \mathcal{H}_t(A\otimes X) \tilde \pi_t( I\otimes L+ C(t)\otimes S + I\otimes L^\ast + C^\dagger(t) \otimes S^\ast ) \} \notag \\ && + (I \otimes f(t) c_f(t) ) \{ \ \tilde \pi_t(A\otimes X) \tilde \pi_t( I\otimes L+ C(t)\otimes S + I\otimes L^\ast + C^\dagger(t) \otimes S^\ast)+ \mathcal{H}_t(A\otimes X) \} ] dt. \notag\end{aligned}$$ Now equating coefficients of $c_f(t)$ and $f(t) c_f(t)$ we solve for $\mathcal{F}_t(A\otimes X)$ and $\mathcal{H}_t(A\otimes X)$ to obtain the filtering equation (\[eq:super-filter\]). We now show that $W(t)$ is a $\mathscr{Y}_t$-martingale, and since $dW(t)dW(t)=dt$, then by Levy’s theorem [@RE82] we have that $W(t)$ is a $\mathscr{Y}_t$-Wiener process. Indeed, for any $K \in \mathscr{Y}_t$ we have $$\begin{aligned} && \mathbb{E}_{\Sigma}[ (I\otimes dW(t) )(I\otimes K) ] \notag \\ &=& \mathbb{E}_{\Sigma}[ (I\otimes dY(t) - \tilde\pi_t(I\otimes L+ C(t)\otimes S + I\otimes L^\ast + C^\dagger(t) \otimes S^\ast) dt)(I\otimes K) ] \notag \\ &=& \mathbb{E}_{\Sigma}[ I\otimes (L(t)+L^\ast(t)) + C(t) \otimes S + C^\ast(t) \otimes S^\ast \notag \\ && - \tilde\pi_t( I\otimes L+ C(t)\otimes S + I\otimes L^\ast + C^\dagger(t) \otimes S^\ast) dt)(I\otimes K) ] dt =0.\end{aligned}$$ This completes the proof. [$\Box$]{} \[thm:filter-cat\] The unconditional expectation $\mu_t(X)$ when the field is in the superposition state $\vert \Psi \rangle$ (defined by (\[eq:super-state\])) is given by $$\pi_t(X) = \frac{ \sum_{jk} \alpha_j^\ast \alpha_k \pi^{jk}_t(X) }{\sum_{jk} \alpha_j^\ast \alpha_k \pi^{jk}_t(I) }, \label{eq:pi-cat-rep}$$ where the conditional quantities $\pi^{jk}_t(X)$ are given by $$\begin{aligned} d \pi^{jk}_t(X) &=& \pi^{jk}_t( \mathcal{G}^{jk}(X) )dt + (\pi^{jk}_t( X(L +Sf_k(t) ) + (L^\ast + S^\ast f_j^\ast(t) ) X) \notag \\ && - \pi^{jk}_t(X) \sum_j \frac{\vert \alpha_j \vert^2}{\vert \alpha \vert^2} \pi^{jj}_t( L+ S f_j(t) + L^\ast + S^\ast f_j^\ast(t) ) ) dW(t) \notag\end{aligned}$$ The innovations process $W(t)$ is a $\mathscr{Y}_t$ Wiener process with respect to the superposition state $\vert \Psi \rangle$ and is given by $$dW(t) = dY(t) - \sum_j \frac{\vert \alpha_j \vert^2}{\vert \alpha \vert^2} \pi^{jj}_t( L+ S f_j(t) + L^\ast + S^\ast f_j^\ast(t) ) dt . \label{eq:innovation-11-cat}$$ The initial conditions are $$\pi^{jk}_0(X)= \langle \eta \vert X \vert \eta \rangle g_{jk}.$$ These assertions follow upon substitution of $$\pi^{jk}_t(X) = \frac{\vert \alpha \vert^2}{\alpha_j^\ast \alpha_k} \tilde\pi_t( e_je_k^\ast \otimes X) .$$ into the relevant expressions from Lemma \[lemma:filter-extended-cat\]. [$\Box$]{} Discussion and Conclusion {#sec:conclusion} ========================= In this paper we have derived the master equation and quantum filter for a class of open quantum systems that are coupled to continuous-mode fields in non-classical states: (i) single photon states, and (ii) superpositions of coherent states. The quantum filter in both of the cases we consider consists of coupled equations that determine the evolution of the conditional state of the system under continuous (weak) measurement performed on the output field, in contrast to the familiar single filtering equation for open Markov quantum systems that are coupled to coherent boson fields. This coupled equations structure of the master and filter equations is a reflection of the non-Markov nature of systems coupled to the non-classical fields. Indeed, a key feature of our approach is the embedding of the system into a larger extended system, a technique often employed in the analysis of non-Markov systems, providing an elegant framework within which to study the the dynamics, both unconditional and conditional, of the system. In contrast to Markovian embeddings [@HPB04], [@GJN11a], [@GJNC11], the extended system (including the field) is initialized in a superposition state. This embedding provides a framework within which the tools of the quantum stochastic calculus may be efficiently applied to determine quantum filtering equations. We expect that the use of suitable embeddings, both Markovian and non-Markovian, could be adapted to study quantum systems that are coupled to other types of highly non-classical fields. Acknowledgement {#acknowledgement .unnumbered} =============== The authors wish to thank J. Hope for helpful discussions and for pointing out reference [@HPB04] to us. 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Email: Matthew.James@anu.edu.au [^3]: Research School of Engineering, Australian National University, Canberra, ACT 0200, Australia. Email: Hendra.Nurdin@anu.edu.au [^4]: This work was supported by the Australian Research Council and the UK Engineering and Physical Sciences Research Council grant EP/G039275/1. [^5]: In terms of annihilation and creation white noise operators $b(t), b^\ast(t)$ that satisfy singular commutation relations $[b(s), b^\ast(t)]=\delta(t-s)$, the fundamental field operators are given by $B(t) = \int_0^t b(s) ds$, $B^\ast(t)= \int_0^t b^\ast(s) ds$, and $\Lambda(t)= \int_0^t b^\ast(s) b(s) ds$. Also, we may write $B(\xi) = \int_0^\infty \xi^\ast(s) dB(s)$. [^6]: Recall $B(t)=B_{in}(t)$ is the input field. [^7]: $\mathscr{Z}_t$ and $\mathscr{Y}_t$ are commutative von Neumann algebras. They are also filtrations, e.g. $\mathscr{Z}_{t_1} \subset \mathscr{Z}_{t_2}$ whenever $t_1 <t_2$.
--- abstract: | Quantum Relativity is supposed to be a new theory, which locally is a deformation of Special Relativity, and globally it is a background independent theory including the main ideas of General Relativity, with hindsight from Quantum Theory. The qubit viewed as a Hopf monopole bundle is considered as a unifying gauge “group”. Breaking its chiral symmetry is conjectured to yield gravity as a deformation of electromagnetism. It is already a quantum theory in the context of Quantum Information Dynamics as a discrete, background independent theory, unifying classical and quantum physics. Based on the above, Quantum Gravity is sketched as an open project. author: - 'Lucian M. Ionescu' date: 'May 15, 2010' title: Quantum Relativity --- \#1[by \#1 ]{} \#1[by \#1 ]{} 0.5in 6.5in 8.5in xypic epsf Preamble ======== The present article aims to present two important ideas in the context of a general conceptual framework called [*Quantum Relativity*]{}, which is the “space-time oriented” companion of Quantum Information Dynamics: 1\) The unifying gauge “group” is the qubit, viewed as the [*Hopf monopole bundle*]{}: $$SU(1)\cong S^1\to SU(2)\cong S^3\to S^2;$$ 2\) “Gravity” is a [*deformation*]{} of Electromagnetism. Therefore in the context of the discrete background independent Quantum Information Dynamics, it is already a “quantum” theory. The focus will be on the physics interface, supported by mathematical ideas from deformation theory. There will be no details nor final formulas, since the present approach is tentative at this stage, occasionally presenting conflicting alternatives as part of the quantum gravity “puzzle” ... Instead, the author hopes that the reader will be interested in pursuing some of the questions raised directly or, quite often indirectly, through claims based on intuition. Introduction ============ From a Computer Science point of view of Mathematical-Physics a new paradigm emerges: [*Quantum Information Dynamics*]{} [@I-MPCS]. It is the mathematical-physics component of the [*Digital World Theory*]{} [@I-DWT], providing a unifying framework for both classical and quantum physics, based on graph complexes and their cohomology [@Kon; @CK; @I-FL; @IF]. The main point is that the role of Electromagnetism was underestimated, even after the remarkable discovery of Aharonov-Bohm a. a. of the close connection between quantum phase and [*classical*]{} EM as a $SU_1$-gauge theory. A closer inspection of [*Special Relativity*]{} reveals that not only time is not “absolute” as a conclusion of a Einstein’s critical analysis of the concept of synchronization, but also the concept of “direction” (parallelism) is subject to the same criticism. Therefore an $SO_3$-connection is mandatory to make sense of “direction correlation”; the assumption of a metric with its induced Levi-Civita connection is clearly an extra assumption, no longer appropriate in a modern physics dominated by quantum theory. In conclusion [@I-MPCS], EM as a $SU_1$-gauge theory is only a Hopf “fiber” of QID as a $SU_2$-Yang-Mills theory on whatever “space” is. The author claims that “space” should be a model for the structure of matter, preferably a dynamical lattice, as argued in [@I-DWT], and “time” per se, with physical meaning is not one dimensional, since it should model the [*change of structure*]{}, preferably as a cobordism [^1]. This yields a nice [*Background Space-Time Independent Theory*]{} (BIT), from which Quantum Mechanics (QM) emerges, with Heisenberg “uncertainty relations” as its trademark due to an averaging over possible space-time coordinate systems, which are embedding in an ambient space-time manifold as in [*String Theory*]{}. From QID also [*Quantum Field Theory*]{} emerges, with its Feynman diagrams and variable number of particles trademark, when taking full advantage of graph homology and loop derivative, as in [*Loop Quantum Gravity*]{}. When we say “emerges”, we mean that a reasonably simple [*interface*]{} exists between the two theories (conceptual level), allowing to incorporate the results of one theory into the other, as if it would be a “plug-in”, once the differences of languages used for their implementations are identified. But then, why is there a “Gravity” to “spoil” this nice picture? In this article we will (try) to prove that “Gravity” is a perturbation of “Electromagnetism” (EM), related to the standard bialgebra deformation technique from quantization [^2]. Although the “bigger picture” seems to indicate that the “speed of light” is in fact an isomorphism (with four eigenvalues) corresponding to a family of deformations of Hodge duality, probably dual to Dirac quantization at the level of infinitesimal symmetries: $$F(G)_c \times U(g)_h\to \mathbf{C},$$ we will restrict our attention to [*deforming Special Relativity*]{}. Indeed, Lorentz group is a one-parameter [*infinitesimal*]{} deformation of Galilei group. In view of General Relativity, it represents the local symmetries of physical processes (classical or quantum). Therefore we believe that one needs to Galilei group, [*not*]{} to “quantize” General Relativity! As hinted above, we claim that “quantization” comes for free, from the discretization process and categorical approach [^3]. The byproduct of deformation of symmetries, via Noether Theorem and Euler-Lagrange Equations, should be a “deformed Lorentz force”, incorporating a gravitational attraction term. Again, once the [*local picture*]{} is taken care of, the [*global picture*]{} is the result of applying the homological algebra machinery: classical information (external space) as a lattice (chains), “time” as cobordisms, quantum information modeled via $SU_2$-coefficients as internal space, and the emergence of classical properties from quantum ones via measurement bases, as well as entanglement due to feedback loops, a consequence off graph (cobordism) extentions and internal-external cohomological duality (IE-duality) [@I-MPCS]. The next section is devoted to the fundamental correspondence [*Quantum Computing - Special Relativity*]{}, used also as a starting point in twistor theory. Section §3 investigates Newton’s law of the gravitational force as a perturbation of the electric force. Since the “issue” is not resolved in such a straightforward manner, alternative “mathematical guns” are thrown into the battle. The concluding section is a preview of the program for developing the new paradigm in physics [@I-MPCS]: QID, and its relations with existing directions of research, notably with String Theory. The background together with the technical material are confined to the Annex. Quantum Computing Model of Special Relativity ============================================= There is a correspondence between the Quantum Computing model, modeling the flow of qubits through the quantum gates of a Quantum Netqork (Q-Net) [@QC], and the Minkowski space $R^{3,1}$ with its Lorentz/Poincare symmetries. It can be traced to the Klein correspondence with its physical interpretation in [*Twistor Theory*]{} [@Twistor-Theory]. It will be denoted by $QC\cong SR$, and hinted to by $2+2\cong 3+1$ (level of dimensions). It is also referred to as the [*Hermitian Model*]{} of SR, since to a 4-vector it associates a hermitian matrix. $SL_2(C)$ and quaternions ------------------------- Quaternions, as a division algebra, can be viewed as the result of a double construction, analogous to the construction of complex numbers: $$H=C\oplus C^*, J:H\to H, J^2=-1.$$ Here the dual of $C$ has conjugate vector space structure, so $J$ is equivalent to a complex conjugate linear map $C\to C$. It is in fact a [*generalized complex structure*]{} (GCS) [@GCS]. Conjugation by non-zero quaternions $C:H^{\times}\to Aut(H)$, with kernel $R^{\times}=<1>^{\times}$, induces the polar decomposition: $$\label{E:SR-CE} Z(H)=R\to H \to g=(R^3,\times).$$ Here $g$ is the [*angular velocity*]{} Lie algebra with cross-product as a Lie bracket. The group of unit norm quaternions is $SU_2$. Under Exponentiation of the ad-action: $$\diagram g \rto^{Ad} & Aut(g) & x \mapsto ad_x\\ SL_2(C) \rto^{C} & Aut(SU_2) \enddiagram$$ yields the action of $SL_2(C)$ on $SU_2$ by conjugation on itself by rotations $SO_3$ [@Kirillov-OM]: $$ad_\omega (v)=\omega \times v.$$ As a central extension of the Lie algebra $g$, it is associated to a 2-cocycle related to the Einstein addition of velocities. It will be called the [*QC/SR central extension*]{}. At the level of groups we have the following short exact sequence of 2:1 covering maps: $$\diagram SU_2\dto^{2:1} \rto & SL_2(C)\dto^{2:1} \\ SO_3 \rto & SO(3,1)^+. \enddiagram$$ Hermitian model --------------- It induces a correspondence between hermitan matrices $\xi$ and quadri-vectors $u=(x,y,z,ct)\in M^{3,1}$ of Minkowski space: $$\mathcal{H}\ni \xi=\left( \begin{array}{cc} z-ct & x-iy \\ x+iy & z+ct \end{array} \right).$$ The conformal structure [@Supersymmetry] corresponds to the Minkowski norm: $$det(\xi)=||u||^2.$$ Wick isomorphism ---------------- $sl_2(C)$ has two isomorphic [*real forms*]{} $sl_2(R)$ and $su_2$ (real structure constants): $$sl_2(R): [K^+,K^-]=S, [K^\pm,S]=\pm S\quad su_2: [J_x,J_y]=J_z\ (cyclic),$$ note also $su_2\cong so_3$ ($\sigma_x$ etc. are Pauli matrices): $$i \sigma_x \mapsto J_x \ etc.$$ and $su_2\otimes C\cong sl_2(C)$. A real vector space isomorphism can be defined in terms of canonical bases (compare with [@Kir-LGLA], p.47): $$WR:sl_2(R)\to su_2, \quad K^\pm \mapsto \sigma_x\pm i\sigma_y, \ S\mapsto \sigma_z,$$ called [*Wick rotation*]{}. Then $sl_2(C)$ can be viewed as $g\oplus g^*$, since $su_2\cong so_3$, with $so_3^*$ the [*angular momentum*]{} Lie algebra (dual to $g$). So $sl_2(C)$ has a canonical generalized complex structure, determined by the Wick rotation $$J:sl_2(C)\to sl_2(C).$$ The real $Re(\xi)$ of a traceless matrix is the projection on the non-compact form $sl_2(R)$, and the imaginary part is the projection on the compact form $su_2$. The Lorentz 2-cocycle --------------------- The QC/SR central extension determines an infinitesimal deformation of the Lie algebra $(g, \times)$ with deformation parameter $c$. Since later we will fully deform $g$, we view the “speed of light” as a [*formal parameter*]{} $\overline{h}=1/c$ $$u=ct +\vec{v}, \quad u/c=t+\vec{v}\overline{h}+\dots\ .$$ In fact the four-speed of light quadri-vector $\mathbf{c}$ consists in the eigenvalues of the Hodge duality isomorphism ($c^2=1/\mu\epsilon$), and therefore we in fact deform the duality, as we should when using the machinery of bialgebra deformation (“quantization”). The relations between fundamental constants, interpreted as topological generators (periods): $$g_m=h/(e/c),\quad g_e=e/c, \quad h=g_m g_e, \quad \alpha=g_m/g_e,$$ together with the duality between the two deformation parameters $\overline{h}=1/c$ and $h$: $$h\cdot 1/c=\alpha (e/c)^2,$$ will be addressed elsewhere. Here we only aim to find a possible relation between Newton’s gravitational constant and the above “fundamental” “constants” (The term “constants” refers to the periods, not to the “running constants” which involve the 2-cocycle which makes them variable with energy, like relativistic mass, electric charge/fine structure constant etc. [@Quarks]; the periods are in some sense the “rest” values: Lie generators/Hopf primitives). Returning to the QC/SR central extension of the angular velocity Lie algebra $$Z(H)\to (H, [,]) \overset{\pi}{\to} (g, \times)$$ where $[q,q']=\pi(q)\times \pi(q')$ since quaternionic product is $$(t,v) (t',v')=tt'-v \cdot v'+ v \times v'.$$ Note that 3D-vectors are interpreted as velocities (elements of the angular velocity Lie algebra $g$), rather then position vectors. Now consider the exponential: $$exp: {\mathcal{H}} \subset sl_2(C)\to SU_2 \subset SL_2(C).$$ Recall that conjugation by $SL_2(C)$ invaries $SU_2$. Let $\gamma:g\to H$ be the trivial section $\gamma(v)=v$. The Einstein velocity addition for parallel velocities [@Ungar] reduces for parallel boosts to an associative and commutative group operation: $$u\oplus v=\frac{u/c+v/c}{1+u/c\cdot v/c}$$ There is an associated 2-cocycle $$F(u,v)=\gamma(u\oplus v)/(\gamma(u)\gamma(v)).$$ We claim that $\oplus$ is an infinitesimal deformation of the Lie algebra $g$ addition of velocities through the logarithm: $$u\oplus v=u+v+ ...$$ and will be addressed later on. Time and projective space ------------------------- “Time” is a parameter for labeling change. In mechanics “change” occurs in the direction of “motion”. So “time” is associated to change in the radial direction of motion. A better adapted model of motion is polar decomposition: $$R^{3x}=S^2\times R_+.$$ With the hidden quantum phase, the role of velocity is that of de Broglie wave vector $\vec{k}$, i.e. rather an angular velocity vector (element of $g$). In QID “motion” is a change of angles (direction correlation) and radial distance (resistence to interaction, or its inverse, capacity of interaction). The 2-cocycle of SR corresponds to a central extension associated to a projective representation. This in turn comes from a representation at the level of projective space as a homogeneous space. It is the base of the Hopf monopole bundle, the [*qubit model of the atom*]{}: $$S^1\to S^3\to S^2 \quad SU_1\to SU_2\to S^2.$$ In passing we mention that this CS-point of view suggests to take the Hopf bundles (monopole and instanton) as the “unifying gauge group” of all interactions [@I-MPCS] ($SU_3$ comes for “free”). More precisely $S^2=CP^1$, $MT=Aut(CP^1)=PSL_2(C)$: $$\xi(z1:z2^*)=(\xi z_1 : \xi^* z_2^*).$$ The Contraction Lorentz to Galilei ================================== The homogeneous Galilei group is the contraction $c\to \infty$ of the Lorentz group [@Kir-LGLA]. a more general setup is the Anti-de Sitter group as a 2-parameter deformation of Galilei group [@SR21]. Galilei group ------------- The Galilei group is the Klein group of the homogeneous space $R^3$, a semi-direct product $$Rot \times Translations$$. As a Klein geometry (kinematics) it is better to view translations as discrete displacements with velocity $v$, i.e. as an affine space / groupoid: $${\mathcal{G}}al: \{ A\to B\}.$$ Without spin there is no angular velocity interpretation, so no (Lie) curvature. Now abandon the classical and deterministic “pointwise physics” and adopt a “bilocal physics” of interactions (quantum mechanics, Cramer’s tranzactional interpretation, Feynman-Wheeler theory, Aharonov-Vaidman bistates quantum mechanics etc.). Translations are banished, every transformation becomes a two fixed point rotation (Feynman paths: I/O-transformations) and space-time is “compactified” as a category of systems and processes morphying one system into another (time as cobordisms). At mathematics level this invites conformal geometry, and at our local model, Moebus transformations (MT). Think that such a MT corresponds to two Bloch vectors (four points: $0, A, \infty, A'$): the “dumbbell picture” of QC (instanton Hopf bundle). Wick isomorphism allows to transport the Lie bracket into a cobracket on $g^*$, yielding a bialgebra structure on $sl_2(C)$. With such a Poisson-Lie structure, one can apply the quantum double construction. Lorentz force ------------- It can be viewed as part of Euler-Lagrange equation. The magnetic term $v\times B$ can be interpreted as a rotation: $Omega(v)$. The Euler equation for $G/g$ is [@Kir-LGLA]: $$\dot{\Omega}=ad^*_v \Omega, \quad v\in g.$$ To understand Gravity as a deformation of EM, consider the whole picture in terms of canonical momentum $P=mv+eA$ versus netwon’s picture separate from Lorentz force: $$d(mv)/dt=q (E + v\times B).$$ Also do not separate particles from fields, as in “pointwise physics”; rather consider the categorical interaction picture (“bilocal physics”). $$d(m_{red} v)=G_N m m'/r^2 + k_e q q'/r^2 + qq' v \times B/q'.$$ here $m_{red}$ is the [*reduced mass*]{}: $$2/m_{red} = 1/m + 1/m'.$$ If $$mc^2=\tau^2, \quad R=log \tau$$ “resistance to change” adds up in “parallel”. This point of view leads to a reinterpretation of Bio-Savart Law as a Gaussian Link; in this sense, as Descartes was saying, every motion is a rotation ... So $q,q'$ are indexes (winding numbers/Chern classes of the two Hopf bundles) and $m,m'$ are associated to the corresponding momentum maps (Hamiltonian circle reduction interpretation of the qubit as a harmonic oscillator [@I-MPCS]). $SL_2(C)$-Gravity ================= We essentially deform $SL_2(C)$-gravity, except we consider Einstein’s equations of General Relativity as expressing a local constraint. It is the curvature of a reductive Lie group: $$Ric + Killing/4=0.$$ When “adding” homology (Q-net chain / QI cochain cohomology with $SU_2$-coefficients), one obtains a non-homogeneous version with a topological RHS which represents matter via 1,2 and 3-periods (fluxons, e/m-charges and spin/action [@Kiehn; @Post; @EMTC; @NLEM]). So the present goal is to justify Newton’s law for Gravity and the associated constant. The parallel between Newton’s constant $G_N$ and Coulomb’s constant $k_e$ clearly indicates that this requires unification of internal and external DOF IE-duality [@I-DWT]), which locally is expressed by the GCS of $sl_2(C)$ (Wick isomorphism): $$\frac{\partial L}{\partial \dot{r}}=P=mv+eA \quad \leftrightarrow Wicki:sl_2(R)\to su_2.$$ here $A$ should be the $SU_2$-yang-Mills connection, as an obvious “upgrade” of EM (it should yield “quark” DOFs for free; see also [@NLEM]). Relation with Higgs mechanism ----------------------------- The breaking of symmetry approach to mass [@Witten] should be adapted to the grand unifying Hopf bundle picture and its associated Gysen sequence. In view of Stashef’s cohomological physics and [@I-FL], it would be a surprise to find that the connecting homomorphism couples EM and Gravity. In the homological algebra formulation of QID interactions should emerge as derived functors of the action (number of action quanta/entropy and harmonics/fluxons). Quantum gravity --------------- So we could say that gravity is “induced” as a “quantum perturbation”, if we compare our approach with the Shacarov’s induced gravity. In fact in QID the Q-net (matter with its classical properties dependent on a space-time coordinate system) emerges dynamically as classical information from internal QI, under IE-(cohomological) duality. Gravity does not emerge as in [@Verlinde] but rather as a “non-linear” (or rather non-commutative) effect; it is the result of deformation via an exponentiation process, to be explained next, being in some sense “quantum”. Mass and electric charge ------------------------ We have focused on gravity, but what is electric charge, and why is it different from mass as a source of “forces” (interactions)? To address this issue we critically review Newton’s law of gravity, with hindsight from electromagnetism. First, we should acknowledge that interactions are relational, so if two systems S1 and S2 have parts (subsystems), then if the separation distance is large compared to the internal substructure, then then the mutual interaction is well modeled by a bipartite graph: $$F_{12}=\sum_{ij} F_{ij}\approx N_1 N_2 F,$$ where $F$ is a typical pair of interacting particles. So in a typical Coulomb law, associated to Newton’s law $ma=F$, LHS will contain a mass coefficient due to system S1, but in the force term in the RHS will be proportional to the products $M_1M2$ and $q_1q_2$ (we disregard spin at this time): $$M_1 r''=Coef. f(M1,M2;q1,q2)/r^2.$$ In fact the essence is the potential of an interacting pair of particle: $$V_{12}=Charge_1 \times Charge_2 \times Conductivity.$$ Here “space” conductivity is $1/r$, i.e. distance is “resistance” to transfer of momentum-energy (interactions). The charges are weights corresponding to the types of particles, via their symmetries (gauge group, Noether theorem). They are in fact the coupling constants, since the total charges are mathematically speaking dimensions or indexes (absolute or net charges, like mass and electric charge). Since we know matter is discrete and mass and charge are additive, we can reduce the global low of interaction to the case of a pair of elementary “particles”. But there is only one: the qubit / Hopf bundle; the electron corresponds to $S^1$ and proton corresponds to $S^2$. Assume that some electrons are “separated” from the protons, by having elongated orbitals shared by the two systems. Let $m^i_\pm$ denote the mass of protons and electrons in the two systems: $$m_i=m^i_++m^i_-, \quad q_i=m^i_+-m^i_-.$$ At a fundamental level, if all known forces are different aspects of one interaction, one cannot distinguish between gravity and electricity, except by an approximation process. So, in the basic Coulomb law for our two types of interacting particles, protons and electrons with masses $m_+$ and $m_-$, let us introduce distinct Coulomb constants for positive and negative charges: $$K_+=k+\delta, K_-=-k, \quad m_\pm r''=K_\pm e^2/r^2,$$ where $e$ denotes the MKS unit of electric charge. Then the force between two neutral H-atoms, i.e. two qubits with preserved $SU_1$-electron symmetry, is the result of four interaction terms, corresponding to all possible combinations of signs: $$G=(K_++K_-)^2=\delta^2.$$ We claim that the hypothesis that $G$ is Newton’s gravitational constant is worth pursuing, even in the context of the well known precision experiments which claim that $e_+=e_-$; experiments are always “theory dependent”, and sometimes the theory might suggest the experiment which rules out new effects not derivable from the given theory. The underlying idea is that ultimately electric charge is a 2-period (integer) [@Post; @Kiehn; @EM], and the coupling constant is chiral, since the Hopf bundle is. In other words, gravity is due to parity violation, which in turn, via Dirac’s equation interpreted in terms of QID, is a topological fact about the Hopf bundle. The “rest” is achieved through deformation theory. At this stage it is not clear if this has to do with the speed of light as a 2nd deformation parameter, but it definitely is related to the absence of right handed neutrinos (see Dirac equation and Weyl spinors). In conclusion, it is reasonable to investigate if mass is an un-oriented version of charge, while the electric charge an oriented version, both related to the 2:1 covering $SU_2\to SO_3$, derived from the Hopf bundle as a unifying “gauge group” and concept: the building block of the universe, the hydrogen atom is modeled as the qubit, the unit of QI, which in turn is the Hopf bundle, with its multiple roles of harmonic oscillator (Hamiltonian reduction) or local Lorentz symmetries of “space-time”. This double function is related to the “amazing duality” [@Arnold] by conformal inversion between the harmonic oscillator and Kepler’s problem: unification of macro and micro-cosmos (Einstein equations would appear locally as the fundamental equation of curvature in a symmetric space $Ricc+Killing/4=0$). Additional arguments regarding the relation between mass and electric charge will be presented later on, in connection with the possible meaning of the vector potential of EM. Chirality and bialgebra deformation ----------------------------------- Now at a technical mathematical level, deformation of EM due to the chirality of the Hopf bundle (Gysen sequence) is similar to the deformation of universal enveloping algebra of a Lie algebra (quantum groups), which can be implemented via BCH-formula. This in turn corresponds to a pull-back of the non-linear group law at the level of the Lie algebra (generators of symmetry, e.g. Lorentz/Poincare Lie algebra), which corresponds to the Birkhoff decomposition of the Poisson-Lie group (R-matrix and all that [@QG-PC]). From this angle one can say that the Birkhoff decomposition of $SU_2$-Yang-Mills yields the gravitational force as a residual Electromagnetic force: $$F_G=F_{EM}^+-F_{EM}^-.$$ But at the level of the charge multiplicative RHS of Newton’s equation, the gravitational constant could conceivably be related to the exponential of another universal constant, which cannot be other then the fine structure constant: $$exp(-1/\alpha)\approx 10^{-59}, \quad G\approx 10^{-38}.$$ Looking for additional confirmations, we will briefly speculate on the meaning of the fundamental constants. We are looking for [*relations*]{} between fundamental constants, corresponding to conceptual aspects; we are [*not*]{} looking for [*numerical coincidences*]{}. On Fundamental Constants ======================== By now there is ample evidence that not only action is quantized (Plank-Einstein), but also magnetic flux and electric charge (fluxons): Aharonov-Bohm effect, quantum Hall and Josephson effects, supraconductivity and Abrikosov lattice (vortex filaments), vorticity of superfluid helium. These quantities were interpreted as periods [@Post; @Kiehn]: $$1-periods: \ fluxons \quad h/e,$$ $$2-periods: \ charge \quad e,$$ $$3-periods: \ action \quad h.$$ From the “mechanical side” of superfluids [@Heliu; @London] another quantum appears: $h/m$. Since what is really conserved is the canonical momentum $P=mv+qA$, where $A$ denotes the EM-vector potential, this is no surprise. In order to understand fundamental constants we need to better understand the meaning of the vector potential. EM-vector potential as a transfer velocity ------------------------------------------ The vector potential (VP) was originally interpreted by Maxwell as momentum per unit of charge [@VP-articles] (see also [@OP]). We go further and interpret it as a transfer velocity, with the following justification. In a BIT the emphasis is on interactions of pair of systems, [*without*]{} the usual separation between particles and felds: $$Pointwise\ physics: \bullet \to, \quad Interaction\ Physics: \bullet\to \bullet.$$ There is an obvious trend towards such a “bilocal physics”: Feynman-Wheeler theory, Cramer’s transactional interpretation, Aharonov-Vaideman two-states quantum mechanics etc., not to mention “categorification of physics” (Stasheff) and the hidden compactification in QM via projectivisation. In order to see the impact on EM, let’s rewrite Biot-Savart law in terms of velocities of the two particles involved ($e_r=r/|r|$): $$F_B=k q_1q_2 (v_1/r)\times ((v_2/r)\times e_r).$$ the $1/r^2$-coulombian factor should be distributed to both particles, showing that in some sense the interaction is a [*linking number*]{} involving the relative “angular momentum” of the two particles: $$F_B\sim \omega_1\times (\omega_2\times e_r)).$$ This does justice to the Neumann mutual potential as the fundamental ingredient of EM as an interaction theory without a background space-time (and in the spirit of String Theory, if abstracting it from a particular embedding; so, physics is conformal after all!). Recalling that the VP is related to current and velocity as in: $$A=\mu/r \ q v, \quad (continuum: \ J=\rho v),$$ then canonical momentum of a particle $S1$ interacting with a 2nd particle $S2$ becomes: $$P_12=m_1v_1+\mu/r\ q_1q_2 v_2.$$ Here we have discarded gravity, to focus on the idea of [*velocity transfer*]{}, as a legitime version of the “ether dragging” theory (dark matter etc.). Recalling Mach’s viewpoint of relativity, the VP $A_1\sim v_2$, which in fact is related to the EM connection 1-form, represents the “correction” of the meaning of direction of $v_1$ due to the existence of the 2-nd system; “space”, as position and direction, is a collective entity due to the presence of matter, as learned from Einstein’s GR. There are other notable sources supporting this important point: 1) QFT and collapsing a subgraph: what is then the “average space”? 2) Cosmology: the Black-hole information paradox. The solution is the IE-duality [@I-DWT]. So, as a provisional conclusion, interpret $A$ as a transfer velocity of particle 2 “correcting” the “proper space” of particle 1; of course, modulo the conversion of units to be discussed in the next section. Notice though, that in view of Descartes point of view and Maxwell’s original point of view [@VP-thoughts], the alternative is to interpret $qA$ (the VP) as a transfered [*angular momentum*]{}: $$P_{12}=m_1v_1+\mu q_1q_2 \omega_2, \quad \omega_2=e_r\times (v_2/r),$$ with a typical VP due to the 2nd charge moving with velocity $v_2$ at $r$ given by: $$A(r)=\mu/r\ v\times e_r, \quad e_r=r/|r|.$$ Regarding IE-duality, cosmic censure-ship etc., there is a need to rewrite mechanics from its current two tiers: point-mechanics, body-mechanics, very reasonable hierarchy when accepting the continuum of space and matter, in to a Mechanics incorporating hierarchy of structure (resolutions), via IE-duality, which automatically would unite it with thermodynamics, including entropy aspects. In this vein, a system is modeled as having external parameters mass, charge and spin (angular momentum), corresponding to internal states “hidden” from the external classical world (C/QI dichotomy), as it was naturally arived at in cosmology (Kerr black-holes etc.). Therefore we take as a working hypothesis that mass measures circulation at a fundamental level: $$External\ circulation: \ h/m, \quad Internal\ circulation(fluxon): h/e.$$ This could corresponds to breaking the symmetry of (splitting) the Hopf bundle: $$S^1\to S^3\to S^2.$$ Think of “free” electron and protons as $S^1$ and $S^2$, with their cohomological 2-period $e$ and 1-period $m$. This would justify to explore the idea of charge as a 2:1 version of mass ($SU_2\to SO3$), as we proposed above ($m_\pm$ etc.). The Hodge structure and its deformations ---------------------------------------- Bialgebra deformation quantization relies on an “gluing isomorphism” (r-matrix), which is similar to Hodge structure. To understand better the relation between the continuum approach to physics via say Lagrangian manifolds and the discrete approach, recall the general framework of QID: $SL_2(C)$-cohomology of a dg-coalgebra of graph cobordisms (“2-graphs”). At chain/homology level we have closed and exact cycles (bounded by a “surface”; vorticity etc.), and cocycles/cohomology (currents, potentials etc.). The Hodge structure and decomposition on graphs is the best substitute for a “space-time” decomposition/signature, and at the level of chains is related cut-cycles decomposition (spanning tree etc.). At cohomological level it implements the [*constitutive relations*]{} of say EM: $$D=\epsilon E, \ H=1/\mu B, \quad \epsilon\mu=1/c^2.$$ It is interpreted by some authors as a “gauge condition” [@Post; @Kiehn], but it is really an additional structure, reducing the symmetry group. It is a compatibility condition similar to the one defining bialgebras via r-matrices or Frobenius algebras etc.; it is part of the duality. Let’s follow [@Post] and consider $\chi:\lambda^2\to \lambda^2$ the middle dimension isomorphism $G=*F$, where $F=dA=(E,B)$ (forms/vectors, if provided with a metric Hodge duality), and $G=(D,H)$ ($=d^*J$?). In a diagonal form: $$\chi=diag[-\epsilon\ I_3, 1/\mu\ I_3].$$ Its invariants are: $$Hall\ impedence: Z_1=-det(\chi)^{1/3}=\epsilon/\mu,$$ $$Gravity: Z_0=1/3\ Tr(\chi)=1/\mu -\epsilon.$$ The “units problem” is ignored for the moment. Now to keep track of the meaning corresponding to the continuum theories, recall that in the lagrangian formalism $\partial^2L/\partial p_i\partial p_j$ has the role of metric, or, if looking at the Euler-Lagrange equation, that of the masses matrix. In the Hermitian Model (QC$\cong$SR), a qubit/spinor corresponds to a Minkovski 4-vector: $$det(Q)=||(ct,x,y,z)||.$$ The metric/mass relation as in GR, is related to the constitutive relations (“material”/matter), say as in: $$ds^2=\epsilon\ dr^2- 1/\mu\ dt^2, \quad (c^2=1/\epsilon\mu).$$ Of course in GR $c$ is [*not*]{} and invariant, and in fact in optical physics we have 4-light velocities material dependent [@Kiehn]. Hodge structure deformation is reminiscent of GR deformation of metric due to matter. We will suggest this correspondence via the trademark solution of GR, the Schwartzschild metric: $$ds^2=dr^2/(1-r_S/r)-(1-r_S/r)dt^2, \quad r_S=Gm/c^2$$ $$\epsilon=\epsilon_0(1+r_S/r), \mu=\mu_0(1+r_S/r).$$ They are equivalent at first order of approximation. Now we return to the cohomological relations between periods, which should be due to the fundamental unit, the qubit/Hopf bundle: $$\quad Ext \quad Int$$ $$1-period: h/m \quad h/e$$ $$2-period: m \quad e$$ $$3-period: h.$$ The IE-duality prompts for a “symmetrization” of the table, taking $\overline{h}=1/c$, the 2nd deformation parameter (besides Plank’s “constant” $h$), as mass-related/external (deformation of Lorentz/Poincare symmetry group: Anti-de Sitter group $SO(3,2)$; corresponding in QC with $SU(2,2)$). The relations between “fundamental constants” should reflect the relations between generators of Chern classes of the cohomology ring of the Hopf bundle as a coefficient “group” for representing the graph cobordism (quiver): $$h= (h/e) \cdot e, \quad h=h/m \cdot m.$$ One may expect a natural duality between the two deformation parameters: $$hc=\alpha e^2, \quad \alpha=\frac{h/e}{e/c}=\frac{q_M}{q_E},$$ as for the duality between the quantum group of observables $F(G)$ a la Woronovich (say $G$ is the Lorentz group) and the quantum group (via bialgebra deformation quantization) corresponding to the universal enveloping algebra of its Lie algebra: $$F(G)_c <-> U_h(g).$$ This is conceivable in view of the invariants of the Hodge constitutive map $\chi$. Note also that quantum Hall effect introduces another “relation”, quantization of $h/e^2$: $$\sigma_{Hall}=\frac{h/e}{e}, \quad \sigma c=1/\alpha.$$ Now recall that on the “external side” of mass-space-time we have: $$Gm_p^2/hc=\beta\approx 10^{-38},$$ and maybe: $$\ln{G}\sim -1/\alpha,$$ a “puzzle” worth staring at for a while. The amazing duality ------------------- The micro and macro-cosmos correspondence via inversion (harmonic oscillator / Kepler problem [@Arnold]) is reminiscent of T-duality from the String Theory [@ST-web-introd]. This is related with the Bohr’s planetary model for the H-atom, which is essentially the harmonic oscillator. Now the relation between Bohr’s radius (Compton wave length) and Schwartzschild radius for H-atom (qubit) is: $$r_S^p/r_B^p=\frac{Gm_p/c^2}{h/m_pc}=Gm_p^2/hc=\beta.$$ Can we relate it with the constitutive relation $\chi$? let’s try ... Hodge invariants and gravitational constant ------------------------------------------- Recall $P=mv+qA$ or in cgs-units $mv+q/c A$. If switching to the bilocal model: $P=m_1v_1+\mu/r q_1q_2 v_2$. With “better” units [@Post] and considering unit charges: $$P_1=m_1v_1+\frac{\mu}{\epsilon} e^2 v_2/r.$$ Now in the bilocal physics angular velocity and angular momentum form a GCS $g\oplus g^*$, while linear momentum should be translated (compactified via projective spaces) into angular momentum (“Descartes motto”), providing additional insight into the distinction between mass and electric charge. Now assume [*distance being quantized*]{} as a multiple of the Bohr radius, with $r/r_B=n$ principal orbital number. Assuming the non-relativistic Bohr model of H-atom (e.g. [@Post-QR], p.156): $$E_n=1/2 m_ec^2\ \alpha^2/n^2,$$ the above coupling coefficient $\mu/\epsilon /r$ becomes (with $m=m_e$): $$Z_{Hall} \frac{e^2}{r_B} \frac{E_n}{\alpha mc^2} =Z_{Hall} \frac{E_n}{E_\infty},$$ where the rest energy was denoted by $E_\infty$. From the above relation may speculate that there could be a relation between $K_\pm$ and $\epsilon$ and $\mu$ of the form $$K_-=-\sqrt{\epsilon}, \quad K_+=1/\sqrt{\mu}$$ yielding a possible meaning to the other invariant of $\chi$: $$\sqrt{G}=\delta=K_++K_-=\frac13\ Tr(\chi).$$ The main idea is that, in view of the interpretation of the EM vector potential, mass and electric charge are directly related, and therefore so are $k_C$ and $G_N$, Coulomb and Newton’s constants. H-atom, protons and electrons ----------------------------- The relation between EM and Gravity, should be a consequence of the relation between mass and electric charge. Their fundamental representatives are the proton and electron, viewed as “free” particles, and the neutral H-atom. Our model is based on the qubit as the fundamental unit (QI, spinor, harmonic oscillator, local Minkovski space). We interpret it as representing the neutral H-atom, with its electron and proton as the Hopf (monopole) fibration: $$SU_1\cong S^1\to SU_2\cong S^3\to S^2.$$ The “free” instances of the electron and proton are represented by sections, or equivalently by the direct product $SU_1\oplus SU_2\cong S^1\times S^2$. Breaking the symmetry could be an alternative mechanism for obtaining gravity. In other words, when viewing an interaction as a link or degree of a bundle morphism, as with the “Neumann” interpretation of the Biot-Savart law (see also [@NLEM]), then the interaction pairings between H-atom $S^3$, proton $S^2$ and electron $S^1$ could yield the different coupling coefficients (matrix), which in turn would reflect into the simplified version $K_+\ne -K_-$. Note that our interpretation of mass and charge amounts to taking the [*even*]{} and [*odd*]{} part of a signed scalar quantity (or relative to some other involution), maybe better with some weights: $$m=m_+ m_p+m_- m_e, q=m_+ e_p-m_-e_e.$$ The current accepted values include $m_p/m_e\approx 1836$, based on the assumption $e_p=e_e$ within most of the theoretical frameworks (e.g. Bohr’s atom or QFT). Compare with the relativistic mass in terms of rapidity $\theta$: $$m=m_0 \cosh{\theta}, \quad \tanh{\theta}=\beta=v/c,$$ and ponder whether its counterpart, the electric charge should vary as the [*odd part*]{} [@Quarks]: $$q=e \sinh{\theta},$$ towards a generalized rotation of Lorentz/symplectic type. Velocity as a Gauge Field ========================= There is enough evidence to interpret velocity as a gauge field, by exploiting the relation with the EM vector potential, especially in the context of a background independent theory (BIT). Recall that the EM-vector potential is thought of as a transfer velocity. It should be (now) thought of as a transfered [*relative*]{} velocity, to gauge away an irrelevant uniform motion (BIT): $$A\sim v_2-v_1,$$ with $v_i$ values of the velocity gauge field at particle 1 and 2 respectively. In other words, [*abandon the coordinate point of view*]{}, as unavoidable in a BIT, and refer to distances as a “potential” via path integration, possibly multi-valued etc. The curvature of velocity, or rather momentum and energy, should be interpreted as acceleration/force related; this at the infinitesimal level. When exponentiated, the resulting monodromy should be the $SL_2(C)$-matrix associated to an exact cycle representing a matter source and acceleration. It should thought of as the the vorticity of the QI-flow through that cycle: $$\Omega(cycle)=\int mv+eA <-> sl_2(C), \quad \Psi=U/B(cycle)=exp(i\Omega/\hbar).$$ Here $mv+eA$ should be replaced by the 4-D relativistic version, corresponding to $sl_2(C)=su_2 \oplus sl_2(R)$ under the Klein correspondence (QC$\cong$ SR), with the two components: unitary “space-like” quantum process and $SL_2(R)$-Lorentz boost (GR: acceleration=gravity?). Again we advocate that chirality of Hopf bundle would imply a Birkhoff decomposition of the spinor, with the corresponding decomposition of curvature/forces via logarithm, possible related to gravity (splitting the Hopf fibration): $$\Psi=\Psi_+ \Psi_-^{-1}, \quad \Omega=\Omega_{EM}-\Omega_G,$$ yielding the “effective” coupling constants $K_\pm$ (or the matrix). Deforming or Quantizing Special Relativity? =========================================== To “deform” Lorentz symmetries, we have the following alternative. The “winner” will be the one yielding a gravitational term when deforming Lorentz force (if any). Deforming quaternions as a Lie algebra: $(1+3)_c$ ------------------------------------------------- Quaternions $\mathbf{Q}$, from the “1+3” SR side, is a central extension of the angular velocity Lie algebra ($[,]$ the cross-product): $$R\to {\mathbf{Q}} \to (R^3, \times), \quad =t + (xL_x+yL_y+zL_z)/c,\quad (L_x=i, L_j=j, L_z=k),$$ with the “speed of light” $c$ as a central element. Viewed as a Lie algebra, it can be canonically deformed via the exponential/logarithm “non-linear” transformations, using BCH-formula. Since we think of $1/c$ as a deformation parameter dual to Plank’s constant $h$, we will denote it with $\overline{h}=1/c$ (not to be confused with $h/2\pi$). In dimensionless formulas $t$ is a number and $|v|/c$ is usually denoted with $\beta$: $$v\overline{h} = v/c, q=(t,v)=t+v\overline{h}.$$ With these notations BCH-formula is: $$q\oplus q'=q+q'+1/2[q,q']\overline{h}+1/6([q,[q,q']]-[q',[q',q]])+\dots .$$ The induced deformed velocity addition should be compared with Einstein’s velocity addition law,which for parallel velocities is: $$v\oplus_E v'=(v+v')/(1+vv'\overline{h}^2)=(v+v')(1-vv'\overline{h}^2+(vv')^2\overline{h}^4-\dots)$$ $$=v+v'-\{v(vv')+v'(v'v)\}\overline{h}^2+\dots$$ Recall that $a\times (b\times c)=(ac)b-(ab)c$. This time the deformed addition [*is*]{} associative for arbitrary velocities, unlike Einstein’s addition, and yielding a “Thomas rotation” time-like term. Quantizing quaternions: $(2+2)_h$ --------------------------------- Since gravity is not velocity dependent, let’s try quantizing the quaternion algebra, viewed from the “harmonic oscillator” side: $${\mathbf{Q}}=(T^*C, \omega).$$ Following the standard procedure for quantizing Heisenberg algebra as a central extension of the symplectic space: $$R\to H\to (T^*,\omega),$$ consider the analog central extension (infinitesimal deformation): $$R\to {\mathbf{Q}}[h]\to {\mathbf{Q}},\quad q=z_0|0>+z_1|1>, \quad [[0>,|1>]=ih.$$ together with the corresponding deformation via BCH-formula (the $UEA(Lie\ algebra)$ side): On the other hand we have a Moyal product or Weyl quantization on its algebra of observables (the $F(G)_h$ side). The resulting Lie algebra/group with one more deformation parameter should be related with the (anti) de Sitter groups [@SR21]. Now the question is weather the correspondence $QC\cong SR: 2+2=1+3$ is compatible with the duality $F(G)_h^*\cong U(g)_{\overline{h}}$. Deforming the Lorentz force ... again ------------------------------------- In the context of Newton’s law, the Lorentz force $F=q(E+v\times B)$ can be interpreted as coming from a potential, via the canonical (“electro-mechanic”) momentum: $$P=mv+qA, V=q(c \phi_E/c-vA), d/dt P=-\nabla V.$$ With electric potential $\Phi_E=1/\epsilon\ q_2/|r|$ in place and transfer angular velocity $A=\mu/r\ q_2 v_2\times e_r$ for the EM vector potential, it becomes: $$P_{12}=m_1v_1+\mu\ q_1q_2\ \omega_2\times e_r, \quad V=q_1 q_2 /|r|\ [\frac1{\epsilon}\ -\mu\ det(r, v_2, v_1)],$$ where the determinant comes from the mixed product $v_1\cdot (r\times v_2)$. The separation of “space-time” components into momentum and energy, is performed at the Lie algebra level, neglecting the non-linearity via exponential. Will this provide the “missing” gravitational term $m_1m_2 G_N /|r|$? In fact a relativistic “potential” energy formula should involve a relativistic canonical momentum: $$P^{rel}=\gamma m_0(c,v)+q(\Phi_E/c, A),$$ $$V^{rel}=P^2/m_0=c^2+q(\Phi_E-vA)+q^2/m_0((\Phi/c)^2-A^2).$$ The 1st term is a constant (discard), the 2nd is the “old” potential, while the 3rd appears as a “correction” term. In fact, in terms of the deformation parameter $\overline{h}=1/c$: $$(P/\overline{h})^2/m_0=1+(q\overline{h})(\Phi_E-vA)+(q\overline{h})^2 ((\Phi_E/c)^2-A^2)/m_0,$$ is an invitation to a deformation series. ... or grading it with external symmetry groups? ------------------------------------------------ On the other hand a gravitational potential seems to come naturally together with the different symmetry groups of the various types of interacting particles”: the “bound” H-atom (neutral qubit) corresponding to $S^3$, and “free” protons and electrons (split qubit) with coupling constants corresponding to $S^2$ and $S^1$. The simplified version was discussed in terms of the effective coupling constants $K_\pm$ (electric charges slightly not equal, or a parity issue due to the Hopf bundle). The electric and magnetic interaction “force” would be expressible in terms of “Neumann potentials” [@FromAtoE] given by linking numbers (see also [@Germain; @Barrett; @EMTC; @NLEM]). The electric force can be represented using a (Gaussian) linking integral $$\Phi_E=\frac{e_1 e_2}{\epsilon} \int_{S^1}\int_{S^1} \frac{dq\cdot dq_2}{|r|},$$ between two “time-like” circle fibers over the space-like charges (Hopf bundle), while the “magnetic force” (with $\mu$ coupling constant in place of $1/\epsilon$), would result from linking two “space-time” circles as “moving charges”. As an interesting possibility, the higher dimensional topological degrees for linking $S^2$ and $S^3$ could be related with weak and cromodynamics interactions. Now if the quaternionic product is deformed as above, there would be corresponding correction terms in the linking integrals. On the other hand, the author inclines to believe that the gravitational and electric forces appear to be unrelated due to splitting the $S^3$-Hopf bundle. At this stage one has to investigate pairs of interacting particles modeled not just as pairs of qubits, but possibly as the Hopf instanton bundle: $$\diagram S^3\dto \rto^{Q-Gate} & S^3 \dto & & Internal\ Space:\ S^3\to S^7 \dto \\ S^2 \rto^{Distance} & S^2 & & External\ Space:\ S^4. \enddiagram$$ The left side corresponds to the discrete picture of QID (quantum networks), while the right side corresponds to a continuum picture with base a compactified Minkovski space-time and fiber gauge group $SU_2$. Conclusions =========== Starting from the general framework of QID as a natural framework for a Background Space-Time Independent Theory, the article presents some alternatives to be investigated starting from the idea that the “unifying gauge group” is in fact the Hopf monopole bundle, underlying quaternion algebra (spinors). Its roles within the physics interface are: 1\) Unit of quantum information (Quantum Computing side: $2+2$), 2\) Harmonic oscillator (Quantum Mechanics side: $2+2^*$), 3\) Minkovski local space-time ($1+3$). Lorentz symmetries and Minkowski space seem to describe the infinitesimal/linear picture. Their deformation into the non-linear/non-commutative realm represents an avenue for obtaining gravitational potential as an exponential correction of the electric force, which together with the magnetic (and possibly weak and strong) appear as given by linking integrals (topological degrees). Together with the dynamics of the quantum networks, as a cohomology theory of dg-coalgebra of graphs, QID seems to provide the general mathematical framework and conceptual interface one would expect from a “Theory of Everything”. Acknowledgments =============== The author thanks ISU for the research support, especially for the sabbatical semester. The author is also grateful for the excellent research conditions at IHES, where some important steps were taken towards the present stage of understanding gravity. I would also like to thank Maxim Kontsevich and Graeme Segal for stimulating conversations. Annex: Quantum Gravity - An Open project ======================================== Knowledge accumulates exponentially, and so do physics theories grow in complexity. This leads to a cumulative development, e.g. the Standard Model as an Expert System in particle physics, similar to the rise of Windows through upgrades and patches. Some say Physics is in “trouble” [@Smolin], and Mathematics has nothing to do with it [@Not-even-wrong]; others are less gentle about the lack of a breakthrough in physics as of the “dark ages of physics” [@CED-mead]. The author believes that a change in the current R&D methodology can trigger a change of paradigm: open math-physics “programming using a web platform such as [*nLab*]{} [@JB-nLab]; a new physics model could be designed top-down, as an Expert System, taking advantage of the existing math-physics “toolkits”, yielding a better and “cleaner” alternative to the SM, similar to Linus as alternative Operating System alternative to Windows. in this article the author sketches a more specific plan of attaking the problem of Quantum Gravity, based on the previous sketches [@I-MPCS], as a follow up of the general principles and incorporated resources from [@I-DWT]. The aim is to formulate a viable Project to be developed according to the above suggested methodology. More specifically, a project for implementing Quantum Gravity as a deformation of information dynamics, as an $SU_2$ “upgrade” of Electromagnetism, to account for the charge-parity violation, is sketched. It is based on the framework of Quantum Relativity, and the associated math-physics tool-box. The unifying “gauge group” being the Hopf bundle, underlying the quaternion algebra as a generalized complex structure. Its quantum group deformation yields Gravity. The discrete formulation renders the resulting theory already quantum, with a particle-wave cohomological duality. The new paradigm of Quantum Information Dynamics is prone for plugging in the main ideas of the Standard Model regarding the weak and strong interactions. The Main Ideas -------------- Gravity is implemented as a deformation of an “Electromagnetism”, in the framework of Quantum Information Dynamics [@I-MPCS; @I-QG]. The framework is that of a category with duality; for example $SL_2(C)$-graph cobordisms, representing quantum processes. The mathematical theory is of the type “loop groupoid representations”; it is a Yang-Mills gauge theory bypassing connections by using Wilson observables directly, corresponding to the connection’s monodromy. In addition to $SL_2(C)$-gravity, we deform the “gauge group” $SL_2(C)$ which here is treated as a bundle enlarging $SU_2$ viewed as the Hopf monopole bundle. The deformation parameter of the resulting quantum group ${{SL_2(C)_{\overline{h}}}}$ [@HA2QG] is the reciprocal of the “speed” of light $\overline{h}=1/c$ [@I-QG]. We develop the mathematical connection between the Hopf algebra structure of $SL_2(C)$ and Hodge structure on the generalized complex structure [@GCS] of the deformed quaternions $Q_{\overline{h}}$, containing ${{SL_2(C)_{\overline{h}}}}$, interpreting it from a physical point of view as a deformation of the “metric” coefficients representing the “curvature” due to matter. The [*quantum net*]{} is thought off as a special material with electric and magnetic permittivity $$\chi=(\epsilon, 1/\mu):\lambda^2\to \Lambda^2 \quad \leftrightarrow \quad ds^2=\epsilon\ dr^2-1/\mu\ dt^2,$$ by using an extended version of the Klein correspondence [@Twistor-Theory] between Minkowski space: the Quantum Computing [@QC] - Special Relativity correspondence $QC\cong SR: 2+2^*=1+3$. The [*constitutive isomorphism*]{} [@Post] is an r-matrix entering the deformation of the Hodge duality structure, and yielding the bialgebra deformation “quantization”, or alternatively, the generalized complex structure deformation (Hodge deformation). Recall that under the $2+2^*=1+3$ above correspondence, the determinant (Kahler form) is mapped onto the Lorentz metric: $$u=(x,y,z,ct)\in (M^{3,1},ds^2) \quad \leftrightarrow \quad {\mathcal{H}}\ni \xi=\left( \begin{array}{cc} a=z-ct & b=x-iy \\ c=x+iy & d=z+ct \end{array} \right)$$ $$\quad \Rightarrow ||u||=\epsilon\ dr^2-1/\mu\ dt^2=det(\xi)=aa^*-bb^*.$$ Here we take advantage of the grading introduced on the central extension of the angular velocity Lie algebra ($[,]$ the cross-product): $$\label{E:CE} R\to {\mathbf{Q}} \to (R^3, \times), \quad =t + (xL_x+yL_y+zL_z)/c,\quad (L_x=i, L_j=j, L_z=k),$$ with the “speed of light” $c$ as a central element; the involution is a $Z_2$-graded complex conjugation: $$*(z+ct)=z-ct, \quad *(x+iy)=x-iy.$$ Charge-conjugation violation ---------------------------- The classical two charges $\pm$ are the eigenvalues of the involution $*$. The quantum deformation yields a [*twist*]{} $*^2=\delta$ [@QG-PC], responsible for breaking the symmetry. In this way we implement the idea that Gravity is due to a charge-conjugation violation, which produces an additional term in Newton’s law with Coulomb’s potential for electric force (static case). in a Lagrangian formulation, with the corresponding Euler-Lagrange equation, the deformed symmetries yield via Noether Theorem a quantum charge which does not obey the standard Fermi statistics, due to a deformed braiding $$\sigma=R\circ Twist, \quad R=Rota-Baxter\ operator;$$ for details on the relation between r-matrix, bialgebra deformation (Poisson-Lie groups, left/right dressing action etc.), Yang-baxter equation, braiding etc. see [@QG-PC]. Deforming $SL_2(C)$, or rather Clifford algebra of quaternions, yields the quantum determinant which is interpreted in [@I-QG] as the Newton-Coulomb coupling constant: $$\label{E:NC} Qdet \left( \begin{array}{cc} q & im'\\ -im& q' \end{array} \right)=qq'-e^{-1/\alpha}\ m m'.$$ The Category with Duality and CPT --------------------------------- The category with duality corresponding to the quantum group implements mathematically the Charge-Parity-Time Theorem, as mentioned in [@I-arrow]. The twist / braiding which breaks the charge-conjugation symmetry also corresponds to the chirality of the “gauge group”, i.e. the chirality of the Hopf monopole bundle. In a precise sense QID posseses not only electric and information charges (qubits), but also [*magnetic charges*]{} as degrees of the hopf bundles “sitting” over the nodes of the Q-net. The interaction picture model consists of a pair of quabits mapped to the Hopf instanton bundle $S^3\to S^7\to S^4$, again a “topological fragment” of the algebraic double $SU(2,2)$. Dirac’s equation ---------------- In the categorical framework, the splitting of the Dirac spinor into the chiral Weyl spinors obeing Dirac’s equation [@Dirac-Weyl] is essentially a Birkhoff decomposition of the representation: $$\Psi=\Phi^+/\Phi^-, \quad D=d+d^*$$ for details see [@Spitzer] and related articles. So the deformation of $SL_2(C)$ induces a parity violation also (see Hermitian categories [@Bakalov]), which allows to account for neutrinos as twist, and the lack of left-handed ones. The quarks and quark charges appear from the interplay between $SU_2$ and $SO(3,C)$, with the corresponding Hopf degrees (monopole and instanton bundles: see the link number interaction model). The Fundamental Constants ------------------------- The two approximate eigenvalues $e^\pm\approx \pm$ of $*$, i.e. the two charges will obey a constraint coming from the central extension of the angular velocity Lie algebra: $$hc = e^+e^-.$$ here $h$ is Plank’s constant, the trademark of the angular velocity Lie algebra $g=(R^3,\times)$, $c=1/\overline{h}$ is the “speed” of light related to the deformation parameter (central element) $\overline{h}$. The constraint satisfied by the deformed involution/antipode of the quantum group is: $$c^2=\frac1{\epsilon\mu}, \quad \overline{h}^2=\epsilon / (1/\mu).$$ The other “quantizations” of magnetic flux, hall conductivity are related to the invariants of the deformed Hodge structure $\chi$: $$Quantum\ flux: g_M=hc/e, \quad Quantum\ Charges: e^+=e-\delta, e^-=-e-\delta,$$ $$Hall\ conductivity=\frac{\epsilon}{\mu}=det(\chi)^{1/3}.$$ Recall that $\delta$ yields the Newton’s gravity term as a correction to Coulomb’s law (\[E:NC\]). The fine structure constant appears as an asymptotic ratio, with the classical approximation for charges $g_E=e$ being used: $$\alpha=e^2/hc=g_E/g_M;$$ for details regarding the interplay of $\epsilon, \mu, \chi, c$ see [@Post-constitutive]. Returning to Dirac equation, as Dirac stated [@Jehle], p.3, electric charge should be derived from $h$ as a square root; that is what we claim above. Splitting the extension yields a double cover and $e=\pm \sqrt{hc}$; in the deformed case, the chirality should be included. The corresponding covering map should be responsible for the “average” yielding the fine structure constant, essentially a zeta value [@I-arrow] combined with a Casimir element/ central charge $$\label{FSC} \alpha=e^2/hc \quad \leftrightarrow \quad ``137''\ h= Det(Q)\ \overline{h}, \quad Det(Q)=e^+e^-.$$ Now on the “mass” side of the canonic momentum $P=mv+qA$, which is implemented in the context of the generalized complex structure $TC\oplus T^*C$ on ${{SL_2(C)_{\overline{h}}}}$, the deformation of charges yields a gravitational constant $$G_N=e^{-\beta}, \beta \leftrightarrow 1/\alpha.$$ The relation should bypass $\alpha$, being expressed in terms of $h,c,e^\pm$. It seems that the charge defect $\delta$ [@I-QG] is related to the other invariant of the deformation $*$: $$G_N=\delta^2 \leftrightarrow Tr(\chi)=\epsilon - 1/\mu, \quad G_N/k_C=e^{-\beta}.$$ The numeric factors were discarded for emphasis of the conceptual meaning, in the sense of dimensional analysis. Harmonic analysis on groupoids ------------------------------ So a quantum process is modeled as a representation of a quantum network, with coefficients in a Hopf algebra (quantum group ${{SL_2(C)_{\overline{h}}}}$). Now in analogy with Fourier analysis, the spectra of graph cobordisms [@Alg-GT; @I-FL] should yield the “vibration modes” of the Q-net. From the Riemann surfaces side of the picture via ribbon graphs and Turaev-Feynman calculus [@Turaev], the modes should correspond to integral periods of the Hodge structure. From the physical point of view, the “frequencies” are due to a resonance condition: $$\omega^2=1/LC \leftrightarrow c^2=1/\epsilon \mu.$$ in a direct analogy with electric circuits as $SU_1$-voltage graphs (circle bundles over chain complexes, Pontryagin duality etc.), inductance corresponds to mass, and “elasticity” of the harmonic oscillator corresponds to capacitance (parameter of the qubit or quantum register viewed as a harmonic oscillator [@I-QG]). Relation to the Standard Model ------------------------------ The relation between the “gauge group of the SM $SU_1\times SU_2\times SU_3$ and the Hopf bundles, monopole and instanton, underlying the main object, the symplectic qubit $X=T^*H$ (reality is One dimensional internally, and potentially infinite dimensional externally), and its processes $G=Aut(T^*H)$, makes the transfer of “technology” to the new paradigm of QID/QG possible. As a previous attempt to implement quarks and mesons using links, we mention [@Jehle-particles]. This attempt was done in the spirit of String Theory, embedding circles in a configuration manifold, without being relativistic. These two important “lessons” can be used to implement the quark model and strong interaction in QID, using [*linking numbers*]{} [@I-QG] (homological algebra invariants and derived functors; dg-coalgebras of graph cobordisms and its equivariant cohomology with coefficients in ${{SL_2(C)_{\overline{h}}}}$ [@I-CFG]). Further Developments -------------------- The Quantum Computing model of Special Relativity, once deformed, seems to be the ideal ground for developing Quantum Gravity as a Grand Unified Theory (Theory of Everything). The Hodge periods/Quantum Groups approach is prone for an abstract formulation in terms of hermitian categories; a Tanaka-Krein duality should be related with the Connes-Kreimer Hopf algebra approach to renormalization [@CK], where Rota-Baxter Algebra with RB-operator $R$ is a different face of an r-matrix and $dd^*$-structure leading to deformation [@I-HREN; @I-LTDT]. A web-based platform such as John Baez’s nLab [@JB-nLab] seems to be ideal for the development of this project. [33]{} Arnold, V.I., Kozlov, V.V., Neishtadt, A.I., Iacob, A., Mathematical Aspects of Classical and Celestial Mechanics, Encyclopaedia of Mathematical Sciences, 3rd ed., 2006. Barrett, T., Topological foundations of EM, World Scientific Series in Contemporary Chemical Physics, Vol. 26, 2008. 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--- abstract: | We consider the two-photon exchange contribution to the $2P-2S$ Lamb shift in muonic deuterium in the framework of forward dispersion relations. The dispersion integrals are evaluated using experimental data on elastic deuteron form factors and inelastic electron-deuteron scattering, both in the quasielastic and hadronic range. The subtraction constant that is required to ensure convergence of the dispersion relation for the forward Compton amplitude $T_1(\nu,Q^2)$ is related to the deuteron magnetic polarizability $\beta(Q^2)$. Based on phenomenological information, we obtain for the Lamb shift $\Delta E_{2P-2S}=2.01\pm0.74$ meV. The main source of the uncertainty of the dispersion analysis is due to lack of quasielastic data at low energies and forward angles. We show that a targeted measurement of the deuteron electrodesintegration in the kinematics of upcoming experiments A1 and MESA at Mainz can help quenching this uncertainty significantly. author: - 'Carl E. Carlson' - Mikhail Gorchtein - Marc Vanderhaeghen title: Nuclear structure contribution to the Lamb shift in muonic deuterium --- Introduction ============ The proton radius puzzle—that the proton radius obtained from the Lamb shift in muonic hydrogen [@pohlmuH; @antogninimuH] is different from what should be the same radius obtained from data involving electrons [@Mohr:2012tt; @bernauer]—has attracted much attention in recent years. The explanation of the problem is not known so far. There are proposals of new dedicated scattering experiments with electrons [@gasparyan; @ISR_A1] and muons [@Gilman:2013eiv]. On the theory side, the discrepancy was addressed in terms of effective non-relativistic QED interactions [@Hill:2011wy], dispersion relations  [@Lorenz:2012tm], exotically large hadronic effects [@Miller:2012ne], or of new physics affecting the muon and electron differently [@TuckerSmith:2010ra; @Batell:2011qq; @Carlson:2012pc].\ Further information can come from measuring the deuteron radius using the Lamb shift in muonic deuterium. The deuteron radius from electron based experiments is already known to good accuracy. The best results come from using the isotope shift, that is, measuring the $1S$-$2S$ splittings in electron-proton ($e$-$H$) and electron-deuteron ($e$-$D$) hydrogen and finding from residual corrections that $$r^2_E(d) - r^2_E(p) = 3.82007(65) {\rm\ fm^2} \,,$$ as quoted in [@Parthey:2010; @huber], where the $r_E(p,d)$ are charge radii. The isotope shift number is so accurate that the uncertainty in the deuteron radius-squared becomes in practice the same as for the proton, and using the CODATA 2010 value for the proton radius one finds [@Mohr:2012tt], $$r_E(d) = 2.1424 (21) {\rm\ fm} \,.$$ The uncertainty is $0.1\%$. In contrast, the current best direct electron-deuteron scattering results yield $r_E(d)=2.128(11)$ fm [@Sick:1996], or $0.5\%$ uncertainty. Planned experiments are expected to reduce this uncertainty [@modkg]. To obtain the charge radius from the Lamb shift requires not only accurate data but also accurate calculation of all corrections that are not hadronic size corrections. Of these, the two-photon correction, which includes the relativistically correct polarizability correction, has drawn continued attention. A deuteron is easily distorted compared to a single proton, and we shall see that the polarizability correction for the $\mu$-$D$ system is about two orders of magnitude larger than for $\mu$-$H$. The requirement that the $\mu$-$D$ polarizability correction be safely smaller than the radius-related energy shift can become quite severe. Of course, without knowing the underlying reason for the proton radius discrepancy, we cannot with certainty predict what the deuteron discrepancy will be. However, we will give the anticipated energy discrepancy in one scenario, and thereby obtain a working number with the expectation that other scenarios would give results similar within a factor of a few. As a reminder, the main energy shift due to finite hadron (or nuclear) size is $$\Delta E_{\rm finite\ size} = \frac{2\pi Z\alpha}{3} \frac{(m_r Z\alpha)^3}{n^3 \pi} r^2_E(h) \,,$$ for a hydrogen-like atom in the $nS$ state, where $m_r$ is the reduced mass. Experiment shows about a $320\,\mu$eV energy discrepancy for the $\mu$-$H$ $2S$ state, compared to expectations based on the CODATA 2010 [@Mohr:2012tt] electron based proton radius. In a scenario where the $\mu$-$H$ energy discrepancy is not actually due to a proton size change but rather to the exchange of a new particle that has a special coupling to the muon and [*e. g.*]{} a dark photon coupling (*i.e.,* a squelched electromagnetic coupling) to other particles, the $\mu$-$D$ energy would be the same as in the $\mu$-$H$ except for the reduced-mass-cubed factor. In this case the anticipated $\mu$-$D$ energy discrepancy is $$\Delta E_{\rm discrepant}(\mu{\rm -}D) \approx 380 \ \mu{\rm{eV}}.$$ Hence the two-photon corrections should be calculated within $100\ \mu$eV or less. As the two-photon corrections are of order $2$ meV, this requires a $5\%$ or better accuracy. There are several currently available polarizability calculations [@friar; @pachuckiD; @rosenfelder; @FriarPayne; @Friar:2013rha; @Ji:2013oba]. Ref. [@pachuckiD] includes the elastic contributions and quotes an uncertainty limit well within the requirement. A more recent calculation of Ref. [@Ji:2013oba] follows the same lines but includes further relativistic corrections. In addition, much of that calculation is supported by a calculation  [@Friar:2013rha] that uses the zero-range approximation which indicates that the bulk of the result is relatively model independent and can be obtained non-relativistically. However, there remain significant energy shifts that are obtained non-relativistically using a potential model. One would like a calculation by an alternative method to verify the results. We here explore a fully relativistic dispersive calculation of the two-photon corrections to $\mu$-$D$. In this type of calculation, the energy shift is obtained from the real part of an amplitude whose imaginary part is related without approximation to physical elastic and inelastic $e$-$D$ scattering. To the extent that the data is accurate and sufficient, the result follows just from the data and the calculation is model independent. The calculation resembles the generally accepted work for the $\mu$-$H$ system [@Pachucki:1996zza; @Martynenko:2005rc; @Carlson:2011zd; @Birse:2012eb; @fesr]. However, for the proton case the corrections are much smaller, and the proton data in the relevant kinematic regions is itself very good. Furthermore, good analytic fits, useful for doing integrals, are already available in the literature [@bostedchristy]. For the deuteron we need elastic and inelastic data over a wide range of energies, including the low energy region, where the data is sparse. There are good analytic fits [@bostedchristy] to the deuteron data above the pion production threshold, but not in the lower energy quasi-elastic region. Part of our effort is devoted to providing such fits. The inelastic data is represented in terms of structure functions or response functions. The overall polarizability effect is sensitive to the structure of the response functions at low energy and low virtual photon masses, which is where the data is sparse. A consequence of this, phrased in terms of fitting procedures, is that small changes in some parameters have big effects on the near threshold behavior but small effect on the quality of the fits to the available data. This leads to a larger than desired uncertainty in the results for the two-photon corrections, when using this method. Future low momentum transfer deuteron breakup data, possibly obtained as background data to elastic scattering experiments [@modkg], can bring about a decisive reduction in the uncertainty limits of the polarizability calculation. We discuss this below with some examples. Our presentation starts with the description of the general dispersive formalism for obtaining energy shifts from elastic and inelastic scattering data and subtraction terms in Section II. Evaluation of the elastic and inelastic dispersion integrals is discussed in Section III where we as well provide details of our global fit to the available quasi-elastic deuteron data. In Section IV we present our results, discuss the uncertainty limits, and address possible help from upcoming experiments. Section V is dedicated to the application of the dispersive, data-based, model-independent polarizability calculation described here to the $e$-$H$ and $e$-$D$ systems and its relevance to the isotope shift measurements. Section VI closes the article with a short summary of our findings. The Basic Formalism =================== The diagram that contains the nuclear and hadronic structure-dependent ${\cal O}(\alpha^5)$ correction to the Lamb shift is shown in Fig. \[tpediag\]. ![(Color online) Two-photon exchange diagram for the ${\cal O}(\alpha^5)$ correction to the Lamb shift.[]{data-label="tpediag"}](TPEdiag.pdf){height="2cm"} The lower part of the diagram, the blob containing the nuclear and hadronic structure dependence is encoded in the forward virtual Compton tensor, &&T\^=d\^4xe\^[iqx]{}p|Tj\^(x)j\^(0)|p\ &&=(-g\^+)T\_1(,Q\^2) + T\_2(,Q\^2),where $\hat p^\mu=p^{\mu}-\frac{p\cdot q}{q^2}q^\mu$, $Q^2=-q^2$, $\nu=(p\cdot q)/M_d$ and $M_d$ is the deuteron mass. Following [@Carlson:2011zd], we can write the contribution of the two-photon exchange diagram to the $n\ell$ energy level as &&E\_[n]{}=\^2\_[n]{}(0)d\^4Q\ &&,where a Wick rotation $q_0=iQ_0$ was made, and $\phi^2_{n\ell}(0)=\mu_r^3\alpha^3/(\pi n^3)\delta_{\ell0}$, $\mu_r=mM/(M+m)$ being the reduced mass. $T_{1,2}(\nu,Q^2)$ are even functions of $\nu$ and their imaginary parts are related to the spin-independent structure functions of lepton-deuteron scattering, T\_1(,Q\^2)&=&F\_1(,Q\^2)\ [Im]{}T\_2(,Q\^2)&=&F\_2(,Q\^2). Given the known high-energy behavior of the structure functions, the two amplitudes obey the following form of dispersion relation, T\_1(q\_0,Q\^2)&=&|T\_1(0,Q\^2)+[Re]{}T\_1\^[pole]{}(q\_0,Q\^2)\ &+&\_[\_[thr]{}]{}\^\ [Re]{}T\_2(q\_0,Q\^2)&=&[Re]{}T\_2\^[pole]{}(q\_0,Q\^2) +\_[\_[thr]{}]{}\^,where for $T_1$ the subtraction at $q_0=0$ was performed with $\bar T_1(0,Q^2)$ the respective subtraction function. Above, we explicitly extracted the contribution of the ground state leading to a pole, $T_{1,2}^{pole}$. This contribution is defined in terms of the deuteron’s electromagnetic vertex &&d(p’)|J\^(q)|d(p)=G\_2(Q\^2)\[[’\^\*]{}\^(q)-\^(’\^\* q)\]\ &&-(p+p’)\^, where $\xi^\mu({\xi'^*}^\mu)$ denote the polarization vector of the initial (final) deuteron with momenta $p(p')$, respectively, and $Q^2=-q^2$ stands for the four-momentum transfer. The form factors $G_{1,2,3}$ are related to the charge, magnetic and quadrupole deuteron form factors as G\_M&=&G\_2,\ G\_C&=&G\_1+\_dG\_Q,\ G\_Q&=&G\_1-G\_2+(1+\_d)G\_3, and $\tau_d=Q^2/(4M_d^2)$. The elastic contribution to the structure functions reads F\_1\^[el]{}&=&(1+\_d)G\_M\^2(1-x\_d),\ F\_2\^[el]{}&=&(1-x\_d),with the Bjorken variable $x_d=Q^2/(2M_d\nu)$. Correspondingly, we distinguish three contributions, $\Delta E_{n0}=\Delta E_{n0}^{subt}+\Delta E_{n0}^{el}+\Delta E_{n0}^{inel}$ where E\_[n0]{}\^[subt]{}&=&\^2\_[n0]{}(0)\_0\^|T\_1(0,Q\^2),\[subtraction\] E\^[el]{}\_[n0]{}&=&\^2\_[n0]{}(0) \_0\^\[elastic\]\ &&{ G\_M\^2 (1+\_d)( -) .\ &&-. (-) }E\^[inel]{}\_[n0]{}&=&-\^2\_[n0]{}(0) \_0\^\_[\_[thr]{}]{}\^\[inelastic\]\ &&.Above, we denote $\tau=\nu^2/Q^2$, $\tau_l=Q^2/(4m^2)$, and the auxiliary functions are given by \_1()&=&(1-2)+2\^[3/2]{}\ \_2()&=&(1+)\^[3/2]{}-\^[3/2]{}-\ \_1(,\_l)&=&\ \_2(,\_l)&=& . Evaluation and Data Fits ======================== Elastic contribution -------------------- We start with the elastic contribution. It can be noted that the integral in Eq. (\[elastic\]) is IR divergent due to an exchange of soft Coulomb photons. Such contributions, however, were already taken into account within the non-relativistic calculations on a pointlike deuteron. Furthermore, the finite size effects were accounted for, as well, and have to be subtracted from the full result of Eq. (\[elastic\]) to avoid double-counting. This subtraction leads to &&|E\^[el]{}\_[n0]{}=\^2\_[n0]{}(0) \_0\^\[elastic\_subt\]\ &&{ G\_M\^2 (1+\_d)( -) .\ &&- (-)\ &&+16M\_d\^2G\_C’(0) }.We evaluate Eq. (\[elastic\_subt\]) with the most recent deuteron form factors’ parametrization from [@abbott]. We use the parametrization I and II of that Ref. to estimate the uncertainty, and list the result with the uncertainty in Table \[tab2\].\ The inelastic contributions contain two parts, E\^[inel]{}\_[n0]{}=E\^[QE]{}\_[n0]{}+E\^[hadr]{}\_[n0]{}, the quasielastic nucleon knock-out ([*QE*]{}) and hadronic excitation spectrum ([*hadr*]{}) that we will treat separately.\ Hadronic contribution --------------------- The part of the deuteron excitation spectrum above the pion production threshold can be dealt with very similarly as it was done in Ref. [@Carlson:2011zd] for the proton case. We use the modern deuteron virtual photoabsorption data that were parametrized in terms of resonances plus non-resonant background by Bosted and Christy in [@bostedchristy]. Since the integration over the energy extend beyond the validity of the fit of Ref. [@bostedchristy], we supplement the correct high-energy behavior by adopting a Regge-behaved background. The Regge fit to world data on the deuteron total photoabsorption cross section was done in [@fesr]. We extend this description to virtual photoabsorption by supplementing a $Q^2$-dependence from generalized VDM, e.g. [@alwall], that provides good description of virtual photoabsorption data at $Q^2\lesssim3$ GeV$^2$. The result for $\Delta E^{hadr}$ is reported in Table \[tab2\].\ Quasielastic contribution ------------------------- In the literature, there exist non-relativistic calculations of the Lamb shift in muonic deuterium with potential models or in zero-range approximations [@friar; @pachuckiD; @rosenfelder; @FriarPayne; @Friar:2013rha; @Ji:2013oba]. In this work we opt for a phenomenological, data-driven approach in the spirit of Ref. [@Carlson:2011zd] where real and virtual photoabsorption data on the proton were utilized to constrain the Lamb shift in the muonic hydrogen. For this purpose we need to fit the quasi-elastic data over the whole kinematical range with an appropriate function of $\nu,\,Q^2$. We start with the plane wave Born approximation (PWBA) that allows to relate the deuteron structure functions in the quasi-elastic kinematics to the elastic nucleon structure functions. In doing this, we Fermi-smear the nucleon virtual Compton tensor, rather than just the structure functions, the result reads F\_1\^[PWBA]{}&=& G\_M\^2 S(,Q\^2)+S\_(,Q\^2)\ F\_2\^[PWBA]{}&=&\ &&. Above, we defined the integrals S(,Q\^2)&=&\_[k\_[min]{}]{}\^[k\_[max]{}]{}dkk(u\^2(p)+w\^2(p)),\ S\_(,Q\^2)&=&\_[k\_[min]{}]{}\^[k\_[max]{}]{}dkk(u\^2(p)+w\^2(p)) k\_\^2 The deuteron wave function is normalized as $\int k^2dk[u^2(p)+w^2(p)]=1$, $u,w(p)$ denote the $s,d$ radial wave function of the deuteron, respectively. The magnitude of the three-momentum of the bound nucleon is denoted by $k=|\vec k|$, and its component perpendicular to the direction of the virtual photon is $k_\perp^2=k^2\sin^2\theta_k$, the angle $\theta_k$ is defined below. ![Quasielastic scattering kinematics.[]{data-label="figQE"}](QEdiag.pdf){width="4.5cm"} The on-shell condition for the external (knock-out) nucleons and 4-momentum conservation M\_d+&=&+, with the average nucleon mass $M\equiv\frac{1}{2}(M_p+M_n)\approx0.938919$ GeV, leads to a delta function for the angle between the three-momenta of the virtual photon and the active nucleon, \_k=. The integral over the Fermi momentum $k$ is constrained between two finite values due to the requirement $-1\leq\cos\theta_k\leq1$, k\_[min]{}&=&|-+|,\ k\_[max]{}&=&+, with $\nu_{min}=Q^2/(2M_d)+\epsilon+{\cal O}(\epsilon^2)$ and $\epsilon=2M-M_d\approx2.224$ MeV. Experimental data on deuteron electrodesintegration feature a sharp peak just above the threshold, that is due to final state interactions between proton and neutron after the break-up [@Durand:1961zz]. We adopt an approximate formula (Eq. (48) of that Ref.) to obtain the following model for the transverse and longitudinal cross sections, \_T\^0&=&\ \_L\^0&=&, with the $n-p$ singlet and triplet scattering lengths entering the FSI part, $a_S=-23.74$ fm, $a_T=5.38$ fm, respectively. Above, we note that the combination $\sqrt{M(\nu-\nu_{min})}=|\vec p|$ corresponds to the three-momentum of the knocked-out nucleon. As a result, we obtain the following representation of the quasielastic structure functions of the deuteron: F\_[1,2]{}\^[d,QE]{}=F\_[1,2]{}\^+F\_[1,2]{}\^[PWBA]{}+F\_[1,2]{}\^[FSI]{},\[model1\] according to the ingredients discussed above. To describe data at arbitrary kinematics, we allow for a rescaling of each ingredient by a function of $Q^2$ that should be obtained from the fit to the deuteron photo- and electrodesintegration data. $$\begin{aligned} F_1^\perp&=&C_\perp\frac{G_E^2+\tau G_M^2}{1+\tau}\frac{1}{2|\vec q|}S_\perp(\nu,Q^2),\nn\\ F_2^\perp&=& \frac{\nu Q^2}{M_d|\vec q|^2}F_1^\perp,\nn\\ F_1^{PWBA}&=&f_T^{PWBA}\!\!(Q^2)\frac{Q^2}{4|\vec q|}G_M^2 S(\nu,Q^2),\nn\\ F_2^{PWBA}&=&f_T^{PWBA}\!\!(Q^2)\frac{\nu Q^4}{M_d|\vec q|^5}\frac{G_E^2+\tau G_M^2}{1+\tau}\nn\\ &\times&S(\nu,Q^2) (M+\frac{\nu}{2})^2,\nn\\ F_1^{FSI}&=&M_d f_T^{FSI}(Q^2)\sigma_T^0,\nn\\ F_2^{FSI}&=&\frac{\nu Q^2}{|\vec q|^2}(f_T^{FSI}(Q^2)\sigma_T^0+f_L^{FSI}(Q^2)\sigma_L^0), \label{model2}\end{aligned}$$ where we adopted the following forms: f\_T\^[PWBA]{}(Q\^2)&=&,\ f\_T\^[FSI]{}(Q\^2)&=&,\ f\_L\^[FSI]{}(Q\^2)&=&.\[fit\] We fitted the available data from $Q^2=0.005$ GeV$^2$ to $Q^2=3$ GeV$^2$, and the resulting values of the parameters are listed in Table \[tab1\]. $a_1$ $b_1$(GeV$^{-2}$) $a_2$(GeV$^{-3}$) ------------------- ------------------- ------------------- 0.995(5) 25.4(6) 215(35) $b_2$(GeV$^{-2}$) $a_3$(GeV$^{-3}$) $b_3$(GeV$^{-2}$) 52(8) 3.5(1.5) 24.5(8.0) : The values of the parameters introduced in Eq. (\[fit\]) as obtained from a fit to the deuteron QE data.[]{data-label="tab1"} ![(Color online) Scaling factors $f_T^{i}(Q^2)$ with uncertainty thereof: $i=PWBA$ (red solid lines), and $i=FSI$ (blue solid line) with uncertainty thereof (thin blue short-dashed lines) plotted vs. QE data as function of $Q^2$. []{data-label="fig1"}](scaling_FTQE_FTFSI_8-01-2014.pdf){width="8.5cm"} ![(Color online) Rescaled PWBA model of Eqs. (\[model1\], \[model2\]) vs data from Ref. [@dytman].[]{data-label="fig2"}](dytman293.pdf){width="8.5cm"} ![(Color online) Same as in Fig. \[fig2\] vs data from Ref. [@parker].[]{data-label="fig3"}](parker220.pdf){width="8.5cm"} ![(Color online) Same as in Fig. \[fig2\] vs data from Ref. [@friedman].[]{data-label="fig4"}](Friedman_175mev_75deg_plot.pdf){width="8.5cm"} ![(Color online) Same as in Fig. \[fig2\] vs data from Ref. [@patterson].[]{data-label="fig5"}](Patterson_Barber_41p5MeV_180deg.pdf){width="8.5cm"} ![(Color online) Same as in Fig. \[fig2\] vs data from Ref. [@ricco].[]{data-label="fig6"}](ricco_q0p35invfm_155deg.pdf){width="8.5cm"} ![(Color online) Same as in Fig. \[fig2\] vs data from Ref. [@yearian] as function of excitation energy $E^x$.[]{data-label="fig7"}](yearian143mev.pdf){width="8.5cm"} ![(Color online) Same as in Fig. \[fig2\] vs. deuteron total photoabsorption data from Refs. [@Bernabei:1986ai; @Birenbaum:1985zz]. Older data compilation can be found in Ref. [@Govaerts:1981xu][]{data-label="fig8"}](sigma_gaD.pdf){width="8.5cm"} The parameter $C_\perp$ is obtained from a fit to real photon data and the value of Baldin sum rule for real photons, \_E\^d+\_M\^d&=&\_[\_[th]{}]{}\^[\_]{} F\^d\_1(,0), where $\alpha_E^d$ and $\beta_M^d$ are the deuteron electric and magnetic polarizabilities, respectively. There exist calculations in chiral EFT by Chen et al. [@chen] and in non-relativistic potential models, [*e.g.*]{} by Friar [@friar], that give close results that can be cast in the following form: $\alpha_E^d=0.633(1)$ fm$^3$, and $\beta_M^d=0.072(5)$ fm$^3$. The uncertainty in the value of the polarizabilities was obtained by averaging over the two calculations. Evaluating Baldin integral with $F_1^\perp$ (the only piece that does not vanish at the real photon point) leads to C\_=1.28(1). Adopting these ingredients, the QE contribution of the two-photon exchange (TPE) to Lamb shift in deuterium can be calculated. We evaluated this contribution with the nucleon form factors in Kelly’s parametrization [@kelly] and using $S(\nu,Q^2),\,S_\perp(\nu,Q^2)$ from Paris WF [@lacomb], and list the result with the uncertainty in Table \[tab2\].\ Subtraction term ---------------- Following Ref. [@Birse:2012eb], we identify |T\_1(0,Q\^2)&=&T\_1\^B(0,Q\^2)-T\_1\^[pole]{}(0,Q\^2)\ &+&\_M\^d(0)F\_(Q\^2), where $T^B$ represents the Born contribution, and where in the polarizability term we explicitly factored out the $Q^2$-dependence. The polarizability contribution to the $nS$-level is given by E\^\_[n0]{}=2\^2\_[n0]{}(0)\_M\^d(0)\_0\^dQ\^2F\_(Q\^2), with $\beta_M^d(0)=0.072(5)$ fm$^3$. The $Q^2$-dependent form factor $F_\beta(Q^2)$ is generally not known. We estimate it by setting $F_\beta(Q^2)=G_C^d(Q^2)/G_C^d(0)$, and to estimate the uncertainty we also try $F_\beta(Q^2)=G_M^d(Q^2)/G_M^d(0)$. The average result and uncertainty is quoted in Tab. 2. Finally, the subtraction function Eq.30 contains the difference between the Born and pole contributions, which results from the contact two-photon deuteron interaction (Thomson term). The pointlike part of it, $-1/4\pi M_d$ was already taken into account in atomic calculations, thus we need to account for &&\[T\_1\^B-T\_1\^[pole]{}\](0,Q\^2)-\[T\_1\^[B,point]{}-T\_1\^[pole,point]{}\](0,Q\^2)\ &&=, thus leading to the shift of an $S$-level E\_[n0]{}\^[Th]{}&=&\^2\_[n0]{}(0)\_0\^dQ\^2 \[thomson\] The result of the numerical evaluation is listed in Table \[tab2\]. $\Delta \bar E^{el}$ – 0.417(2) ---------------------- -------------- $\Delta E^{PWBA}$ – 1.616(739) $\Delta E^{FSI}$ – 0.391(44) $\Delta E^{\perp}$ – 0.322(3) $\Delta E^{hadr}$ – 0.028(2) $\Delta E^{\beta}$ 0.740(40) $\Delta E^{Th}$ 0.023(1) $\Delta E_{total}$ – 2.011(740) : TPE corrections to the $2S_{1/2}$ energy level in muonic deuterium in units of meV.[]{data-label="tab2"} Discussion of Results and Impact of Further Scattering Experiments ================================================================== The total result for the $2P-2S$ Lamb shift obtained from the sum of all terms ${\cal O}(\alpha^5)$ due to two-photon exchange amounts to E \_[2P-2S]{}&=&2.01(74). The uncertainty of our result comes from three sources: elastic deuteron form factors, inelastic hadronic excitations and nuclear (quasi-elastic) contributions. The deuteron elastic form factors have been measured over a wide $Q^2$-range with good precision, and the error associated with different parametrizations of these data amounts to $2\mu$eV or relative 2% uncertainty. The hadronic part contribution is constrained to a relative 7%, however fortunately the contribution itself is rather small, so this somewhat large relative uncertainty translates in $2\mu$eV absolute uncertainty. At the moment, for the calculation of the subtraction contribution we rely on the $Q^2$-dependence for the magnetic polarizability obtained from a model. A direct calculation of $\bar T_1(0,Q^2)$, for instance in chiral EFT would help reducing the corresponding uncertainty. The largest contribution and the source of the largest uncertainty is the quasielastic piece, in particular the $Q^2$-dependence of the inelastic structure function $F_2(\nu,Q^2)$ in the range $\nu\leq10$ MeV, $Q^2\leq0.01$ GeV$^2$ from which the dominant contribution to the Lamb shift stems. A dedicated measurement at Mainz with the existing A1 apparatus at $E_0=180$ MeV and angles $\theta_{lab}\geq15^\circ$ is planned [@A1], and it would help somewhat to constrain the uncertainty with $Q^2\gtrsim2.2\times10^{-3}$ GeV$^2$. Going to lower energies will be possible with the new linear accelerator machine MESA at Mainz, and we include a few plots demonstrating the sensitivity to the parameter $a_1$ in several representative kinematics in Fig. \[sensitivity\]. ![(Color online) Sensitivity to the variation of the parameter $a_1$ entering $f_T^{PWBA}$ in the range \[0.99, 1\] and $a_2$ entering $f_T^{FSI}$ in the range \[180, 250\] is shown by the dashed and solid lines, respectively, in the kinematics relevant for the MAMI A1 apparatus [@A1] (three upper panels), and for MESA at 80 MeV (lower panel).[]{data-label="sensitivity"}](sigma180mev_30deg.pdf "fig:"){width="7.5cm"} ![(Color online) Sensitivity to the variation of the parameter $a_1$ entering $f_T^{PWBA}$ in the range \[0.99, 1\] and $a_2$ entering $f_T^{FSI}$ in the range \[180, 250\] is shown by the dashed and solid lines, respectively, in the kinematics relevant for the MAMI A1 apparatus [@A1] (three upper panels), and for MESA at 80 MeV (lower panel).[]{data-label="sensitivity"}](sigma180mev_22deg.pdf "fig:"){width="7.5cm"} ![(Color online) Sensitivity to the variation of the parameter $a_1$ entering $f_T^{PWBA}$ in the range \[0.99, 1\] and $a_2$ entering $f_T^{FSI}$ in the range \[180, 250\] is shown by the dashed and solid lines, respectively, in the kinematics relevant for the MAMI A1 apparatus [@A1] (three upper panels), and for MESA at 80 MeV (lower panel).[]{data-label="sensitivity"}](Sensitivity_a1_180mev_16deg.pdf "fig:"){width="7.5cm"} ![(Color online) Sensitivity to the variation of the parameter $a_1$ entering $f_T^{PWBA}$ in the range \[0.99, 1\] and $a_2$ entering $f_T^{FSI}$ in the range \[180, 250\] is shown by the dashed and solid lines, respectively, in the kinematics relevant for the MAMI A1 apparatus [@A1] (three upper panels), and for MESA at 80 MeV (lower panel).[]{data-label="sensitivity"}](Sensitivity_a1_80mev_16deg.pdf "fig:"){width="7.5cm"} To bring the discussion to a more quantitative level, we list the projected impact of a $d(e,e')pn$ measurement in several kinematics of A1 and MESA for the uncertainty of the dispersion calculation of the Lamb shift in Table \[tab3\]. For this analysis, we assumed for simplicity that the uncertainty of the fit will be equal to the precision of the data.[^1] For the kinematics $E_{lab}=80$ MeV, $\theta=16^\circ$ the uncertainty of the quasielastic contribution is reduced by a factor of 15 and the theory uncertainty starts being dominated by that due to the subtraction constant (estimated to be 40 $\mu$eV). It can be seen that already the next MAMI A1 runs at the lowest energy of 180 MeV and the most forward angle of 16$^\circ$ with a 2% precision have the potential to reduce the uncertainty of our dispersion calculation by at least factor of 4. The sensitivity to the value of the parameter $a_1$ is further enhanced at a lower energy as can be seen in the lower panel of Fig. \[sensitivity\]. Future measurements will allow to test other theoretical frameworks, such as potential models and EFT, as well. $E_{lab},\,\theta_{lab}$ Exp. precision $\begin{array}{c} \delta(\Delta E_{2S-2P}^{\mu D}) \\ {\rm in}\;\mu {\rm eV} \end{array}$ $\begin{array}{c} \delta(\Delta E_{1S-2S}^{e D}) \\ {\rm in\;kHz} \end{array}$ -------------------------- ---------------- ------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------- 180 MeV, 30$^\circ$ 2% 740 12 1% 370 6 180 MeV, 22$^\circ$ 2% 390 6.32 1% 195 3.16 180 MeV, 16$^\circ$ 2% 211 3.36 1% 110 1.68 80 MeV, 16$^\circ$ 2% 67 1.08 1% 48 0.78 : Impact of future measurements of the deuteron electrodesintegration at MAMI A1 and MESA (kinematics in the first column and experimental precision in the second column) on the theoretical uncertainty of the TPE contribution to the Lamb shift in muonic deuterium (third column) and the ($1S-2S$) splitting in electronic deuterium (fourth column).[]{data-label="tab3"} Contribution This work [@pachuckiD] [@Friar:2013rha] [@Ji:2013oba] [@rosenfelder] [@martynenkofaustov] -------------- ------------ -------------- ------------------ --------------- ---------------- ---------------------- Elastic 0.394(2) – – – – 0.37 Hadronic 0.028(2) 0.043 – – – – Nuclear 1.589(740) 1.637(16) – – 1.5 – Total 2.011(740) 1.680(16) 1.942 1.698 – – : Nuclear and nucleon structure-dependent ${\cal{O}}(\alpha^5)$ contributions to the $2P-2S$ Lamb shift in muonic deuterium as calculated by different groups, in units of meV. In case of Refs. [@pachuckiD; @Friar:2013rha; @Ji:2013oba] the separation of the result into “elastic" and “nuclear" contributions is not possible, and the sum of the two is quoted. []{data-label="tab4"} Our result should be compared to those obtained by other groups: [@pachuckiD; @martynenkofaustov; @Borie:2012zz; @rosenfelder; @krutov; @friar; @Friar:2013rha; @Ji:2013oba]. Note that [@martynenkofaustov; @Borie:2012zz; @krutov] did not perform a complete calculation and take, for instance, the nuclear polarizability correction from other works. To facilitate the comparison, we list our results along with those obtained by other groups in Table \[tab4\]. To make a sensible comparison possible we reorganized the various contributions listed in Table II as follows: “Elastic” denotes $\Delta\bar E^{el}+\Delta E^{Th}$, and “Nuclear” is sum all nuclear contributions, $\Delta E^{PWBA}+\Delta E^{FSI}+\Delta E^\perp+\Delta E^\beta$. Ref. [@rosenfelder] quotes 1.500 meV $2P-2S$ correction due to the deuteron nuclear electric dipole polarizability; in Ref. [@pachuckiD] a result of 1.680(16) meV is obtained by considering the electric polarizability (and various corrections thereto), elastic and hadronic contributions, and magnetic polarizability. Ref. [@pachuckiD] furthermore obtains the sum of the proton and neutron intrinsic polarizabilities to the Lamb shift in muonic deuterium by rescaling the [*total*]{} Lamb shift for muonic hydrogen, $\Delta E_{\mu H}=36.9\,\mu$eV obtained in Ref. [@Carlson:2011zd] with the ratio $(\mu_r^D/\mu_r^H)^3$ with the result $\Delta E_{\mu D}^{hadr.}=43(3)\,\mu$eV. This estimate is not correct because the main contribution to $\Delta E_{\mu H}$ is due to the elastic contribution, and only about a third of it, $13.5\,\mu$eV comes from polarizabilities. Since proton and neutron electric polarizabilities are very close, $\alpha_p\approx\alpha_n$, one should expect that the result for the deuteron should be roughly equal to their sum, $\Delta E_{\mu D}^{hadr}\sim 2\Delta E_{\mu H}=27\,\mu$eV. Indeed, our result (third entry in Table \[tab2\]) is consistent with this simple estimate, $\Delta E_{\mu D}^{hadr}=28(2)\,\mu$eV. This suggests that after correction the full result of Ref. [@pachuckiD] should be 1.665(16) meV. On the other hand, Ref. [@Friar:2013rha] estimates the Lamb shift in the zero-range approximation to be 1.912 meV (1.942 with further corrections), and quotes the result of Ref. [@pachuckiD] in that approximation as 1.899 meV. These numbers are close to each other, nevertheless, we point out that the differences are not small, especially compared to the uncertainty of $16\,\mu$eV claimed in Ref. [@pachuckiD]. As mentioned above, the correct account of the nucleon polarizability corrections alone shifts the result of Ref. [@pachuckiD] by $15\,\mu$eV that exhausts the claimed precision of the calculation. In Ref. [@Ji:2013oba] the calculation of the polarizability correction is reexamined and higher-order relativistic corrections from longitudinal and transverse two-photon exchanges were included, leading to an additional contribution of $18\,\mu$eV. Electronic Hydrogen =================== To complete the discussion, we assess the nuclear polarizability correction for the $nS$-levels in the usual (electronic) deuterium, too. In particular, the isotopic shift measurement of $1S-2S$ splitting of Ref. [@Parthey:2010] relies on the theoretical estimate according to Ref. [@FriarPayne], E\_[2S-1S]{}\^[e-D]{}&=&19.04(7)[kHz]{}, where the polarizability correction of 18.58(7) kHz and the elastic contribution of 0.46 kHz were added together. Ref. [@milshtein] gives a somewhat different result, E\_[2S-1S]{}\^[e-D]{}&=&19.25[kHz]{}, with the Coulomb contribution 17.24 kHz, the magnetic contribution 2.28 kHz and the magnetic polarizability correction -0.27 kHz. Our evaluation for the $1S-2S$ splitting in deuterium is E\_[2S-1S]{}\^[e-D]{}&=&28.812.0[kHz]{}, that is the sum of the elastic (0.53(1) kHz), inelastic (33.4(12.0) kHz) and subtraction (-4.60(3) kHz) contributions. The uncertainty is about a half of the full result. Since for the electronic deuterium the integrals over structure functions are even more strongly weighted at low values of $Q^2$ where no experimental information is available, the large uncertainty does not come unexpectedly. We show in Table \[tab3\] (fourth column) how future electron-deuteron scattering measurements can help improving on this estimate. Note that this uncertainty estimate exceeds the one in Eq. (34) by two orders of magnitude. However, the main uncertainty in the isotope shift given in [@Parthey:2010] is actually due to uncertainties in other theoretical corrections, largely caused by uncertainties in parameters such as particle masses. The total radius-related energy uncertainty in [@Parthey:2010] is $0.89$ kHz. The uncertainty from the dispersive polarizability calculation is still an order of magnitude larger; using it would change the radius difference result to r\_E\^2(d) - r\_E\^2(p) = 3.8274(88) [fm]{}\^2, increasing the uncertainty by a factor of $\sim10$ as compared to Eq. (1). Using the CODATA value for the proton charge radius $r_E(p)=0.8775(51)$ fm leads us to a new extracted value of the deuteron radius, r\_E(d)=2.1442(29) [fm]{}, that should be compared to the previous extraction [@Mohr:2012tt], $r_E(d)=2.1424(21)$ fm. Thus, in the electron case the increase in the polarizability uncertainty for the isotope shift makes it comparable to the existing uncertainty in the proton radius-squared. Using it merely increases the uncertainty in the inferred deuteron radius by a factor of $\sqrt2$. Conclusion ========== We conclude that in the case of deuterium, model independence that is the main objective of our approach comes at a high price. Scattering data do not constrain the behavior of structure functions, especially the longitudinal one, at low values of the momentum transfer. Microscopical nuclear calculations do a much better job in terms of intrinsic precision that typically is of order of fractions of a per cent. However, in absence of data this claimed precision is not warranted, and once new low-$Q^2$ electrodisintegration data will be available they will serve as a useful cross check for nuclear calculations, as well. We are grateful to M. Distler, Z.-E. Meziani, V. Pascalutsa, S. Karshenboim, D.R. Phillips, J. Yang, H. Griesshammer, K. Pachucki, J.L. Friar and S. Bacca for useful discussions. 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--- abstract: 'We offer a generalization of a formula of Popov involving the Von Mangoldt function. Some commentary on its relation to other results in analytic number theory is mentioned as well as an analogue involving the m$\ddot{o}$bius function.' author: - Alexander E Patkowski title: 'On Popov’s formula involving the Von Mangoldt function' --- Introduction and Main result ============================ In Titchmarsh’s famous text on the Riemann zeta function \[5\], we find a reference to A.I. Popov’s 1943 note \[4\]. Therein, we find the curious formula $x>1,$ $$\sum_{n>x}\frac{\Lambda(n)}{n^2}\left(\{\frac{n}{x}\}-\{\frac{n}{x}\}^2\right)=\frac{2-\log(2\pi)}{x}+\sum_{\rho}\frac{x^{\rho-2}}{\rho(\rho-1)}+\sum_{k\ge1}\frac{k+1-2k\zeta(2k+1)}{2k(k+1)(2k+1)}x^{-2k-2},$$ where $\Lambda(n)$ is the Von Mangoldt function \[3\], $\zeta(s)$ is the Riemann zeta function \[3, 5\], and $\{x\}$ is the fractional part of $x.$ As usual, the $\rho$ denotes the non-trivial zeros of the Riemann zeta function \[3\]. Following this formula in \[4\] are some interesting corollaries, but ultimately no proof. It is interesting to note that arithmetic series involving the fractional part function have also been studied by H. Davenport. We mention one of his identities \[1\] $$\sum_{n\ge1}\frac{\Lambda(n)}{n}\{nx\}=-\frac{1}{\pi}\sum_{n\ge1}\frac{\log(n)}{n}\sin(2\pi nx).$$ Now it does not appear that a proof has been offered of (1.1) to the best of our knowledge. The purpose of this note is to offer a generalization of (1.1) and in doing so we offer what appears to be the first proof of Popov’s result. For $x>1,$ and $r\ge1,$ we have $$\frac{1}{2}\sum_{n>x}\frac{\Lambda(n)}{n^{r+1}}\left(\{\frac{n}{x}\}-\{\frac{n}{x}\}^2\right)=h_r(x)+\sum_{\rho}\left(\frac{r+1-\rho}{2(\rho+1-r)(\rho-r)}-\frac{\zeta(r-\rho)}{r-\rho} \right)\frac{x^{\rho-r-1}}{r+1-\rho}$$ $$-\sum_{k\ge1}\left(\frac{2(k+1)+r-1}{2(1-2k-r)(2k+r)}+\frac{\zeta(2k+r)}{2k+r} \right)\frac{x^{-2k-r-1}}{2k+r+1},$$ where $$h_1(x)=\frac{2-\log(2\pi)}{2x},$$ and for $r>1,$ $$h_r(x)=\left(\frac{r}{2(2-r)(1-r)}-\frac{\zeta(1-r)}{r-1}\right)\frac{x^{-r}}{r}.$$ First, it is well-known that \[5\] $$\int_{1}^{\infty}t^{-s-1}\left(\{t\}-\frac{1}{2}\right)dt=\frac{s+1}{2s(s-1)}-\frac{\zeta(s)}{s},$$ for $\Re(s)>1.$ Hence, using Mellin inversion and integrating over $[0, u],$ we have for $a>1,$ $$\frac{1}{2}\left(\{u\}^2-\{u\}\right)=\frac{1}{2\pi i}\int_{(a)}\left(\frac{s+1}{2s(s-1)}-\frac{\zeta(s)}{s}\right)\frac{u^{s+1}}{s+1}ds,$$ provided that $u>1.$ Now this integral has simple poles at $s=0$ and $s=1,$ both of which have residues of $0.$ Hence, for $-1<b<0,$ $$\frac{1}{2}\left(\{u\}^2-\{u\}\right)=\frac{1}{2\pi i}\int_{(b)}\left(\frac{s+1}{2s(s-1)}-\frac{\zeta(s)}{s}\right)\frac{u^{s+1}}{s+1}ds.$$ Now we may invert the desired series with $u=\frac{n}{x}>1,$ to obtain $$\sum_{n>x}\frac{\Lambda(n)}{n^{r+1}}\left(\{\frac{n}{x}\}-\{\frac{n}{x}\}^2\right)=\frac{1}{2\pi i}\int_{(b)}\left(\frac{s+1}{2s(s-1)}-\frac{\zeta(s)}{s}\right)\frac{x^{-s-1}\zeta'(r-s)}{\zeta(r-s)(s+1)}ds,$$ provided that $r\ge1.$ Replace $s$ with $1-s$ to get the integral for $1<c<2,$ $$\frac{1}{2\pi i}\int_{(c)}\left(\frac{2-s}{2s(s-1)}-\frac{\zeta(1-s)}{1-s}\right)\frac{x^{s-2}\zeta'(s+r-1)}{\zeta(s+r-1)(2-s)}ds,$$ and we see there is a double pole at $s=1$ when $r=1,$ a simple pole at $s=2-r$ when $r>1,$ simple pole at the non trivial zeros $s=\rho+1-r,$ and simple pole at the trivial zeros $s=-2k-r+1.$ The residue at the double pole $s=1,$ $r=1$ is the $h_1(x),$ and the residue at the simple pole $s=2-r$ when $r>1,$ is the $h_r(x),$ $r>1,$ part of the theorem. The residue at the non-trivial zeros $s=\rho+1-r$ gives the sum over $\rho$ in the theorem. Lastly, the residue at the trivial zeros $s=-2k-r+1$ gives the sum over $k$ on the far right side of the theorem. After these residues have been computed the remaining integral is $0$ and the theorem follows. Further Comments ================ We do not know specifically where Popov encountered (1.1), but we believe it is interesting to note that the sum involving the roots $\rho$ appears in \[2, pg.69\], which shows a connection with $\psi(x)=\sum_{n\le x}\Lambda(n),$ through the integral $\int_{0}^{x}\psi(t)dt/t^2.$ It is not difficult to make small adjustments to our proof to include other arithmetic functions as well. In closing we offer the analogue for Merten’s function $M(x)=\sum_{n\le x}\mu(n)$ \[5, pg. 370\], involving $\int_{0}^{x}M(t)dt/t^2.$ We have for $x>1,$ $$\frac{1}{2}\sum_{n>x}\frac{\mu(n)}{n^2}\left(\{\frac{n}{x}\}-\{\frac{n}{x}\}^2\right)=\sum_{\rho}\frac{x^{\rho-2}}{\zeta'(\rho)\rho(\rho-1)}+\sum_{k\ge1}\frac{k+1-2k\zeta(2k+1)}{2k(k+1)(2k+1)}\frac{\pi^{2k}2^{2k+1}(-1)^k}{(2k)!\zeta(2k+1)}x^{-2k-2}.$$ The proof is identical to the $r=1$ case of Theorem 1.1 with the notable difference that the residue at the pole $s=1$ is 0, and at the trivial zeros we apply the formula \[5\] $$\frac{1}{\zeta'(-2n)}=\frac{\pi^{2n}2^{2n+1}}{(-1)^n\zeta(2n+1)(2n)!},$$ for natural numbers $n>0.$ The remaining details are left for the reader. [9]{} H. Davenport, *On some infinite series involving arithmetic function,* Quarterly Journal of Mathematics, 8 (1937), pp. 8–13. H. M. Edwards. *Riemann’s Zeta Function,* 1974. Dover Publications. H. Iwaniec and E. Kowalski, *Analytic number theory,* American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. A. I. Popov, *Several series containing primes and roots of $\zeta(s),$* C. R. Acad. Sri. U.R.S.S., N.S. 41 (1943), 362-3 E. C. Titchmarsh, *The theory of the Riemann zeta function,* Oxford University Press, 2nd edition, 1986. 1390 Bumps River Rd.\ Centerville, MA 02632\ USA\ E-mail: alexpatk@hotmail.com
--- abstract: 'It is well-known that selling different goods in a single bundle can significantly increase revenue, even when the valuations for the goods are independent. However, bundling is no longer profitable if the goods have high production costs. To overcome this challenge, we introduce a new mechanism, Pure Bundling with Disposal for Cost (PBDC), where after buying the bundle, the customer is allowed to return any subset of goods for their production cost. We derive both distribution-dependent and distribution-free guarantees on its profitability, which improve previous techniques. Our distribution-dependent bound suggests that the firm should never price the bundle such that the profit margin is less than 1/3 of the expected welfare, while also showing that PBDC is optimal for a large number of independent goods. Our distribution-free bound suggests that on the distributions where PBDC performs worst, individual sales perform well. Finally, we conduct extensive simulations which confirm that PBDC outperforms other simple pricing schemes overall.' author: - 'Will Ma[^1]' - 'David Simchi-Levi[^2]' bibliography: - 'bibliography.bib' title: Reaping the Benefits of Bundling under High Production Costs --- Introduction ============ We study the monopolist pricing problem of a firm selling $n$ heterogeneous items. For each item, customers have a valuation, which is their maximum willingness-to-pay, drawn from a known distribution. A customer wants at most one of each item. The firm offers take-it-or-leave-it prices for every subset of items, and the customer chooses the subset maximizing her surplus (valuation for the subset minus price), with the no-purchase option always being available. We assume the customer’s valuation for a subset is additive over the items in the set. The objective of the firm is to maximize expected per-customer revenue. In the full generality of the problem, the firm has $2^n-1$ prices to set. However, it is important to find profitable yet *simple* pricing schemes that are determined by a small number of prices. Two such schemes are *Pure Components* (PC), where items are priced separately (and the price of a subset is understood to be the sum of its constituent prices), and *Pure Bundling* (PB), the strategy of only offering all the items together. A third scheme that generalizes both PC and PB is *Mixed Bundling* (MB), which offers individual item prices as well as a bundle price for all the items. MB can be seen as a form of price discrimination, where customers who highly value an item can buy it for its individual price, while customers with lower valuations still have a chance of buying it as part of a discounted bundle. The efficacy of simple pricing schemes is of immense importance in retail, and has been studied over the past few decades in the economics literature, the operations research/marketing interface literature, and more recently, the computer science literature. For a single item, the solution is immediate: choose the price $p$ maximizing $p(1-F(p))$, where $F$ is the CDF of the valuation (see [@Mye81]). However, for two items, even if their valuations are independent, bundling can be better than individual sales. For example, suppose we have two products with IID valuations, each of which is $1$ half the time, and $2$ half the time. If we sell the items individually, we can always get a sale for $1$, or get a sale half the time for $2$. In either case, the combined expected revenue is $2$. However, if we sell the items as a bundle for $3$, then the bundle will be purchased $\frac{3}{4}$ of the time, yielding an expected revenue of $\frac{9}{4}$. The key observation is that the valuation of the bundle is more concentrated around its mean than the valuation of the individual items, which causes less consumer heterogeneity, and we can choose a price that is the highest willingness-to-pay for a larger fraction of customers. This makes it easier to reduce deadweight loss, which is revenue lost from pricing a customer with positive valuation out of the market, and consumer surplus, which is revenue lost from offering a customer a better price than necessary. The power of bundling is even greater when valuations are negatively correlated—consider two products with marginal valuations that are uniform on $[0,1]$ but correlated in a way such that they always sum to $1$. In this case, offering the bundle at the price of $1$ will always get a sale, extracting the entire welfare, while selling the items individually yields at most $\frac{1}{2}$, half the available surplus. These effects have long been known in the economics literature, following the pioneering work of [@Sti63], [@AY76], [@Sch84], and [@MMW89]. Of course, bundling is not always superior to individual sales—this is especially true once we consider production costs. For example, suppose we have two goods with IID valuations that are uniform on $[0,3]$, but each cost $2$ to produce. Selling them individually at price $\frac{5}{2}$ will yield a profit of $\frac{1}{12}$ per item and is better than selling them as a bundle—these are low-profit-margin items that are only valuable to a small fraction of the population, and by bundling them we may force a customer into consuming a good for which her valuation is less than the production cost. Over the decades, a lot of work has been done to compare the profit of Pure Bundling versus Pure Components. [@AY76] write, “The chief defect of Pure Bundling is its difficulty in complying with Exclusion,” where Exclusion refers to the principle that a transfer is better off not occuring when the consumer’s valuation is below the producer’s cost. It is observed in [@Sch84] for the case of bivariate normal valuations that Pure Bundling is better when mean valuations are high compared to costs. [@BB99] prove that bundling a large number of goods can extract almost all of the available surplus, but this is crucially dependent on the items being “information goods”, i.e. goods with no production costs. [@FN06] characterize conditions under which Pure Bundling outperforms Pure Components for a fixed number of items, and all of their conditions imply low costs. [@LFCK13] define a measure of consumer heterogeneity that increases with costs, and present computational results showing Pure Bundling performs poorly relative to Pure Components as their measure of consumer heterogeneity increases. The indisputable conclusion from all this work is that high costs are the greatest impediment to the magic of bundling. However, we argue that there is a simple way to enjoy the effects of bundling while allowing for the flexibility of individual sales—sell all of the items as a bundle, but allow the customer to return any subset of items for a refund equal to their total production cost. We call this scheme *Pure Bundling with Disposal for Cost* (PBDC). It is a strict improvement of Pure Bundling where the customer’s valuations that were below the cost have been *truncated* to equal the cost. Meanwhile, the firm is indifferent between producing an item for its cost or returning its cost to the customer, but PBDC makes it easier to sell the bundle because customers with low valuations for specific items won’t be priced out of the market. Importantly, there is great flexibility in how to present PBDC to the customer in a transparent and attractive way. In fact, we show that PBDC has a few equivalent formulations which can already be seen in the market. One formulation is a *tariff* to enter the market, after which all products are sold at cost. Alternatively, PBDC can be introduced with an individual price for each item and a *per-item discount* for each item purchased beyond the first. From a marketing point of view, the tariff strategy is more attractive when the number of items is large, while the discount strategy is more attractive when the number of items is small. Our scheme can be compared to that of [@HC05], who recognized the need for a middle ground between Pure Bundling and Pure Components. They introduced the scheme *Customized Bundling*, which prices each bundle based only on its size, and not which items are included. [@CLS08][^3] perform extensive numerical experiments for the same scheme, calling it *Bundle-Size Pricing* (BSP), showing that it can extract $99\%$ of the optimal profit in their simulations. PBDC can be seen as orthogonal to BSP—while BSP imposes symmetric pricing across items but allows non-linear pricing based on quantity, PBDC allows asymmetric pricing across items based on cost but imposes additive pricing once the customer pays the tariff to enter the market. When all item costs are identical, PBDC is a simplified version of BSP, because instead of having $n$ prices to decide, there is only one price to decide, be it thought of as the bundle price, the tariff, or the discount. However, since we are able to relate PBDC to Pure Bundling, it is much easier to analyze. Our work provides the first theoretical explanation for some of the empirical successes in [@CLS08]—indeed, in their simulations, costs are either equal, or insignificant (equal to half of the product’s mean valuation). We present two types of theoretical bounds in this paper. Both require that items have independent valuations, which is a standard and often necessary assumption in the bundling literature. Both also rely on a transformation from costs to negative valuations which as far as we know is new. First, we prove that PBDC, with an appropriately chosen bundle price, extracts the entire welfare as the number of items approaches infinity, so long as the valuations have uniformly bounded variances. This type of result is based on the law of large numbers, which says that the sum of cost-truncated valuations, which we denote with the random variable $X$, lies within $(1\pm{\varepsilon}){\mathbb{E}}[X]$ with high probability. Therefore, the bundle price of $(1-{\varepsilon}){\mathbb{E}}[X]$ will be accepted by a $(1-{\varepsilon})$-fraction of the customers, profiting almost the expected welfare, ${\mathbb{E}}[X]-C$ ($C$ is the total cost of producing all the items), which is an upper bound on profit. [@BB99] have already proven this result for the case without costs, and [@Arm99] has proven this result for a cost-based two-part tariff which we show is equivalent to PBDC, so this result in itself is not new. However, our analysis introduces the use of Cantelli’s stronger, one-sided concentration inequality in bundling, recovering previous bounds asymptotically and achieving a better bound when the coefficient of variation of $X$ is large. In the latter case, both of the previous works recommend a bundle price of $C+{\varepsilon}({\mathbb{E}}[X]-C)$, earning negligible profit, whereas our analysis never recommends a bundle price below $C+\frac{1}{3}({\mathbb{E}}[X]-C)$ and guarantees a non-zero profit. Furthermore, we recommend PBDC even when the number of items is small—the second type of theoretical bound we present is problem-independent, unaffected by the number of items or their variances. We prove that the expected profit of the best PBDC pricing is at least $\frac{1}{5.2}$ of the expected profit of the optimal mechanism, except in detectable pathological cases, where the best PC pricing achieves this guarantee. The benchmark in this theorem is the maximum expected profit that could be achieved from explicitly pricing all subsets of items[^4]. This is a tighter benchmark than expected welfare, which could be infinite without distributional assumptions. We use tools from the computer science literature to bound this benchmark, most notably from [@BILW14], who prove in the costless case that the better of PB and PC earns at least $\frac{1}{6}$ of the optimal revenue. We improve their bound from $\frac{1}{6}$ to $\frac{1}{5.2}$ by using Cantelli’s inequality, and enhancing their *core-tail decomposition* technique in analyzing the core and tail together. We also construct an example improving the upper bound from $\frac{12}{13}\approx\frac{1}{1.08}$ to $\frac{3+\ln2}{3+2\ln2}\approx\frac{1}{1.19}$, where the previous best-known example is from [@HN12]. Finally, we generalize the result of Babaioff et al. to the case with costs, where PBDC is needed instead of PB. We should point out that when the benchmark is the optimal mechanism, one cannot simply truncate all valuations from below by cost, because the optimal mechanism could exploit low valuations to reduce the cannibalization of higher-profit options. In general, profit is *non-monotone*, i.e. increasing customer valuations can decrease the optimal profit, as reported in [@HN12]. In addition to the theoretical considerations, we provide a continuation of the numerical experiments from [@CLS08], extensively testing the efficacy of PBDC on a finite number of items. We use the same independent demand distributions with the same parameters as [@CLS08]. In their setting where costs are low and identical across items, PBDC is a special case of BSP. However, it still attains between $97.5\%$ and $100\%$ of the (nearly optimal) BSP profit. If we allow costs to vary and be more significant, then PBDC drastically outperforms other simple mechanisms (PC, PB, BSP), demonstrating its robustness under different scenarios. In fact, the worst case for PBDC is the aforementioned setting where it attains $97.5\%$ of the profit of the best simple mechanism; contrast this with $79.9\%$, $16.8\%$, $59.5\%$ for PC, PB, BSP, respectively, in their wost-case settings. In addition to being profit-maximizing, PBDC also achieves excellent global surplus in our simulations. Finally, we show that PBDC attains between $96.6\%$ and $99.4\%$ of the optimal profit (which prices all subsets) when $n=3$, and scales well as $n$ increases. The general goal of our work is to dispel the myth that high costs should drive a firm away from bundling and toward individual sales. PBDC allows the firm to reap the benefits of bundling while preventing items from being consumed for utility below cost. We should point out that there do exist costless examples with independent valuations on which PBDC performs poorly relative to the optimal mechanism (which is why it is necessary to include PC in the statement of the second theoretical result). Here is a list, along with why PBDC (equivalent to PB) is ill-advised for each instance: - Example 15 from [@HN12]: there are various different valuations, each of which realizes to an exorbitant value with a small probability; bundling is ineffective because the probability that more than one valuation is non-zero is infinitessimal - Examples 1 and 2 from [@Rub16]: there is a need to *partition* the items, i.e. split them into groups, and offer each group as a different bundle - Example \[upper\_bound\_example\] from Section \[finite\_bounds\] of our paper: there is a need to price-discriminate, i.e. offer high individual prices and a discounted bundle price However, our numerical experiments demonstrate that over “average” instances, PBDC performs far better than these pathological constructions and the worst-case bound of $\frac{1}{5.2}\approx19.2\%$ suggest. Indeed, once PBDC has eliminated the effect of costs, selling everything under one bundle leaves very little to be desired. Literature Review ----------------- Bundling has been the focus of many papers in three different disciplines: economics, computer science, and operations research/marketing science. In general, the literature can be classified into three categories: papers that provide insights, papers that suggest approximate algorithms with attractive worst-case bounds, and papers that develop computationally efficient algorithms. In this subsection we attempt to highlight the most important contributions to the bundling literature, independent of discipline. **Two Items.** In the economics literature, the earliest recognition of bundling being able to increase the revenue from selling two items is usually attributed to [@Sti63]; other early research for two products includes [@AY76; @Sch84; @MMW89]. Since then, [@VK03; @MRT07] have established conditions for bundling being optimal for two potentially correlated goods. **Simple Mechanisms.** For more than two items, there is a great practical interest in finding simple pricing schemes that are both profitable and easy to explain to the customer; for surveys on how bundling has affected marketing practice see [@ST02; @VM09]. However, the only concrete, general pricing scheme we have found in this literature, other than the classical PC and PB, is the BSP proposed by [@HC05] and [@CLS08]. Our scheme, PBDC, attempts to add to this literature by providing a transparent, easy-to-compute heuristic. Most of the attempts to prove that simple pricing schemes are indeed capturing most of the optimal profit have been restricted to special cases (see [@MV06; @MV07]), or empirical evidence, as in the case of BSP, where its great experimental success has been unexplained. That’s where we turn to the computer science literature. There has been more general work on auctions with multiple buyers, or valuation functions where the valuation for a subset may not be additive over the items in the set, for which we refer to [@RW15; @Yao15] and the references therein. We focus on the case of a single buyer with additive valuations, which is the bundling problem. In this special case, [@HN12] introduce performance guarantees for simple mechanisms, which are further studied in [@HR12; @HN13]. One line of work ([@LY13; @BILW14]) culminated in a proof that either PB or PC must be within $\frac{1}{6}$ of optimal, for arbitrary independent valuations. By relating PBDC to PB, and improving upon their techniques, we are able to prove that either PBDC or PC must be within $\frac{1}{5.2}$ of optimal for the independent case with costs. When all costs are equal, PBDC is a special case of BSP, so our work provides the first theoretical explanation for some of the successes in [@CLS08]. Recently, mechanisms where items are partitioned before bundling have also been advocated as simple in [@CH13; @Rub16]. Our bound improves the theoretical guarantee for the partitioning scheme in [@Rub16] from $\frac{1}{6}$ to $\frac{1}{5.2}$. The same core-tail decomposition of [@BILW14] has also been recently used in [@BDHS15; @Yao15], so our new bound improves guarantees from those works as well. **Computational Solutions.** Others have tried to tackle the problem with more items by giving up on simplicity and computing an explicit optimal or near-optimal solution using optimization techniques. A mixed integer programming formulation was first seen in [@HM90], and recently in the mechanism design literature, explicit polynomial-time solutions were provided via linear programming in [@CDW12; @CH13]. As far as computing the optimal prices for simple mechanisms, [@WHCA08] use non-linear mixed integer programming to solve for the optimal BSP prices, while [@Rub16] gives a PTAS for the optimal partitioning. Computation is another benefit of PBDC—like PB, it only requires calculating one price, which can be done via convolution. **Large Number of Items.** Yet another line of work addresses the complexity of many items by claiming that PB is guaranteed to be optimal as the number of items approaches infinity, assuming independence and uniformly bounded variances. Traditionally, this line of work has dealt with information goods which have no marginal costs (see [@BB99; @BB00]), or showed that costs have a substantial effect on the efficacy of PB (see [@FN06; @IW10]). [@Arm99] advocates that the same result can be achieved with costs by using a cost-based two-part tariff, which we prove is equivalent to PBDC. Our research strengthens this line of work by using Cantelli’s one-sided concentration inequality to get a tighter problem-dependent bound. Furthermore, we advocate for PBDC even on a small number of items, both with our problem-independent bound, and our numerical experiments. **Closed-form Solutions.** There is also interest in finding analytical closed-form solutions for the optimal pricing under simple cases of the problem. In the case of two independent valuations, one of which is uniform on $[0,b_1]$ and the other which is uniform on $[0,b_2]$, [@Eck10] derives elementary equations for the optimal Mixed Bundling prices. [@Bha13] shows that the equations involve roots of high-degree polynomials once costs are introduced, and uses a linear approximation to record solutions. Our transformation in Section \[sect\_setup\] shows that the problem with costs is equivalent to the problem for distributions uniform on $[a_1,b_1]$ and $[a_2,b_2]$, where $a_1$ and $a_2$ could be negative. The difficulty of analytical solutions in general is discussed in [@Wil93; @Arm96; @PVM10]. Summary of Contributions and Outline of Paper --------------------------------------------- - We introduce the idea of PBDC, eliminating the problem costs pose to bundling (Section \[sect\_setup\]): - We show that PBDC has equivalent formulations in the *cost-based two-part tariff* that exists in the theoretical literature, as well as a new *per-item discount* scheme - The idea of PBDC motivates a transformation from costs to negative valuations, enabling the analysis in subsequent sections - We improve “large-number-of-items” bounds for the performance of PBDC, using Cantelli’s inequality (Section \[asymptotic\_bounds\]): - We recover existing bounds asymptotically and achieve a better bound when the coefficient of variation is large - Our bound suggests that the firm should not price the bundle such that the profit margin is less than 1/3 of the expected welfare - We provide finite-item, distribution-free bounds for the performance of PBDC (Section \[finite\_bounds\]): - We generalize existing bounds to the case with costs, where PBDC is needed instead of PB - We improve existing bounds in both directions (upper and lower bound) - We compare this type of performance guarantee to that in the previous section - We provide a continuation of the numerical experiments from [@CLS08], demonstrating the efficacy of PBDC for a finite number of items (Section \[sect\_numerical\_experiments\]) Set-up and Equivalence Propositions {#sect_setup} =================================== A firm has $n$ different items for sale. For each $i$, the cost incurred by the firm for selling item $i$ is $c_i$, a non-negative real number. $c_i$ can be thought of as an instantaneous production cost, the opportunity cost of saving the inventory for someone else, or the value of the item to the seller. Each of the firm’s customers has a valuation vector $x\in{\mathbb{R}}^n$ for the items. A customer wants at most one of each item, and her utility for a subset of items $S$ is $\sum_{i\in S}x_i$. $x$ can be thought of as a random vector drawn from $D$, the distribution of valuation vectors across the population. The firm is risk-neutral and its objective is to maximize the expected per-customer profit. In the full generality of the problem, the firm’s mechanism for selling the items is a *menu* ${\mathcal{M}}$ of *entries* $(q,s)$, where $q\in[0,1]^n$ is the *allocation* indicating the fraction of each item transferred to the customer, and $s$ is the *payment* that must be made for this allocation. If $q$ has fractional entries, then the allocation can be thought of as a lottery where the customer only gets some items with a certain probability. The customer is also risk-neutral and chooses the entry maximizing her expected surplus, $q^Tx-s$. We will assume that for every entry, the payment covers the expected cost for the firm to produce that allocation, i.e. $s\ge q^Tc$, where $c=(c_1,\ldots,c_n)$. Simultaneously removing all entries in the menu with $s<q^Tc$ cannot decrease the profit, since this can only force a customer who previously selected an entry with negative profit to select an entry with non-negative profit. Let ${\mathcal{X}}$ denote the support of $D$. Given a menu ${\mathcal{M}}$, for all $x\in{\mathcal{X}}$, let $q_{{\mathcal{M}}}(x)$ denote the allocation chosen by a customer with valuation vector $x$, and $s_{{\mathcal{M}}}(x)$ denote the corresponding payment. We will omit the subscript ${\mathcal{M}}$ when the context is clear. $(q_{{\mathcal{M}}}(x),s_{{\mathcal{M}}}(x))$ must maximize the surplus of the customer among all entries of ${\mathcal{M}}$[^5] (the mechanism is *incentive-compatible*), and one of these entries must be the no-purchase option with $q=0,s=0$ (the mechanism is *individually rational*). The firm’s profit maximization problem can be written as $\max_{{\mathcal{M}}}{\mathbb{E}}_{x\sim D}[s_{{\mathcal{M}}}(x)-q_{{\mathcal{M}}}(x)^Tc]$. However, the optimization over general menus is intractable, and the resulting menu may not be practical. A *pricing scheme* is a restricted class of menus, implied by a compact way of communicating the menu to the customer. It is assumed that the optimization problem over the restricted class of menus can be solved efficiently. The following pricing schemes are frequently referenced throughout this paper: 1. Pure Components (PC): items are offered individually, at respective non-negative prices ${P^{\mathsf{PC}}}_1,\ldots,{P^{\mathsf{PC}}}_n$. 2. Pure Components with Uniform Discount (PCUD): items are offered individually, at respective prices ${P^{\mathsf{PCUD}}}_1,\ldots,{P^{\mathsf{PCUD}}}_n$. There is an absolute discount of ${P^{\mathsf{PCUD}}}_0$ which is applied to each item bought beyond the first. For all $i$, ${P^{\mathsf{PCUD}}}_i\ge{P^{\mathsf{PCUD}}}_0\ge0$ is imposed. 3. Pure Bundling (PB): all of the items are offered in a single bundle at non-negative price ${P^{\mathsf{PB}}}_0$, and there are no separate sales. 4. Pure Bundling with Disposal (PBD): all of the items are offered in a single bundle at price ${P^{\mathsf{PBD}}}_0$, with the agreement that if the customer buys the bundle, she can return any subset of items $S$ for a refund equal to $\sum_{i\in S}{P^{\mathsf{PBD}}}_i$. For all $i$, ${P^{\mathsf{PBD}}}_i\ge0$ is imposed. Furthermore, ${P^{\mathsf{PBD}}}_0\ge\sum_{i=1}^n{P^{\mathsf{PBD}}}_i$ is imposed. 5. Tariff Pricing (TP): there is a membership fee of ${P^{\mathsf{TP}}}_0$ to enter the market. If the customer enters the market, she can buy up to one unit of each item $i$ for the price of ${P^{\mathsf{TP}}}_i$. ${P^{\mathsf{TP}}}_0,{P^{\mathsf{TP}}}_1,\ldots,{P^{\mathsf{TP}}}_n$ are all imposed to be non-negative. 6. Bundle-Size Pricing (BSP): the price of a subset $S$ is ${P^{\mathsf{BSP}}}_{|S|}$, which is dependent only on the number of items in $S$, and not which items are in $S$. $0={P^{\mathsf{BSP}}}_0\le{P^{\mathsf{BSP}}}_1\le\ldots\le{P^{\mathsf{BSP}}}_n$ is imposed. PC and PB were introduced by [@AY76]. PCUD and PBD are variations of PC and PB, respectively, and to the best of our knowledge, PCUD and PBD are new to our paper. BSP was introduced by [@HC05; @CLS08], while the idea of TP could be seen in [@Arm99]. Note that PC corresponds to the degenerate case of PBD where ${P^{\mathsf{PBD}}}_0=\sum_{i=1}^n{P^{\mathsf{PBD}}}_i$. Our first observation is the following: \[equivalence\] PCUD, PBD, and TP represent the same class of menus. Specifically, given a PBD representation with prices $$({P^{\mathsf{PBD}}}_0,{P^{\mathsf{PBD}}}_1,\ldots,{P^{\mathsf{PBD}}}_n)$$ the corresponding PCUD representation is $$\label{repn_pcud} ({P^{\mathsf{PCUD}}}_0={P^{\mathsf{PBD}}}_0-\sum_{i=1}^n{P^{\mathsf{PBD}}}_i,{P^{\mathsf{PCUD}}}_1={P^{\mathsf{PBD}}}_1+{P^{\mathsf{PCUD}}}_0,\ldots,{P^{\mathsf{PCUD}}}_n={P^{\mathsf{PBD}}}_n+{P^{\mathsf{PCUD}}}_0)$$ and the corresponding TP representation is $$\label{repn_tp} ({P^{\mathsf{TP}}}_0={P^{\mathsf{PBD}}}_0-\sum_{i=1}^n{P^{\mathsf{PBD}}}_i,{P^{\mathsf{TP}}}_1={P^{\mathsf{PBD}}}_1,\ldots,{P^{\mathsf{TP}}}_n={P^{\mathsf{PBD}}}_n).$$ The proofs of propositions are deferred to Appendix A. Hereinafter, we will always refer to PBD instead of PCUD and TP for the analysis; however, the existence of PCUD and TP gives the firm additional flexibility in how to describe these menus to the customer. Specifically, when the number of items is small, PCUD should be used instead of TP, since it does not sound so enticing for one to pay a surcharge in order to buy a small number of items. On the other hand, when the number of items is large, ${P^{\mathsf{PCUD}}}_0$ tends to be large, causing the individual items to be marked at exorbitant prices should PCUD be used. In this case, paying a membership fee to have access to all the items does not sound so bad. We are especially interested in the following subclass of PBD, where ${P^{\mathsf{PBD}}}_i=c_i$ for all $i$. Similar subclasses can also be defined for PCUD and TP, following Proposition \[equivalence\]. Setting the refund for each item equal to its cost is a logical restriction to put on PBD. To see why, consider the following definitions: \[economics\_figures\] The *welfare* generated by a customer with valuations $(x_1,\ldots,x_n)$ is $\sum_{i=1}^n\max\{x_i-c_i,0\}$, which is realized when every item valued above cost is transferred and no other items are transferred. Welfare can be split up as follows: - The *total surplus* is the welfare realized from transfers that occurred, equal to $\sum_{i=1}^nq_i(x)(x_i-c_i)$. Total surplus can be further split up depending on the price charged: - The *producer surplus* is another term for the profit earned by the firm, equal to $s(x)-\sum_{i=1}^nq_i(x)c_i$. - The *consumer surplus* is the utility gained by the customer, equal to $\sum_{i=1}^nq_i(x)x_i-s(x)$. - The *deadweight loss* is the welfare lost because an item valued above cost was not transferred, equal to $\sum_{i:x_i>c_i}(1-q_i(x))(x_i-c_i)$. - The *overinclusion loss*[^6] is the welfare lost because items were consumed for utility below cost, equal to $\sum_{i:x_i<c_i}q_i(x)(c_i-x_i)$. It is clear from the equations that the sum of producer surplus, consumer surplus, deadweight loss, and overinclusion loss is $\sum_{i:x_i>c_i}(x_i-c_i)$, equal to welfare. Also, the fact that the consumer surplus is non-negative (since the customer can always choose the no-purchase option) implies that the profit is no greater than the total surplus, which in turn is no greater than the welfare. PBDC (and thus PBD) is strictly better than PB in the following sense: \[dominance\] Given a PB menu with price ${P^{\mathsf{PB}}}$ which is at least $c_1+\ldots+c_n$, consider instead the PBD menu with prices $({P^{\mathsf{PBD}}}_0={P^{\mathsf{PB}}},{P^{\mathsf{PBD}}}_1=c_1,\ldots,{P^{\mathsf{PBD}}}_n=c_n)$. For all $x$: - The producer surplus is no less than before. - The consumer surplus is no less than before. - The deadweight loss is no more than before. - The overinclusion loss is no more than before. Note that the preceding statements are not only in expectation; for *every* valuation vector $x$ both the firm and the customer are better off. There is no reason to use PB if PBDC can be used instead, because PBDC is effectively PB where all valuations $x_i$ have been replaced by $\max\{x_i,c_i\}$. This observation leads us to the following lemma: \[transformation\] The firm’s problem of maximizing expected profit with distribution $D$ and costs $c$ is equivalent to the transformed problem of maximizing expected revenue with distribution $D'$, where $D'$ is the distribution $D$ shifted downward by $c_i$ in every dimension $i$. Furthermore, for any menu in the original problem, the corresponding menu in the transformed problem has the payment for each allocation reduced by the cost of producing that allocation. Proposition \[transformation\] is stated in precise mathematical language and proven in Appendix A. If the original optimization problem was over a restricted class of menus, then the class restriction in the transformed setting can be found via the second statement in Proposition \[transformation\]. For the remainder of this paper, we focus on bounding the revenue of PBDC in the transformed setting, which is more amenable to analysis. PBDC becomes the class of menus that offer the same price $P$ for any non-empty subset of items (see Remark \[pbdc\_transformation\] in Appendix A for a technical proof of this). The customer makes a purchase if and only if her non-negative valuations (corresponding to valuations no less than cost) sum to at least $P$, in which case the firm earns $P$. It may be tempting to truncate all negative customer valuations to $0$ and claim that after this further transformation, PBDC is identical to PB. However, in Section \[finite\_bounds\], we bound the performance of the best PBDC menu relative to the *optimal* menu (with no restriction to a pricing scheme), which can be designed to exploit negative valuations to reduce the cannibalization of higher-priced menu entries. In general, revenue is *non-monotone*, i.e. increasing customer valuations can decrease the optimal revenue—see [@HN12]. Asymptotic Performance Bounds {#asymptotic_bounds} ============================= In this section we analyze the performance of PBDC with a large number of items, whose costs have been transformed into negative valuations according to the previous section. We assume that the valuations for different items are *independent* random variables. Also making some assumptions on the means and variances of the individual distributions, PBDC is asymptotically optimal as the number of items becomes large. [@Arm99] has already proven this result for *Cost-based Tariff Pricing* (TP with the additional restriction that ${P^{\mathsf{TP}}}_i=c_i$ for all $i$), which is equivalent to PBDC via Proposition \[equivalence\]. However, our analysis works under weaker assumptions, by employing Cantelli’s inequality, along with other tools. To our knowledge, we are the first to use Cantelli’s one-sided concentration inequality to get an improved performance bound for bundling; previous works by [@BB99; @Arm99; @FN06] all use the weaker Chebyshev’s inequality. The analysis also motivates our finite-item, distribution-free bounds in Section \[finite\_bounds\], where we again make improvements using Cantelli’s inequality. \[cantelli\] (Cantelli’s Inequality) Let $X$ be a random variable with (finite) mean $\mu$ and variance $\sigma^2$. Let $t$ be an arbitrary non-negative real number. Then $${\mathbb{P}}[X-\mu\le-t]\le\frac{\sigma^2}{\sigma^2+t^2}$$ We refer the reader to [@Lug09] for a proof, as well as more background. Our main result in this section is the following: \[mr1\] Suppose a firm is selling items to a customer with valuation vector $x$ drawn from distribution $D$. Let ${\textsc{Val}^+}(D)$ denote the mean of the welfare, equal to ${\mathbb{E}}_{x\sim D}[\sum_i\max\{x_i,0\}]$, and assume that $0<{\textsc{Val}^+}(D)<\infty$. Furthermore, let ${\textsc{Cv}^+}(D)$ denote the coefficient of variation of the welfare, equal to $\frac{\sqrt{{\mathrm{Var}}_{x\sim D}[\sum_i\max\{x_i,0\}]}}{{\mathbb{E}}_{x\sim D}[\sum_i\max\{x_i,0\}]}$, and assume that ${\textsc{Cv}^+}(D)<\infty$. Then for all ${\varepsilon}\in[0,1]$, the expected revenue of the PBDC menu with price $(1-{\varepsilon}){\textsc{Val}^+}(D)$ is at least $\frac{{\varepsilon}^2-{\varepsilon}^3}{{\varepsilon}^2+({\textsc{Cv}^+}(D))^2}\cdot{\textsc{Val}^+}(D)$. In particular, if $$\label{benchmark_price} {\varepsilon}=\frac{2({\textsc{Cv}^+}(D))^{\frac{2}{3}}}{3({\textsc{Cv}^+}(D))^{\frac{2}{3}}+2},$$ then the expected revenue is at least $$\label{ugly_bound} \frac{4}{4+24({\textsc{Cv}^+}(D))^{\frac{2}{3}}+45({\textsc{Cv}^+}(D))^{\frac{4}{3}}+27({\textsc{Cv}^+}(D))^2}\cdot{\textsc{Val}^+}(D)$$ which in turn is at least $$\label{cvgence_rate} (1-6({\textsc{Cv}^+}(D))^{\frac{2}{3}})\cdot{\textsc{Val}^+}(D).$$ (\[cvgence\_rate\]) shows that when the coefficient of variation is close to 0, ${\varepsilon}$ scales with $({\textsc{Cv}^+}(D))^{\frac{2}{3}}$ and earns a $\big(1-\Theta(({\textsc{Cv}^+}(D))^{\frac{2}{3}})\big)-$fraction of the expected welfare, recovering the result from [@BB99] and [@Arm99]. However, for larger ${\textsc{Cv}^+}(D)$, we still get a non-zero revenue guarantee in (\[ugly\_bound\]), and interestingly our analysis never recommends offering the bundle below the price of $(1-\frac{2}{3}){\textsc{Val}^+}(D)=\frac{{\textsc{Val}^+}(D)}{3}$. Contrast this to the previous analyses, which recommend ${\varepsilon}=1$ when ${\textsc{Cv}^+}(D)\ge1$, earning zero revenue. The value of ${\varepsilon}$ in (\[benchmark\_price\]), recommended by our analysis, is useful even when the firm has the resources to compute the optimal value of ${\varepsilon}$ from $D$—both as a managerial reference point, as well as in situations where the firm knows the mean and variance in demand but is uncertain about the exact distribution. Theorem \[mr1\] treats the welfare as an abstract random variable, but the revenue guarantee is weak if the coefficient of variation is large. Independence is important in allowing the “law of large numbers” to control ${\textsc{Cv}^+}(D)$ when the number of items $n$ is large. \[cr1\] Suppose a firm is selling $n$ items to a customer with independent valuations $x_1,\ldots,x_n$ forming product distribution $D$. Let ${\mu_{\min}}$ be a lower bound on ${\mathbb{E}}[\max\{x_i,0\}]$, and let ${\sigma_{\max}}^2$ be an upper bound on ${\mathrm{Var}}[\max\{x_i,0\}]$, over $i=1,\ldots,n$. Suppose ${\mu_{\min}}>0$, ${\sigma_{\max}}<\infty$, and $n>(\frac{{\sigma_{\max}}}{{\mu_{\min}}})^2$. Then the expected revenue of an optimal menu within PBDC is at least $$(1-6(\frac{{\sigma_{\max}}}{{\mu_{\min}}})^{\frac{2}{3}}\frac{1}{\sqrt[3]{n}})\cdot{\textsc{Val}^+}(D).$$ Taking $n\to\infty$, we get the result that PBDC extracts the entire welfare. Note that truncating the random variables $x_i$ from below by $0$ can only increase the mean and decrease the variance, so any lower bound on ${\mathbb{E}}[x_i]$ and upper bound on ${\mathrm{Var}}[x_i]$ would also satisfy the conditions in Corollary \[cr1\]. Let $X=\sum_{i=1}^n\max\{x_i,0\}$ be a single random variable representing the welfare of a valuation vector drawn from $D$. As additional shorthand, let $\mu={\textsc{Val}^+}(D)$ denote the mean of $X$, $\sigma={\textsc{Val}^+}(D)\cdot{\textsc{Cv}^+}(D)$ denote the standard deviation of $X$, and $C={\textsc{Cv}^+}(D)$ denote the coefficient of variation of $X$. We would like to bound the probability that $X<(1-{\varepsilon})\mu$ from above. Applying Cantelli’s inequality with $t={\varepsilon}\mu$, this probability is at most $\frac{\sigma^2}{\sigma^2+{\varepsilon}^2\mu^2}$. Therefore, our expected revenue is at least $$(1-{\varepsilon})\mu\cdot(1-\frac{\sigma^2}{\sigma^2+{\varepsilon}^2\mu^2})=\mu\cdot\frac{(1-{\varepsilon}){\varepsilon}^2\mu^2}{\sigma^2+{\varepsilon}^2\mu^2}.$$ The fraction of expected welfare earned is $$\begin{aligned} \frac{{\varepsilon}^2-{\varepsilon}^3}{{\varepsilon}^2+C^2} & \ge & \frac{{\varepsilon}^2-{\varepsilon}^3}{\frac{2}{3}y^3C^{-\frac{2}{3}}+\frac{1}{3}C^{\frac{4}{3}}+C^2} \label{original} \\ & \ge & \frac{{\varepsilon}^2-(1+\frac{2}{3}C^{-\frac{2}{3}}){\varepsilon}^3}{\frac{1}{3}C^{\frac{4}{3}}+C^2}. \label{derivative}\end{aligned}$$ The first inequality uses the *weighted arithmetic mean–geometric mean inequality* (see [@Zha08] for a reference), which yields $\frac{2y^3+C^2}{3}\ge(y^6C^2)^{\frac{1}{3}}=y^2C^{\frac{2}{3}}$. The second inequality is because for a fraction $\frac{a}{b}$ with $0<a\le b$, subtracting the same amount less than $b$ from both the numerator and the denominator can only decrease the fraction. Now, if we choose ${\varepsilon}=\dfrac{2C^{\frac{2}{3}}}{3C^{\frac{2}{3}}+2}$ (this is motivated by setting the derivative of (\[derivative\]) to zero), then the LHS of (\[original\]) becomes $$\begin{aligned} \frac{4C^{\frac{4}{3}}(1-\frac{2}{3})}{(3C^{\frac{2}{3}}+2)^2(\frac{1}{3}C^{\frac{4}{3}}+C^2)} & = & \frac{\frac{4}{3}}{(2+3C^{\frac{2}{3}})^2(\frac{1}{3}+C^{\frac{2}{3}})} \\ & = & \frac{4}{4+24C^{\frac{2}{3}}+45C^{\frac{4}{3}}+27C^2} \\ & = & 1-6C^{\frac{2}{3}}\left(\frac{4+\frac{15}{2}C^{\frac{2}{3}}+\frac{9}{2}C^{\frac{4}{3}}}{4+24C^{\frac{2}{3}}+45C^{\frac{4}{3}}+27C^2}\right) \\ & \ge & 1-6C^{\frac{2}{3}}\end{aligned}$$ where the inequality holds because the expression in parentheses is less than 1. This establishes both (\[ugly\_bound\]) and (\[cvgence\_rate\]), completing the proof of Theorem \[mr1\]. By independence, ${\mathrm{Var}}[\sum_{i=1}^n\max\{x_i,0\}]=\sum_{i=1}^n{\mathrm{Var}}[\max\{x_i,0\}]$ which is at most $n{\sigma_{\max}}^2$. Furthermore, ${\mathbb{E}}[\sum_{i=1}^n\max\{x_i,0\}]\ge n{\mu_{\min}}$. Therefore, ${\textsc{Cv}^+}(D)$ is upper bounded by $\frac{{\sigma_{\max}}}{{\mu_{\min}}\sqrt{n}}$, and it is easy to see from the proof of Theorem \[mr1\] that all of its statements continue to hold when ${\textsc{Cv}^+}(D)$ is replaced by an upper bound on ${\textsc{Cv}^+}(D)$. The condition $n>(\frac{{\sigma_{\max}}}{{\mu_{\min}}})^2$ ensures that ${\textsc{Cv}^+}(D)<1$, and the result follows immediately from substituting ${\textsc{Cv}^+}(D)\le\frac{{\sigma_{\max}}}{{\mu_{\min}}\sqrt{n}}$ into (\[cvgence\_rate\]). Finite-item, Distribution-free Performance Bounds {#finite_bounds} ================================================= In this section we analyze the performance of PBDC with only the independence assumption on the items, whose costs have been transformed into negative valuations according to Section \[sect\_setup\]. All proofs are deferred to Appendices B–C, but we sketch the techniques needed to handle arbitrary distributions. \[mr2\] Suppose a firm is selling items to a customer with independent (and potentially negative) valuations forming product distribution $D$. Let ${\textsc{Rev}}(D)$ denote the expected revenue of an optimal menu (along with tie-breaking rules) for distribution $D$. Then the expected revenue of either the optimal menu within PBDC or the optimal menu within PC is at least $$\frac{1}{5.2}\cdot{\textsc{Rev}}(D).$$ In the previous section, we showed that with assumptions on the number of items and their variances, PBDC can earn almost all of the expected welfare, ${\textsc{Val}^+}(D)$. However, this is clearly false without distributional assumptions—${\textsc{Val}^+}(D)$ can be infinite. To recover some guarantee on performance, we need to use the *core-tail decomposition*, a technique developed through [@HN12; @LY13; @BILW14]. The idea of the core-tail decomposition is to split off from each independent distribution all the valuations above a large positive cutoff (the “tail”). The remaining valuations (the “core”) are bounded, and it can be shown using a concentration inequality that PB (in our case PBDC) performs well relative to the welfare of the core. Meanwhile, PC can be shown to perform well relative to the *optimal mechanism* in the tail. Finally, the core bound (relative to the expected welfare of the core) and the tail bound (relative to the optimal expected revenue for the tail) can be combined to get a performance guarantee relative to the optimal expected revenue on $D$. Theorem \[mr2\] improves upon the main result of [@BILW14] by increasing the guarantee from $\frac{1}{6}$ to $\frac{1}{5.2}$, and allowing for negative valuations. The differences in our analysis can be summarized as follows: - We analyze the core and tail together, and show that the worst case for PBDC in the core and worst case for PC in the tail cannot simultaneously occur - We use Cantelli’s inequality instead of Chebyshev’s inequality in the core bound - We show that the core bound and the tail bound can still be combined to upper-bound the optimal revenue on $D$ when the optimal mechanism can exploit negative valuations Table \[comparison\] compares Theorem \[mr2\] to the type of bound in the previous section, in particular Corollary \[cr1\]. Essentially, to accommodate arbitrary distributions, we have to settle for a constant fraction of the optimum, compare against an optimum that is convoluted, and also allow ourselves to use PC in pathological cases. [\*The advantages of each bound are bolded.]{} One additional point worth mentioning is that it is unclear from Theorem \[mr2\] what the optimal prices for PBDC or PC are. It is assumed that the firm, knowing distribution D, can compute the optimal prices for both PBDC and PC and use the scheme with higher expected revenue, with the knowledge that it will be within $\frac{1}{5.2}$ of optimal. Meanwhile, Theorem \[mr1\], with its simpler analysis, has an explicit benchmark price of $\frac{({\textsc{Cv}^+}(D))^{2/3}+2}{3({\textsc{Cv}^+}(D))^{2/3}+2}\cdot{\textsc{Val}^+}(D)$ for the bundle in PBDC. Finally, we address the tightness of Theorem \[mr2\]. First we present a theoretical upper bound. \[upper\_bound\_example\] Consider an instance with $2$ costless items, which have IID valuations distributed as follows. There is a point mass of size $1-\rho$ at $0$, a point mass of size $\frac{\rho}{2}$ at $2$, and the remaining $\frac{\rho}{2}$ mass distributed in an *equal-revenue* fashion on $[1,2)$, i.e. selling individually at any price in $[1,2)$ results in the same revenue. Formally, if $Y$ is a random variable with this distribution, then $${\mathbb{P}}[Y\ge y]=\begin{cases} 1 & y=0 \\ \rho & 0<y\le1 \\ \frac{\rho}{y} & 1\le y\le2 \end{cases}$$ where the value of $\rho$ is optimized to be $\frac{3}{3+\ln2}\approx0.81$. \[thm\_upper\_bound\] Consider the instance in Example \[upper\_bound\_example\]. The best possible PC revenue is $2\rho$, attained by selling individual items at any price in $[1,2]$. The best possible PB revenue is also $2\rho$, attained by selling the bundle at the price of $2$ or $3$. The optimal revenue is at least $2\rho(2-\rho)$; this value can be achieved by selling individual items at the price of $2$, and the bundle at the discounted price of $3$. Therefore, neither PC nor PB can obtain more than $\frac{3+\ln2}{3+2\ln2}\cdot{\textsc{Rev}}(D)$ which is approximately $$\frac{1}{1.19}\cdot{\textsc{Rev}}(D).$$ In Example \[upper\_bound\_example\], both PC and PB perform poorly because there is a need to *price-discriminate*, i.e. allow customers who highly value an item to buy it for its individual price, but still give customers with lower valuations a chance of buying it as part of a discounted bundle. Very recently, [@Rub16] constructed an example where both PC and PB perform poorly because there is a need to *partition* the items, i.e. split them into groups, and offer each group as a different bundle. In his example, the better of PC and PB can only obtain $\frac{1}{2}+\varepsilon$ of the revenue via partitioning, which is smaller than our bound. However, our example exhibits the worst-known loss from not price-discriminating, where even partitioning performs poorly relative to the optimal mechanism. Our example also only requires two IID items, following the examples of [@HN12; @HR12]; the example in [@Rub16] requires a large number of distinct items. Nonetheless, there is a large gap between the best-known lower bound from Theorem \[mr2\] and the best-known upper bounds, and furthermore, being guaranteed only $\frac{1}{5.2}\approx19.2\%$ of the optimal profit does not sound so enticing. However, this bound arises from a worst-case analysis that needs to address pathological instances, on which PBDC does not obtain $\frac{1}{5.2}$ of the optimum, but PC does. In the next section, we test the performance of PBDC over “average” instances. Numerical Experiments {#sect_numerical_experiments} ===================== In this section we conduct a continuation of the numerical experiments from [@CLS08] where PBDC is included as an additional pricing scheme. As a disclaimer, we should quote [@CLS08] on the limitations inherent to this kind of numerical analysis: > “Although we attempt to cover a large space of parameter values, the results clearly depend on the specific parameters we choose (i.e., the choice of grid). Further, there is no way for us to know whether we are under- or oversampling the relevant (i.e., empirically plausible) combinations of parameters. So, for example, when we describe average outcomes, these should certainly not be interpreted as outcomes that would be expected in an empirical sense—they should be interpreted narrowly as the average of the experiments we performed.” Procedure --------- For consistency, we follow the setup from [@CLS08] as closely as possible. We use the same five families of valuation distributions commonly used to model demand—Exponential, Logit, Lognormal, Normal, and Uniform. We also use the same ranges of parameters for these families, as outlined in Table \[ranges\_of\_parameters\]. The parameters were calibrated so that valuations across different families have similar means on average, and the highest means are around $10$ times the lowest means. We allow for free disposal, just like [@CLS08]—all negative valuations are converted to $0$. We assume that valuations are independent across items. [\*Note that [@CLS08] have two separate families of Normal distributions, one with varying mean and one with varying variance. For convenience, we allow both to vary at the same time.]{} As far as costs, we consider three scenarios: 1. *Heterogeneous Items*: valuation distributions fluctuate in accordance with Table \[ranges\_of\_parameters\], while costs are low. The cost of each item is set to $0.2$, except in the case of Uniform distributions, where it is set to half the item’s mean valuation. These are the same numbers used in [@CLS08], so this scenario is a duplicate of some of their experiments. 2. *Heterogeneous Costs*: valuation distributions are identical, while costs fluctuate. The costs are chosen uniformly from $[0,2.5]$, approximately the same range as the means. In the case of the bounded Uniform distribution, the costs are chosen uniformly from $0$ to $0.75$ times the maximum valuation, so that there always are some customers who value the item above cost. The fixed valuation distributions are disclosed in Table \[numerical\_summary\]—we choose a mean that is on the high end of the range to avoid degenerate instances, where the welfare in the system is near $0$ when costs are high. 3. *Heterogeneous Items and Costs*: both valuation distributions and costs are allowed to fluctuate (independently) according to the preceding scenarios. The parameters and costs are summarized in Table \[numerical\_summary\]. We compare the four simple pricing schemes—PC, PB, BSP, and PBDC. Unlike [@CLS08], we do not compute the optimal deterministic profit with $2^n-1$ prices, since it is hard to compute, difficult to implement in practice, and could be far off from the optimal profit of a randomized mechanism anyway. Skipping this expensive computation allows us to consider $n$ from $2$ up to $6$. For each combination of the $3$ cost scenarios, $5$ demand distributions, and $5$ options for $n$, we randomly generate $200$ instances, resulting in $15000$ total instances. [@CLS08] were able to discretize the parameter space for each combination and generate $220$ instances in a grid. While generating instances in a grid is more reliable than generating instances randomly, we simply have too many combinations, because we allow costs to vary independently, allow for larger $n$, and in the case of Normal distributions, also allow variances to vary independently. Our randomized approach has the advantage of being scalable, and not depending on the exact grid chosen. Furthermore, we have verified that $200$ instances per combination is enough, in that repeating the experiments does not cause the reported observations to change by any significance. Observations ------------ First, we report the performance of the simple pricing schemes separated by scenario. For each instance (out of the $15000$), we compute which of PC, PB, BSP, PBDC earns the most profit on that instance, and record the performance of every pricing scheme as a *fraction* of this optimum. For each scenario (out of the $3$), we report the median performance as well as $10$’th percentile performance of every pricing scheme across the $1000$ instances of each distribution family ($200$ for each of $n=2,\ldots,6$), in Table \[all\_scenarios\]. We also count the number of instances on which each pricing scheme was best, in Table \[winrates\]. [\*For each scenario, the best performance in each row is **bolded**. The overall worst median performance of each pricing scheme is *italicized*.]{} We know from [@CLS08] that BSP is within $1\%$ of the deterministic optimum in most of their settings, so there is minimal room for improvement under scenario 1. In fact, PBDC is a special case of BSP when all costs are identical, and very similar to PB when costs are low. However, as one can see in Table \[all\_scenarios\], PBDC still extracts close to $100\%$ of the BSP profit under this scenario, hence it also extracts close to $100\%$ of the deterministic optimum. For Uniform valuations, PBDC is no longer a special case of BSP, since costs vary proportionally with means. PBDC actually outperforms BSP in this setting—indeed, this is by far the worst setting for BSP listed in [@CLS08 tbl. 5], where it only extracts $91\%$ of the deterministic optimum. Scenario 2, where valuation distributions are identical but costs are allowed to fluctuate, really exhibits the power of PBDC, which allows customers to consume only the items they value above cost via self-selection. PC loses out on not bundling similar items that differ only in cost, while BSP is forced to compromise between charging cheap prices where high-cost items may be consumed for utility below cost, or charging expensive prices that result in a lot of deadweight loss in the low-cost items. In Appendix D, we show an instance that exemplifies why BSP performs so poorly when the costs in the setup from [@CLS08] are increased. When both valuation distributions and costs are allowed to vary under scenario 3, PBDC is still the best strategy by a significant margin. However, the benefits of bundling have decreased when items can be drastically different, so PC has gained ground. It seems intuitive to hypothesize that the performance of PC is inflated by the small values of $n$ we are using. In the next subsection, we organize our reports separated by $n$, under scenario 3 (where both valuation distributions and costs are allowed to fluctuate). Separation by $n$ and Effects on Welfare ---------------------------------------- In this subsection, we allow both valuation distributions and costs to vary, and report averages across demand distributions, separated by $n$ (instead of medians over the different choices for $n$, separated by demand distribution). Since the distribution families we’re amalgamating were calibrated to have similar means over their ranges of parameters, it makes sense in this subsection to report average absolute profits, instead of median fractions. We also report the figures defined in Definition \[economics\_figures\], in the same way as [@CLS08]. In Table \[economics\_table\], we report the expected values of these figures across the $1000$ instances for each $n$. The main conclusions are best summarized in Figures \[breakdown\_of\_welfare\_graph\]-\[increase\_with\_n\_graph\]. ![Breakdown of Welfare for each Pricing Scheme, averaged over $n$[]{data-label="breakdown_of_welfare_graph"}](breakdownOfWelfareGraph.jpg){height="2in"} The first graph (Figure \[breakdown\_of\_welfare\_graph\]) shows that although PBDC optimizes from the perspective of a selfish monopolist interested only in Producer Surplus, it has a similar advantage in Total Surplus. There is no Overinclusion Loss, and the monopolist is encouraged to choose a low tariff price so that most customers can enter the market. PC also incurs no Overinclusion Loss, but incurs more Deadweight Loss because it does not bundle. PB incurs significantly more Overinclusion Loss than any other strategy, forcing the customer into buying every item at once. All in all, PBDC is equally attractive from the standpoint of an altruistic policymaker interested in maximizing Total Surplus. ![Average Profit of each Pricing Scheme, as a function of $n$[]{data-label="increase_with_n_graph"}](increaseWithNGraph.jpg){height="4in"} The second graph (Figure \[increase\_with\_n\_graph\]) shows the profits of each pricing scheme as $n$ increases. The PC profits increase linearly with $n$, since items are sold separately. Both the PB and the BSP profits are concave in $n$—that is, the marginal gain from having one more item to sell is decreasing. Indeed, PB is burdened with adding to its grand bundle another item that could be valued below cost, while BSP is burdened with an additional distinct item to consider in its item-symmetric cost structure. PBDC is the only pricing scheme where the profit is convex in $n$, as each item creates additional incentive for the customer to enter the market, and makes their total utility from entering the market more concentrated. This confirms the hypothesis that while Table \[all\_scenarios\] reports a small gap between PC and PBDC under scenario 3, this gap quickly widens as $n$ increases. Grid Instances and Comparing with the Deterministic Optimum for $n=3$ --------------------------------------------------------------------- In this subsection, we generate instances in a grid where both valuation distributions and costs are allowed to vary, for the $n=3$ case. There are $3$ possibilities for distribution mean and $3$ possibilities for cost for each of $3$ different items, resulting in a total of $3^6=729$ instances. This is repeated over the $5$ different demand distributions. The grid is outlined in Table \[numerical\_summary2\]; we centered the grid around the values from Table \[numerical\_summary\]. We report the performance of each simple pricing scheme over these $729$ instances in the same manner as Table \[all\_scenarios\], except this time every number is recorded as a fraction of the optimal deterministic profit, which is at least the profit of any simple pricing scheme. The results are displayed in Table \[grid\_instances\]. In the median case, PBDC obtains between $96.6\%$ to $99.4\%$ of the deterministic optimum across the different demand distributions. This confirms both that PBDC is performing well relative to the optimal deterministic profit and not just other simple mechanisms, and that our earlier numbers with random instances are consistent. To summarize our numerical experiments, we considered both scenarios with low costs and scenarios with high costs, and reported median performances over $n=2,\ldots,6$ for different demand distributions. When costs are low, PC can earn as little as $79.9\%$ of the profit of the optimal simple mechanism. When costs are high, PB can earn as little as $16.8\%$ of the profit of the optimal simple mechanism, BSP can earn as little as $59.5\%$, and PC also falls behind as $n$ increases. PBDC has the highest percentages overall, and is by far the most robust over different cost scenarios, always obtaining at least $97.5\%$ of the profit of the optimal simple mechanism. We should point out that throughout our simulations, PBDC was also computationally much faster than BSP, requiring an optimization over $1$ price instead of $n$. Conclusion and Open Questions {#sect_conclusion} ============================= In this paper, we propose a simple strategy for the multi-product pricing problem: Pure Bundling with Disposal for Cost, or PBDC. We prove that PBDC is asymptotically optimal. When there are only a small number of items, we still guarantee that either PBDC or PC earns at least $\frac{1}{5.2}\approx19.2\%$ of the optimal profit, and our simulations suggest that this is closer to 96.6%-99.4% in the average case, and that PC is not needed. While this is worse than the 99% achieved by [@CLS08] for BSP in their experiments with lower costs, the pricing problem becomes much harder when costs are significant, and the existing simple pricing schemes (including BSP) fall behind PBDC by a great deal. Yet, production costs exceeding mean valuations is a common occurrence in industry, where only a small fraction of a company’s customers may have interest in any particular item. One caveat with PBDC is that the prices do reveal production costs to the customer. If this is undesired, a potential remedy is optimizing prices over the larger class of Tariff Pricing (TP) strategies, which has $n+1$ degrees of freedom and is guaranteed to be at least as profitable as PBDC. We believe that using TP instead of PBDC is very reasonable in practice, so long as the firm can accept the significant increase in computation time and decrease in the manager’s ability to interpret the pricing. However, the true demand distribution is never known, and must be constructed from data. When the given demand is prone to error, we hypothesize that there is additional benefit in choosing strategies that optimize one price at a time (such as PC, PB, PBDC) over strategies that optimize $\Theta(n)$ prices together (such as BSP, TP, MB). Besides, the theoretical guarantee for PBDC is no worse than that for TP, and PBDC is optimal as the number of items approaches infinity. We find it particularly interesting that as $n$ increases and there are *more* potential prices to optimize, the benefit of optimizing only *one* price is greater. All in all, PBDC captures the concentration effects of bundling and the selection effects of individual sales in a single heuristic that is computationally minimal and highly marketable. We hope our work on PBDC will have an impact on both the theory and practice of bundling, and be viewed as an effort to tie together the streams of research from three different disciplines: economics, computer science, and operations research. Appendix A: Proofs from Section \[sect\_setup\] =============================================== By the definition of PBD, the customer can purchase any non-empty subset $S$ of items for the price of ${P^{\mathsf{PBD}}}_0-\sum_{i\notin S}{P^{\mathsf{PBD}}}_i$. Of course, the customer can also choose not to make a purchase. Altogether, the class of menus represented by PBD is $$\label{class_of_menus} \Big\{\{({\mathbbm{1}}_S,{P^{\mathsf{PBD}}}_0-\sum_{i\notin S}{P^{\mathsf{PBD}}}_i):S\neq\emptyset\}\cup\{(0,0)\}:{P^{\mathsf{PBD}}}_1\ge0,\ldots,{P^{\mathsf{PBD}}}_n\ge0,{P^{\mathsf{PBD}}}_0\ge{P^{\mathsf{PBD}}}_1+\ldots+{P^{\mathsf{PBD}}}_n\Big\}$$ where ${\mathbbm{1}}_S\in\{0,1\}^n$ is the indicator vector for items belonging to $S$. Now, note that (\[repn\_pcud\]) defines a valid menu within PCUD since for all $i$, ${P^{\mathsf{PCUD}}}_i={P^{\mathsf{PBD}}}_i+{P^{\mathsf{PCUD}}}_0\ge{P^{\mathsf{PCUD}}}_0={P^{\mathsf{PBD}}}_0-\sum_{j=1}^n{P^{\mathsf{PBD}}}_j\ge0$. The class of menus represented by (\[repn\_pcud\]) is $$\begin{aligned} & & \Big\{\{({\mathbbm{1}}_S,\sum_{i\in S}{P^{\mathsf{PCUD}}}_i-(|S|-1){P^{\mathsf{PCUD}}}_0):S\neq\emptyset\}\cup\{(0,0)\}:{P^{\mathsf{PBD}}}_1\ge0,\ldots,{P^{\mathsf{PBD}}}_n\ge0,{P^{\mathsf{PBD}}}_0\ge{P^{\mathsf{PBD}}}_1+\ldots+{P^{\mathsf{PBD}}}_n\Big\} \\ & = & \Big\{\{({\mathbbm{1}}_S,\sum_{i\in S}{P^{\mathsf{PBD}}}_i+({P^{\mathsf{PBD}}}_0-\sum_{i=1}^n{P^{\mathsf{PBD}}}_i)):S\neq\emptyset\}\cup\{(0,0)\}:{P^{\mathsf{PBD}}}_1\ge0,\ldots,{P^{\mathsf{PBD}}}_n\ge0,{P^{\mathsf{PBD}}}_0\ge{P^{\mathsf{PBD}}}_1+\ldots+{P^{\mathsf{PBD}}}_n\Big\} \\ & = & \Big\{\{({\mathbbm{1}}_S,{P^{\mathsf{PBD}}}_0-\sum_{i\notin S}{P^{\mathsf{PBD}}}_i):S\neq\emptyset\}\cup\{(0,0)\}:{P^{\mathsf{PBD}}}_1\ge0,\ldots,{P^{\mathsf{PBD}}}_n\ge0,{P^{\mathsf{PBD}}}_0\ge{P^{\mathsf{PBD}}}_1+\ldots+{P^{\mathsf{PBD}}}_n\Big\}\end{aligned}$$ which is identical to (\[class\_of\_menus\]). Furthermore, it is easy to see that the relation defined by (\[repn\_pcud\]) is a bijection between (\[class\_of\_menus\]) and $$\Big\{\{({\mathbbm{1}}_S,\sum_{i\in S}{P^{\mathsf{PCUD}}}_i-(|S|-1){P^{\mathsf{PCUD}}}_0):S\neq\emptyset\}\cup\{(0,0)\}:{P^{\mathsf{PCUD}}}_i\ge{P^{\mathsf{PCUD}}}_0\ge0\ \forall i\in[n]\Big\}.$$ Similarly, note that (\[repn\_tp\]) defines a valid menu within TP since ${P^{\mathsf{TP}}}_0={P^{\mathsf{PBD}}}_0-\sum_{i=1}^n{P^{\mathsf{PBD}}}_i\ge0$, and for all $i$, ${P^{\mathsf{TP}}}_i={P^{\mathsf{PBD}}}_i\ge0$. The class of menus represented by (\[repn\_tp\]) is $$\begin{aligned} & & \Big\{\{({\mathbbm{1}}_S,{P^{\mathsf{TP}}}_0+\sum_{i\in S}{P^{\mathsf{TP}}}_i):S\neq\emptyset\}\cup\{(0,0)\}:{P^{\mathsf{PBD}}}_1\ge0,\ldots,{P^{\mathsf{PBD}}}_n\ge0,{P^{\mathsf{PBD}}}_0\ge{P^{\mathsf{PBD}}}_1+\ldots+{P^{\mathsf{PBD}}}_n\Big\} \\ & = & \Big\{\{({\mathbbm{1}}_S,({P^{\mathsf{PBD}}}_0-\sum_{i=1}^n{P^{\mathsf{PBD}}}_i)+\sum_{i\in S}{P^{\mathsf{PBD}}}_i):S\neq\emptyset\}\cup\{(0,0)\}:{P^{\mathsf{PBD}}}_1\ge0,\ldots,{P^{\mathsf{PBD}}}_n\ge0,{P^{\mathsf{PBD}}}_0\ge{P^{\mathsf{PBD}}}_1+\ldots+{P^{\mathsf{PBD}}}_n\Big\} \\ & = & \Big\{\{({\mathbbm{1}}_S,{P^{\mathsf{PBD}}}_0-\sum_{i\notin S}{P^{\mathsf{PBD}}}_i):S\neq\emptyset\}\cup\{(0,0)\}:{P^{\mathsf{PBD}}}_1\ge0,\ldots,{P^{\mathsf{PBD}}}_n\ge0,{P^{\mathsf{PBD}}}_0\ge{P^{\mathsf{PBD}}}_1+\ldots+{P^{\mathsf{PBD}}}_n\Big\}\end{aligned}$$ which is identical to (\[class\_of\_menus\]). Furthermore, it is easy to see that the relation defined by (\[repn\_tp\]) is a bijection between (\[class\_of\_menus\]) and $$\Big\{\{({\mathbbm{1}}_S,{P^{\mathsf{TP}}}_0+\sum_{i\in S}{P^{\mathsf{TP}}}_i):S\neq\emptyset\}\cup\{(0,0)\}:{P^{\mathsf{TP}}}_0\ge0,{P^{\mathsf{TP}}}_1\ge0,\ldots,{P^{\mathsf{TP}}}_n\ge0\Big\}.$$ This completes the proof of Proposition \[equivalence\]. Consider any valuation vector $x\in{\mathbb{R}}^n$. First suppose the customer bought the bundle with all the items for ${P^{\mathsf{PB}}}$. Under the PBD menu, the customer will still buy the bundle, since it is non-negative utility even if she keeps all the items. However, she will choose to return any items $i$ with $x_i<c_i$. Let $S$ denote the set of such items, which is possibly empty. - The producer surplus under PB is ${P^{\mathsf{PB}}}-\sum_{i=1}^nc_i$. The producer surplus under PBD is $({P^{\mathsf{PB}}}-\sum_{i\in S}c_i)-\sum_{i\notin S}c_i$, which is identical. - The consumer surplus under PB is $\sum_{i=1}^nx_i-{P^{\mathsf{PB}}}$. The consumer surplus under PBD is $\sum_{i\notin S}x_i-({P^{\mathsf{PB}}}-\sum_{i\in S}c_i)=\sum_{i=1}^n\max\{x_i,c_i\}-{P^{\mathsf{PB}}}$ which can only be greater than the consumer surplus under PB. - The deadweight loss is $0$ in both cases: under PB every item is transferred, whereas under PBD every item valued above cost is still transferred. - The overinclusion loss under PB is $\sum_{i\in S}(c_i-x_i)\ge0$. The overinclusion loss under PBD is $0$, since items in $S$ are not transferred. On the other hand, suppose the customer did not buy the bundle with all the items for ${P^{\mathsf{PB}}}$. - The producer surplus under PB is $0$. The producer surplus under PBD is either $0$ or ${P^{\mathsf{PB}}}-\sum_{i=1}^nc_i$ (if the return option allowed the customer to enter the market), which is non-negative. - The consumer surplus under PB is $0$. The consumer surplus under PBD cannot be negative, since the customer is rational and the no-purchase option is always available. - The deadweight loss under PB is $\sum_{i:x_i>c_i}(x_i-c_i)$, which is the maximum possible. Therefore, the deadweight loss under PBD cannot be greater. - The overinclusion loss under PB is $0$. The overinclusion loss under PBD is always $0$ when ${P^{\mathsf{PBD}}}_i=c_i$ for all $i$, since items valued below cost are never transferred. In both cases, we have proven that the statements in Proposition \[dominance\] hold. The firm’s problem is to find a menu along with tie-breaking rules which maximize profit. Note that this is equivalent to finding functions $q,s$ defined on ${\mathcal{X}}$ which are incentive-compatible, individually rational, and profit-maximizing. Formally, the firm’s problem is $$\begin{array}{rrclr} \max & {\mathbb{E}}_{x\sim D}[s(x)-q(x)^Tc] & & & \\ s.t. & q(x)^Tx-s(x) & \ge & q(y)^Tx-s(y) & \forall x,y\in{\mathcal{X}}\\ & q(x)^Tx-s(x) & \ge & 0 & \forall x\in{\mathcal{X}}\end{array}$$ which can be rewritten as $$\begin{array}{rrclr} \max & {\mathbb{E}}_{x\sim D}[s(x)-q(x)^Tc] & & & \\ s.t. & q(x)^T(x-c)-(s(x)-q(x)^Tc) & \ge & q(y)^T(x-c)-(s(y)-q(y)^Tc) & \forall x,y\in{\mathcal{X}}\\ & q(x)^T(x-c)-(s(x)-q(x)^Tc) & \ge & 0 & \forall x\in{\mathcal{X}}\end{array}$$ Now, define $x':=x-c$, $y':=y-c$, $q'(x):=q(x+c)$, and $s'(x):=s(x+c)-q(x+c)^Tc$. Let ${\mathcal{X}}':=\{x-c:x\in{\mathcal{X}}\}$, and similarly let $D'$ be the distribution $D$ shifted $c_i$ units downward in dimension $i$ for every $i\in[n]$. We can see that the above is equivalent to $$\begin{array}{rrclr} \max & {\mathbb{E}}_{x'\sim D'}[s'(x')] & & & \\ s.t. & q'(x')^Tx'-s'(x') & \ge & q'(y')^Tx'-s'(y') & \forall x',y'\in{\mathcal{X}}' \\ & q'(x')^Tx'-s'(x') & \ge & 0 & \forall x'\in{\mathcal{X}}' \end{array}$$ which is identical to the original problem without costs on this new distribution $D'$. Now suppose there was a restriction on the menu ${\mathcal{M}}=\{(q^{(1)},s^{(1)}),(q^{(2)},s^{(2)}),\ldots\}$ to belong to some class $\mathscr{M}$ in the original problem. The menu after the transformation, ${\mathcal{M}}'$, looks like $\{(q^{(1)},s^{(1)}-(q^{(1)})^Tc),(q^{(2)},s^{(2)}-(q^{(2)})^Tc),\ldots\}$. Therefore, ${\mathcal{M}}'$ is restricted to the class $$\mathscr{M}':=\{\{(q^{(1)},s^{(1)}-(q^{(1)})^Tc),(q^{(2)},s^{(2)}-(q^{(2)})^Tc),\ldots\}:\{(q^{(1)},s^{(1)}),(q^{(2)},s^{(2)}),\ldots\}\in\mathscr{M}\}.$$ By assumption that $s-q^Tc\ge0$ for all menu entries, the payments in ${\mathcal{M}}'$ are non-negative. Throughout this paper, it will be clear whether we are in the context of the original problem or the transformed problem, and we will omit the superscripts used in the preceding proof. \[pbdc\_transformation\] As a concrete example of this transformation, consider the pricing scheme PBDC. ${\mathcal{M}}$ is restricted to be of the form $\{({\mathbbm{1}}_S,{P^{\mathsf{PBD}}}_0-{\mathbbm{1}}_{[n]\setminus S}^Tc):\emptyset\neq S\subseteq[n]\}\cup\{(0,0)\}$ where ${\mathbbm{1}}_S\in\{0,1\}^n$ is the indicator vector for items belonging to $S$. Hence ${\mathcal{M}}'$ is restricted to be of the form $$\{({\mathbbm{1}}_S,{P^{\mathsf{PBD}}}_0-{\mathbbm{1}}_{[n]\setminus S}^Tc-{\mathbbm{1}}_S^Tc):\emptyset\neq S\subseteq[n]\}\cup\{(0,0)\}=\{({\mathbbm{1}}_S,{P^{\mathsf{PBD}}}_0-{\mathbbm{1}}_{[n]}^Tc):\emptyset\neq S\subseteq[n]\}\cup\{(0,0)\}.$$ Put in words, ${\mathcal{M}}'$ must belong to the class of menus that offer the same price for any non-empty subset of items. The fact that the customer can choose to take a subset of items instead of taking all the items is important, because valuations $x'_i$ can be negative ($x'_i$ is equal to the original valuation $x_i$ subtract the cost $c_i$). Appendix B: Proof of Theorem \[mr2\] ==================================== We will WOLOG normalize the valuations so that the optimal PC revenue is $1$ (we can do this so long as the original optimal revenue was positive; if it was $0$ then the statement of the theorem is trivial). The Core-Tail Decomposition --------------------------- We use the core-tail decomposition of [@BILW14], with the original idea coming from [@LY13]. We will cut up the domain of the joint distribution and consider the conditional distributions on the smaller subdomains. Below, we introduce the notation for working with these distributions on smaller subdomains. One should get comfortable with the idea that some of the distributions defined could be the null distribution, if they were distributions conditioned on a set of measure $0$, or a product over an empty set of distributions. The product of a null distribution with any other distribution is still a null distribution. We make the following definitions for this appendix. - For all $i\in[n]$, let $r_i$ denote the optimal revenue earned by selling item $i$ individually (by our normalization, $\sum_{i=1}^nr_i=1$). - Let $D_i^C$ (the “core” of $D_i$) denote the conditional distribution of $D_i$ when it lies in the range $(-\infty,1]$. - Let $D_i^T$ (the “tail” of $D_i$) denote the conditional distribution of $D_i$ when it lies in the range $(1,\infty)$. - Let $p_i:={\mathbb{P}}_{x_i\sim D_i}[x_i>1]$, the probability item $i$ lies in its tail. - Let $A\subseteq[n]$ represent a subset of items, usually the items whose valuations lie in their tails. - Let $D_A^T:=\times_{i\in A}D_i^T$, the product distribution of only items in their tails. - Let $D_A^C:=\times_{i\notin A}D_i^C$, the product distribution of only items in their cores. - Let $D_A:=D_A^C\times D_A^T$, the conditional distribution of $D$ when exactly the subset $A$ of items lie in their tails. Let $p_A$ be the probability this occurs, which is equal to $(\prod_{i\notin A}(1-p_i))(\prod_{i\in A}p_i)$, by independence. - Let $x_i^+:=\max\{x_i,0\}$. - For any valuation distribution $S$, let ${\textsc{Val}^+}(S):=\sum_i{\mathbb{E}}_{x\sim S}[x_i^+]$, which is the expected welfare after the transformation from costs to negative valuations. Note that the sum is only over the admissible $i$ if $S$ is a distribution on a smaller subdomain. - Let ${\textsc{Rev}}(S)$ denote the optimal revenue obtainable from valuation distribution $S$ via any Incentive Compatible and Individually Rational mechanism, which could include lotteries. - Let ${\textsc{SRev}}(S)$ denote the optimal revenue of any pricing scheme falling under the class of separate sales (Pure Components). - Let ${\textsc{BdcRev}}(S)$ denote the optimal revenue of any pricing scheme falling under the class of PBDC. (It is understood that ${\textsc{Val}^+},{\textsc{Rev}},{\textsc{SRev}},{\textsc{BdcRev}}$ are $0$ when evaluated on the null distribution.) Lemmas for Negative Valuations {#lemmas} ------------------------------ We need to modify the statements of lemmas from [@HN12], [@LY13], and [@BILW14] to handle negative valuations. While their proofs can be extended to negative valuations in a straight-forward manner, we provide full self-contained proofs here for ease of exposition. (Marginal Mechanism) \[marginal\_mechanism\] Let $S,S'$ be (potentially negative) valuation distributions over disjoint sets of items. Then $${\textsc{Rev}}(S\times S')\le{\textsc{Val}^+}(S)+{\textsc{Rev}}(S')$$ The Marginal Mechanism tells us that when selling a group of independent items, we cannot do better than breaking off some items individually, extracting the entire welfare from those items, and selling the remaining items as a group. Consider the following mechanism for selling to a buyer with valuations drawn from $S'$. First, sample a value $v\sim S$, and reveal to the buyer these make-believe valuations for the items in $S$. Then run a mechanism obtaining ${\textsc{Rev}}(S\times S')$ on this buyer, with the modification that whenever the buyer would have received an item $i$ from the support of $S$, instead she will receive (or pay) money equal to $v_i$. By independence, this modified mechanism on the buyer with valuations drawn from $S'$ is IC and IR (a buyer with valuations $S'$ will choose the same menu entry under the modified mechanism as a buyer with valuations $S\times S'$ would have chosen under the original mechanism) and we will obtain ${\textsc{Rev}}(S\times S')$, but then have to settle for the items in $S$. The most we stand to lose in the settlement is $\sum_iv_i^+$ (each item $i$ in $S$ is transferred in full whenever $v_i\ge0$, and not transferred when $v_i<0$), so this amount is upper bounded in expectation by ${\textsc{Val}^+}(S)$. Therefore, the optimal revenue from $S'$ is at least ${\textsc{Rev}}(S\times S')-{\textsc{Val}^+}(S)$, completing the proof of the lemma. (Subdomain Stitching) \[subdomain\_stitching\] Let $S$ be a product distribution over valuations, with support ${\mathcal{X}}\subseteq{\mathbb{R}}^m$ for some $m\in{\mathbb{N}}$. Let ${\mathcal{X}}_1,\ldots,{\mathcal{X}}_k$ form a partition of ${\mathcal{X}}$ inducing conditional distributions $S^{(1)},\ldots,S^{(k)}$, respectively, and let $s_j={\mathbb{P}}_{x\sim S}[x\in{\mathcal{X}}_j]$. Then $${\textsc{Rev}}(S)\le\sum_{j=1}^ks_j{\textsc{Rev}}(S^{(j)})$$ Intuitively, Subdomain Stitching says that revenue can only increase if we sell to each subdomain separately, since we can use a different mechanism for each subdomain that specializes in extracting the welfare from that customer segment. Let $M$ be an optimal mechanism obtaining ${\textsc{Rev}}(S)$, and for any valuation distribution $S'$, let ${\textsc{Rev}}_M(S')$ denote the expected revenue obtained from mechanism $M$ when the buyer’s valuation is drawn from $S'$. Clearly ${\textsc{Rev}}(S)=\sum_{j=1}^ks_j{\textsc{Rev}}_M(S^{(j)})$, and furthermore for all $j\in[k]$, ${\textsc{Rev}}_M(S^{(j)})\le{\textsc{Rev}}(S^{(j)})$ since M is an IC-IR mechanism for selling to $S^{(j)}$, completing the proof of the lemma. \[subdomain\_reverse\] Let $S$ be a product distribution over valuations, with support ${\mathcal{X}}\subseteq{\mathbb{R}}^m$ for some $m\in{\mathbb{N}}$. Let ${\mathcal{X}}'$ be a subset of ${\mathcal{X}}$ inducing conditional distribution $S'$, and let $s'={\mathbb{P}}_{x\sim S}[x\in{\mathcal{X}}']$. Then $${\textsc{Rev}}(S)\ge s'{\textsc{Rev}}(S')$$ While Subdomain Stitching places an upper bound on ${\textsc{Rev}}(S)$, Lemma \[subdomain\_reverse\] places a lower bound on ${\textsc{Rev}}(S)$ based on the optimal revenue of any single subdomain. Consider an optimal mechanism for $S'$, and extend this to an IC-IR mechanism on $S$ by allowing the buyer to report a value in ${\mathcal{X}}'$ maximizing her utility. With probability $s'$, the buyer’s valuation will actually be drawn from $S'$ and we will obtain revenue ${\textsc{Rev}}(S')$; otherwise, we still earn a non-negative revenue, since the mechanism never admits a negative payment. Therefore, the optimal revenue for $S$ is at least $s'{\textsc{Rev}}(S')$, completing the proof of the lemma. \[number\_of\_items\] Let $S$ be a product distribution over $m$ independent (potentially negative) valuations, for some $m\in{\mathbb{N}}$. Then $${\textsc{Rev}}(S)\le m\cdot{\textsc{SRev}}(S)$$ While selling $m$ items together can definitely be better than selling them separately, this lemma tells us it can be no more than $m$ times better. We proceed by induction. The statement is trivial when $m=1$. Now, suppose we have proven the statement for $m$ valuations, and we will prove it for $m+1$ valuations. Partition the support ${\mathcal{X}}\subseteq{\mathbb{R}}^{m+1}$ of $S$ into ${\mathcal{X}}_1$ and ${\mathcal{X}}_2$, where ${\mathcal{X}}_1:=\{x\in{\mathcal{X}}:x_1\ge\max\{x_j,0\}\ \forall\ j=2,\ldots,m+1\}$ and ${\mathcal{X}}_2:={\mathcal{X}}\setminus{\mathcal{X}}_1$. Let $s_1$ denote the probability a value sampled from $S$ lies in ${\mathcal{X}}_1$, and let $S_1$ be its distribution conditioned on this event. Define $s_2,S_2$ respectively. Subdomain stitching tells us ${\textsc{Rev}}(S)\le s_1{\textsc{Rev}}(S^{(1)})+s_2{\textsc{Rev}}(S^{(2)})$. Our goal is to separately show that $s_1{\textsc{Rev}}(S^{(1)})\le(m+1){\textsc{SRev}}(S_1)$ and $s_2{\textsc{Rev}}(S^{(2)})\le(m+1){\textsc{SRev}}(S_{-1})$. Now, applying Marginal Mechanism on $S^{(1)}$ and multiplying both sides of the inequality by $s_1$, we get $s_1{\textsc{Rev}}(S^{(1)})\le s_1{\textsc{Val}^+}(S^{(1)}_{-1})+s_1{\textsc{Rev}}(S^{(1)}_1)$. By considering a distribution that samples $v\sim S$ but only outputs $v_1$, we can use Lemma \[subdomain\_reverse\] to show that $s_1{\textsc{Rev}}(S^{(1)}_1)\le{\textsc{Rev}}(S_1)$. To bound ${\textsc{Val}^+}(S^{(1)}_{-1})$, consider the following mechanism for selling just item $1$: sample $v_{-1}\sim S_{-1}$, and set the price to be $\max_{i=2}^{m+1}\{\max\{v_i,0\}\}$. Since the buyer’s valuation is drawn from $S_1$, by independence, we get a sale with probability exactly $s_1$. Furthermore, $\max_{i=2}^{m+1}\{\max\{v_i,0\}\}\ge\frac{1}{m}\sum_{i=2}^{m+1}\max\{v_i,0\}$, so conditioned on us getting a sale, the expected payment is at least $\frac{1}{m}{\textsc{Val}^+}(S^{(1)}_{-1})$. We have proven ${\textsc{Rev}}(S_1)\ge\frac{s_1}{m}{\textsc{Val}^+}(S^{(1)}_{-1})$, hence $s_1{\textsc{Rev}}(S^{(1)})\le(m+1){\textsc{Rev}}(S_1)=(m+1){\textsc{SRev}}(S_1)$, as required. It remains to bound $s_2{\textsc{Rev}}(S^{(2)})$, and using Marginal Mechanism and Lemma \[subdomain\_reverse\] in the same way as before, we obtain that it is no more than $s_2{\textsc{Val}^+}(S^{(2)}_1)+{\textsc{Rev}}(S_{-1})$. Consider the following mechanism for selling items $2,\ldots,m+1$: sample $v_1\sim S_1$, and set the individual price for each item $2,\ldots,m+1$ to be $\max\{v_1,0\}$. Note that the probability of getting at least one sale is less than $s_2$, since even when there is some $j=2,\ldots,m+1$ such that $v_1<\max\{x_j,0\}$, it is possible for both $v_1,x_j$ to be negative. However, in this case $\max\{v_1,0\}=0$, so not getting a sale is still equivalent to getting at least one sale for $\max\{v_1,0\}$. Therefore, we can think of it as we get at least one sale with probability $s_2$, in which case we earn in expectation at least ${\textsc{Val}^+}(S_1^{(2)})$. We have proven that $s_2{\textsc{Val}^+}(S^{(2)}_1)\le{\textsc{SRev}}(S_{-1})$, and by the induction hypothesis ${\textsc{Rev}}(S_{-1})\le m\cdot{\textsc{SRev}}(S_{-1})$, so $s_2{\textsc{Rev}}(S^{(2)})\le(m+1){\textsc{SRev}}(S_{-1})$. Putting everything together, we have ${\textsc{Rev}}(S)\le(m+1)({\textsc{SRev}}(S_1)+{\textsc{SRev}}(S_{-1}))=(m+1){\textsc{SRev}}(S)$, completing the induction and the proof of the lemma. Using these lemmas, we decompose the revenue of the initial distribution $D$ in the same way as [@BILW14]: $$\begin{aligned} {\textsc{Rev}}(D) & \le & \sum_{A\subseteq[n]}p_A{\textsc{Rev}}(D_A) \\ & \le & \sum_{A\subseteq[n]}p_A\big({\textsc{Val}^+}(D_A^C)+{\textsc{Rev}}(D_A^T)\big) \\ & \le & \sum_{A\subseteq[n]}p_A{\textsc{Val}^+}(D_\emptyset^C)+\sum_{A\subseteq[n]}p_A{\textsc{Rev}}(D_A^T) \\ & = & {\textsc{Val}^+}(D_\emptyset^C)+\sum_{A\subseteq[n]}p_A{\textsc{Rev}}(D_A^T) \label{bound_on_rev}\end{aligned}$$ where the first inequality is Subdomain Stitching, the second inequality is Marginal Mechanism, the third inequality is immediate from the definition of $D_A^C$, and the equality is a consequence of $\sum_{A\subseteq[n]}p_A=1$. Now, for all $A\subseteq[n]$ such that $p_A>0$, Lemma \[number\_of\_items\] tells us that ${\textsc{Rev}}(D_A^T)\le|A|{\textsc{SRev}}(D_A^T)=|A|\sum_{i\in A}{\textsc{SRev}}(D_i^T)$. Lemma \[subdomain\_reverse\] tells us that ${\textsc{SRev}}(D_i^T)\le\frac{r_i}{p_i}$, where $p_i\neq0$ since $p_A>0$, so $$\begin{aligned} \sum_{A\subseteq[n]}p_A{\textsc{Rev}}(D_A^T) & \le & \sum_{A\subseteq[n]}p_A|A|\sum_{i\in A}\frac{r_i}{p_i} \\ & = & \sum_{i=1}^nr_i\sum_{A\ni i}|A|\frac{p_A}{p_i} \\\end{aligned}$$ $\sum_{A\ni i}|A|\frac{p_A}{p_i}$ is the expected number of items in their tails conditioned on item $i$ being in its tail, so it is equal to $1+\sum_{j\neq i}p_j$. Thus $$\begin{aligned} \sum_{A\subseteq[n]}p_A{\textsc{Rev}}(D_A^T) & \le & \sum_{i=1}^nr_i\Big(1+\sum_{j\neq i}p_j\Big) \\ & = & 1+\sum_{j=1}^np_j\sum_{i\neq j}r_i \\ & = & 1+\sum_{j=1}^np_j(1-r_j)\end{aligned}$$ We will use $\tau$ to denote the quantity $\sum_{i=1}^np_i(1-r_i)$. It is immediate that $\tau\le\sum_{i=1}^np_i\le1$, but we can get a stronger bound for the welfare of the core if we don’t immediately apply the inequality $\tau\le1$. We have $$\label{tau_bound_on_rev} {\textsc{Rev}}(D)\le{\textsc{Val}^+}(D_\emptyset^C)+1+\tau$$ Before we proceed, one final lemma we will need later is: \[variance\_lemma\] Let $Y$ be a random variable distributed over $[0,1]$ and suppose $y(1-F(y))$ is upper bounded by some value $v\in[0,1]$. Then $\mathrm{Var}(Y)\le2v$. $$\begin{aligned} \mathrm{Var}(Y) & = & {\mathbb{E}}[Y^2]-{\mathbb{E}}[Y]^2 \\ & \le & {\mathbb{E}}[Y^2] \\ & = & \int_0^1{\mathbb{P}}[Y^2\ge y]dy \\ & = & \int_0^1{\mathbb{P}}[Y\ge\sqrt{y}]dy \\ & \le & \int_0^1\frac{v}{\sqrt{y}}dy \\ & = & 2v\end{aligned}$$ where the second inequality uses the fact that the Myerson revenue for $Y$ is upper bounded by $v$. A Tighter Bound for the Welfare of the Core ------------------------------------------- The main observation behind our improvement is that for $\tau$ to be large (and the above bound to be weak), the tail probabilities must be large. However, we will choose the price of the grand bundle, $P_t$, to be at most $2$, so that whenever $2$ or more valuations lie in their tails, the customer is guaranteed to want to buy the bundle (and dispose of items for which her valuation is negative). Thus $$\begin{aligned} {\mathbb{P}}[\scriptstyle\sum x_i^+<P_t\displaystyle] & = & p_\emptyset\cdot{\mathbb{P}}_{x\sim D_\emptyset}[\scriptstyle\sum x_i^+<P_t\displaystyle]+\sum_{|A|=1}p_A\cdot{\mathbb{P}}_{x\sim D_A}[\scriptstyle\sum x_i^+<P_t\displaystyle]+\sum_{|A|\ge2}p_A\cdot(0) \nonumber\\ & \le & \Big(p_\emptyset+\sum_{|A|=1}p_A\Big)\cdot{\mathbb{P}}_{x\sim D_\emptyset^C}[\scriptstyle\sum x_i^+<P_t\displaystyle] \nonumber\\ & = & \Big(\prod_{i=1}^n(1-p_i)+\sum_{i=1}^np_i\prod_{j\neq i}(1-p_j)\Big)\cdot{\mathbb{P}}_{x\sim D_\emptyset^C}[\scriptstyle\sum x_i^+<P_t\displaystyle] \label{tighter_bound_welfare_core}\end{aligned}$$ where the inequality comes from the fact that the probability of $\sum x_i^+$ being less than the bundle price is greater conditioned on no items being in the tail, than conditioned on some item being in the tail. We used independence to compute the probabilities in the final expression, which we will bound in the following way: \[key\_inequality\] Let $p_1,\ldots,p_n$, $r_1,\ldots,r_n$ be real numbers satisfying $0\le p_i\le r_i$ and $\sum_{i=1}^nr_i=1$. Let $\tau=\sum_{i=1}^np_i(1-r_i)$. Then $$\prod_{i=1}^n(1-p_i)+\sum_{i=1}^np_i\prod_{j\neq i}(1-p_j)\le\frac{\frac{5}{4}+\tau}{e^\tau}$$ This is the key inequality that enables our improved ratio and its proof requires new analysis. Note that we do indeed have the condition $p_i\le r_i$ in our case, since by Lemma \[subdomain\_reverse\] $r_i\ge p_i{\textsc{Rev}}(D_i^T)$, and ${\textsc{Rev}}(D_i^T)$ must be at least $1$ when $D_i^T$ is distributed over $(1,\infty)$. We will first prove $$\label{three_quarters} \frac{3}{4}\cdot\prod_{i=1}^n(1-p_i)+\sum_{i=1}^np_i\prod_{j\neq i}(1-p_j)\le\frac{1+\tau}{e^\tau}$$ Assume that $p_i<1$ for all $i\in[n]$; the lemma is trivially true otherwise because we would have $\mathrm{LHS}=1$ and $\tau=0$. Since $\tau=\sum_{i=1}^np_i(1-r_i)$ and $1-x\le e^{-x}$, it suffices to prove $$\frac{3}{4}\cdot\prod_{i=1}^n(1-p_i)+\sum_{i=1}^np_i\prod_{j\neq i}(1-p_j)\le\Big(1+\sum_{i=1}^np_i(1-r_i)\Big)\prod_{i=1}^n(1-p_i(1-r_i))$$ which is equivalent to $$\frac{3}{4}+\sum_{i=1}^n\frac{p_i}{1-p_i}\le\Big(1+\sum_{i=1}^n(p_i-p_ir_i)\Big)\prod_{i=1}^n(1+\frac{p_ir_i}{1-p_i})$$ Observe that the RHS is at least $$\begin{aligned} & & \Big(1+\sum_{i=1}^n(p_i-p_ir_i)\Big)\Big(1+\sum_{i=1}^n\frac{p_ir_i}{1-p_i}\Big) \\ & = & 1+\sum_{i=1}^n\frac{(p_i-p_ir_i)(1-p_i)+p_ir_i}{1-p_i}+\Big(\sum_{i=1}^np_i(1-r_i)\Big)\Big(\sum_{i=1}^n\frac{p_ir_i}{1-p_i}\Big) \\ & = & 1+\sum_{i=1}^n\frac{p_i}{1-p_i}-\sum_{i=1}^n\frac{p_i^2(1-r_i)}{1-p_i}+\Big(\sum_{i=1}^np_i(1-r_i)\Big)\Big(\sum_{i=1}^n\frac{p_ir_i}{1-p_i}\Big) \\ & = & 1+\sum_{i=1}^n\frac{p_i}{1-p_i}-\sum_{i=1}^n\frac{p_i^2(1-r_i)^2}{1-p_i}+\sum_{i\neq j}p_i(1-r_i)\cdot\frac{p_jr_j}{1-p_j} \\\end{aligned}$$ so it remains to prove $$\sum_{i=1}^n\frac{p_i^2(1-r_i)^2}{1-p_i}-\sum_{i\neq j}p_i(1-r_i)\cdot\frac{p_jr_j}{1-p_j}\le\frac{1}{4}$$ But $p_i\le r_i$ for all $i\in[n]$, so the LHS is at most $\sum_{i=1}^np_i^2(1-p_i)$, which can be seen to be at most $\frac{1}{4}$, since $p_i(1-p_i)$ is always at most $\frac{1}{4}$ and $\sum_{i=1}^np_i\le1$. Also, since $\tau\le\sum_{i=1}^np_i$, $e^{-\tau}\ge\exp(-\sum_{i=1}^np_i)\ge\prod_{i=1}^n(1-p_i)$. Multiplying by $\frac{1}{4}$ and adding to (\[three\_quarters\]), we complete the proof of the lemma. Applying Cantelli’s Inequality ------------------------------ To bound ${\mathbb{P}}_{x\sim D_\emptyset^C}[\sum x_i^+<P_t]$, we want to show that $\sum x_i^+$ concentrates around its mean, where valuation $x_i$ is drawn from its conditional core distribution $D_i^C$ for all $i\in[n]$. Note that $y(1-F_{x_i}(y))$ is bounded above by $r_i$ for all $y\in[0,1]$; otherwise ${\textsc{SRev}}(D_i^C)>r_i\implies{\textsc{SRev}}(D_i)>r_i$ which is a contradiction. Hence $y(1-F_{x_i^+}(y))$ is also bounded above by $r_i$ and we can invoke Lemma \[variance\_lemma\] to get $\mathrm{Var}_{x_i\sim D_i^C}(x_i^+)\le2r_i$ for all $i\in[n]$. By independence, $\mathrm{Var}_{x\sim D_\emptyset^C}(\sum x_i^+)=\sum_{i=1}^n\mathrm{Var}_{x\sim D_\emptyset^C}(x_i^+)\le\sum_{i=1}^n2r_i=2$ and we have successfully bounded the variance of the quantity we are interested in. At this point, it is common in the literature to see an application of Chebyshev’s inequality (e.g. [@BB99; @FN06; @HN12; @BILW14]). However, since we are only interested in the lower tail, we can actually use Cantelli’s one-sided inequality (Lemma \[cantelli\]), which optimizes a shift parameter to obtain an improved bound for a single tail. Now, note that ${\mathbb{E}}_{x\sim D_\emptyset^C}[\sum_{i=1}^nx_i^+]={\textsc{Val}^+}(D_\emptyset^C)$ by definition. Also, it will be convenient to write the bundle price as $P_t=\alpha\cdot{\textsc{Val}^+}(D_\emptyset^C)$, for some $\alpha\in[0,1]$ (we would never want $\alpha>1$ since then the price would be greater than the mean and it would be impossible to use Cantelli). Then $$\begin{aligned} {\mathbb{P}}_{x\sim D_\emptyset^C}[\scriptstyle\sum x_i^+<P_t\displaystyle] & = & {\mathbb{P}}_{x\sim D_\emptyset^C}\Big[\sum_{i=1}^nx_i^+-{\textsc{Val}^+}(D_\emptyset^C)<-(1-\alpha){\textsc{Val}^+}(D_\emptyset^C)\Big] \\ & \le & \frac{\mathrm{Var}_{x\sim D_\emptyset^C}(\sum x_i^+)}{\mathrm{Var}_{x\sim D_\emptyset^C}(\sum x_i^+)+(1-\alpha)^2{\textsc{Val}^+}(D_\emptyset^C)^2} \\ & \le & \frac{2}{2+(1-\alpha)^2{\textsc{Val}^+}(D_\emptyset^C)^2} \\\end{aligned}$$ where the first inequality is Cantelli’s inequality, and the second inequality comes from our variance bound above. So long as we choose $P_t\le2$, we can use (\[tighter\_bound\_welfare\_core\]), and combined with Lemma \[key\_inequality\] we get $${\mathbb{P}}[\scriptstyle\sum x_i^+<P_t\displaystyle]\le\min\big\{\frac{1.25+\tau}{e^\tau},1\big\}\cdot\frac{2}{2+(1-\alpha)^2{\textsc{Val}^+}(D_\emptyset^C)^2}$$ and hence the expected revenue from selling the grand bundle at price $\alpha\cdot{\textsc{Val}^+}(D_\emptyset^C)$ is at least $$\alpha\cdot{\textsc{Val}^+}(D_\emptyset^C)\cdot\Big(1-\min\big\{\frac{1.25+\tau}{e^\tau},1\big\}\cdot\frac{2}{2+(1-\alpha)^2{\textsc{Val}^+}(D_\emptyset^C)^2}\Big)$$ Recall from (\[tau\_bound\_on\_rev\]) that ${\textsc{Rev}}(D)\le{\textsc{Val}^+}(D_\emptyset^C)+1+\tau$. While $\tau$ could take on any value in $[0,1]$, we can choose the price of the bundle based on $\tau$ and ${\textsc{Val}^+}(D_\emptyset^C)$ by adjusting $\alpha\in[0,1]$. **Case 1.** If ${\textsc{Val}^+}(D_\emptyset^C)\le3.2$, then ${\textsc{Rev}}(D)\le3.2+1+1=5.2\cdot{\textsc{SRev}}(D)$ is immediate and we can just sell the items individually. **Case 2.** If $3.2<{\textsc{Val}^+}(D_\emptyset^C)\le4$, then we will choose $\alpha=\frac{1}{2}$ which guarantees $P_t\le2$. Thus $${\textsc{BdcRev}}(D)\ge{\textsc{Val}^+}(D_\emptyset^C)\cdot\frac{1}{2}\Big(1-\min\big\{\frac{1.25+\tau}{e^\tau},1\big\}\cdot\frac{2}{2+(1-\frac{1}{2})^2(3.2)^2}\Big)$$ It can be shown with calculus (or numerically) that: For all $\tau\in[0,1]$, $2\Big(1-\min\left\{\frac{1.25+\tau}{e^\tau},1\right\}\cdot\frac{2}{2+(1-\frac{1}{2})^2(3.2)^2}\Big)^{-1}+(1+\tau)<5.2$, with the maximum of $\approx5.1952$ occuring at the unique positive $\tau$ satisfying $\frac{1.25+\tau}{e^\tau}=1$. Hence ${\textsc{Val}^+}(D_\emptyset^C)\le(4.2-\tau){\textsc{BdcRev}}(D)$. Substituting into (\[tau\_bound\_on\_rev\]), we get $$\begin{aligned} {\textsc{Rev}}(D) & \le & (4.2-\tau){\textsc{BdcRev}}(D)+(1+\tau){\textsc{SRev}}(D) \\ & \le & 5.2\cdot\max\{{\textsc{SRev}}(D),{\textsc{BdcRev}}(D)\}\end{aligned}$$ as desired. **Case 3.** If $4<{\textsc{Val}^+}(D_\emptyset^C)$, then we will still choose $\alpha=\frac{1}{2}$. We no longer have $P_t\le2$, so we have to use the weaker bound ${\mathbb{P}}_{x\sim D}[\sum x_i^+<P_t]\le{\mathbb{P}}_{x\sim D_\emptyset^C}[\sum x_i^+<P_t]$. However, applying Cantelli yields $${\mathbb{P}}_{x\sim D_\emptyset^C}[\scriptstyle\sum x_i^+<P_t\displaystyle]\le\frac{2}{2+(1-\frac{1}{2})^2(4)^2}=\frac{1}{3}$$ so ${\textsc{BdcRev}}(D)\ge{\textsc{Val}^+}(D_\emptyset^C)\cdot\frac{1}{2}(1-\frac{1}{3})$. We get ${\textsc{Rev}}(D)\le3\cdot{\textsc{BdcRev}}(D)+(1+\tau){\textsc{SRev}}(D)<5.2\cdot\max\{{\textsc{SRev}}(D),{\textsc{BdcRev}}(D)\}$, completing the proof of Theorem \[mr2\]. Appendix C: Proof of Theorem \[thm\_upper\_bound\] ================================================== It is immediate that the optimal revenue from PC is $2\rho$, attained by selling individual items at any price in $[1,2]$. Next, we would like to argue that the optimal revenue from PB is also $2\rho$. If we offer the bundle at $2$, it is guaranteed to get bought if either valuation realizes to $2$ or both valuations realize to a positive number, and won’t get bought otherwise. Therefore the revenue is $2(\rho^2+2(1-\rho)\frac{\rho}{2})=2\rho$. We can do equally well by offering the bundle at $3$, and any other price is inferior. \[upper\_bound\] The optimal revenue from PB is $2\rho$, attained by setting a bundle price of $2$ or $3$. Let $z$ denote the price of the bundle. We will systematically analyze all the cases over $1\le z\le4$ and show that the maximum revenue of $2\rho$ is attained at $z=2$ and $z=3$. **Case 1.** Suppose $1\le z\le2$. Let us condition on the realization $y$ of the first valuation. If $y=0$, then we get a sale with probability $\frac{\rho}{z}$. If $y\in[1,z)$, then we get a sale so long as the second valuation realizes to a positive number, which occurs with probability $1-\rho$. If $y\ge z$, then the first valuation alone is enough to guarantee a bundle sale. The expected revenue is $$z\left((1-\rho)\frac{\rho}{z}+(\rho-\frac{\rho}{z})\rho+\frac{\rho}{z}\right)=2\rho+(z-2)\rho^2$$ which is clearly maximized at $z=2$, in which case the revenue is $2\rho$. **Case 2.** Suppose $2<z\le3$. Let us condition on the realization $y$ of the first valuation. If $y=0$, then we have no chance of selling the bundle. If $y\in[1,z-1]$, then we get a sale when the other valuation is at least $z-y$. Since $z-y\in[1,2]$, the probability of this occurring is $\frac{\rho}{z-y}$. If $y\ge z-1$, then we get a sale so long as the other valuation realizes to a positive number, which occurs with probability $\rho$. The total probability of getting a sale is $$\int_1^{z-1}\frac{\rho}{y^2}\frac{\rho}{z-y}dy+\frac{\rho}{z-1}\rho$$ where the PDF of $Y$ satisfies $f(y)=\frac{\rho}{y^2}$ over $[1,2)$. Using partial fractions, the antiderivative of $\frac{1}{y^2(z-y)}$ can be computed to be $$\frac{1}{z}\left(\frac{\ln y-\ln(z-y)}{z}-\frac{1}{y}\right)$$ as demonstrated in the proof of [@HN12 lem. 6]. Therefore, the definite integral evaluates to $$\rho^2\left(\frac{2\ln(z-1)}{z^2}+\frac{2}{z}-\frac{1}{z-1}\right)$$ and the expected revenue is $$z\rho^2\left(\frac{2\ln(z-1)}{z^2}+\frac{2}{z}-\frac{1}{z-1}+\frac{1}{z-1}\right)=2\rho^2\left(\frac{\ln(z-1)}{z}+1\right)$$ However, $\frac{\ln(z-1)}{z}$ is a strictly increasing function on $(2,3]$, so this expression is uniquely maximized at $z=3$ where it equals $2\rho^2(\frac{\ln2}{3}+1)=2\rho$. **Case 3.** Suppose $3\le z\le4$. Let us condition on the realization $y$ of the first valuation. If $y<z-2$, then we have no chance of selling the bundle. Otherwise, the probability of getting a sale is $\frac{\rho}{z-y}$, since $z-y\in[1,2]$. The total probability of getting a sale is $$\int_{z-2}^2\frac{\rho}{y^2}\frac{\rho}{z-y}+\frac{\rho}{2}\frac{\rho}{z-2}$$ and the integral evaluates to $$\rho^2\left(\frac{2\ln2-2\ln(z-2)}{z^2}+\frac{1}{z(z-2)}-\frac{1}{2z}\right)$$ Therefore, the expected revenue is $$z\rho^2\left(\frac{2\ln2-2\ln(z-2)}{z^2}+\frac{1}{z(z-2)}-\frac{1}{2z}+\frac{1}{2(z-2)}\right)=2\rho^2\left(\frac{\ln2-\ln(z-2)}{z}+\frac{1}{z-2}\right)$$ $\frac{\ln2-\ln(z-2)}{z}+\frac{1}{z-2}$ is a strictly decreasing function on $[3,4]$, so this expression is uniquely maximized at $z=3$. Now, consider the strategy of offering either item for $2$ or the bundle for the discounted price of $3$. Note that if buying the bundle is non-negative utility for the customer, then buying either individual item cannot be higher utility, since the price savings is one and the value of the item lost is at least one (recall that the firm gets to break ties in a way that favors itself). Hence there is no cannibalization of bundle sales from individual sales and we earn revenue at least $2\rho$. However, when exactly one valuation realizes to a positive number (in which case we have no chance of selling the bundle), we still have a $\frac{1}{2}$ conditional probability of selling that individual item. Hence the revenue from Mixed Bundling is $2\rho+2(2(1-\rho)\frac{\rho}{2})=2\rho(2-\rho)$. The relative gain over both the PC revenue and the PB revenue is $2-\rho=\frac{3+2\ln2}{3+\ln2}$, completing the proof of Theorem \[thm\_upper\_bound\]. A motivating example for our construction is a small modification of the earlier best-known example from [@HN12]: consider a distribution that takes on values $0,1,2$ with probabilities $\frac{1}{9},\frac{4}{9},\frac{4}{9}$, respectively. Let $D$ be the instance consisting of two independent copies of this distribution. Then it can be shown that the optimal PC revenue is $\frac{16}{9}$ (attained at individual prices $1$ or $2$), the optimal PB revenue is $\frac{16}{9}$ (attained at bundle price $2$ or $3$), and the optimal revenue is at least $\frac{160}{81}$ (attained at individual prices $2$ and bundle price $3$), achieving a ratio of $\frac{9}{10}$. [@HN12] had the probabilities be $\frac{1}{3},\frac{1}{3},\frac{1}{3}$ instead, achieving a ratio of $\frac{12}{13}$. Appendix D: Example where BSP Performs Poorly {#video_game_example} ============================================= Consider a firm that is bundling a higher-profit-margin, lower-valuation good with a low-profit-margin, high-valuation good. This is a common occurrence, for example when video games are bundled with a console, which we will hereinafter refer to as item 1 and item 2, respectively. Item 1 costs zero to produce and has a valuation uniform on \[0,1\]; item 2 costs 4.5 to produce and has a valuation uniform on \[0,5\] and independent from item 1. Most of the welfare comes from the lower-valuation item: the expected welfare for item 1 and item 2 are 0.5 and 0.025, respectively. The optimal deterministic profit is $\approx0.265$, attained by offering item 1 at 0.51, item 2 at 4.83, and the bundle at the discounted price of 5.13. The optimal BSP pricing charges 4.83 for a single item and 5.03 for both items, earning only 19% of the deterministic optimum. This example highlights the issue with BSP: it cannot afford to charge a low price for a single item if any item has a high production cost. However, most of the potential profit could be coming from offering lower-valuation items at low prices! [@CLS08] bypass such examples in their numerical experiments, assuming that all items have a low cost compared to its mean valuation. PBDC offers item 1 at 0.51, item 2 at 5.01, and the bundle at 5.01—which is the right idea and earns 99.1% of the deterministic optimum. Interestingly, even the analytical solution provided by [@Bha13], which computes the optimal deterministic pricing when there are two independent uniform distributions and costs, is less effective than PBDC on this example. The solution from [@Bha13] only attains $97.5\%$ of the deterministic optimum for this example, because it requires a bit of linear approximation. Optimal bundling is an intricate problem even in the case of two independent uniform distributions, so a simple pricing heuristic as robust as PBDC is invaluable. In fact, for this example PBDC recommends *Partial Mixed Bundling*, which is a Mixed Bundling scheme where one of the items, in this case item 2 (the high-cost low-welfare item), is never sold individually. This matches the intuition that the seller should add item 1 (the low-cost high-welfare item) to item 2 in order to increase the total amount customer is willing to pay (see Proposition 1 in [@Bha13]). BSP, on the other hand, does not perform well: it recommends a Partial Mixed Bundling scheme where item 1 is never sold individually. [^1]: Operations Research Center, Massachusetts Institute of Technology, `willma@mit.edu`. [^2]: Institute for Data, Systems, and Society, Department of Civil and Environmental Engineering, and Operations Research Center, Massachusetts Institute of Technology, `dslevi@mit.edu`. [^3]: See [@CLS11] for the journal version. [^4]: Technically, the optimal mechanism is also allowed to offer lotteries of items, which have been shown to be necessary for achieving the optimum, in [@HR12]. [^5]: In the case there are multiple maxima, the firm can choose between them; this can always be achieved by small perturbations. [^6]: We use this terminology because [@AY76] refer to the act of not incurring this loss as *exclusion*.
--- abstract: 'Arnol’d showed the uniqueness of the complex analytic structure of a small neighborhood of a non-singular elliptic curve embedded in a non-singular surface whose normal bundle satisfies Diophantine condition in the Picard variety. We show an analogue of this theorem for a neighborhood of a cycle of rational curves.' address: 'Department of Mathematics, Graduate School of Science, Osaka City University 3-3-138, Sugimoto, Sumiyoshi-ku Osaka 558-8585 Japan ' author: - Takayuki Koike title: 'Arnol’d’s type theorem on a neighborhood of a cycle of rational curves' --- Introduction ============ Let $C$ be a cycle of rational curves (i.e. a reduced singular complex curve with only nodes such that the dual graph is a cycle graph and each complement of the normalization is biholomorphic to the projective line $\mathbb{P}^1$) holomorphically embedded in a non-singular complex surface $S$. Assume that the normal line bundle $N_{C/S}:=[C]|_C$ is topologically trivial, where $[C]$ is the holomorphic line bundle on $S$ defined by the divisor $C$. Denote by $t(N_{C/S})\in\mathbb{C}^*$ the complex number which corresponds to $N_{C/S}$ via the natural identification of Picard variety ${\rm Pic}^0(C)$ of $C$ with $H^1(C, \mathbb{C}^*) = \mathbb{C}^*$, where $\mathbb{C}^*=\mathbb{C}\setminus\{0\}$ ([@U91 Lemma 1], see also §\[section:Pic\]). We show the following theorem on the uniqueness of the complex analytic structure of a small neighborhood of $C$ under a Diophantine type condition for the normal bundle. \[thm:arnold\_for\_nodal\_rational\] Let $C$ be a cycle of rational curves, and $i\colon C\to S$ and $i'\colon C\to S'$ be holomorphic embeddings into non-singular complex surfaces $S$ and $S'$ respectively. Assume that $t(N_{i(C)/S})=t(N_{i'(C)/S'})=e^{2\pi\sqrt{-1}\theta}$ holds for a Diophantine irrational number $\theta\in\mathbb{R}$ (i.e. there exist positive constants $\alpha$ and $A$ such that $|n\cdot \theta-m|\geq A\cdot n^{-\alpha}$ holds for any integer $m$ and any positive integer $n$). Then there exists a biholomorphism $f\colon V\to V'$ between a neighborhood $V$ of $i(C)$ in $S$ and $V'$ of $i'(C)$ in $S'$ with $f|_{i(C)}=i'\circ i^{-1}$. Theorem \[thm:arnold\_for\_nodal\_rational\] can be regarded as an analogue of Arnol’d’s theorem [@A], which states that the conclusion of the theorem holds for a non-singular elliptic $C$ embedded in a non-singular surface $S$ under the assumption that $N_{C/S}$ satisfies the Diophantine type condition in the Picard variety. Note that, in our notation, $C$ is a cycle of rational curves with only one irreducible component when $C$ is a rational curve with a node. Neighborhoods of a rational curve with a node embedded in a surface was first investigated by Ueda in [@U91] when $t(N_{C/S})\in \mathbb{C}^*\setminus {\rm U}(1)$, where ${\rm U}(1):=\{t\in\mathbb{C}^*\mid |t|=1\}$. In [@K6], we slightly generalized his results to the case where, for example, $C$ is a cycle of rational curves [@K6 Theorem 1.6]. In that paper, we also treated the case where $t(N_{C/S})\in {\rm U}(1)$, which is the case that $N_{C/S}$ is a ${\rm U}(1)$-flat line bundle: i.e. $N_{C/S}$ admits a $C^\omega$ Hermitian metric with flat curvature. In this case, we showed the existence of a pseudoflat neighborhoods system of $C$ under the assumption that $t(N_{C/S})=e^{2\pi\sqrt{-1}\theta}$ holds for a Diophantine irrational number $\theta\in\mathbb{R}$ [@K6 Theorem 1.4], which can be regarded as an analogue of Ueda’s theorem for a non-singular compact curve embedded in a surface [@U83 Theorem 3]. Here we remark that Theorem \[thm:arnold\_for\_nodal\_rational\] is also regarded as an improved version of [@K6 Theorem 1.4]. In the proof of Theorem \[thm:arnold\_for\_nodal\_rational\], we compare the complex structure of a small neighborhood $V$ of $C$ with that of [*the standard model*]{} we describe in Example \[ex:standard\_model\] or \[ex:standard\_model\_general\]. We consider the cohomology class $\alpha=\alpha(C, V)\in H^1(V, \mathcal{O}_V)$ which corresponds to the difference of them. Then one can see that it is sufficient to show that the restriction $\alpha|_{V^*}$ of this class to a small neighborhood $V^*$ of $C$ in $V$ is equal to zero in $H^1(V^*, \mathcal{O}_{V^*})$. Note that this class satisfies $\alpha|_C=0\in H^1(C, \mathcal{O}_C)$. Therefore the problem is reduced to showing the injectivity of the restriction morphism $\lim_{V^*\to} H^1(V^*, \mathcal{O}_{V^*})\to H^1(C, \mathcal{O}_C)$, where $V^*$ runs all the neighborhoods of $C$ in $V$ (Proposition \[prop:H\^1\_main\_general\]). We show this by using a complex dynamical technique originated from [@S], which is also used in the proofs of [@U83 Theorem 3] and [@K6 Theorem 1.4]. The main motivation of the present paper comes from [@T] and [@KK3]. In [@KK3], as an application of Arnol’d’s theorem [@A], we constructed a K3 surface by holomorphically gluing two open complex surfaces obtained as the complements of tubular neighborhoods of non-singular elliptic curves embedded in the blow-ups of the projective planes at appropriate nine points. As described in [@KK3 §4.1.1], this construction can be regarded as a concrete description of a general fiber of a degeneration of K3 surfaces of type II. Theorem \[thm:arnold\_for\_nodal\_rational\] can also be applied to nodal curves embedded in the blow-ups of the projective planes at appropriate nine points (Examples \[ex:ABU\], \[ex:ABU\_2\], and \[ex:ABU\_3\]). Toward a concrete description of a general fiber of a degeneration of K3 surfaces of type III, we will investigate these examples precisely. The organization of the paper is as follows. In §2, we correct some fundamental facts on cycles of rational curves. Here we also fix coordinates on a neighborhood of each irreducible component of a cycle of rational curves by using Grauert’s theorem [@G] intrinsically. In §3, we show the injectivity of the morphism $\lim_{V^*\to} H^1(V^*, \mathcal{O}_{V^*})\to H^1(C, \mathcal{O}_C)$, where $V^*$ runs all the neighborhoods of $C$. In §4, we prove Theorem \[thm:arnold\_for\_nodal\_rational\]. In §5, we investigate Examples \[ex:ABU\], \[ex:ABU\_2\], and \[ex:ABU\_3\] precisely. The author would like to give heartful thanks to Prof. Tetsuo Ueda whose enormous support and insightful comments were invaluable. He thanks Dr. Takahiro Matsushita and Dr. Yuta Nozaki who gave him many valuable comments on the topological aspects of Levi-flat hypersurfaces which is treated in §5. He is supported by Leading Initiative for Excellent Young Researchers (No. J171000201). Preliminaries ============= In this section, we collect some fundamental facts and fix some notation on a cycle of rational curves. Let $C$ be a cycle of rational curves embedded in a non-singular surface $S$ with Diophantine condition as in Theorem \[thm:arnold\_for\_nodal\_rational\]. Take an open covering $\{U_j\}_j$ of $C$, and a small neighborhood $V_j$ of $U_j$ in $S$ with $V_j\cap C=U_j$ for each $j$. Denote by $V$ the neighborhood $\bigcup_jV_j$ of $C$. It follows from [@K6 Proposition 2.5 (2)] that the pair $(C, S)$ is of infinite type in the sense of [@K6]. Therefore, from [@K6 Theorem 1.4], we have that there exists a defining function $w_j$ of $U_j$ in $V_j$ for each $j$ such that $w_j=t_{jk}w_k$ holds for some $t_{jk}\in {\rm U}(1)$ on each $V_{jk}:=V_j\cap V_k$ when $\{U_j\}$ and $\{V_j\}$ are sufficiently fine. Preliminaries on a rational curve with a node --------------------------------------------- ### Notation {#section:prelim_notation} Let $C$ be a rational curve with a node. In this case, we choose open coverings $\{U_j\}_j$ and $\{V_j\}_j$ such that the index set is $\{0, 1\}$ as follows: Let $U_0$ be a small neighborhood of the nodal point of $C$ and $U_1$ be the regular part $C_{\rm reg}:=C\setminus \{\text{nodal point}\}$ of $C$. By taking $V_j$ as a sufficiently small neighborhood of $U_j$, we may assume that $V_0\cap V_1$ consists of two connected component $V^+$ and $V^-$. Let $t_{\pm}$ be elements of ${\rm U}(1)$ such that $$w_1= \begin{cases} t_+\cdot w_0 & (\text{on}\ V^+) \\ t_-\cdot w_0 & (\text{on}\ V^-). \end{cases}$$ Note that $t:=t_+/t_-=t(N_{C/S})\in {\rm U}(1)\subset \mathbb{C}^*=H^1(C, \mathbb{C}^*)$, see §\[section:Pic\]. Let $z$ be a non-homogeneous coordinate of the normalization $\widetilde{C}\cong \mathbb{P}^1$ of $C$ such that the preimage of the nodal point is $\{0, \infty\}$. As we will see in §\[section:V\_tilde\], we can extend the function $z|_{U_1}$ to $V_1$, where we are identifying $\widetilde{C}\setminus\{0, \infty\}$ with $U_1$ (see also [@Siu]). The resulting holomorphic function on $V_1$ is also denoted by $z$. Take coordinates $(x, y)$ of $V_0$ such that $x\cdot y$ is a defining function of $U_0$ in $V_0$. These functions $(x, y)$ will also be chosen by more careful argument in §\[section:V\_tilde\] in actual. Denote by $U^+_0$ the subset $\{(x, y)\in V_0\mid y=0\}$ and $U^-_0$ the subset $\{(x, y)\in V_0\mid x=0\}$. We may assume that $U^+:=V^+\cap U_0$ coincides with $U^+_0\setminus\{\text{nodal point}\}$, and that $U^-:=V^-\cap U_0$ coincides with $U^-_0\setminus\{\text{nodal point}\}$. ### Picard variety and some cohomologies {#section:Pic} Let $L\in {\rm Pic}^0(C)$ be a topologically trivial line bundle on $C$. Then there is a uniquely determined complex constant $t=t(L)\in \mathbb{C}^*$ with $$L=[\{(U^+, t), (U^-, 1)\}]\in \check{H}^1(\{U_j\}, \mathcal{O}_C^*) =H^1(C, \mathcal{O}_C^*)$$ where we are using the notation in the previous section (see the arguments around [@U91 Lemma 1]). In particular, it is observed that $L$ admits $\mathbb{C}^*$-flat structure: i.e. $L$ admits a flat connection. From this fact, one have that ${\rm Pic}^0(C)$ is naturally identified with $H^1(C, \mathbb{C}^*)=\mathbb{C}^*$. When $t(L)\in {\rm U}(1)\setminus \{1\}$, $L$ is a non-trivial ${\rm U}(1)$-flat line bundle. In this case, one can obtain by considering the long exact sequence comes from $0\to \mathcal{O}_C(L)\to i_*\mathcal{O}_{\widetilde{C}}(i^*L)\to \mathcal{O}_{\{\text{the nodal point}\}}\to 0$ that $H^0(C, L)=H^1(C, L)=0$, where $i\colon\widetilde{C}\to C$ is the normalization (see [@K6 p. 852]). By the same argument, one also have that $H^0(C, \mathcal{O}_C)\cong H^1(C, \mathcal{O}_C)\cong\mathbb{C}$. ### Standard model of a neighborhood of a rational curve with a node and some examples The following example can be regarded as the standard model of a neighborhood of a rational curve with a node. \[ex:standard\_model\] Let $\widetilde{V}$ be a neighborhood of the zero section $\widetilde{C}$ of the line bundle $\pi\colon\mathcal{O}_{\mathbb{P}^1}(-2)\to \mathbb{P}^1$. Let $S$ be a non-homogeneous coordinate of $\mathbb{P}^1$. We also use the non-homogeneous $T:=S^{-1}$ especially when we observe a neighborhood of $\{S=\infty\}$. Let $\xi_0$ and $\xi_\infty$ be fiber coordinates of $\mathcal{O}_{\mathbb{P}^1}(-2)$ defined in a neighborhood of $\{S=0\}$ and $\{T=0\}$, respectively. We may assume that $\xi_\infty=\xi_0\cdot S^2$ holds in the fibers over $\mathbb{P}^1\setminus\{S=0, \infty\}$. Fix a constant $0<\varepsilon <1$ and let us consider subsets $$\widetilde{V}_0^+:=\{(S, \xi_0)\in \pi^{-1}(\mathbb{C})\mid |S|<\varepsilon,\ |\xi_0|<\varepsilon\}$$ and $$\widetilde{V}_0^-:=\{(T, \xi_\infty)\in \pi^{-1}(\mathbb{P}^1\setminus\{0\})\mid |T|<\varepsilon,\ |\xi_\infty|<\varepsilon\}$$ of $\widetilde{V}$. By shrinking $\widetilde{V}$ if necessary, we may assume that $\pi^{-1}(\{|S|<\varepsilon\})\cap \widetilde{V}=\widetilde{V}_0^+$ and $\pi^{-1}(\{|T|<\varepsilon\})\cap \widetilde{V}=\widetilde{V}_0^-$. Define a biholmorphism $F\colon \widetilde{V}_0^+\to \widetilde{V}_0^-$ by $F^*(T, \xi_\infty):=(t\cdot \xi_0, S)$ (i.e. $F^*T:=T\circ F:=t\cdot \xi_0$ and $F^*\xi_\infty:=\xi_\infty\circ F:=S$), where $t\in {\rm U}(1)$ is a constant. Denote by $i\colon \widetilde{V}\to V$ the quotient by the relation induced by $F$. Then $V$ is a non-singular surface and the compact analytic subset $C:=i(\widetilde{C})$ is a rational curve with a node such that $t(N_{C/S})=t$. Next example is an analogue of Arnol’d–Ueda–Brunella’s example [@A] [@U83] [@B]. \[ex:ABU\] Take a plane cubic $C_0\subset \mathbb{P}^2$ which admits only one nodal point, and nine points $Z\subset \{p_1, p_2, \dots, p_9\}\subset (C_0)_{\rm reg}$, where $(C_0)_{\rm reg}$ is the non-singular locus of $C_0$. Denote by $\pi\colon S\to \mathbb{P}^2$ the blow-up at $Z$ and by $C$ the strict transform $(\pi^{-1})_*C_0$. Then it is known that, by taking a normalization $i\colon \mathbb{P}^1\to C_0$ with $i^{-1}((C_0)_{\rm sing})=\{0, \infty\}$ appropriately ($(C_0)_{\rm sing}:=C_0\setminus (C_0)_{\rm reg}$), the complex constant $t=t(N_{C/S})\in\mathbb{C}^*$ can be calculated by $t = \prod_{\nu=1}^9i^{-1}(p_\nu) \in \mathbb{C}^* = H^1(C_0, \mathbb{C}^*)$, where we are identifying $C_0$ and $C$ via $\pi$. Especially, each point of ${\rm Pic}^0(C_0)$ is attained by choosing appropriate nine points configuration $Z$. Finally, we give a counter example of Theorem \[thm:arnold\_for\_nodal\_rational\] when $N_{C/S}$ does not satisfy Diophantine condition. Let $\{(\widetilde{V}, \widetilde{V}_0^\pm, S, T, \xi_0, \xi_\infty)\}$ be those in Example \[ex:standard\_model\]. Denote by $\widetilde{W}_0^+$ the subset $\{(S, \xi_0)\in \pi^{-1}(\mathbb{C})\cap\widetilde{V}\mid |S|<1\}$ and by $\widetilde{W}_0^-$ the subset $\{(T, \xi_\infty)\in \pi^{-1}(\mathbb{P}^1\setminus\{0\})\cap\widetilde{V}\mid |T|<1\}$ of $\mathcal{O}_{\mathbb{P}^1}(-2)$. Note that $\widetilde{V}_0^+\subset \widetilde{W}_0^+$ and $\widetilde{V}_0^-\subset \widetilde{W}_0^-$. For sufficiently small positive constant $\delta$, set $\widetilde{W}_1:=\{(S, \xi_0)\in \widetilde{V}\mid 1/2<|S|<2,\ |\xi_0|<\delta\}$. We may assume that $\widetilde{V}_0^+\cap \widetilde{W}_1=\emptyset$ and $\widetilde{V}_0^-\cap \widetilde{W}_1=\emptyset$ hold. Take a univalent holomorphic function $\varphi$ defined on $\{w\in\mathbb{C}\mid |w|<\delta\}$ such that $\varphi(0)=0$ and $\lambda:=\varphi'(0)\in{\rm U}(1)$ hold. Denote by $\Phi_+\colon\widetilde{W}_1\cap \widetilde{W}_0^+\to \widetilde{W}_0^+$ the map defined by $(\Phi_+)^*(S, \xi_0)=(S, \varphi(S\cdot \xi_0)\cdot S^{-1})$ and by $\Phi_-\colon\widetilde{W}_1\cap \widetilde{W}_0^-\to \widetilde{W}_0^-$ the natural injection. Define a surface $W$ by $W:=(\widetilde{W}_0^+\amalg \widetilde{W}_1\amalg \widetilde{W}_0^-)/\sim$, where $\sim$ is the relation generated by $$\begin{cases} \widetilde{W}_0^+\ni p\sim F(p) \in \widetilde{W}_0^- & \text{if}\ p\in\widetilde{V}_0^+\\ \widetilde{W}_1\ni p\sim \Phi_+(p) \in \widetilde{W}_0^+ & \text{if}\ p\in\left\{(S, \xi_0)\in \widetilde{W}_1\left| \frac{1}{2}<|S|<1\right.\right\}\\ \widetilde{W}_1\ni p\sim \Phi_-(p)\in \widetilde{W}_0^-& \text{if}\ p\in\{(S, \xi_0)\in \widetilde{W}_1\mid 1<|S|<2\}, \end{cases}$$ where $F$ is the one in Example \[ex:standard\_model\] with $t=1$. Denote by $C$ the image of $\widetilde{C}$ by the quotient map. Note that $C$ is a compact leaf of the holomorphic foliation $\mathcal{F}$ on $W$ whose leaves are defined by $$\begin{cases} \{S\cdot \xi_0=\text{constant}\} & (\text{on}\ \widetilde{W}_0^+) \\ \{S\cdot \xi_0=\text{constant}\} & (\text{on}\ \widetilde{W}_1) \\ \{T\cdot \xi_\infty=\text{constant}\} & (\text{on}\ \widetilde{W}_0^-). \end{cases}$$ Assume that $\varphi$ is the one as in [@U83 p. 606]. Then $t(N_{C/S})=\varphi'(0)$ is a non-torsion element of ${\rm U}(1)$, and any small neighborhood $W^*\subset W$ of $C$ includes a compact leaf of $\mathcal{F}$ which is biholomorphic to an elliptic curve and has no intersection with $C$. As it follows from the same argument as in [@U83 §5.3] that there is no compact subvariery $W^*\setminus C$ for sufficiently small $W^*$ if $C$ admits pseudoflat neighborhoods system, we have that $C$ does not admit a neighborhood as in Example \[ex:standard\_model\] in this example. Definition of the covering map $\widetilde{V}\to V$ and outline of the proof of Theorem \[thm:arnold\_for\_nodal\_rational\] {#section:V_tilde} ---------------------------------------------------------------------------------------------------------------------------- Here we use the notation in §\[section:prelim\_notation\]. Take a copy $\widetilde{V}_1$ of $V_1$ and two copies $\widetilde{V}_0^+$ and $\widetilde{V}_0^-$ of $V_0$. Denote by $\widetilde{V}$ the manifold constructed by patching $\widetilde{V}_0^+$, $\widetilde{V}_1$ and $\widetilde{V}_0^-$ by considering the natural injections $$\xymatrix{ \widetilde{V}_0^+& &\widetilde{V}_1 & & \widetilde{V}_0^- \\ & V^+\ar[ur]\ar[ul]& & V^-\ar[ur]\ar[ul] &\\ }$$ of $V^\pm$. Note that $\widetilde{V}$ can be regarded as an open submanifold of the universal covering of $V$. Denote by $i\colon \widetilde{V}\to V$ the natural map. In what follows, we regard $\widetilde{V}^\pm_0$ and $\widetilde{V}_1$ as subsets of $\widetilde{V}$. Then $i|_{\widetilde{C}}$ is a normalization of $C$, where $\widetilde{C}\subset i^{-1}(C)$ is the irreducible component which is compact. By identifying $\widetilde{C}$ and $\mathbb{P}^1$, we may assume that the preimage of the nodal point is $\{0, \infty\}$. Denote by $D_0$ and $D_\infty$ the other two irreducible components of $i^{-1}(C)$ which intersects $\widetilde{C}$ at $0$ and $\infty$, respectively. Define the defining function $\widetilde{w}$ of the divisor $i^*C=\widetilde{C}+D_0+D_\infty$ of $\widetilde{V}$ by $$\widetilde{w}:= \begin{cases} (\widetilde{V}_0^+\to V_0)^*(t_+\cdot w_0) & (\text{on}\ \widetilde{V}_0^+) \\ (\widetilde{V}_1\to V_1)^*w_1 & (\text{on}\ \widetilde{V}_1) \\ (\widetilde{V}_0^+\to V_0)^*(t_-\cdot w_0) & (\text{on}\ \widetilde{V}_0^-), \end{cases}$$ where $\widetilde{V}_0^+\to V_0$, $\widetilde{V}_1\to V_1$ and $\widetilde{V}_0^+\to V_0$ be the natural biholomorphisms. By a simple argument, we have that ${\rm deg}\,N_{\widetilde{C}/\widetilde{V}}=-2$. Therefore, it follows from Grauert’s theorem [@G] that $\widetilde{V}$ can be holomorphically embedded in the total space of the line bundle $\mathcal{O}_{\mathbb{P}^1}(-2)\to \mathbb{P}^1$ by shrinking $\widetilde{V}$ (see also [@CM Theorem 2.5.2]). In what follows, we regard $\widetilde{V}$ as a subset of $\mathcal{O}_{\mathbb{P}^1}(-2)$ and identify $\widetilde{C}$ with the zero section via this embedding. Take a strictly pseudoconvex neighborhood $\mathcal{V}$ of $\widetilde{C}$ in $\widetilde{V}$. It follows from Ohsawa’s vanishing theorem [@O Theorem 4.5] that $H^1(\mathcal{V}, \mathcal{O}_{\mathcal{V}})=0$. Thus we have that the line bundle on $\widetilde{V}$ corresponds to the divisor $D_0-D_\infty$ is holomorphically trivial. Therefore there exists a holomorphic map $p\colon \mathcal{V}\to \mathbb{P}^1$ such that $p^*(\{0\}-\{\infty\})=D_0-D_\infty$ holds as divisors. By shrinking $\widetilde{V}$ so that $\widetilde{V}\subset \mathcal{V}$, we may assume that the map $p$ is defined on $\widetilde{V}$. Let $S=T^{-1}$ be non-homogeneous coordinate of $\mathbb{P}^1$. Denote also by $S$ and $T$ the meromorphic functions $p^*S$ and $p^*T$ on $\widetilde{V}$, respectively. Then we have that $D_0=\{S=0\}=\{T=\infty\}$ and $D_\infty=\{S=\infty\}=\{T=0\}$. Setting $\xi_0:=\widetilde{w}\cdot S^{-1}$ on a neighborhood of $D_0$ and $\xi_\infty:=\widetilde{w}\cdot T^{-1}$ on a neighborhood of $D_\infty$, we regard $(S, \xi_0)$ and $(T, \xi_\infty)$ as coordinates of a neighborhood of $D_0$ and $D_\infty$, respectively. Denote by $F\colon \widetilde{V}^+_0\to \widetilde{V}^-_0$ the biholomorphism such that $i^{-1}(i(p))=\{p, F(p)\}$ for each $p\in \widetilde{V}^+_0$. As it hold that $F^*\widetilde{w}=t\cdot \widetilde{w}$, $F^*D_\infty=\widetilde{C}\cap \widetilde{V}^+_0$ and that $F^*(\widetilde{C}\cap \widetilde{V}^-_0)=D_0$, we have that $$F^*(T, \xi_\infty)= \left(\frac{t\cdot \xi_0}{G(S, \xi_0)},\ G(S, \xi_0)\cdot S\right)$$ holds for a nowhere vanishing holomorphic function $G$ defined on $\widetilde{V}^+_0$, where $t=t_+/t_-$. By changing the scaling of $\widetilde{w}$, we may assume that $G(0, 0)=1$. In §\[section:prf\], we will prove Theorem \[thm:arnold\_for\_nodal\_rational\] by showing that one may assume that $G\equiv 1$ by changing coordinate functions appropriately. Preliminaries on a cycle of rational curves in general {#prelim:general} ------------------------------------------------------ Let $C$ be a cycle of rational curves in general. Denote by $n=n(C)$ the number of the irreducible components of $C$. Here we treat the case where $n\geq 2$. Denote by $\{C_{(\nu)}\}_{\nu=1}^n$ the set of all irreducible components of $C$. We sometimes use the notation $C_{(0)}:=C_{(n)}$. We may assume that $C_{(\nu)}\cap C_{(\mu)}\not=\emptyset$ if and only if $\nu-\mu=\pm 1$ modulo $n$. It holds that $H^1(C, \mathcal{O}_C)=\mathbb{C}$ also in this case, since $H^1(C, i_*\mathcal{O}_{\widetilde{C}})=H^1(\widetilde{C}, \mathcal{O}_{\widetilde{C}})=0$ follows from the same exact sequence as we considered in §\[section:Pic\], where $i\colon\widetilde{C}\to C$ is the normalization (Note that the higher direct images vanish for $i$, since $i$ is a finite morphism). Thus ${\rm Pic}^0(C)$ is naturally identified with $H^1(C, \mathbb{C}^*)=\mathbb{C}^*$ also in this case. The following example can be regarded as the standard model of a neighborhood of $C$. \[ex:standard\_model\_general\] Let $\{(\widetilde{V}_{(\nu)}, \widetilde{C}_{(\nu)}, \widetilde{V}_{0 (\nu)}^\pm, S_{(\nu)}, T_{(\nu)}, \xi_{0 (\nu)}, \xi_{\infty (\nu)})\}_{\nu=1}^n$ be $n$-copies of $(\widetilde{V}, \widetilde{C}, \widetilde{V}_{0}^\pm, S, T, \xi_{0}, \xi_{\infty})$ in Example \[ex:standard\_model\]. Define a biholomorphism $F_{\nu+1, \nu}\colon \widetilde{V}_{0 (\nu+1)}^+\to \widetilde{V}_{0 (\nu)}^-$ by $(F_{\nu+1, \nu})^*(T_{(\nu)}, \xi_{\infty (\nu)})=(t_{\nu+1, \nu}\cdot \xi_{0 (\nu)}, S_{(\nu)})$ for $\nu=0, 1, \dots, n-1$, where $t_{\nu+1, \nu}\in {\rm U}(1)$ and $$(\widetilde{V}_{(0)}, \widetilde{C}_{(0)}, \widetilde{V}_{0 (0)}^\pm, S_{(0)}, T_{(0)}, \xi_{0 (0)}, \xi_{\infty (0)}):=(\widetilde{V}_{(n)}, \widetilde{C}_{(n)}, \widetilde{V}_{0 (n)}^\pm, S_{(n)}, T_{(n)}, \xi_{0 (n)}, \xi_{\infty (n)}).$$ Let $i\colon \widetilde{V}:=\coprod_{\nu=1}^n\widetilde{V}_\nu\to V$ be the quotient by the relation induced by $F_{\nu+1, \nu}$’s. Denote by $C$ the image $i(\widetilde{C})$, where $\widetilde{C}:=\coprod_{\nu=1}^n\widetilde{C}_{(\nu)}$. Then $C$ is a cycle of $n$ rational curves embedded in $V$ with $t(N_{C/V})=\prod_{\nu=0}^{n-1}t_{\nu+1, \nu}$. \[rmk:V\_tilde\_general\] It follows from the same argument as in §\[section:V\_tilde\] that one can construct a finite covering map $i\colon \widetilde{V}\to V$ as in Example \[ex:standard\_model\_general\] for a small neighborhood $V$ of $C$ also in the case where $C$ consists of $n$ irreducible components ($n\geq 2$). In this case, $\widetilde{V}$ is the disjoint union of a neighborhood $\widetilde{V}_{(\nu)}$ of each irreducible component $C_{(\nu)}$ of $C$ with the same local coordinates $(S_{(\nu)}, T_{(\nu)}, \xi_{0 (\nu)}, \xi_{\infty (\nu)})$ as in Example \[ex:standard\_model\_general\] (Here we use Grauert’s theorem [@G] again). In general, the gluing morphism $F_{\nu+1, \nu}\colon \widetilde{V}_{0 (\nu+1)}^+\to \widetilde{V}_{0 (\nu)}^-$ needs not to coincides with the one in Example \[ex:standard\_model\_general\]. From the same argument as in §\[section:V\_tilde\] by using [@K6 Theorem 1.4], it follows that, by choosing $S_{(\nu)}, T_{(\nu)}, \xi_{0 (\nu)}$ and $\xi_{\infty (\nu)}$ suitably, we may assume that $$(F_{\nu+1, \nu})^*(T_{(\nu)}, \xi_{\infty (\nu)})=\left(\frac{t_{\nu+1, \nu}\cdot \xi_{0 (\nu)}}{G_{\nu+1, \nu}(S_{(\nu)}, \xi_{0 (\nu)})},\ G_{\nu+1, \nu}(S_{(\nu)}, \xi_{0 (\nu)})\cdot S_{(\nu)}\right)$$ holds for a nowhere vanishing holomorphic function $G_{\nu+1, \nu}$ defined on $\widetilde{V}_{0 (\nu+1)}^+$ and a constant $t_{\nu+1, \nu}\in{\rm U}(1)$. In §\[section:prf\], we will prove Theorem \[thm:arnold\_for\_nodal\_rational\] by showing that one may assume that $G_{\nu+1, \nu}\equiv 1$ by changing the coordinate functions appropriately. \[ex:ABU\_2\] Fix a plane cubic $C_0\subset \mathbb{P}^2$ which admits only nodal singularities and consists of two irreducible components, say $C_0^{(1)}$ and $C_0^{(2)}$. One may assume that $C_0^{(\nu)}$ is of degree $\nu$ for $\nu=1, 2$. Take three points $\{p_1, p_2, p_3\}\subset C_0^{(1)}\cap (C_0)_{\rm reg}$ and six points $\{p_4, p_5, \dots, p_9\}\subset C_0^{(2)}\cap (C_0)_{\rm reg}$. Denote by $\pi\colon S\to \mathbb{P}^2$ the blow-up at $Z:=\{p_1, p_2, \dots, p_9\}$ and by $C$ the strict transform $(\pi^{-1})_*C_0$. Then it is known that, by taking a normalization $i\colon \mathbb{P}^1\amalg\mathbb{P}^1\to C_0$ with $i^{-1}((C_0)_{\rm sing})= \{0, \infty\}$ appropriately, the complex constant $t=t(N_{C/S})\in\mathbb{C}^*$ can be calculated by $t = \prod_{\nu=1}^9i^{-1}(p_\nu) \in \mathbb{C}^* = H^1(C_0, \mathbb{C}^*)$, where we are identifying $C_0$ and $C$ via $\pi$. \[ex:ABU\_3\] Fix a plane cubic $C_0\subset \mathbb{P}^2$ which admits only nodal singularities and consists of three irreducible components, say $C_0^{(1)}$, $C_0^{(2)}$ and $C_0^{(3)}$. Each $C_0^{(\nu)}$ is a line for $\nu=1, 2, 3$. Take three points $\{p_1, p_2, p_3\}\subset C_0^{(1)}\cap (C_0)_{\rm reg}$, $\{p_4, p_5, p_6\}\subset C_0^{(2)}\cap (C_0)_{\rm reg}$, and $\{p_7, p_8, p_9\}\subset C_0^{(3)}\cap (C_0)_{\rm reg}$. Denote by $\pi\colon S\to \mathbb{P}^2$ the blow-up at $Z:=\{p_1, p_2, \dots, p_9\}$ and by $C$ the strict transform $(\pi^{-1})_*C_0$. Then it is known that, by taking a normalization $i\colon \mathbb{P}^1\amalg\mathbb{P}^1\amalg\mathbb{P}^1\to C_0$ with $i^{-1}((C_0)_{\rm sing})= \{0, \infty\}$ appropriately, the complex constant $t=t(N_{C/S})\in\mathbb{C}^*$ can be calculated by $t = \prod_{\nu=1}^9i^{-1}(p_\nu) \in \mathbb{C}^* = H^1(C_0, \mathbb{C}^*)$, where we are identifying $C_0$ and $C$ via $\pi$. Note that each point of ${\rm Pic}^0(C_0)$ is attained by choosing appropriate nine points configuration $Z$ in Examples \[ex:ABU\_2\] and \[ex:ABU\_3\] (as in Example \[ex:ABU\]). Injectivity of the restriction $\lim_{V^*\to} H^1(V^*, \mathcal{O}_{V^*})\to H^1(C, \mathcal{O}_C)$ =================================================================================================== We will show Theorem \[thm:arnold\_for\_nodal\_rational\] in §\[section:prf\] by the strategy as we mentioned in §\[section:V\_tilde\] and Remark \[rmk:V\_tilde\_general\]. When $C$ is a rational curve with a node, for example, we will choose suitable coordinates of $\widetilde{V}$ so that $G\equiv 1$ holds. Consider the composition $g$ of the natural biholomorphism $V_0\to \widetilde{V}^+_0$ and the branch of $\frac{1}{2\pi\sqrt{-1}}\log G$ such that $g(0, 0)=0$. By the arguments we will explain the details in §\[section:prf\], the problem can be reduced to showing that the cohomology class $\alpha:=[\{(V^+, -g|_{V^+}), (V^-, 0)\}] \in \check{H}^1(\{V_j\}, \mathcal{O}_V)$ is trivial. As it is easily observed that $\alpha|_C=0\in H^1(C, \mathcal{O}_C)$ (see the proof of “Proposition \[prop:H\^1\] $\Rightarrow$ Proposition \[prop:H\^1\_main\]" below), it is sufficient to show the injectivity of the restriction $H^1(V, \mathcal{O}_V)\to H^1(C, \mathcal{O}_C)$ by shrinking $V$ in a suitable sense. For such a purpose, we will show the following: \[prop:H\^1\_main\_general\] Let $C$ be a cycle of a curve embedded in non-singular surface $V$ such that the normal bundle $N_{C/V}$ is topologically trivial and satisfies Diophantine condition as in Theorem \[thm:arnold\_for\_nodal\_rational\]. For any element $\alpha$ of the kernel of the restriction $H^1(V, \mathcal{O}_V)\to H^1(C, \mathcal{O}_C)$, there exists a neighborhood $V^*$ of $C$ such that $\alpha|_{V^*}=0\in H^1(V^*, \mathcal{O}_{V^*})$. Proof of Proposition \[prop:H\^1\_main\_general\] when $C$ is a rational curve with a node ------------------------------------------------------------------------------------------ ### Notation and statement in this case Assume that $C$ is a rational curve with a node. Then we can use the notation as in §\[section:V\_tilde\]. By a simple argument, Proposition \[prop:H\^1\_main\_general\] can be reworded as follows: \[prop:H\^1\_main\] Let $F_+$ and $F_-$ be holomorphic functions defined on $V^+$ and $V^-$, respectively, such that $\{(U^\pm, F_\pm|_{U^\pm})\}$ extends to a holomorphic function defined on $U_0$. Then there exists a neighborhood $V^*$ of $C$ such that the class $$\alpha:= [\{(V^*\cap V^+, F_+|_{V^*\cap V^+}), (V^*\cap V^-, F_-|_{V^*\cap V^-})\}]\in \check{H}^1(\{V^*\cap V_j\}, \mathcal{O}_{V^*})$$ is trivial. As will be proven immediately after, Proposition \[prop:H\^1\_main\] follows from: \[prop:H\^1\] Let $F_+$ and $F_-$ be holomorphic functions defined on $V^+$ and $V^-$, respectively, such that $F_{\pm}|_{U^\pm}\equiv 0$. Then there exists a neighborhood $V^*\subset V$ of $C$ such that the Čech cohomology class $$[\{(V^*\cap V^+, F_+|_{V^*\cap V^+}), (V^*\cap V^-, F_-|_{V^*\cap V^-})\}]\in \check{H}^1(\{V^*\cap V_j\}, \mathcal{O}_{V^*}(-C))$$ is trivial. Denote by $g_0$ the extension of $\{(U^\pm, F_\pm|_{U^\pm})\}$ to $U_0$. As $V_0$ is coverd by a Stein neighborhood of $U_0$, we obtain a holomorphic function $G_0$ on $V_0$ such that $G_0|_{U_0}=g_0$. By using a function $G_1$ on $V_1$ defined by $G_1\equiv 0$, consider $$\beta:=[\{(V^+, (G_0-G_1)|_{V^+}), (V^-, (G_0-G_1)|_{V^-})\}]\in \check{H}^1(\{V_j\}, \mathcal{O}_{V}).$$ Then it follows from Proposition \[prop:H\^1\] that the class $(\alpha-\beta)\in \check{H}^1(\{V^*\cap V_j\}, \mathcal{O}_{V^*}(-C))$ is trivial for a neighborhood $V^*$ of $C$, which proves Proposition \[prop:H\^1\_main\]. Here we first give some notation which will be used in the proof of Proposition \[prop:H\^1\]. Let $V_0^*$ be a small neighborhood of the nodal point in $V_0$. Denote by $x$ the holomorphic function obtained by pulling back the function $S$ by the natural biholomorphism $V_0\to \widetilde{V}_0^+$, and by $y$ the one obtainded by pulling back the function $T$ by the natural biholomorphism $V_0\to \widetilde{V}_0^-$. We regard $(x, y)$ as coordinates of a neighborhood of $\overline{V_0^*}$. Note that $x\cdot y$ is a local defining function of $C$ in this locus. For sufficiently small positive constants $\varepsilon$ and $\delta$, we may assume that $V^*_0:=\{(x, y)\in V_0\mid \max\{|x|,\ |y|\}<2\varepsilon,\ |w_0|<\delta\}$. Denote by $U^*_0$ the subset $V^*_0\cap C$: i.e. $U^*_0=\{(x, y)\in V^*_0\mid |x|<2\varepsilon,\ y=0\}\cup\{(x, y)\in V^*_0\mid x=0,\ |y|<2\varepsilon\}$. In what follows we always assume that $\varepsilon$ and $\delta$ are sufficiently small so that $V_0^*$ is a relatively compact subset of $V_0$. Next we give a definition of a relatively compact subset $V_1^*$ of $V_1$. Denote by $z$ the holomorphic function obtained by pulling back the function $S$ by the natural biholomorphism $V_1\to \widetilde{V}_1$. Denote by $V_1^*$ the subset $\{(z, w_1)\in V_1\mid \varepsilon<|z|<1/\varepsilon,\ |w_1|<\delta\}$, where we are regarding $(z, w_1)$ as coordinates of this locus. Let $U^*_1$ be the subset of $U_1$ defined by $U^*_1:=V^*_1\cap C$: i.e. $U^*_1=\{(z, w_1)\in V^*_1\mid \varepsilon<|z|<1/\varepsilon,\ w_1=0\}$. Set $U^*_+:=U^+_0\cap U^*_1=\{(x, y)\in V^*_0\mid \varepsilon<|x|<2\varepsilon,\ y=0\}$ and $U^*_-:=U^-_0\cap U^*_1=\{(x, y)\in V^*_0\mid x=0,\ \varepsilon<|y|<2\varepsilon\}$. Denote by $V^*_\pm$ the connected components of $V^*_0\cap V^*_1$ which includes $U^*_\pm$ respectively, and by $V^*$ the subset $V^*_0\cup V^*_1=i(\{|\widetilde{w}|<\delta\})$. In what follows, we fix $\varepsilon$ and do not vary this value any more, whereas we will shrink $\delta$ as necessary. ### Outline of the proof of Proposition \[prop:H\^1\] We will construct holomorphic functions $F_j$ on $V^*_j$ for each $j=0, 1$ such that $F_0|_{V^*_\pm}-F_1|_{V^*_\pm}=F_\pm|_{V^*_\pm}$ holds on each $V^*_\pm$ by shrinking $\delta$. Actually, it is sufficient to construct such $\{(V_j^*, F_j)\}$, since we can construct $\widehat{F}_j\colon V^*\cap V_j\to \mathbb{C}$ such that $\delta\{(V^*\cap V_j, \widehat{F}_j)\} =\{(V^*\cap V^\pm, F_\pm|_{V^*\cap V^\pm})\}$ from them as follows: Set $\widehat{F}_j(p):=F_j(p)$ for $p\in V^*_j$. For $p\in (V^*\cap V_j)\setminus V^*_j$, set $$\widehat{F}_j(p):= \begin{cases} F_{1-j}(p)+(-1)^j\cdot F_+(p) & (\text{if}\ p\in V^+) \\ F_{1-j}(p)+(-1)^j\cdot F_-(p) & (\text{if}\ p\in V^-). \end{cases}$$ Note that $p\in V^*_{1-j}$, and thus it holds that $p\in V_j\cap V^*_{1-j}\subset V_0\cap V_1=V^+\cup V^-$ in this case. ### Proof of Proposition \[prop:H\^1\] (Step 1: Construction of $F_j$’s as formal power series) In this step, we will construct $F_j$’s in the form of $$F_0(x, y)=\sum_{\nu=1}^\infty a_{0, \nu}(x, y)\cdot w_0^\nu$$ and $$F_1(z, w_1)=\sum_{\nu=1}^\infty a_{1, \nu}(z)\cdot w_1^\nu$$ formally. Here $a_{1, \nu}$ is a function defined on $U_1^*$, which is also be regarded as a function on $V_1^*$ by pulling back the natural projection $(z, w_1)\mapsto z$. Similarly, $a_{0, \nu}$ is a function defined on $U_0^*$ with $$a_{0, \nu} = \begin{cases} p_\nu + r_\nu & (\text{if}\ p\in U^+\cap U^*_0) \\ q_\nu + r_\nu & (\text{if}\ p\in U^-\cap U^*_0), \end{cases}$$ where $p_\nu(x)$ is a holomorphic function on $U_+^*$ with $p_\nu(0)=0$, $q_\nu(y)$ is a holomorphic function on $U_-^*$ with $q_\nu(0)=0$, and $r_\nu\in\mathbb{C}$ is a constant. We also regard $a_{0, \nu}$ as a function defined on $V_0^*$ by setting $a_{0, \nu}(x, y):=p_\nu(x)+q_\nu(y)+r_\nu$, where $p_\nu$ and $q_\nu$ are extended by considering the pull-back by the projection $(x, y)\mapsto x$ and $(x, y)\mapsto y$, respectively. Denote by $$F_\pm (z, w_1)=\sum_{\nu=1}^\infty b_{\pm, \nu}(z)\cdot w_1^\nu$$ the expansion of $F_\pm$ by $w_1$ on $V^*_\pm$. First, let us construct $\{a_{j, 1}\}_{j=0, 1}$. As $N_{C/S}$ is non-torsion, it holds that $\check{H}^1(\{U_j\}, N_{C/S}^{-1})=0$ (see §\[section:Pic\]). Therefore, by considering the $1$-cocycle $[\{(U_\pm^*, b_{\pm, 1})\}]\in \check{H}^1(\{U_j^*\}, N_{C/S}^{-1})$, one can take $\{a_{j, 1}\}$ such that $$\begin{cases} t_+^{-1}a_{0, 1}(z)-a_{1, 1}(z)=b_{+, 1}(z) & (\text{on}\ U^*_+) \\ t_-^{-1}a_{0, 1}(z)-a_{1, 1}(z)=b_{-, 1}(z) & (\text{on}\ U^*_-). \end{cases}$$ Note that such $\{a_{j, 1}\}$ is unique since $H^1(C, N_{C/S}^{-1})=0$. By letting $r_1$ be that value of $a_{0, 1}$ at the nodal point, $p_1$ and $q_1$ are uniquely determined. We here remark that, for any choice of the other coefficients $\{a_{j, \nu}\}_{j=0, 1, \nu\geq 2}$, we have that $$F_0-F_1= \begin{cases} F_++O(w_1^2) & (\text{on}\ V^*_+) \\ F_-+O(w_1^2) & (\text{on}\ V^*_-) \end{cases}$$ holds as $w_1\to 0$. Next, we construct $\{a_{j, n+1}\}$ by assuming that $\{a_{j, \nu}\}_{j=0, 1, \nu\leq n}$ is determined so that the following inductive assumption holds: for any choice of $\{a_{j, \nu}\}_{j=0, 1, \nu\geq n+1}$, $$F_0-F_1= \begin{cases} F_++O(w_1^{n+1}) & (\text{on}\ V^*_+) \\ F_-+O(w_1^{n+1}) & (\text{on}\ V^*_-) \end{cases}$$ holds as $w_1\to 0$. In what follows, we regard $\{a_{j, \nu}\}_{j=0, 1, \nu\geq n+1}$ as unknown functions. Denote by $$p_\nu(x(z, w_1))= \begin{cases} p_\nu(x(z)) & (\text{on}\ V^*_+) \\ \sum_{\lambda=1}^\infty P^-_{\nu, \lambda}(z)\cdot w_1^\lambda & (\text{on}\ V^*_-) \end{cases}$$ and $$q_\nu(y(z, w_1))= \begin{cases} \sum_{\lambda=1}^\infty Q^+_{\nu, \lambda}(z)\cdot w_1^\lambda & (\text{on}\ V^*_+) \\ q_\nu(y(z)) & (\text{on}\ V^*_-) \end{cases}$$ the expansion of $p_\nu$ and $q_\nu$ by $w_1$ respectively (Note that $x=x(z, w_1)$ and $y=y(z, w_1)$ do not depend on $w_1$ on $V^*_+$ and $V^*_-$, respectively, in our coordinates, and that $q_\nu|_{U^+}\equiv q_\nu(0)=0$ and $p_\nu|_{U^-}\equiv p_\nu(0)=0$). On $V_+$, one can expand $F_0|_{V^*_+}$ as follows: $$F_0|_{V^*_+} = \sum_{\nu=1}^\infty a_{0, \nu}(x, y)\cdot w_0^\nu = \sum_{\nu=1}^\infty t_+^{-\nu}\cdot \left( p_\nu(x(z)) +\sum_{\lambda=1}^\infty Q^+_{\nu, \lambda}(z)\cdot w_1^\lambda +r_\nu \right)\cdot w_1^\nu.$$ By setting $$h^+_m(z):= \sum_{\nu=1}^{m-1} t_+^{-\nu}\cdot Q^+_{\nu, m-\nu}(z),$$ we have that the coefficient of $w_1^m$ in the expansion of $F_0|_{V^*_+}$ is $h^+_m(z)+t_+^{-m} \left(p_m(x(z))+r_m\right)$. The function $h^+_m$ can be regarded as a function obtained by pulling back a function on $U^*_+$ by the local projection $(z, w_1)\mapsto z$, which coincides with $(x, y)\mapsto x$ in this locus. Note that $\{h^+_m\}_{m\leq n}$ are regarded as known functions, since $h^+_m$ depends only on the data $\{q_\nu\}_{\nu=1}^{m-1}$. By a simple observation, it turns out that one should construct $a_{j, n+1}$’s so that $$b_{+, n+1}(z)=h^+_{n+1}(z)+t_+^{-n-1}\left(p_{n+1}(x(z))+r_{n+1}\right)-a_{1, n+1}(z)$$ holds on $U^*_+$ in order for the inductive assumption to hold for $n+1$. Similarly, we have that $$F_0|_{V^*_-} = \sum_{\nu=1}^\infty a_{0, \nu}(x, y)\cdot w_0^\nu = \sum_{\nu=1}^\infty t_-^{-\nu}\cdot \left( \sum_{\lambda=1}^\infty P^-_{\nu, \lambda}(z)\cdot w_1^\lambda +q_\nu(y(z)) +r_\nu \right)\cdot w_1^\nu$$ on $V^*_-$. By setting $$h^-_m(z):= \sum_{\nu=1}^{m-1} t_-^{-\nu}\cdot P^-_{\nu, m-\nu}(z),$$ we have that the coefficient of the expansion of $h^-_m(z)$ in $w_1^m$ is $h^-_m(z)+t_-^{-m}\left(q_m(y(z))+r_m\right)$. The function $h^-_m$ can be regarded as a function obtained by pulling back a function on $U^*_-$ by the local projection $(z, w_1)\mapsto z$, which coincides with $(x, y)\mapsto y$ in this locus. Note that $\{h^-_m\}_{m\leq n}$ are regarded as known functions, since $h^+_m$ depends only on the data $\{p_\nu\}_{\nu=1}^{m-1}$. By a simple observation, it turns out that one should construct $a_{j, n+1}$’s so that $$b_{-, n+1}(z)=h^-_{n+1}(z)+t_-^{-n-1}\cdot \left(q_{n+1}(y(z))+r_{n+1}\right)-a_{1, n+1}(z)$$ holds on $U^*_-$ in order for the inductive assumption to hold for $n+1$. By the observations above, we have that $b_{\pm, n+1}(z)-h^\pm_{n+1}(z)$ is known function after we finish defining $\{a_{j, \nu}\}_{j=0, 1, \nu\leq n}$. Therefore, we can define $\{(U^*_0, a_{0, n+1}(x, y)=p_{n+1}(x)+q_{n+1}(y)+r_n), (U^*_1, a_{1, n+1}(z))\}$ by considering the equations $$\begin{cases} t_+^{-n-1}a_{0, n+1}(z)-a_{1, n+1}(z)=b_{+, n+1}(z)-h^+_{n+1}(z) & (\text{on}\ U^*_+) \\ t_-^{-n-1}a_{0, n+1}(z)-a_{1, n+1}(z)=b_{-, n+1}(z)-h^-_{n+1}(z) & (\text{on}\ U^*_-). \end{cases}$$ As $H^0(C, N_{C/S}^{-n})=H^1(C, N_{C/S}^{-n})=0$ (see §\[section:Pic\]), we actually have the unique solution. ### Proof of Proposition \[prop:H\^1\]: (Step 2: Estimate of the coefficient functions) As $V^*_\pm\Subset V^\pm$, there exists a constant $M$ such that $$\max\left\{ \sup_{V^*_+}|F_+|,\ \sup_{V^*_-}|F_-| \right\} <M.$$ In what follows, we assume that $M>1$. Fix a positive constant $R$ sufficiently larger than $1/\delta, 1/\varepsilon$, $\sup_{V^*_+}|w_0/y|$, $\sup_{V^*_+}|w_1/y|$, $\sup_{V^*_+}|w_1/y|$, $\sup_{V^*_-}|w_0/x|$, $\sup_{V^*_-}|w_1/x|$, and the inverses of these. Then we may assume that $$\{(z, w_1)\mid \varepsilon<|z|<2\varepsilon ,\ |w_1|=1/R\}\subset V^*_+$$ and $$\{(z, w_1)\mid 1/(2\varepsilon)<|z|<1/\varepsilon ,\ |w_1|=1/R\}\subset V^*_-$$ hold (see also Remark \[rmk:R\]). Let $B(X)=X+\sum_{\nu=2}^\infty B_\nu X^\nu$ be the formal power series defined by $$\label{eq:func_eq_B} \sum_{\nu=2}^\infty |1-t^{\nu-1}|\cdot B_\nu X^\nu=KRM\frac{B(X)^2}{1-RB(X)},$$ where the constant $K$ is a positive real number as in Lemma \[lem:ueda\_type\_estim\]. Note that it follows from the argument in [@S] that $B(X)$ has a positive radius of convergence (see also [@U83 Lemma 5]). Define a convergent power series $A(X)=\sum_{\nu=1}^\infty A_\nu X^\nu$ by $A_{n}:=B_{n+1}$ ($n\geq 1$): i.e. $B(X)=X+XA(X)$. In this step, we show that $$\label{eq:A_nu} \max_{j=0, 1}\sup_{p\in U_j^*}|a_{j, \nu}(p)|\leq A_\nu$$ holds for each $\nu$ by induction. First, by Cauchy’s inequality, we have that $$\sup_{z\in U^*_\pm}|b_{\pm, 1}(z)| \leq M\cdot R.$$ Therefore, the inequality (\[eq:A\_nu\]) for $\nu=1$ follows from Lemma \[lem:ueda\_type\_estim\] below. Next we show the inequality (\[eq:A\_nu\]) for $\nu=n+1$ by assuming that it holds for $\nu=1, 2, \dots, n$. As it holds that $|h^+_{n+1}(z)|\leq \sum_{\nu=1}^{n} |Q^+_{\nu, n+1-\nu}(z)|$, we have that $$\sup_{U^*_+}|Q^+_{\nu, \lambda}| \leq A_\nu\cdot R^\lambda$$ holds by Cauchy’s inequality. Therefore it follows that $$\sup_{U^*_+}|h^+_{n+1}(z)|\leq \sum_{\nu=1}^{n} A_\nu\cdot R^{n+1-\nu} =\text{the coefficient of}\ X^{n+1}\ \text{in the expansion of}\ \frac{RXA(X)}{1-RX}.$$ Note that the same estimate holds also for $h^-_{n+1}$. As it holds that $$\sup_{z\in U^+\cap U^*_\pm}|b_{\pm, n+1}(z)| \leq MR^{n+1} =\text{the coefficient of}\ X^{n+1}\ \text{in the expansion of}\ \frac{MRX}{1-RX},$$ it follows from Lemma \[lem:ueda\_type\_estim\] that $$\max_{j=0, 1}\sup_{U^*_j}|a_{j, n+1}|\leq \text{the coefficient of}\ X^{n+1}\ \text{in the expansion of}\ \frac{1}{|1-t^{n+1}|}\cdot \frac{KRX(A(X)+M)}{1-RX}.$$ As $M\geq 1$, we have that $$\begin{aligned} &&\text{the coefficient of}\ X^{n+1}\ \text{in the expansion of}\ \frac{1}{|1-t^{n+1}|}\cdot \frac{KRX(A(X)+M)}{1-RX}\nonumber \\ &\leq& \text{the coefficient of}\ X^{n+1}\ \text{in the expansion of}\ \frac{1}{|1-t^{n+1}|}\cdot \frac{KRMB(X)}{1-RX}\nonumber \\ &=& \text{the coefficient of}\ X^{n+2}\ \text{in the expansion of}\ \frac{1}{|1-t^{n+1}|}\cdot \frac{KRMXB(X)}{1-RX}\nonumber \\ &\leq& \text{the coefficient of}\ X^{n+2}\ \text{in the expansion of}\ \frac{KRM}{|1-t^{n+1}|}\cdot \frac{B(X)^2}{1-RB(X)}. \nonumber \end{aligned}$$ Thus we have the inequality (\[eq:A\_nu\]) for $\nu=n+1$ by the equation (\[eq:func\_eq\_B\]). ### Proof of Proposition \[prop:H\^1\] (Step 3: Convergence of $F_j$’s) Let us shrink $\delta$ so that it is smaller than the radius of convergence of the poser series $A(X)$. Then it clearly holds that $\sup_{V^*_1}|a_{1, \nu}|\leq A_\nu$ when we regard $a_{1, \nu}$ as a function $V^*_1$ by the rule we mentioned above. For $(x, y)\in V^*_0$, $$\begin{aligned} |a_{0, \nu}(x, y)| &=&|p_\nu(x)+q_\nu(y)+r_\nu| \leq |p_\nu(x)+r_\nu|+|q_\nu(y)+r_\nu|+|r_\nu| \nonumber \\ &\leq&\sup_{x\in U^+\cap U^*_0}|a_{0, \nu}(x)|+\sup_{y\in U^-\cap U^*_0}|a_{0, \nu}(y)|+|a_{0, \nu}(0, 0)| \leq 3A_\nu.\nonumber \end{aligned}$$ Thus we can regard $F_j$ as a holomorphic function defined on $V_j^*$. By construction, we have that $F_0|_{V^*_\pm}-F_1|_{V^*_\pm}=F_\pm|_{V^*_\pm}$ holds on $V^*_\pm$. \[rmk:R\] In Step 2 of the proof above, we applied Cauchy’s inequality in several times, in which we used the fact that the circle $\{(z, w_1)\in V^*_1\mid z=z_0,\ |w_1|=1/R\}$ is included in $V^*_0$ for each $z_0\in U^*_\pm$. For this, we need to choose $V^*_j$’s and its coordinates appropriately as we did in §\[section:V\_tilde\] and at the beginning of the proof. One of the most important property of our coordinates is that the projection $(z, w_1)\mapsto z$ coincides with $(x, y)\mapsto x$ on $V^*_+$ and with $(x, y)\mapsto y$ on $V^*_-$. On the other hand, we used an open covering of a neighborhood of $C$ taken by using a general theory (Siu’s theorem [@Siu]) in [@K6]. Here we had to refine and shrink the open sets in order to take $R$ as a constant, see also [@K6 Remark 4.3]. We here remark that one can slightly simplify the proof of [@K6 Theorem 1.4] by replacing the open covering with $\{V^*_j\}$ we used in the present paper. \[lem:ueda\_type\_estim\] Let $n$ be a positive integer, $b_\pm$ a holomorphic function on $U_\pm^*$, and $a_j$ be a function on $U_j^*$ for $j=0, 1$ such that $$\begin{cases} t_+^{-n}\cdot a_0-a_1= b_+ & (\text{on}\ U_+^*) \\ t_-^{-n}\cdot a_0-a_1= b_- & (\text{on}\ U_-^*). \end{cases}$$ Then there exists a constant $K=K(C, \{U^*_j\})$ which does not depend on neither $n$, $a_j$ nor $b_\pm$ such that $$\max_{j=0, 1}\sup_{U^*_j}|a_j| \leq \frac{K}{|1-t^n|}\cdot \max\left\{\sup_{x\in U_+^*}|b_+(x)|,\ \sup_{y\in U_-^*}|b_-(y)|\right\}$$ holds. In the rest of this subsection, we give a proof of Lemma \[lem:ueda\_type\_estim\] for the convenience of the reader, although its statement is nothing but a summary of some arguments in [@K6 §4.2.3, 4.2.4] intrinsically. Note that $t^n\not=1$ and $H^0(C, N_{C/S}^{-n})=H^1(C, N_{C/S}^{-n})=0$ hold (as we mentioned in §\[section:Pic\]), since $N_{C/S}$ is non-torsion. Therefore, $a_j$’s are uniquely determined by $b_\pm$. Set $M:=\max\left\{\sup_{x\in U_+^*}|b_+(x)|,\ \sup_{y\in U_-^*}|b_-(y)|\right\}$. Let $r$ be the value of $a_0$ at the nodal point. Then there uniquely exists a function $p$ on $U^+\cap U^*_0$ and $q$ on $U^-\cap U^*_0$ such that $$a_0=\begin{cases} p+r & (\text{on}\ U^+\cap U^*_0) \\ q+r & (\text{on}\ U^-\cap U^*_0). \end{cases}$$ Define $1$-forms $\omega_0$ and $\omega_1$ by $$\omega_0:=da_0=\begin{cases} p'(x)dx & (\text{on}\ U^+\cap U^*_0) \\ q'(y)dy & (\text{on}\ U^-\cap U^*_0) \end{cases}$$ and $\omega_1:=da_1=a_1'(z)dz$. By the assumption, we have that $t_\pm^{-n}\cdot \omega_0-\omega_1= db_\pm (=b_\pm'(z)dz)$ on $U_\pm^*$. Define a new open covering $\{U_j^{**}\}$ by $$U^{**}_0:=\left\{(x, y)\in U^*_0\left| \max\{|x|, |y|\}<\frac{5\varepsilon}{3}\right.\right\},\ U^{**}_1:=\left\{(z, 0)\in U^*_1\left| \frac{4\varepsilon}{3} < |z| < \frac{3}{4\varepsilon}\right.\right\}.$$ As $U^{**}_j\Subset U^*_j$, we have that $$\sup_{z\in U^\pm\cap U^{**}_{01}}|b_\pm'(z)|\leq K_1\cdot M$$ holds on a constant $K_1>0$, where $U^{**}_{01}:=U^{**}_0\cap U^{**}_1$. By Lemma \[lem:KS\_type\_estim\], we have that $$\max\left\{ \sup_{x\in {U^{**}_0}\cap {U^+}}|p'(x)|, \sup_{y\in {U^{**}_0}\cap {U^-}}|q'(y)|, \sup_{z\in {U^{**}_1}}|a_1'(z)| \right\} \leq K_0K_1M$$ holds for a constant $K_0$. By considering the path integral from the nodal point, we have that $$\max\left\{ \sup_{x\in {U^{**}_0}\cap {U^+}}|p(x)|, \sup_{y\in {U^{**}_0}\cap {U^-}}|q(y)| \right\} \leq \frac{5\varepsilon}{3} K_0K_1M.$$ By fixing point $z_\pm$ from $U^{**}_{01}\cap U^\pm$ and letting $C_\pm:=b_\pm(z_\pm)$ and $C_1:=a_1(z_+)$, respectively, we have that $$b_\pm(z)=C_\pm+\int_{z_\pm}^zb_\pm'(\zeta)d\zeta,\ a_1(z)=C_1+\int_{z_+}^za_1'(\zeta)d\zeta.$$ Note that $$\sup_{z\in U^{**}_1}\left|\int_{z_+}^za_1'(\zeta)d\zeta\right| \leq K_2\cdot \sup_{z\in {U^{**}_1}}|a_1'(z)| \leq K_0K_1K_2M$$ holds for a constant $K_2$ which depends only on the diameter of $U^{**}_1$ (or equivalently, only on $\varepsilon$). As it follows $$\begin{cases} t_+^{-n}\cdot (-p(z_+)+r)-C_1=C_+ \\ t_-^{-n}\cdot (-q(z_-)+r)-\left(\int_{z_+}^{z_-} a_1'(z)dz + C_1\right)=C_-, \end{cases}$$ we have that $r=\frac{1}{t_+^{-n}-t_-^{-n}}\cdot\left(D_+-D_-\right)$ and $C_1=\frac{1}{t_+^{-n}-t_-^{-n}}\cdot\left(t_-^{-n}D_+-t_+^{-n}D_-\right)$, where $D_+:=t_+^{-n}p(z_+)+C_+$ and $D_-:=t_-^{-n}q(z_-)+C_-+\int_{z_+}^{z_-}a_1'(\zeta)d\zeta$. Note that $$|D_+|\leq |p(z_+)|+|C_+|\leq\left(1+\frac{5\varepsilon}{3} K_0K_1\right)M$$ and $$|D_-|\leq |q(z_-)|+|C_-|+\sup_{z\in U^{**}_1}\left|\int_{z_+}^za_1'(\zeta)d\zeta\right|\leq \left(1+\frac{5\varepsilon}{3} K_0K_1+K_0K_1K_2\right)M.$$ Let us denote by $K_3$ the constant $2+\frac{10\varepsilon}{3} K_0K_1+K_0K_1K_2$. Then it follows from the arguments above that $$\sup_{z\in U^{**}_1}\left|a_1(z)\right| \leq |C_1|+ \sup_{z\in U^{**}_1}\left|\int_{z_+}^za_1'(\zeta)d\zeta\right| \leq K_3\cdot \left(1+\frac{1}{|1-t^n|}\right)\cdot M$$ and $$\sup_{z\in U^{**}_0}\left|a_0(z)\right| \leq K_3\cdot \left(1+\frac{1}{|1-t^n|}\right)\cdot M.$$ Thus we have $$\max_{j=0, 1}\sup_{U^{**}_j}|a_j|< \frac{3K_3}{|1-t^n|}\cdot M.$$ When $z\in U^*_1\setminus U^{**}_1$, it holds that $z\in U^+\cap U^{**}_0$ or $z\in U^-\cap U^{**}_0$. In the former case, we have that $$|a_1(z)|=|t_+^{-n}a_0(z)-b_+(z)|\leq |a_0(z)|+|b_+(z)|\leq \left(1+\frac{3K_3}{|1-t^n|}\right)\cdot M.$$ By the same arguments for the other cases, the lemma follows by letting $K:=2+3K_3$. \[lem:KS\_type\_estim\] Let $n$ be a positive integer and $i\colon\widetilde{C}\to C$ be the normalization such that the preimage of the nodal point is $\{0, \infty\}\subset \mathbb{P}^1=\widetilde{C}$. Denote by $\widetilde{U^{**}_j}$ the preimage $i^{-1}(U^{**}_j)$ and $\widetilde{U^\pm}$ the preimage $i^{-1}(U^{\pm})$. Let $\eta_{\pm}$ be $1$-forms on $\widetilde{U^\pm}\cap \widetilde{U^{**}_{01}}$ such that the Čech cohomology class $[\{(\widetilde{U^\pm}\cap \widetilde{U^{**}_{01}},\eta_{\pm})\}]\in \check{H}^1(\{\widetilde{U^{**}_j}\}, K_{\widetilde{C}}\otimes i^*N_{C/S}^{-n})$ is trivial. Denote by $\omega_j$ the $1$-form on $\widetilde{U_j^{**}}$ for $j=0, 1$ uniquely determined by $$\begin{cases} t_+^{-n}\cdot \omega_0-\omega_1= \eta_+ & (\text{on}\ \widetilde{U^+}\cap \widetilde{U^{**}_{01}}) \\ t_-^{-n}\cdot \omega_0-\omega_1= \eta_- & (\text{on}\ \widetilde{U^-}\cap \widetilde{U^{**}_{01}}). \end{cases}$$ Then there exists a constant $K_0=K_0(C, \{U^{**}_j\})$ which does not depend on neither $n$ nor $\eta_\pm$ such that $$\begin{aligned} &&\max\left\{ \sup_{x\in \widetilde{U^{**}_0}\cap \widetilde{U^+}}|g^+_0(x)|, \sup_{y\in \widetilde{U^{**}_0}\cap \widetilde{U^-}}|g^-_0(y)|, \sup_{z\in \widetilde{U^{**}_1}}|g_1(z)| \right\} \nonumber \\ &\leq& K_0\cdot \max\left\{\sup_{z\in \widetilde{U^+}\cap \widetilde{U^{**}_{01}}}|h_+(z)|,\ \sup_{z\in \widetilde{U^-}\cap \widetilde{U^{**}_{01}}}|h_-(z)|\right\}, \nonumber\end{aligned}$$ where $\omega_1=g_1(z)dz$, $$\omega_0=\begin{cases} g^+_0(x)dx & (\text{on}\ \widetilde{U^+}\cap \widetilde{U^{**}_{01}}) \\ g^-_0(y)dy & (\text{on}\ \widetilde{U^-}\cap \widetilde{U^{**}_{01}}), \end{cases}$$ and $\eta_\pm=h_\pm(z)dz$. By replacing $\omega_0$ with $$\begin{cases} t_+^{-n}\cdot \omega_0 & (\text{on}\ \widetilde{U^+}\cap \widetilde{U^*_{01}}) \\ t_-^{-n}\cdot \omega_0 & (\text{on}\ \widetilde{U^-}\cap \widetilde{U^*_{01}}), \end{cases}$$ the proof of the lemma is reduced to the case of $n=0$, which follows from [@KS Lemma 2]. \[rmk:ueda\_type\_estim\_general\] Lemma \[lem:ueda\_type\_estim\] also holds in the case where $C$ is a cycle of multiple rational curves (see [@K6 §4.2.3, 4.2.4] for details). Note that [@K6 Lemma 4.2] is used for the estimate of the constants appears in the proof of the general statement which corresponds to the constant $C_1$ and $r$ in the proof above. Proof of Proposition \[prop:H\^1\] when $C$ is a cycle of multiple rational curves ---------------------------------------------------------------------------------- Let $C$ be a cycle of rational curves consists of $n$ irreducible components ($n\geq 2$). As Proposition \[prop:H\^1\] for this $C$ is shown by intrinsically the same arguments as in the previous section, here we only explain the outline. Denote by $C_{(1)}, C_{(2)} \dots, C_{(n-1)}, C_{(n)}=C_{(0)}$ the irreducible components of $C$. For $\nu=0, 1, 2, \dots, n-1$, Fix a small neighborhood $V_\nu$ of $C_{(\nu)}\cap C_{\rm reg}$ and $V_{\nu, \nu+1}$ of $C_{(\nu)}\cap C_{(\nu+1)}$. We may assume that $V_\nu\subset i(\widetilde{V}_{(\nu)})$, and that $V_{\nu, \nu+1}$ is included in the image of $\widetilde{V}_{0 (\nu+1)}^+=F_{\nu+1, \nu}^{-1}(\widetilde{V}_{0 (\nu)}^-)$ by $i$, where we are using the notation in Remark \[rmk:V\_tilde\_general\]. Define coordinates $(z_\nu, w_\nu)$ of $V_\nu$ by $i^*z_\nu=S_{(\nu)}$ and $i^*w_\nu=S_{(\nu)}\cdot \xi_{0 (\nu)}=T_{(\nu)}\cdot \xi_{\infty (\nu)}$, and $(x_{\nu}, x_{\nu+1})$ of $V_{\nu, \nu+1}$ by $i^*x_{\nu+1}=S_{(\nu+1)}$ and $i^*x_\nu=F_{\nu+1, \nu}^*T_{(\nu)}$. Let $$F^+(z_\nu, w_\nu)=\sum_{n=1}^\infty b^+_{\nu, \nu+1, n}(z_\nu)\cdot w_{\nu}^n$$ be a holomorphic function defined on $V_\nu\cap V_{\nu, \nu+1}$, and $$F^-(z_{\nu+1}, w_{\nu+1})=\sum_{n=1}^\infty b^-_{\nu, \nu+1, n}(z_{\nu+1})\cdot w_{\nu+1}^n$$ be a holomorphic function defined on $V_{\nu+1}\cap V_{\nu, \nu+1}$. Then it is sufficient to find a holomorphic function $F_\nu$ on $V_\nu$ and $F_{\nu, \nu+1}$ on $V_{\nu, \nu+1}$ such that $$\begin{cases} F_{\nu, \nu+1}-F_\nu = F^+ & (\text{on}\ V_\nu\cap V_{\nu, \nu+1})\\ F_{\nu, \nu+1}-F_{\nu-1} = F^- & (\text{on}\ V_{\nu+1}\cap V_{\nu, \nu+1}) \end{cases}$$ by shrinking $V$. $F_\nu$ is constructed in the form of $$F_\nu(z_\nu, w_\nu)=\sum_{n=1}^\infty a_{\nu, n}(z_\nu)\cdot w_{\nu}^n,$$ and $F_{\nu, \nu+1}$ is of $$F_{\nu, \nu+1}(x_\nu, x_{\nu+1})=\sum_{n=1}^\infty a_{\nu, \nu+1, n}(x_\nu, x_{\nu+1})\cdot w_{\nu, {\nu+1}}(x_\nu, x_{x_\nu, x_{\nu+1}})^n,$$ where $w_{\nu, {\nu+1}}$ is the function defined by $i^*w_{x_\nu, x_{\nu+1}}=S_{(\nu)}\cdot \xi_{0 (\nu)}$, and the functions $a_{\nu, n}(z_\nu)$ and $a_{\nu, \nu+1, n}$ are holomorphic functions defined on $C\cap V_\nu$ and $C\cap V_{\nu, \nu+1}$, respectively. Let $p^{\nu+1}_{\nu, n}(x_\nu)$ be a function on $C_{(\nu)}\cap V_{\nu, \nu+1}$ and $p^{\nu}_{\nu+1, n}(x_{\nu+1})$ be a function on $C_{(\nu+1)}\cap V_{\nu, \nu+1}$ such that $$a_{\nu, \nu+1, n}(x_\nu, x_{\nu+1})=\begin{cases} p^{\nu+1}_{\nu, n}(x_\nu)+r_{\nu, \nu+1, n} & (\text{on}\ C_{(\nu)}\cap V_{\nu, \nu+1}) \\ p^{\nu}_{\nu+1, n}(x_{\nu+1})+r_{\nu, \nu+1, n} & (\text{on}\ C_{(\nu+1)}\cap V_{\nu, \nu+1})\\ \end{cases}$$ holds, where $r_{\nu, \nu+1, n}:=a_{\nu, \nu+1, n}(0, 0)$. The function $a_{\nu, \nu+1, n}$ is also regarded as a function defined on $V_{\nu, \nu+1}$ by $a_{\nu, \nu+1, n}(x_\nu, x_{\nu+1}):=p^{\nu+1}_{\nu, n}(x_\nu)+p^{\nu}_{\nu+1, n}(x_{\nu+1})+r_{\nu, \nu+1, n}$. By setting $t^+_{\nu, \nu+1}:=1$ and $t^+_{\nu, \nu+1}:=t_{\nu+1, \nu}$, we have that $$\begin{cases} w_\nu=t^+_{\nu, \nu+1}\cdot w_{\nu, \nu+1} & (\text{on}\ C_{(\nu)}\cap V_{\nu, \nu+1}) \\ w_{\nu+1}=t^-_{\nu, \nu+1}\cdot w_{\nu, \nu+1} & (\text{on}\ C_{(\nu+1)}\cap V_{\nu, \nu+1})\\ \end{cases}$$ holds. By the same argument as in Step 1 of the proof of Proposition \[prop:H\^1\], it follows that one should define $a_{\nu, n}$’s and $a_{\nu, \nu+1, n}$’s by $$\begin{cases} (t_{\nu, \nu+1}^+)^{-n}a_{\nu, \nu+1, n+1}-a_{\nu, n}=b^+_{\nu, \nu+1, n}-h^+_{\nu, \nu+1, n} & (\text{on}\ C_{(\nu)}\cap V_{\nu, \nu+1}) \\ (t_{\nu, \nu+1}^-)^{-n}a_{\nu, \nu+1, n+1}-a_{\nu, n+1}=b^-_{\nu, \nu+1, n}-h^-_{\nu, \nu+1, n}(z) & (\text{on}\ C_{(\nu+1)}\cap V_{\nu, \nu+1}). \end{cases}$$ Here the functions $h^\pm_{\nu, \nu+1, n}(z_\nu)$ are defined by $$h^+_{\nu, \nu+1, n}(z_\nu) =\sum_{m=1}^{n-1}(t_{\nu, \nu+1}^+)^{-m}\cdot P^{\nu}_{\nu+1, n, n-m}(z_\nu)$$ and $$h^-_{\nu, \nu+1, n}(z_{\nu+1}) =\sum_{m=1}^{n-1}(t_{\nu, \nu+1}^-)^{-m}\cdot P^{\nu+1}_{\nu, n, n-m}(z_{\nu+1}),$$ where $$p^{\nu}_{\nu+1, n}(x_{\nu+1}(z_\nu, w_\nu)) =\sum_{\lambda=1}^\infty P^{\nu}_{\nu+1, n, \lambda}(z_\nu)\cdot w_\nu^\lambda$$ and $$p^{\nu+1}_{\nu, n}(x_{\nu}(z_{\nu+1}, w_{\nu+1})) =\sum_{\lambda=1}^\infty P^{\nu+1}_{\nu, n, \lambda}(z_{\nu+1})\cdot w_{\nu+1}^\lambda.$$ As one can estimate $|a_{\nu, n}|$ and $|a_{\nu, \nu+1, n}|$ by the same argument as in Step 2 of the proof of Proposition \[prop:H\^1\], the proposition holds (see also Remark \[rmk:ueda\_type\_estim\_general\]). Proof of Theorem \[thm:arnold\_for\_nodal\_rational\] {#section:prf} ===================================================== Proof of Theorem \[thm:arnold\_for\_nodal\_rational\] when $C$ is a rational curve with a node ---------------------------------------------------------------------------------------------- Let $C$ be a rational curve with a node embedded in $S$ such that the normal bundle satisfies the Diophantine assumption in Theorem \[thm:arnold\_for\_nodal\_rational\]. We the notation in §\[section:V\_tilde\]. Then it is sufficient to show that we may assume $G\equiv 1$ by changing the coordinates such as $S$ and $T$. Let $g(S, \xi_0):=\frac{1}{2\pi\sqrt{-1}}\log G(S, \xi_0)$ be the branch such that $g(0, 0)=0$. By applying Proposition \[prop:H\^1\_main\] to $F_+:=-(V_0\to\widetilde{V}^+_0)^* g$ and $F_-:=0$, we have that, by shrinking $\widetilde{V}$ if necessary, there exist holomorphic functions $h_+\colon \widetilde{V}^+_0\to \mathbb{C}$, $h_1\colon \widetilde{V}_1\to \mathbb{C}$ and $h_-\colon \widetilde{V}^-_0\to \mathbb{C}$ such that $$\begin{cases} h_+-h_1=-g & (\text{on}\ \widetilde{V}_1\cap \widetilde{V}^+_0) \\ h_--h_1=0 & (\text{on}\ \widetilde{V}_1\cap \widetilde{V}^-_0) \end{cases}$$ holds (Set $h_+:=(\widetilde{V}^+_0\to V_0)^*F_0$, $h_-:=(\widetilde{V}^-_0\to V_0)^*F_0$ and $h_1:=(\widetilde{V}_1\to V_1)^*F_1$, for the solution $\{(V_j, F_j)\}$ in Proposition \[prop:H\^1\_main\]). Define a function $h$ on $\widetilde{V}$ by $$h:= \begin{cases} h_++g & (\text{on}\ \widetilde{V}_0^+) \\ h_1 & (\text{on}\ \widetilde{V}_1) \\ h_- & (\text{on}\ \widetilde{V}_0^-). \end{cases}$$ As clearly it holds that $F^*h_-=h_+$ by definition, we have that $F^*(h|_{\widetilde{V}_0^-})=h|_{\widetilde{V}_0^+}+g$. Denote by $H$ the function $e^{2\pi\sqrt{-1}h}$. Define a new coordinate function $\widehat{S}$ on $\widetilde{V}_0^+\cup \widetilde{V}_1$ by $\widehat{S}:=S\cdot H^{-1}$, $\widehat{T}$ on $\widetilde{V}_0^-\cup \widetilde{V}_1$ by $\widehat{T}:=T\cdot H$, $\widehat{\xi}_0$ on a neighborhood of $D_0$ by $\widehat{\xi}_0:=\widetilde{w}\cdot \widehat{S}^{-1}=\xi_0\cdot H$, and $\widehat{\xi}_\infty$ on a neighborhood of $D_\infty$ by $\widehat{\xi}_\infty:=\widetilde{w}\cdot \widehat{T}^{-1}=\xi_\infty\cdot H^{-1}$. Then, as it follows $F^*(H|_{\widetilde{V}_0^-})=H|_{\widetilde{V}_0^+}\cdot G$ by the construction, we have that $$F^*\widehat{T}=(F^*T)\cdot (F^*H)= \frac{t\cdot \xi_0}{G}\cdot (H\cdot G) =t\cdot (\xi_0 H) =t\cdot \widehat{\xi}_0$$ and $$F^*\widehat{\xi}_\infty =(F^*\xi_\infty)\cdot (F^*H)^{-1} =(G\cdot S)\cdot (H\cdot G)^{-1} =S\cdot H^{-1} =\widehat{S}$$ on $F^{-1}(\widetilde{V}^-_0\cap \widetilde{V}_1)$. Therefore, by replacing $(S, \xi_0)$ and $(T, \xi_\infty)$ with $(\widehat{S}, \widehat{\xi}_0)$ and $(\widehat{T}, \widehat{\xi}_\infty)$ respectively, we have that $F(S, \xi_0)=(t\cdot \xi_0, S)$ holds, which proves the theorem. Proof of Theorem \[thm:arnold\_for\_nodal\_rational\] when $C$ is a cycle of multiple rational curves {#section:prf_general} ----------------------------------------------------------------------------------------------------- Here we use the notation in Remark \[rmk:V\_tilde\_general\]. First, we show that we may assumes that $G_{\nu+1, \nu}\equiv 1$ holds for $\nu=1, 2, n-2$ by changing the coordinates appropriately. Let $\{(\widetilde{V}_{(\nu)}', \widetilde{C}_{(\nu)}', (\widetilde{V}_{0 (\nu)}')^\pm, S_{(\nu)}', T_{(\nu)}', \xi_{0 (\nu)}', \xi_{\infty (\nu)}')\}_{\nu=1}^n$ be the $n$-copies of $(\widetilde{V}, \widetilde{C}, \widetilde{V}_{0}^\pm, S, T, \xi_{0}, \xi_{\infty})$ in Example \[ex:standard\_model\]. Denote by $i'\colon\coprod_{\nu=1}^n\widetilde{V}_{(n)}'\to \widetilde{V}'$ the quotient by the relation generated by the maps $(\widetilde{V}_{0 (\nu+1)}')^+\to (\widetilde{V}_{0 (\nu)}')^-$ naturally induced by $\widetilde{F}_{\nu+1, \nu}$’s for $\nu=1, 2, \dots, n-1$. In what follows, we regard $\widetilde{V}_{(\nu)}'$ as a subset of $\widetilde{V}'$. Note that $\widetilde{V}_{(1)}'\cap \widetilde{V}_{(n)}'=\emptyset$ holds as subset of $\widetilde{V}'$. Then it follows from a simple observation that the quotient $\widetilde{C}'$ of $\coprod_{\nu=1}^n\widetilde{C}_{(n)}'$ is a tree of rational curves with intersection matrix $$\left( \begin{array}{ccccccc} -2 & 1 & 0 &\ldots & 0 & 0 & 0 \\ 1 & -2& 1 &\ldots & 0 & 0 & 0 \\ 0 & 1 & -2 & \ddots& & \vdots& \vdots \\ 0 &0 & \ddots&\ddots &\ddots & \vdots& \vdots \\ \vdots & \vdots & & \ddots& -2& 1& 0 \\ 0 &0 & 0 &\ldots & 1& -2& 1 \\ 0 &0 & 0 & \ldots & 0 & 1 & -2 \end{array} \right).$$ As this matrix is negative definite, it follows from [@L Theorem 4.9] and Grauert’s theorem [@G] that $\widetilde{C}'$ admits a strictly pseudoconvex neighborhood $\widetilde{V}'$ whose maximal compact analytic subset is $\widetilde{C}'$. Note that, by the arguments as in §\[prelim:general\], it holds that $H^1(\widetilde{C}', N_{\widetilde{C}'/\widetilde{V}'}^{-m})=0$ holds for each $m\geq 0$. Thus, it follows the same argument as in the proof of [@K2 Proposition 3.1] that the restriction $H^1(\widetilde{V}', \mathcal{O}_{\widetilde{V}'})\to H^1(\widetilde{C}', \mathcal{O}_{\widetilde{C}'})$ is injective. As $H^1(\widetilde{C}', \mathcal{O}_{\widetilde{C}'})=0$, it follows from the same arguments as in the previous subsection that we may assume $G_{\nu+1, \nu}\equiv 1$ for $\nu=1, 2, n-2$. Therefore, the problem is reduced to showing that we may assume $G_{1, n}\equiv 1$ by changing the coordinates. By replacing $\xi_{0(0)}$ with $G_{1, n}(0, 0)^{-1/n}\cdot \xi_{0(0)}$, we may assume that $G_{1, n}(0, 0)=1$. Then the theorem follows by the same argument as in the previous subsection. Toward the gluing construction of K3 surfaces corresponding to degenerations of K3 surfaces of type III ======================================================================================================= Let $(C, S)$ be the example as in Example \[ex:ABU\], \[ex:ABU\_2\] or \[ex:ABU\_3\]. Assume that the normal bundles $N_{C/S}$ is a ${\rm U}(1)$-flat line bundles which satisfies Diophantine condition. Then it follows from Theorem \[thm:arnold\_for\_nodal\_rational\] that one can take a neighborhood $V$ of $C$ in $S$ as in Example \[ex:standard\_model\] or Example \[ex:standard\_model\_general\] with $n=2$ or $3$. Define a function $\Phi\colon V\to\mathbb{R}$ by $i^*\Phi=|\widetilde{w}|$ when $C$ is a rational curve with a node, and by $(i^*\Phi)|_{\widetilde{V}_{(\nu)}}=|S_{(\nu)}\cdot \xi_{0 (\nu)}|=|T_{(\nu)}\cdot \xi_{\infty (\nu)}|$ when $C$ consists of two or three irreducible components. Fix positive constants $\delta$ and $R$ such that $R>1$ and $\delta<<1$. By the same argments as in [@KK3] we may assume $W:=\{p\in V\mid \Phi(p)<\delta R\}$ are relatively compact subsets of $V$ by shrinking $V$ and changing the scaling of the coordinates. Denote by $W^*$ the subset $\{p\in V\mid \delta/R<\Phi(p)<\delta R\}$ of $W$ and set $\widetilde{W}:=i^{-1}(W)$ and $\widetilde{W}^*:=i^{-1}(W^*)$, where $i\colon \widetilde{V}\to V$ is as in Example \[ex:standard\_model\] or \[ex:standard\_model\_general\]. The set $W^*$ is a subset of $M:=S\setminus \{p\in V\mid \Phi(p)\leq \delta/R\}$. Define a meromorphic $2$-form $\eta_{\widetilde{W}}$ on $\widetilde{W}$ by $$\eta_{\widetilde{W}}:=\frac{dS\wedge d\xi_0}{S\cdot \xi_0}=-\frac{dT\wedge d\xi_\infty}{T\cdot \xi_\infty}$$ when $n=1$, and by $$\eta_{\widetilde{W}}|_{\widetilde{W}\cap \widetilde{V}_{(\nu)}}:=\frac{dS_{(\nu)}\wedge d\xi_{0 (\nu)}}{S_{(\nu)}\cdot \xi_{0 (\nu)}}=-\frac{dT_{(\nu)}\wedge d\xi_{\infty (\nu)}}{T_{(\nu)}\cdot \xi_{\infty (\nu)}}$$ for each $\nu$ when $n\geq 2$. As it holds that $$F^*\eta_{\widetilde{W}}=-F^*\frac{dT}{T}\wedge F^*\frac{d\xi_\infty}{\xi_\infty} =-\frac{d(t\xi_0)}{t\xi_0}\wedge \frac{dS}{S} =\frac{dS\wedge d\xi_0}{S\cdot \xi_0}=\eta_{\widetilde{W}}$$ when $n=1$ and $$(F_{\nu+1, \nu})^*\eta_{\widetilde{W}}=-(F_{\nu+1, \nu})^*\frac{dT_{(\nu)}}{T_{(\nu)}}\wedge (F_{\nu+1, \nu})^*\frac{d\xi_{\infty (\nu)}}{\xi_{\infty (\nu)}} =-\frac{d(t\xi_{0 (\nu)})}{t\xi_{0 (\nu)}}\wedge \frac{dS_{(\nu)}}{S_{(\nu)}} =\frac{dS_{(\nu)}\wedge d\xi_{0 (\nu)}}{S_{(\nu)}\cdot \xi_{0 (\nu)}}=\eta_{\widetilde{W}}$$ when $n\geq 2$, it follows that there exists a meromorphic $2$-form $\eta_W$ on $W$ with $i^*\eta_W=\eta_{\widetilde{W}}$ in both the cases. Now we have the following: \[prop:mero1form\] $S$ admits a meromorphic $2$-form $\eta$ which has no zero and has poles only along $C$ such that $\eta|_{W}=\eta_W$ holds. Proposition \[prop:mero1form\] is shown by the same argument as in the proof of [@KK3 Proposition 3.1]. Here we use the fact that any leaf of a compact Levi-flat hypersurface of $W^*$ defined by $\{\widetilde{w}=\text{constant}\}$ is dense (Therefore, it follows that $H^0(W, \mathcal{O}_{W})\cong \mathbb{C}$ by the same arguments as in the proof of [@KK3 Lemma 3.2]). \[prop:K3\_pi1\] It holds that $H_1(M, \mathbb{C})=0$. Take a real number $r$ with $\delta /R<r<\delta R$. As it is clear that $W^*$ is homotopic to $H_r:=\Phi^{-1}(r)$, it follows from Lemma \[lem:H\] below that $H_1(W^*, \mathbb{C})\cong \mathbb{C}^2$. By Mayer–Vietoris sequence corresponds to the open covering $\{W, M\}$ of $S$, we obtain an exact sequence $$H_2(S, \mathbb{C})\to H_1(W^*, \mathbb{C})\to H_1(W, \mathbb{C})\oplus H_1(M, \mathbb{C})\to H_1(S, \mathbb{C}).$$ As it is easily observed that the image of the map $H_2(S, \mathbb{C})\to H_1(W^*, \mathbb{C})$ is isomorphic to $\mathbb{C}$, we have that $H_1(M, \mathbb{C})=0$ (Note that $H_1(W, \mathbb{C})\cong \mathbb{C}$ and $H_1(S, \mathbb{C})=0$). \[lem:H\] The Levi-flat manifold $H_r:=\Phi^{-1}(r)$ is $C^\omega$-diffeomorphic to $(\mathbb{C}^*\times{\rm U}(1))/\sim_{r, n}$ for sufficiently small $r$, where $\sim_{r, n}$ is the relation generated by $$(\eta, \lambda)\sim_{r, n} (r^n\cdot \lambda^n\cdot \eta,\ t(N_{C/S})\cdot \lambda)$$ for $(\eta, \lambda)\in \mathbb{C}^*\times{\rm U}(1)$. Let $\{(\widehat{V}_{(\nu)}, \widehat{C}_{(\nu)}, \widehat{V}_{0 (\nu)}^\pm, S_{(\nu)}, T_{(\nu)}, \xi_{0 (\nu)}, \xi_{\infty (\nu)})\}_{\nu=-\infty}^\infty$ be copies of $(\widetilde{V}, \widetilde{C}$, $\widetilde{V}_{0}^\pm$, $S$, $T$, $\xi_{0}$, $\xi_{\infty})$ in Example \[ex:standard\_model\]. Define a biholomorphism $F_{\nu+1, \nu}\colon \widehat{V}_{0 (\nu+1)}^+\to \widehat{V}_{0 (\nu)}^-$ by $(F_{\nu+1, \nu})^*(T_{(\nu)}, \xi_{\infty (\nu)})=(\xi_{0 (\nu)}, S_{(\nu)})$ for each $\nu$. Define $\widehat{V}$ by gluing $\widehat{V}_{(\nu)}$’s by $F_{\nu+1, \nu}$’s. Note that there is the natural covering map $\widehat{V}\to V$, which can be regarded as the universal covering. Consider the map $\widehat{g}_\nu\colon\{(\eta, \lambda)\in \mathbb{C}^*\times {\rm U}(1)\mid 2r^{-\nu+1}<|\eta|<2r^{-\nu-1}\}\to \widehat{V}_{(\nu)}$ defined by $$(\widehat{g}_\nu)^*(S_{(\nu)}, \xi_{0 (\nu)})=\left(r^\nu\cdot \lambda^\nu\cdot \eta,\ \frac{1}{r^{\nu-1}\cdot \lambda^{\nu-1}\cdot \eta}\right).$$ Then, by a simple argument, it follows that $\widehat{g}_\nu$’s glue together to define an embedding $\widehat{g}\colon \mathbb{C}^*\times {\rm U}(1)\to \widehat{V}$. As it follows that $\widehat{g}$ induces an embedding $g\colon (\mathbb{C}^*\times{\rm U}(1))/\sim_{r, n}\to V$ and the image clearly coincides with $H_r$, the lemma follows. Note that, by Lemma \[lem:H\], $H_r$ is diffeomorphic to $T^2_{g_n}$, which is the fiber bundle over ${\rm U}(1)$ whose fiber is $T^2:={\rm U}(1)\times {\rm U}(1)$ and the monodromy is $g_n\colon T^2\ni (p, q)\mapsto (pq^n, q)\in T^2$. From this, one can have that $H_1(H_r, \mathbb{Z})\cong\mathbb{Z}\oplus \mathbb{Z}\oplus (\mathbb{Z}/n\mathbb{Z})$. We emphasize that $H_r$ is not homeomorphic to $T^3:={\rm U}(1)\times {\rm U}(1)\times {\rm U}(1)$, since $H_1(T^3, \mathbb{Z})\cong \mathbb{Z}^{\oplus 3}$. By Proposition \[prop:mero1form\] and \[prop:K3\_pi1\], it seems to be natural to expect that one can construct a K3 surface by holomorphically gluing such models $\{(M_\nu, W^*_\nu, \eta_\nu|_{M_\nu})\}_\nu$ as obtained by the same construction as $(M, W^*, \eta|_M)$ (cf. [@KK3]). \[q:K3\] Does there exist a non-singular K3 surface $X$ with holomorphic $2$-form $\sigma$ which admits an open covering $X=\bigcup_\nu M_\nu$ such that $\sigma|_{M_\nu}=\eta_\nu|_{M_\nu}$ and $M_\nu\cap M_\mu\subset W^*_\nu$ for each $\nu\not=\mu$, where $(M_\nu, W^*_\nu, \eta_\nu|_{M_\nu})$’s are as above? By considering the limit as the tab for gluing $\bigcup_\nu W^*_\nu$ goes to the set of zero measure (i.e. as $R\to 1$ and $\delta\to 0$), the K3 surfaces $X$ should degenerate to a singular K3 surface which is the union of rational surfaces and whose singular part is the union of a cycle of rational curves. We remark that the affirmative answer to Question \[q:K3\] implies the existence of a K3 surface which includes a Levi-flat hypersurface which is diffeomorphic to $T^2_{g_n}$ (and thus is not homeomorphic to $T^3$). [99]{} <span style="font-variant:small-caps;">V. I. Arnol’d</span>, Bifurcations of invariant manifolds of differential equations and normal forms in neighborhoods of elliptic curves, Funkcional Anal. i Prilozen., 10-4 (1976), 1–12 (English translation : Functional Anal. Appl., 10-4 (1977), 249–257). <span style="font-variant:small-caps;">C. Camacho, H. 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J, [**66**]{}, 3 (2017), 845–876. <span style="font-variant:small-caps;">T. Koike</span>, Complex K3 surfaces containing Levi-flat hypersurfaces, arXiv:1703.03663. <span style="font-variant:small-caps;">H. B. Laufer</span>, Normal two-dimensional singularities, Ann. Math. Stud., [*71*]{}. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo (1971). <span style="font-variant:small-caps;">T. Ohsawa</span>, Vanishing theorems on complete Kähler manifolds, Publ. Res. Inst. Math. Sci. [**20**]{} (1984), no. 1, 21–38. , Iterations of analytic functions, Ann. of Math., [**43**]{} (1942), 607–612. , Every Stein subvariety admits a Stein neighborhood, Invent. Math., [**38**]{} (1976), pp. 89–100. , Complex structures on $S^3\times S^3$, Tohoku Math. J. (2), [**36**]{}, 3 (1984), 351–376. <span style="font-variant:small-caps;">T. Ueda</span>, On the neighborhood of a compact complex curve with topologically trivial normal bundle, Math. 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--- abstract: 'Let $A^2$ be the affine plane over a field $K$ of characteristic $0$. Birational morphisms of $A^2$ are mappings $A^2 \to A^2$ given by polynomial mappings $\varphi$ of the polynomial algebra $K[x,y]$ such that for the quotient fields, one has $K(\varphi(x), \varphi(y)) = K(x,y)$. Polynomial automorphisms are obvious examples of such mappings. Another obvious example is the mapping $\tau_x$ given by $x \to x, ~y \to xy$. For a while, it was an open question whether every birational morphism is a product of polynomial automorphisms and copies of $\tau_x$. This question was answered in the negative by P. Russell (in an informal communication). In this paper, we give a simple combinatorial solution of the same problem. More importantly, our method yields an algorithm for deciding whether a given birational morphism can be factored that way.' --- [**Birational morphisms of the plane**]{} [**Vladimir Shpilrain**]{} and [**Jie-Tai Yu**]{}$^{\ast}$ [**1. Introduction** ]{} Let $ K[x, y]$ be the polynomial algebra in two variables over a field $K$ of characteristic $0$, and $A^2$ the affine plane over $K$. Birational morphisms of $A^2$ are mappings $A^2 \to A^2$ given by polynomial mappings $\varphi$ of the algebra $K[x,y]$ such that for the quotient fields, one has $K(\varphi(x), \varphi(y)) = K(x,y)$. Polynomial automorphisms are obvious examples of such mappings. Another obvious example is the mapping $\tau_x$ given by $x \to x, ~y \to xy$. Call either of those mappings a [*simple affine contraction*]{}. For a while, it was an open question whether every birational morphism is a product of simple affine contractions. This question was answered in the negative by Russell (in an informal communication). The paper [@Daigle] by Daigle contains an elaboration of Russell’s methods. In this paper, we use an altogether different method of “peak reduction" to easily establish the same result. More importantly, our method gives an algorithm for deciding whether a given birational morphism can be factored that way: [**Theorem 1.1.**]{} Let $K$ be an algebraically closed field. Then there is an algorithm for deciding whether a given birational morphism of $A^2$ over $K$ is a product of simple affine contractions. Here we assume that we are able to perform calculations in the ground field $K$, which basically means that, given (arithmetic expressions for) two elements of $K$, we can decide whether or not they are equal. Our method can also be applied to fields that are not algebraically closed to prove that a particular birational morphism is [*not*]{} a product of simple affine contractions. [**Proposition 1.2.**]{} Let $K$ be any field of characteristic $0$, and $\phi : x \to u, ~y \to v$ a birational morphism of $A^2$ over $K$. If $\phi$ is a product of simple affine contractions, then the sum of the degrees of $u$ and $v$ can be decreased by a single generalized simple affine contraction. Generalized simple affine contractions are morphisms of the types (ET1)–(ET3), defined in Section 2. For example, the following birational morphism of $A^2$ over ${\bf R}$ is not a product of simple affine contractions (see Example 2 in the end of Section 2): $x \to x, ~y \to yx^2 + y$. To find a birational morphism of $A^2$ over ${\bf C}$ that is not a product of simple affine contractions, is somewhat more difficult. The following birational morphism is a simplification of an example due to Cassou-Nogues and Russell [@CN] : $x \to x^4 y^2 -2x^3y+x^2+xy, \\ y \to x^6y^3 -3x^5y^2 +3x^4 y + 2x^3y^2 - x^3 - 3x^2y +x +y$. We explain this example in Section 2, after the proof of Theorem 1.\ [**2. Peak reduction** ]{} The “peak reduction" method is a simple but rather powerful combinatorial technique with applications in many different areas of mathematics as well as theoretical computer science. It was introduced by Whitehead, who used it to solve an important algorithmic problem concerning automorphisms of a free group (see e.g. [@Lyndonbook]). Since then, this method has been used to solve various problems in group theory, topology, combinatorics, and probably in some other areas as well. In general, this method is used to find some kind of canonical form of a given object $P$ under the action of a given group (or a semigroup) $T$ of transformations. The idea behind the method is rather simple: one chooses the [*complexity*]{} of an object $P$ one way or another, and declares a canonical form of $P$ an object $P'$ whose complexity is minimal among all objects $t(P), ~t \in T$. To actually find a canonical form, or a “canonical model", $P'$ of a given object $P$, one tries to arrange a sequence of sufficiently simple transformations so that the complexity of an object decreases [*at every step*]{}. To prove that such an arrangement is possible, one uses “peak reduction"; that means, if in some sequence of simple transformations the complexity goes up (or remains unchanged) before eventually going down, then there must be a pair of [*consecutive*]{} simple transformations in the sequence (a “peak") such that one of them increases the complexity (or leaves it unchanged), and then the other one decreases it. Then one tries to prove that such a peak can always be reduced. In the commutative algebra context, objects are polynomials; their complexity is their degree; the group of transformations is the group of polynomial automorphisms; simple transformations are elementary and linear automorphisms. (An elementary automorphism is a one that changes just one variable.) We have used this technique in our earlier paper [@ShYu] to contribute toward a classification of two-variable polynomials having classified, up to an automorphism, polynomials of the form $ax^n + by^m + \sum_{im+jn \le mn} c_{ij} x^i y^j$ (i.e., polynomials whose Newton polygon is either a triangle or a line segment). Later, Wightwick [@Wightwick] used the idea of peak reduction in combination with splice diagrams technique due to Eisenbud and Neumann [@EN] to classify [*all*]{} two-variable polynomials over ${\bf C}$ up to an automorphism. More details and results can be found in our survey [@ShYusurvey]. Here we use this technique to prove our statements. [**Proof of Theorem 1.1 and Proposition 1.2.**]{} Consider the direct product $K[x,y] \times K[x,y]$ of two copies of the polynomial algebra $K[x,y]$, and introduce the following elementary transformations (ET) that can be applied to elements of this direct product: [**(ET1)**]{} $(u, ~v) \longrightarrow (\frac{u+a}{c\cdot v+b}, ~v)$ for arbitrary $a, b, c \in K$, with the denominator not 0. [**(ET1$'$)**]{} $(u, ~v) \longrightarrow (u, ~\frac{v+a}{c\cdot u+b})$ for arbitrary $a, b, c \in K$, with the denominator not 0. [**(ET2)**]{} $(u, ~v) \longrightarrow (u + q(v), ~v)$ for an arbitrary polynomial $q(v)$. [**(ET2$'$)**]{} $(u, ~v) \longrightarrow (u, ~v+ q(u))$ for an arbitrary polynomial $q(u)$. [**(ET3)**]{} $(u, ~v) \longrightarrow (v, ~u)$. Transformations (ET1) or (ET1$'$) are only applied if the corresponding ratio is a polynomial. It is clear that a birational morphism $x \to u, ~y \to v$ is a product of polynomial automorphisms and mappings of the form $x \to x, ~y \to x\cdot y$ if and only if the pair $(u, ~v)$ can be taken to $(x, ~y)$ by a sequence of elementary transformations (ET1)–(ET3). We are now going to use “peak reduction" to show that, if the sum of the degrees of a pair of polynomials can be reduced by a sequence of elementary transformations (ET1)–(ET3), then it can be reduced by a single elementary transformation. Obviously, (ET3) cannot change the sum of the degrees; also, a single (ET1) or (ET1$'$) can only decrease the sum of the degrees, unless $c=0$. Thus, up to a symmetry, there are only the following possibilities for a “peak": [**(1)**]{} (ET3) followed by one of the (ET1), (ET1$'$), (ET2), or (ET2$'$). [**(2)**]{} (ET2) followed by (ET2$'$). [**(3)**]{} (ET2) followed by (ET1) or (ET1$'$). The first possibility is trivial: for example, (ET3) followed by (ET1) is the same as (ET1$'$) followed by (ET3). Therefore, if the sum of the degrees could be reduced by (ET3) followed by (ET1), then it could also be reduced by (ET1$'$) followed by (ET3), hence by just (ET1$'$). The second possibility was handled in [@ShYu]; it turns out to be easy, too. In (3), if (ET2) is followed by (ET1$'$), we get the pair $(u+q(v),~\frac{v+a}{c\cdot [u+q(v)]+b})$. Since we are under the assumption that (ET2) does not decrease the sum of the degrees, we have $deg(u+q(v)) \ge deg(u)$. Therefore, we have two possibilities: [**(i)**]{} $deg(u+q(v)) > deg(u)$. In this case, $deg(u+q(v)) = deg(q(v))= deg(q) \cdot deg(v)$, and we conclude that the polynomial $q$ must be linear; otherwise, $v+a$ cannot be divisible by $c\cdot [u+q(v)]+b$. Moreover, $q$ has to be a constant since otherwise, the ratio $~\frac{v+a}{c\cdot [u+q(v)]+b}$ is a constant, but no pair of the form $(u, ~c), ~c \in K$, can be taken to $(x, ~y)$ by a sequence of (ET1)–(ET3). But if $q(v)$ is a constant, then (ET1$'$) cannot change the sum of the degrees. [**(ii)**]{} $deg(u+q(v)) = deg(u)$. In this case, $deg(u) \ge deg(q(v))$, so again, the polynomial $q$ must be linear; otherwise, $v+a$ cannot be divisible by $c\cdot [u+q(v)]+b$. As in (i) above, we conclude that $q$ has to be a constant, in which case (ET1$'$) cannot change the sum of the degrees. Thus, we are left with the crucial possibility where (ET2) is followed by (ET1) with $c \ne 0$. The result of applying this pair of elementary transformations to $(u, v)$ is $(\frac{u+q(v)+a}{c\cdot v+b}, ~v)$. The polynomial $q(v)$ can be written as $r(c\cdot v+b)$ for some other polynomial $r$. Then $$(\frac{u+q(v)+a}{c\cdot v+b}, ~v) = (\frac{u+r(c\cdot v+b)+a}{c\cdot v+b}, ~v) = (\frac{u+r_1(c\cdot v+b)+a_1}{c\cdot v+b}, ~v),$$ where the polynomial $r_1$ has zero constant term. Since $r_1(c\cdot v+b)$ is divisible by $c\cdot v+b$, we get $$(\frac{u+r_1(c\cdot v+b)+a_1}{c\cdot v+b}, ~v) = (\frac{u+a_1}{c\cdot v+b}+r_2(c\cdot v+b), ~v).$$ This means, in particular, that $u+a_1$ is divisible by $c\cdot v+b$. But then a single (ET1) would reduce the sum of the degrees of the pair $(u, v)$. This completes the “peak reduction". To get now an algorithm claimed in the statement of Theorem 1, we first have to show that one can effectively determine whether a single elementary transformation can reduce the sum of the degrees of a given pair of polynomials. For (ET2) or (ET2$'$), this is well known (see e.g. [@PMCohn Theorem 6.8.5]). For (ET1) or (ET1$'$), the procedure is quite straightforward. Suppose we want to find out whether, say, $u(x,y)+a$ is divisible by $v(x,y)+b$ for some $a, b \in K$ (we can clearly assume that $c=1$). We apply the usual “long division" algorithm with respect to some fixed term ordering (see e.g. [@AL]). In the end, the divisibility condition translates into a system of polynomial equations over $K$ in a single variable $b$, complemented by an equation $a=p(b)$ for some polynomial $p$. The solvability of such a system over an algebraically closed field can be decided by using Gröbner bases technique (see e.g. [@AL]). (Of course, an actual solution may not be found in general.) If the system has no solutions, then a single elementary transformation cannot reduce the sum of the degrees of a given pair of polynomials, and we conclude that our birational morphism is not a product of simple affine contractions. If the system has a solution, then we apply the corresponding elementary transformation, keeping our parameters $a, b$, etc. as variables, because we do not have explicit values for them. Then, for the new pair of polynomials, we do the same thing: we check if a single elementary transformation can reduce the sum of the degrees. This will yield another system of polynomial equations over $K$, that expands the previous system by introducing new variables and new equations. Again, it can be decided whether or not this system has a solution. This procedure will obviously terminate in a number of steps not exceeding the sum of the degrees of the original pair of polynomials. This completes the proof. $\Box$ [**Example 1.**]{} The birational morphism $x \to u(x,y) = x^4 y^2 -2x^3y+x^2+xy, \\ y \to v(x,y) = x^6y^3 -3x^5y^2 +3x^4 y + 2x^3y^2 - x^3 - 3x^2y +x +y$, mentioned in the Introduction, is constructed as follows. Start with the birational morphism $(xy+1, ~x^2y+x)$,  apply $x \to x+y, ~y \to y$ to get $((x+y)y+1, ~(x+y)^2y+(x+y))$. Then apply $x \to \frac{1}{x}, ~y \to -x+x^2y$ (this is a bijective morphism of $K(x,y)$) to get the above example. A single transformation (ET2$'$) cannot reduce the sum of the degrees of the pair $(u, ~v)$ because the degree of $v$ is not divisible by the degree of $u$ (cf. [@ShYu]). Neither can a single (ET1$'$) : a straightforward check shows that $v(x,y)+a$ is not divisible by $c\cdot u(x,y)+b$ for any $a, b, c \in K$. Therefore, by Proposition 1.2, this birational morphism is not a product of simple affine contractions. [**Example 2.**]{} The birational morphism $x \to x, ~y \to yx^2 + y$ is not a product of simple affine contractions over ${\bf R}$, by Proposition 1.2. However, over ${\bf C}$, a single (ET1$'$) can reduce the degree of $yx^2 + y$ since $yx^2 + y = y(x^2 + 1)=y(x+i)(x-i)$. It is now easy to see that over ${\bf C}$, this birational morphism is a product of simple affine contractions.\ [**Acknowledgement**]{} We are grateful to Andrew Campbell for helpful comments. 11 pt [ABC]{} W. Adams and P. Loustaunau, [*An introduction to Gröbner bases*]{}. American Mathematical Society, Providence, 1994. P. Cassou-Nogues and P. Russell, [*On some birational endomorphisms of the affine plane*]{}, preprint. P. M. Cohn, [*Free Rings and Their Relations. 2nd Ed.*]{} Academic Press, London, 1985. D. Daigle, [*Birational endomorphisms of the affine plane*]{}, J. Math. Kyoto Univ. [**31**]{} (1991), 329–358. D. Eisenbud and W. D. Neumann, [*Three-dimensional link theory and invariants of plane curve singularities.*]{} Ann. Math. Stud. [**110**]{}, Princeton. Princeton Univ. Press (1985). R. Lyndon, P. Schupp, [*Combinatorial Group Theory*]{}, (Reprint of the 1977 edition). In [*Classics in Mathematics*]{}, Springer-Verlag, Berlin (2001). P. Wightwick, [*Equivalence of polynomials under automorphisms*]{}, J. Pure Appl. Algebra [**157**]{} (2001), 341–367. V. Shpilrain and J.-T. Yu, [*Embeddings of curves in the plane*]{}, J. Algebra [**217**]{} (1999), 668–678. V. Shpilrain and J.-T. Yu, [*Peak reduction technique in commutative algebra: a survey*]{}, in: Combinatorial and computational algebra (Hong Kong, 1999), 237–247, Contemp. Math. [**264**]{}, Amer. Math. Soc., Providence, RI, 2000.\ Department of Mathematics, The City College of New York, New York, NY 10031 [*e-mail address*]{}:  shpil@groups.sci.ccny.cuny.edu [*http://www.sci.ccny.cuny.edu/\~shpil/*]{}\ Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong [*e-mail address*]{}:  yujt@hkusua.hku.hk [*http://hkumath.hku.hk/\~jtyu*]{}
--- address: 'Departamento de Física Teórica C-XI and Instituto de Física Teórica C-XVI, Universidad Autónoma de Madrid, Cantoblanco, E-28049 Madrid, Spain' author: - 'E. Arganda. M.J. Herrero' title: 'Lepton flavour violation in constrained MSSM-seesaw models' --- FTUAM/08-18\ IFT-UAM/CSIC-08-58\ LFV within SUSY-seesaw models {#intro} ============================= The current knowlegde of neutrino mass differences and mixing angles clearly indicates that lepton flavour number is not a conserved quantum number in Nature. However, the lepton flavour violation (LFV) has so far been observed only in the neutrino sector. One challenging task for the present and future experiments will then be to test if there is or there is not LFV in the charged lepton sector as well. Here we focus in the Minimal Supersymmetric Standard Model (MSSM) enlarged by three right-handed neutrinos and their SUSY partners where potentially observable LFV effects in the charged lepton sector are expected to occur. We further assume a seesaw mechanism for neutrino mass generation and use, in particular, the parameterisation proposed in [@Casas:2001sr] where the solution to the seesaw equation is written as $m_D =\,Y_\nu\,v_2 =\,\sqrt {m_N^{\rm diag}} R \sqrt {m_\nu^{\rm diag}}U^{\dagger}_{\rm MNS}$. Here, $R$ is defined by $\theta_i$ ($i=1,2,3$); $v_{1(2)}= \,v\,\cos (\sin) \beta$, $v=174$ GeV; $m_{\nu}^\mathrm{diag}=\, \mathrm{diag}\,(m_{\nu_1},m_{\nu_2},m_{\nu_3})$ denotes the three light neutrino masses, and $m_N^\mathrm{diag}\,=\, \mathrm{diag}\,(m_{N_1},m_{N_2},m_{N_3})$ the three heavy ones. $U_{\rm MNS}$ is given by the three (light) neutrino mixing angles $\theta_{12},\theta_{23}$ and $\theta_{13}$, and three phases, $\delta, \phi_1$ and $\phi_2$. With this parameterisation is easy to accommodate the neutrino data, while leaving room for extra neutrino mixings (from the right-handed sector). It further allows for large Yukawa couplings $Y_\nu \sim \mathcal{O}(1)$ by choosing large entries in $m^{\rm diag}_N$ and/or $\theta_i$. The predictions in the following are for two different constrained MSSM-seesaw scenarios, with universal and non-universal Higgs soft masses and with respective parameters (in addition to the previous neutrino sector parameters): 1) CMSSM-seesaw: $M_0$, $M_{1/2}$, $A_0$ $\tan \beta$, and sign($\mu$), and 2) NUHM-seesaw: $M_0$, $M_{1/2}$, $A_0$ $\tan \beta$, sign($\mu$), $M_{H_1}=M_0(1+\delta_1)^{1/2}$ and $M_{H_2}=M_0(1+\delta_2)^{1/2}$. All the predictions presented here include the full set of SUSY one-loop contributing diagrams and we do not use the Leading Logarithmic (LLog) nor the mass insertion approximations. The hadronisation of quark bilinears is performed within the chiral framework, using $\chi$PT and R$\chi$T. This is a very short summary of several publications [@Arganda:2005ji; @Antusch:2006vw; @Arganda:2007jw; @Arganda:2008jj] to which we refer the reader for more details. Results and Discussion {#results} ====================== We focus on the dependence on the most relevant parameters which, for the case of hierarchical (degenerate) heavy neutrinos, are: the neutrino mass $m_{N_3}$ ($m_N$), $\tan\beta$, $\theta_1$ and $\theta_2$. We also study the sensitivity of the BRs to $\theta_{13}$. The other input seesaw parameters $m_{N_1}$, $m_{N_2}$ and $\theta_3$, play a secondary role since the BRs do not strongly depend on them. The light neutrino parameters are fixed to: $m_{\nu_2}^2= \Delta m_{\rm sol}^2 + m_{\nu_1}^2$, $m_{\nu_3}^2= \Delta m_{\rm atm}^2 + m_{\nu_1}^2$, $\Delta m^2_{\rm sol}=8\times 10^{-5}\,{\rm eV}^2$, $\Delta m^2_{\rm atm}=2.5\times 10^{-3}\,{\rm eV}^2$, $m_{{\nu}_1}=10^{-3}\,{\rm eV}$, $\theta_{12}=30^\circ$, $\theta_{23}=45^\circ$, $\theta_{13}\lesssim 10^\circ$ and $\delta=\phi_1=\phi_2=0$. \[figurita1\] The results for the CMSSM-seesaw scenario are collected in Figs. 1 through 5. In Fig. 1, we display the predictions of BR$(\tau \to \mu \gamma)$ and CR($\mu -e$, Ti) as a function of the heaviest neutrino mass $m_{N_3}$ for the various SPS points, and for the particular choice $\theta_i=0$ ($i=1,2,3$) and $\theta_{13}=5^\circ$. We have also considered the case of degenerate heavy neutrino spectra (not shown here). In both scenarios for degenerate and hierarchical heavy neutrinos, we find a strong dependence on the the heavy neutrino masses, with the expected behaviour $~|m_N \log m_N|^2$ of the LLog approximation, except for SPS 5 point, which fails by a factor of $\sim 10^4$. The rates for the various SPS points exhibit the following hierarchy, BR$_{4}$ $>$ BR$_{\rm 1b}$ $\gtrsim$ BR$_{\rm 1a}$ $> $ BR$_{3}$ $\gtrsim$ BR$_{2}$ $>$ BR$_{5}$. This behaviour can be understood in terms of the growth of the BRs with $\tan \beta$, and from the different mass spectra associated with each point. Most of the studied processes reach their experimental limit at $m_{N_3} \in [10^{13}, 10^{15}]$ which corresponds to $Y_\nu^{33,32} \sim 0.1 - 1$. At present, the most restrictive one is $\mu \to e \gamma$ (which sets bounds for SPS 1a of $m_{N_3} < 10^{13}-10^{14}$ GeV), although $\mu - e$ conversion will be the best one in future, with a sensitivity to $m_{N_3} > 10^{12}$ GeV. [ccc]{} & &\ \ & & \[figurita2\] Fig. 2 shows the behaviour of the six considered LFV $\tau$ and $\mu$ decays, for SPS 4 point, as a function of $|\theta_1|$, for various values of arg$\theta_1$. We see clearly that the BRs for $0 < |\theta_1| < \pi$ and $0 < {\rm arg} \theta_1 < \pi/2$ can increase up to a factor $10^2 - 10^4$ with respect to $\theta_i = 0$. Similar results have been found for $\theta_2$, while BRs are nearly constant with $\theta_3$ in the case of hierarchical neutrinos. The behaviour of CR($\mu -e$, Ti) with $\theta_i$ is very similar to that of BR($\mu \to e \gamma$) and BR($\mu \to 3e$). For instance, Fig. 3 shows the dependence of CR($\mu -e$, Ti) with $\theta_2$, and illustrates that for large $\theta_2$, rates up to a factor $\sim 10^4$ larger than in the $\theta_i=0$ case can be obtained. \[figurita3\] In Fig. 4 we show the dependence of $\mu \to e \gamma$, $\mu \to 3e$ and $\mu-e$ conversion on the light neutrino mixing angle $\theta_{13}$. These figures clearly manifest the very strong sensitivity of their rates to the $\theta_{13}$ mixing angle for hierarchical heavy neutrinos. Indeed, varying $\theta_{13}$ from 0 to $10^\circ$ leads to an increase in the rates by as much as five orders of magnitude. -- -- -- -- -- -- \[figurita4\] \[figurita5\] On the other hand, since $\mu \to e\,\gamma$ is very sensitive to $\theta_{13}$, but BR($\tau \to \mu\,\gamma$) is clearly not, and since both BRs display the same approximate behaviour with $m_{N_3}$ and $\tan \beta$, one can study the impact that a potential future measurement of $\theta_{13}$ and these two rates can have on the knowledge of the otherwise unreacheable heavy neutrino parameters. The correlation of these two observables as a function of $m_{N_3}$, is shown in Fig. 5 for SPS 1a. Comparing these predictions for the shaded areas along the expected diagonal “corridor”, with the allowed experimental region, allows to conclude about the impact of a $\theta_{13}$ measurement on the allowed/excluded $m_{N_3}$ values. The most important conclusion from Fig. 5 is that for SPS 1a, and for the parameter space defined in the caption, an hypothetical $\theta_{13}$ measurement larger than $1^\circ$, together with the present experimental bound on the BR($\mu \to e\,\gamma$), will have the impact of excluding values of $m_{N_3} \gtrsim 10^{14}$ GeV. Moreover, with the planned MEG sensitivity, the same $\theta_{13}$ measurement could further exclude $m_{N_3} \gtrsim 3\times 10^{12}$ GeV. -- -- -- -- \[figurita6\] The numerical results for the NUHM-seesaw scenario as a function of $M_0=M_{1/2}=M_{\rm SUSY}$ are collected in Figs. 6 and 7. The behaviour of the predicted $m_{H^0}$ as a function of $M_{\rm SUSY}$ is shown in Fig. 6 (left panel). The most interesting solutions with important phenomenological implications are found for negative $\delta_1$ and positive $\delta_2$. Notice that, for all the explored $\delta_{1,2}$ values, we find a value of $m_{H^0}$ that is significantly smaller than in the universal case ($\delta_{1,2}=0$). In Fig. 6 (right panel) the various contributions from the $\gamma$-, $Z$-, Higgs mediated penguins and box diagrams as a function of $M_{\rm SUSY}$ are shown. Here, we choose $\delta_1 = -1.8$ and $\delta_2 = 0$. We observe a very distinct behaviour with $M_{\rm SUSY}$ of the Higgs-mediated contributions compared to those of the CMSSM case. In fact, the Higgs-mediated contribution can equal, or even exceed that of the photon, dominating the total conversion rate in the large $M_0 = M_{1/2}$ region. These larger Higgs contributions are the consequence of their exclusive SUSY non-decoupling behaviour for large $M_{\rm SUSY}$, and of the lighter Higgs boson mass values encountered in this region, as previously illustrated in Fig. 6. \[figurita7\] In Fig. 7 we display the predicted $\mu-e$ conversion rates for other nuclei, concretely Al, Ti, Sr, Sb, Au and Pb, as a function of $M_{\rm SUSY}$. We clearly see that CR($\mu -e$, Sb) $>$ CR($\mu -e$, Sr) $>$ CR($\mu -e$, Ti) $>$ CR($\mu -e$, Au) $>$ CR($\mu -e$, Pb) $>$ CR($\mu -e$, Al). The most important conclusion from Fig. 7 is that we have found predictions for Gold nuclei which, for the input parameters in this plot, are above its present experimental bound throughout the explored $M_{\rm SUSY}$ interval. Finally, althought not shown here for shortness, we have also found an interesting loss of correlation between the predicted CR($\mu -e$, Ti) and BR($\mu \to e \gamma$) in the NUHM-seesaw scenario compared to the universal case where these are known to be strongly correlated. This loss of correlation occurs when the Higgs-contributions dominate the photon-contributions and could be tested if the announced future sensitivities in these quantities are reached. -- -- -- -- \[figurita8\] The corresponding predictions for $\theta_2 = 2.9 e^{i\pi/4}$ of the nine LFV semileptonic $\tau$ decays studied in this work as a function of $M_{\rm SUSY}$ are shown in Fig. 8. In this case, we work with $\delta_1 = -2.4$ and $\delta_2 = 0.2$, that drive us to Higgs boson masses around 150 GeV even for heavy SUSY spectra. In this Fig. 8 we can see that, the choice of $\theta_2$ increase all the rates about two orders of magnitude respect to the case $\theta_i = 0$, not shown here for brevitiy. BR$(\tau \to \mu \pi^+ \pi^-)$ and BR$(\tau \to \mu \rho)$ get the largest rates and, indeed, the predictions of these two latter channels reach their present experimental sensitivities at the low $M_{\rm SUSY}$ region, below 200 GeV and 250 GeV respectively, for this particular choice of input parameters. In Fig. 9 we plot finally the predictions for BR$(\tau \to \mu K^+K^-)$ and BR$(\tau \to \mu \eta)$ as a function of one the most relevant parameters for these Higgs-mediated processes which is the corresponding Higgs boson mass. -- -- -- -- \[figurita9\] Firstly, we see that the approximate (see the approximate formulae in [@Arganda:2008jj]) and exact results of the Higgs contribution agree within a factor of two for both channels, but the agreement of the full result with respect to the Higgs contribution is clearly worse in the case of $\tau \to \mu K^+ K^-$ than in $\tau \to \mu \eta$. In the latter, the agreement is quite good because the $Z$-mediated contribution is negligible, and this holds for all $M_{\rm SUSY}$ values in the studied interval, 250 GeV $<M_{\rm SUSY}<$ 750 GeV . In the first, it is only for large $M_{\rm SUSY}$ that the $H$-mediated contribution competes with the $\gamma$-mediated one and the Higgs rates approach the total rates. For instance, the predictions for BR($\tau \to \mu K^+K^-$) shows that for $M_{\rm SUSY}=750$ GeV and $m_{H^0}=160$ GeV the total rate is about a factor 2 above the Higgs rate, but for $m_{H^0}=240$ GeV it is already more than a factor 5 above. In this figure we have also explored larger values of $m_{N_3}$ and $\tan \beta$, by using in those cases the approximate formula, and in order to conclude about the values that predict rates comparable with the present experimental sensitivity. We can conclude then that, at present, it is certainly $\tau \to \mu \eta$ the most competitive LFV semileptonic tau decay channel. The paremeter values that provide rates being comparable to the present sensitivities in this channel are $\tan \beta = 60$ and $m_{N_3}= 10 ^{15}$ GeV which correspond to $|\delta_{32}| \simeq 2$. Interestingly, the most competitive channels to explore simultaneously LFV $\tau-\mu$ transitions and the Higgs sector are $\tau \to \mu \eta$, $\tau \to \mu \eta'$ and also $\tau \to \mu K^+K^-$. Otherwise, the golden channels to tackle the Higgs sector are undoubtly $\tau \to \mu \eta$ and $\tau \to \mu \eta^\prime$. On the other hand, the rest of the studied semileptonic channels, $\tau \to \mu \pi^+\pi^-$, etc., will not provide additional information on LFV with respect to that provided by $\tau \to \mu \gamma$. In conclusion, we believe that a joint measurement of the LFV branching ratios, the $\mu - e$ conversion rates, $\theta_{13}$ and the SUSY spectrum will be a powerful tool for shedding some light on the otherwise unreachable heavy neutrino parameters. Futhermore, in the case of a NUHM scenario, it may also provide interesting information on the Higgs sector. It is clear from this study that the connection between LFV and neutrino physics will play a relevant role for the searches of new physics beyond the SM. [*We aknowledge*]{} Ana M. Teixeira, Stefan Antusch and Jorge Portolés for their participation in our works. E. Arganda thanks the organizors for his invitation to this fruitful conference. [999]{} J. A. Casas and A. Ibarra, Nucl. Phys. B 618 (2001) 171 \[arXiv:hep-ph/0103065\]. E. Arganda and M. J. Herrero, Phys. Rev.  D [**73**]{} (2006) 055003 \[arXiv:hep-ph/0510405\]. S. Antusch, E. Arganda, M. J. Herrero and A. M. Teixeira, JHEP [**0611**]{} (2006) 090 \[arXiv:hep-ph/0607263\]. E. Arganda, M. J. Herrero and A. M. Teixeira, JHEP [**0710**]{} (2007) 104 \[arXiv:0707.2955 \[hep-ph\]\]. E. Arganda, M. J. Herrero and J. Portoles, JHEP [**0806**]{} (2008) 079 \[arXiv:0803.2039 \[hep-ph\]\].
--- author: - 'Jothi Prasanna Shanmuga Sundaram, , Wan Du, , Zhiwei Zhao, ' bibliography: - 'ref.bib' title: 'A Survey on LoRa Networking: Research Problems, Current Solutions and Open Issues' --- [^1] [^1]: Wan Du and Jothi Prasanna Shanmuga Sundaram are with the Department of Computer Science and Engineering, the University of California, Merced. E-mail: {wdu3, jshanmugasundaram}@ucmerced.edu, Zhiwei Zhao is with the School of Computer Science and Engineering, University of Electronic Science and Technology of China. Email: zzw@uestc.edu.cn. Wan Du is the first corresponding author of this article and Zhiwei Zhao is the second corresponding author.
--- abstract: 'The Weak Gravity Conjecture (WGC) asserts a powerful consistency condition on gauge theories coupled to gravity, and it is widely believed that its proof will shed light on the quantum origin of gravitational interactions. Holography, and in particular the AdS/CFT correspondence, is a well-suited tool by means of which it is possible to explore the world of gravity. From the holographic perspective gravity is an emergent statistical phenomenon, and the laws of gravitation can be recast as the laws of thermodynamics. It is interesting to ask whether the WGC can be formulated in terms of the AdS/CFT correspondence. A positive answer is given: The WGC in the bulk is linked to the thermalization properties of the CFT living on the boundary. The latter is related to the Sachdev-Ye-Kitaev model of strange metals. In the thermodynamic picture, the validity of the WGC is verified.' --- \ [**Alfredo Urbano$^{a}$**]{}\ [*$^a$ INFN, sezione di Trieste, SISSA, via Bonomea 265, 34136 Trieste, Italy.*]{}\ Introduction {#sec:intro} ============ The Weak Gravity Conjecture (WGC) [@ArkaniHamed:2006dz] imposes a powerful consistency condition on gauge theories coupled to gravity. In its original formulation (sometimes dubbed “electric” WGC), it can be stated as follows. \  [*A $U(1)$ gauge symmetry coupled consistently to gravity requires the existence of at least one state with charge larger than its mass $\mu$ in Planck units*]{} $$\label{eq:WGC} {\fcolorbox{gray}{Gray}{~$\displaystyle qe > \frac{\mu}{M_{\rm P}} $~}}$$ where $e$ is the $U(1)$ gauge coupling and $q$ the charge of the state in units of $e$.[^1] A rigorous mathematical proof of the WGC so far escaped the net. However, there are many pieces of evidence that make us think it is correct. - [*Trouble for remnants*]{}. Consider a charged black hole with charge $Q$ and mass $M$ decaying into particles with mass $\mu$ and charge $q e$. Conservation of charge gives the number of particles in the final state, $N = Q/qe$. Conservation of energy implies $\mu N= \mu(Q/qe) < M$ from which it follows that a charged black hole decays if there exist lighter states with higher charge-to-mass ratio. For an extremal black hole with $Q = M/M_{\rm P}$, this argument implies $\mu/M_{\rm P} < qe$. Said equivalently, if the WGC fails extremal black holes are exactly stable remnants, a situation that seems to imply some pathologies [@Giddings:1992hh; @Susskind:1995da]. - [*No global symmetries can exist in a theory of quantum gravity*]{}. The WGC poses an obstruction to the limit of vanishing gauge couplings. This is in agreement with the conjecture according to which there are no global continuous symmetries in models of quantum gravity [@Banks:2010zn]. If the WGC fails, it will be possible to emulate a global symmetry by taking the limit $e \to 0$. - [*Infrared consistency*]{}. A violation of the WGC could induce some pathologies in the infrared dynamics describing photons and gravitons [@Cheung:2014ega]. - [*Absence of explicit counterexamples*]{}. The WGC was verified in a handful of explicit string theory constructions, in particular in simple heterotic setups [@ArkaniHamed:2006dz] and in the framework of F-theory [@Lee:2018urn]. We remark that the WGC says nothing about the spin of the state in eq. (\[eq:WGC\]). Furthermore, for the state satisfying eq. (\[eq:WGC\]) we can say that “gravity is the weakest force” because the Coulomb-like repulsion (proportional to $q^2e^2$) overcomes the gravitational attraction (proportional to $\mu^2/M_{\rm P}^2$). The WGC plays a central role in the so-called “Swampland program” that is the possibility to distinguish the Landscape (that is defined by the set of consistent low-energy effective field theories that are compatible with string theory) from the Swampland (that is defined by the set of consistent-looking low-energy effective field theories which are actually not compatible with string theory) using infrared data [@Vafa:2005ui]. In this respect the WGC, if true, would provide, at least conceptually, a powerful discriminatory tool: Remarkably, by simply inspecting the low-energy spectrum, it might be possible to catalogue a given effective field theory in one or the other set. There are many other interesting conjectures – related or complementary to the original WGC – that aim at populating the swampland with effective field theories that are consistent-looking at low-energy but pathological as a consequence of their violation. Needless to say, increasingly strong constraints on low-energy effective field theories require more elaborated conjectures that are, consequently, more difficult to justify on the basis of simple black hole arguments. Of particular relevance are some proposed extensions of the WGC (motivated by the fact that the WGC is not stable under dimensional reduction) which imply that there must be not just one charged particle of appropriate mass but a whole tower of increasingly heavy charged states fulfilling the WGC, $\mu_i < q_i e M_{\rm P}$ [@Heidenreich:2015nta; @Heidenreich:2016aqi; @Andriolo:2018lvp]. This (much) stronger version of the WGC turns out to be closely connected to the Swampland Distance Conjecture [@Ooguri:2006in] and the Completeness Conjecture [@Polchinski:2003bq]. Besides theoretical implications, there are also intriguing phenomenological consequences. In [@Cheung:2014vva], connections between the WGC and naturalness in theories with fundamental scalar were discussed. Even more ambitiously, in [@Ooguri:2016pdq] it was conjectured that the WGC (to be more precise, its generalized version in terms of $p$-form gauge fields [@ArkaniHamed:2006dz]) implies that non-supersymmetric anti-de Sitter (AdS) vacua supported by fluxes are unstable. If true, this extension of the WGC would imply interesting constraints on neutrino masses, the cosmological constant, and generic beyond the Standard Model physics [@Ibanez:2017kvh]. All these arguments motivate the necessity to prove the WGC beyond the folklore theorems that justify it. Interesting attempts in this direction are related to black hole entropy [@Cottrell:2016bty; @Fisher:2017dbc; @Cheung:2018cwt], the Cosmic Censorship [@Crisford:2017gsb], the anti-de Sitter/Conformal Field Theory (AdS/CFT) factorization problem [@Harlow:2015lma], unitarity and causality arguments [@Hamada:2018dde], and the universal relaxation bound [@Hod:2006jw; @Hod:2017uqc]. The proposal in ref. [@Hod:2017uqc] is particularly intriguing because it puts forward the possibility to link the WGC to thermalization properties of black holes. Inspired by this connection, in this paper we explore the WGC using the language of the AdS/CFT correspondence [@Maldacena:1997re; @Gubser:1998bc; @Witten:1998qj]. The bottom line of this work is the following. The AdS/CFT correspondence is a natural tool for analyzing thermodynamic aspects of gravity. From the AdS/CFT perspective, gravity is an emergent statistical phenomenon, and it is possible to reformulate the laws of gravitations in terms of the laws of thermodynamics. It is therefore interesting to ask [*i)*]{} whether the WGC admits an AdS/CFT interpretation and [*ii)*]{} if the dual picture sheds some light on the validity of the WGC. We shall give a positive answer to these questions in section \[sec:WGC\] and section \[sec:Spin\]. Our starting point is the simple observation that argument \#1 enumerated before makes use of extremal charged black holes to motivate the WGC. It is, therefore, a good idea to start our discussion by giving a closer look at the geometry of these objects. This will be the topic of the next section. Near-extremal Reissner-Nordström black holes {#sec:NRS} ============================================ In General Relativity, a black hole with mass $M$ and charge $Q$ is known as the Reissner-Nordström (RN) black hole. In four space-time dimensions with coordinates $(t,r,\theta,\varphi)$, the metric is given by the line element[^2] $$\label{eq:RN} ds^2 = - f(r) dt^2 + \frac{dr^2}{f(r)} + r^2\left( d\theta^2 + \sin^2\theta d\varphi^2 \right)~,~~~~~ f(r) = \left( 1 - \frac{2M}{r} + \frac{Q^2}{r^2} \right)~,$$ where we used a geometrized unit system with $c = G_N = 1$. The RN spacetime has two horizons located at $r_{\pm} \equiv M \pm \sqrt{M^2 - Q^2}$, the outer at $r = r_+$ being the event horizon of the black hole.[^3] The area of the event horizon is $A_{\rm RN} = 4\pi r_+^2$, and the Bekenstein-Hawking entropy $S_{\rm RN} = A_{\rm RN}/4 = \pi r_+^2$. The black hole temperature is $$T_{\rm RN} = \frac{r_+ - r_-}{4\pi r_+^2} = \frac{\sqrt{M^2 - Q^2}}{2\pi\left[ M + \sqrt{M^2 - Q^2} \right]^2}~.$$ For a Schwarzschild black hole $T_{\rm Sch} = \lim_{Q\to 0}T_{\rm RN} = 1/8\pi M$. The expression of the temperature can be obtained by Wick rotating and compactifying the Euclidean time, and identifying the period with the inverse temperature. More physically, it is easy to show that the first law of thermodynamics $$\label{eq:Thermo1} d M = T dS + \xi dQ~,~~~~~ T = \left( \frac{\partial M}{\partial S} \right)_Q~,~~~\xi = \left( \frac{\partial M}{\partial Q} \right)_S = \frac{Q}{r_+}~,$$ is satisfied for the above expressions of temperature and entropy. In eq. (\[eq:Thermo1\]), $\xi$ is the electric potential and it plays the role of chemical potential. The gauge field for the black hole solution in eq. (\[eq:RN\]) takes the form $A_{\mu}dx^{\mu} = A(r)dt$ with[^4] $$A(r) = Q\left( -\frac{1}{r} + \frac{1}{r_+} \right)~,~~~~A(r_+) = 0~.$$ The chemical potential of the black hole is the value of $A(r)$ at $r\to \infty$, and its expression matches that of $\xi$ in eq. (\[eq:Thermo1\]). To avoid the presence of a naked singularity, the black hole charge must satisfy the bound $Q \leqslant M$. The extremal RN black hole has $Q = M$. In this limit, the two horizons $r_{\pm}$ coincide. This is an interesting limit since an extremal RN black hole behaves like a thermodynamic system with ground state degeneracy: It has finite entropy $S_{\rm RN}^{\rm {\small \,ext}} = \pi M^2$ with vanishing temperature $T_{\rm RN}^{\rm {\small \,ext}} = 0$. Another quantity of interest is the behavior of the heat capacity as a function of the black hole charge. We find (cf. fig. \[fig:Schematic\]) $$\label{eq:Heat} C_{\rm RN} = T\left( \frac{\partial S}{\partial T} \right)_Q = \frac{2\pi r_+^2\sqrt{M^2 - Q^2}}{M - 2\sqrt{M^2 - Q^2}}~.$$ Far from extremality, the heat capacity is negative for $Q < \sqrt{3}M/2$. This property can seem paradoxical at first – a RN black hole with $Q < \sqrt{3}M/2$ gets hotter as it radiates energy – but it turns out to be rather ordinary for black holes with asymptotically flat spacetimes (including the simplest case of the Schwarzschild black hole for which $C_{\rm Sch} = -8\pi M^2$) as well as for some gravitationally bound systems that do not meet the strict definition of thermodynamic equilibrium, such as stars. This is not, indeed, the interesting part of the story. What is most surprising is that for a RN black hole the specific heat diverges at $Q = \sqrt{3}M/2$ and becomes positive for $Q > \sqrt{3}M/2$. Going towards the extremal limit, therefore, some sort of phase transition takes place and the black hole turns from a thermodynamic oddity into a more ordinary object. A RN black hole with positive heat capacity is to all effects a well-behaved thermodynamic system since it can be in equilibrium with a surrounding heat bath. $$\includegraphics[width=.485\textwidth]{HeatCap.pdf} \qquad \includegraphics[width=.485\textwidth]{NearHorGeo.pdf}$$ These arguments motivate a more quantitative analysis of the extremal limit. Let us start considering the extremal case with $Q = M$. We define the new variables $$\label{eq:Near} r = Q\left( 1+ \frac{\lambda}{z} \right)~,~~~~ t = \frac{Q\tau}{\lambda}~,$$ where $\lambda$ is an arbitrary small parameter. The position of the horizon corresponds to $z \to \infty$. From eq. (\[eq:RN\]), by taking the limit $\lambda \to 0$, we find the [*near-horizon metric of the extremal RN black hole*]{} $$\label{eq:NEBN} {\fcolorbox{gray}{Gray}{~$\displaystyle ds^2 = \underbrace{\frac{Q^2}{z^2}\left( -d\tau^2 + dz^2 \right)}_{AdS_2\,\,{\rm (Poincare\,patch)}} + \underbrace{Q^2\left( d\theta^2 + \sin^2\theta d\varphi^2 \right)}_{S^2}~, ~~~~A_{\tau}(z) = \frac{Q}{z} $~}}$$ A new geometry has emerged [@Guica:2008mu]. The extremal limit is a limit of enhanced symmetry since the metric in eq. (\[eq:NEBN\]) is $AdS_2 \times S^2$. The extremal RN background in the near-horizon limit enjoys an $SO(3)$ isometry acting on the sphere $S^2$. This is nothing but the spherical symmetry of the original black hole solution. In addition, the metric also has an $SO(2,1)$ isometry acting on the $AdS_2$ space that was not present in the original solution. For an extremal RN black hole, in the near-horizon limit an effective two-dimensional AdS geometry is realised in the time-radial direction.[^5] Notice that for an extremal black hole there is only one scale, the charge $Q$ corresponding to both the radius of the sphere $S^2$ and the radius of the $AdS_2$ factor. The presence of the $AdS_2$ factor in eq. (\[eq:NEBN\]) suggests the possible existence of an AdS/CFT correspondence. Technically speaking, the key point is that the boundary of the near-horizon geometry, located at $z\to 0$, inherits the full conformal group from the isometries of $AdS_2$. We are facing a special case of the AdS/CFT correspondence with a one-dimensional conformal field theory (a.k.a. conformal quantum mechanics): The boosts and rotation in $AdS_2$ close, in the limit $z\to 0$, the one-dimensional conformal algebra[^6] generated by dilatations, translations in $\tau$, and special conformal transformations (cf. [@Zaanen:2015oix]). Notice that the limit $z\to 0$ does not cover the full range of the original radial coordinate (sketch in the right panel of fig. \[fig:Schematic\]). The above construction can be generalized to the more interesting case of finite temperature. In addition to the variables defined in eq. (\[eq:Near\]), we shift the position of the horizon from the extremal limit $r_+ = Q$ to $$r_+ = Q\left( 1 + \frac{\lambda}{z_0} \right)~.$$ We obtain the [*near-extremal near-horizon metric of the RN black hole*]{} $$\label{eq:NENH} {\fcolorbox{gray}{Gray}{~$\displaystyle ds^2 = \underbrace{\frac{Q^2}{z^2}\left[ -\left( 1-\frac{z^2}{z_0^2} \right)d\tau^2 + \left( 1-\frac{z^2}{z_0^2} \right)^{-1}dz^2\right]}_{{\rm finite\,temperature}\,AdS_2\,{\rm factor\,\,(Rindler\,patch)}} + Q^2 d\Omega_2^2~,~~~~~ A_{\tau}(z) = \frac{Q}{z}\left( 1-\frac{z}{z_0} \right) $~}}$$ We remark that both eq.s (\[eq:NEBN\],\[eq:NENH\]) are actual solutions of the Einstein-Maxwell field equations, as verified by explicit computation. The temperature associated to the metric in eq. (\[eq:NENH\]) is $T = 1/2\pi z_0$. Notice that this is a dimensionless quantity since it refers to the variable $\tau$ which is defined in units of $Q$. In eq. (\[eq:NENH\]) the coordinate $z$ ranges from $z = z_0$ at the horizon to $z= 0$ at the CFT boundary. The non-zero value of the temperature acts as a cut-off for the limit $z \to \infty$, and for $z\ll z_0$ the metric is essentially $AdS_2 \times S^2$. We are interested in the near-extremal limit of the black hole – that is at temperatures $T\ll 1$ – and we shall study its response at low energies, $\omega \ll 1$ ($T\ll 1/Q$ and $\omega\ll 1/Q$ if we restore the $Q$-dimension of the $\tau$ variable) $$\label{eq:Condition} {\fcolorbox{gray}{Gray}{~$\displaystyle T \ll \frac{1}{Q}~,~~~~~\omega \ll \frac{1}{Q}~,~~~~~T\sim \omega $~}}$$ Once the background metric is defined, a natural question is to determine its response to small perturbations. Isolated black holes in equilibrium are indeed idealized objects. For instance, during the first moments after its formation due to gravitational collapse of matter, a newly born black hole is in a perturbed state. Generally, black holes always have complex distributions of matter around them such as accretion disks, and they actively interact with their surroundings. These simple considerations motivate the study of black hole perturbations. Of course, it is hard to expect that electrically charged black holes play any fundamental role in observational astrophysics due to charge neutrality of the Universe. However, we remind that we are considering a generic $U(1)$ local symmetry, a situation in which it is possible to envisage phenomenologically viable scenarios where charged black holes – even close to extremality – are allowed [@Cardoso:2016olt]. Apart from these phenomenological motivations, studying the stability of solutions of Einstein field equations is essential to determine their validity. Clearly, all the arguments that support the WGC listed in section \[sec:intro\] assume that RN black holes are valid solutions all the way up to the extremal limit. In our analysis, we shall, therefore, impose stability of the RN metric against perturbations. In particular, we shall work under the assumption that the near-extremal near-horizon metric of the RN black hole is stable. Let us first set the problem in general fashion. We consider the metric $g_{\mu\nu}$ and matter fields $\Phi_i$ as a sum of the unperturbed background values and the actual perturbations, $g_{\mu\nu} = g_{\mu\nu}^{0} + \delta g_{\mu\nu}$ and $\Phi_i = \Phi_i^{0} + \delta \Phi_i$. Solving the perturbation dynamics is a highly non-linear problem since in Einstein field equations variations in the energy-momentum tensor imply an alteration of space-time geometry, which in turn involves a matter redistribution. However, assuming small perturbations, we can neglect terms of order $O(\delta g_{\mu\nu}^2)$, $O(\delta \Phi_i^2)$, $O(\delta g_{\mu\nu}\delta \Phi_i)$ or higher. Under this assumption one finds, using the Einstein field equations and the equations of motion for the matter fields, a set of linear equations for the perturbations $\delta g_{\mu\nu}$ and $\delta \Phi_i$. Let us now focus on the case of RN black holes. The RN background solution is not sourced by any matter fields (i.e. $\Phi_i^0 = 0$, cf. discussion below eq. (\[eq:GenericAction\])), and the field perturbations are not coupled to the perturbations of the metric. The former are, therefore, equivalent to the dynamics of test fields in the black hole background. At the linear level perturbations of the RN space-time can be performed in two ways: by studying the dynamics of test fields in the black hole background or by perturbing the black hole metric itself. In the next section we shall study the dynamics of test fields in the RN background. Emergent CFT and the WGC {#sec:WGC} ======================== We are in the position to study the propagation of generic perturbations in the background geometries in eq.s (\[eq:NEBN\],\[eq:NENH\]). For illustrative purposes, we start from the near-horizon metric of the extremal RN black hole, and we focus on a charged scalar $\Phi$ (which is, therefore, necessarily complex) with charge $q$ in units of $e$ and mass $\mu$. The action for $\Phi$ is $$\mathcal{S} = - \int d^4x\sqrt{-g}\left[ \left( D^{\mu}\Phi \right)^*\left( D_{\mu}\Phi \right) + \mu^2 \Phi^*\Phi \right]~,$$ where $D^{\mu}\Phi = (\partial^{\mu} - igA^{\mu})\Phi$, with $g\equiv qe$.[^7] The equations of motion is given by $$\label{eq:MasterPert} \left[ \left( \nabla^{\nu} - igA^{\nu} \right) \left( \nabla_{\nu} - igA_{\nu} \right) - \mu^2 \right]\Phi = 0~,$$ where the covariant derivative acts on a generic vector according to $\nabla_{\mu}v_{\nu} = \partial_{\mu}v_{\nu} - \Gamma^{\alpha}_{\mu\nu}v_{\alpha}$. The metric allows for the separation of variables, and we use the ansatz $$\label{eq:Separation} \Phi(\tau,z,\theta,\varphi) = e^{-i\omega \tau}\sum_{l,m} e^{im\varphi}\phi_l(z)S_l(\theta)~.$$ The angular part $S(\theta)$ solves the eigenvalue equation for the associated Legendre polynomials $$\label{eq:legendre} \left[ \sin\theta \frac{d}{d\theta}\left( \sin\theta\frac{d}{d\theta} \right) + l(l+1)\sin^2\theta - m^2 \right]S_l(\theta) = 0~,$$ and regularity at the poles $\theta = 0,\pi$ imposes the usual quantization conditions on the azimuthal $l$ and magnetic $m$ quantum numbers $l=0,1,\dots$ with $-l\leqslant m\leqslant +l$. The function $\phi_l(z)$ solves the differential equation $$-\frac{d^2\phi_l}{dz^2} + \left\{ \frac{1}{z^2}\left[ \mu^2 Q^2 + l(l+1) \right] - \left( \omega + \frac{gQ}{z} \right)^2 \right\}\phi_l = 0~.$$ This equation admits an exact analytical solution in terms of the Whittaker functions $$\begin{aligned} \phi_l(z,\omega) &=& C_{\rm out}\mathcal{W}_{-igQ,\nu_l}(2i\omega z) + C_{\rm in}\mathcal{W}_{igQ,\nu_l}(-2i\omega z)~, \label{eq:Whitt}\\ \nu_l &\equiv& \sqrt{\frac{1}{4} + Q^2(\mu^2 - g^2) +l(l+1)}~.\label{eq:ConfDim}\end{aligned}$$ Physical solutions are characterized by ingoing boundary condition at the horizon. The latter is located at $z\to \infty$, see eq. (\[eq:Near\]). The Whittaker functions feature the asymptotic behavior $\mathcal{W}_{\kappa,\nu}(\zeta) \stackrel{\zeta \to \infty}{\approx} e^{-\zeta/2}\zeta^{\kappa}$. Consequently, the only solution in eq. (\[eq:Whitt\]) that is ingoing at the horizon is $\phi_l(z,\omega) = C_{\rm in}\mathcal{W}_{igQ,\nu_l}(-2i\omega z)$. The crucial aspect of the computation is the behavior of the function $\phi_l(z)$ at the boundary of the near-horizon geometry, $z\to 0$. From the properties of the Whittaker functions we have $$\label{eq:WittExp} \mathcal{W}_{\kappa,\nu}(\zeta) \stackrel{\zeta \to 0}{\approx} \frac{\Gamma(2\nu)}{\Gamma(1/2 + \nu -\kappa)}\,\zeta^{1/2-\nu}+ \frac{\Gamma(-2\nu)}{\Gamma(1/2 - \nu -\kappa)}\,\zeta^{1/2+\nu}~.$$ We, therefore, find $$\begin{aligned} \phi_l(z,\omega) &\stackrel{z \to 0}{\approx}& C_{\rm in}\left[ \frac{\Gamma(2\nu_l)(-2i\omega)^{1/2-\nu_l}}{\Gamma(1/2 + \nu_l - igQ)}\,z^{1/2-\nu_l}+ \frac{\Gamma(-2\nu_l)(-2i\omega)^{1/2+\nu_l}}{\Gamma(1/2 - \nu_l - igQ)}\,z^{1/2+\nu_l} \right]\nonumber \\ &\equiv& \mathcal{A}_l(\omega)z^{1-\Delta_l} + \mathcal{B}_l(\omega)z^{\Delta_l}~,\label{eq:Boundary}\end{aligned}$$ with $\Delta_l$ defined as $$\Delta_l \equiv \frac{1}{2} + \underbrace{\sqrt{ \frac{1}{4} + Q^2\left\{ \mu^2\left[ 1 + \frac{l(l+1)}{\mu^2 Q^2} \right] - g^2 \right\} }}_{\equiv~\nu_l\,{\rm in\,eq.~(\ref{eq:ConfDim})}}~.\label{eq:ConfDimension}$$ Notice that the expansion in eq. (\[eq:Separation\]) is equivalent to a Kaluza-Klein (KK) reduction from $AdS_2 \times S^2$ to $AdS_2$. The hallmark of any KK reduction on a compact manifold is the appearance of an infinite but discrete tower of modes of increasing mass. In our case the field $\Phi$ propagating in $AdS_2 \times S^2$ gives rise to a tower of fields $\phi_l(z)$ in $AdS_2$ with KK spectrum $$\label{eq:KKmodes} M_{\rm KK}^{(l)} = \mu^2 + \frac{l(l+1)}{Q^2}~,~~~~l=0,1,\dots~.$$ The lightest mode with $l=0$ has mass $\mu$. The non-zero KK modes start at $\sim 1/Q$, where the radius $Q$ of the sphere $S^2$ plays the role of compactification scale. As in any other extra-dimensional scenario, in the low energy effective theory below the energy $1/Q$ the KK modes can be neglected. We shall return on this point later. According to the AdS/CFT dictionary there exists a correspondence between the scalar field $\phi_l$ propagating in the AdS bulk and the scalar operator $\mathcal{O}_l$ belonging to the CFT at the boundary. The conformal dimension of the scalar operator $\mathcal{O}_l$ is given by $\Delta_l$. The scalar operator $\mathcal{O}_l$ is dubbed [*irrelevant*]{} if $\Delta_l > 1$, [*relevant*]{} if $\Delta_l < 1$ and [*marginal*]{} if $\Delta_l = 1$. By imposing the condition that the conformal dimension stays real we find the Breitenlohner-Freedman bound $$\label{eq:BFbound} Q^2\left( \mu^2 - q^2 \right) + \left( l + \frac{1}{2} \right)^2 > 0~.$$ Below the Breitenlohner-Freedman bound the AdS/CFT construction is unstable. On the gravity side, the instability is related to the Schwinger effect caused by the spontaneous production of charged particle/anti-particle pairs in the RN spacetime [@Chen:2012zn]. On the CFT side, the instability is related to causality violation. We will not contemplate this possibility any further in this section but it is important to remark that the presence of the instability is crucially related to the fact that we are dealing with a scalar perturbation. Bose-Einstein statistics plays indeed an important role. On the gravity side, the number of boson pairs produced spontaneously in a given state is not limited by statistics, while such limitation does exist for fermions due to Pauli blocking [@Man; @Hansen:1980nc]. This physical argument is the same that prohibits the existence of superradiance for fermions [@Brito:2015oca]. On the CFT side, causality violation manifests itself for scalars but not for spinors (e.g., cf. [@Liu:2009dm] for the $AdS_4/CFT_3$ case). From eq. (\[eq:Boundary\]) it is evident that the term proportional to $\mathcal{A}_l(\omega)$ diverges as $z \to 0$ while the term $\mathcal{B}_l(\omega)$ is regular.[^8] The former (latter) is called leading (sub-leading) term. According to the rules of the AdS/CFT correspondence the CFT two-point correlation function for the scalar operator $\mathcal{O}_l$ can be expressed in terms of the ratio of the coefficients of the sub-leading and leading asymptotes of the corresponding AdS massive scalar wave in eq. (\[eq:Boundary\]) $$\label{eq:TwoPoint} \langle \mathcal{O}_l(-\omega)\mathcal{O}_l(\omega) \rangle = \frac{\mathcal{B}_l(\omega)}{\mathcal{A}_l(\omega)}~.$$ More formally, it can be shown that the leading behavior of the solution $\phi_l(z,\omega)$ at the boundary acts as a local source $J$ for the operator $\mathcal{O}_l$, and the sub-leading term corresponds to the expectation value $\langle\mathcal{O}_l\rangle_J$ in the presence of the source. The ratio in eq. (\[eq:TwoPoint\]) thus describes the reaction of the field to a source, and can be identified – up to an overall constant which is independent of $\omega$ – with the retarded Green’s function $\mathcal{G}_R^{(l)}(\omega) = \mathcal{B}_l(\omega)/\mathcal{A}_l(\omega)$. In our discussion the poles of the retarded Green’s function play a central role. They are defined by the values of $\omega$ such that $\mathcal{A}_l(\omega) = 0$. According to eq. (\[eq:Boundary\]) and to the previous discussion, the condition $\mathcal{A}_l(\omega) = 0$ corresponds to the absence of a source term at the boundary $z \to 0$. On the gravitational side, this setup has a neat physical interpretation. Solutions of eq. (\[eq:MasterPert\]) with ingoing boundary condition at the horizon and outgoing boundary condition at infinity are characterized by a discrete set of eigenfrequencies that are called quasi-normal modes. Quasi-normal modes describe the dissipative properties of a black hole, and for such reason they are defined by the absence of inward-directed waves generated by some source at spatial infinity (the presence of a non-zero source at infinity would keep the system perturbed, contrary the very same definition of dissipative dynamics). The eigenfrequencies corresponding to the quasi-normal modes have both a real and an imaginary part. The latter is negative, and gives the inverse of the relaxation time $\tau_{d}$ of the corresponding mode, $\omega = \mathbb{Re}[\omega] + i \mathbb{Im}[\omega] \equiv \mathbb{Re}[\omega] - i2\pi/\tau_{d}$. From the $\tau$-dependence of the solution, $e^{-i\omega \tau}$, a negative imaginary part describes an exponential damping with characteristic time-scale set by $\tau_{d}$. The quasi-normal modes control the black hole ringdown, that is the decay of the perturbed black hole toward its hairless state. This is a thermalization process: The relaxation towards thermal equilibrium after a perturbation. The quasi-normal mode with the smallest value of $-\mathbb{Im}[\omega]$ (equivalently, with the largest value of $\tau_d$) is the less damped mode, and it dominates the thermalization time-scale. The ringdown of the black hole is dual to the thermalization process of the corresponding CFT [@Horowitz:1999jd]. In the CFT, the quasi-normal modes manifest themselves as Ruelle resonances, that appear as poles in the Fourier transform of the retarded Green’s function. The interpretation of black hole quasi-normal modes as poles of the retarded Green’s function in the dual CFT was first pointed out for BTZ black holes in [@Birmingham:2001pj]. From eq. (\[eq:Boundary\]) we find $$\label{eq:ScalarGreenT0} \mathcal{G}^{(l)}_R(\omega) = e^{-i\pi\nu_l}\,\frac{\Gamma(-2\nu_l)\Gamma(1/2 + \nu_l -igQ)}{ \Gamma(2\nu_l)\Gamma(1/2 - \nu_l -igQ)}\,(2\omega)^{2\nu_l}~.$$ We note that in this case there are no poles in the retarded Green’s function. This means that there are no quasi-normal modes for an extremal RN black hole associated with the dynamics of a test charged scalar field. This is not surprising since this is the limit in which the black hole temperature vanishes. In order to have some interesting dynamics, we shall now move to consider the near-extremal near-horizon metric of the RN black hole with non-vanishing temperature $T = 1/2\pi z_0$. We have to solve eq. (\[eq:MasterPert\]) in the background given by eq. (\[eq:NENH\]). It is again possible to exploit the separation of variables, and for the angular part we find again the associated Legendre polynomials in eq. (\[eq:legendre\]). The equation for $\phi(z)$ is given by $$\frac{d^2\phi}{dz^2} + \left(\frac{2z}{z^2 - z_0^2}\right)\frac{d\phi}{dz} +\left\{ -\frac{\left[Q^2\mu^2 + l(l+1)\right]}{z^2 - z^4/z_0^2} + \frac{\left[ \omega + gQ\left( \frac{1}{z}-\frac{1}{z_0} \right) \right]^2}{1- 2z^2/z_0^2 + z^4/z_0^4} \right\}\phi = 0~.$$ This is a second-order ordinary differential equation with three regular singular points at $z = 0,z_0,\infty$ and, as such, it can be transformed into the hypergeometric differential equation. We find the two independent solutions $$\begin{aligned} \phi_l(z,\omega) &\sim& \left( \frac{1}{z}-\frac{1}{z_0} \right)^{-\frac{1}{2}\mp \nu_l}\left( \frac{z_0 + z}{z_0 - z} \right)^{\frac{i\omega z_0}{2} - igQ} \times \\ && {}_2F_1\left( \frac{1}{2}\pm\nu_l +i\omega z_0 -igQ, \frac{1}{2} \pm \nu_l -igQ, 1\pm2\nu_l;\frac{2z}{z-z_0} \right)~.\nonumber\end{aligned}$$ The upper sign gives the ingoing solution at the horizon. As done before, we have to consider the behavior of the solution at the AdS boundary in order to extract the retarded Green’s function as the the quotient of the sub-leading to leading term. Using the properties of the hypergeometric functions, we find $$\label{eq:GreenScalar} \mathcal{G}_R^{(l)}(\omega) = (4\pi T)^{2\nu_l}\,\frac{\Gamma(-2\nu_l)\Gamma(1/2 +\nu_l -i\omega/2\pi T + i gQ)\Gamma(1/2 + \nu_l - igQ)} {\Gamma(2\nu_l)\Gamma(1/2 - \nu_l -i\omega/2\pi T + i gQ)\Gamma(1/2 - \nu_l - igQ)}~,$$ where we used the explicit definition of $T$ instead of $z_0$. The quasi-normal frequencies are, therefore, dictated by the poles of the Gamma function, and for the imaginary part we find $$\label{eq:MasterScalar} {\fcolorbox{gray}{Gray}{~$\displaystyle \mathbb{Im}[\omega_{n,l}] = -2\pi T\left(\frac{1}{2} + n + \nu_l\right)_{n=0,1,\dots}~,~~~~~~~\nu_l = \sqrt{ \frac{1}{4} + Q^2\left\{ \mu^2\left[ 1 + \frac{l(l+1)}{\mu^2 Q^2} \right] - g^2 \right\}} $~}}$$ We note that this result agrees with the expression for the quasi-normal modes obtained in [@Hod:2010hw]. In [@Hod:2010hw] the quasi-normal modes were computed considering the full metric outside the black hole horizon instead of limiting the computation to the near-horizon geometry as done in this paper. The agreement between the two results is due to the fact that in the low-energy limit in eq. (\[eq:Condition\]) only the near-horizon geometry is relevant. The thermalization time-scale is set by the less damped mode with $n=0$, and we find (remember our definition $\tau_d \equiv -2\pi/\mathbb{Im}[\omega]$) $$\label{eq:RelTimescale} \tau_d^{(n,l)} = \frac{1}{T\left(1/2 + n + \nu_l\right)} ~~~~ \stackrel{n = 0}{\Rightarrow} ~~~~ \tau_d^{(0,l)} = \frac{1}{T\left(1/2 + \nu_l\right)}~.$$ We are interested in the response of the system at low frequencies, $\omega \ll 1/Q$ (cf. eq. (\[eq:Condition\])). As discussed below eq. (\[eq:KKmodes\]), in this limit the KK modes do not participate to the low-energy dynamics and the thermalization time-scale is $$\tau_d \equiv \tau_d^{(0,0)} = \frac{1}{T\left(1/2 + \nu_0\right)}~,~~~~ \nu_0 = \sqrt{ \frac{1}{4} + Q^2\left( \mu^2 - g^2 \right)}~.$$ We are now in the position to discuss the connection with the WGC. Notice that $\tau_d > 0$. Negative $\tau_d$ would correspond to an exponentially-growing unstable fundamental mode that would threat the stability of the black hole geometry. Most importantly, this would clash against the thermodynamic interpretation of quasi-normal modes according to which $\tau_d$ must be positive. From this perspective, it is totally reasonable to expect not just $\tau_d > 0$ but the actual existence of a lower bound on $\tau_d$ since the thermalization process cannot be arbitrarely fast. An educated guess is $\tau_d \gtrsim 1/T$.[^9] This intuition is supported by many explicit examples. On the gravity side, in [@Horowitz:1999jd] the relation $\tau_d \gtrsim 1/T$ was obtained for scalar perturbations of Schwarzschild-AdS black holes in four, five, and seven dimensions. Furthermore, on the CFT side, there exists a lower bound on $\tau_d$ as $T\to 0$ in all many-body quantum systems that admit an AdS/CFT description, $\tau_d > c/T$ where $c$ is a temperature-independent positive constant. A remarkable example that is close to the construction proposed in this paper is the Sachdev-Ye-Kitaev (SYK) model [@Sachdev:1992fk; @Kit; @Maldacena:2016hyu], a maximally chaotic model of strongly-interacting Majorana fermions. In [@Eberlein:2017wah] a numerical study found a thermalization rate consistent with the expectation $\tau_d > c/T$. We shall discuss in more detail the connection with the SYK model in section \[sec:Con\]. All in all, we impose the condition $\nu_0 < 1/2$ to ensure the condition $\tau_d > 1/T$ on the thermalization time-scale and, after restoring units of $M_{\rm P}$, we find $$\label{eq:WGC2} qe > \frac{\mu}{M_{\rm P}}$$ that is precisely the inequality envisaged by the WGC. The crucial question is: Does the result derived in this section represent a proof of the WGC? To answer this question, we remind once again that the WGC asserts that with $qe > \mu/M_{\rm P}$. This means that any attempt for its proof must involve some robust argument that makes mandatory the existence of a state satisfying eq. (\[eq:WGC2\]). At first sight, our result does not imply [*per se*]{} the existence of such state – at face value it says that a charged scalar field appearing in the action in eq. (\[eq:GenericAction\]) must satisfy eq. (\[eq:WGC2\]) in order to be consistent with the laws of black hole thermodynamics but it does not guarantee the presence of such charged scalar in the spectrum. However, there is more. In our case what makes mandatory the existence of a state with $\mu < qeM_{\rm P}$ is the condition $\tau_d > c/T$ on the thermalization time-scale. In the black hole literature, this is known as the universal relaxation bound [@Hod:2006jw; @Hod:2017uqc; @Gruzinov:2007ai]. As we discussed before, the universal relaxation bound can be beautifully related – in the spirit of the AdS/CFT correspondence – to the thermalization properties of the CFT living on the boundary of the black hole near-horizon geometry. Let us elaborate more on this crucial point, along the lines of [@Hod:2006jw]. As we discussed at the end of section \[sec:NRS\], there are two kinds of perturbations, decoupled at the linear level: those that involve perturbations of the background metric (including the electromagnetic field in the case of a RN black hole) and those related to test fields propagating in the black hole background. In principle the two sets of perturbations have different thermalization time-scales, and at least one of the two must satisfy the condition $\tau_d > c/T$. Background metric perturbations are characterized by $\tau_d^{\rm BG} \sim M$. In the case of a Schwarzschild black hole, we have $T = 1/8\pi M$ and we can conclude that these perturbations satisfy the scaling $\tau_d^{\rm BG} \sim 1/T$. However, the relation $T = 1/8\pi M$ is only true for ordinary black holes with negative heat capacity (cf. section \[sec:NRS\]) since they get hotter and hotter as they radiate (that is for decreasing mass $M$). On the contrary, in the near-extremal limit RN black holes do not behave this way since their heat capacity becomes positive, and the relation $M \sim 1/T$ breaks down. Background metric perturbations still behave according to $\tau_d^{\rm BG} \sim M \sim Q$ [@Andersson:1996xw] but now $\tau_d^{\rm BG} \sim Q \ll 1/T$ as $T \to 0$, and, therefore, these perturbations decay too fast to satisfy the condition $\tau_d > c/T$. With background metric perturbations out of the way, we conclude that for a near-extremal RN black hole there must exist at least one particle whose perturbations set the thermalization rate $\tau_d > c/T$. For such state, our computation shows that eq. (\[eq:WGC2\]) must be satisfied. This concludes our attempt to demonstrate the WGC. The fact that we derived our result for the specific case of a charged scalar field may raise some eyebrows (the WGC, according to its original formulation, does not seem to prefer any specific spin). In the next section, we shall discuss the case of spin-$1/2$ particles. What about the spin? {#sec:Spin} ==================== As mentioned in section \[sec:intro\], the WGC says nothing about the spin of the state whose mass and charge are involved in eq. (\[eq:WGC\]). This could indicate that whatever physical argument is supporting the WGC it should be independent from the spin. In section \[sec:WGC\] we derived the WGC considering the specific case of a scalar particle. It is interesting to investigate higher-spin dynamics. If the thermodynamic interpretation of the WGC proposed in this paper is correct, we expect to obtain the same result considering particles with different spin, and in this section we shall discuss the case of a spin-$1/2$ particle. We have to solve the dynamics of a charged Dirac fermion in the near-extremal near-horizon metric of the RN black hole, eq. (\[eq:NENH\]). The logic of the computation follows the same steps explained in section \[sec:WGC\] with extra technical complications due to the presence of the spin. The reader interested in the final result can directly jump to eq. (\[eq:MasterFermionQNM\]). To solve the Dirac equation in curved space we use the vierbein formalism [@Brill:1957fx]. The Dirac equation is $$\label{eq:CurvedDirac} \left[ \gamma^{(a)}e_{(a)}^{~~\rho}\left( \partial_{\rho} + \Gamma_{\rho} - igA_{\rho} \right) + \mu \right]\Psi = 0~,$$ where $\mu$ is the mass and $g = qe$ the charge of $\Psi$. In four space-time dimensions the spinor $\Psi$ has four complex components. At each space-time point $x$ it is possible to define a locally inertial system of coordinates by introducing the tetrad $e^{(a)}_{~~\alpha}$ by means of $$g_{\alpha\beta}(x) = e^{(a)}_{~~\alpha}(x)e^{(b)}_{~~\beta}(x)\eta_{ab}~,$$ with Minkowski metric $\eta_{ab}=(-1,+1,+1,+1)$. Latin (greek) indices are raised and lowered with the Minkowski (general non-inertial) metric, $e_{(a)}^{~~\alpha} = e^{(b)}_{~~\beta}\eta_{ab}g^{\alpha\beta}$. Using this construction, we introduce generally covariant Dirac matrices $\gamma^{\alpha}$ defined as $\gamma^{\alpha} \equiv \gamma^{(a)}e_{(a)}^{~~\alpha}$, where the flat-space Dirac matrices $\gamma^{(a)}$ satisfy the usual relations $\gamma^{(a)}\gamma^{(b)} + \gamma^{(b)}\gamma^{(a)} = 2\eta^{ab}$. From the definition of the tetrad, we have $\gamma^{\alpha}\gamma^{\beta} + \gamma^{\beta}\gamma^{\alpha} = 2g^{\alpha\beta}$ that generalizes the algebra of the Dirac matrices in curved space. The spin connection $\Gamma_{\rho}$ entering in the definition of the covariant derivative in eq. (\[eq:CurvedDirac\]) is given by $$\Gamma_{\rho} = \frac{1}{8}\left[ \gamma^{(a)},\gamma^{(b)} \right]g_{\mu\nu}e_{(a)}^{~~\mu}\nabla_{\rho}e_{(b)}^{~~\nu}~,$$ where the covariant derivative acts on the tetrad according to $\nabla_{\mu}e_{(b)}^{~~\alpha} = \partial_{\mu}e_{(b)}^{~~\nu} + \Gamma_{\mu\nu}^{\alpha}e_{(b)}^{~~\alpha}$. We solve the Dirac equation in the so-called rotation frame defined by the tetrad $$e^{(\tau)}_{~~\tau} = \frac{Q}{z}\left( 1-\frac{z^2}{z_0^2} \right)^{1/2}~,~~~~~e^{(z)}_{~~z} = \frac{Q}{z}\left( 1-\frac{z^2}{z_0^2} \right)^{-1/2}~,~~~~~e^{(\theta)}_{~~\theta} = Q ~,~~~~~e^{(\varphi)}_{~~\varphi} = Q\sin\theta~.$$ This choice lends itself particularly well to separate variables [@Villalba:1994mv]. Eq. (\[eq:CurvedDirac\]) becomes $$\label{eq:DiracEl} \left\{\gamma^{(0)} \frac{z}{\sqrt{f(z)}}\left[ \partial_{\tau} - \frac{igQ}{z}\left( 1-\frac{z}{z_0} \right) \right] + \gamma^{(1)}z\sqrt{f(z)}\partial_{z} + \gamma^{(2)}\partial_{\theta} + \gamma^{(3)}\frac{1}{\sin\theta}\partial_{\varphi} +Q\mu\right\}\Psi = 0~,$$ where we used the short-hand notation $f(z) \equiv 1-z^2/z_0^2$ and where we rescaled the spinor field according to $\Psi \to \Psi/\sqrt{\sin\theta}$. It is now possible to separate variables. The total angular momentum is $\vec{J} = \vec{L} + \vec{S}$. We combine spherical harmonics, which are eigenstates of $\vec{L}^2$ and $L_z$, and spinors, which are eigenstates of $\vec{S}^2$ and $S_z$, to form eigenstates of $\vec{J}^2$ and $J_z$. The latter are the so-called spherical spinors. The ansatz analogue to eq. (\[eq:Separation\]) is $$\label{eq:SpinorAnsatz} \Psi(\tau,z,\theta,\varphi) = e^{-i\omega\tau}\left[ Z_+(z)\Phi^+_{\kappa,m}(\theta,\varphi) + Z_-(z)\Phi^-_{\kappa,m}(\theta,\varphi) \right]~,$$ where $\Phi^{\pm}_{\kappa,m}(\theta,\varphi)$ are spherical spinors in the rotation frame.[^10] We obtain the radial equation $$\left[ z\sqrt{f(z)}\partial_z \pm \kappa \right]Z_{\pm}(z) - i\left\{ \frac{z}{\sqrt{f(z)}}\left[ \omega + \frac{gQ}{z}\left( 1-\frac{z}{z_0} \right) \right] \pm Q\mu \right\}Z_{\mp}(z) = 0~.$$ By introducing the combinations $\mathcal{Z}_{\pm} \equiv Z_+ \pm Z_-$, we find $$\label{eq:Zint} \left\{ z\sqrt{f(z)}\partial_z \mp i\frac{z}{\sqrt{f(z)}}\left[ \omega + \frac{gQ}{z}\left( 1-\frac{z}{z_0} \right) \right] \right\}\mathcal{Z}_{\pm}(z) + \left( \kappa \pm iQ\mu \right)\mathcal{Z}_{\mp}(z) = 0~.$$ These equations can be recast in matrix form by introducing the two-component vector $\tilde{\mathcal{Z}} \equiv (\tilde{\mathcal{Z}}_+,\tilde{\mathcal{Z}}_-)^{\rm T}$. We find $$\label{eq:MasterSpinor} \left\{\partial_z - \frac{i}{f(z)}\sigma_3\left[ \omega + \frac{gQ}{z}\left( 1-\frac{z}{z_0} \right) \right] \right\}\tilde{\mathcal{Z}} = \frac{Q}{z\sqrt{f(z)}} \left( \sigma_2 \mu - \sigma_1 \frac{\kappa}{Q} \right)\tilde{\mathcal{Z}}~,$$ where $\sigma_{i=1,2,3}$ are the usual Pauli matrices acting on the two components of $\tilde{\mathcal{Z}}$. Let us pause for a moment to comment on this result. The ansatz in eq. (\[eq:SpinorAnsatz\]) made possible the dimensional reduction of a spinor $\Psi$ propagating in $AdS_2 \times S^2$ to a tower of spinor fields $\tilde{\mathcal{Z}}_{\kappa}$ in $AdS_2$, each one solution of eq. (\[eq:MasterSpinor\]) for a fixed value of $\kappa$. In two dimensions, a Dirac spinor has indeed two complex components that are precisely $\tilde{\mathcal{Z}}_{\pm}$ introduced before. Eq. (\[eq:MasterSpinor\]) can be decoupled and admit analytical solutions [@Faulkner:2011tm]. It is instructive to consider first the extremal case $z_0 \to \infty$ (that is the near-horizon limit of an extremal RN black hole). The general solution of eq. (\[eq:MasterSpinor\]) in this limit is $$\begin{aligned} \tilde{\mathcal{Z}}_{\kappa}(z,\omega) &=& \frac{1}{\sqrt{z}}\left[ C_{\rm out}\mathcal{W}_{-\frac{\sigma_3}{2} - igQ,\nu_{\kappa}}(2i\omega z) \left( \begin{array}{c} \tilde{\mu} \\ -1 \end{array} \right) + C_{\rm in} \mathcal{W}_{\frac{\sigma_3}{2}+igQ,\nu_{\kappa}}(-2i\omega z) \left( \begin{array}{c} -1 \\ \tilde{\mu}^* \end{array} \right) \right]~,\nonumber\\ \nu_{\kappa} &\equiv& \sqrt{Q^2\left( \mu^2 - g^2 \right) +\kappa^2}~,\label{eq:MasterSpinorSol}\end{aligned}$$ where $\sigma_3 = \pm 1$ when acting on the upper and lower component of the spinor and where we used the short-hand notation $\tilde{\mu} \equiv -\kappa/Q - i\mu$. Notice the close analogy with eq.s (\[eq:Whitt\],\[eq:ConfDim\]). The ingoing boundary condition at the horizon selects the solution $\propto \mathcal{W}_{\sigma_3/2+igQ,\nu_{\kappa}}(-2i\omega\nu_{\kappa})$. We now turn our attention to the $AdS_2$ boundary at $z\to 0$. In this limit eq. (\[eq:MasterSpinor\]) takes the form $$\label{eq:MatrixAsy} z\left(\partial_z \tilde{\mathcal{Z}}\right) = U\tilde{\mathcal{Z}}~,~~~~U \equiv \left( \begin{array}{cc} igQ & -iQ\mu -\kappa \\ iQ\mu -\kappa & -igQ \end{array} \right)~.$$ The matrix $U$ can be diagonalized, and we find $$\label{eq:Eigen} U v_{\pm} = \pm \nu_{\kappa}v_{\pm}~, ~~~~~~~~~~v_{\pm} = \left( \begin{array}{c} \frac{1}{Q}\left( igQ \pm \nu_{\kappa} \right) \\ -\frac{\kappa}{Q} +i\mu \equiv \tilde{\mu}^* \end{array} \right)~.$$ Using this result, it is simple to verify that eq. (\[eq:MatrixAsy\]) is solved by $$\tilde{\mathcal{Z}}_{\kappa}(z,\omega)\stackrel{z \to 0}{\approx} \mathcal{A}_{\kappa}(\omega) v_- z^{-\nu_{\kappa}} + \mathcal{B}_{\kappa}(\omega) v_+ z^{\nu_{\kappa}}~.$$ Notice that in eq. (\[eq:Eigen\]) we fixed the arbitrary normalization of the eigenstates of $U$ to match the bottom component of the incoming solution in eq. (\[eq:MasterSpinorSol\]). We can, therefore, easily extract (up to a $\omega$-independent normalization) the functions $\mathcal{A}_{\kappa}(\omega)$ and $\mathcal{B}_{\kappa}(\omega)$ directly from the asymptotic expansion of eq. (\[eq:MasterSpinorSol\]) (obtained by means of eq. (\[eq:WittExp\])). We find $$\mathcal{A}_{\kappa}(\omega) = \frac{\Gamma(2\nu_{\kappa})}{ \Gamma\left( 1+\nu_{\kappa} - igQ \right) }(-2i\omega)^{1/2-\nu_{\kappa}}~,~~~ \mathcal{B}_{\kappa}(\omega) = \frac{\Gamma(-2\nu_{\kappa})}{ \Gamma\left( 1 - \nu_{\kappa} - igQ \right) }(-2i\omega)^{1/2 + \nu_{\kappa}}~.$$ Using the definition of Green’s function as the quotient of the sub-leading to leading term, we find $$\mathcal{G}_{R}^{(\kappa)}(\omega) = e^{-i\pi\nu_{\kappa}}\, \frac{\Gamma(-2\nu_{\kappa})\Gamma(1 + \nu_{\kappa} -igQ)}{ \Gamma(2\nu_{\kappa})\Gamma(1 - \nu_{\kappa} -igQ)}\,(2\omega)^{2\nu_{\kappa}}~.$$ Notice the similarities with eq. (\[eq:ScalarGreenT0\]). According to the AdS/CFT dictionary, a bulk Dirac spinor $\tilde{\mathcal{Z}}_{\kappa}$ with charge $q$ is mapped to a fermionic operator $\mathcal{O}_{\kappa}$ with the same charge in the CFT at the boundary.[^11] The conformal dimension of $\mathcal{O}_{\kappa}$ is given by $$\label{eq:ScalingFermions} \Delta_{\kappa} = \frac{1}{2} + \underbrace{\sqrt{Q^2\left[ \mu^2\left( 1+\frac{\kappa^2}{\mu^2 Q^2} \right) - g^2 \right]}}_{\equiv \nu_{\kappa}\,{\rm in\,eq.~(\ref{eq:MasterSpinorSol})}}~.$$ In analogy with the scalar case, there are no poles in the retarded Green’s function in the extremal limit, and we need to investigate the near-extremal limit at non-zero temperature. The general solution of eq. (\[eq:MasterSpinor\]) at finite temperature is a combination of hypergeometric functions (we do not report here the corresponding lengthy expressions). Using standard properties of the hypergeometric functions it is possible to identify the source term at the boundary and the quotient of the sub-leading to leading term. In close analogy with eq. (\[eq:GreenScalar\]), we find for the retarded Green’s function (see also [@Faulkner:2011tm]) $$\label{eq:GreenFermion} \mathcal{G}_R^{(\kappa)}(\omega) = (4\pi T)^{2\nu_{\kappa}} \,\frac{\Gamma(-2\nu_{\kappa})\Gamma(1/2 +\nu_{\kappa} -i\omega/2\pi T + i gQ)\Gamma(1 + \nu_{\kappa} - igQ)} {\Gamma(2\nu_{\kappa})\Gamma(1/2 - \nu_{\kappa} -i\omega/2\pi T + i gQ)\Gamma(1 - \nu_{\kappa} - igQ)}~.$$ The quasi-normal frequencies are dictated by the poles of the Gamma function, and for the imaginary part we find $$\label{eq:MasterFermionQNM} {\fcolorbox{gray}{Gray}{~$\displaystyle \mathbb{Im}[\omega_{n,\kappa}] = -2\pi T\left(\frac{1}{2} + n + \mathbb{Re}[\nu_{\kappa}]\right)_{n=0,1,\dots}~,~~~~~\nu_{\kappa} = \sqrt{ Q^2\left\{ \mu^2\left[ 1 + \frac{\kappa^2}{\mu^2 Q^2} \right] - g^2 \right\}} $~}}$$ This result is remarkably similar to eq. (\[eq:MasterScalar\]). Following the discussion in section \[sec:WGC\] we concentrate on the lowest mode with $n=0$ and $l=0$. Because of the non-zero spin, we have $\kappa = -1$, as discussed in eq. (\[eq:Kappa\]). The thermalization time-scale is given by $$\tau_d \equiv \tau_d^{(0,\kappa = -1)} = \frac{1}{T\left(1/2 + \nu_{\kappa = -1}\right)}~,~~~~ \nu_{\kappa = -1} = \sqrt{ 1 + Q^2\left( \mu^2 - g^2 \right)}$$ The condition $\nu_{\kappa = -1} < 1/2$ guarantees the condition $\tau_d > 1/T$ on the thermalization time-scale and, after restoring units of $M_{\rm P}$, we find $$\label{eq:WGC3} qe > \frac{\mu}{M_{\rm P}}$$ that is the WGC. We conclude that the same proof of the WGC put forward in section \[sec:WGC\] for a charged scalar particle remains true also in the case of a Dirac fermion. Discussion and conclusions {#sec:Con} ========================== The WGC asserts a powerful consistency condition on gauge theories coupled to gravity. However, little is known about the physics from which it originates. In this paper we investigated the WGC from a thermodynamic perspective, and the main motivation to pursue this route can be summarized as follows. The original arguments that motivated the WGC were based on black hole physics, and the basic properties of black holes can be expressed as a fairly simple set of rules known as black hole thermodynamics. More ambitiously, the connection between gravity and thermodynamics lies at the heart of the AdS/CFT correspondence in which the thermodynamics of black holes is identified with thermodynamics of CFTs. All these arguments bring out the idea that gravity and thermodynamics are intimately linked. It is natural to ask whether the WGC admits a thermodynamic formulation. In this paper, a positive answer is given: [*the WGC is related to the thermalization dynamics governing the relaxation process after a perturbation, and its validity is guaranteed by the existence of a lower bound on the thermalization time-scale*]{}. The latter is known as the universal relaxation bound. This thermodynamic interpretation of the WGC was already proposed in [@Hod:2017uqc]. The novelty we added in this paper is that we came to the same conclusion by studying the dynamics of test fields – both scalar and fermions – in the near-extremal near-horizon geometry of RN black hole. This limit is particularly interesting since the isometry group of the black hole is enhanced to $AdS_2 \times S^2$ thus allowing, after KK reduction on $S^2$, the formulation of the dynamics in terms of an AdS/CFT-type correspondence. This is interesting since CFTs duals to RN black holes admit a lower bound on the thermalization time-scale of the same form of the one used to prove the WGC, thus corroborating the validity of the positivity constraints used in section \[sec:WGC\], and providing a dual description for the universal relaxation bound. A final comment about the AdS/CFT correspondence is in order. First of all, it is more correct to talk about a “near-AdS/near-CFT” (nAdS/nCFT) correspondence. In our construction, this is because the $AdS_2$ metric is obtained by carrying out a KK reduction over the sphere $S^2$. The latter has compactification radius equal to the black hole charge $Q$. Consequently, $1/Q$ characterizes the scale at which corrections to the AdS geometry become significant, and the pure AdS/CFT limit only applies at low energy and temperature. We did not explore in depth the duality in this paper (for related studies, cf. [@Nayak:2018qej; @Moitra:2018jqs]) but it is worth emphasizing that it shares remarkable similarities with the SYK model of strange metals. We stress once again that the SYK model admits a lower bound on the thermalization time-scale of the same form of the one used to prove the WGC. The SYK model realizes a $nAdS_2/nCFT_1$ correspondence between a one-dimensional quantum mechanical system consisting of $N$ Majorana fermions with random four-fermion interactions (the nCFT side of the correspondence) and a two-dimensional dilaton-gravity theory with a negative cosmological constant first studied by Jackiw and Teitelboim (JT) [@Teitelboim:1983ux; @Jackiw:1984je] (the nAdS side of the correspondence). In the SYK model the attribute near- is due to the fact that the conformal symmetry consisting of all time-reparametrization is both spontaneously and explicitly broken. On the gravity side, the $AdS_2$ geometry is broken by a non-constant dilaton field [@Maldacena:2016upp]. The connection with the setup studied in this paper is due to the fact that the near-horizon geometry of near-extremal RN black holes shares some properties with the JT model. To be more specific, the JT theory describes the s-wave sector ($l=0$) of higher dimensional theories featuring an $AdS_2$ throat geometry after compactification (e.g., cf. [@Gaikwad:2018dfc]). This is the case of the reduction from $AdS_2 \times S^2$ to $AdS_2$ exploited in this paper. In our setup, the role of the dilaton is played by the radius of the sphere $S^2$, and a non-constant profile can be obtained by introducing small fluctuations around it [@Nayak:2018qej; @Moitra:2018jqs]. In light of the computation proposed in this paper, it is interesting to explore more the connection with the JT model. We hope to clarify some of these aspects in the near future. Assuming our speculations are correct, a proper identification of the nAdS/nCFT correspondence valid for near-extremal RN black holes will put on more solid ground the universal relaxation bound $\tau_d > c/T$ that is crucial for the proof of the WGC. 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[^2]: This is an exact solution of the Einstein-Maxwell theory in which the source term in the Einstein field equations is the energy-momentum tensor of the electromagnetic field of a point charge. Consider the action $$\label{eq:GenericAction} \mathcal{S}_{\rm EM} = \frac{1}{16\pi G_N}\int d^4 x \sqrt{-g} \mathcal{R} - \frac{1}{16\pi}\int d^4 x \sqrt{-g}F_{\mu\nu}F^{\mu\nu} + \int d^4 x \sqrt{-g} \mathcal{L}_{\rm M}[\Phi_i(x),\partial_{\mu}\Phi_i(x)]~,$$ for a $U(1)$ gauge theory coupled to gravity. $\mathcal{R}$ is the Ricci scalar and $\mathcal{L}_{\rm M}$ describes, in full generality, all matter fields $\Phi_i$ appearing in the theory with equations of motion $\delta \mathcal{L}_{\rm M}/\delta \Phi_i =0$. The RN is a solution of Einstein field equations $\mathcal{R}_{\mu\nu} - \mathcal{R}g_{\mu\nu}/2 = 8\pi T_{\mu\nu}$ coupled to the Maxwell equations $\nabla_{\mu}F^{\mu\nu}=0$, with $F_{\mu\nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$. The source term is $T_{\mu\nu} = (g^{\rho\sigma}F_{\mu\rho}F_{\nu\sigma} - g_{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}/4)/4\pi$ and there is no contribution from the matter fields, $T_{\mu\nu}^{\rm M} = -\frac{2}{\sqrt{-g}}\frac{\partial \sqrt{-g}\mathcal{L}_{\rm M}}{ \partial g^{\mu\nu} } = 0$. Notice that in the geometric unit system the charge $Q$ has dimension $[Q] = [L]$ while it is dimensionless in natural units where the metric function takes the form $f(r) = 1 - 2G_N M/r + G_N Q^2/r^2$. [^3]: The inner horizon is a Cauchy horizon that is the boundary of the region which contains closed time-like geodesics. [^4]: In geometric units the function $A(r)$ is dimensionless. [^5]: Taking the near-horizon limit of the black hole metric is geometrically equivalent to a low-energy limit. This can be understood by noting that the redshift factor $f(r)$ in eq. (\[eq:RN\]) is non-constant as a function of $r$. As a consequence the energy $E$ of a particle measured by an observer at constant radial position $r$ differs from the energy of the same object measured by an observer at infinity, $E_{\infty} = E\sqrt{f(r)}$. In the near-horizon region, $f(r)$ is strictly zero as $\lambda \to 0$, and from the point of view of an observer at infinity everything in the near-horizon region is infinitely redshifted. Said differently, the redefinition of the time variable in eq. (\[eq:Near\]) implies that finite values of $\tau$ correspond to the long-time limit of the original coordinate $t$. Thinking in terms of conjugate variables, therefore, the near horizon geometry only applies to the limit of low frequencies. [^6]: In $D$ dimensions, the conformal group has $(D+1)(D+2)/2$ generators. [^7]: In our convention for the geometric units $[\mu] = [g] = [1/L]$. The combinations $\mu Q$ and $gQ$ are, therefore, dimensionless. [^8]: In the AdS background given by eq. (\[eq:NEBN\]), the boundary contributions to the norm of $\phi$ are $\lim_{z\to 0} [\mathcal{A}_l^2 z^{1-2\Delta_l} + \mathcal{B}_l^2 z^{2\Delta_l - 1} + \mathcal{A}_l\mathcal{B}_l]$, with an extra factor $1/z$ coming from the determinant of the induced AdS metric on the boundary. If $1-2\Delta_l <0 \Rightarrow \Delta_l > 1/2$, as guaranteed by the Breitenlohner-Freedman bound, the term proportional to $\mathcal{A}_l^2$ always diverges in the limit $z\to 0$ while the term proportional to $\mathcal{B}_l^2$ always vanishes. For this reason the leading term $\mathcal{A}_l$ (sub-leading term $\mathcal{B}_l$) in eq. (\[eq:Boundary\]) is also called non-normalisable (normalisable) solution. [^9]: There are two possibilities, based on dimensional analysis: $\tau_d \sim M$ and $\tau_d \sim 1/T$. For a Schwarzschild black hole, $T_{\rm Sch} = 1/8\pi M$ and the two scales are related. This is not true for a RN black hole close to the extremal limit where $M\sim Q \ll 1/T$, and we expect the thermalization time-scale to be dominated by the scaling $\tau_d \sim 1/T$. [^10]: For spin $s=1/2$ the possible values of $j$ are $j = l\pm1/2$ for $l=1,2,\dots$ and $j = 1/2$ if $l=0$. As customary, we define the relativistic angular momentum quantum number $\kappa$ as $$\label{eq:Kappa} \kappa = \left\{ \begin{array}{cc} -l -1 & j=l+1/2 \\ l & j=l-1/2 \end{array} \right.~~~~~\Longrightarrow~~~~~ \left\{ \begin{array}{cccc} s & l=0 & j=1/2 & \kappa = -1 \\ p_{1/2} & l=1 & j=1/2 & \kappa = 1 \\ p_{3/2} & l= 1 & j=3/2 & \kappa = -2 \\ \dots & & & \end{array} \right.$$ The values of $\kappa$ can be summarized as $\kappa = \mp (j+1/2)$ for $j=l\pm 1/2$. The spherical spinors in the Cartesian frame $\bar{\Phi}_{\kappa,m}^{\pm}$ are defined by [@Villalba:1994mv] $$\bar{\Phi}_{\mp(j+1/2),m}^{+} = \left( \begin{array}{c} i\Omega_{j \mp 1/2}^m \\ 0 \end{array} \right)~,~~~~~\bar{\Phi}_{\mp(j+1/2),m}^{-} = \left( \begin{array}{c} 0 \\ \Omega_{j \pm 1/2}^m \end{array} \right)~,$$ with $$\Omega_{j-1/2}^m = \left( \begin{array}{c} \sqrt{\frac{j+m}{2j}}Y_{j-1/2}^{m-1/2} \\ \sqrt{\frac{j-m}{2j}}Y_{j-1/2}^{m+1/2} \end{array} \right)~,~~~~~ \Omega_{j+1/2}^m = \left( \begin{array}{c} \sqrt{\frac{j-m+1}{2j+2}}Y_{j+1/2}^{m-1/2} \\ -\sqrt{\frac{j+m+1}{2j+2}}Y_{j+1/2}^{m+1/2} \end{array} \right)~,$$ and $-l\leqslant m\leqslant l$. The spherical spinors in the rotation frame $\Phi_{\kappa,m}^{\pm}$ are related to $\bar{\Phi}_{\kappa,m}^{\pm}$ by means of a similarity transformation [@Villalba:1994mv], and they satisfy the relations $\gamma^{(0)}\Phi_{\kappa,m}^{\pm} = \mp i \Phi_{\kappa,m}^{\pm}$, $\gamma^{(1)}\Phi_{\kappa,m}^{\pm} = \pm i \Phi_{\kappa,m}^{\mp}$, $\mathcal{K}\Phi_{\kappa,m}^{\pm} = i\kappa \Phi_{\kappa,m}^{\mp}$ with $\mathcal{K} = \gamma^{(2)}\partial_{\theta} + \gamma^{(3)}\frac{1}{\sin\theta}\partial_{\varphi}$. [^11]: It is instructive to count components of fermions. In dimensions $d= 2n$ (even) and $d= 2n+ 1$ (odd) a Dirac spinor has $2^n$ complex components and $2^{n-1}$ complex degrees of freedom. Thus in $d=4$ ($n=2$) a Dirac fermion has 4 complex components and 2 complex degrees of freedom (spin up and spin down for the particle, and further two for the anti-particle). In the $AdS_2$ bulk we have $d=2$ ($n=1$) and a Dirac fermion has 2 complex components and 1 complex degree of freedom. The mismatch between components and degrees of freedom is due to the first order nature of the Dirac action. From the Dirac Lagrangian, it follows that the momentum conjugate to the spinor $\psi$ is given by $\pi_{\psi} = i\psi^{\dag}$. The phase space of the Dirac fermion has $2\times 2^n$ real dimensions and correspondingly the number of real degrees of freedom is $2^n = 2\times 2^{n-1}$. In the context of the $AdS_2/CFT_1$ correspondence, it means that only half of the components of $\tilde{\mathcal{Z}}_{\kappa}$ correspond to a boundary fermionic operator $\mathcal{O}_{\kappa}$.
--- abstract: 'TiO$_2$ anatase doped with Co has been recently reported to exhibit room-temperature ferromagnetism. $Ab$ $initio$ study on substitutional Co doping, however, yielded much larger magnetic moment for Co than experiment. Our calculations based on density-functional theory show that the substitutional Co ions incorporated into TiO$_2$ anatase tend to cluster and then the neighboring interstitial tetrahedral sites become energetically favorable for Co to reside, yielding a local environment more like Co$_3$O$_4$ than CoTiO$_3$. The interstitial Co destroys the spin-polarization of the surrounding substitutional Co but enhances the stability of the ferromagnetism significantly. In the absence of carriers, this room-temperature ferromagnetism can only be accounted for by superexchange interaction.' author: - 'W. T. Geng$^{1,2}$' - 'Kwang S. Kim$^1$' bibliography: - 'ti.bib' title: 'Room-Temperature Ferromagnetism in Co-Doped TiO$_2$ Anatase: Role of Interstitial Co' --- The discovery of ferromagnetism in III-V semiconductors and the successful control of spin coherence of electrons injected from a magnetic semiconductor into a nonmagnetic semiconductor suggests the possibility of harnessing both charge and spin for new functionalities[@1; @2; @3]. The limiting factor that represents a serious bottleneck for their practical spintronic applications is the fact that both the observed ferromagnetism and the attractive injection phenomena are essentially limited to low temperature. The development of new materials grows rapidly. Room-temperature ferromagnetism has been observed, for instance, in Mn doped ternary compounds such as CdGeP$_2$[@CdGeP2], ZnSnAs$_2$[@ZnSnAs2], and ZnGeP$_2$[@ZnGeP2]. Using combinatorial pulsed-laser-deposition (PLD) molecular beam epitaxy (MBE) technique, Matsumoto $et$ $al$.[@science] have fabricated Co-doped anatase thin films with different Co contents on SrTiO$_3$(001) surface. At a concentration of 0.07, the film was reported to be ferromagnetic at room-temperature with a magnetic moment of 0.32 $\mu_B$ per Co. In a more recent work, Chambers $et$ $al$.[@apl] employed oxygen-plasma-assisted (OPA) MBE to grow Co-doped anatase films on the same substrate. They observed a significantly larger magnetic moment for Co ions, i.e., 1.26 $\mu_B$, which is much closer to the value of the low spin state of Co. From the comparison of the surface-sensitive X-ray photoemission spectrascopy (XPS) and bulk-sensitive X-ray absorption and emission spectroscopy (XAS) of Co-doped TiO$_2$ anatase and various other reference compounds such as Co, Co$_2$O$_3$, Co$_3$O$_4$, and CoTiO$_3$, Co ions in anatase were judged to be substitutional and in the +2 formal oxidation state. Also, they found the Co distribution, and therefore the magnetic properties depend critically on the growth condition. To gain a microscopic understanding of the magnetism of this noval materials, Park $et$ $al$.[@prb] carried out a systematic computational study on Ti$_{0.9325}$$M_{0.0625}$O$_2$ ($M$=Co, Mn, Fe, and Ni) using the linearized muffin-tin orbital (LMTO) method both with local-spin-density approximation (LSDA) and with LSDA + $U$ incorporating the on-site Coulomb correlation interaction $U$. Their calculations showed that the spin magnetic moment of Co was about 1$\mu_B$, indicating a low spin state. But they also obtained an orbital moment for Co as large as 0.9$\mu_B$, so the total magnetic moment per Co was 1.90$\mu_B$, about six times as high as the PLD-MBE result[@science], or, 50$\%$ higher than the OPA-MBE experiment[@apl]. This discrepancy is obviously too large to be acceptable. Recent density-functional computations on various diluted magnetic semiconductors such as Mn-doped GaAs[@t1; @t2] and ZnGeP$_2$[@ZnGeP2], for instance, have reproduced a magnetic moment of Mn in good agreement with experiment[@3; @ZnGeP2]. Although LSDA is not appropriate to describe strongly correlated systems, LSDA+$U$ has proved quite successful to yield a valid magnetic structure for such systems[@CoO]. Thus, the marked disagreement between experiment and theory casts some doubt on the understanding of the magnetism and its connection to the structure of Co-doped anatase. Motivated by the recent discovery that defects could play a key role in the magnetic structure of the diluted magnetic semiconductors[@zunger; @eriksson], we conjecture that clustering of substitutional Co and/or interstitial Co, if exists, might be responsible for the reduction of Co magnetization. In an attempt to clarify this point, we have performed $ab$ $initio$ density-functional calculations to study the structural, electronic, and magnetic properties of Co-doped TiO$_2$ anatase. We show that the substitutional Co ions tend to cluster, but this has no remarkable effect on the magnetization of Co. We also find that a (seriously flattened) tetrahedral interstitial site will become energetically favorable for a Co atom (in reference to bulk Co) when two of its neighboring substitutional (octahedral) sites are occupied by Co. The interstitial Co is also in the low spin state (1.04$\mu_B$), but it exerts strong detrimental effect on the magnetization of its neighboring Co (0.69$\mu_B$ $\rightarrow$ $-0.04\mu_B$) and therefore reduces the magnetic moment Co in this magnetic semiconductor. While substitutional Co ions turn anatase half-metallic, the interstitial ones bring the material back to a semiconductor. In absence of carriers, the exchange interaction stablizing the ferrimagnetism is of superexchange type. In the following, we give a brief outline of our theoretical procedure, a discussion of the results based on bonding characteristics and its relevance to our understanding of the magnetism in this Co-doped oxide. The calculations were done using a spin-polarized version of the Vienna [*ab initio*]{} Simulation Package (VASP)[@vasp1]. Fully nonlocal Vanderbilt-type ultrasoft pseudopotentials[@pp] were used to represent the electron-ion interaction. Exchange correlation interactions were described by the Perdew-Wang 1991 generalized gradient approximation (GGA)[@pw91]. Wave functions and augmentation charges were expanded in terms of plane waves with an energy cutoff of 396 and 700 eV, respectively. The Brillouin zone integration was performed within Monkhorst-Pack scheme[@k] using a (2$\times$2$\times$2) $k$ mesh for geometry optimization and a (4$\times$4$\times$4) mesh for plotting the density of states(DOS). For each system, the geometry was relaxed until the atomic forces were smaller than 0.03 eV/Å. To gain a confidence for applying pseudopotentials to such an oxide with open structure, we first conducted a comparitive study of the structural properties of non-doped anatase (see Figure 1a) using both VASP and the highly precise all-electron full-potential linearized augmented plane wave (FLAPW) method[@flapw] with WIEN97 implementation[@wien]. FLAPW is considered to be more accurate but also more computationally expensive. The same GGA functional and $k$ mesh ($8\times 8\times 4$) were used in both calculations. The optimized structural parameters for anatase are listed in Table I. Also listed are experimental values[@tio2] and a previous FLAPW + LDA result[@asahi]. It is evident that VASP agrees fairly well with WIEN97, and that GGA results are more consistent with experiment than LDA. For Co-doped anatase, we mainly dealt with a supercell containing 16 primitive unit cells ($2a\times 2a\times c$, see Figure 1b) with one or two Ti atoms (site B1, or, sites B1 and B2) replaced by Co and with/without an interstitial Co (site A). In view of the fact that the calculated $c/a$ value, 2.46, of a hyperthetical CoO$_2$ in anatase structure is close to that of TiO$_2$ anatase, we fixed $c/a$ at the latter value when optimizing the volume of Co-doped anatase to save computational effort. To see whether substitutional Co (Co$^S$) cluster or not, we first minimized the free energy of the cell Ti$_{15}$Co$^S_{B1}$O$_{32}$. Upon Co-doping, the volume of such a unit cell contracted by 1.1$\%$, in accordance with the fact that Co has a smaller covalent radius than Ti[@pauling]. Then we doubled this cell along $x$ direction and brought the two Co$^S$ together to form Ti$_{30}$Co$^S_{B1}$Co$^S_{B2}$O$_{64}$ ($4a\times 2a\times c$). We assumed that there was no volume change upon clustering, and optimized only the internal freedoms for the 96 atom cell. We find the latter configuration is more stable by 0.14eV/Co, indicating that Co clustering does occur. The calculated formation energy of an interstitial Co (Co$^I$) as a neutral defect, $E$(Ti$_{16}$Co$^I_A$O$_{32}$)-$E$(Ti$_{16}$O$_{32}$)-$E$(Co), is +2.07 eV, implying a sole Co$^I$ is highly unstable in reference to bulk Co. This can be understood in the bond-order band-strength picture. Ti:O + Co:O is much weaker than O:Ti:O, as is evident from the comparison of the formation heat of TiO$_2$, TiO, and CoO[@CRC]. Nonetheless, with the presence of Co$^S_{B1}$, the formation energy of Co$^I$, i.e., $E$(Ti$_{15}$Co$^S_{B1}$Co$^I_A$O$_{32}$)-$E$(Ti$_{15}$Co$^S_{B1}$O$_{32}$)-$E$(Co), drops drastically to +0.18 eV. According to our calculations, the formation heat of CoO$_2$ in a hypothetical anatase structure is $-3.66$ eV, slightly higher than two times of that of CoO ($-1.59$ eV). This means that Co:O + Co:O is only slightly weaker than O:Co:O , thus Co$^I$ becomes much less unstable. Further calculations show that when both B1 and B2 site are occupied by Co upon clustering, site A becomes favorable for a Co$^I$ with a formation energy of $-0.49$ eV. Once again, this can be explained by the higher formation heat of Co$_3$O$_4$ ($-9.57$ eV) than two times of that of CoO$_2$ ($-3.66$ eV). Our calculations thus reveal a 2Co$^S$+Co$^I$ local structure in Co-doped TiO$_2$ anatase. Table II displays the calculated Co-O bond length in different local environments. The overall feature is that the Co-O bond is shorter than Ti-O bond. As mentioned above, this is because both octahedral and tetrahedral Co ions have a radius smaller than octahedral Ti ions. Co-O bond lengths along the $y+$ and $y-$ directions, $d_{y+}$ and $d_{y-}$, do not change upon clustering in the case that Co(B1) and Co(B2) are in the same $x-z$ plane; whereas along the $x$ and $z$ directions the bond lengths are contracted by $0.02-0.05$ Å. The presence of an interstitial Co pushes the surrounding atoms a bit further apart and increases the Co-O bond length in the case of Co(B1) and Co(B2). The local lattice distortion induced by Co$^S$ or Co$^I$, however, is rather small. It is worth noting that the volume effect on chemical bonding is only marginal. A sole Co$^I$ yields a volume expansion of about 1.0$\%$ to Ti$_{16}$O$_{32}$, and the formation heat of Ti$_{16}$Co$^I_A$O$_{32}$ at the volume of Ti$_{16}$O$_{32}$ is only 0.05 eV lower than the optimized value. To examine the effect of clustering on magnetism, we performed calculations on both intra-cell FM and intra-cell AFM alignments of the Co pair in a Ti$_{14}$Co$^S_{B1}$Co$^S_{B2}$O$_{32}$ unit cell (For simplicity, we denote this cell as Ti$_{14}$Co$_2$O$_{32}$ in the following) while keeping the inter-cell coupling ferromagnetic. The FM phase is found to be lower in total energy by 0.09 eV than the AFM phase. When the two Co ions are far apart, this energy difference is 0.08 eV[@prb]. Apparently, clustering has negligible effect on the spin-polarization of this magnetic oxide. In the case of 2Co$^S$+Co$^I$, we checked all the three possible intra-cell spin alignments of the three Co ions. All of them converge to the same final state, with both Co$^I$ and Co$^S$ in the low spin state. We then double the unit cell along $x$ direction and computed the inter-cell AFM coupling between two 2Co$^S$+Co$^I$ complexes. It is found to be 0.35 eV higher in energy than the FM coupling. Obviously, with a much enhanced FM-AFM energy difference, the formation of 2Co$^S$+Co$^I$ will raise remarkably the Curie temperature of Co-doped anatase, in accordance with the observed room-temperature ferromagnetism[@science; @apl]. The calculated total spin magnetic moment in a unit cell of Ti$_{15}$Co$_1$O$_{32}$, Ti$_{14}$Co$_2$O$_{32}$, and Ti$_{14}$Co$_3$O$_{32}$, and that in the atomic sphere of Co are listed in Table III. It is seen that in all three cases, Co ions are in the low spin state, regardless to their oxidation states. The spin moment of Co$^S$ does not change much upon clustering. Co$^I$, on the other hand, behaves rather differently. When coupled with Co$^S$, Co$^I$ kills almost entirely the spin moments of Co$^S$. As a result, the average spin magnetic moment of Co is reduced to 0.32 $\mu_B$, less than a half of the non-Co$^I$ case. This means that if there are a remarkable amount of 2Co$^S$+Co$^I$ complexes in Co-doped anatase, the magnetic moment per Co atom would be reduced significantly and this could explain the disagreement between experiments[@science; @apl] and the previous first-principles calculations[@prb]. We note that the oxygen-rich condition during OPA-MBE growth in Chambers $et$ $al.$’s work[@apl] prevented the formation of extensive 2Co$^S$+Co$^I$ complexes, hence a much higher magnetic moment for Co than that in the film grown by Matsumoto $et$ $al.$[@science] with PLD-MBE; first-principles calculations[@prb] gave a even higher magnetic moment for Co$^S$, as a result of the absence of Co$^I$. In oxygen-poor condition, on the other hand, the oxygen vacancies help the diffusion of the Co ions, resulting in the formation of Co metal clusters[@cluster]. Figure 2 displays the calculated density of states for non-doped and various cases of Co-doped anatase. Clearly, the Co states are mainly located in the energy-gap region and the O and Ti states are not much affected by Co doping. A comparison of Ti$_{15}$Co$_1$O$_{32}$ and Ti$_{14}$Co$_2$O$_{32}$ indicates clustering does not change the half-metallic electronic structure, and as a consequence the magnetic interaction between Co ions remains to be carrier-mediated Zener-like.[@zener] Sole Co$^I$ ions turn the material into a metallic system with a highly lifted fermi level. Very interestingly, our calculations show a complex of 2Co$^S$+Co$^I$ brings the system back to a semiconductor, in consistence with the experimental observation[@science; @apl]. To gain a deeper insight into the electronic structure of Co, we compare $d$ DOS of Co in anatase with those of Co in other Co oxides with different oxidation states such as CoO$_2$ (+4), CoTiO$_3$ (+2), and Co$_3$O$_4$ (+3 and +2). CoO$_2$, the end member of LiCoO$_2$, has recently been isolated[@coo2_1]. It seems to be of CdCl$_2$ type with a monoclinic distortion, but positions of O remain to be determined. In this regard, we believe a study on a hypothetical CoO$_2$ in anatase structure is meaningful in illustrating the electronic structure of Co$^{4+}$. As mentioned above, we have optimized all of its structural parameters. For CoTiO$_3$ and Co$_3$O$_4$, experimental lattice constants[@wyckoff] were adopted and only internal freedoms were optimized. The calculated $d$ DOS of Co in CoO$_2$, CoTiO$_3$, and Co$_3$O$_4$, and that in Co-doped anatase are plotted in Figure 3. We observe that although much more localized, $d$ DOS of Co$^S$ in Ti$_{15}$Co$_1$O$_{32}$ and Ti$_{14}$Co$_2$O$_{32}$ resembles that of Co in CoO$_2$ in both exchange splitting and crystal field splitting, suggesting they are in +4 oxidation state, as was also proposed in Ref.\[9\]. In an octahedral field, $t_{2g}$ is lower than $e_g$ and thus Co$^{4+}$ will have a spin magnetic moment of 1$\mu_B$. On the other hand, Co$^I$ in Ti$_{16}$Co$_1$O$_{32}$ reminds us of Co in CoTiO$_3$. This resemblance reveals Co$^I$ is in +2 oxidation state in a Ti$_{16}$O$_{32}$ environment. In a normal tetrahedral field, $e_g$ level is lower than $t_{2g}$. Co$^I$, nevertheless, is in a seriously flattened tetrahedral field, thus the crystal field splitting states is comparable to the exchange splitting, yielding an intermediate spin state for Co$^I$. As for the combination 2Co$^S$+Co$^I$, an even better similarity can be found between Co$^S$ (Co$^I$) in anatase and Co$^B$ (Co$^A$) in Co$_3$O$_4$. It thus provides a strong evidence that Co$^S$ and Co$^I$ in such a combination are in +3 and +2 oxidation states, respectively. Similar to Co$^B$ in Co$_3$O$_4$, Co$^S$ in an octahedral crystal field has a vanishing spin-polarization, and the contribution to magnetism comes only from Co$^I$. To conclude, our prediction of the formation of 2Co$^S$+Co$^I$ resolves the disagreement between experiment and theory on the magnetic moment of Co in Co-doped TiO$^2$ anatase. The insulating property is reproduced without employing an on-site Coulomb correlation interaction $U$. Superexchange, rather than carrier-mediated Zener-like magnetic interaction, should be responsible for room-temperature ferromagnetism. The novel strucure discoverd in this work may also exist in other transition elements such as Mn- and Fe-doped anatase and may thus resolve the discrepancy between experiment[@ass] and theory[@prb]. Finally, we want to note that the formation of more complicated (Co$^S$, Co$^I$) combinations is also possible depending on the growth condition, but the formation probability should be lower than 2Co$^S$+Co$^I$. Work in Korea was supported by Korea Institute of Science and Technology Evaluation and Planning (Creative Research Initiative) and in Germany by Max-Planck-Society Fellowship. W.T.G. is grateful to Professor B. I. Min for helpful discussions. --------- -------- ------- ------- ------- WIEN97 VASP Expt. FLAPW (GGA) (GGA) (LDA) $a$ (Å) 3.826 3.825 3.782 3.692 $c$ (Å) 9.706 9.678 9.502 9.471 $c/a$ 2.537 2.530 2.512 2.566 $u$ 0.207 0.207 0.207 0.206 --------- -------- ------- ------- ------- : \[tab:table1\] Optimized structural parameters for anatase TiO$_2$: A comparison of all-electron full-potential method (WIEN97), pseudopotential method (VASP), experiment[@tio2], and a previous FLAPW LDA study[@asahi]. ---------- -------- -------- -------- -------- -------- ------- Co(B1) Co(B1) Co(B2) Co(B1) Co(B2) Co(A) $d_{x+}$ 1.91 1.85 1.93 1.93 1.98 1.92 $d_{x-}$ 1.91 1.93 1.85 1.88 1.93 1.91 $d_{y+}$ 1.91 1.91 1.91 1.96 1.97 1.91 $d_{y-}$ 1.91 1.91 1.91 1.90 1.93 1.85 $d_{z+}$ 1.89 1.84 1.95 1.93 1.98 - $d_{z-}$ 1.89 1.94 1.84 1.96 1.98 - ---------- -------- -------- -------- -------- -------- ------- : \[tab:table2\] Optimized Co-O bond lengths (Å) in Co doped anatase TiO$_2$. It is denoted as $d_{x+}$ ($d_{y+}$, $d_{z+}$) when O is in the $x+$ direction of Co, and as $d_{x-}$ ($d_{y-}$, $d_{z-}$) when O is in the $x-$ direction of Co. Co(B1), Co(B2), and Co(A) denote Co ions occupying sites B1, B2, and A (see Figure 1b). Unit cell Co(B1) Co(B2) Co (A) Total -------------------------- -------- -------- -------- ------- Ti$_{15}$Co$_1$O$_{ 32}$ 0.73 - - 1.00 Ti$_{14}$Co$_2$O$_{ 32}$ 0.69 0.69 - 2.00 Ti$_{14}$Co$_3$O$_{ 32}$ -0.01 -0.07 1.04 1.00 : \[tab:table3\] Claculated spin magnetic moment ($\mu_B$) in the atomic sphere of Co (radius = 1.10 Å) and that for a whole unit cell.
--- author: - 'Shengling Wang,  Peizi Ma, Qin Hu, Xiuzhen Cheng,  and Weifeng Lv' bibliography: - 'IEEEabrv.bib' title: 'Egoistic Incentives Based on Zero-Determinant Alliances for Large-Scale Systems\' --- Introduction {#sec:intro} ============ Many human decisions occur in situations where the results of one’s own decisions are interdependent with those of others. Such interdependence situations tend to breed social dilemmas, which has two traits: 1) each individual who makes social defective choices gains higher profits no matter what other individuals do; 2) comparing with the situation where everyone cooperates, if everyone chooses to defect, then all individuals get lower returns [@Dawes1980Social]. Therefore, a social dilemma is essentially an inherent conflict between defection and cooperation in a system where the former is the dominant strategy for each individual but the latter can maximize the overall social welfare. Social dilemmas exist in various scenarios such as data delivery in mobile opportunistic networks [@wangicdcs2017], file sharing in peer-to-peer networks [@hu2009budget], and autonomous vehicle programming [@science]. A typical issue resulted from a social dilemma is [*free-riding*]{}, an individually rational but socially defecting choice which can lead to a collective fiasco. Moreover, such a [*tragedy of the commons*]{} would be aggravated as the system expands because the behavior of an individual becomes more difficult to track so that the influence on others diminishes [@pnus]. Hence, there is a pressing need to induce cooperation, especially in a large-scale system. Cooperation in a social dilemma is often explained in terms of [*egoistic incentives*]{} [@zhao]. The state-of-the-art egoistic incentive mechanisms realize their aims by transforming a social dilemma game into one not involving a dilemma, which can be categorized into two types: reputation-based [@Lu2010Pi; @uddin2010relics; @hu2009budget; @chen2015multicent; @Ning15] and pricing-based [@ning13; @chen; @Koutsopoulos; @Dejunmobicom]. Reputation-based schemes employ reputation or quasi-reputation to evaluate a node’s contribution to others, based on which reacts to its service requests, while pricing-based mechanisms treat service provision as a transaction, taking advantage of monetary incentives to stimulate cooperation. Existing incentive mechanisms have two common traits: they all are *individual-based* and involve *state (reputation or transaction status) maintenance and management*. The first trait comes from the underlying concept in which the success of cooperation inducement to each individual inevitably results in that of the whole system. Such a case-by-case based approach is obviously inefficient as the size of the system enlarges. The second trait exists due to the state-dependent nature of the state-of-the-art methods. Regretfully, the cost of maintaining and managing states would soar in a large-scale system. In this paper, we realize large-scale egoistic incentives from the perspective of [*the network*]{} rather than [*an individual*]{}. Our idea stems from the following observation: each person in this world is more or less socially connected to others, forming a so-called [*social community*]{}. Within such an environment, cooperation and competition among all members jointly determine the utility of an individual. Since each individual is utility-driven, cooperation inducement requires us to analyze the game among all players in the social community, which involves the structure of their social ties. Hence, it is reasonable to consider incentives from the [*connected*]{} (network) perspective rather than the [*isolated*]{} (individual) perspective to stimulate cooperation. Moreover, directly incentivizing the whole system has the potential to bring high efficiency to a large-scale system. Our egoistic incentive approach has two desired properties: [*statelessness*]{} and [*stability*]{}. These merits are obtained by taking advantage of the zero-determinant (ZD) strategy [@press2012iterated] whose adopter (the ZD player) can control its opponent’s payoff in a unilateral way. Thus, through rewarding cooperation and negatively sanctioning defection, a ZD player can stimulate cooperation of its co-players. After proving cooperation is the dominant strategy of the ZD players, we optimize their deployment, facilitating cooperation over the whole system. Our method is stateless because it does not need to manage or maintain any state; it is stable since the deployment of the ZD players only depends on the social ties among the players, which are usually steady. The statelessness and stability contribute to the sound scalability of our egoistic incentive. Another contribution of this paper is the derivation of a ZD alliance strategy for sequential multiple-player repeated games, where a ZD alliance refers to a group of players taking the same ZD strategy. Such a derivation brings two benefits. First, the alliance strategy widens the range of a co-player’s utility that ZD players can set, implying that ZD players can achieve higher controllable leverage through allying. The increased controlling power of a ZD alliance breeds an environment where cooperation thrives. Second, our derivation enriches the theoretical system of ZD strategies, broadening their application domain. We conduct extensive simulations based on real-world trace data as well as synthetic data, which represent different types of typical network topologies including [*star*]{}, [*ring*]{}, [*tree*]{}, and [*mesh*]{} structures. These simulation results demonstrate the effectiveness of our egoistic incentive approach. The rest of the paper is organized as follows. The most related work is investigated in Section \[sec:related\]. Section \[sec:formulation\] presents our problem formulation. The ZD alliance strategy for a sequential multiple-player repeated game is deduced in Section \[sec:zd\], and our incentive algorithm is proposed in Section \[sec:op\]. We report our simulation results on the proposed algorithm in Section \[sec:simu\], and Section \[sec:conclusion\] concludes the paper. Related Work {#sec:related} ============ Over the last two decades, research on cooperation induction has made considerable progress, which can be categorized into two types: reputation-based and pricing-based. Reputation-based methods employ the concept of reputation or quasi-reputation to evaluate the behaviors of a node based on its contributions to others, which is also a criterion for reacting to the service requests of this node. For example, in [@Lu2010Pi], relay nodes would get good reputation values for their cooperation, which can build other nodes’ confidence on them thus helping forward their bundles. The in-network realization of incentives was proposed in [@uddin2010relics] to attach an explicit ranking to a node in light of its transit behaviors and translate the rank into message priority. Hu [*et al.*]{} [@hu2009budget] proposed a budget-based self-optimized incentive search protocol for unstructured P2P file sharing systems, motivating selfish nodes to earn more credits by providing services to others. As a game theoretical incentive, [*Multicent*]{} [@chen2015multicent] assigns credits for packet forwarding/storage in proportional to the priorities specified in the routing strategy. The concept of [*virtual credit*]{} was adopted in [@Ning15] to encourage selfish nodes to cooperate in data forwarding. Pricing-based methods take advantage of monetary incentives to stimulate cooperation. For instance, Ning [*et al.*]{} [@ning13] introduced the concept of [*virtual checks*]{} to pay the cooperation of selfish nodes for ad dissemination in autonomous mobile social networks. Chen [*et al.*]{} [@chen] proposed an auction-based incentive mechanism for paid content offloading considering the dual identity of service providers. Koutsopoulos [@Koutsopoulos] cast participatory sensing in the context of optimal reverse auction design, offering reasonable payments for data contributors. Yang [*et al.*]{} [@Dejunmobicom] also proposed an auction-based mechanism to incentivize participants of mobile phone sensing while allowing users to have more control over the payment they can receive. Both reputation-based and pricing-based egoistic incentives aim to shape the behaviors of individuals and involve maintaining and managing the reputation or transaction states, leading to low efficiency in a large-scale system. In this paper, we take a dramatically different approach from the perspective of [*a network*]{} rather than [*an individual*]{} to achieve good scalability by employing a ZD strategy. ZD strategies [@press2012iterated] were firstly proposed by Press and Dyson in 2012, providing us a revolutionary understanding of simultaneous-move repeated games. ZD strategies can enforce a fixed linear relationship between the expected payoffs of two players, indicating that a ZD player can control its opponent’s payoff in a unilateral way. Since ZD strategies were proposed, several studies have been carried out to enrich their theoretical hierarchy. For example, the application of ZD strategies was extended from a two-player simultaneous-move game to a multi-player one in [@zhoutao]. The concept of [*ZD alliance*]{} was first proposed in [@pnus], whose strategy was explored for simultaneous-move multi-player games. In this paper, we employ a different theoretical approach to deduce the strategy of a ZD alliance for sequential multi-player games to serve our problem scenario. Problem Formulation {#sec:formulation} =================== To analyze the behaviors of the players from a network perspective, we employ an indirect graph $G=(V,E)$ to describe a large-scale system, where $V$ is the set of nodes representing the players in the system and $E$ is the set of edges denoting the social ties of the players. Each player has two choices in the game, namely [*cooperation*]{} $c$ and [*defection*]{} $d$. For example, in mobile opportunistic networks, $c$ means a player’s willingness to transmit data for others and $d$ implies rejecting relay services. Even though $c$ and $d$ have different meanings in different scenarios, they own common traits: the former is driven by social interests while the latter involves that of an individual. In practice, an application may involve multiple rounds and each player may choose $c$ or $d$ within a round; and some players may move first while the others take actions after observing the first movers. As a result, we focus on a sequential repeated game with multiple players in this paper. In any game, a rational player aims to maximize its utility, which depends on not only the action of itself but also those of others. In this paper, we adopt a method similar to that in [@pnus; @zhoutao] to calculate the utility $U_i(a_i|j)$ of a player $i$ when the number of cooperators among its neighbors is $j$: $$\label{eq:u} U_i(a_i|j) = \frac{r(N_i)\big(j+a_i\big)}{N_i+1}+(1-a_i), ~a_i\in \{0,1\},$$ where $a_i=1$ means player $i$ selects cooperation while $a_i=0$ implies defection; $N_i$ is the number of neighbors of player $i$, and obviously $0 \leq j\leq N_i$; and $r(N_i)$ is the profit proportional to $N_i+1$. We use $r(N_i)=r$ in sequel for simplicity. \[dilemma\] A social dilemma happens when $r>1$. According to [@pnas32], a social dilemma occurs when the utility of each player possesses the following properties: 1) a player, regardless its decision, can get a higher utility when more co-players select cooperation; 2) the utility of a defector is higher than that of a cooperator; and 3) mutual cooperation outperforms mutual defection. As a result, the following inequalities should be satisfied, which are built based on the above three properties: $$\begin{aligned} \begin{cases} U_i(a_i|j+1) \geq U_i(a_i|j),~ 0 \leq j \leq N_i-1, \notag \\ U_i(0|j+1)>U_i(1|j),~ 0 \leq j \leq N_i-1, \notag \\ U_i(1|N_i)>U_i(0|0). \notag \end{cases}\end{aligned}$$ According to (\[eq:u\]), the first two inequalities obviously hold without any constraint while for the last one, it is sufficient that $r>1$. Hence, the theorem is proved. According to Theorem \[dilemma\], a defector can obtain a higher utility than a cooperator. However, once all players adopt defection to act as [*free riders*]{}, each of which only gets the utility of 1 according to (\[eq:u\]), which is lower than the utility of $r$ when all players cooperate because $r>1$, leading to the tragedy of the commons. To deter free riders, a natural idea is to teach players to get the cognition that cooperation outperforms defection, synchronizing self-interest with the common one. This requires an egoistic incentive by an explicit side payment in the form of rewarding cooperation while punishing defection. However, it is inefficient to employ centralized methods for the egoistic incentives especially in a large-scale system due to the lack of flexibility, robustness, and [*the single point of failure*]{} issue. Hence, in this paper, we resort to a distributed one. To realize a distributed egoistic incentive, we take advantage of ZD strategies [@press2012iterated]. A ZD player can set its opponent’s payoff irrespective of what action the opponent would take. Such capability is very valuable for shaping the behavior of free riders. Hence, in this paper, we address the social dilemma problem through the deployment of ZD players forming an alliance in the network. We call players not adopting the ZD strategy [*regular players*]{}. Then, as shown in Fig. \[fig:tt\], in a network with heterogeneous players, the utility $\widetilde{U_i}$ of any regular player $i$ may consist of two components: the one obtained from the game playing with other regular players and the other from playing with the ZD alliance. The former game is called [*a regular game*]{} while the latter one is called [*a ZD game*]{}. Particularly, $$\label{eq:newu} \widetilde{U_i}(a_i|n_i,n^{\mathcal{A}}) = U_i(a_i|n_i)+ u_i(a_i|n^{\mathcal{A}}), \\~a_i\in \{0,1\}.$$ In (\[eq:newu\]), $U_i(a_i|n_i)$ is the utility obtained from the regular game when player $i$ acts $a_i \in \{0,1\}$ playing with $n_i$ cooperators among the regular neighbors, which can be calculated by (\[eq:u\]); and $u_i(a_i|n^{\mathcal{A}})$ is the utility set by the ZD players when player $i$ acts $a_i \in \{0,1\}$ and there are $n^{\mathcal{A}}$ ZD players as its neighbors. ![Two types of games involving any regular player.[]{data-label="fig:tt"}](twotype){width="30.00000%"} Let the probability of a player choosing $c$ or $d$ be its strategy. Being rational, a player chooses a strategy according to the utility brought by cooperation or defection, which is also affected by the co-players’ moves. Hence, we use the following equation, similar to the one in [@zhao], to determine the strategy of player $i$ at each round: $$\label{eq:newstrategy} q_i(c) = \frac{e^{\widetilde{U_i}(1|n_i, n^{\mathcal{A}})-\widetilde{U_i}(0|n_i+1,n^{\mathcal{A}})}}{1+e^{\widetilde{U_i}(1|n_i,n^{\mathcal{A}})-\widetilde{U_i}(0|n_i+1,n^{\mathcal{A}})}}.$$ In (\[eq:newstrategy\]), $q_i(c)$ is the probability that player $i$ cooperates. Obviously, the probability that player $i$ defects is $1-q_i(c)$. Here, $q_i(c)$ depends on the difference between $\widetilde{U_i}(1|n_i, n^{\mathcal{A}})$ and $\widetilde{U_i}(0|n_i+1,n^{\mathcal{A}})$, the utilities of player $i$ adopting $c$ and $d$, respectively, when the number of regular cooperators involving in the game is $n_i+1$ and there are $n^{\mathcal{A}}$ ZD players in the social community[^1]. Note that both the number of regular cooperators and that of ZD players can be obtained according to historical information of games. As a result, our optimization problem turns out to be the following: [*how to deploy ZD players to maximize the overall cooperation probability in the whole system, where each regular player’s cooperation probability is calculated by (\[eq:newstrategy\])?*]{} ZD alliance strategy in sequential multiple-player repeated game {#sec:zd} ================================================================ To solve our optimization problem, we need to determine $u_i(a_i|n^{\mathcal{A}})$ in (\[eq:newu\]), the utility set by $n^{\mathcal{A}}$ ZD players when a regular player $i$ acts $a_i\in \{0,1\}$. Note that for any player $i$, there might exist more than one ZD players in its neighborhood. In this case, multiple ZD players must take the same action for the same move made by player $i$ because they should abide by consistent rules for being fair-minded regulators in our approach. The ZD players taking the same action form a [*ZD alliance*]{}. In this section, we need to analyze how a ZD alliance sets the utility of a regular player to determine $u_i(a_i|n^{\mathcal{A}})$. Particularly, when $n^{\mathcal{A}}=1$, a ZD alliance strategy regresses to a classical ZD strategy. Note that, the original or variant ZD strategies [@press2012iterated; @zhoutao; @pnus] are only applicable to simultaneous-move repeated games with two or more players, which are not suitable for our scenario. Hence, we first extend the application domain of ZD strategies for a sequential multiple-player repeated game. In a sequential multiple-player repeated game, players are classified into two types: the first-move players and the second-move ones. We call them [*leaders*]{} and [*followers*]{} respectively in this paper. No matter which type a player belongs to, its utility is calculated according to (\[eq:u\]), which is affected by the co-players’ moves. Because a leader has to move first, its strategy is made based on all players’ actions in the previous round. Let $\mathbf{p}^{i}$ be the strategy of leader ${i}$. We have $$\begin{aligned} \label{p} \mathbf{p}^{i}=(&p_{c,n^{\mathcal{L}}-1,N-n^{\mathcal{L}}}^{i},\cdots,p_{c,n^{\mathcal{L}}-1,0}^{i},\cdots,p_{c,0,0}^{i},\\ &p_{d,n^{\mathcal{L}}-1,N-n^{\mathcal{L}}}^{i},\cdots,p_{d,n^{\mathcal{L}}-1,0}^{i},\cdots,p_{d,0,0}^{i}),\\ &{i}\in \{1,2,\ldots, n^{\mathcal{L}}\}, \end{aligned}$$ where each element can be represented in the form of $p^i_{s_i,x, y}$, the probability of leader $i$ choosing $c$ given that it played $s_i \in \{c,d\}$ previously and the numbers of cooperators among other leaders and followers were respectively $x$ and $y$ in the previous round; and $n^{\mathcal{L}}$ and $N$ are respectively the numbers of leaders and all players in this game. Since a follower can observe the behavior of the leaders in the current round, its strategy depends on the leaders’ moves. Let $\mathbf{q}^{j}$ be the strategy of follower $j$, which can be represented as $$\label{q} \mathbf{q}^{j}=(q_{n^{\mathcal{L}}}^{j},q_{{n^{\mathcal{L}}}-1}^{j},\cdots,q_{0}^{j}), j \in \{n^{\mathcal{L}}+1,n^{\mathcal{L}}+2,\ldots, N\},$$ where each element can be written in the form of $q_{z}^{j}$, the probability of follower $j$ choosing $c$ when there are $z$ cooperators among the leaders. With the definitions of $\mathbf{p}^{i}$ and $\mathbf{q}^{j}$, $i \in \{1,2,\ldots, n^{\mathcal{L}}\}$ and $j \in \{n^{\mathcal{L}}+1,n^{\mathcal{L}}+2,\ldots, N\}$, we can construct the following Markov matrix $$\mathbf{M}=[M_{vw}]_{2^N\times2^N}, \nonumber$$ where each element $M_{vw}$ denotes the one-step transition probability from state $v$ to $w$. Here, a state denotes the moves made by all players in this round. The state transition probability is essentially a joint one that can be obtained by $$M_{vw}=\prod_{i=1}^{n^{\mathcal{L}}}\alpha_{i}\prod_{j=n^{\mathcal{L}}+1}^{N}\beta_{j}, \nonumber$$ in which $\alpha_i$ and $\beta_j$ respectively denote the probabilities of actions made by leader $i$ and follower $j$ in state $w$ and they can be calculated as $$\label{a} \alpha_i=(p^i_{s_i,x,y})^{a_{i}}(1-p^i_{s_i,x,y})^{1-a_i}$$ and $$\label{b} \beta_{j}=(q_{z}^{j})^{a_j}(1-q_{z}^{j})^{1-a_j}.$$ In (\[a\]), $p^i_{s_i,x,y}$ also denotes the probability of leader $i$ cooperating when it played $s_i \in \{c,d\}$ and the numbers of cooperators among other leaders and followers were respectively $x$ and $y$ in the last round (state $v$); and $a_i \in \{0,1\}$ is the action of leader $i$ made in state $w$. In (\[b\]), $q_{z}^{j}$ is also the cooperation probability of follower $j$ when there are $z$ cooperators among the leaders in state $w$ and $a_j \in \{0,1\}$ is its action in state $w$. Define a matrix $\mathbf{M'}=\mathbf{M}-\mathbf{I}$, where $\mathbf{I}$ is the unitary matrix. Let $\mathbf{v}$ be the stable vector of the transition matrix $\mathbf{M}$; then we have $\mathbf{v}^\mathrm{T} \mathbf{M} = \mathbf{v}^\mathrm{T}$; hence $\mathbf{v}^\mathrm{T} \mathbf{M'} = 0$. Denote $[M_1, M_2, \ldots, M_m, \ldots, M_{2^N}]$ by $\mathbf{M'}$, where $M_i$ is the $i^{th}$ column vector. In light of Cramer’s rule, $Adj(\mathbf{M'})\mathbf{M'}=det(\mathbf{M'})\mathbf{I}=\mathbf 0$, where $Adj(\mathbf{M'})$ is the adjugate matrix of $\mathbf{M'}$. Combining this equation with $\mathbf{v}^\mathrm{T} \mathbf{M'} = 0$, we know that $\mathbf{v}$ is proportional to each row of $Adj(\mathbf{M'})$ [@pnus]. Therefore, the dot product of the stable vector $\mathbf{v}$ and any vector $\mathbf{f}=(f_1,f_2,\cdots,f_{2^{N}-1},f_{2^{N}})$ is $$\mathbf{v} \cdot \mathbf{f} = \text{det}~[M_1, M_2, \ldots, M_m, \ldots, M_{2^N-1},\mathbf{f}]. \nonumber$$ Now we apply an elementary column transformation to $[M_1, M_2, \ldots, M_m, \ldots, M_{2^N-1},\mathbf{f}]$: find a column corresponding to a state where only one leader cooperates (i.e., the element of that column is in the form of $p^i_{s_i,x,y}\prod_{j\neq i}(1-p^j_{s_j,x,y})\prod_k(1-q_1^k),~ i\in \{1,2,\ldots, n^{\mathcal{L}}\}$, $j\in \{1,2,\ldots, n^{\mathcal{L}}\}$, $k \in \{n^{\mathcal{L}}+1,n^{\mathcal{L}}+2,\ldots, N\}$) and a set of columns (denoted as $\Theta$) corresponding to the states where player $i$ and at least one co-player cooperates (i.e., each element of those columns is represented as $p^i_{s_i,x,y}\prod_{j\neq i}\alpha_j\prod_k\beta_{k}$, $i\in \{1,2,\ldots, n^{\mathcal{L}}\}$, $j\in \{1,2,\ldots, n^{\mathcal{L}}\}$, $k \in \{n^{\mathcal{L}}+1,n^{\mathcal{L}}+2,\ldots, N\}$, $\exists j$, $a_j=1$ $\vee$ $\exists k$, $a_{k}=1$); then add all columns in $\Theta$ to the first column we found to form a new column whose element is $p^i_{s_i,x,y}\mathcal{T}-1$ if a diagonal element of $\mathbf{M'}$ is added to this entry and otherwise is $p^i_{s_i,x,y}\mathcal{T}$, where $\mathcal{T}=\prod_{j\neq i}(1-p^j_{s_j,x,y})\prod_k(1-q_1^k)+\sum_{\Theta}\prod_{j\neq i}\alpha_j\prod_k\beta_{k}$. Obviously, $\mathcal{T}=1$. After the elementary column transformation, the matrix $[M_1, M_2, \ldots, M_m, \ldots, M_{2^N-1},\mathbf{f}]$ has a new form, i.e., $$\label{vf} [M_1, M_2, \ldots, M'_m, \ldots, M_{2^N-1},\mathbf{f}], \nonumber$$ where $M'_m$ is the new column formed according to the above method. More specifically, $$\begin{aligned} \label{juz} & ~[M_1, M_2, \ldots, M'_m, \ldots, M_{2^N-1},\mathbf{f}]\\ & = \left[ \begin{matrix} \cdots&\ p^i_{s_i,x,y}-1&\cdots&f_{1}\\ \cdots&\ p^i_{s_i,x,y}-1&\cdots&f_{2}\\ \ddots &\vdots&\ddots&\vdots\\ \cdots&\ p^i_{s_i,x,y}&\cdots&f_{2^{N}-1}\\ \cdots&\ p^i_{s_i,x,y}&\cdots&f_{2^N} \end{matrix} \right]. \end{aligned}$$ It is worthy of noting that the new column can be located at any place of the original $[M_1, M_2, \ldots, M_m, \ldots, M_{2^N-1},\mathbf{f}]$ indicating a state where only one leader cooperates. Here, we assume leader $i$ is such a player, and the corresponding column is located at the $m^{th}$ position. Notably, the $m^{th}$ column in (\[juz\]) is only related to the strategy of leader $i$, denoted as $$\mathbf{\widetilde{p}}=(p^i_{s_i,x,y}-1,p^i_{s_i,x,y}-1,\cdots,p^i_{s_i,x,y},p^i_{s_i,x,y}). \nonumber$$ If we let $\mathbf{\widetilde{p}}=\phi\mathbf{f}$, where $\phi\neq 0$ is a coefficient, the $m^{th}$ column is proportional to the last one, resulting in $$\label{vf} \mathbf{v} \cdot \mathbf{f}=0.$$ To set a suitable $\mathbf{f}$, we divide the players in the game into two groups : [*alliance members*]{} ($\mathcal{A}$) and [*non-alliance members*]{} ($\mathcal{-A}$), with the former being a subset of the leaders taking the same strategy as leader $i$ while the latter including the rest of the leaders and all followers. Let vectors $\mathbf{g^{\mathcal{A}}}=(g_{s, b}^{\mathcal{A}})$ and $\mathbf{g^{\mathcal{-A}}}=(g_{s,b}^{\mathcal{-A}})$ be respectively the average payoffs of alliance members and the non-alliance members under all possible outcomes, where each element $g_{s,b}^{\mathcal{A}}$ ($g_{s,b}^{\mathcal{-A}}$) is the average payoff of a member from the set $\mathcal{A}$ ($\mathcal{-A}$) when the alliance adopts $s \in \{c,d\}$ and there are $b$ cooperators in total. Denote by $n^{\mathcal{A}}$ the number of alliance members. According to (\[eq:u\]), we have $$\begin{aligned} \begin{cases} \label{g} g_{c,b}^{\mathcal{A}}=\frac{rb}{N},~ n^{\mathcal{A}} \leq b \leq N , \\ g_{d,b}^{\mathcal{A}}=\frac{rb}{N}+1,~ 0 \leq b \leq N- n^{\mathcal{A}}, \\ g_{c,b}^{\mathcal{-A}}=\frac{(b-n^{\mathcal{A}})\frac{rb}{N}+(N-b)(\frac{rb}{N}+1)}{N-n^{\mathcal{A}}},~ n^{\mathcal{A}}\leq b \leq N, \\ g_{d,b}^{\mathcal{-A}}=\frac{b\frac{rb}{N}+(N-n^{\mathcal{A}}-b)(\frac{rb}{N}+1)}{N-n^{\mathcal{A}}}, 0 \leq b \leq N- n^{\mathcal{A}}. \end{cases}\end{aligned}$$ In (\[g\]), all alliance members have the same utility because their behaviors are the same. This is different from the non-alliance members whose utilities depend on not only their actions but also the number of cooperators and defectors from both sets of $\mathcal{A}$ and $\mathcal{-A}$. Based on the above analysis, one can set $$\mathbf{f}=\chi(\mathbf{g^{\mathcal{A}}}-l\cdot\mathbf{1})-(\mathbf{g^{\mathcal{-A}}}-l\cdot\mathbf{1}), \nonumber$$ where $\chi$ and $l$ are coefficients, and $\mathbf{1}$ is a vector with each element being 1. Let $\mathbf{\widetilde{p}}=\phi\mathbf{f}=\phi[\chi(\mathbf{g^{\mathcal{A}}}-l\cdot\mathbf{1})-(\mathbf{g^{\mathcal{-A}}}-l\cdot\mathbf{1})]$. According to (\[vf\]), we have $$\begin{aligned} \label{eq:I} \mathbf{v}\cdot \mathbf{f}& = & \mathbf{v}\cdot \big(\chi(\mathbf{g^{\mathcal{A}}}-l\cdot\mathbf{1})-(\mathbf{g^{\mathcal{-A}}}-l\cdot\mathbf{1})\big) \nonumber \\ & = & \chi(\mathbf{\mathbf{v}\cdot g^{\mathcal{A}}}-l\cdot\mathbf{v}\cdot\mathbf{1})-(\mathbf{v}\cdot\mathbf{g^{\mathcal{-A}}}-l\cdot\mathbf{v}\cdot\mathbf{1}) \nonumber \\ &= & \chi(\pi^{\mathcal{A}} -l)-(\pi^{\mathcal{-A}}-l) =0.\end{aligned}$$ In (\[eq:I\]), $\pi^{\mathcal{A}}$ and $\pi^{\mathcal{-A}}$ are respectively the expected utilities of alliance members and non-alliance members. Recall that $\mathbf{v}$ is the stable vector of the transition matrix; hence $\mathbf{\mathbf{v}\cdot g^{\mathcal{A}}}=\pi^{\mathcal{A}}$, $\mathbf{\mathbf{v}\cdot g^{\mathcal{-A}}}=\pi^{\mathcal{-A}}$ and $\mathbf{v}\cdot\mathbf{1}=1$. Obviously, (\[eq:I\]) shows that when leader $i$ sets its strategy to $\mathbf{\widetilde{p}}$, it can enforce a linear relationship between $\pi^{\mathcal{A}}$ and $\pi^{\mathcal{-A}}$, i.e., $$\label{l} \pi^{\mathcal{-A}} = \chi\mathbf{\pi^{\mathcal{A}}}+(1-\chi)l,$$ which means that leader $i$ acts as a ZD player in this situation. Since leader $i$ is a member of the alliance and all alliance members should take the same action, this alliance takes a ZD strategy and is termed a ZD alliance. We call the non-ZD alliance members [*outsiders*]{}. Then, when $\chi=0$, the ZD alliance can set the expected utility of the outsiders with $\mathbf{\pi^{\mathcal{-A}}}=l$. \[avut\] When $\frac{r-1}{2r}N < n^{\mathcal{A}} \leq \frac{r-1}{r}N$, the expected utility of the outsiders satisfies $\frac{r(N-n^{\mathcal{A}})}{N} \leq \pi^{\mathcal{-A}} \leq \frac{rn^{\mathcal{A}}}{N}+1$. According to (\[l\]), if $\chi\neq 1$, $l=\frac{\pi^{\mathcal{-A}}- \chi\mathbf{\pi^{\mathcal{A}}}}{1- \chi}$, which is proportional to $-\pi^{\mathcal{A}}$ and $\pi^{\mathcal{-A}}$. Because the ZD alliance moves first, we start by analyzing the range of $l$ from $\pi^{\mathcal{A}}$. Given that there are $b$ cooperators, according to the nature of social dilemma, we have $$\label{pia} g_{c,b}^{\mathcal{A}}=\frac{rb}{N} \leq \pi^{\mathcal{A}} \leq g_{d,b}^{\mathcal{A}}=\frac{rb}{N}+1.$$ In other words, the expected utility of the ZD alliance is higher when it defects rather than cooperates. When the ZD alliance defects, the average utility of the outsiders is $g_{d,b}^{\mathcal{-A}}=\frac{b\frac{rb}{N}+(N-n^{\mathcal{A}}-b)(\frac{rb}{N}+1)}{N-n^{\mathcal{A}}}$; otherwise, it is $g_{c,b}^{\mathcal{-A}}=\frac{(b-n^{\mathcal{A}})\frac{rb}{N}+(N-b)(\frac{rb}{N}+1)}{N-n^{\mathcal{A}}}$. Combining the range of $\pi^{\mathcal{A}}$ given in (\[pia\]), we can deduce $$\begin{aligned} \max_{0\leq b \leq N-n^{\mathcal{A}}}&\{\frac{\vartheta \cdot b}{N(N-n^{\mathcal{A}})(1-\chi)}+1\}\\ & \leq l \leq \min_{n^{\mathcal{A}}\leq b \leq N}\{\frac{\vartheta \cdot b+N^{2}}{N(N-n^{\mathcal{A}})(1-\chi)}\}, \nonumber \end{aligned}$$ where $\vartheta=r(N-n^{\mathcal{A}})(1-\chi)-N$. When $\frac{r-1}{2r}N < n^{\mathcal{A}} \leq \frac{r-1}{r}N$ and $\chi=0$, we can obtain the expected utility range of the outsiders. According to Theorem \[avut\], when $n^{\mathcal{A}}=1$, which implies that there is only one ZD alliance member, the controllable range of $\pi^{\mathcal{-A}}$ shrinks to $[\frac{r(N-1)}{N}, \frac{r}{N}+1]$. This demonstrates that a ZD alliance can obtain higher regulation leverage compared to a single ZD player. $$\begin{minipage}[c]{0.5\textwidth} \centering \includegraphics[width=1.0\textwidth]{cdc_ddc_all} \end{minipage} \begin{aligned} \label{examp} \ \ \ \ \ \ \text{Sta}&\text{te}:\{ccc,ccd,cdc,cdd,dcc,dcd,ddc,ddd\} \\ \ \ \ \ \ \ U_1&=[r, \frac{2r}{3}, \frac{2r}{3}, \frac{r}{3}, \frac{2r}{3}+1, \frac{r}{3}+1, \frac{r}{3}+1, 1 ] \\ U_2&=[r, \frac{2r}{3}, \frac{2r}{3}+1, \frac{r}{3}+1, \frac{2r}{3}, \frac{r}{3}, \frac{r}{3}+1, 1 ] \\ U_3&=[r, \frac{2r}{3}+1, \frac{2r}{3}, \frac{r}{3}+1, \frac{2r}{3}, \frac{r}{3}+1, \frac{r}{3}, 1 ] \\ \mathbf{p}^{1}&=[p_{c,1,1}^{1},p_{c,1,0}^{1},p_{c,0,1}^{1},p_{c,0,0}^{1},p_{d,1,1}^{1},p_{d,1,0}^{1},p_{d,0,1}^{1},p_{d,0,0}^{1}] \\ \mathbf{p}^{2}&=[p_{c,1,1}^{2},p_{c,1,0}^{2},p_{d,1,1}^{2},p_{d,1,0}^{2},p_{c,0,1}^{2},p_{c,0,0}^{2},p_{d,0,1}^{2},p_{d,0,0}^{2}] \\ \mathbf{q}^{3}&=[q_{2}^{3},q_{1}^{3},q_{0}^{3}] \nonumber \hfill \end{aligned}$$ $$\begin{aligned} \label{examp} &\mathbf{v} \cdot \mathbf{f} =D(\mathbf{p}^{1},\mathbf{p}^{2},\mathbf{q}^{3}, \mathbf{f}) = \\ &{\text det} \left[ \begin{matrix} p_{c,1,1}^{1}p_{c,1,1}^{2}q_{2}^{3}-1 & p_{c,1,1}^{1}p_{c,1,1}^{2}(1-q_{2}^{3}) & p_{c,1,1}^{1}(1-p_{c,1,1}^{2})q_{1}^{3} & \textcolor[rgb]{1.00,0.00,0.00}{p_{c,1,1}^{1}-1} & (1-p_{c,1,1}^{1})p_{c,1,1}^{2}q_{1}^{3} & \textcolor[rgb]{0.00,0.50,1.00}{p_{c,1,1}^{2}-1} & (1-p_{c,1,1}^{1})(1-p_{c,1,1}^{2})q_{0}^{3} & f_{1} \\ p_{c,1,0}^{1}p_{c,1,0}^{2}q_{2}^{3} & p_{c,1,0}^{1}p_{c,1,0}^{2}(1-q_{2}^{3})-1 & p_{c,1,0}^{1}(1-p_{c,1,0}^{2})q_{1}^{3} & \textcolor[rgb]{1.00,0.00,0.00}{p_{c,1,0}^{1}-1} & (1-p_{c,1,0}^{1})p_{c,1,0}^{2}q_{1}^{3} & \textcolor[rgb]{0.00,0.50,1.00}{p_{c,1,0}^{2}-1} &(1-p_{c,1,0}^{1})(1-p_{c,1,0}^{2})q_{0}^{3} & f_{2} \\ p_{c,0,1}^{1}p_{d,1,1}^{2}q_{2}^{3} & p_{c,0,1}^{1}p_{d,1,1}^{2}(1-q_{2}^{3}) &p_{c,0,1}^{1}(1-p_{d,1,1}^{2})q_{1}^{3}-1 & \textcolor[rgb]{1.00,0.00,0.00}{p_{c,0,1}^{1}-1} & (1-p_{c,0,1}^{1})p_{d,1,1}^{2} q_{1}^{3} & \textcolor[rgb]{0.00,0.50,1.00}{p_{d,1,1}^{2}} & (1-p_{c,0,1}^{1})(1-p_{d,1,1}^{2})q_{0}^{3} & f_{3} \\ p_{c,0,0}^{1}p_{d,1,0}^{2}q_{2}^{3} & p_{c,0,0}^{1}p_{d,1,0}^{2}(1-q_{2}^{3}) & p_{c,0,0}^{1}(1-p_{d,1,0}^{2})q_{1}^{3} & \textcolor[rgb]{1.00,0.00,0.00}{p_{c,0,0}^{1}-1}& (1-p_{c,0,0}^{1})p_{d,1,0}^{2}q_{1}^{3} & \textcolor[rgb]{0.00,0.50,1.00}{p_{d,1,0}^{2}} & (1-p_{c,0,0}^{1})(1-p_{d,1,0}^{2})q_{0}^{3} & f_{4} \\ p_{d,1,1}^{1}p_{c,0,1}^{2}q_{2}^{3} & p_{d,1,1}^{1}p_{c,0,1}^{2}(1-q_{2}^{3}) & p_{d,1,1}^{1}(1-p_{c,0,1}^{2})q_{1}^{3} & \textcolor[rgb]{1.00,0.00,0.00}{p_{d,1,1}^{1}} & (1-p_{d,1,1}^{1})p_{c,0,1}^{2}q_{1}^{3}-1 & \textcolor[rgb]{0.00,0.50,1.00}{p_{c,0,1}^{2}-1} & (1-p_{d,1,1}^{1})(1-p_{c,0,1}^{2})q_{0}^{3} & f_{5} \\ p_{d,1,0}^{1}p_{c,0,0}^{2}q_{2}^{3} & p_{d,1,0}^{1}p_{c,0,0}^{2}(1-q_{2}^{3}) & p_{d,1,0}^{1}(1-p_{c,0,0}^{2})q_{1}^{3} & \textcolor[rgb]{1.00,0.00,0.00}{p_{d,1,0}^{1}} & (1-p_{d,1,0}^{1})p_{c,0,0}^{2}q_{1}^{3} & \textcolor[rgb]{0.00,0.50,1.00}{p_{c,0,0}^{2}-1} & (1-p_{d,1,0}^{1})(1-p_{c,0,0}^{2})q_{0}^{3} & f_{6} \\ p_{d,0,1}^{1}p_{d,0,1}^{2}q_{2}^{3} & p_{d,0,1}^{1}p_{d,0,1}^{2}(1-q_{2}^{3}) & p_{d,0,1}^{1}(1-p_{d,0,1}^{2})q_{1}^{3} & \textcolor[rgb]{1.00,0.00,0.00}{p_{d,0,1}^{1}} & (1-p_{d,0,1}^{1})p_{d,0,1}^{2}q_{1}^{3} & \textcolor[rgb]{0.00,0.50,1.00}{p_{d,0,1}^{2}} & (1-p_{d,0,1}^{1})(1-p_{d,0,1}^{2})q_{0}^{3}-1 & f_{7} \\ p_{d,0,0}^{1}p_{d,0,0}^{2}q_{2}^{3} & p_{d,0,0}^{1}p_{d,0,0}^{2}(1-q_{2}^{3}) & p_{d,0,0}^{1}(1-p_{d,0,0}^{2})q_{1}^{3} & \textcolor[rgb]{1.00,0.00,0.00}{p_{d,0,0}^{1} }& (1-p_{d,0,0}^{1})p_{d,0,0}^{2}q_{1}^{3} & \textcolor[rgb]{0.00,0.50,1.00}{p_{d,0,0}^{2}} & (1-p_{d,0,0}^{1})(1-p_{d,0,0}^{2})q_{0}^{3} & f_{8} \nonumber \end{matrix} \right] \end{aligned}$$ We use an example shown in Fig. \[examp\] to explain key terms and definitions involved in this section. Assume that there are three players in the game, among which players 1 and 2 are the leaders while player 3 is the follower. The states of the game include all possible outcomes, namely $$\{ccc, ccd, cdc, cdd, dcc, dcd, ddc, ddd\} \nonumber.$$ Accordingly, the utility of any player in each outcome can be calculated according to (\[eq:u\]). The strategy of each leader and the follower can be represented in light of (\[p\]) and (\[q\]). For instance, given a previous outcome $cdc$, the conditional probability under which player 1 adopts $c$ in the current round is $p_{c,0,1}^{1}$ because it acted $c$ and there was no other cooperative leader but one cooperative follower in the previous round; similarly, player 2 adopts $c$ with the conditional probability of $p_{d,1,1}^{2}$ in the current round since it acted $d$ and both the remaining leader and the follower cooperated in the previous round; player 3 is a follower, whose strategy space’s cardinality reduces to $n^\mathcal{L}+1$, which is much lower than that of any leader (i.e., $2^N$). The conditional probability of player 3 adopting $c$ is $q_{z}^{3}$, depending on the number ($z$) of cooperators among the leaders in the current round. As a result, the probability from state $cdc$ to any other state can be derived. For example, the transition probability to state $ddc$ is $(1-p_{c,0,1}^{1})(1-p_{d,1,1}^{2})q_{0}^{3}$. All state transition probabilities contribute to the transition matrix $\mathbf{M}$. Let $\mathbf{M'}=\mathbf{M}-\mathbf{I}$, which can be denoted as $$[M_1, M_2, M_3, M_4, M_5, M_6, M_7, M_8]. \nonumber$$ Next, we perform an elementary column transformation on matrix $[M_1, M_2, M_3, M_4, M_5, M_6, M_7, \mathbf{f}]$, identifying a column corresponding to a state where only one leader cooperates (e.g., player 1) and adding to it all the other columns whose states indicate that player 1 cooperates and at least one of players 2 and 3 also selects $c$. The determinant after the elementary column transformation becomes $$[M_1, M_2, M_3, M_4', M_5, M_6, M_7, \mathbf{f}], \nonumber$$ where the fourth column, denoted by the red color in Fig. \[examp\], is formed according to the above method. We can find that $M_4'$ is only dependent on the strategy of player 1. If we apply an elementary column transformation corresponding to player 2, we can obtain the sixth column (denoted by the blue color in Fig. \[examp\]) that only relies on player 2. Let player 1 be the ZD player and ally with player 2. Both can set their strategies as $\mathbf{p}^{1}=\mathbf{p}^{2}=\phi[\chi(\mathbf{g^{\mathcal{A}}}-l\cdot\mathbf{1})-(\mathbf{g^{\mathcal{-A}}}-l\cdot\mathbf{1})]$, where $\mathbf{g^{\mathcal{A}}}= [g_{c,2}^{\mathcal{A}}, g_{c,3}^{\mathcal{A}}, g_{d,0}^{\mathcal{A}},g_{d,1}^{\mathcal{A}}]=[\frac{2r}{3}, r, 1, \frac{r}{3}+1]$ and $\mathbf{g^{\mathcal{-A}}}= [g_{c,2}^{\mathcal{-A}}, g_{c,3}^{\mathcal{-A}}, g_{d,0}^{\mathcal{-A}},g_{d,1}^{\mathcal{-A}}]=[\frac{2r}{3}+1, r, 1, \frac{r}{3}]$. According to Theorem \[avut\], the ZD alliance including players 1 and 2 can set the expected utility of player 3 ranging from $\frac{r}{3}$ to $\frac{2r}{3}+1$. Now, it is the time to answer the question proposed at the beginning of this section. Taking advantage of Theorem \[avut\], we can set $$\begin{aligned} \label{ui} u_i(a_i|n^{\mathcal{A}})= \begin{cases} \frac{rn^{\mathcal{A}}}{N}+1,~ a_i=1, \\ \frac{r(N-n^{\mathcal{A}})}{N},~ a_i=0, \end{cases}\end{aligned}$$ which means a ZD alliance would reward the cooperation of the outsiders with the highest expected utility while punish their defection with the lowest utility. This extreme reward-punishment incentive mechanism can help to facilitate cooperation as much as possible. ZD alliance for egoistic incentive {#sec:op} ================================== Based on our analysis in Section \[sec:formulation\], one can see that a rational player $i$ would take an action according to the difference between the utilities of adopting $c$ and $d$, which are related to the co-players’ moves among its social community. Due to the heterogeneous nature of the members in the social community, the utility of player $i$ may come from the regular game and the ZD game, with the former being calculated by (\[eq:u\]) while the latter solved by (\[ui\]). Note that Theorem \[avut\] in Section \[sec:zd\] can serve for a more general case, where the number of outsiders is arbitrary. However, in our consideration, for any player $i$, the corresponding ZD game only involves the ZD alliance and the player itself as shown in Fig. \[fig:tt\], implying that the outsider is just player $i$ and hence we have $N-n^{\mathcal{A}}=1$. From the perspective of a single player, one can realize egoistic incentives by rewarding more expected utility when the player chooses $c$ and punishing it if it defects. However, to achieve this goal from the network perspective, we need to optimize the deployment of ZD players so that the overall cooperation probability in the whole system can be maximized. To optimize the above problem, we need to answer such a question: according to (\[l\]), $\frac{\pi^{\mathcal{A}}-l}{\pi^{\mathcal{-A}}-l}=\frac{1}{\chi}$, implying that a ZD alliance has the ability to carry out the extortionate strategy by enforcing a ratio between its expected utility and that of the outsider. Moreover, the smaller the $\chi$, the more the expected utility that can be transferred to the ZD alliance. In this case, as a dominant player, can the ZD alliance push the outsider to cooperate while it intends to defect for obtaining more expected utility? We use the following theorem to answer this question: \[domi\] The dominant strategy of the ZD alliance is cooperation irrespective of the strategy adopted by the outsider. In the ZD game, when player $i$ takes $a_i \in \{0,1\}$, the utilities of the ZD alliance when it cooperates and defects are respectively $\frac{r(n^{\mathcal{A}}+a_i)}{N}$ and $\frac{ra_i}{N}+1$ according to (\[g\]). To prove cooperation is the dominant strategy of the ZD alliance, we should prove $\frac{r(n^{\mathcal{A}}+a_i)}{N}>\frac{ra_i}{N}+1$, which is equivalent to proving $(r-1)n^{\mathcal{A}}>1$ due to $N-n^{\mathcal{A}}=1$ in our scenario. In light of Theorem \[avut\], the condition under which a ZD alliance can control others is $\frac{r-1}{2r}N < n^{\mathcal{A}} \leq \frac{r-1}{r}N$, which can be simplified as $r>N$ due to $N-n^{\mathcal{A}}=1$. Because $N \geq 2$, we have $(r-1)n^{\mathcal{A}}>1$. As a conclusion, this theorem holds. When cooperation is the dominant strategy of the ZD alliance, maximizing the overall cooperation probability of the whole system is equivalent to maximizing those of all regular players. Hence, our optimization problem can be written as: $$\begin{aligned} \label{max} &\max \ \sum_{i=1}^V q_i(c)x_i,\\ &s.t. \ \sum x_i=V-K, \\ & \ \ x_i \in \{0,1\}, \end{aligned}$$ where $x_i=0$ denotes that player $i$ adopts the ZD strategy and $x_i=1$ means that the player is a regular one; and $K$ and $V$ are respectively the number of ZD players we deploy and the number of total players in the system. To find an optimal solution for (\[max\]), we resort to genetic algorithms (GA) because GA is efficient in dealing with problems with nonlinear and multi-constraint properties. As GA has been well-studied, we refer the interested readers to [@Mahmud2012Genetic] and omit the algorithm description here. The basic idea of our egoistic incentive mechanism is to take advantage of the ZD alliance to shape the behaviors of the regular players. According to (\[eq:newstrategy\]), the cooperation probability of a regular player $i$ depends on the number of ZD players ($n^{\mathcal{A}}$) and that of cooperators ($n_i$) among the regular neighbors in its social community. However, due to the nature of social dilemma [@pnas32], we can obtain $U_i(1|n_i)-U_i(0|n_i+1)=\frac{r(n_i+1)}{\widetilde{N}}-(\frac{r(n_i+1)}{\widetilde{N}}+1)=-1$, where $\widetilde{N}$ is the total number of regular players in the social community. This equation reflects that a player $i$ can definitely obtain a higher payoff if it defects rather than cooperates, no matter how many cooperators there exist (i.e., $n_i$) in its neighborhood. This makes the strategy of a regular player $i$ only relies on $n^{\mathcal{A}}$, the number of ZD players around $i$. While $n^{\mathcal{A}}$ is determined by the solution of the optimization problem (\[max\]), which is closely related to the topology of network $G$. Because $G$ is constructed based on the nodes’ social ties, which are relatively steady, $n^{\mathcal{A}}$ is steady. Hence, our method is stable, and does not need to maintain and manage any state; thus achieving the property of statelessness introduced in Section \[sec:intro\]. Performance Simulation {#sec:simu} ====================== In this section, we evaluate the performance of our proposed mechanism using real and synthetic data. The real data we adopt is the [*iMote data*]{} [@cambridge-haggle], including traces of Bluetooth sightings by groups of users carrying small devices (iMotes) for a number of days in different offices, conferences, and cities. We use all iMote nodes (54 in total) and their links to perform our simulation. Synthetic data is also employed since we want to test the performance of our approach under different network topologies. The types of networks in our simulation are [*star*]{}, [*ring*]{}, [*tree*]{}, and [*mesh*]{}, each of which includes 80 nodes. More specifically, all nodes in the star network are individually connected to a central node; in the ring network, each node connects to exactly two other nodes, forming a closed loop; each node has two neighbors except the leaf nodes in the tree network, making its shape a binary tree; and in the mesh network, each node has at least two randomly selected neighbors. All simulations have been repeated 30 times to obtain the average values with statistical confidence. Fig. \[fig:total\] demonstrates how the average cooperation probability of our mechanism changes with the number of ZD players ($K$) deployed in different networks. In this study, we set $r=2n+3$, where $n$ is the total number of players in a game. Note that each regular player may involve in two kinds of games, the regular game and the ZD game as shown in Fig. \[fig:tt\]. Hence, $n$ varies in different games. According to Fig. \[fig:total\], one can see that the mesh network has the best performance, realizing fast cooperation over the whole system using fewer ZD players. Although the performance of the real network is slightly lower than that of the mesh network, both have similar trends. This is because compared to other networks, the nodes in these two networks have higher degrees; thus fewer ZD players are needed to control more regular players. Fig. \[fig:total\] also shows that in the star network, once a ZD player is deployed, the average cooperation probability can rocket to $73\%$ and keep this level even though the number of ZD players increases. The reason behind this lies in that each node has only one neighbor except the central node in the star network, and a single ZD player is definitely needed in the central position to control all the other nodes while other ZD players completely lose their control power due to the lack of connections to other regular players. However, because of only one ZD player functions, the controllable range of the outsider’s expected utility shrinks to $[\frac{r}{2}, \frac{r}{2}+1]$, making the cooperation probability to be $\frac{e}{1+e}\approx 73\%$. This is different from the situations in the mesh network and the real network, where ZD players can achieve higher control leverage by forming alliances and finally obtain the desired cooperation probabilities. Finally, one can see from Fig. \[fig:total\] that the growth rates of cooperation probabilities in the tree and the ring networks are the slowest. This is because the average degrees in the tree and ring networks are respectively 1.95 and 2, which are about $\frac{1}{20}$ and $\frac{1}{10}$ of those in the mesh and the real networks. Although the tree and ring networks have almost the same average degree as the star network, the ZD player located at the central place in the star network has the largest degree value, empowering its control. ![Average cooperation probabilities in different networks when $r=2n+3$.[]{data-label="fig:total"}](xianxing){width="48.00000%"} Fig. \[fig:total1\] illustrates the average cooperation probabilities of our method under different numbers of ZD players ($K$) deployed in different networks when $r=2n^2+3$, which implies a nonlinear relationship between $r$ and $n$. According to this figure, one can see that the results are similar to those shown in Fig. \[fig:total1\]. Note that we have done extensive simulations under other forms of $r(n)$ and obtained results with very similar trends; hence the corresponding results are omitted in this paper to avoid redundancy. ![Average cooperation probability in different networks when $r=2n^2+3$.[]{data-label="fig:total1"}](feixianxing){width="48.00000%"} Figs. \[fig:mesh\], \[fig:real\], \[fig:star\], \[fig:tree\], and \[ring\] further demonstrate the performance of the proposed egoistic incentive mechanism for the mesh, real, star, tree, and ring networks, respectively, when $r=fn+3$, where $f$ is a coefficient. The subfigures (a) indicate the ratio of cooperators varying with $f$ and the number of ZD players $K$ in different networks while all subfigures (b) show the optimal deployments of ZD players in the corresponding networks when $f=2$ and $K=10$. The red lines in the subfigures (b) identify the social ties (edges) of the ZD players. We can clearly observe that [*the emergence of cooperation*]{} phenomenon happens in the mesh network from Fig. \[fig:mesh\](a). That is, a large number of cooperators arise after more than 3 ZD players are deployed. When the number of ZD players is higher than 5, almost all regular players choose to cooperate. Fig. \[fig:mesh\](b) indicates the positions of 10 ZD players. The statistical data reveals that these ZD players are popular nodes because their average degree is 51.7, compared to 39, the average degree of all nodes in the mesh network. Fig. \[fig:real\](a) demonstrates that the real network has similar performance as the mesh network. However, a further analysis on Fig. \[fig:real\](b) indicates that the ZD players are not hot spots since their average degree is 16.8, which is lower than that of the whole network, i.e., 21.8. Contrarily, the ZD strategy is adopted by the nodes with high betweenness whose average value is 67.8888, almost twice as much as the average betweenness of the whole system. Here, the betweenness of a node is defined as the total number of times a node acts as a relay along the shortest path between any two other nodes. The reason behind this phenomenon may lie in that there exist sparse subnetworks in the real network that cannot be controlled by the popular nodes but can be regulated by nodes with a high betweenness value because betweenness can be employed to evaluate a node’s contribution to the network connectivity, quantifying to what extent this node can impact on others. Fig. \[fig:star\](a) indicates that there still exists an emergence of cooperation in the star network after deploying one ZD player. Just as we have expected, this ZD player is located at the central place indicated in Fig. \[fig:star\](b), which has the highest popularity as well as connectivity. As we have analyzed earlier, even though we put more ZD players in other positions, they would not function since there exist no regular players as their neighbors, leading to low regulation power because no ZD alliance can be formed in this case. Hence, we just label the position of the single ZD player in Fig. \[fig:star\](b). According to Fig. \[fig:tree\](a), there is no emergence of cooperation phenomenon in the tree network. Moreover, the ratio of cooperators in the network rises steadily and relatively slowly due to the low average degree of the tree network. From subfigure (b), one can see that all ZD players are deployed in the nodes with degree being 3, higher than the root node and the leaf nodes whose degree values are respectively 2 and 1. Higher popularity makes ZD players possess more regular players as their neighbors, increasing their regulation range. Fig. \[ring\](a) demonstrates that the performance of our mechanism in the ring network is similar to that in the tree network: no emergence of cooperation and low growth ratio of the number of cooperators. However, there exists a difference between them if we compare Fig. \[fig:tree\](a) with Fig. \[ring\](a). Specifically, when the number of ZD players is smaller than 30, the tree network outperforms the ring network; but when there are more than 30 ZD players deployed, the ring network wins. The underlying reason can be found from Fig. \[fig:tree\](b) and Fig. \[ring\](b). The cause of the first phenomenon lies in that each ZD player in the tree network has a degree of 3, which is higher than 2, the degree of a ZD player in the ring network; the second phenomenon happens because ZD alliances can be formed more easily in the ring network than in the tree network and when the number of ZD players increases, the power of ZD alliance can stimulate more cooperation. For example, as indicated by Fig.\[fig:tree\](b) and Fig. \[ring\](b), when 10 ZD players are deployed to shape the behaviors of others, no ZD alliance exists in the tree network while several ZD alliances are formed in the ring network. For instance, in the ring network, node 35 can ally with not only node 33 to regulate node 34, but also node 37 to control node 36. Conclusion {#sec:conclusion} ========== This paper proposes a mechanism to realize large-scale egoistic incentives via optimally deploying ZD players to reward cooperation and punish defection. We further derive a ZD alliance strategy for sequential multiple-player repeated games to speed up cooperation. Our approach has the traits of statelessness and stability, making it scalable and suitable for large-scale systems. The simulation results demonstrate that mesh and real networks can achieve the best performance where [*the emergence of cooperation*]{} phenomenon happens when only a few ZD players are deployed. Although such phenomenon also exists in a star network, its topological trait makes only one ZD player function, pinning the cooperation probability to be $73\%$. The tree and ring networks perform the worst due to a low average degree. However, compared to the tree network, ZD alliances are more easily formed in the ring network, making it perform better as the number of ZD players increases. Acknowledgment {#acknowledgment .unnumbered} ============== The authors would like to thank the support from the National Natural Science Foundation of China under grants 61772080 and 61472044, and the National Science Foundation of the US under grants CNS-1704397 and IIS-1741279. [Shengling Wang]{} is an associate professor in the College of Information Science and Technology, Beijing Normal University. She received her Ph.D. in 2008 from Xi’an Jiaotong University. After that, she did her postdoctoral research in the Department of Computer Science and Technology, Tsinghua University. Then she worked as an assistant and associate professor from 2010 to 2013 in the Institute of Computing Technology of the Chinese Academy of Sciences. Her research interests include mobile/wireless networks, game theory, crowdsourcing. [Peizi Ma]{} received her B.S. degree in Computer Science and Technology from Beijing Normal University in 2016. She is currently pursuing her M.S. degree in Computer Science at Beijing Normal University. Her research interests include wireless network, network science and game theory. [Qin Hu]{} received her Ph.D. degree in Computer Science from the George Washington University in 2019. She is currently an Assistant Professor in the department of Computer and Information Science, Indiana University - Purdue University Indianapolis. Her research interests include wireless and mobile security, crowdsourcing/crowdsensing and blockchain. [Xiuzhen Cheng]{} \[F\] received her M.S. and Ph.D. degrees in computer science from the University of Minnesota Twin Cities in 2000 and 2002, respectively. She is a professor in the Department of Computer Science, George Washington University, Washington, DC. Her current research interests focus on privacy-aware computing, wireless and mobile security, dynamic spectrum access, mobile handset networking systems (mobile health and safety), cognitive radio networks, and algorithm design and analysis. She has served on the Editorial Boards of several technical publications and the Technical Program Committees of various professional conferences/workshops. She has also chaired several international conferences. She worked as a program director for the U.S. National Science Foundation (NSF) from April to October 2006 (full time), and from April 2008 to May 2010 (part time). [Weifeng Lv]{} received his Ph.D. degree in computer science from Beihang University. His research interests include massive information system, urban cognitive computing, swarm intelligence, and smart cities. He is a professor of computer science, the dean of the School of Computer Science and Engineering, and the vice director of the State Key Laboratory of Software Development Environment, at Beihang University. He also serves as the Secretary-General of the China Software Industry Association and the director of National Engineering Research Center for Science and Technology Resources Sharing Service. He obtained multiple internationally renowned awards, including the second prize of the 2016 China National Science and Technology Invention Award and the first prize of the 2010 Beijing Science and Technology Award. [^1]: $1|n_i$ means player $i$ adopts cooperation when the number of other players’ adopting $c$ is $n_i$ and hence the total number of cooperators is $n_i+1$ in this case.
--- abstract: 'We investigate a generalized stochastic model with the property known as mean reversion, that is, the tendency to relax towards a historical reference level. Besides this property, the dynamics is driven by multiplicative and additive Wiener processes. While the former is modulated by the internal behavior of the system, the latter is purely exogenous. We focus on the stochastic dynamics of volatilities, but our model may also be suitable for other financial random variables exhibiting the mean reversion property. The generalized model contains, as particular cases, many early approaches in the literature of volatilities or, more generally, of mean-reverting financial processes. We analyze the long-time probability density function associated to the model defined through a Itô-Langevin equation. We obtain a rich spectrum of shapes for the probability function according to the model parameters. We show that additive-multiplicative processes provide realistic models to describe empirical distributions, for the whole range of data.' author: - 'C. Anteneodo' - 'R. Riera' title: 'Additive-multiplicative stochastic models of financial mean-reverting processes ' --- Introduction ============ Accurate statistical description of the stochastic dynamics of stock prices is fundamental to investment, option pricing and risk management. In particular, a relevant quantity is the volatility of price time series[@volatility], that quantifies the propensity of the market to fluctuate. Since volatility represents a measure of the risk associated to the fluctuating dynamics of prices, it is crucial to develop suitable models to predict its complex intermittent behavior. There is empirical evidence that it fluctuates following a stochastic dynamics subjacent to that of prices, whose dynamics, in turn, depends on the time evolving volatility. Many approaches are based on that assumption[@prices], although others propose the existence of a reciprocal feedback between both processes[@leverage]. Our approach builds on the development of a simple Langevin equation to characterize the stochastic process of volatility. The equation provides an unifying description that generalizes widely discussed models in the literature. We analyze the shape of the long-time probability density function (PDF) associated to the stochastic differential equation that characterizes each particular case of the generalized model. Most previous results focus on the tails of the PDFs. In fact, for stochastic variables, such as volatilities presenting fat-tailed PDFs[@liu; @volat1], it is specially important to reproduce extreme events in a realistic model. Now we go a step further and aim to predict the PDFs in the whole range of events. One of the main features observed in the dynamics of some financial variables, such as volatilities, stock volumes or interest rates, is their tendency to permanently relax, towards a reference level $\theta$, a property known as mean reversion. Another feature is the multiplicative market processing of random news, whose strength becomes modulated by a function of the stochastic variable itself. These two properties are modeled by means of a nonlinear mean-reverting force and nonlinear multiplicative noise. They are discussed in detail in Sect. \[Sec:props\]. In Sect. \[Sec:mean\_rev\], we discuss the shapes of the PDFs that such family of models yields. Despite being of a general form, they give rise to PDFs that, decay exponentially fast, either above the mode, below it, or both, in disagreement with empirical observations. For instance, log-normal behavior has been reported for volatility computed from global data of S&P500[@liu], at intermediate values. However, at high values, a power-law behavior, with exponent outside the stable Lévy range, was observed. The same analysis performed for individual companies[@liu] yields also power-law tails. But in that case, the results show a variation slower than log-normal below the mode, suggesting a power-law also in the limit of small values. The volatility of capitalized stocks traded in US equity markets exhibits similar features[@micciche]. Other variables with mean-reversion, such as volume of transactions (number of trades) present akin distributions. Power-law tails out of the Lévy range have been reported for the PDFs of normalized NYSE stock volumes[@volumes]. More recently, studies of normalized volumes, performed over high resolution data (1-3 minutes) of NYSE and NASDAQ[@vol1] (see also [@vol2]), display PDFs with power-law behavior both at large and small values. We will show that the class of multiplicative processes considered in Sect. \[Sec:mean\_rev\], although general enough, is not able to reproduce, for any value of its parameters, these empirical PDFs in the whole range. In a realistic model, we must deal with various sources of fluctuations acting upon the collective variable. Then, we propose to include a component that is lacking to suitably model many real processes, that is the presence of fluctuations that act additively, besides the multiplicative noise already taken into account. The latter originates from the internal correlated behavior of the market, representing a sort of endogenous feed-back effect, while additive noise concerns fluctuations of purely external origin or random speculative trading. Then, in Sect. \[Sec:add\_mult\], we present a further generalization that consists in incorporating an independent additive source of noise. Depending on the parameters of the process, the additive or multiplicative contributions will play the dominant role. This gives rise to a rich spectrum of PDF shapes, in particular, a subclass with two-fold power-law behavior, both above and below the mode, providing a general realistic framework for describing the shape of empirical distributions. A comparison with experimental results is presented in Sect. \[Sec:empirical\]. Finally, Sect. \[Sec:final\] contains the main conclusions and general remarks. Mean reversion and multiplicative fluctuations {#Sec:props} ============================================== The reversion to the mean is one of the basic ingredients to describe the dynamics of several stochastic variables of interest in economy. It is fundamental since it concerns the behavior around a central value $\theta$ and reflects the global market response to deviations from a consensus or equilibrium level. It depends on monetary unit, market size, degree of risk aversion, etc., hence, it is characteristic of each market. The aversion to deviations from the mean needs not be linear, specially when large deviations are involved. Similarly, a nonlinear mechanism due to the cooperative behavior of traders, rules the way the market modulates the amplitude of fluctuations (mainly external) giving rise to innovations. We consider the general class of stochastic differential equations given by $$\label{ILE0} {\rm d} x \;=\;-\gamma[x-\theta]x^{r-1}{\rm d}t + \mu x^s{\rm d}w \;,$$ where, $\theta, \gamma,\mu>0$, $r,s\in\mathbb{R}$, and $w$ is a Wiener process, such that $\langle {\rm d}w\rangle=0$ and $\langle ({\rm d}w)^2\rangle=2{\rm d}t$. The definition of the stochastic process is completed by the Itô prescription. This class generalizes well-known models employed to describe the dynamics of mean-reverting financial variables[@juros]. In particular, some traditional processes for modeling volatilities or, mainly, squared volatilities are the Hull-White ($r=1,s=1$)[@hw] and the Heston ($r=1,s=1/2$)[@heston] models, the latter also known either as Cox-Ingersoll-Ross[@cir] or Feller process[@feller]. The arithmetic ($r=1,s=0$) and geometric ($r=2,s=1$) Ornstein-Ulhenbeck processes are particular cases too. Moreover, several other models employed in the literature of volatilities are related to this class[@others; @micciche]. Different values of $r$ in Eq. (\[ILE0\]) represent different possible relaxation mechanisms of amplitude $\gamma$, determined, amongst other factors, by constraints, flux of information, stock liquidity and risk aversion, which are particular of a given market. Notice that the restoring force in Eq. (\[ILE0\]) corresponds to a confining potential, with minimum at $\theta$, for all $r\in\mathbb{R}$. The larger $r$, the more attractive the potential for large $x$, but the less attractive for vanishing $x$. Similarly, different values of $s$ specify the market informational connectivity, which conditions the degree of coherent multiplicative behavior. Models in the literature typically set $s\geq 0$, meaning that the effective amplitude of fluctuations increases with $x$. Negative $s$ makes multiplicative fluctuations grow with decreasing $x$, thus it mainly reflects a cooperative reaction to quiescence. Although it does not seem to reflect a realistic steady state of the market, it may occur as a transient, driven by speculative trading. The two mechanisms are complementary. If $r<0$ the restoring force decreases for increasing $x$ above the reference level, in particular, for $r<-1$, the force tends to zero in the limit of large $x$. Thus, decreasing $r$ represents markets that, become less able to recover the reference level by means of the deterministic tendency alone. However, a strong multiplicative response to large fluctuations (positive $s$) could still compensate that inability and restore the market historical level. Concerning the response to small values, the restoring force diverges at the origin if $r<1$, while for $r>1$, it vanishes at $x=0$, meaning that this point becomes an unstable equilibrium state. This corresponds to a market indifferent to low levels of trading activity. Again, this effect can be balanced by the multiplicative contribution (with a small value of parameter $s$). In early works, only very particular values of $(r,s)$ have been considered. However, this may be sometimes owed more to reasons of mathematical solvability, than to econophysical ones. Following the above discussion, $(r,s)$ may be non-universal, depending on the particular nature of a market or its agents. Therefore, we will not discard any possibility [*a priori*]{}. Generalized multiplicative process with mean reversion {#Sec:mean_rev} ====================================================== We consider the simple class of stochastic multiplicative differential equations given by Eq. (\[ILE0\]), that generalizes many processes usually found in the literature of volatilities. We investigate, in this Section, the long-time PDFs that this class of processes yields. The Fokker-Planck equation associated to Eq. (\[ILE0\]), following standard methods[@books], is $$\label{FP0} \partial_t\rho = \gamma \partial_x( [x-\theta]x^{r-1}\rho) + \mu^2 \partial^2_{xx} [x^{2s}\rho] \;.$$ Its long-term solution is relevant in connection to the assumption that the process can be treated as quasi-stationary. In that case the PDF obtained from an actual data series will coincide with the stationary solution. Considering reflecting boundary conditions at $x=0$ and $x\to\infty$[@books], the steady state solution of Eq. (\[FP0\]) reads: $$\label{ss0} \rho(x)=\frac{\rho_o}{x^{2s}} \exp\left( -\gamma_\mu\Bigl[ \frac{x^{p+1}}{p+1} - \theta\frac{x^{p}}{p} \Bigr]\right),$$ with $p\equiv r-2s\neq0,-1$, where $\rho_o$ is a normalization constant and $\gamma_\mu\equiv\gamma/\mu^2$ an effective restoring amplitude, such that $\gamma$ (a parameter associated to order) becomes reduced by the amplitude of multiplicative noise (associated to disorder). ![ Diagram of the asymptotic behavior of the PDF given by Eq. (\[ss0\]), in ($s,r$)-plane. Unshadowed regions and dotted borders identify regions excluded by the normalization condition. At the positive $r$-axis, the PDF is finite at the origin. Tilted lines denote the marginal cases $r=2s$ ($p=0$), with pure exponential tail and power-law growth at the origin, and $r=2s-1$ ($p=-1$), with power-law tail and exponential of $1/x$ growth at the origin (the threshold points of these lines have coordinates $([1+\gamma_\mu\theta]/2,0)$ and $([1-\gamma_\mu]/2,0)$, respectively). Parameter $a>0$, in the exponential formulas, as well as the power-law exponents, depend on model parameters. Symbols correspond to the special processes: Hull-White (HW), Heston (H), Ornstein-Ulhenbeck (OU) and geometric OU (GOU). []{data-label="fig:m"}](vfigure1.eps){width="40.00000%"} The class of processes described by Eq. (\[ss0\]) thus generically yields asymptotic exponential-like behaviors for small or/and large values. As soon as $p+1>0$, a [*stretched exponential decay*]{} is obtained for large enough $x$, such that the argument of the exponential is proportional to $-x^{p+1}$. If $p<0$, a [*stretched exponential of the inverse argument*]{} ($-1/x^{|p|}$) is obtained for vanishing $x$. Therefore, for $p\in(-1,0)$, the PDF presents dominant exponential-like behavior both for low and large values, without any restriction on the value of $s$. Outside that interval, the [*power law*]{} $x^{-2s}$ in (\[ss0\]) asymptotically dominates, for either small (if $p>0$) or large (if $p<-1$) argument. Then, normalization in $[0,\infty)$ restricts the possible values of $s$ according to: $s<1/2$ (if $p>0$), $x>1/2$ (if $p<-1$). In the marginal cases, Eq. (\[ss0\]) explicitly is:  I: For $p\equiv r-2s=-1,$ $$\label{caseI} \rho(x)=\frac{\rho_o}{x^{2s+\gamma_\mu}}\; \exp( -\gamma_\mu\theta/x ) \;,$$ with $2s>1-\gamma_\mu$ for nomalizability. II: For $p\equiv r-2s=0$, $$\label{caseII} \rho(x)=\rho_o x^{\gamma_\mu\theta-2s} \exp( -\gamma_\mu x ) \;,$$ with $2s<\gamma_\mu\theta+1$ for normalization, but $2s<\gamma_\mu\theta$ to avoid the divergence at the origin. Fig. \[fig:m\] displays the possible PDF asymptotic shapes in $(s,r)$ space. Notice that the $s=0$ axis gives the solution for mean-reverting models with purely additive fluctuations. Let us analyze some special cases. In the trivial case $(s,r)=(0,1)$, corresponding to the Ornstein-Ulhenbeck process [@books] $$\label{OU} {\rm d} x \;=\;-\gamma[x-\theta]{\rm d}t + \mu {\rm d}z \;,$$ the noisy contribution becomes additive and the stationary PDF is Gaussian (truncated at $x=0$). Although we are dealing with $\theta>0$, it is worth of mention the case $(\theta,s,r)=(0,1,1)$, corresponding to the geometric Brownian process, that leads to the [*log-normal*]{} distribution. For type I ($r=2s-1$), notice that the PDF decays as a power law, for large $x$, and goes to zero faster than power law, for vanishing $x$. The power-law exponent is controlled by $s$ and $\gamma_\mu$, that is, all the model parameters, except $\theta$ are involved. In the particular case $r=s=1$, one recovers the Hull-White process[@hw] $$\label{HW} {\rm d} x \;=\;-\gamma[x-\theta] {\rm d}t + \mu x{\rm d}z \;.$$ In case II ($r=2s$), observe that the PDF has opposite behavior: it increases at the origin as a power law and decays exponentially for large $x$. All the model parameters, including $\theta$ are buried in the power-law exponent. In particular, if $r=2s=1$, one gets the Heston model[@heston] $$\label{H} {\rm d} x \;=\;-\gamma[x-\theta]{\rm d}t + \mu \sqrt{x}{\rm d}z \;.$$ If $r=2s=2$, the geometric Ornstein-Uhlenbeck process is obtained $$\label{Geo} {\rm d} x \;=\;-\gamma[x-\theta]x{\rm d}t + \mu x{\rm d}z \;.$$ Diverse other models proposed in the literature can also be thought as particular instances of our generalized model. For example, the one proposed by Micciché et al. [@micciche] is in correspondence with the Hull-White model (\[HW\]), with $x$ representing volatility $v$, whereas in the latter $x\equiv v^2$. Also a family of multiplicative models, studied before in the context of a wide spectrum of physical processes[@schenzle], belongs to the class here considered, through the transformation $x\to x^\beta$. Summarizing, from Eqs. (\[ss0\])-(\[caseII\]), in general, the asymptotic behaviors below and above the mode are tied, such that, in a log-log scale, if one flattens the other changes rapidly. This explains why models of this class fail to describe empirical volatilities in the whole range of observed data, even under the transformation $v^2\mapsto v$. Generalized model with additive-multiplicative structure {#Sec:add_mult} ======================================================== We analyze in this section, processes that take into account the presence of some additional source of noise. Previous works[@multi1; @multi2; @multi3] show that additive-multiplicative stochastic processes constitute an ubiquitous mechanism leading to fat-tailed distributions and correlated sequences. This extra noise represents a quite realistic feature, since, besides noise modulated by the market, other fluctuations may act directly, additively. From the stream of news, represented by a noisy signal, some are amplified or reduced by cooperative actions, others incorporated unaltered. Related ideas has been discussed in Ref. [@sornette]. Also, a model of financial markets that leads to additive-linear-multiplicative processes has been recently proposed[@spins], where the noises are identified with the fluctuating environment and fluctuating interaction network, respectively. In general, the two white noises are considered uncorrelated. However, they may even correspond to identical time-series as soon as they are shifted with a time lag greater than the correlation time. In such case, the endogenous noise is expected to act with a delay due to its very nature of feedback process, whereas, the additive noise is incorporated immediately, free of signal processing. By including purely exogenous fluctuations, in the process defined by Eq. (\[ILE0\]), we obtain the following Itô-Langevin equation (ILE) $$\label{ILE} {\rm d} x \;=\;-\gamma[x-\theta]x^{r-1}{\rm d}t + \mu x^s{\rm d}w + \alpha {\rm d}z \;,$$ where $w,z$ are two [*independent*]{} standard Wiener processes, defined as above, and $\mu,\alpha$ their respective amplitudes. The corresponding Fokker-Planck equation reads $$\label{FP} \partial_t\rho = \gamma \partial_x( [x-\theta]x^{r-1}\rho) + \partial_{xx}( [\mu^2 x^{2s} + \alpha^2]\rho) \; .$$ Its steady state solution with reflecting boundary conditions is $$\label{FPss} \rho(x) = \frac{\rho_o}{1+\lambda^2x^{2s}} \exp[-\gamma_\alpha \int^x \frac{ y^{r-1}(y-\theta) }{1+\lambda^2y^{2s}}{\rm d}y] \; ,$$ with $\rho_o$ a normalization constant, $\gamma_\alpha\equiv \gamma/\alpha^2$, $\lambda^2\equiv (\mu/\alpha)^2\equiv\gamma_\alpha/\gamma_\mu$. In most cases the integral can be written in terms of hypergeometric functions $_2F_1$[@abram], through $$\label{int} \int^x \frac{y^{\beta-1}}{1+\lambda^2y^{2s}}{\rm d}y \equiv \frac{\beta}{x^\beta} \,_2F_1(c,1,c+1,-\lambda^2x^{2s})$$ with $c\equiv\beta/(2s)\neq -1,-2,\ldots$, whereas, in the marginal case $\beta=0$, we will use $$\label{intmarg} \int^x \frac{y^{-1}}{1+\lambda^2y^{2s}}{\rm d}y\;\equiv\; \ln x\,-\,\ln(1+\lambda^2x^{2s})/(2s).$$ By means of these definitions and their asymptotic formulas[@abram; @formula], we obtain the possible PDF shapes, in $(s,r)$-space, as schematized in Fig. \[fig:am\]. The marginal cases $r=0$ and $r=-1$ will be considered latter. In general, sufficiently large positive $s$ is required in order to yield power-law tails, otherwise, stretched exponential tails prevail, as for the processes considered in Sect. \[Sec:mean\_rev\]. The additive noise does not add new domains with power-law tails, although regions with stretched exponential law are excluded or included by the normalization condition. For vanishing $x$, the main difference with purely multiplicative processes is that, for positive both $s$ and $r$, the PDF is truncated at the origin. Notice that, as the PDF is finite at the origin, then, if $x$ is identified with the squared volatility ($x\equiv v^2$), the PDF for $v$ increases linearly at the origin. Let us analyze, in more detail, the marginal cases $r=0$ and $r=-1$ that can yield power-laws in both asymptotic limits. From Eqs. (\[FPss\])-(\[intmarg\]), we obtain [**A**]{}: For $r=0$, the PDF has the form $$\label{IIm} \rho(x) = \rho_o \frac{x^{\gamma_\alpha\theta} \Theta(x)} {[1+\lambda^2x^{2s}]^{\gamma_\alpha\theta/(2s)+1} } \; ,$$ where $\Theta(x)\equiv\exp[-\gamma_\alpha x \,_2F_1({\scriptstyle \frac{1}{2s},1,\frac{1}{2s}+1},-\lambda^2x^{2s})]$ is a smooth function of $x$, such that $\Theta(0)$ is finite, hence it does not spoil the power-law growth at the origin. For large $x$, it may present different asymptotic behaviors depending on the value of $s$: [**A.1**]{}: If $s\leq 0$, $\Theta(x)$ decays as pure [*exponential*]{} of $x$. Therefore, the asymptotic decay is finally dominated by this exponential factor. [**A.2**]{}: If $0<s<1/2$, $\Theta(x)$ behaves asymptotically as a [*stretched exponential*]{} with argument $x^{1-2s}$. That is, the tail, although a power law for moderate $x$, becomes asymptotically dominated by a stretched exponential decay. [**A.3**]{}: If $s>1/2$, $\Theta(x)$ tends to a positive value, therefore, in this instance, the tail remains [*power-law*]{}. There, by switching $s$, one tunes the tail type, being a power-law for $s\ge 1/2$. In the threshold case $s=1/2$, we have $_2F_1(1,1,2,-z)\equiv\ln(1+z)/z$, then we get the explicit expression $$\label{A} \rho_{\rm A}(x) = \rho_o \frac{x^{\gamma_\alpha\theta}} {[1+\lambda^2x]^{\gamma_\alpha\theta+\gamma_\mu+1} } \; .$$ Thus, the case $r=0$, $s\ge1/2$ allows one to model empirical PDFs with twofold power-law behavior. [**B**]{}: In the case $r=-1$, the normalization condition requires: $s\ge 1/2$, or also, if $\gamma_\alpha>1$, $s\le 0$ is allowed. The PDF has the form $$\label{IIIm} \rho(x) = \rho_o \frac{x^{-\gamma_\alpha} \Theta(x)} {[1+\lambda^2x^{2s}]^{-\gamma_\alpha/(2s)+1} } \; ,$$ where, $\Theta(x)\equiv\exp[-\gamma_\alpha\theta\,_2F_1({\scriptstyle -\frac{1}{2s}, 1,-\frac{1}{2s}+1},-\lambda^2x^{2s})/x]$ tends to a finite value for large $x$, therefore, the tail is a [*power-law*]{}. The asymptotic behavior of $\Theta(x)$ for small $x$, depends on $s$. [**B.1**]{}: For $s>1/2$, it behaves as an [*exponential*]{} of $-1/x$, that dominates the low $x$ behavior of the PDF. [**B.2**]{}: For $-1/2<s\le 0$, $\Theta(x)$ behaves as an [*exponential*]{} of $-1/x^{1+2s}$, that dominates the asymptotic behavior. [**B.3**]{}: However, $\Theta(x)$ takes asymptotically a finite value, if $s<-1/2$; hence, the complete expression increases at the origin as a [*power-law*]{}. At the threshold value $s=-1/2$, by employing again the explicit expression for $_2F_1(1,1,2,-z)$, one obtains $$\label{B} \rho_{\rm B}(x) = \rho_o\frac{x^{\gamma_\mu\theta+1}} {[1+x/\lambda^2]^{\gamma_\alpha+\gamma_\mu\theta+1} }, \;\;\;\;\mbox{if $\gamma_\alpha>1$}\; .$$ Thus, the case $r=-1$, $s\le -1/2$ also provides twofold power-law distributions. ![ Diagram of the asymptotic behavior of the PDF defined by Eq. (\[FPss\]), in $(s,r)$-plane. Unshadowed regions and dotted borders are regions excluded by the normalization requirement. At both positive semi-axes, the growth at the origin is power law. On dark gray lines, tails are power law, with the tilted line corresponding to $r=2s-1$ ($p=-1$). Dashed lines correspond to pure exponential tails, with the tilted line corresponding to $r=2s$ ($p=0$). In the formulas, $a>0$, as well as the power-law exponents, generically depend on model parameters, moreover $p\equiv r-2s$. Symbols correspond to the special processes [**A**]{} \[Eq. (\[A\])\] and [**B**]{} \[Eq. (\[B\])\]. []{data-label="fig:am"}](vfigure2.eps){width="40.00000%"} In general, the class of asymptotic behavior is ruled by $(s,r)$ that determine the form of market laws. This holds, of course, as soon as the remaining parameters assume moderate values. For instance, the factor $\lambda^2\equiv\gamma_\alpha/\gamma_\mu$ accompanies $x^{2s}$ in the formula for $\rho(x)$ \[Eqs. (\[FPss\])-(\[intmarg\])\], then, extreme values of $\lambda$ will change the asymptotic regime. In fact, in the limit $\alpha=0$ (negligible additive noise, corresponding to $\lambda\to \infty$), different laws arise, as we have seen in the precedent Section. Summarizing, we have shown the whole picture of asymptotic behaviors that a general class of additive-multiplicative processes produce. As a consequence of the extra additive noise, new types of asymptotic behaviors emerge. Specially interesting solutions arise in the marginal cases $r=0,-1$ where two-fold power-law PDFs are found. Moreover, additive-multiplicative processes lead to higher richness of crossover behaviors, with respect to purely multiplicative processes. Therefore, the appearance of new PDF shapes exceeds the one resulting from the mere analysis of the asymptotic regimes. This is specially important because depending on the values of the parameters, the true asymptotic regime might fall outside the observable range. Comparison with empirical distributions {#Sec:empirical} ======================================= Let us consider, as paradigm of the PDFs with two-fold power-law behavior, Eqs. (\[A\]) and (\[B\]), that have a simple exact expression. Actually they have the same functional form, via redefinition of parameters ($\theta,\gamma_\alpha,\gamma_\mu$). This expression has been recently proposed in the literature as an ansatz for fitting the distribution of high-frequency stock-volumes [@vol1], under the form $$\label{fitting} \rho(x) = \rho_o \frac{(x/x_o)^\nu} {[1+(q-1)x/x_o]^\frac{1}{q-1} } \; ,$$ where, in that specific application, $x$ is identified with normalized stock volume. Therefore, identification of the process for real volumes with one of the models above, may allow an econophysical interpretation of the fitting parameters. Table I presents the correspondence between the parameters of Eq. (\[fitting\]) and those of processes [**A**]{} and [**B**]{}, given by Eqs. (\[A\]) and (\[B\]), respectively. ------------------- ------------------------------------ ------------------------------------ [**A**]{} [**B**]{} Eq. (\[fitting\]) Eq. (\[A\]) Eq. (\[B\]) \[3mm\] $1/(q-1)$ $1+\gamma_\alpha\theta+\gamma_\mu$ $1+\gamma_\alpha+\gamma_\mu\theta$ \[3mm\] $\nu$ $\gamma_\alpha\theta$ $1+\gamma_\mu\theta$ \[3mm\] $x_o$ $(q-1)\gamma_\mu/\gamma_\alpha$ $(q-1) \gamma_\alpha/\gamma_\mu$ \[3mm\] ------------------- ------------------------------------ ------------------------------------ : Correspondence amongst model parameters.[]{data-label="table"} Recall that $\gamma_\alpha\equiv\gamma/\alpha^2$ and $\gamma_\mu\equiv\gamma/\mu^2$ ($\lambda^2\equiv\gamma_\alpha/\gamma_\mu$), thus, the power-law exponent for small values of $x$, given by $\nu$ (see Table), increases with $\gamma$ and $\theta$, and is reduced by either one of the two noise amplitudes: the additive noise in process [**A**]{} and the multiplicative one in process [**B**]{}. The power-law decay (with exponent $1/(q-1)-\nu$) for large values of $x$ is ruled by either one of the effective coefficients $\gamma_\mu$ (in [**A**]{}) or $\gamma_\alpha$ (in [**B**]{}) (see Table). That is, the tail is fatter, the larger the corresponding noise amplitude. While in process [**A**]{} the multiplicative noise affects the tail, in model [**B**]{} it is affected by the additive noise, oppositely to what happens for small values. This is related to the sign of $s$, indicating higher multiplicative feedback for either increasing ([**A**]{}) or decreasing ([**B**]{}) values of $x$. Besides the good agreement already observed for volumes [@vol1; @vol2], we tested this functional form to daily data of volatilities reported in the literature[@micciche]. The results are shown in Fig. \[fig:data\]. In the models we are generalizing, the variable $x$ is usually identified with the variance or squared volatility ($x=v^2$). Then, the resulting PDF for $v$ is $$\label{fitvol} P(v)\;=\;\rho_o \frac{(v/v_o)^{2\nu+1}} {[1+(q-1)(v/v_o)^2]^\frac{1}{q-1} } \; ,$$ with $\rho_o={\scriptstyle 2(2-q)^\nu(q-1)\Gamma(\frac{1}{q-1}-1)/ [v_o\Gamma(\nu+1)\Gamma(\frac{1}{q-1}-\nu-1)]}$. Fig. \[fig:data\] shows an excellent agreement between theoretical and empirical PDFs, for the full range of data. Notice that the very central part of the distribution is parabolic in the log-log plot, then a poor statistics at the tails may mislead to think that the distribution is log-normal. ![PDF of normalized volatility of stocks traded in US equity market (data from Ref. [@micciche]). The full line corresponds to a fitting by the theoretical PDF given by expression (\[fitvol\]). Fitting parameters are ($q,\nu,v^2_o)\simeq(1.178,2.20,0.097)$. Insets: linear-linear and log-log representation of the same data. []{data-label="fig:data"}](vfigure3.eps){width="50.00000%"} Underlying dynamics {#underlying-dynamics .unnumbered} ------------------- The satisfactory agreement between the empirical data and Eq. (\[fitvol\]) suggests that processes similar to either [**A**]{} or [**B**]{} may rule squared-volatility evolution. Hence, let us look at the explicit form of the ILEs associated to processes [**A**]{} and [**B**]{}: $$\label{ILEA} \mbox{\bf A:}\;\;\;\; {\rm d} x =-\gamma[x-\theta]\frac{1}{x}{\rm d}t + \mu \sqrt{x}{\rm d}z + \alpha {\rm d}w \,.$$ $$\label{ILEB} \mbox{\bf B:}\;\;\; {\rm d} x =-\gamma[x-\theta]\frac{1}{x^2}{\rm d}t + \mu \frac{1}{\sqrt{x}}{\rm d}z + \alpha {\rm d}w \,.$$ The first term in each ILE represents the deterministic restoring force with respect to the level $\theta$. It derives from a confining potential of the form $\gamma[x-\theta \ln x]$ ([**A**]{}) or $\gamma[\ln x +\theta/x]$ ([**B**]{}). In both cases, the potential has a minimum located at $x=\theta$ and is divergent at $x=0$. Average values are $$\label{averages} \langle x \rangle_{\bf A} =\frac{\theta+1/\gamma_\alpha}{1-1/\gamma_\mu} \;\;\;\mbox{and}\;\;\; \langle x \rangle_{\bf B} =\frac{\theta+2/\gamma_\mu}{1-2/\gamma_\alpha}\;,$$ both averages are greater than $\theta$ and coincide only in the limit of relatively small noise amplitudes ($\gamma >>\alpha^2,\mu^2$). Moments $\langle x^n \rangle$ are finite only if $\gamma_\mu>n$ ([**A**]{}) or $\gamma_\alpha>n+1$ ([**B**]{}). In particular, the second moment is $$\label{s2a} \langle x^2 \rangle_{\bf A} =\frac{\gamma_\alpha(\theta+1/\gamma_\alpha)(\theta+2/\gamma_\alpha)} {\gamma_\mu(1-1/\gamma_\mu)(1-2/\gamma_\mu)} \;,$$ $$\langle x^2 \rangle_{\bf B} =\frac{\gamma_\mu(\theta+2/\gamma_\mu)(\theta+3/\gamma_\mu)} {\gamma_\alpha(1-2/\gamma_\alpha)(1-3/\gamma_\alpha)}\;.$$ In model [**A**]{}, increasing(decreasing) amplitude of the additive(multiplicative) noise, increases the width of the distribution, whereas model [**B**]{} presents opposite behavior. Thus, for instance, the additive noise has a confining effect in process [**A**]{}, opposite to the effect observed in processes with null $\theta$[@multi1]. On the other hand, the distribution has a maximum at $$\label{maxima} x^{max}_{\bf A} =\frac{\theta}{1+1/\gamma_\mu} \;\;\;\mbox{and}\;\;\; x^{max}_{\bf B} =\theta+\frac{1}{\gamma_\mu}\;.$$ Notice that the additive noise does not affect the mode, as expected. The most probable value of distribution [**A**]{} shifts to the right with increasing multiplicative amplitude, while in distribution [**B**]{} the opposite tendency occurs. From Eqs. (\[averages\]) and (\[maxima\]), $x^{max}_{\bf A}<\theta_{\bf A}<\langle x\rangle_{\bf A}$, while $\theta_{\bf B}<x^{max}_{\bf B}<\langle x\rangle_{\bf B}$. That is, in model [**A**]{}, the reference value $\theta$ represents a typical value comprised between two central measures, which does not hold in model [**B**]{}. This observation, in addition to the positivity of $s$, point to model [**A**]{} as a more realistic long-term process. The fitting parameters in Fig. \[fig:data\] lead to ($\theta,\gamma_\mu,\gamma_\alpha)_{\rm \bf A}\simeq(0.50,2.4,4.4)$ or ($\theta,\gamma_\mu,\gamma_\alpha)_{\rm \bf B}\simeq(0.19,6.3,3.4)$. In both cases, $\gamma_\mu,\gamma_\alpha>1$, as expected for regulated markets. While $\langle v \rangle =1$, because empirical volatility is normalized, $\langle x \rangle =\langle v^2 \rangle \simeq 1.3$ and the mode is $x^{max}\simeq 0.35$, consistently with Eqs. (\[averages\])-(\[maxima\]). Numerical integration of ILEs (\[ILEA\]) and (\[ILEB\]), by standard methods[@books], shows that both processes produce time series with bursting or clustering effects, as observed in real sequences. However, process [**B**]{} may present, for some values of the parameters, a kind of ergodicity breaking, with large jumps to a state basically governed by additive noise. This occurs because, once $x$ jumps to a high value, both the restoring force and the effective amplitude of multiplicative noise become small as to pull $x$ back to its reference level. Then, relaxation is slowed down and the regime of high volatility persists for long time stretches. Although a process with $s<0$ is not expected to be a realistic model for very long time intervals, it can model, for instance, the transient behavior of the market around crashes. In fact, process [**B**]{} yields akin crises. After the crash occurs, this drastic event might switch the system back to a $s\ge 0$ regime. Final remarks {#Sec:final} ============= We have analyzed stochastic models of a quite general form, with algebraic restoring force and algebraic multiplicative noise. A further generalization with the inclusion of an extra source of noise, of standard Wiener type, has also been analyzed. These additive-multiplicative processes are built on the basis of realistic features: The multiplicative noise describes innovations generated by endogenous mechanisms that amplify or attenuate a random signal, depending on the internal state of the system. Whereas, the additive noise encodes a direct influence of external random fields such as news or spontaneous fluctuations due to speculative trading. One of the goals of this work was to study systematically the PDF asymptotic solutions of these generalized models. We have shown that the inclusion of additive noise gives rise to new PDF shapes, with a richer spectrum of cross-over behaviors and, in particular, two-fold power-law decays. The shapes of the PDFs are governed by the effective market rules parametrized by $r$ and $s$. These parameters describe the algebraic nature of the global mean-reverting strength of the market and the informational coupling among the traders, respectively. On the other hand, power-law exponents and coefficients of exponential-like functions depend also on the reduced parameters $\gamma_\mu$, $\gamma_\alpha$ and on $\theta$. This means that one may expect universal behavior among markets that share similar rules (same $r$ and $s$) and same rescaled restoring parameters, for a properly normalized reference level $\theta$. Summarizing, the additive-multiplicative processes given by Eq. (\[ILE\]) provide a general realistic framework to describe the shape of empirical distributions for financial, as well as, for physical systems. An illustrative application to empirical volatility data was presented in Sect. \[Sec:empirical\], showing excellent results. The statistical description of a market should include its dynamical properties such as the temporal decay of correlations. In real time series of volatilities[@liu] and volumes[@volumes], power-law decaying correlations have been observed. It is worth noting that stochastic processes with additive-multiplicative structure (without mean reversion) are being currently studied in connection with a generalization of standard (Boltzmann-Gibbs) statistical mechanics, recently proposed by C. Tsallis [@tsallis]. The PDFs associated to this new formalism generalize the exponential weights, namely, $\exp_q(-x) \equiv (1-[1-q]x)^\frac{1}{1-q}$ \[entering as a factor in Eq. (\[fitting\])\]. The time series arising from additive-multiplicative processes without mean reversion present strong correlations that prevent convergence to either Gauss or Lévy limits[@multi3] and lead to $q$-Gaussian distributions. This suggests that similar correlations may persist in mean-reverting processes with the additive-multiplicative character. Once Eq. 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--- abstract: 'We show that there is no algorithm which, provided a polynomial number of random points uniformly distributed over a convex body in $\RR^n$, can approximate the volume of the body up to a constant factor with high probability.' author: - 'Ronen Eldan [^1]' title: --- Introduction ============ Volume-related properties of high-dimensional convex bodies is one of the main topics of convex geometry in research today. Naturally, calculating or approximating the volume of a convex body is an important problem. Starting from the 1980’s, several works have been made in the area of finding a fast algorithm for computing the volume of a convex body (see for example [@B],[@BF],[@LS],[@DFK],[@LV] and references therein).\ These algorithms usually assume that the convex body $K \subset \RR^n$, is given by a certain oracle. An oracle is a “black box” which provides the algorithm some information about the body. One example of an oracle is the **membership oracle**, which, given a point $x \in \RR^n$, answers either “$x \in K$” or “$x \notin K$”. Another example, is the **random point oracle**, which generates random points uniformly distributed over $K$.\ All volume computing algorithms, known to the author, which appear in the literature use the membership oracle. This note deals with a question asked by L. Lovász about the random point oracle. It has been an open problem for a while whether or not it is possible to find a fast algorithm which computes the volume of $K$ provided access to the random point oracle ([@GR], [@L0]).\ We answer this question negatively. In order to formulate our main result, we begin with some definitions.\ An algorithm which uses the random point oracle is a (possibly randomized) function whose input is a finite sequence of random points generated according to the uniform measure on $K$ and whose output is number, which is presumed be an approximation for the volume of $K$. The complexity of the algorithm will be defined by the length of the sequence of random points. We are interested in the existence of algorithms with a complexity which depends polynomially on the dimension $n$.\ We say that an algorithm is correct up to $C$ with probability $p$, if for any $K \subset \RR^n$, given the sequence of random points from $K$, the output of the algorithm is between $\frac{Vol(K)}{C}$ and $C Vol(K)$, with probability at least $p$.\ We prove the following theorem: \[maingen\] There does not exist constants $C,p,\kappa>0$ such that for any dimension $n$, there exists an algorithm with complexity $O(n^\kappa)$ which is correct in estimating the volume of convex bodies in $\RR^n$ up to $C$ with probability $p$. It is important to emphasize that this result is not a result in complexity theory. In this note we show that a polynomial number of points actually does not contain enough information to estimate the volume, regardless of the number of calculations, and hence, it is of information-theoretical nature.\ For convex geometers, the main point in this study may be the additional information on volume distribution in convex bodies it provides. We suggest the reader to look this result in view of the recent results concerning the distribution of mass in convex bodies. In particular, results regarding thin-shell concentration and the Central Limit Theorem for Convex bodies, proved in the general case by B. Klartag, show that essentially all of the mass of an isotropic convex body $K$ is contained in a very thin-shell around the origin, and that almost all of the marginals are approximately gaussian. This may suggest that, in some way, all convex bodies, when neglecting a small portion of the mass, behave more or less the same as a euclidean ball in many senses. Philosophically, one can also interpret these results as follows: provided a small number of points from a logarithmically-concave measure, one cannot distinguish it from a spherically symmetric measure. For definitions and results see [@K]. One of the main stages of our proof is to show that one cannot distinguish between the uniform distribution over certain convex bodies, which are geometrically far from a euclidean ball, and some spherically symmetric distribution, when the number of sample points is at most polynomially large.\ Here is a more quantitative formulation of what we prove: \[main\] There exists $\eps > 0$ and a number $N \in \mathbb{N}$ such that for all $n > N$, there does not exist an algorithm whose input is a sequence of length $e^{n^\eps}$ of points generated randomly according to the uniform measure in a convex body $K \subset \mathbb{R}^n$, which determines $Vol(K)$ up to $e^{n^{\eps}}$ with probability more than $e^{-n^\eps}$ to be correct. **Remark.** After showing that the volume of a convex body cannot be approximated, one may further ask: what about an algorithm that estimates the **volume radius** of a convex body, defined by $VolRad(K) = Vol(K)^{\frac{1}{n}}$? A proof which shows that it is also impossible has to be far more delicate than our proof. For example, under the hyperplane conjecture, it is easy to estimate the volume radius of a convex body up to some $C>0$.\ One may also compare this result to the recent result of N.Goyal and L.Rademacher ([@GR]). They show that in order to **learn** a convex body, one needs at least $2^{c \sqrt \frac{n}{\eps}}$ random points. Learning a convex body rougly means finding a set having at most $\eps$ relative symmetric difference with the actual body (see [@GR]).\ The general idea of the proof is as follows. Let $\{ K_\alpha \}_{\alpha \in I_1 }$ and $\{K_\alpha \}_{\alpha \in I_2}$ be two families of convex bodies. For $i=1,2$, a probability measure $\mu_i$, on the set of indices $I_i$ induces a probability measure $P_i$ on the set of sequences of points in $\RR^n$ in the following simple way: first generate an index $\alpha$ according to $\mu_i$, and then generate a sequence of uniformly distributed random points from $K_{\alpha}$. Suppose that the two families have the following properties:\ 1. For every $\alpha_1 \in I_1$ and $\alpha_2 \in I_2$, the ratio between $Vol(K_{\alpha_1})$ and $Vol(K_{\alpha_2})$ is large.\ 2. There exist two probability measures $\mu_1, \mu_2$ on $I_1, I_2$ respectively such that $P_1$ and $P_2$ are very similar in a sense described later on.\ A simple application of Yao’s lemma will help us assume that the algorithm is deterministic. A deterministic algorithm is actually a function $F : \mathbb{R}^{n^{\kappa + 1}} \to \mathbb{R}$ which takes a sequence of points and returns the volume of the body. If the total variation distance between the probabilities $P_1$ and $P_2$ defined above is small, then, there exist a set $A \subset \mathbb{R}^{n^{\kappa + 1}}$ which has a high probability with respect to both $P_1$ and $P_2$. Obviously, for all $x \in A$, $F(x)$ is wrong in approximating the volume of at least one of the families.\ In section 2, we will describe how we build these families of bodies, $\{K_{\alpha} \}$, using a random construction which starts from a euclidean ball, to which deletions which cut out parts of it, generated by some Poisson process, are applied. Then, using elementary properties of the Poisson process and some concentration of measure properties of the ball, we will see that the correlation between different points in polynomially long sequence of random points generated uniformly from the body will be very weak (with respect to the generation of the body itself). Using this fact, we will only have to inspect the distribution of a single random point. The construction will have a spherically-symmetric nature, so the density of a single random point will only depend on its distance from the origin, and therefore we will only have to care about the distribution of the distance of a point from the origin in the generated bodies. The role of the following section, which is more technical but fairly delicate, will be to calibrate this construction so that these families have different volumes, yet, approximately the same distribution of distance from the origin.\ \ Before we proceed to the proof, let us introduce some notation. In this note the number $n$ will always denote a dimension. For an expression $f(n)$ which depends on $n$, we agree that $f(n) \ll \Xi(n)$ means: there exists some $n_0 \in \mathbb{N}$ and $\epsilon > 0$ such that for all $n > n_0$, $f(n) < e^{-n^{\epsilon}}$. Also write $f(n) \asymp g(n)$ for $\left |\frac{f(n)}{g(n)} - 1 \right | \ll \Xi(n)$ and $f(n) \approx g(n)$ for $|f(n) - g(n)| \ll \Xi(n)$. The notation $f(n) \lesssim g(n)$ and $f(n) \gtrsim g(n)$ will be interpreted as $f(n) < g(n)$ and $f(n) > g(n)$ for $n$ large enough. Moreover, we decide that $N=N(n)$, denotes the length of the sequence of random points. All throughout this note we assume that there exists a universal constant $\eps > 0$, such that $N(n) < e^{n^\eps}$.\ \ **Acknowledgements** I am deeply grateful to my supervisor Prof. Boäz Klartag for very useful discussions and encouragement all along my work on the subject. I would also like to express my thanks to my supervisor, Prof Vitali Milman for introducing me to this question and encouraging me to work on it. The Deletion Process {#sec2} ==================== In this section we will describe the construction of the random bodies which will later be used as counter-examples. Our goal, after describing the actual construction, will be to prove, using some simple properties of the Poisson distribution, a weak-correlation property between different points generated from the body.\ Denote by $D_n$ the $n$ dimensional euclidean ball of unit radius, centered at the origin, and by $\omega_n$ its Lebesgue measure.\ Recall that for two probability measures $P_1, P_2$ on a set $\Omega$, the total variation distance between the two measures is defined by $$d_{TV}(P_1,P_2) = \sup_{A \subseteq \Omega} |P_1(A) - P_2(A)|$$ One can easily check that if these measures are absolutely continuous with respect to some third measure $Q$, then it is also equal half the $L_1(Q)$ distance between the two densities.\ Define $r_0 = n^{-\frac{1}{3}}$, and $$T_0(\theta) = D_n \cap \{ x; \langle x, \theta \rangle \leq r_0 \}.$$ Let $T$ be a function from the unit sphere to the set of convex bodies, such that for every $\theta \in \Sph$, $T(\theta)$ satisfies $T_0(\theta) \subseteq T(\theta) \subseteq D_n$. (Recall that most of the mass of the euclidean ball is contained in $\{ x_1 \in [-1, C n^{-\frac 1 2}] \}$. So $T(\theta)$ contains almost all the mass of the euclidean ball). Moreover let $m>0$. We will now describe our construction of a random convex body, $K_{T, m}$. First, suppose that $m \in \mathbb{N}$. Let $\Theta = (\theta_1, \theta_2, ..., \theta_m)$ be $m$ independent random directions, distributed according to the uniform measure on $\Sph$. We define $K_{T,m}$ as, $$K_{T,m} = D_n \bigcap_i T(\theta_i).$$ Finally, instead of taking a fixed $m \in \mathbb{N}$, we take $\zeta$ to be a a Poisson random variable with expectation $m$, independent of the above. We can now define $ K_{T, \zeta} $ in the same manner.\ Let us denote the probability measure on the set of convex bodies induced by the process described above by $\mu$. After generating the body $K_{T, m}$, which, from now on will be denoted just by $K$ wherever there is no confusion caused, we consider the following probability space: let $\Omega = (D_n)^N$ be the set of sequences of length $N$ of points from $D_n$. Denote by $\lambda$ the uniform probability measure on $\Omega$, and for a convex body $K$ denote by $\lambda_K$ the uniform probability measure on $K^N = \prod_{1 \leq i \leq N} K \subseteq \Omega$. Finally, define a probability measure $P=P_{T,m}$ on $\Omega$ as follows: for $A \subseteq \Omega$, $$P(A) = \int \lambda_K(A) d \mu(K) = \int \frac{Vol(K^N \cap A)}{Vol(K^N)} d \mu(K)$$ (The measure $P$ describes the following process: first generate the random set $K$ according to construction described above, and then generate $N$ i.i.d random points, independent of the above, according to the uniform measure on $K$). Moreover, for $p = (x_1, ..., x_N) \in \Omega$, define $\pi_i(p) = x_i$, the projections onto the $i$-th copy of the euclidean ball.\ It it easy to check that $P$ is absolutely continuous with respect to $\lambda$. We define the following function on $\Omega$: $$\label{deff} f_{T,m}(p) = \P(p \in K_{T,m}^N) = \P(\forall 1 \leq i \leq N, \pi_i(p) \in K_{T, m}).$$ As we will see later, the function $f$ is related in a simple way to $\frac{dP}{d \lambda}$. Namely, we will have, $$\frac{dP}{d \lambda}(p) \asymp \frac{f(p)}{\int_{\Omega} f}$$ for all $p$ in some subset of $\Omega$ with measure close to 1. Let us also agree that from now on $f_{T, m}$ is just denoted by $f$.\ \ We start with some simple geometric observations regarding $\Omega$. Define, for $p \in \Omega$, $1 \leq i \leq N$, $$\label{defai} A_i(p) = \{ \theta \in \Sph; \pi_i(p) \notin T(\theta) \}$$ For $1 \leq i,j \leq N$, let $F_{i,j} \subset \Omega_N$ be the event, defined by $$F_{i,j} = \left \{p; ~~ \frac{\sigma(A_{i}(p) \cap A_{j}(p))}{\sigma(A_{i}(p))} < e^{-n^{0.1}} \right \}$$ and let, $$F = \bigcap_{1 \leq i \neq j \leq N} F_{i,j}$$ (which should be understood as “no two points are too close to each other”, and, as we will see, will imply that points are weakly correlated). We start with the following simple lemma. \[mutualcaps\] Under the above notations:\ (i) $\lambda(F) \approx 1 $.\ (ii) There exists some $\eps_0 > 0$ such that: if we assume that following condition holds, $$\label{volume} \P_\mu (Vol(K) < \omega_n e^{-n^{\eps_0}}) < e^{-n^{\eps_0}}$$ (hence, the volume of $K$ is typically not much smaller than the volume of $D_n$). Then we have $P(F) \approx 1$.\ **Proof:**\ (i) Let $p$ be uniformly distributed in $\Omega$. Denote $x_i = \pi_i(p)$, so $x_1, x_2$ are independent points uniformly distributed in $D_n$. Let us calculate $\lambda(F_{1,2})$.\ First, for a fixed $\theta \in \Sph$, one has $$\P(x_1 \notin T(\theta)) \leq \P(x_1 \notin T_0(\theta)) = \P(\{\langle x_1, \theta \rangle \geq r_0\})$$ Recalling that $r_0 = n^{-\frac{1}{3}} \gg n^{-\frac{1}{2}}$, by elementary calculations regarding marginals of the euclidean ball, one gets $$\P(x_1 \notin T(\theta)) \lesssim e^{-n^{0.2}}$$ Now, fix $x_2' \in D_n$. We agree that $A_i:=A_i(p)$. One has, $$\EE (\sigma (A_1 \cap A_2) | x_2 = x_2') = \int_{A_2} \P({\theta \in A_1}) d \sigma(\theta) = \int_{A_2} \P(x_1 \notin T(\theta)) d \sigma(\theta) \lesssim \sigma(A_2) e^{-n^{0.2}}$$ And so, $$\EE (\frac{\sigma (A_1 \cap A_2)}{\sigma(A_2)} | x_2 = x_2') \lesssim e^{-n^{0.2}}$$ Now, this is true for every choice of $x_2'$, so integrating over $x_2'$ gives $$\EE \frac{\sigma (A_1 \cap A_2)}{\sigma(A_2)} \lesssim e^{-n^{0.2}}$$ Now we use Markov’s inequality to get $$\label{lambdaf} \lambda(F_{1,2}^C) = \lambda(\left \{ \frac{\sigma(A_1\cap A_2)}{\sigma(A_2)} > e^{-n^{0.1}} \right \}) \ll \Xi(n)$$ A union bound completes the proof of (i).\ **Proof of (ii)** First, we can condition on the event $\{ Vol(K) > \omega_n e^{\eps_0} \}$ (with $\eps_0$ to be chosen later). (\[volume\]) ensures us that it will happen with probability $\approx 1$. Observe that for any event $E \subset \Omega$ which is measurable by the $\sigma$-field generated by $\pi_1, \pi_2$, we have $$\label{twocoords} \lambda_K (E) = \frac{\omega_n^2 \lambda((K \times K \times D_n \times ... \times D_n) \cap E)}{Vol(K)^2} \leq \frac{\omega_n^2 \lambda(E)}{Vol(K)^2}$$ Now, taking $E=F_{1,2}^C$, choosing $\eps_0$ to be small enough and using (\[lambdaf\]) and (\[twocoords\]) along with (\[volume\]), one gets $$P(F_{1,2}) \approx 1.$$ Applying a union bound finishes the proof.\ \ We can now turn to the lemma which contains the main ideas of this section: : \[indep\] There exist $\eps_0, \eps_1 > 0$ and $n_0$ such that for every $n > n_0$, the following holds: Whenever $m$ is small enough such that the following condition is satisfied: $$\label{vol2} \P (\{\theta \in K \}) > e^{-n^{\eps_0}}, ~~\forall \theta \in \Sph$$ (hence, we are not removing too much volume, in expectation, even from the outer shell). Then:\ (i) We have, $$\label{volconc} P(|Vol(K) - \EE(Vol(K))| > e^{-n^{\eps_1}} \EE(Vol(K)) \ll \Xi(n)$$ and also (\[volume\]) holds.\ (ii) For all $p \in F$, we have $$f(p) \asymp \prod_{j=1}^N \P(\pi_j(p) \in K)$$ In other words, if we define $\tilde f : D_n \to \RR$ as, $$\label{deftildef} \tilde f (x) = \P(x \in K)$$ then $$\label{fftilde} f(p) \asymp \prod_i \tilde f(\pi_i(p)), \forall p \in F.$$ and, $$\mbox{(iii)} ~~~ \frac{\EE(Vol(K^N \cap F))}{(\EE Vol(K))^N} - 1 \ll \Xi(n)$$ **Proof**: We begin by proving (ii).\ Fix $p \in F$. Define $x_i = \pi_i(p)$, and $A_i = A_i(p) \subset \Sph$ as in (\[defai\]). Also define $G_j = \bigcap_{i \leq j} \{x_i \in K \}$. Fix $2 \leq j \leq N$. Let us try to estimate $ P(G_j | G_{j-1})$.\ When conditioning on the event $G_{j-1}$, we can consider our Poisson process as a superposition of three “disjoint” Poisson processes: the first one, with intensity $\lambda_s$, only generates deletions that cut $x_j$, but leave all the $x_i$’s for $i < j$ intact. The second one, with intensity $\lambda_u$ deletes $x_j$ along with one of the other $x_i$’s, and the third one is the complement (hence, deletions that do not affect $x_j$). We have, recalling that the the expectation of the number of deletions is $m$, $$\lambda_s(\Sph) + \lambda_u(\Sph) = m \sigma(A_j)$$ Moreover, $$\lambda_u(\Sph) \leq m \sum_{i < j} \sigma(A_i \cap A_j)$$ (in the above formula we are including, multiple times, deletions that cut more than two points, hence the inequality rather than equality).\ Now, using the definition of $F$ one gets $$\frac{\lambda_u(\Sph)}{\lambda_s(\Sph) + \lambda_u(\Sph)} \ll \Xi(n).$$ Note that (\[vol2\]) implies $$e^{-(\lambda_s(\Sph) + \lambda_u(\Sph))} \geq e^{-m \sigma(\{\theta; \frac{x_j}{|x_j|} \notin T(\theta)\} )} \geq e^{-n^{\eps_0}}$$ (the first inequality follows from the fact that $T(\theta)$ are star-shaped). The last two inequalities give, $$\lambda_u(\Sph) \ll \Xi(n).$$ It follows that, $$\label{telescope1} \left | \frac{P(G_j | G_{j-1})}{P(\{x_j \in K \})} - 1 \right | = \frac{e^{-\lambda_s(\Sph)}}{e^{-(\lambda_s(\Sph) + \lambda_u(\Sph))}} - 1 \ll \Xi(n)$$ Moreover, one has $$\label{telescope2} P(G_N) = \prod_{j} P(G_j | G_{j-1}) = \prod_{j} \left (\frac{P(G_j | G_{j-1})}{P(\{x_j \in K \})} P(\{x_j \in K \}) \right)$$ Using (\[telescope1\]) and (\[telescope2\]) we get $$\label{finalindep} f(p) = P(G_N) \asymp \prod_j P(\{x_j \in K \})$$ This proves (ii).\ **Proof of (i):** Showing that (\[volume\]) holds is just a matter of noticing that $\P(x \in K)$ is monotone decreasing with respect to $|x|$ and taking $\eps_0$ to be small enough. We turn to estimate $\EE(Vol(K)^2)$. We have $$\EE(Vol(K)^2) = \int_{D_n \times D_n} \P(\{x_1 \in K\} \cap \{x_2 \in K\}) dx1 dx2 =$$ $$\int_{(D_n \times D_n) \cap F_{1,2}} \P(\{x_1 \in K\} \cap \{x_2 \in K\}) dx1 dx2 +$$ $$\int_{(D_n \times D_n) \cap F_{1,2}^C} \P(\{x_1 \in K\} \cap \{x_2 \in K\}) dx1 dx2$$ (we will later see that the second summand is negligible). Now, (\[finalindep\]) gives $$\int_{(D_n \times D_n) \cap F_{1,2}} \P(\{x_1 \in K\} \cap \{x_2 \in K\}) dx1 dx2 \asymp$$ $$\int_{(D_n \times D_n) \cap F_{1,2}} \P(\{x_1 \in K\}) \P(\{x_2 \in K\}) dx1 dx2,$$ which also implies that $$\int_{(D_n \times D_n) \cap F_{1,2}} \P(\{x_1 \in K\} \cap \{x_2 \in K\}) dx1 dx2 > \frac{1}{2} e^{-2n^{\eps_0}}$$ Recall that $\lambda(F_{1,2}^C) \ll \Xi(n)$ (as a result of the previous lemma). Taking $\eps_0$ to be small enough, we will get $$\EE(Vol(K)^2) \asymp \int_{(D_n \times D_n) \cap F_{1,2}} \P(\{x_1 \in K\} \cap \{x_2 \in K\}) dx1 dx2 \asymp$$ $$\int_{(D_n \times D_n) \cap F_{1,2}} \P(\{x_1 \in K\}) \P(\{x_2 \in K\}) dx1 dx2.$$ On the other hand, $$\EE (Vol(K))^2 = \int_{(D_n \times D_n)} \P(\{x_1 \in K\}) \P(\{x_2 \in K\}) dx1 dx2.$$ Using the same considerations as above, the part of the integral over $F_{1,2}^C$ can be ignored, hence, $$\EE (Vol(K))^2 \asymp \int_{(D_n \times D_n) \cap F_{1,2}} \P(\{x_1 \in K\}) \P(\{x_2 \in K\}) dx1 dx2.$$ So we finally get $$\EE (Vol(K)^2) \asymp \EE(Vol(K))^2$$ Recalling that we assume (\[vol2\]), using Chebishev’s inequality, this easily implies (i), which finishes (ii).\ For the proof of (iii), $$\EE (Vol(K^N \cap F)) = \int_F \P(p \in K^N) dp \asymp \int_F \prod_i \P(\pi_i(p) \in K) \leq (\EE Vol(K))^N.$$\ \ Consider the density $\frac{dP}{d \lambda}$. Our next goal is to find a connection between this density and the function $f$. Let $A \subseteq F \subset \Omega$. Using the concentration properties of $Vol(K)$, we will prove the following, $$\label{fandp} P(A) \approx \frac{\int_A f(p) dp}{(\int_{D_n} \tilde f(x))^N}.$$ where $f, \tilde f$ are defined in equations (\[deff\]) and (\[deftildef\]).\ We have, $$\label{eqng0} P(A) = \EE_{\mu} \left ( \frac{Vol(K^N \cap A)}{Vol(K^N)} \right )= \EE_{\mu} \left (\frac{Vol(K^N \cap A)}{Vol(K)^N} \right).$$ By Fubini, $$\label{fubg} \EE_{\mu} Vol(K^N \cap A) = \int_A f(p) dp.$$ Consider the event $$G := \left \{ \left | \frac{Vol(K)^N}{\EE(Vol(K))^N} - 1 \right | < e^{-n^{\frac{\eps_1}{2}}} \right \}$$ (where $\eps_1$ is the constant from lemma \[indep\]). We have by the definition of $G$, $$\label{eqng1} \int_{G} \frac{Vol(K^N \cap A) }{Vol(K)^N} d \mu(K) \approx \frac{\int_{G} Vol(K^N \cap A) d \mu(K)}{\EE(Vol(K))^N}.$$ It follows from part (i) of lemma \[indep\] that, $$\mu(G) = \P(\left| (\frac{Vol(K)}{\EE(Vol(K))})^N - 1 \right | \leq e^{-n^{\frac{\eps_1}{2}}}) \geq$$ $$\P(\left| \frac{Vol(K)}{\EE(Vol(K))} - 1 \right | \leq 2 N e^{-n^{\frac{\eps_1}{2}}}) \geq \P(\left| \frac{Vol(K)}{\EE(Vol(K))} - 1 \right | \leq e^{-n^{\eps_1}}) \approx 1.$$ So $\mu(G) \approx 1$ which gives, $$\label{eqng2} \int_{G^C} \frac{Vol(K^N \cap A)}{Vol(K)^N} d \mu (K) \leq \mu(G^C) \ll \Xi(n).$$ We will also need: $$\label{eqng3} \frac{ \int_{G^C} Vol(K^N \cap A) d \mu(K)}{(\EE Vol(K))^N} \ll \Xi(n).$$ To prove this, first recall that $A \subseteq F$. This gives, $$\label{eqng4} \frac{ \int_{G^C} Vol(K^N \cap A) d \mu(K)}{(\EE Vol(K))^N} \leq \frac{ \int_{G^C} Vol(K^N \cap F) d \mu (K)}{(\EE Vol(K))^N} =$$ $$\frac{ \EE_\mu Vol(K^N \cap F)}{(\EE Vol(K))^N} - \frac{ \int_{G} Vol(K^N \cap F) d \mu (K)}{(\EE Vol(K))^N}.$$ Now, $$\int_G \frac{Vol(K^N \cap F)}{Vol(K^N)} d \mu(K) \approx \EE_\mu \frac{Vol(K^N \cap F)}{Vol(K^N)} = P(F) \approx 1$$ so, $$\label{eqng5} \frac{ \int_{G} Vol(K^N \cap F) d \mu (K)}{(\EE Vol(K))^N} \approx 1$$ Using part (iii) of lemma \[indep\] along with (\[eqng4\]) and (\[eqng5\]) proves $(\ref{eqng3})$.\ Plugging together (\[eqng0\]), (\[eqng1\]), (\[eqng2\]) and (\[eqng3\]) imply $$\label{fandpi} P(A) = \EE_{\mu} \frac{Vol(K^N \cap A)}{Vol(K)^N} \approx \int_G \frac{Vol(K^N \cap A)}{Vol(K)^N} d \mu (K)$$ $$\approx \frac{\int_G Vol(K^N \cap A) d \mu (K)}{\EE(Vol(K))^N} \approx \frac{\EE_{\mu} Vol(K^N \cap A) }{\EE(Vol(K))^N}$$ Recall that, as a result of Fubini’s theorem, $$\label{expvolf} \EE_\mu (Vol(K)) = \int_{D_n} \tilde f(x) dx.$$ Plugging (\[fandpi\]), (\[expvolf\]) and (\[fubg\]) proves (\[fandp\]). We would now like to use the result of lemma \[indep\], to replace $f$ with $\tilde f$. Let $A' \subseteq \Omega$. Define $A=A' \cap F$, $$P(A') = P(A) + P(A' \cap F^C).$$ Part (ii) of lemma \[mutualcaps\] with (\[fandp\]) gives $$P(A') \approx P(A) \approx \frac{\int_{A} f(p) dp}{(\int_{D_n} \tilde f(x)dx)^N}.$$ We can now plug in (\[fftilde\]) to get $$P(A') \approx \frac{\int_{A} \prod_i \tilde f(\pi_i(p)) d p}{(\int_{D_n} \tilde f(x))^N}.$$ So, finally defining $$\frac{d \tilde P}{d p} = \frac{\mathbf{1}_{\{ p \in F \}} \prod_i \tilde f(\pi_i(p))}{(\int_{D_n} \tilde f(x))^N} = \mathbf{1}_{\{ p \in F \}} \prod_i \frac{\tilde f(\pi_i(p))}{\int_{D_n} \tilde f(x) dx}$$ we have proved the following lemma: \[ptilde\] Suppose that the condition (\[vol2\]) from Lemma \[indep\] holds. Then one has $$d_{TV} (P, \tilde P) \ll \Xi(n)$$ Note that the measure $\tilde P$ is not, in general, a probability measure. The lemma, however, ensures us that $\tilde P(\Omega)$ is very close to 1.\ Recall that our plan is to find two families of convex bodies, which are achieved by two pairs $(T_1, m_1)$ and $(T_2, m_2)$, such that $d_{TV} (P_1, P_2)$ is small, even though their volumes differ.\ The above lemma motivates us to try to find such pairs with $\frac{\tilde f_1}{\int \tilde f_1} \approx \frac{\tilde f_2}{\int \tilde f_2}$. We formulate this accurately in the following lemma. \[sec1final\] Suppose there exist two pairs $(T_i, m_i)$ for $i=1,2$ such that (\[vol2\]) is satisfied, and in addition, defining $\tilde f_1$ and $\tilde f_2$ as in (\[deftildef\]), $$\label{l1dist} \left | \left | \frac{\tilde f_1}{\int_{D_n} \tilde f_1} - \frac{\tilde f_2}{\int_{D_n} \tilde f_2} \right | \right |_{L_1(D_n)} \ll \Xi(n)$$ Then $d_{TV}(P_1, P_2) \ll \Xi(n)$. **Proof**:\ Using the previous lemma, it is enough to show that $d_{TV}(\tilde P_1, \tilde P_2) \ll \Xi(n)$. Define $g_i = \frac{\tilde f_i}{\int_{D_n} \tilde f_i}$. We have $$d_{TV} (\tilde P_1, \tilde P_2) \leq \int_{\Omega} \left | \prod_{1 \leq i \leq N} g_1(\pi_i(p)) - \prod_{1 \leq i \leq N} g_2(\pi_i(p)) \right | \leq$$ $$\sum_{1 \leq j \leq N} \int_{\Omega} \left |\prod_{1 \leq i \leq j} g_1(\pi_i(p)) \prod_{j+1 \leq i \leq N} g_2(\pi_i(p)) - \prod_{1 \leq i \leq j+1} g_1(\pi_i(p)) \prod_{j+2 \leq i \leq N} g_2(\pi_i(p)) \right | =$$ $$N \int_{D_n} |g_1(x) - g_2(x)| \ll \Xi(n)$$ In the next section we deal with how to calibrate $T_i$ and $m_i$ so that (\[l1dist\]) holds. Building the two profiles ========================= For a measurable body $L \subset \RR^n$, define $$\label{tvoldist} g_{L}(r) = 1 - \sigma(\frac{1}{r} L \cap \Sph),$$ This function should be understood as the “profile” of mass of the complement of $L$, which will eventually be the ratio of mass which a single deletion removes, in expectation, as a function of the distance from the origin.\ Our goal in this section is to build convex bodies with a prescribed profile, as formulated in the following lemma: \[bodies\] For every dimension $n$, there exist two convex bodies $T_1, T_2 \subset \RR^n$, satisfying the following: $$\label{tsupset} \mbox{(i) } D_n \supseteq T_i \supseteq D_n \cap \{x; \langle x, e_1 \rangle \leq n^{-\frac{1}{3}} \}, ~~ i=1,2$$ (ii) Defining $g_i(r) = g_{T_i}(r)$ as in (\[tvoldist\]), we have $$\label{derratio} g_1(1) = g_2(1) \neq 0, ~~ \mbox{ and } ~~ g_1'(r) = 2 g_2'(r) ~~ \forall r \in [1 - n^{-0.99}, 1]$$ To achieve this, we begin by describing the following construction: Define $\delta_0 = n^{-\frac{1}{4}}$, $\delta_1 = n^{-0.99}$. For every two constants $a, b$ such that $a \in [2,200]$ and $b \in [-1000,1000]$, let $f=f_{a,b}$ be the linear function with negative slope which satisfies: $$\label{lin1} f (\delta_0 (1 + \delta_1b)) = \sqrt{1 - (\delta_0 (1 + \delta_1b))^2}$$ and, $$\label{lin2} \min_{x \in \RR} \sqrt{x^2 + f^2(x)} = a \delta_0$$ (hence, it is a line of distance $a \delta_0$ from the origin which meets the unit circle at $x=\delta_0(1 + b \delta_1)$. Note that there exists such a linear function with negative slope since $a \delta_0 \gg \delta_0(1 + b \delta_1)$). We define a convex body $T_{a,b}$ by, $$T_{a,b} = D_n \cap \left \{(x, \vec y) \in \RR \times \RR^{n-1} = \RR^n ; |y| \leq f(x) \right \}$$ (an intersection of the ball with a cone defined by a linear equation the coefficients of which depend of $a,b$).\ Recall that we require that $a > 2$ and $b > -1000$. First of all, it follows directly from requirement (\[lin1\]) and from the fact that the slope of $f$ is negative, that $T_{a,b}$ satisfies (\[tsupset\]) (since $\delta_0 \gg n^{-1/3}$).\ Define $g_{a,b}(r) = g_{T_{a,b}}(r)$ as in (\[tvoldist\]). Let us find an expression for $g_{a,b}(r)$. First, a simple calculation shows that (\[lin2\]) implies that the function $f_{a,b}$ intersects the $x$ axis at $x < 2 a \delta_0$. This shows that $T_{a,b} \cap r \Sph$ has only one connected component for all $r > \frac{1}{2}$ (hence, the vertex of the cone is inside the sphere).\ Consider the intersection $\frac{1}{r} T_{a,b} \cap \Sph$. If $r > \frac{1}{2}$, it must be a set of the form $\Sph \cap \{x_1 < x(a,b,r)\}$, for some function $x(a,b,r)$. Let us try to find the expression for this function. Equation (\[lin2\]) shows that $T_{a,b}$ is an intersection of $D_n$ with halfspaces at distance $a \delta_0$ from the origin. This implies that $x(a,b,r)$ must satisfy $$x(a,b,r) = \sin(\arcsin(\frac{a \delta_0}{r}) + c)$$ for some constant $c$ (draw a picture). To find the value of $c$, we use (\[lin1\]) to get $x(a,b,1) = \delta_0 (1 + b \delta_1)$, and so $$\label{finalxabr} x(a,b,r) = \sin(\arcsin(\frac{a \delta_0}{r}) - \arcsin(a \delta_0) + \arcsin(\delta_0 (1 + b \delta_1)).$$ Next, define $$\Psi(x) = \frac{1}{\omega_n} \int_{\min(x,1)}^1 (1 - t^2)^{\frac{n-3}{2}} dt,$$ the surface area measure of a cap the base of which has distance $x$ from the origin. We have finally, $$g_{a,b}(r) = \sigma(\Sph \cap \{x_1 \geq x(a,b,r) \}) = \Psi(x(a,b,r)).$$ Given a subset $I' \subseteq \RR \times \RR$, we define $$\label{constrk} K_{I'} = \bigcap_{(a,b) \in I'} T_{a,b}.$$ Clearly $$g_{I'}(r) := g_{K_{I'}}(r) = \sup_{(a,b) \in I'} g_{a,b}(r)$$ Our goal is to choose such a subset so that (\[derratio\]) is fulfilled. We will use the following elementary result: \[createconv\] Let $c > 0$, and let $\{ f_{\alpha} \}_{\alpha \in I}$ be a family of twice-differentiable functions defined on $[x_1,x_2]$ such that for every triplet $(x, y, y') \in [x_1,x_2] \times [y_1,y_2] \times [y_1',y_2']$, there exists $\alpha \in I$ such that $$f_{\alpha}(x) = y, ~~ f_{\alpha}(x)' = y', ~~ f''(t) \leq c, \forall t \in [x_1,x_2]$$ then for every twice differentiable function $g: [x_1,x_2] \to [y_1,y_2]$ with $$\label{eqg} g'(x) \in [y_1',y_2'], ~~ g''(x) > c,$$ there exists a subset $I' \subset I$ such that $$g(x) = \sup_{\alpha \in I'} f_{\alpha}(x)$$ In view of the above lemma, we would like to show that by choosing appropriate values of $a,b$, one can attain functions $g_{a,b}$ which, for a fixed $r_0$, have prescribed values $g_{a,b}(r_0), g_{a,b}'(r_0)$, and a small enough second derivative. Define $r(u) = 1 - \delta_1 u$. Note that substituting $r \to u$, almost all of the mass of the euclidean ball is contained in $u \in [0,1]$ (the thin shell of the euclidean ball). We now turn to prove the following lemma: \[uglylemma\] Suppose that $(u, g_0, g_0')$ satisfy $0 \leq u \leq 1$, $$\Psi(\delta_0) - 100 \delta_0 \delta_1 \Psi'(\delta_0) \leq g_0 \leq \Psi(\delta_0) + 100 \delta_0 \delta_1 \Psi'(\delta_0),$$ $$10 \delta_0 \delta_1 \Psi'(\delta_0) \leq g_0' \leq 100 \delta_0 \delta_1 \Psi'(\delta_0).$$ There exist constants $a \in [2, 200], b \in [-1000, 1000]$ such that $g_{a,b}(r(u)) = g_0$, $(g_{a,b}(r(u)))' = g_0'$ and $g_{a,b}(r(t))'' \leq \delta_0 \delta_1 \Psi'(\delta_0), \forall 0 \leq t \leq 1$. **Proof:** Throughout this proof we always assume $u \in [0,1], a \in [2,200]$ and $b \in [-1000, 1000]$.\ Let us inspect the function $x(a,b,r))$ defined in (\[finalxabr\]). Differentiating it twice, while recalling that $a \delta_0 \ll \frac{1}{2}$, gives us the following fact: there exists $C > 0$ independent of $n$, such that $ |\frac{\partial^2}{\partial r^2} x(a,b,r)| < C $. Consider $x(a,b,u) := x(a,b,r(u))$. One has, $$x_{uu}(a,b,u) = O(\delta_1^2)$$ (here and afterwards, by “$O$”, we mean that the term is smaller than some universal constant times the expression inside the brackets, which is valid as long as $u,a,b$ attain values in the intervals defined above). This implies that for all $u \in [0,1]$, $$x_{u}(a,b,u) = x_{u}(a,b,0) + O(\delta_1^2) =$$ $$a \delta_0 \delta_1 \sin'(\arcsin(\delta_0 (1 + b \delta_1)))(1 + O(\delta_0)) + O(\delta_1^2)=$$ $$a \delta_0 \delta_1 (1 + O(\delta_0))$$ and so, $$x(a,b,u) = x(a,b,0) + a \delta_0 \delta_1 u (1 + O(\delta_0)) = \delta_0 + \delta_0 \delta_1(a u + b)(1 + O(\delta_0))$$ Let us now define $w(a,b,u) = \frac{1}{\delta_1} (\frac{x(a,b,r(u))}{\delta_0} - 1)$. So, $$w(a,b,u) = (au + b)(1 + O(\delta_0))$$ and, $$\label{dersw} w_u(a,b,u) = a(1 + O(\delta_0)), ~~ w_{uu}(a,b,u) = O(\frac{\delta_1}{\delta_0}).$$ Next, we consider $g_{a,b}(r(u)) = \Psi(x(a,b,u)) = \Psi(\delta_0(1 + \delta_1 w(a,b,u))$. We have, $$\Psi(x(a,b,u)) = \Psi(\delta_0) + \delta_0 \delta_1 \Psi'(\delta_0) w(a,b,u) + \frac{\delta_0^2 \delta_1^2}{2} \Psi''(t)w(a,b,u)^2$$ for some $t \in [\delta_0, x(a,b,u)]$. But, note that the following holds, $$\label{secondderiv} (\log \Psi'(v))' = \frac{\Psi''(v)}{\Psi'(v)} = - \frac{2v\frac{n-3}{2} (1 - v^2)^{\frac{n-5}{2}}}{(1 - v^2)^{\frac{n-3}{2}}} = - \frac{ v (n-3)}{(1 - v^2)},$$ and for all $v \in [\frac{\delta_0}{2}, 2 \delta_0]$, $$(\log \Psi'(v))' = O(n \delta_0).$$ Integration of this inequality yields that for $t$ such that $t - \delta_0 = O(\delta_0 \delta_1)$, one has $$\log \Psi'(t) - \log \Psi'(\delta_0) = O(n \delta_0^2 \delta_1)$$ or, $$\label{firstderivt} \Psi'(t) = \Psi'(\delta_0) (1 + O(n \delta_0^2 \delta_1))$$ Combining (\[secondderiv\]) and (\[firstderivt\]) gives $$\label{secondderfinal} \delta_0^2 \delta_1^2 \Psi''(t) = O(\Psi'(\delta_0) \delta_1^2 n \delta_0^3) = o(\Psi'(\delta_0) \delta_0 \delta_1).$$ This finally gives, $$\label{psifinal} g_{a,b}(r(u)) = \Psi(x(a,b,u)) = \Psi(\delta_0) + (\delta_0 \delta_1 \Psi'(\delta_0) w(a,b,u)) (1 + o(1))$$ $$= \Psi(\delta_0) + \delta_0 \delta_1 \Psi'(\delta_0)(au+b)(1 + o(1))$$ Next we try to estimate the derivative of $\Psi(x(a,b,u))$. We have, $$\frac{\partial }{ \partial u} g_{a,b}(r(u)) = \frac{\partial}{\partial u} \Psi(x(a,b,u)) =$$ $$\Psi'(x(a,b,u)) x_u(a,b,u) = \Psi'(x(a,b,u)) \delta_0 \delta_1 w_u(a,b,u)$$ And using (\[firstderivt\]), $$\label{derfinal} \frac{\partial}{\partial u} \Psi(x(a,b,u)) = \Psi'(\delta_0) (1 + o(1)) \delta_0 \delta_1 w_u(a,b,u) =$$ $$(a \delta_0 \delta_1 \Psi'(\delta_0))(1 + o(1)).$$ Using the continuity and $\Psi$ and $x(a,b,u)$, we can now conclude the following: for any fixed $b \in [-1000,1000]$ and $u \in [0,1]$, an inspection of equation (\[derfinal\]) teaches us that when $a$ varies in $[2,200]$, $\frac{\partial}{\partial u} \Psi(x(a,b,u))$ can attain all values in the range $[3 \delta_0 \delta_1 \Psi'(\delta_0), 100 \delta_0 \delta_1 \Psi'(\delta_0)]$. An inspection of equation (\[psifinal\]) shows that afterwards, by letting $b$ vary in $[-1000,1000]$, $g_{a,b}(r(u))$ will attain all values in $[\Psi(\delta_0) - 100 \delta_0 \delta_1 \Psi'(\delta_0), \Psi(\delta_0) + 100 \delta_0 \delta_1 \Psi'(\delta_0)]$. To estimate the second derivative, $g_{a,b}''$, we write $$\frac{\partial^2}{\partial u^2} \Psi(x(a,b,u)) = \Psi''(x(a,b,u)) \delta_0^2 \delta_1^2 w_u^2(a,b,u) + \delta_0 \delta_1 \Psi'(x(a,b,u)) w_{uu}(a,b,u) w_{u}(a,b,u)$$ (using (\[dersw\]) and (\[secondderfinal\])) $$= o(\delta_0 \delta_1 \Psi'(\delta_0)) + O(\delta_1^2 \Psi'(\delta_0)) = o(\delta_0 \delta_1 \Psi'(\delta_0))$$ This completes the lemma. We are now ready to prove the main lemma of the secion.\ **Proof of lemma \[bodies\]**\ Define: $$f_i(r) = \Psi(\delta_0) + C_i \delta_0 \delta_1 \Psi'(\delta_0) (u + 1)^2$$ with $C_1 = 20, C_2 = 40$. Usage of lemmas (\[uglylemma\]) and (\[createconv\]) shows that there exists two subsets $I_1, I_2$ of $[2,200] \times [-1000,1000]$ such that the bodies $T_i = T_{I_i}$ that we constructed in (\[constrk\]) satisfy (\[tvoldist\]). Also, (\[tsupset\]) is satisfied, since it is satisfied for $T_{a,b}$ for all $(a,b) \in [2,200] \times [-1000,1000]$, as we have seen. Proof of the main result ======================== **Proof of theorem \[main\]**:\ Use lemma \[bodies\] two build the two bodies $T_i$. Let $U_{\theta}$ be an orthogonal transformation which sends $e_1$ to $\theta$. Define $T_i(\theta) = U_{\theta}(T_i)$ (note that the choice of orthogonal transformation does not matter because $T_i$ are bodies of revolution around $e_1$). Define the functions $g_i = g_{T_i}$ as in (\[tvoldist\]). Let $m_1 = \frac{n^{\eps}}{g_1(1)}$, with $\eps > 0$ to be chosen later. Define $m_2 = 2 m_1$. So, (\[derratio\]) implies that $$\label{volratio} m_2 g_2(r) = m_1 g_1(r) + m_1 g_1(1), ~~ \forall r \in [1 - n^{-0.99}, 1].$$ Now, let $K_i = K_{T_i, m_i}$ be the random bodies we constructed in section 2.\ For a fixed $x \in D_n$, we have $$\label{pxi} \tilde f_i(x) = P(x \in K_i) = e^{-m_i \sigma(\{x \notin T_i(\theta)\})} = e^{- m_i g_i(|x|)}.$$ Now, (\[volratio\]) and (\[pxi\]) give $$\label{ratioff} \frac{\tilde f_1(x)}{\tilde f_2(x)} = e^{m_1 (g(1) )} = e^{n^{\eps}}$$ for all $x$ with $|x| \in [1 - n^{-0.99}, 1]$.\ Let us choose $\eps$ to be small enough so that $$m_2 g_2(1) < n^{\eps_0}$$ where $\eps_0$ is the constant from (\[vol2\]). Clearly, that ensures that (\[vol2\]) holds for both random bodies $K_i$. Now, $\eps$ can be made further smaller, so that concentration properties of the euclidean ball will give us, $$\label{thinshellf} \int_{D_n} \tilde f_i \asymp \int_{D_n \setminus (1 - n^{-0.99}) D_n} \tilde f_i$$ for $i=1,2$. Clearly, the above can still be satisfied for some universal constant $\eps > 0$ as long as $n$ is large enough. Next, (\[ratioff\]) and (\[thinshellf\]) imply that $$\frac{\int_{D_n} \tilde f_1}{\int_{D_n} \tilde f_2} \asymp e^{n^\eps}$$ and so (again, taking $\eps$ to be small enough) one gets $$\int_{D_n} \left |\frac{\tilde f_1}{\int_{D_n} \tilde f_1} - \frac{\tilde f_2}{\int_{D_n} \tilde f_2} \right | dx \approx \int_{(1 - n^{-0.99}) D_n} \left |\frac{\tilde f_1}{\int_{D_n} \tilde f_1} - \frac{\tilde f_2}{\int_{D_n} \tilde f_2} \right | dx \ll \Xi(n).$$ Now use Lemma \[sec1final\] to get that $$\label{totalvarsmall} d_{TV} (P_1, P_2) \ll \Xi(n).$$ Denote $R = \frac{1}{2} e^{n^{\eps}}$. Then, $$\EE(Vol(K_1)) \asymp 2 R \EE(Vol(K_2)).$$ Suppose by negation that there exists a classification function $F: \omega \to \mathbb{R}$ that determines the volume of a body $K$ up to a constant $e^{n^{\eps_2}}$ with probability $0.52$. Denote $L = [\frac{\EE(Vol(K_1))}{R}, R \EE(Vol(K_1))]$. Note that using (\[volconc\]), the “correctness” of the function implies that $$P_1(F(p) \in L) \geq 0.51$$ Denote $A \subset \Omega$ as $A = \{p \in \Omega; F(p) \in L\}$. Then $P_1(A) > 0.51$, and (\[totalvarsmall\]) implies that also $P_2(A) > 0.51$. But this means that $$P_2(F(p) \in L) > 0.5$$ But clearly, again, (\[volconc\]) implies that with probability $\approx 1$, the volume of $K_2$ is not in $L$. This contradicts the existence of such a function $F$.\ We still have to generalize the above in two aspects: for an even smaller probability of estimating the volume, and the possibility that the algorithm is non-deterministic. Upon inspection of the proof above, we notice that it can be easily extended in the following way: instead of taking just two families of random bodies, $K_1$ and $K_2$, one may take $d$ different families, $d>2$, which are all indistinguishable by the algorithm, and have different volumes. The proof can be stretched as far as $d=e^{n^{\frac{\eps}{2}}}$. To deal with non-determinsitic algorithms, we will use Yao’s lemma (See [@RV], Lemma 11). Let us generate an index $i$ uniformly distributed in $\{1,...,d\}$, then a body $K$ from the family $K_i$, and then a sequence of uniformly distributed random points on $K$. Following the lines of the above proof, we see that every deterministic algorithm, given this sequence, will be incorrect in estimating the volume of $K$ with probability (at least) $\approx 1 - \frac{1}{d}$. It follows from Yao’s lemma that every non-deterministic algorithm will be incorrect with the same probability for at least one of the families $K_i$. This finishes the theorem.\ \ [GGM]{} B.Bollobás [*Volume Estimates and Rapid Mixing*]{} Flavors of Geometry, MSRI Publications, Vol 31 (1997). I. Bárány and Z. Füredi. [*Computing the volume is difficult.*]{} Discrete & Computational Geometry 2 (1987) 319-326 M. E. Dyer, A. M. Frieze, and R. Kannan. [*A random polynomial time algorithm for approximating the volume of convex bodies.*]{} J. ACM, 38,1 (1991) 1-17 N. Goyal, L.Rademacher. [*Learning convex bodies is hard*]{} (Submitted manuscript, 2008). B. Klartag. [*Power-law estimates for the central limit theorem for convex sets.*]{} J. Funct. Anal., 245, (2007), 284–310. L. Lovász. private communication. L. Lovász, M.Simonovits. [*Random Walks in a Convex Body and an Improved Volume Algorithm*]{} Random Structures Algorithms 4 (1993), no. 4, 359–412. L. Lovász, S. Vempala. [*Simulated Annealing in Convex Bodies and an $0^*(n^4)$ Volume Algorithm*]{} FOCS 2003: 650-672 L.Rademacher, S. Vempala. [*Dispersion of Mass and the Complexity of Randomized Geometric Algorithms*]{} Proc. of the 47th IEEE Symposium on Foundations of Computer Science (2006) [^1]: Partially supported by the Israel Science Foundation
--- author: - Kate Ponto bibliography: - 'trace.bib' date: | \ The author was supported by a National Science Foundation postdoctoral fellowship. title: Relative Fixed Point Theory ---
--- abstract: 'We report the results of a theoretical and experimental study of a spherical gravitational wave antenna. We show that it is possible to understand the data from a spherical antenna with 6 radial resonant transducers attached to the surface in the truncated icosahedral arrangement. We find that the errors associated with small deviations from the ideal case are small compared to other sources of error, such as a finite signal-to-noise ratio. An [*in situ*]{} measurement technique is developed along with a general algorithm that describes a procedure for determining the direction of an external force acting on the antenna, including the force from a gravitational wave, using a combination of the transducer responses. The practicality of these techniques was verified on a room-temperature prototype antenna.' address: 'Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803' author: - 'Stephen M. Merkowitz[^1] and Warren W. Johnson' title: The TIGA technique for detecting gravitational waves with a spherical antenna --- Introduction ============ Techniques to directly detect gravitational waves have been under study for more than 25 years [@Thorne_300y]. Two different methods are aggressively being pursued today: large laser interferometers, such as the proposed LIGO [@Abramovici_Science_1992], and cryogenic resonant mass antennas, such as the operating ALLEGRO [@Mauceli_PRD_1996] and NAUTILUS [@Astone_APP_1997] detectors. While most past work on resonant antennas has been with the original Weber bar type [@Weber_PR_1960], large spherical antennas have recently been proposed and become of interest [@OMNI_1997]. Several characteristics of a spherical antenna make it a unique and interesting instrument. First, it is omnidirectional, capable of detecting gravitational waves from all directions and polarizations. In addition, only a single spherical antenna is necessary for determining the direction of an incident gravitational wave. A sphere has a larger cross section than an equivalent bar [@Wagoner_Pavia_1976]. A sphere can measure all the tensorial components of a gravitational wave, thus it is capable of testing different metric theories of gravity [@Bianchi_CQG_1996]. Finally, almost all of the more than 25 years of experience gained on bar antennas (cryogenics, resonant transducers, suspension,…) can be applied to a spherical antenna. We begin this paper by reviewing the interaction between an elastic sphere and a gravitational wave. We start in Sec. \[sec:quadrupole\_gravity\] by describing how the gravitational field can be decomposed into 5 quadrupole components that will have a one-to-one correspondence with the quadrupole modes of a sphere as described in Sec. \[sec:bare\_sphere\]. When we first began this problem [@Johnson_PRL_1993; @Merkowitz_PRD_1995] we developed a model for a spherical antenna with 6 resonant mass motion sensors attached to the sphere surface at special locations. This model was limited because it put relatively strong constraints on the motion sensors. In order to further investigate the behavior of a more realistic antenna, it was necessary to generalize the model. In Sec. \[sec:sphere\_with\_res\] we develop a more general description of the detector by keeping the number, tuning, and arrangement of the motion sensors arbitrary. Section \[sec:mode\_channels\] shows how the response of the motion sensors can be used to observe the 5 quadrupole modes of the sphere and in Sec. \[sec:direction\_finding\] we describe how this information can be used to determine the direction of an external excitation, including that from a gravitational wave. In Sec. \[sec:ideal\_TI\] we show how the general equations of motion can be greatly simplified if a special arrangement of motion sensors is used. This system was the basis for our original model which we called a truncated icosahedral gravitational wave antenna (TIGA) [@Johnson_PRL_1993]. The arrangement of resonators was similarly called the truncated icosahedral (TI) arrangement [@Merkowitz_PRD_1995]. Other arrangements of transducers have been suggested [@Lobo_EPL_1996; @Zhou_PRD_1995], however, we feel the TI arrangement is advantageous not only because it simplifies the equations of motion but because it maintains equal sensitivity to gravitational waves from all directions and polarizations. We have also found that it facilitates in the interpretation of the signal from the motion sensors. In addition, it has been shown that in the presence of noise the use of 6 resonant transducers in the TI arrangement, compared to that of one for bar antenna, do not reduce the overall sensitivity of the antenna [@Johnson_PRL_1993; @Stevenson_Amaldi_1995] and that the arrangement is fairly robust to the failure of a single motion sensor [@Stevenson_OMNI_1997; @Stevenson_PRD_1997]. In an actual antenna, it may be difficult to achieve a perfect TI arrangement. The motion sensors can be misplaced, mistuned, etc. To account for this we develop a measurement technique in Sec. \[sec:real\_TI\] that takes into account any small deviations from the ideal arrangement. The results of a numerical simulation are also presented that show this technique to be accurate within reasonable levels of precision set for the detector components. We begin discussion of experiments performed on a prototype spherical antenna in Sec. \[sec:prototype\]. The results of the prototype without resonators attached is reviewed in Sec. \[sec:uncoupled\_prototype\]. The behavior of the prototype with resonant transducers attached is presented in Sec. \[sec:coupled\_prototype\]. To demonstrate the validity of the TIGA techniques in a single test, we show that from the response of the motion sensors, we can determine the location of an impulse excitation applied to the prototype’s surface. This same procedure can be applied to determine the direction of a gravitational wave as discussed in Sec. \[sec:direction\_finding\]. Quadrupole decomposition of the gravitational field {#sec:quadrupole_gravity} =================================================== A gravitational wave is a traveling time-dependent deviation of the metric tensor, denoted by $h_{\mu\nu}$. We follow a common textbook development for the metric deviation of a gravitational wave, which finds that only the spatial components, $h_{ij}$, are non-zero, and further can be taken to be transverse and traceless [@Thorne_300y]. The tensor is simplified if we initially write it in the “wave-frame”, denoted by primed coordinates and indices. It is a coordinate frame with origin at the center of mass of the detector, and the $z'\text{-axis}$ aligned with the propagation direction of the wave. Since we restrict ourselves to detectors much smaller than the gravitational wavelength, only the time dependence of $h_{i' j'}$ will have significant physical effects. Thus, the most general possible form for the spatial components of the metric deviation in the wave-frame can be written as: $$h_{i' j'}(t) = \left[ \begin{array}{ccc} h'_+(t) & h'_\times(t) & 0 \\ h'_\times(t) & -h'_+(t) & 0 \\ 0 & 0 & 0 \end{array} \right]$$ where $h'_+(t)$ and $h'_\times(t)$ are the time-dependent gravitational wave amplitudes for the two allowed states of linear polarization, and are called the plus and cross amplitudes. The detector is more easily described in the “lab-frame”, denoted by unprimed coordinates and indices, with origin also at the center of mass of the detector, and $z\text{-axis}$ aligned with the local vertical. In this frame, the primary physical effect of a passing gravitational wave is to produce a time dependent “tidal” force density $f^{\text{GW}}(\bbox{x},t)$ on material at coordinate location $x_i$ with mass density $\rho$, which is related to the metric perturbation by $$f^{\text{GW}}_i(\bbox{x},t) = \frac{1}{2} \rho \sum_{j} \frac{\partial^2 h_{ij}(t)}{\partial t^2} x_j. \label{eqn:accel_metric_pert}$$ We notice that this force can be written as the gradient of a time-dependent scalar potential: $$f_i^{\text{GW}}(\bbox{x},t) = \nabla_i \Phi(\bbox{x},t) = \nabla_i \left( {\sum\limits_{j,k} {\frac{1}{4} \rho x_j \ddot{h}_{jk}(t) x_k}} \right). \label{eqn:metric_pot}$$ This scalar potential is a quadratic form in the spatial coordinates. It is natural to look for an alternate expression that separates the coordinate dependence into radial and angular parts. Because the tensor $h_{ij}$ is traceless, the angular expansion can be done completely with the five ordinary spherical harmonics of order 2, which we denote by $Y_m(\theta,\phi)$ or $Y_m$. We call the resulting time dependent expansion coefficients, denoted by $h_m(t)$, the “spherical gravitational amplitudes.” They are a complete and orthogonal representation of the cartesian metric deviation tensor $h_{ij}(t)$. They depend only on the two wave-frame amplitudes and the direction of propagation, and are defined by $$\Phi(\bbox{x},t) = \sqrt{\frac{\pi}{15}} \rho r^2 \sum\limits_m{\ddot{h}_m(t) Y_m}.$$ The spherical harmonics $Y_m$ can be any linear combination or rotation of the standard spherical harmonics of order 2, as long as the orthogonality between them is maintained. The advantage of not using the standard spherical harmonics will become apparent later, but for an example of an alternative set and their relation to a lab frame see reference [@Merkowitz_PRD_1995]. The 5 orthogonal spherical amplitudes $h_m$ are a complete set of measurable quantities of the local gravitational field. Once a proper relation to the lab coordinate system is defined, the determination of the source direction follows immediately by inversion of this relationship [@Nadja_RAS_1995]. By examining Eq. (\[eqn:metric\_pot\]) closely, we note that because the force is a quadratic it is the equation of an ellipsoid. Therefore, we can picture a gravitational wave as a time-dependent ellipsoidal deformation of the local coordinates. While this may seem obvious, we note it here as this is a very useful visual tool for understanding the interaction between a gravitational wave and the detector. We discuss this idea in more detail below to show the connection with other quadratic quantities. The uncoupled sphere {#sec:bare_sphere} ==================== The mechanics of a general antenna can be described by ordinary elastic theory. Forces acting on the body will cause a deformation described by the displacement vector $\bbox{u}(\bbox{x},t)$, where $\bbox{x}$ is the equilibrium position of a mass element. The equations of motion are then $$\rho \frac{\partial ^2 \bbox{u}}{\partial t^2} = (\lambda+\mu) \bbox{\nabla} (\bbox{\nabla} \cdot \bbox{u}) + \mu \bbox{\nabla} ^2 \bbox{u} + \sum \bbox{f}, \label{eqn:general_eom}$$ where the Lamé coefficients $\lambda$ and $\mu$ specify the elastic stiffness of the material and $\sum \bbox{f}$ represents the sum of external force densities acting on the body [@LL_v7]. In this paper, we include two forces in $\sum \bbox{f}$. First, the signal or gravitational force density $\bbox{f}^{\text{GW}}$ from Eq. (\[eqn:accel\_metric\_pert\]). Second, if objects are attached to the antenna, there will exist a reaction force between the object and the surface of the antenna. Thus we choose to express the coupling to other objects, such as secondary resonators, as if they were external forces in Eq. (\[eqn:general\_eom\]). This device lets us partition the equations of motion in a convenient way. A solution to the differential Eq. (\[eqn:general\_eom\]) can be found by the standard eigenfunction expansion. This allows a separation of the spatial and time dependence of the displacement vector $$\bbox{u}(\bbox{x}_i,t) = \sum\limits_m {a_m\left( t \right) \bbox{\Psi}_m\left({\bbox{x}_i} \right)}.$$ Each spatial eigenfunction, $\bbox{\Psi}_m\left( {\bbox{x}} \right)$, is the time independent part of the solution for unforced harmonic oscillation at the eigenfrequency $\omega _m$, and is found by solving $$-\rho \omega_m^2\bbox{\Psi}_m\ =\ (\lambda+\mu) \bbox{\nabla} (\bbox{\nabla} \cdot \bbox{\Psi}_m)\ +\ \mu \nabla ^2\bbox{\Psi}_m,$$ subject to the time-stationary boundary conditions, which for a sphere require that the total force per unit area at the surface vanish in the direction normal to the surface. The quantity $a_m(t)$ is the time-dependent mode amplitude. The mode index, m, enumerates the discrete set of modes, which obey the usual orthogonality property $$\int_V{\bbox{\Psi}_m(\bbox{x}) \cdot \bbox{\Psi}_n(\bbox{x}) d^3x} = N_m \delta_{mn}.$$ The normalization constant $N_m$ is arbitrary, however, in the case of a sphere of radius $R$ we define it to be $$N_m \equiv \frac{4}{3} \pi R^3. \label{eqn:normalization}$$ Combining the equations above, and using orthogonality to eliminate the summation, we find the standard result, one forced harmonic oscillator equation for each mode amplitude, $$\ddot a_m\left(t\right)+\omega_m^2a_m\left(t\right) = {1 \over {\rho N_m}}\int {\bbox{\Psi}_m(\bbox{x})\cdot \sum \bbox{f}\left( {\bbox{x},t} \right)d^3x}. \label{mode_amplitude}$$ The mode amplitudes are a complete set of collective coordinates for the description of the antenna motion. All the interactions with the outside world, including gravitation, can be included as separate terms in the “effective force” on each mode. An efficient approximation scheme will use only those modes needed for an accurate description of the antenna. Only a few of the “overlap integrals” with $\bbox{f}^{\text{GW}}$ in Eq. (\[mode\_amplitude\]) are large, therefore, only a few of the mode amplitudes are strongly coupled to gravitational waves. Let us consider the case a perfectly homogeneous and isotropic sphere uncoupled from the outside world. This should give a reasonable approximation to the behavior of a sphere where all the external forces are small. The eigenfunctions for this case were found over a hundred years ago [@Jaerisch_1880; @Lamb_1882], however, more elegant derivations, using modern notation are available [@Wagoner_Pavia_1976; @Ashby_PRD_1975]. The eigenfunctions of a sphere can be described in terms of spherical harmonics $Y_{\ell m}\left( {\theta, \phi } \right)$. Looking at the overlap integral in Eq. (\[mode\_amplitude\]), we see that we need only consider odd-parity modes. For a sphere of radius $R$ the eigenfunctions are written: $$\bbox{\Psi}_{\ell m} = \left[ {\alpha_\ell(r) \bbox{\hat r} + \beta_\ell(r) R \bbox{\nabla} } \right] Y_{\ell m}(\theta, \phi),\ \ell \ \text{even.} \label{eqn:sphere_eigf}$$ The radial eigenfunctions $\alpha _\ell \left( r \right)$ and $\beta _\ell \left( r \right)$ determine the motion in the radial and tangential directions respectively. There are 5 quadrupole modes of vibration which strongly couple to the force density of a gravitational wave, and are all degenerate, having the same angular eigenfrequency $\omega _o$. They are distinguished only by their angular dependence. For the remainder of this discussion we only consider the quadrupole ($\ell =2$) modes so we drop the $\ell $ in our notation. The radial eigenfunctions are given by Ashby and Dreitlein [@Ashby_PRD_1975]: $$\begin{aligned} \alpha(r) & = & cR \frac{\partial}{\partial r} j_2(qr) + 6dR \frac{1}{r} j_2(kr) \label{eqn:alpha_ef_def} \\ \beta(r) & = & c j_2(qr) + d \frac{\partial}{\partial r} \left[ r j_2(kr) \right], \label{eqn:beta_ef_def}\end{aligned}$$ where $j_2$ is the spherical Bessel function of order 2. The longitudinal and transverse wave vectors are given by $q^2 = \rho \omega_o^2/(\lambda + 2\mu)$ and $k^2 = \rho \omega_o^2/\mu$ respectively. The boundary conditions $$\left. c\frac{d}{dr} \left[{ \frac{j_2(qr)}{r} }\right] + d\left[{ \frac{5}{r^2} - \frac{k^2}{2} - \frac{1}{r}\frac{d}{dr} }\right] j_2(kr) \right|_{r=R} = 0,$$ $$\left. c\left[{ \frac{6}{r^2} - \frac{k^2}{2} - \frac{2}{r} \frac{d}{dr} }\right] j_2(qr) + 6d\frac{d}{dr} \left[{ \frac{j_2(kr)}{r} }\right] \right|_{r=R} = 0,$$ determine the uncoupled mode frequency $\omega_o$. Inclusion of the normalization condition Eq. (\[eqn:normalization\]) determines the constants $c$ and $d$. These coefficients specify the shape of the eigenfunctions and are all weakly dependent on Poisson’s ratio [@Wagoner_Pavia_1976; @Merkowitz_PRD_1995]. The gravitational effective force for mode m of the sphere, $F_m^S$, from Eq. (\[mode\_amplitude\]) is $$F_m^S \equiv \int_{V_o} {\bbox{\Psi}_m \cdot \bbox{f}^{\text{GW}}\,d^3x}.$$ Solving the integrals, using Eqs. (\[eqn:metric\_pot\]) and (\[eqn:sphere\_eigf\]), we find $$\begin{aligned} F_m^S(t) & = & \sqrt{\frac{4\pi}{15}} \rho \ddot{h}_m(t) R^4 \left[ {cj_2\left( {qR} \right)+3dj_2\left( {kR} \right)} \right] \nonumber \\ & = & \frac{1}{2} \ddot{h}_m(t) \, m_S \, \chi R . \label{eqn:eff_force}\end{aligned}$$ Thus we find that each spherical component of the gravitational field determines uniquely the effective force on the corresponding mode of a sphere, and they are all identical in magnitude. We can interpret the effective force $F^S_m$ in each mode as the product of: the physical mass of the sphere $m_S$, an effective length $\chi R$, and the gravitational acceleration $\frac{1}{2}\ddot{h}_m$. The factor $\chi$ is a weak function of Poisson’s Ratio [@Merkowitz_PRD_1995]. We now see why it was convenient to write the gravitational wave amplitudes in terms of spherical harmonics: we have a clear way to make the connection between the gravitational strain $h_m$, the force they apply to the sphere modes $F_m$, and the amplitudes of the sphere’s quadrupole modes $a_m$. There is a one-to-one correspondence between these three quantities when the same set of spherical harmonics are used. Once we know any of these quantities, we can immediately infer the other two. Later in this paper we will add one more quantity to this list, “mode channels” which are constructed from the observables of the antenna to have a one-to-one correspondence with the above quantities. As in the case of the quadratic form of the gravitational field discussed above, the spherical harmonics $Y_m$ can be any linear combination or rotation of the standard spherical harmonics of order 2, as long as orthogonality between them is maintained. The advantage here is that if the 5 quadrupole modes are not degenerate, but have “fixed” themselves in a particular orientation, one can choose an appropriate set of spherical harmonics that match the actual orientation of the quadrupole modes relative to the lab frame. This basis set can then be used to describe the gravitational field to maintain the one-to-one connection between the spherical amplitudes and the 5 quadrupole modes of the sphere. Also analogous to the spherical amplitudes, the deformation of the sphere due to the excitation of a quadrupole normal mode can be described by the quadratic equation of an ellipsoidal surface. We recall that the geometry of an ellipsoid can be visualized by the principal axis theorem. It shows that the general ellipsoid has three orthogonal axes that pierce the surface at three principal radii, two of which are extremal points on the surface. The orientation of the axes are described by 3 parameters, such as Euler angles. The shape is described by the relative size of the principal radii. If we call $dr_1$, $dr_2$, and $dr_3$ the deviation of these radii from their average, then a true sphere has $dr_1=dr_2=dr_3=0$, an oblate (or prolate) ellipsoid has $dr_2=dr_3=-2dr_1$, and a triaxial ellipsoid has $dr_1>dr_2>dr_3$. Thus 6 parameters completely describe the geometry. However, there is one restriction on the ellipsoids describing the quadrupole modes: they are isovolumetric with the sphere, which requires $dr_1+dr_2+dr_3=0$, so that 5 parameters suffice. Since the superposition of any ellipsoid is another ellipsoid, the eigenfunctions $Y_{1} \ldots Y_{5}$ form a complete and orthogonal basis set for a 5 dimensional abstract vector space that describes all possible isovolumetric ellipsoids and all possible quadrupolar vibrations of the sphere. Sphere coupled to an arbitrary number of resonators {#sec:sphere_with_res} =================================================== We have just shown that measurement of the quadrupole modes of a sphere measures all of the tensorial components of the gravitational field, but a simple spherical resonator is not a practical detector. One requirement for practicality is a set of secondary modes or mechanical resonators. All current bar antennas use resonators that interact only with the vector component of antenna motion normal to the surface on which they are mounted, thus it seems natural to restrict our consideration to resonators of this type. We choose here to describe the sphere’s quadrupole modes in the coupled system using the eigenfunctions derived above for the uncoupled sphere. Lobo and Serrano showed this approximation to be valid when the ratio of the mass of the sphere to the mass of a resonator is much less than one [@Lobo_EPL_1996]. This approximation allows us to use a much more simple mathematical framework, without loss of generality, as all the proposed detectors [@OMNI_1997] satisfy this requirement. We look now at $J$ number of resonators attached to the sphere surface at arbitrary angular positions $\left({\theta_j,\phi_j} \right)$. The values of the relative radial displacements of the sphere surface at the resonator locations can be grouped together into a “pattern vector” for a particular mode. These column vectors in turn may be collected together to form a “pattern matrix” $B_{mj}$ defined by $$B_{mj} \equiv \frac{1}{\alpha} \hat{\bbox{r}} \cdot \bbox{\Psi}_m\left({\theta_j,\phi_j}\right), \label{eqn:pattern_matrix_def}$$ where $\alpha$ is the radial eigenfunction given by Eq. (\[eqn:alpha\_ef\_def\]) evaluated at the surface of the sphere. From Eq. (\[eqn:sphere\_eigf\]) we find $$\begin{aligned} B_{mj} = Y_m\left({\theta _j,\phi _j} \right). \end{aligned}$$ Because the eigenfunctions are invariant to reflection through the origin, we may restrict the location of resonators to one hemisphere, without loss of generality. By mechanical resonator we mean a small elastic system that has one of its own normal modes tuned to be resonant with the frequency of the antenna. The antenna surface motion excites this mode, and there is resonant transfer of momentum between the resonator and the antenna. Hence it acts as a resonant mechanical transformer, turning small motions of the large antenna into large motions of the small resonator. Each resonator $j$ is constructed to obey a one-dimensional harmonic oscillator equation: $$m^r_j {\ddot q}_j(t) + m^r_j \sum\limits_m {\alpha B_{mj} {\ddot a}_m(t)} + k^r_j {q}_j(t) = F^r_j(t). \label{eqn:resonator_eom}$$ The displacement of a resonator, relative to the sphere surface, is denoted by $q_j$. Any random or noise forces that act between the small resonator and the sphere are included in $F^r_j$. Under ideal circumstances we would assume that the resonators are identical, such that the mass $m^r_j$ and spring constant $k^r_j$ of each are tuned to match the frequency of the degenerate 5 sphere modes, however, at this point we wish to keep the equations general so we [*do not*]{} put any restrictions on these parameters. Combining the above, we find the coupled equations of motion for the sphere modes are $$m^s_m {\ddot a}_m(t) + k^s_m a_m(t) - \sum\limits_j{\alpha B_{mj} k^r_j q_j(t)} = - \sum\limits_j{\alpha B_{mj} F^r_j(t)} + F^s_m(t). \label{eqn:coupled_eom}$$ Again, under ideal circumstances we would assume that the 5 quadrupole modes of the sphere are degenerate so that the mass $m^s_m$ and spring constant $k^s_m$ of each mode are identical, however, at this point we wish to keep the equations general so we [*do not*]{} put any restrictions on these parameters. It is convenient to combine Eqs. (\[eqn:resonator\_eom\]) and (\[eqn:coupled\_eom\]) into a matrix notation. We denote matrices by a double underscore and column vectors by a single underscore. We begin by defining the following diagonal matrices: $$\begin{aligned} \begin{array}{ll} M^s_{jm} \equiv \delta_{jm} m^s_m, \;& M^r_{jm} \equiv \delta_{jm} m^r_j, \\ K^s_{jm} \equiv \delta_{jm} k^s_m, \;& K^r_{jm} \equiv \delta_{jm} k^r_j. \end{array}\end{aligned}$$ The complete set of coupled equations of motion can now be written: $$\left[{ \begin{array}{cc} \underline{\underline M}^s & \underline{\underline 0} \\ \alpha \underline{\underline M}^r \underline{\underline B}^T & \underline{\underline M}^r \end{array} }\right] \left[ \begin{array}{c} \underline{\ddot a}(t) \\ \underline{\ddot q}(t) \end{array} \right] + \left[ \begin{array}{cc} \underline{\underline K}^s & -\alpha \underline{\underline B} \, \underline{\underline K}^r \\ \underline{\underline 0} & \underline{\underline K}^r \end{array} \right] \left[ \begin{array}{c} \underline a(t) \\ \underline q(t) \end{array} \right] = \left[ \begin{array}{cc} \underline{\underline I} & -\alpha \underline{\underline B} \\ \underline{\underline 0} & \underline{\underline I} \end{array} \right] \left[ \begin{array}{c} \underline F^s(t) \\ \underline F^r(t) \end{array} \right]. \label{eqn:eom_matrix}$$ The column vector $\underline{a}$ has 5 components, one for each sphere mode, and the column vector $\underline{q}$ has $J$ components, one for each resonator. The dimensions of the constant matrices can be inferred from these two column vectors. The matrix $\underline{\underline{0}}$ is defined to have all elements equal to zero, and $\underline{\underline{I}}$ is the identity matrix. These equations should give a good account of the mechanics of the system for arbitrary numbers and locations of resonators. We do not include terms which represent the “dissipation” part of friction, which can be shown to be negligible for the calculations we do here, however, we do include the “fluctuation” part of friction, within the random driving forces in $\underline{F}^s$ and $\underline{F}^r$. We also do not include any deviations to the shape of the quadrupole modes. One possible cause for changes in shape is the the attachment of the resonators. This would obviously become a problem if very large resonators were used. However, as shown by Lobo and Serrano [@Lobo_EPL_1996], if we limit ourselves to resonators with mass less than 1% of the sphere mass this effect becomes negligible. A second possible cause for a change in mode shape is a hole drilled through the sphere for suspension. However, finite element analysis of a sphere with a hole [@Merkowitz_Thesis] as well as experiments [@Merkowitz_PRD_1996; @Coccia_PLA_1996] have shown the mode shapes to be changed by less than 1% due to the suspension hole. It is clear that Eq. (\[eqn:eom\_matrix\]) represents a set of elastically coupled harmonic oscillators with driving forces. The apparent peculiarities (off-diagonal terms in the mass matrix and asymmetry in the elastic matrix) are simply artifacts of use of the non-inertial coordinates $\underline{q}$. However, we can greatly simplify these equations by transforming to a normal coordinate system. We begin by noting that Eq. (\[eqn:eom\_matrix\]) is of the form $$\underline{\underline{M}} \, \underline{\underline{\gamma}} \, \underline{\ddot{y}}(t) + \underline{\underline{K}} \, \underline{\underline{\gamma}} \, \underline{y}(t) = \underline{\underline{R}} \, \underline{F}(t), \label{eqn:eom_form}$$ where we have defined $$\begin{aligned} \underline{\underline{M}} & \equiv & \left[{ \begin{array}{cc} \underline{\underline M}^s & \underline{\underline 0} \\ \alpha \underline{\underline M}^r \underline{\underline B}^T & \underline{\underline M}^r \end{array} }\right], \\ \underline{\underline{K}} & \equiv & \left[ \begin{array}{cc} \underline{\underline K}^s & -\alpha \underline{\underline B} \, \underline{\underline K}^r \\ \underline{\underline 0} & \underline{\underline K}^r \end{array} \right], \\ \underline{\underline{R}} & \equiv & \left[ \begin{array}{cc} \underline{\underline I} & -\alpha \underline{\underline B} \\ \underline{\underline 0} & \underline{\underline I} \end{array} \right],\end{aligned}$$ and for convenience we have transformed to mass weighted coordinates $\underline{y}$ with the matrix $\underline{\underline{\gamma}}$. We can rewrite Eq. (\[eqn:eom\_form\]) as: $$\underline{\ddot{y}}(t) + \underline{\underline{X}} \, \underline{y}(t) = \underline{\underline{\gamma}}^{-1} \underline{\underline{M}}^{-1} \underline{\underline{R}} \, \underline{F}(t), \label{eqn:eom_form2}$$ where we have defined $$\underline{\underline{X}} \equiv \underline{\underline{\gamma}}^{-1} \underline{\underline{M}}^{-1} \underline{\underline{K}} \, \underline{\underline{\gamma}}. \label{eqn:X_matrix}$$ We may diagonalize $\underline{\underline{X}}$ using the transformation $\underline{\underline{D}} = \underline{\underline{U}}^{-1} \underline{\underline{X}} \, \underline{\underline{U}}$. We now define our normal coordinates as $\underline{\eta}(t) \equiv \underline{\underline{U}}^{-1} \underline{y}(t)$. For convenience, we also define a transformation matrix $\underline{\underline{V}} \equiv \underline{\underline{\gamma}} \, \underline{\underline{U}}$. Substituting these into Eq. (\[eqn:eom\_form2\]) and multiplying the entire expression by $\underline{\underline{U}}^{-1}$ we find $$\underline{\ddot{\eta}}(t) + \underline{\underline{D}} \, \underline{\eta}(t) = \underline{\underline{V}}^{-1} \underline{\underline{M}}^{-1} \underline{\underline{R}} \, \underline{F}(t). \label{eqn:nc_eom}$$ The problem has now been reduced to $5+J$ decoupled harmonic oscillator equations. To solve them we begin by taking the Fourier transform of Eq. (\[eqn:nc\_eom\]) $$\underline{\underline{G}}^{-1}(\omega) \underline{\eta}(\omega) = \underline{\underline{V}}^{-1} \underline{\underline{M}}^{-1} \underline{\underline{R}} \, \underline{F}(\omega),$$ where we have defined $$\underline{\underline{G}}^{-1}(\omega) \equiv \underline{\underline D} - \omega^2 \underline{\underline I}.$$ Because $\underline{\underline{D}}$ is diagonal, $\underline{\underline{G}}^{-1}(\omega)$ is also diagonal, so its inverse is just the diagonal elements inverted. We can now easily solve for the normal coordinates: $$\underline{\eta}(\omega) = \underline{\underline{G}}(\omega) \underline{\underline{V}}^{-1} \underline{\underline{M}}^{-1} \underline{\underline{R}} \, \underline{F}(\omega). \label{eqn:nc_solution}$$ To return to the original coordinates we reverse the transformations: $$\begin{aligned} \left[\begin{array}{c} \underline{a}(\omega) \\ \underline{q}(\omega) \end{array} \right] & = & \underline{\underline{\gamma}} \, \underline{y}(\omega) \\ & = & \underline{\underline{V}} \, \underline{\eta}(\omega) \label{eqn:V_eta} \\ & = & \underline{\underline{V}} \, \underline{\underline{G}}(\omega) \, \underline{\underline{V}}^{-1} \underline{\underline{M}}^{-1} \underline{\underline{R}} \, \underline{F}(\omega).\end{aligned}$$ Note that Eq. (\[eqn:V\_eta\]) provides a convenient way to transform to normal modes where the frequency response is simple. The matrix $\underline{\underline{V}}$ is always invertible as we know the inverse of $\underline{\underline{U}}$ and $\underline{\underline{\gamma}}$ exist, thus making it possible to transform in both directions. This transformation will be important in the final analysis of the detector discussed below. Mode channels {#sec:mode_channels} ============= In our original TIGA model [@Johnson_PRL_1993] we showed it was possible to combine the observable resonator displacements $\underline{q}(t)$ into a quantity which we called “mode channels” because they have a one-to-one correspondence with the quadrupole modes of a sphere, and thus the spherical amplitudes of a gravitational wave. It is desirable at this point to develop a general expression for the equivalent of mode channels for any number and arrangement of radial resonant transducers. We begin by taking the Fourier transform of Eqs. (\[eqn:resonator\_eom\]) and (\[eqn:coupled\_eom\]): $$\left[ \underline{\underline{K}}^r - \omega^2 \underline{\underline{M}}^r \right] \underline{q}(\omega) - \alpha \omega^2 \underline{\underline{M}}^r \underline{\underline{B}}^T \underline{a}(\omega) = \underline{F}^r(\omega) \label{eqn:resonator_eom_ft2}$$ $$\left[ \underline{\underline{K}}^s - \omega^2 \underline{\underline{M}}^s \right] \underline{a}(\omega) - \alpha \underline{\underline{B}} \, \underline{\underline{K}}^r \underline{q}(\omega) = -\alpha \underline{\underline{B}} \, \underline{F}^r(\omega) + \underline{F}^s(\omega) \label{eqn:coupled_eom_ft2}$$ For the moment, we are only interested in the force of the gravitational wave acting on the sphere, so we will assume to have a high signal-to-noise ratio, thus we can ignore the external forces on the resonators and set $\underline{F}^r(\omega) = 0$. We can now solve for $\underline{F}^s(\omega)$ in terms of $\underline{q}(\omega)$ $$\begin{aligned} \underline{F}^s(\omega) & = & \left[ \frac{1}{\alpha \omega^2} \underline{\underline{K}}^s \left( \underline{\underline{B}} \, \underline{\underline{M}}^r \underline{\underline{B}}^T \right)^{-1} \underline{\underline{B}} \, \underline{\underline{K}}^r \right. - \frac{1}{\alpha} \underline{\underline{K}}^s \left( \underline{\underline{B}} \, \underline{\underline{M}}^r \underline{\underline{B}}^T \right)^{-1} \underline{\underline{B}} \, \underline{\underline{M}}^r \nonumber \\ & & \left. - \frac{1}{\alpha} \underline{\underline{M}}^s \left( \underline{\underline{B}} \, \underline{\underline{M}}^r \underline{\underline{B}}^T \right)^{-1} \underline{\underline{B}} \, \underline{\underline{K}}^r + \frac{\omega^2}{\alpha} \underline{\underline{M}}^s \left( \underline{\underline{B}} \, \underline{\underline{M}}^r \underline{\underline{B}}^T \right)^{-1} \underline{\underline{B}} \, \underline{\underline{M}}^r - \alpha \underline{\underline{B}} \, \underline{\underline{K}}^r \right] \underline{q}(\omega) \label{eqn:soln_F_full}\end{aligned}$$ Eq. (\[eqn:soln\_F\_full\]) gives us the means, using the observable resonator displacements $\underline{q}$, to infer the force on the quadrupole modes applied by a gravitational wave. However, the complicated frequency response will make its implementation difficult because all the parameters on the right hand side must be known. While it may be possible to determine all of these parameters (see the Appendix for an example) we would prefer a technique that is not so strongly dependent upon measuring these. In addition, we would prefer an arrangement where the frequency response can be simplified. In the following sections we propose an alternative technique that does not require one to know all the parameters of the detector to high accuracy. Using a special symmetric arrangement of resonators, the above equations can be simplified. Along with a special procedure, possibly unique to this arrangement, we can obtain all the information about the external forces without knowing all the parameters of the system to high accuracy. Direction finding technique {#sec:direction_finding} =========================== General technique ----------------- It is very desirable to demonstrate a general algorithm for finding the location of an arbitrary excitation solely from the mode channel amplitudes. We were able to find and successfully test such an algorithm, one suggested by the ellipsoidal picture for the shape of the modes discussed above. The measured amplitudes of the quadrupole modes directly tell us the relative amounts of each of the 5 basis ellipsoids that must be superimposed to get the net ellipsoidal deformation. We denote the 5 ellipsoidal amplitudes by $h_m$, and call them the vibration amplitudes in the “spherical representation.” We can also define a matrix of the quadratic form $h_{ij}$ whose elements form a complete set of amplitudes in what we call the “cartesian representation.” The connection between representations is easily found to be $$\begin{aligned} h_{ij}(t) & = & \left[\begin{array}{ccc} h_{xx} & h_{xy} & h_{xz} \\ h_{yx} & h_{yy} & h_{yz} \\ h_{zx} & h_{zy} & h_{zz} \end{array} \right] \nonumber \\ & = & \left[\begin{array}{ccc} h_{1} - \frac{1}{\sqrt{3}}h_{5} & h_{2} & h_{4} \\ h_{2} & -h_{1} - \frac{1}{\sqrt{3}}h_{5} & h_{3} \\ h_{4} & h_{3} & \frac{2}{\sqrt{3}} h_{5} \end{array} \right]. \label{eqn:cartesian_strain_tensor}\end{aligned}$$ The connection between this representation and the geometry of ellipsoids comes again from the principal axes theorem, which states that the three eigenvectors of the matrix $h_{ij}$ are parallel to the three principal axes, and the radial deviations $dr_i$ are the corresponding eigenvalues of $h_{ij}$. An excitation can be classified by the shape and orientation of the ellipsoid it produces. Once this shape is realized, one need only solve for the eigenvalues and eigenvectors of the matrix $h_{ij}$ to determine the direction of the excitation. The exact interpretation of the eigenvalues and eigenvectors will of course depend upon the expected ellipsoidal deformation. Inferring a gravitational wave’s direction ------------------------------------------ The method described in the previous section can be used to determine the direction of an incident gravitational wave. As shown in Sec. \[sec:quadrupole\_gravity\], the gravitational field can also be represented by an ellipsoid derived from the electric components of the Riemann tensor [@Eardley_PRL_1973; @Eardley_PRD_1973]. It describes the relative acceleration of gravity that causes an ellipsoidal deformation of an initially spherical group of free test particles. In a conventional gauge, that ellipsoid is also described by the 9 spatial components of the gravitational strain tensor $h_{ij}$ in cartesian coordinates. It can easily be shown that this tensor is in exact one-to-one correspondence to the cartesian amplitudes of vibration of the sphere, so in this paper we have used the same symbol $h_{ij}$ for both. Now, the direction problem in gravitation requires only knowledge of what sort of ellipsoid is produced by a gravitational wave. By examining the conventional description of the strain tensor of a wave according to General Relativity [@MTW], we find that one principal axis of the ellipsoid is aligned with the direction of propagation, and that the corresponding radial deviation is zero. Therefore, we need only determine the wave’s ellipsoid, and then we know the eigenvector of the zero eigenvalue points at the source. (This position determination is unique only within a hemisphere; sources in diametrically opposite directions are indistinguishable.) Note that this method does not require intensive calculations, such as those used to compute the maximum likelihood estimates performed by Zhou and Michelson [@Zhou_PRD_1995]; however, its effectiveness in the presence of noise still needs to be evaluated. Inferring the direction of a radial impulse ------------------------------------------- Since a laboratory source of gravitational waves does not exist, we need an alternative type of excitation for testing this technique. We find that a radial impulse to the surface of a sphere is a good substitute. We present here a simple picture of the antenna’s response to a radial impulse. Suppose a sphere has a degenerate quadrupole mode multiplet, so we are free to choose a basis set with [*arbitrary*]{} orientation to describe it. If we choose an orientation with the $z'\text{-axis}$ of the mode frame to be along the direction of the impulse, then only a [ *single*]{} mode ($Y_5$) in that frame will be excited (all of the other modes have a vanishing radial component of their eigenfunctions at this location, which makes their “overlap” integral with the impulse vanish). The corresponding ellipsoid produced is an oblate spheroid which has maximum radial deviation at the location of the impulse, and two half-size radial deviations of opposite sign in the orthogonal directions. Therefore, the location of the impulse is given by the eigenvector with the largest eigenvalue. This also provides a check for the assumed shape (in a measurement with a high signal-to-noise ratio) as the other two eigenvalues should be equal to each other, but half the size and opposite in sign of the first. The ideal truncated icosahedral arrangement {#sec:ideal_TI} =========================================== Symmetry -------- When we began this problem [@Johnson_PRL_1993] we introduced a special arrangement of 6 resonators which we termed a Truncated Icosahedral Gravitational Wave Antenna (TIGA) shown in Fig. \[fig:TI\_arrangement\]. We proposed using a truncated icosahedron as an approximation to a sphere, however the only requirement for the Truncated Icosahedral (TI) arrangement was that the resonant transducers be placed at positions on the surface of a sphere at the center of six non-antipodal pentagon faces of an imaginary truncated icosahedron (or dodecahedron) concentric to the sphere. The original TIGA model [@Merkowitz_PRD_1995] assumed perfect symmetry of the sphere as well as the tuning and placement of the resonant transducers. While the effects of deviations from perfect symmetry on a sphere’s uncoupled quadrupole modes have been studied [@Lobo_EPL_1996; @Merkowitz_PRD_1996; @Coccia_PLA_1996], we still need to investigate the effect of asymmetries on our ability to properly interpret the signals from resonant motion sensors. This is why we have kept the equations general until now. It is possible to investigate alternative arrangements of radial resonators, such as the one proposed by Lobo and Serrano [@Lobo_EPL_1996], with the above framework, however, we will limit ourselves here to the TI arrangement for reasons stated in the introduction. The symmetry of a TI, shown in Fig. \[fig:TI\_symmetry\], greatly simplifies various aspects of the problem; not only in the calculations that follow, but also in the construction of such a device. A TI has 32 flat surfaces suitable for mounting transducers, calibrators, balancing weights and suspension attachments. In addition, the symmetry makes machining the solid TI relatively simple [@Merkowitz_Thesis]. However, as stated above, the only requirement is on the placement of the resonators, not on the shape of the “spherical” mass. The high symmetry of the TI arrangement becomes apparent when you examine its pattern matrix $\underline{\underline{B}}$. Each pattern vector is orthogonal to the others, and each has the same magnitude, $\sqrt{\frac{3}{2\pi}}$, or in other words: $$\underline{\underline B} \, \underline{\underline B}^T = \frac{3}{2\pi} \underline{\underline I}. \label{eqn:BB=I}$$ This property causes the cross terms between sphere modes in the normal mode eigenfunctions to vanish. In addition to the orthogonality, the sum of the components of each pattern vector vanishes: $$\underline{\underline B}\, \underline{1} = \underline{0}. \label{eqn:B1=0}$$ The $6 \times 1$ column vector $\underline{1}$ is defined to have all elements equal to unity, while the $5 \times 1$ column vector $\underline{0}$ has all elements equal to zero. Eigenfunction solution ---------------------- The symmetry of the pattern matrix also suggested that there might be an analytic solution for the collection of eigenvectors $\underline{\underline{U}}$ and the eigenvalue matrix $\underline{\underline{D}}$ of Eq. (\[eqn:nc\_eom\]). Examination of the numerical results suggested a likely form for $\underline{\underline{U}}$, and substitution in the equations verified that it was a solution and determined the values of the constants. The details of this solution can be found elsewhere [@Merkowitz_PRD_1995; @Merkowitz_Thesis]. It is convenient to divide the resulting set of eigenvectors, $\underline{\underline{U}}$, into three groups. The first two groups each contain 5 column eigenvectors and we denote them by $\underline{\underline{U}}_+$ and $\underline{\underline{U}}_-$: $$\underline{\underline{U}}_\pm = n_\pm \left[ \begin{array}{c} \underline{\underline{I}} \\ c_\pm \underline{\underline{B}}^T \end{array} \right]. \label{eqn:U_pm}$$ The physical interpretation of these is simple: each coupled eigenmode “mimics” the motion of one of the uncoupled sphere eigenmodes. In other words, each coupled resonator’s radial motion is proportional to the uncoupled sphere eigenfunctions at that resonator’s location. This amplified version of a mode’s pattern vector is either in-phase and down-shifted in frequency, or anti-phase and up-shifted in frequency. The frequency shifts are all identical, so that the quintuplet of degenerate bare sphere-modes has bifurcated into up-shifted and down-shifted degenerate quintuplets of modes. The amount of frequency shifting is given by the eigenvalues of $\underline{\underline{U}}$, which are the diagonal elements of the matrix $\underline{\underline{D}}$. The identity matrix in Eq. (\[eqn:U\_pm\]) is an indication that energy will [*not*]{} be transferred from one sphere mode to another. The $\pm$ notation has been used on the dimensionless constants $n_\pm$ and $c_\pm$ as well to refer to the up ($+$) or down ($-$) shifting of the frequencies. The remaining single eigenvector is $$\underline U_o = \left[ \begin{array}{c} \underline{0} \\ n_o \underline{1} \end{array} \right]. \label{eqn:U_o}$$ This mode is at the original sphere frequency and does not strongly interact with a gravitational wave. All the resonators move in unison and the sphere modes do not move at all. Ideal mode channels ------------------- Now let us see what the mode channels look like for the ideal TI arrangement: $$\begin{aligned} \underline{\underline{M}}^s = m_s \underline{\underline{I}} &, \;& \underline{\underline{M}}^r = m_r \underline{\underline{I}} , \\ \underline{\underline{K}}^s = k_s \underline{\underline{I}} &, \;& \underline{\underline{K}}^r = k_r \underline{\underline{I}},\end{aligned}$$ thus $$\underline{F}^s(\omega) = \left[ \frac{2 \pi}{3 \alpha \omega^2 m_r} \left(k_s - \omega^2 m_s \right) \left(k_r - \omega^2 m_r \right) - \alpha k_r \right] \underline{\underline{B}} \, \underline{q}(\omega). \label{eqn:ideal_mc}$$ What is striking about Eq. (\[eqn:ideal\_mc\]) is that all the complicated frequency dependence has been separated from the matrices and a simple linear combination of the resonator responses can be made to obtain all the information about the external forces. We, therefore, define a quantity $\underline{g}$ which does not contain the complicated frequency dependence, but still maintains the one-to-one correspondence with the quadrupole components of the external force acting on the sphere: $$\underline{g} \equiv \underline{\underline{B}}\,\underline{q}. \label{eqn:original_mc}$$ The components of the mode channels $g_m$ can be used as a substitute for the amplitudes $h_m$ in order to solve for the directional information of an excitation, such as a gravitational wave. A practical application of this technique for an impulse excitation to the surface of a prototype antenna will be described later in Sec. \[sec:uncoupled\_prototype\]. Resonator ellipsoids -------------------- From the equations of motion of an ideal TIGA, we found that the eigenfunctions of the coupled modes were such that the motion of the resonators mimicked the ellipsoidal deformation of the sphere’s surface either in phase or anti-phase. We, therefore, can picture the collective motion of the six resonators to describe six “resonator ellipsoids”, 5 of which are mimicking the “quadrupole ellipsoids” of the sphere. The sixth resonator ellipsoid is just a sphere, where the six resonators are moving in unison with equal amplitude and phase, and the sphere surface does not move at all, as described by Eq. (\[eqn:U\_o\]). Each individual resonator now represents a superposition of the point radial deformation of the 6 resonator ellipsoids at a particular position. The transformation between point radial deformations $\underline{q}$ and ellipsoidal amplitudes $\underline{g}$ is given by the pattern matrix $\underline{\underline{B}}$ defined by the positions of the resonators and the orientation of the 5 quadrupole ellipsoids relative to a fixed lab frame: $$\underline{g} = \underline{\underline{B}} \; \underline{q}. \label{eqn:pattern_matrix}$$ Note that Eq. (\[eqn:pattern\_matrix\]) is identical to Eq. (\[eqn:original\_mc\]) for transforming to mode channels. Through this discovery we realize that in the case of the TI arrangement, we can think of mode channels as the result of a linear coordinate transformation from resonator displacements to ellipsoidal deformations. This relationship is [*not*]{} general! In other arrangements of resonators, the equivalent resonator ellipsoids do not, in general, mimic the quadrupole ellipsoids, thus are not the same as mode channels. To produce mode channels for other arrangements, one would have to introduce the complicated frequency response described by Eq. (\[eqn:soln\_F\_full\]). A nearly truncated icosahedral arrangement {#sec:real_TI} ========================================== We have seen how simple things become when the TI arrangement is used, but what happens if the system is not ideal? Using a numerical model, described below, we investigated the effects on the eigenfunctions due to perturbations of the system parameters. We found that small deviations of the various parameters (of the order 1%) from the ideal TI case did not significantly change the resonator ellipsoids. We therefore will discuss a situation where the tolerance on the parameters is relaxed to be of the order a few percent. In an actual experiment this is a rather poor level of precision; one expects to be able to do much better. Normal mode coordinates ----------------------- While all the signal information is contained in the resonator ellipsoids, it is useful to be able to transform the data to normal mode coordinates using Eq. (\[eqn:V\_eta\]) where the frequency response is simple. While not important for transforming between point radial coordinates to ellipsoidal coordinates, the symmetry breaking can be significant when transforming to normal mode coordinates. To overcome this, we developed an [*in situ*]{} measurement technique [@Merkowitz_OMNI_1997] to determine the transformation matrix $\underline{\underline{V}}$. The transformation matrix can be measured by applying a continuous sinusoidal force anywhere on the sphere’s surface at the frequency of one of the normal modes. (Note that this technique requires the normal modes to be [ *non-degenerate*]{}, thus it is actually preferable to have a small amount of symmetry breaking.) The frequency response of the resonators will be simple because they are being driven at a single frequency. From Eq. (\[eqn:V\_eta\]) we see that the amplitude and phase of their response make up a single column of $\underline{\underline{V}}$. By exciting each normal mode in turn, the complete $\underline{\underline{V}}$ matrix can be measured. The only assumption made in calculating $\underline{\underline{V}}$ is that the resonators are “close” to the TI arrangement so that an ideal pattern matrix $\underline{\underline{B}}$ can be used and the quadrupole ellipsoids can be replaced by Eq. (\[eqn:pattern\_matrix\]). Once $\underline{\underline{V}}$ is measured the antenna can be operated to observe gravitational waves. The response of the resonators can be recorded and transformed to normal mode coordinates: $$\underline{\eta} = \underline{\underline{V}}^{-1} \left[\left[ \begin{array}{c} \underline{\underline{B}}Ê\; \underline{q}_{-} \\ \underline{q}_{-} \end{array} \right] + \left[ \begin{array}{c} -\underline{\underline{B}} \; \underline{q}_{+} \\ \underline{q}_{+} \end{array} \right] \right]. \label{eqn:nm_transformation}$$ Note that the resonator response must be bandpass filtered to separate the in-phase ($-$) and anti-phase ($+$) resonator ellipsoids. Since the frequency response of the normal modes is simple they can easily be fit for various parameters such as phase and amplitude. This information can then be transformed to mode channels using using Eq. (\[eqn:V\_eta\]). From the mode channels the direction and amplitude information of a possible gravitational wave event can be calculated as described above. Numerical simulation of errors ------------------------------ ### Parameters Now that we have written down the solutions to the equations of motion we can look at what are the parameters of the system and how uncertainties and deviations of them will affect a measurement. For the TIGA case we have the following parameters: $$\begin{array}{ccll} 6 & \times & k^r_j & \text{resonator spring constants} \\ 5 & \times & k^s_m & \text{sphere mode spring constants} \\ 6 & \times & \phi_j & \text{resonator positions} \\ 6 & \times & \theta_j & \text{resonator positions} \\ 5 & \times & \beta_m & \text{sphere mode orientations} \\ 5 & \times & \gamma_m & \text{sphere mode orientations} \\ 6 & \times & m^r_j & \text{resonator masses} \\ 5 & \times & m^s_m & \text{quadrupole mode masses} \\ 6 & \times & \epsilon^r_j & \text{resonator radial couplings} \\ 6 & \times & \epsilon^\theta_j & \text{resonator transverse couplings} \\ 6 & \times & \epsilon^\varphi_j & \text{resonator transverse couplings} \\ \hline 62 & \multicolumn{3}{l}{\text{total parameters.}} \end{array}$$ The angles $\beta_m$ and $\gamma_m$ describe the orientation of the quadrupole modes relative to a fixed lab frame, and will be discussed further below. The parameters $\epsilon^r_j$, $\epsilon^\theta_j$, and $\epsilon^\varphi_j$ are a measure of the coupling of the transducers to the radial and transverse motion of the sphere surface. In the above model we assumed $\epsilon^r_j=1$ and $\epsilon^\theta_j=\epsilon^\varphi_j=0$ because we felt their inclusion was unnecessary as transducers are currently available that do not strongly couple to transverse motion. However, it is useful to test this assumption and determine how strong a requirement should be set for the actual instrument. We include them here by replacing the pattern matrix defined by Eq. (\[eqn:pattern\_matrix\_def\]) with $$B_{mj} \equiv \frac{1}{\alpha} \left( \epsilon^r_j \hat{\bbox{r}} + \epsilon^\theta_j \hat{\bbox{\theta}} + \epsilon^\varphi_j \hat{\bbox{\varphi}} \right) \cdot \bbox{\Psi}_m\left({\theta_j,\phi_j}\right)$$ This should give a good approximation of the effects of transverse coupling, without the need of changing the model of the resonators from one dimensional harmonic oscillators. We considered these parameters as independent from each other. One might relate them with a parameter such as the angle between the transducer axis and the normal to the sphere surface, however we do not do this because for an actual resonant transducer there are several other mechanisms that can lead to transverse coupling, thus keeping these parameters independent seems reasonable. Some of the above parameters can potentially be measured directly, such as the masses, however, we include them here for generality. In addition, the 5 quadrupole mode masses $m^s_m$ would normally be set equal to each other and to the physical mass of the sphere, however, again for generality we kept it as a parameter. In the appendix we describe a procedure to measure most of these parameters, however the resonator ellipsoid method described above makes this unnecessary. ### Simulation results The transformation matrix $\underline{\underline{V}}$ can be measured to very high accuracies, but our assumption that the resonator ellipsoids still mimic the sphere ellipsoids will have some error associated with it. This error will propagate through the analysis and into the results of a measurement. We, therefore, studied the effects of small perturbations to the above parameters on our ability to accurately determine the direction of an excitation. We developed a Monte Carlo type simulation where we added a small random perturbation (uniform distribution) to the above parameters within a specified tolerance. We then simulated an excitation and calculated the direction using the resonator ellipsoid method. The direction calculation assumed the ideal case: it was not given knowledge of the true values of the parameters. As shown in Fig. \[fig:direction\_error\], the results of the numerical simulation indicate that a direction calculation becomes unreliable only after the tolerance of all the parameters exceeds about 3%. This is certainly an obtainable level of precision. Fig. \[fig:tiga\_error\_sang\] shows the solid angle estimation error $\Delta\Omega$ [@Gursel_PRD_1989] for several tolerances. We also varied the location of the excitation, but found no significant difference in the results. Note that these are systematic errors due to the analyses technique, not random errors as the figures might imply. To put these results into perspective, we compared these systematic errors to the random error due to a finite signal-to-noise ratio as calculated by Zhou and Michelson [@Zhou_PRD_1995]. We find that one would need a signal-to-noise ratio of about 1000 in energy before our systematic errors become significant (choosing a tolerance of 2%). While one might hope to observe sources at this level, the most optimistic predictions lead to considerably smaller signal-to-noise ratios [@Finn_OMNI_1997]. We, therefore, feel that the systematic errors are sufficiently low that there is no need to develop an alternative technique that requires precise knowledge of the parameters. Looking at the individual contribution to the errors from each parameter we determined which were most dominant. We found that the most significant errors came from the resonator positions $\phi_j$ and $\theta_j$. All the other parameters had associated errors at least one order of magnitude in $\Delta\Omega$ lower than those from the resonator positions for reasonable tolerance levels. This included the errors associated with the coupling parameters $\epsilon^r_j$, $\epsilon^\theta_j$, and $\epsilon^\varphi_j$, thus justifying our earlier decision to omit them from the model. Perturbations to the sphere mode orientations $\beta_m$ and $\gamma_m$ did not lead to [*any*]{} errors! We expected that simple linear combination of the quadrupole modes do not lead to any error, however, what is surprising is that a direction calculation’s ignorance of the true orientation in a non-degenerate system does not lead to any errors. This is important as it tells us that it is unnecessary to measure the mode orientation before equipping the sphere with resonant transducers (as was done on the prototype TIGA for other reasons discussed below). While these parameters may not completely describe the effects of deviations of the quadrupole modes from an ideal sphere, we found from measurements [@Merkowitz_PRD_1996] that they are the dominant effect of symmetry breaking. The fact that they do not contribute [*at all*]{} to the errors on a measurement also frees us from putting strong constraints on the spherical mass. This allows us, for example, to put a hole through the center of the sphere for suspension purposes, or use a TI (or some other “spherical” shape) instead of a sphere. The LSU prototype TIGA {#sec:prototype} ====================== The above model outlines a clear algorithm for obtaining the gravitational amplitudes from a spherical antenna. However, like most models, we must evaluate its worth with an actual experiment. We therefore constructed a room temperature prototype TIGA. In the following sections we describe how the prototype was used to: first, verify that a TI has the same mode structure as a sphere; second, determine the effects of asymmetries on the sphere modes, such as a hole drilled through the center for suspension; third, verify the mode channel and ellipsoidal theories; and finally, verify the direction finding algorithms. The prototype TI was machined from a bar of aluminum alloy 6063 that had previously been used as a cylindrical gravitational wave detector and was known to have good mechanical properties. Some key dimensions of the TI are shown in Fig. \[fig:TI\_shop\_drawing\]. The prototype had a center of mass suspension. A hole was bored along a diameter that started from the center of a hexagon face. The hole changed diameter just above the center of mass, and a thin titanium suspension rod, which widened to a cone at one end to mate with the hole’s change in diameter, was inserted from the large diameter side. The prototype was first suspended and tested without mechanical resonators attached. This testing gave many insights into the differences between an ideal sphere and a real one. The results of this testing was summarized elsewhere [@Merkowitz_PRD_1996] but we include here some of the important results that are needed to describe the coupled system. Once the uncoupled tests were completed, resonant transducers were attached, and the coupled system was studied. Preliminary results of these tests were also summarized elsewhere [@Merkowitz_OMNI_1997; @Merkowitz_PRL_1997], but we report here the completed work in detail. While we have attempted to report here as many of the important aspects of the experiment as possible, we have omitted many of the specifics of this particular apparatus; for those details we refer the reader to reference [@Merkowitz_Thesis]. The uncoupled prototype {#sec:uncoupled_prototype} ======================= Normal Mode Frequencies ------------------------ The measured frequency spectrum of the uncoupled TI is shown in reference [@Merkowitz_PRD_1996]. We were able to identify most of the predominant modes using solutions to the elastic equations of a sphere [@Lobo_PRD_1995] and a finite element model of a TI [@Merkowitz_Thesis]. We found the degeneracy of all multiplets to be lifted by a small amount: 1% or less in frequency. We were able to match the measured frequencies of most of the multiplets to the theory for a sphere to better than 1%. The modes of most interest for gravitational wave detection are the 5 members of the lowest quadrupole mode multiplet near 3235 Hz. For a homogeneous isotropic sphere, those modes are exactly degenerate. We found that this quintuplet was split into two doublets and a singlet, spread over a range of 0.8% in frequency, as shown in Fig. \[fig:quadrupole\_spec\]. Additional data (not shown), confirmed that the two peaks labeled as doublets are each composed of two modes split by about 1 Hz. Upon reflection, we realized that the suspension hole bored through the TI must be the primary cause for the splitting of the quintuplet. It breaks the spherical symmetry, but preserves cylindrical symmetry about the hole axis. The specific identification of the multiplets shown in Fig. \[fig:quadrupole\_spec\] was surmised on physical grounds, and confirmed by measurements described below. We have not attempted to calculate the magnitude of the splitting caused by the hole, so we cannot make a comparison with the data, however, this effect has subsequently been confirmed by others [@Coccia_PLA_1996]. Monopole Mode Calibration {#sec:calibration} ------------------------- This experiment dealt with high signal-to-noise ratios, and absolute energy calibration was unnecessary. However, it was important to know the relative sensitivity of the motion sensors and correct for any differences. The monopole, or breathing, mode of a sphere (which for this TI had a frequency near 6880 Hz) is a spherically symmetric radial expansion and contraction of the surface. The TI had no other modes close in frequency to the monopole mode. This made it ideal to measure the relative sensitivity of the motion sensors. We excited vibrations of the TI with radial impulses from a hammer at various locations on the surface. Shown in Fig. \[fig:monopole\], we found that the responses of the six motion sensors, at the monopole frequency, were identical in phase, and independent of the position of the impulse, but differed systematically in amplitude by up to 10%. These amplitude differences were due to the quality of the attachment of the motion sensors as well as gain differences in the electronics chain. These measured gain deviations were then used to correct the amplitudes in all the subsequent measurements. This method proved to be very convenient as the motion sensors did not have to be removed or remounted, which was found to change their sensitivity. Simple mode channels -------------------- We observed the quadrupole mode multiplet by sampling the motion of the TI at 6 discrete positions, using small, non-resonant, accelerometers waxed to the surface in the TI arrangement as shown in Fig. \[fig:TI\_arrangement\]. According to the standard normal mode picture of vibrational mechanics, the free motion at these points, or any point on the surface, can be viewed as the combination, or superposition, of the response of the normal modes. Thus each motion sensor will record a different linear superposition of the responses of all the modes. A hammer was used to impulsively excite vibrations of the sphere. Narrow-band filtering was used to isolate the quadrupole modes. The measured response of each motion sensor is shown in reference [@Merkowitz_PRD_1996]. As expected, the non-degeneracy of the modes caused the individual sensor outputs to display the various modes beating against each other, making it difficult to make a [*direct*]{} estimate of the amplitude of each normal mode. We showed above that the desired mode amplitudes could be separated out by combining the outputs of all the sensors in special linear combinations, whose coefficients were grouped together into the pattern matrix $\underline{\underline{B}}$. We called these combinations “mode channels” to indicate they had a one-to-one correspondence with the quadrupole normal mode amplitudes of the sphere. For the case of the uncoupled prototype, we do not need to included the measurement of the matrix $\underline{\underline{V}}$ to convert to normal modes, as the uncoupled sphere quadrupole modes [*are*]{} the normal modes, thus their frequency response is simple. In addition, for the case of [*this prototype*]{}, we could not use this procedure because several of the modes were nearly degenerate, thus exciting them individually with a simple sine-wave excitation was impossible. To obtain nearly perfect mode channels, we rotated the spherical harmonics that determined the pattern matrix, until we found the best fit to a single frequency in each mode channel. We chose to use the y-convention for the Euler angles [@Goldstein_1980] to perform the rotations. The rotation $\alpha$ about the $z\text{-axis}$ was not used because it had little effect on the fit. The $\beta$ rotation about the new $y\text{-axis}$ mixed mode 5 with the other 4 modes, while maintaining orthogonality. The $\gamma$ rotation about the new $z\text{-axis}$ mixed the new modes 1 with 2, and 3 with 4, but not 1 with 3 or 4, etc. Therefore, these rotation angles could be different for the two pairs and still maintain orthogonality. Mode 5 was unaffected by any $\gamma$ rotation. The best fit values for the rotation angles from the lab coordinate system shown in Fig. \[fig:TI\_arrangement\] were $\gamma_{12} = -0.1$, $\gamma_{34} = -7.2$, and $\beta = 1.0$. Each mode channel was well separated from the others and behaved as expected, an exponentially decaying sine wave. By examining the power spectrum, we determined that the residual amplitude of the “wrong” modes present in a channel was less than 2%. This small residual admixture may, or may not, be due to imprecise positioning of the accelerometers. Simple impulse test ------------------- As a final test of the uncoupled system, we applied several radial impulses to the center of nine different faces of the TI, and then calculated the locations from the algorithm described above. The results of this comparison are shown in reference [@Merkowitz_PRD_1996]. The locations calculated from the data were very consistent; with three hits at each location, the overall standard deviation from the mean was $\sim 0.4^\circ$. The calculated locations were all within $\sim 3\%$ of the values expected from the measured geometric position of the impulse hammer. The deviation from the expected values is apparently a systematic error, perhaps from imprecise placement of the accelerometers or the impulse hammer. Below we describe the results of a similar test, but with the accelerometers replaced by resonant transducers. Insights from the uncoupled prototype ------------------------------------- Experiments on the uncoupled prototype showed that the departures from perfect spherical behavior and symmetry were not large. The quadrupole modes were no longer degenerate, but the source of their frequency splitting is understood. The eigenfunctions of the uncoupled TI were found to be unchanged in shape from those of a perfect sphere by an amount less than 2%; the main effect of the symmetry breaking was to fix them in a particular orientation. The simple impulse test confirmed the practicality of the direction finding technique. From these results, we conclude that a TI represents a good approximation for a sphere and is sufficient for use as an omnidirectional gravitational wave antenna. Knowing these results, we were confident enough to instrument the prototype TI with resonant transducers to fully test the TIGA theory. The prototype with resonant transducers {#sec:coupled_prototype} ======================================= Transducer design and attachment -------------------------------- Section \[sec:sphere\_with\_res\] lists the rudimentary requirements for a resonator, but practical considerations require a more extensive list. First, the “transducer mode” must be reasonably easy to tune to the quadrupole frequency. Second, the transducer mode must be purely radial, so that it couples strongly only to the radial motion of the quadrupole modes. Third, there should not be any other modes of the resonator nearby in frequency. Fourth, there must be a practical method of attachment with sufficient mechanical Q. The design we adopted for the prototype, shown in Fig. \[fig:fathead\], approximates a lumped mass and a spring. The lumped mass, or “head”, is attached to a thin stem, or “neck.” The neck is fixed to a base which is then attached to the surface of the prototype. These three parts were machined from a single piece of aluminum. The transducer mode is such that radial motion of the head compresses and extends the neck against the base. While the neck is relatively rigid in the radial direction, it is relatively flexible in the transverse directions, which decouples the transducer mode from transverse motions. While designing the resonator, the length and diameter of the neck can be adjusted to move the rocking and torsional modes of the resonator well below the transducer mode frequency. A piezoelectric strain gauge was epoxied to the neck of each resonator. The strain induced in the crystal will be proportional to the change in length of the neck, thus providing an efficient way of observing the motion of the resonators. The output of the strain gauges were first demodulated to low frequency using 6 separate lock-in amplifiers, using the same reference, and then recorded on a high speed data acquisition system. The resonators and measurement system are described in detail in reference [@Merkowitz_Thesis]. We used finite element analysis to fix the final parameters of the resonator. The final tuning of the resonators was done while attached to the TI. The equations of motion for this system are given in Sec. \[sec:sphere\_with\_res\] taking the number of resonators equal to one. We measured the coupled mode frequencies of the prototype and one resonator and compared them to the eigenvalue solution of Eq. (\[eqn:eom\_matrix\]) to determine the spring constant of the small resonator. We had two practical options for tuning the resonator: reduce the mass of the head, or lower the spring constant by reducing the diameter of the neck. We attached the resonators to the prototype with epoxy. While this method may not lead to the best mechanical Q, we found it was sufficient. The coupled modes had a Q of about $10^3$ in vacuum while the uncoupled sphere modes had a Q of about $10^4$. We suspect this difference is due to the method of attachment. In air the coupled modes had a very poor Q, thus we felt it was necessary to perform all test of the coupled system under vacuum. The resonators were attached to the prototype one at a time, and the frequencies of the coupled modes were measured after each change. The calculated and measured quadrupole mode frequencies were fairly consistent with each other. The non-degeneracy of the prototype’s quadrupole modes did not introduce much deviation from a perfectly degenerate system. It was also found that neither the toroidal modes nor the monopole mode of the sphere were shifted by more than 1 Hz when the resonators were added. Fig. \[fig:coupled\_data\] shows the results of the frequency measurements of the coupled modes for each addition of a resonator. The results are compared with what is expected from the eigenvalue solution of Eq. (\[eqn:eom\_matrix\]) beginning with the measured uncoupled eigenfrequencies. The two sets are consistent within 0.2%. While we consider this very good agreement, Lobo and Serrano found slightly better agreement (possibly due to better numerical precision), with this data, by applying the equations of motion in an equivalent, but different form [@Lobo_EPL_1996]. Transformation to normal modes ------------------------------ With the 6 resonators attached, we were ready to observe the sphere modes using the resonator ellipsoid technique. The first step was to measure the transfer function $\underline{\underline{V}}$. We attached a simple non-resonant piezoelectric shaker to the surface of the prototype. The frequencies of the 11 normal modes were accurately measured by driving the shaker with a single frequency sine-wave, adjusting the frequency until a maximum response of a single normal mode was observed. With this system in equilibrium, we recorded the response of the 6 resonators to the continuous wave excitation. The frequency of the excitation was then changed to measure the next normal mode frequency and the above steps repeated. As shown in Fig. \[fig:cw\_raw\], the frequency response of the resonators was simple because we were driving at a single frequency. The amplitudes of their response made up a single column of $\underline{\underline{V}}$. By exciting each normal mode in turn, the complete transformation matrix was measured. We repeated this measurement for several different locations of the shaker, all of which gave consistent results. Shown in Fig. \[fig:cw\_nm\] is the results of using the measured transformation matrix $\underline{\underline{V}}$ to calculate the response of the normal modes to the normal mode excitation of Fig. \[fig:cw\_raw\]. For this case the TI was driven with a continuous wave excitation at the frequency of the fifth normal mode. As shown in the figure, only the fifth normal mode was excited, as expected. Again, this experiment had high signal-to-noise ratios, thus the essentially flat lines of the non-excited modes actually represent a “leakage” level of about 5% in amplitude. Impulse direction test ---------------------- To combine the entire TIGA technique into a single test, we applied an impulse excitation to the surface of the prototype TIGA to determine if the location of the impulse can be measured from the response of the resonators. An impulsive force was applied to the surface of the TI by sending a short electrical pulse to a non-resonant piezoelectric shaker attached to the surface. Shown in Fig. \[fig:burst\_raw\] is a typical response of the six transducers to an impulsive excitation. Following the technique described above, this data can be transformed to normal coordinates using the matrix $\underline{\underline{V}}$. The results of this transformation to the data of Fig. \[fig:burst\_raw\] is shown in Fig. \[fig:burst\_nm\]. As expected, the data separated into 11 channels, each containing a single frequency representing the response of a single normal mode. Since each channel contains only a single frequency, it is relatively easy to fit them for their phase and amplitude at the time of excitation. Once these quantities are found we can transform them to mode channels and compute the location of the impulse as described above. The results of this analysis for the various impulse locations is shown in Fig. \[fig:burst\_location\]. The locations calculated from the data were very consistent; with several impulses at each location, the overall standard deviation from the mean was $0.1^\circ$. On average, the calculated locations were all within $2.7^\circ$ of the values expected from the geometrically measured position of the center of the shaker. The deviation from the expected values can be accounted for by the accuracy of the excitation method. The shaker used to apply the impulsive force did not actually apply a “point” impulse, but rather one that was distributed over a ring about the circumference of the shaker. By systematically repositioning the shaker, we determined that the “true” location of the impulse was anywhere within $2.5^\circ$ of the geometric center of the shaker. A more precise way of exciting the prototype would have been preferred, however, we found this method to be adequate to verify the principle of the technique. Summary ======= Experiments on the prototype TIGA showed that the departures from ideal behavior were not large. In every case, ways could be found to handle the asymmetries without major difficulty, and to some extent they actually simplify the problem. A technique for determining the location of an external excitation, including that of a gravitational wave, from the motion sensor data was developed which, except for some bandpass filtering, is simply linear algebra. This makes its implementation simple in an automated data analysis system. The [*in situ*]{} measurement technique takes into account most deviations from perfect symmetry and the resulting transformation matrices enable the data to be transformed to a space where the frequency complications can be easily handled. The algorithm was tested on the prototype TIGA and was found to be consistent with the measured results within the accuracy of the experiment. Since all the techniques described can be applied [*in situ*]{}, they are directly applicable for use on a real spherical antenna searching for gravitational waves. We thank W. O. Hamilton for many years of essential advice and support on this project. The final data analysis and writing of this paper was done at Eindhoven University of Technology and at the INFN Laboratori Nazionali di Frascati; S. M. M. thanks A. T. A. M. de Waele, E. Coccia, G. Pizzella and the rest of the GRAIL and ROG collaborations for their support during this time. This research was supported by the National Science Foundation under Grant No. . Resonator channels {#resonator-channels .unnumbered} ================== To measure all the parameters of a spherical antenna [*in situ*]{} without introducing any new parameters we could do the following. Excite one of the resonators with an external force and measure the response of all the resonators. If we were to excite at some other location, this would introduce new unknown parameters such as the location of the excitor. We assume that we have a high signal-to-noise ratio so that we can ignore any external forces acting directly on the sphere modes. We therefore can set the forces $\underline{F}^s = 0$. We now write Eqs. (\[eqn:resonator\_eom\_ft2\]) and (\[eqn:coupled\_eom\_ft2\]) as $$\left( \underline{\underline{K}}^r - \omega^2 \underline{\underline{M}}^r \right) \underline{q}(\omega) - \alpha \omega^2 \underline{\underline{M}}^r \underline{\underline{B}}^T \underline{a}(\omega) = \underline{F}^r(\omega) \label{eqn:rc_resonator_eom}$$ $$\left( \underline{\underline{K}}^s - \omega^2 \underline{\underline{M}}^s \right) \underline{a}(\omega) - \alpha \underline{\underline{B}} \, \underline{\underline{K}}^r \underline{q}(\omega) = -\alpha \underline{\underline{B}} \, \underline{F}^r(\omega). \label{eqn:rc_coupled_eom}$$ Solving for $\underline{F}^r(\omega)$ in terms of the observable $\underline{q}(\omega)$ we find $$\underline{F}^r(\omega) = \left[ \underline{\underline{I}} -\alpha^2 \omega^2 \underline{\underline{M}}^r \underline{\underline{B}}^T \left(\underline{\underline{H}}^s(\omega)\right)^{-1} \underline{\underline{B}} \right]^{-1} \left[ \underline{\underline{H}}^r(\omega) -\alpha^2 \omega^2 \underline{\underline{M}}^r \underline{\underline{B}}^T \left(\underline{\underline{H}}^s(\omega)\right)^{-1} \underline{\underline{B}} \, \underline{\underline{K}}^r \right] \underline{q}(\omega). \label{eqn:resonator_channels}$$ For the case of an ideal TIGA Eq. (\[eqn:resonator\_channels\]) can be further simplified: $$\underline{F}^r(\omega) = \frac{ \left(k_s - \omega^2 m_s \right) \left(k_r - \omega^2 m_r \right) - \frac{3}{2 \pi} \alpha^2 \omega^2 m_{r} k_{r} }{ \left(k_s - \omega^2 m_s \right) - \frac{3}{2 \pi} \alpha^2 \omega^2 m_{r} } \underline{q}(\omega)$$ One can imagine performing this experiment and then fitting the resulting data for the various parameters. 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Finn, in [*Proceedings of the OMNI-1 workshop*]{} [@OMNI_1997]. S. M. Merkowitz and W. W. Johnson, submitted to Physics Letters A (unpublished). J. A. Lobo, Physical Review D [**52**]{}, 591 (1995). H. Goldstein, [*Classical Mechanics*]{}, 2nd ed. (Addison-Wesley Publishing Company, Reading, Massachusetts, 1980). \[fig:TI\_arrangement\] [^1]: Present affiliation ROG collaboration and the INFN Laboratori Nazionali di Frascati, Italy
--- abstract: 'The relationship between the GNS representations associated to states on a quasi \*-algebra, which are [*local modifications*]{} of each other (in a sense which we will discuss) is examined. The role of local modifications on the spatiality of the corresponding induced derivations describing the dynamics of a given quantum system with infinite degrees of freedom is discussed.' --- [**Representations and derivations of quasi \*-algebras induced by [*local modifications*]{} of states**]{}\ [F. Bagarello]{}\ Dipartimento di Metodi e Modelli Matematici, Fac. Ingegneria, Università di Palermo, I-90128 Palermo, Italy\ e-mail: bagarell@unipa.it\ [A. Inoue]{}\ Department of Applied Mathematics, Fukuoka University, Fukuoka 814-0180, Japan\ email: a-inoue@fukuoka-u.ac.jp\ \ Dipartimento di Matematica ed Applicazioni, Università di Palermo,\ I-90123 Palermo (Italy)\ e-mail: trapani@unipa.it\ Introduction and preliminaries ============================== In two recent papers, [@bit1; @bit2], we have investigated the role of derivations of quasi \*-algebras and the possibility of finding a certain symmetric operator which [*implements the derivation*]{}, in the sense that in a suitable representation the derivation can be written as a commutator with an operator which in the physical literature is usually called the [*effective hamiltonian*]{}. This is useful for physical applications and produces an algebraic framework in which the time evolution of some physical model can be analyzed, [@fbrev]. Here we continue our analysis, taking inspiration again from physical motivations: it is known [@sew] that in a physical context [*local modifications do not affect much the main physical results*]{}. Our interest here is to understand this statement more in detail, mainly in the framework of quasi \*-algebras, [@schmu; @ait_book], which, as we have discussed in several other places, see [@ait_book; @fbrev; @ctrev], in our opinion play an important role in the mathematical description of quantum mechanical systems with infinite degrees of freedom. Just as an introductory example, let us consider a C\*-algebra ${{\cal A}}$ with unit $e$, and let $\omega$ and $\omega'$ be two (different) positive linear functionals on ${{\cal A}}$. Let further $(\pi_\omega,\xi_\omega,{{\cal H}}_\omega)$ and $(\pi_{\omega'},\xi_{\omega'},{{\cal H}}_{\omega'})$ be their associated GNS-representations. An interesting problem is the following: under which conditions on $\omega$ and $\omega'$ are the representations $\pi_{\omega}$ and $\pi_{\omega'}$ unitarily equivalent? It is somehow more convenient to consider first the following preliminary problem: [*how must $\omega$ and $\omega'$ be related for $\pi_{\omega'}$ to be unitarily equivalent to a sub \*-representation of $\pi_{\omega}$*]{} ? An easy proof shows that [*$\pi_{\omega'}$ is unitarily equivalent to a sub \*-representation of $\pi_\omega$ if, and only if, there exists a sequence $\{b_n\}$ of elements of ${{\cal A}}$ such that $\omega'(a)=\lim_{n\to\infty}\omega(b_n^* a b_n)$ $\forall\,a\in{{\cal A}}$, and the sequence $\{\pi_\omega(b_n)\xi_\omega\}$ converges in ${{\cal H}}_\omega$.*]{} We refer to [@sew] for the physical implications of this result. Here we observe that, in particular, if $\omega$ is a positive linear functional on ${{\cal A}}$, and $b\in{{\cal A}}$ a fixed element such that $\omega(b^*b)\neq 0$, then the GNS-representation associated to $\omega_b(\cdot)=\omega(b^*\cdot b)$ is unitarily equivalent to a sub \*-representation of $\pi_\omega$. This means that there exists a subspace ${{\cal H}}_\omega^b$ of ${{\cal H}}_\omega$ and a unitary operator $U: {{\cal H}}_{\omega_b}\to{{\cal H}}_\omega^b$ such that $\pi_{\omega_b}(a)=U^{*}\pi_\omega^b(a)U$ for all $a\in{{\cal A}}$, where $\pi_\omega^b(a):=\pi_{\omega}(a)\upharpoonright _{{{\cal H}}_\omega^b}$. Going back to our original question, i.e. to the unitary equivalence of $\pi_\omega$ and $\pi_{\omega'}$, we will postpone this analysis to the next section, where the more relevant case of quasi \*-algebras is discussed. Let now $\delta$ be a \*-derivation on ${{\cal A}}$ and let us define $\delta_{\pi_{\omega}^b} (a)=\pi_{\omega}^b(\delta(a))$ and $\delta_{\pi_{\omega_b}} (a)=\pi_{\omega_b}(\delta(a))$, $a\in{{\cal A}}$. The first obvious remark is that, under our assumptions, $$\delta_{\pi_{\omega_b}}(a)=\pi_{\omega_b}(\delta(a))=U^*\pi_{\omega}^b(\delta(a))U=U^*\delta_{\pi_{\omega}^b} (a)U.$$ Secondly, if $\delta_{\pi_{\omega}^b}(a)$ is spatial, i.e. there exists an element $H_{\pi_{\omega}^b}\in B({{\cal H}}_{\omega}^b)$ such that $\delta_{\pi_{\omega}^b}(a)=i[H_{\pi_{\omega}^b},\pi_{\omega}^b(a)]$, $a\in{{\cal A}}$, then $\delta_{\pi_{\omega_b}}$ is also spatial and the implementing operator is $H_{\pi_{\omega_b}}=U^*H_{\pi_{\omega}^b}U$, which belongs to $B({{\cal H}}_{\omega_b})$. &gt;From a physical point of view we can interpret this result as follows: it is well known that no hamiltonian operator exists in general which implements the dynamics of an infinitely extended system, [@sew]. For this reason one has to consider a finite-volume approximation of the system, for which a self-adjoint energy operator $H_V$ can be defined. Associated to $H_V$ we can introduce a finite-volume derivation $\delta_V(X)=i[H_V,X]$, for each observable $X$ localized in $V$, and a time evolution $\alpha_V^t(X)=e^{iH_Vt}Xe^{-iH_Vt}$. However, usually, neither $\delta_V(X)$ nor $\alpha_V^t(X)$ converge in the uniform, strong or weak topology. One usually has to consider some representation of the abstract algebra and, as in [@bit1], the corresponding family of [*effective derivations*]{}, i.e. derivations in the given representation. This net of derivations may now be converging and, under suitable conditions, it still defines a derivation whose implementing operator is the [ effective hamiltonian]{}. Therefore the choice of the representations in this procedure is crucial. Our results show that, in fact, there is no essential difference between the effective hamiltonians that we obtain starting from two different representations, at least if they are GNS generated by a fixed positive linear functional $\omega$ and by a different positive linear functional $\omega'=\omega_b$, for each possible choice of $b\in{{\cal A}}$. In particular this implies that, if $b$ is a local observable (meaning by this that it belongs to some of the ${{\cal A}}_V$’s which produce the quasi local C\*-algebra, [@sew; @fbrev]), then the two related derivations are unitarily equivalent and, consequently, the two effective hamiltonians are unitarily equivalent as well. Hence their physical content is essentially the same, as claimed before. The case of quasi \*-algebras ============================= We begin this section with recalling briefly the definitions of quasi \*-algebras and their \*-representations and sub \*-representations. More details can be found in [@schmu; @ait_book]. Let ${{\cal A}}$ be a complex vector space and ${{\cal A}_0}$ a $^\ast$ -algebra contained in ${{\cal A}}$. We say that ${{\cal A}}$ is a quasi $^\ast$ -algebra with distinguished $^\ast$ -algebra ${{\cal A}_0}$ (or, simply, over ${{\cal A}_0}$) if - the left multiplication $ax$ and the right multiplication $ xa$ of an element $a$ of ${{\cal A}}$ and an element $x$ of ${{\cal A}}_0$ which extend the multiplication of ${{\cal A}}_0$ are always defined and bilinear; - $x_1 (x_2 a)= (x_1x_2 )a$ and $x_1(a x_2)= (x_1 a) x_2$, for each $x_1, x_2 \in {{\cal A}}_0$ and $a \in {{\cal A}}$; - an involution $*$ which extends the involution of ${{\cal A}}_0$ is defined in ${{\cal A}}$ with the property $(ax)^*= x^*a^*$ and $(xa)^ * =a^* x^*$ for each $x \in {{\cal A}}_0$ and $a \in {{\cal A}}$. Let now ${{\cal D}}$ be a dense subspace of a Hilbert space ${{\cal H}}$. We denote by $ {{\cal L}}^\dagger({{\cal D}},{{\cal H}}) $ the set of all (closable) linear operators $X$ such that $ {{{\cal D}}}(X) = {{{\cal D}}},\; {{{\cal D}}}(X^*) \supseteq {{{\cal D}}}.$ The set $ {{\cal L}}^\dagger({{\cal D}},{{\cal H}}) $ is a partial \*-algebra with respect to the following operations: the usual sum $X_1 + X_2 $, the scalar multiplication $\lambda X$, the involution $ X \mapsto X^\dagger = X^* {\upharpoonright}{{{\cal D}}}$ and the *(weak)* partial multiplication $X_1 {\,{\scriptstyle \square}\,}X_2 = {{X_1}^\dagger}^* X_2$, defined whenever $X_2$ is a weak right multiplier of $X_1$ (we shall write $X_2 \in R^{\rm w}(X_1)$ or $X_1 \in L^{\rm w}(X_2)$), that is, iff $ X_2 {{{\cal D}}} \subset {{{\cal D}}}({{X_1}^\dagger}^*)$ and $ X_1^* {{{\cal D}}} \subset {{{\cal D}}}(X_2^*).$ Let ${{\cal L}}^\dagger({{\cal D}})$ be the subspace of ${{\cal L}}^\dagger({{\cal D}},{{\cal H}})$ consisting of all its elements which leave, together with their adjoints, the domain ${{\cal D}}$ invariant. Then ${{\cal L}}^\dagger({{\cal D}})$ is a \*-algebra with respect to the usual operations. Let $({{\cal A}},{{\cal A}_0})$ be a quasi \*-algebra with identity $e$ and ${{\cal D}}_\pi$ a dense domain in a certain Hilbert space ${{\cal H}}_\pi$. A linear map $\pi$ from ${{\cal A}}$ into ${{\cal L}}^\dagger({{\cal D}}_\pi, {{\cal H}}_\pi)$ such that: \(i) $\pi(a^*)=\pi(a)^\dagger, \quad \forall a\in {{\cal A}}$, \(ii) if $a\in {{\cal A}}$, $x\in {{\cal A}_0}$, then $\pi(a)$[$\Box$]{} $\pi(x)$ is well defined and $\pi(ax)=\pi(a)$[$\Box$]{} $\pi(x)$, is called a \*-representation of ${{\cal A}}$. Moreover, if \(iii) $\pi({{\cal A}_0})\subset {{\cal L}}^\dagger({{\cal D}}_\pi)$, then $\pi$ is said to be a \*-representation of the quasi \*-algebra $({{\cal A}},{{\cal A}_0})$. If $\pi$ is a \*-representation of $({{\cal A}}, {{\cal A}_0})$, then the [*closure*]{} $\widetilde{\pi}$ of $\pi$ is defined, for each $x \in {{\cal A}}$, as the restriction of $\overline{\pi(x)}$ to the domain $\widetilde{{{\cal D}}_\pi}$, which is the completion of ${{\cal D}}_\pi$ under the [*graph topology*]{} $t_\pi$ [@schmu] defined by the seminorms $\xi \in {{\cal D}}_\pi \to \|\pi(a)\xi\|$, $a\in {{\cal A}}$. If $\pi=\widetilde{\pi}$ the representation is said to be [*closed*]{}. The adjoint of a \*-representation $\pi$ of a quasi \*-algebra $({{\cal A}}, {{\cal A}_0})$ is defined as follows: $${{\cal D}}_{\pi^*} \equiv \bigcap_{x \in {{\cal A}}} {{\cal D}}(\pi(x)^* ) \text{ and } \pi^* (x) = \pi(x^* )^*\, \upharpoonright {{\cal D}}_{\pi^*}, \quad x \in {{\cal A}}.$$ The representation $\pi$ is said to be [*self-adjoint*]{} if $\pi=\pi^*$. The representation $\pi$ is said to be [*ultra-cyclic*]{} if there exists $\xi_0\in{{\cal D}}_\pi$ such that ${{\cal D}}_\pi=\pi({{\cal A}_0})\xi_0$, while is said to be [*cyclic*]{} if there exists $\xi_0\in{{\cal D}}_\pi$ such that $\pi({{\cal A}_0})\xi_0$ is dense in ${{\cal D}}_\pi$ w.r.t. $t_\pi$. Let $\pi$ be a \*-representation of ${{\cal A}}$. A subspace ${{\cal M}}\subset {{\cal D}}_\pi$ is said to be [[*quasi-invariant*]{}]{} for $\pi$ if $\pi({{\cal A}_0}){{\cal M}}\subset{{\cal M}}$ and $\pi({{\cal A}}){{\cal M}}\subset\overline{{{\cal M}}}$, the closure of ${{\cal M}}$ in the Hilbert norm of ${{\cal H}}_\pi$. Moreover the quasi-invariant subspace ${{\cal M}}$ is called [*ultra-cyclic*]{} if there exists $\xi_0\in{{\cal M}}$ such that ${{\cal M}}=\pi({{\cal A}_0})\xi_0$. ${{\cal M}}$ is called [*cyclic*]{} if there exists $\xi_0\in{{\cal M}}$ such that $\pi({{\cal A}_0})\xi_0$ is dense in ${{\cal M}}$ w.r.t. $t_\pi$. Let $\pi$ be a \*-representation of ${{\cal A}}$ and ${{\cal M}}$ a quasi-invariant subspace of ${{\cal D}}_\pi$ for $\pi$. We put $$\left\{ \begin{array}{l} {{\cal D}}_{\pi{\upharpoonright}{{\cal M}}} := {{\cal M}}, \\ (\pi{\upharpoonright}{{\cal M}})(x):=\pi(x){\upharpoonright}{{\cal M}},\quad x\in{{\cal A}}. \\ \end{array} \right.$$ Then $\pi{\upharpoonright}{{\cal M}}$ is a \*-representation of ${{\cal A}}$ with domain ${{\cal M}}$ in $\overline{{{\cal M}}}$. Let $\pi_{{\cal M}}$ denote the closure of $\pi{\upharpoonright}{{\cal M}}$. Then \(i) if ${{\cal M}}$ is ultra-cyclic then $\pi{\upharpoonright}{{\cal M}}$ is ultra-cyclic and $\pi_{{\cal M}}$ is cyclic; \(ii) if ${{\cal M}}$ is cyclic then $\pi{\upharpoonright}{{\cal M}}$ and $\pi_{{\cal M}}$ are cyclic. In the sequel we will also need the following definitions: Let $\rho$ and $\pi$ be \*-representations of ${{\cal A}}$ respectively on ${{\cal D}}_\rho\subset{{\cal H}}_\rho$ and ${{\cal D}}_\pi\subset{{\cal H}}_\pi$. Then $\rho$ and $\pi$ are unitarily equivalent if there exists a unitary operator $U:{{\cal H}}_\rho\rightarrow{{\cal H}}_\pi$ such that $U{{\cal D}}_\rho={{\cal D}}_\pi$ and $\rho(x)=U^*\pi(x)U$, for all $x\in{{\cal A}}$. Let $\pi$ be a \*-representation of ${{\cal A}}$. Then $\pi'$ is a sub \*-representation of $\pi$ if and only if $\pi'=\pi{\upharpoonright}{{\cal M}}$, for a certain quasi-invariant subspace ${{\cal M}}$ of ${{\cal D}}_\pi$. Furthermore $\pi'$ is a closed sub \*-representation of $\pi$ if and only if $\pi'=\pi_{{\cal M}}$, for a certain quasi-invariant subspace ${{\cal M}}$ of ${{\cal D}}_\pi$. The following proposition, proved by one of us in [@ct_ban], extends the GNS construction to quasi \*-algebras. Let $\omega$ be a linear functional on ${{\cal A}}$ satisfying the following requirements: (L1) $\omega(a^*a)\geq 0$ for all $a\in{{\cal A}_0}$; (L2) $\omega(b^*x^*a)=\overline{\omega(a^*xb)}$, $\forall\,a,b\in{{\cal A}_0}$, $x\in{{\cal A}}$; (L3) $\forall x\in{{\cal A}}$ there exists $\gamma_x>0$ such that $|\omega(x^*a)|\leq \gamma_x\,\omega(a^*a)^{1/2}$. Then there exists a triple $(\pi_{\omega}, \lambda_{\omega}, {{\cal H}}_{\omega})$ such that $\bullet$ $\pi_{\omega}$ is a ultra-cyclic \*-representation of ${{\cal A}}$ with ultra-cyclic vector $\xi_\omega$; $\bullet$ $\lambda_{\omega}$ is a linear map of ${{\cal A}}$ into ${{\cal H}}_{\omega}$ with $\lambda_{\omega}({{\cal A}_0})={{\cal D}}_{\pi_\omega}$, $\xi_\omega=\lambda_{\omega}(e)$ and $\pi_{\omega}(x)\lambda_{\omega}(a)=\lambda_{\omega}(xa)$, for every $x \in{{\cal A}},\, a \in {{\cal A}_0}$; $\bullet$ $\omega(x)={\langle {\pi_{\omega}(x)\xi_\omega}|{\xi_\omega}\rangle}$, for every $x \in {{\cal A}}$. The representation $\pi_\omega$ satisfies the properties: (1) $\pi_{\omega_0}=\pi_\omega\upharpoonright_{{{\cal A}_0}}$; (2) $\pi_\omega(x)\lambda_\omega(a)=\lambda_\omega(xa)$, $x\in {{\cal A}}, \, a\in {{\cal A}_0}$ and (3) $\pi_\omega^*(a)\lambda_\omega(x)=\lambda_\omega(ax)$, $x\in {{\cal A}}, \, a\in {{\cal A}_0}$. Here $\pi_\omega^*$ denotes the adjoint representation of $\pi$, see [@schmu; @ait_book]. For shortness, a linear functional $\omega$ on ${{\cal A}}$ satisfying (L1)-(L3) will be called a [*representable*]{} functional on ${{\cal A}}$. If $\omega$ is representable, $(\pi_\omega, \lambda_\omega, {{\cal H}}_\omega)$ will be called, as usual, the GNS construction for $\omega$. It is possible to check that conditions (L1)-(L3) are stable under the map $\omega\rightarrow \omega_b$, with $b \in{{\cal A}_0}$. This means that, if $\omega$ is representable, then $\omega_b$ is representable, for every $b \in {{\cal A}_0}$. We only prove (L3) since (L1) and (L2) are trivial. We have $$\left|\omega_b(x^*a)\right|=\left|\omega((xb)^*)ab\right|\leq \gamma_{xb}\omega((ab)^*ab)^{1/2}=\gamma_{xb}\omega_b(a^*a)^{1/2}.$$ Hence $\omega_b$ produces a GNS representation as well, so that it is worth comparing the two representations arising from $\omega$ and $\omega_b$, in view of extending to quasi \*-algebras what we discussed in the first section for C\*-algebras. We start with considering the following question: [*when a representable linear functional $\omega'$ can be written as $\omega'=\omega_b$, for some $b\in{{\cal A}_0}$?*]{} To answer this question we give the following Let $\omega'$ and $\omega$ be representable linear functionals on ${{\cal A}}$. Then $\omega'=\omega_b$ for some $b\in{{\cal A}_0}$ if and only if $\pi_{\omega'}$ is unitarily equivalent to a sub \*-representation of $\pi_\omega$. Suppose first that $\omega'=\omega_b$ for some $b\in{{\cal A}_0}$. For every $x \in {{\cal A}}$ and $a,c \in {{\cal A}_0}$, we have $$\label{U1} \omega_b(c^*xa)={\langle {\pi_{\omega_b}(x)\lambda_{\omega_b}(a)}|{\lambda_{\omega_b}(c)}\rangle}.$$ On the other hand, $$\label{U2}\omega_b(c^*xa)= \omega(b^*c^*xab)= {\langle {\pi_\omega(x)\pi_\omega(a)\lambda_{\omega}(b)}|{\pi_\omega(c)\lambda_{\omega}(b)}\rangle}.$$ Now put ${{\cal H}}_\omega^b := \overline{\pi_\omega({{\cal A}_0})\lambda_{\omega}(b)}$. Then, from equality , it follows that there exists a unitary operator $U:{{\cal H}}_\omega^b \to{{\cal H}}_{\omega_b}$ such that $$U\pi_\omega(a)\lambda_{\omega}(b)= \lambda_{\omega_b}(a), \quad \forall a\in {{\cal A}_0}.$$ &gt;From we deduce that, for every $a \in {{\cal A}}$ and $a,c \in{{\cal A}_0}$, $$\begin{aligned} {\langle {\pi_\omega(x)\pi_\omega(a)\lambda_{\omega}(b)}|{\pi_\omega(c)\lambda_{\omega}(b)}\rangle} &=& {\langle {\pi_{\omega_b}(x)\lambda_{\omega_b}(a)}|{\lambda_{\omega_b}(c)}\rangle} \\ &=& {\langle {\pi_{\omega_b}(x)U\pi_\omega (a)\lambda_{\omega}(b)}|{U\pi_\omega (c)\lambda_{\omega}(b)}\rangle} \\ &=& {\langle {U^*\pi_{\omega_b}(x)U\pi_\omega(a)\lambda_{\omega}(b)}|{\pi_\omega(c)\lambda_{\omega}(b)}\rangle}.\end{aligned}$$ This implies that $$\pi_\omega^b(x):= \pi_\omega(x)_{\upharpoonright \pi_\omega({{\cal A}_0})\lambda_{\omega}(b) }= U^*\pi_{\omega_b}(x)U \upharpoonright \pi_\omega({{\cal A}_0})\lambda_{\omega}(b).$$ Hence, ${\pi_\omega({{\cal A}_0})\lambda_{\omega}(b)}$ is a quasi-invariant subspace for $\pi_\omega$, that is, $\pi_\omega({{\cal A}}){\pi_\omega({{\cal A}_0})\lambda_{\omega}(b)}\subseteq \overline{\pi_\omega({{\cal A}_0})\lambda_{\omega}(b)}$ and so $\pi_\omega^b$ is a sub \*-representation of $\pi_\omega$ with ultra-cyclic vector $\lambda_{\omega}(b)$, and it is unitarily equivalent to $\pi_{\omega_b}$. Conversely, suppose that $\pi_{\omega'}$ is unitarily equivalent to a sub \*-representation of $\pi_\omega$. Then there exists a quasi-invariant subspace ${{\cal M}}$ of ${{\cal D}}_{\pi_\omega}$, and a unitary operator $U: {{\cal H}}_{\omega'}\rightarrow \overline{{{\cal M}}}\subset {{\cal H}}_\omega$ such that $U\lambda_{\omega'}({{\cal A}_0})={{\cal M}}\subset\lambda_{\omega}({{\cal A}_0})={{\cal D}}_{\pi_\omega}$ and $\pi_{\omega'}(x)=U^*(\pi{\upharpoonright}{{\cal M}})(x)U$, $\forall x\in{{\cal A}}$. Since $U\lambda_{\omega'}(e)\in{{\cal M}}\subset\lambda_{\omega}({{\cal A}_0})$, then there exists $b\in{{\cal A}_0}$ such that $U\lambda_{\omega'}(e)=\lambda_{\omega}(b)$. Thus, for every $x\in{{\cal A}}$, $$\begin{aligned} \omega'(x)={\langle {\pi_{\omega'}(x)\lambda_{\omega'}(e)}|{\lambda_{\omega'}(e)}\rangle} &=& {\langle {\pi_{\omega}(x)U\lambda_{\omega'}(e)}|{U\lambda_{\omega'}(e)}\rangle} \\&=& {\langle {\pi_{\omega}(x)\lambda_{\omega}(b)}|{\lambda_{\omega}(b)}\rangle} = \omega_b(x).\end{aligned}$$ We now consider a slightly generalized problem, looking for conditions under which a representable linear functional $\omega'$ on ${{\cal A}}$ can be written as $\omega'=\lim_\alpha\omega_{b_\alpha}$ for some net $\{b_\alpha\}$ in ${{\cal A}_0}$. Let $\omega'$ and $\omega$ be representable linear functionals on ${{\cal A}}$. Then $\omega'=\lim_{\alpha}\omega_{b_\alpha}$ for some net $\{b_\alpha\}$ in ${{\cal A}_0}$ such that $\{\pi_\omega(b_\alpha)\xi_\omega\}$ converges w.r. to $t_{\pi_\omega}$ if, and only if, $\pi_{\omega'}$ is unitarily equivalent to a sub \*-representation of $\tilde\pi_\omega$. Suppose that $\omega'=\lim_\alpha\omega_{b_\alpha}$, for some net $\{b_\alpha\}$ in ${{\cal A}_0}$ such that $\{\pi_\omega(b_\alpha)\xi_\omega\}$ converges w.r. to $t_{\pi_\omega}$. Then, it is easily shown that ${{\cal M}}:= \tilde{\pi}_\omega({{\cal A}_0})\xi_0$ is a quasi-invariant subspace of ${{\cal D}}_{\tilde{\pi}_\omega}$, where $\xi_0:= t_{\pi_\omega}-\lim_\alpha \pi_\omega(b_\alpha)\xi_\omega$. For every $x \in {{\cal A}}$ and every $a,c \in {{\cal A}_0}$, we have $$\begin{aligned} {\langle {\pi_{\omega'}(x)\lambda_{\omega'}(a)}|{\lambda_{\omega'}(c)}\rangle}&=& \omega' (c^*xa)\nonumber\\ &=& \lim_\alpha \omega(b_\alpha^*c^*xab_\alpha)\nonumber\\ &=& \lim_\alpha{\langle {\pi_\omega(xa)\lambda_\omega(b_\alpha)}|{\pi_\omega(c)\lambda_\omega(b_\alpha)}\rangle}\nonumber\\ &=& {\langle {\tilde{\pi}_\omega(xa)\xi_0}|{\tilde{\pi}_\omega(c)\xi_0}\rangle}\nonumber\\ &=& {\langle {(\tilde{\pi}_\omega{\upharpoonright}{{\cal M}})(x)\tilde{\pi}_\omega(a)\xi_0}|{\tilde{\pi}_\omega(c)\xi_0}\rangle}. \label{ex1}\end{aligned}$$ Here we put $$U\tilde\pi_\omega(a)\xi_0=\lambda_{\omega'}(a), \quad a\in{{\cal A}_0}.$$ Then $U$ extends to a unitary operator of $\overline{{{\cal M}}}$ onto ${{\cal H}}_{\omega'}$, which we denote with the same symbol, such that $U{{\cal M}}=\lambda_{\omega'}({{\cal A}_0})={{\cal D}}_{\pi_{\omega'}}$. Furthermore, by (\[ex1\]), we have $$\begin{aligned} {\langle {\pi_{\omega'}(x)\lambda_{\omega'}(a)}|{\lambda_{\omega'}(c)}\rangle} &=& {\langle {(\tilde\pi_{\omega}{\upharpoonright}{{\cal M}})(x)\tilde\pi_{\omega}(a)\xi_0}|{\tilde\pi_{\omega}(c)\xi_0}\rangle} \\&=& {\langle {(\tilde\pi_{\omega}{\upharpoonright}{{\cal M}})(x)U^*\lambda_{\omega'}(a)}|{U^*\lambda_{\omega'}(c)}\rangle}\\ &=& {\langle {U(\tilde\pi_{\omega}{\upharpoonright}{{\cal M}})(x)U^*\lambda_{\omega'}(a)}|{\lambda_{\omega'}(c)}\rangle},\end{aligned}$$ for each $a,c\in{{\cal A}_0}$ and $x\in{{\cal A}}$, which implies that $$\pi_{\omega'}(x)=U(\tilde\pi_{\omega}{\upharpoonright}{{\cal M}})(x)U^*, \qquad \forall x\in{{\cal A}}.$$ Thus $\pi_{\omega'}$ is unitarily equivalent to a sub \*-representation $\tilde\pi_\omega{\upharpoonright}{{\cal M}}$ of $\tilde\pi_\omega$. Conversely, suppose $\pi_{\omega'}$ is unitarily equivalent to a sub \*-representation of $\tilde\pi_\omega$. Then, there exists a quasi-invariant subspace of ${{\cal D}}_{\tilde\pi_\omega}$, ${{\cal M}}$, and a unitary operator $U: {{\cal H}}_{\omega'}\rightarrow \overline{{{\cal M}}}$ such that $U\lambda_{\omega'}({{\cal A}_0})={{\cal M}}\subset{{\cal D}}_{\tilde\pi_\omega}$ and $\pi_{\omega'(x)}=U^*(\pi_\omega{\upharpoonright}{{\cal M}})(x)U$, $\forall x\in{{\cal A}}$. Since $U\lambda_{\omega'}(e)\in{{\cal M}}\subset{{\cal D}}_{\tilde\pi_\omega}$, there exists $\{b_\alpha\}\subset{{\cal A}_0}$ such that $\lambda_{\omega}(b_\alpha)=\pi_\omega(b_\alpha)\xi_\omega\rightarrow U\lambda_{\omega'}(e)$, in the topology $t_{\pi_\omega}$. Hence, $$\begin{aligned} \omega' (x) &=& {\langle {\pi_{\omega'}(x)\lambda_{\omega'}(e)}|{\lambda_{\omega'}(e)}\rangle}\\ &=& {\langle {\pi_\omega(x)U\lambda_{\omega'}(e)}|{U\lambda_{\omega'}(e)}\rangle}\\ &=& \lim_\alpha {\langle {\pi_\omega(x)\lambda_{\omega}(b_\alpha)}|{\lambda_{\omega}(b_\alpha)}\rangle} \\ &=& \lim_\alpha \omega_{b_\alpha}(x),\end{aligned}$$ for every $x\in{{\cal A}}$. The previous propositions, and in particular Proposition 6, show that, for every $b\in{{\cal A}_0}$ such that $\omega(b^*b)\neq0$, $\omega$ and $\omega_b$ produce [*close*]{} GNS representations and the same physical considerations given in Section I can also be repeated here, with no major change. In particular we consider now some consequences of our results on the theory of spatial derivations in the quasi \*-algebraic setting discussed in [@bit1; @bit2]. To keep the paper self-contained, let us first recall few definitions. Let $({{\cal A}},{{\cal A}_0})$ be a quasi \*-algebra. A [*-derivation of*]{} ${{\cal A}_0}$ is a map $\delta: {{\cal A}_0}\rightarrow {{\cal A}}$ with the following properties: - $\delta(a^*)=\delta(a)^*, \; \forall a \in {{\cal A}_0}$; - $\delta(\alpha a+\beta b) = \alpha \delta( a)+\beta\delta( b), \; \forall a,b \in {{\cal A}_0}, \forall \alpha,\beta \in \mathbb{C}$; - $\delta(ab) = a\delta( b)+\delta( a)b, \; \forall a,b \in {{\cal A}_0}$. Further, let $\pi$ be a \*-representation of $({{\cal A}},{{\cal A}_0})$. As in [@bit1] we will always assume that whenever $a\in {{\cal A}_0}$ is such that $\pi(a)=0$, then $\pi(\delta(a))=0$ as well. Under this assumption, the linear map $ \delta_\pi(\pi(a))=\pi(\delta(a)), \quad a\in {{\cal A}_0}$, is well-defined on $\pi({{\cal A}_0})$ with values in $\pi({{\cal A}})$ and it is a \*-derivation of $\pi({{\cal A}_0})$. We call $\delta_\pi$ the \*-derivation [*induced*]{} by $\pi$. Given such a representation $\pi$ and its dense domain ${{\cal D}}_\pi$, we consider the usual graph topology $t_\dagger$ generated by the seminorms $ \xi\in{{\cal D}}_\pi \rightarrow \|A\xi\|, \quad A\in {{\cal L}}^\dagger({{\cal D}}_\pi)$. If ${{\cal D}}_\pi'$ denotes the conjugate dual of ${{\cal D}}_\pi$, we get the usual rigged Hilbert space ${{\cal D}}_\pi[t_\dagger] \subset {{\cal H}}_\pi \subset {{\cal D}}_\pi'[t_\dagger']$, where $t_\dagger'$ is the strong dual topology of ${{\cal D}}_\pi'$. As usual, we denote by ${{\cal L}}({{\cal D}}_\pi,{{\cal D}}_\pi')$ the space of all continuous linear maps from ${{\cal D}}_\pi[t_\dagger]$ into ${{\cal D}}_\pi'[t_\dagger']$. In this case, ${{\cal L}}^\dagger({{\cal D}}_\pi)\subset {{\cal L}}({{\cal D}}_\pi,{{\cal D}}_\pi')$. Each operator $A\in {{\cal L}}^\dagger({{\cal D}}_\pi)$ can be extended to the whole ${{\cal D}}_\pi'$ by putting $$<\hat A\xi',\eta>=<\xi',A^\dagger \eta>, \quad \forall \xi'\in {{\cal D}}_\pi', \quad \eta\in {{\cal D}}_\pi,$$ where $<\cdot, \cdot>$ denotes the form which puts ${{\cal D}}_\pi$ and ${{\cal D}}_\pi'$ in conjugate duality. Hence the multiplication of $X\in{{\cal L}}({{\cal D}}_\pi,{{\cal D}}_\pi')$ and $A\in{{\cal L}}^\dagger({{\cal D}}_\pi)$ can always be defined. Indeed, [@bit1], $(X\circ A)\xi=X(A\xi), \mbox{ and } (A\circ X)\xi=\hat A(X\xi)$, $\forall \xi\in {{\cal D}}_\pi$. With these definitions, however, $({{\cal L}}({{\cal D}}_\pi,{{\cal D}}_\pi'),{{\cal L}}^\dagger({{\cal D}}_\pi))$ may fail to be a quasi \*-algebra, since the operator $X\circ A$ need not be continuous from ${{\cal D}}_\pi[t_\dagger]$ into ${{\cal D}}_\pi'[t_\dagger']$, unless some additional condition, like the reflexivity of ${{\cal D}}_\pi[t_\dagger]$, is fulfilled. From now on, we will assume that ${{\cal D}}_\pi[t_\dagger]$ is a reflexive space. This assumption (which was missing in [@bit1]) even though restrictive, is fulfilled in most of the physical models considered so far, [@fbrev]. Given a derivation $\delta$ of $({{\cal A}},{{\cal A}_0})$ and a \*-representation $\pi$ of $({{\cal A}},{{\cal A}_0})$, that we suppose to be cyclic with cyclic vector $\xi_0$, the induced derivation $\delta_\pi$ is spatial if there exists [ $H_\pi=H_\pi^\dagger\in {{\cal L}}({{\cal D}}_\pi,{{\cal D}}_\pi')$]{} such that $H_\pi\xi_0\in {{\cal H}}_\pi$ and $$\delta_\pi(\pi(x))=i\{H_\pi\circ\pi(x)-\pi(x)\circ H_\pi\},\quad\forall x\in{{\cal A}_0}.$$ Let $({{\cal A}}, {{\cal A}_0})$ be a locally convex quasi \*-algebra with locally convex topology $\tau$. In [@bit1] we have found necessary and sufficient conditions for an induced derivation to be spatial. One of these conditions is the following: [*there exists a positive linear functional $f$ on ${{\cal A}_0}$ such that: $$f(a^*a)\leq p(a)^2, \quad \forall a\in {{\cal A}_0}, \label{21}$$ for some continuous seminorm $p$ of $\tau$ and, denoting with $\tilde f$ the continuous extension of $f$ to ${{\cal A}}$, the following inequality holds: $$|\tilde f(\delta(a))|\leq C(\sqrt{f(a^*a)}+\sqrt{f(aa^*)}), \quad \forall a\in {{\cal A}_0}, \label{22}$$ for some positive constant $C$.*]{} Suppose now that $\omega_0$ is a positive linear representable functional on ${{\cal A}_0}$ satisfying condition . Let $\omega:=\widetilde{\omega_0}$ be the continuous extension of $\omega_0$ to ${{\cal A}}$, that is $$\omega(x)=\lim_\alpha \omega_0(a_\alpha), \quad x \in {{\cal A}},$$ where ${a_\alpha}$ is a net in ${{\cal A}_0}$ which converges to $x$ w. r. to $\tau$. Then $\omega$ automatically satisfies conditions (L1), (L2) and (L3). Indeed, (L1) is clear since $\omega_0$ is positive by assumption. As for (L2), let $x\in{{\cal A}}$ and $\{x_\alpha\}\subset{{\cal A}_0}$ be a net $\tau$-converging to $x$. Since $\omega_0$ is hermitian we have $\omega_0(b^*x_\alpha^*a)=\overline{\omega_0(a^*x_\alpha b)}$, for all $a,b\in{{\cal A}_0}$. Because of (\[21\]), taking the limit on $\alpha$ of this equality we get (L2). To prove (L3) we first use the Schwarz inequality on ${{\cal A}_0}$: $|\omega_0(x_\alpha a)|\leq \omega_0(x_\alpha^* x_\alpha)^{1/2}\,\omega_0(a^* a)^{1/2}$. But $\omega_0(x_\alpha^* x_\alpha)^{1/2}\leq p(x_\alpha)^2\rightarrow p(x)^2$ so that $$|\omega(xa)|=\lim_\alpha|\omega_0(x_\alpha a)|\leq p(x)\,\omega(a^* a)^{1/2}$$ which is (L3). Suppose that $\omega_0$ is a positive linear representable functional on ${{\cal A}_0}$ satisfying both conditions (\[21\]) and (\[22\]). Then we consider the question as to whether ${(\omega_0)}_b$ satisfies these same conditions. This is important for the following reason. If both $\omega_0$ and $(\omega_0)_b$ satisfy (\[21\]) and (\[22\]), then they have continuous extensions $\omega$ and $\widetilde{(\omega_0)_b}$ respectively to ${{\cal A}}$ and it turns out that $\widetilde{(\omega_0)_b}=\omega_b$. Thus $\omega_b$ satisfies conditions (L1), (L2) and (L3) and both $\delta_{\pi_\omega}$ and $\delta_{\pi_{\omega_b}}$ are spatial. Hence a relation between the effective hamiltonians can be found. First we notice that, because of the continuity of the multiplication, we have $${(\omega_0)}_b(a^*a)=\omega_0((ab)^*ab)\leq p(ab)^2\leq q(a)^2, \quad a \in {{\cal A}_0}$$ for some continuous seminorm $q$ of $\tau$. Thus we have the following Let $({{\cal A}},{{\cal A}_0})$ be a locally convex quasi \*-algebra with locally convex topology $\tau$, $\delta$ a \*-derivation of $({{\cal A}},{{\cal A}_0})$and $\omega_0$ a positive linear functional on ${{\cal A}_0}$. \(1) Suppose that $\omega_0$ satisfies the condition $$\omega_0(a^*a)\leq p(a)^2,\qquad \forall\,a\in{{\cal A}_0}$$ for some continuous seminorm $p$ of $\tau$. Then the continuous extension $\omega:=\tilde\omega_0$ of $\omega_0$ to ${{\cal A}}$ and every $\omega_b$, $b\in{{\cal A}_0}$, produce the ultra-cyclic GNS-representations $\pi_\omega$ and $\pi_{\omega_b}$. \(2) Furthermore, suppose that $$|\omega(\delta(a))|\leq C\left(\sqrt{\omega(a^*a)}+\sqrt{\omega(aa^*)}\right), \quad \forall\,a\in{{\cal A}_0}$$ for some positive constant $C$. Then the \*-derivation $\delta_{\pi_\omega}$ induced by $\pi_\omega$ is spatial. If is bounded, in particular in the case where ${{\cal A}_0}$ is a C\*-algebra, then the \*-derivation $\delta_{\pi_{\omega_b}}$ induced by $\pi_{\omega_b}$ is also spatial for every $b\in{{\cal A}_0}$. We need only to prove the last statement in (2). For this we notice that if $b\in{{\cal A}_0}$ is such that $\pi_\omega(b)$ is bounded, then $\omega_b$ satisfies (\[22\]). Indeed, taking into account that, for every $a\in{{\cal A}_0}$, the equality $b^*\delta(a)b=\delta(b^*ab)-\delta(b^*)ab-b^*a\delta(b)$ holds, we have $$| \omega_b(\delta(a))|=\left| \omega(b^*\delta(a)b)\right|\leq \left| \omega(\delta(b^*ab))\right|+\left| \omega(\delta(b^*)ab)\right| +\left| \omega(b^*a\delta(b))\right|.$$ Using (\[22\]) for the first and introducing $\pi_\omega$ for the second and the third contributions above, we find that, for every $a \in {{\cal A}_0}$, $$\begin{aligned} | \omega_b(\delta(a))|&\leq& C\left( \omega(b^*a^*bb^*ab)^{1/2}+ \omega(b^*abb^*a^*b)^{1/2}\right)\\&+& \left|{\langle {\lambda_\omega(ab)}|{\lambda_\omega(\delta(b))}\rangle}\right|+ \left|{\langle {\lambda_\omega(\delta(b))}|{\lambda_\omega(a^*b)}\rangle}\right| \\ & =&C\left(\|\pi_\omega(b)^*\lambda_\omega(ab)\| +\|\pi_\omega(b)^*\lambda_\omega(a^*b)\|\right)\\ &+& \left|{\langle {\lambda_\omega(ab)}|{\lambda_\omega(\delta(b))}\rangle}\right|+ \left|{\langle {\lambda_\omega(\delta(b))}|{\lambda_\omega(a^*b)}\rangle}\right|\\ &\leq & \left(C\|\overline{\pi_\omega(b)}\|+\|\lambda_\omega(\delta(b)\|\right) \left(\omega_b(a^*a)^{1/2}+\omega_b(aa^*)^{1/2}\right),\end{aligned}$$ The conclusion is therefore that, under mild conditions on $\pi_\omega$, and therefore on $\omega$, both $\delta_{\pi_{\omega}}$ and $\delta_{\pi_{\omega_b}}$ turn out to be spatial so that two different effective hamiltonians $H_\omega$ and $H_{\omega_b}$ do exist, and they are related as in Section I. Once again, the physical contents of the two representations is essentially the same. We end this section with some further results on the GNS representations of a quasi \*-algebra $({{\cal A}},{{\cal A}_0})$. Let $({{\cal A}},{{\cal A}_0})$ be a locally convex quasi \*-algebra, $\omega_0$ a positive linear functional on ${{\cal A}_0}$ satisfying (\[21\]) and $\omega=\widetilde{\omega_0}$ its continuous extension on ${{\cal A}}$. As we have shown, both $\omega$ and $\omega_b$, $b\in{{\cal A}_0}$, satisfy conditions (L1), (L2) and (L3), and so the GNS-constructions $(\pi_\omega,\lambda_\omega,{{\cal H}}_\omega)$ and $(\pi_{\omega_b},\lambda_{\omega_b},{{\cal H}}_{\omega_b})$ are defined. Let $\tilde\pi_\omega$ and $\tilde\pi_{\omega_b}$ be the closures of $\pi_\omega$ and $\pi_{\omega_b}$, respectively. In this section we find conditions which imply that $\tilde\pi_\omega$ is unitarily equivalent to the direct sum of a family of $\tilde\pi_{\omega_b}$, $b\in{{\cal A}_0}$. \[lemma10\] Let $x\in{{\cal A}}$ and $\{x_\alpha\}\subset{{\cal A}_0}$ such that $\tau-\lim_\alpha x_\alpha =x$, then $\lambda_{\omega}(x_\alpha)=\lambda_{\omega_0}(x_\alpha)\rightarrow \lambda_\omega(x)$. We begin with proving that $\{\lambda_\omega(x_\alpha)\}$ is a Cauchy net in the Hilbert space ${{\cal H}}_\omega$: $$\|\lambda_\omega(x_\alpha)-\lambda_\omega(x_\beta)\|^2= \omega((x_\alpha-x_\beta)^*(x_\alpha-x_\beta))\leq p(x_\alpha-x_\beta)^2\rightarrow 0.$$ Therefore there exists a vector $\xi\in{{\cal H}}_\omega$ such that $\lambda_\omega(x_\alpha)\rightarrow \xi$. We now prove that $\xi=\lambda_\omega(x)$. Indeed we have, for every $c\in{{\cal A}_0}$, ${\langle {\lambda_\omega(x_\alpha)}|{\lambda_\omega(c)}\rangle}\rightarrow {\langle {\xi}|{\lambda_\omega(c)}\rangle}$ and, on the other hand, ${\langle {\lambda_\omega(x_\alpha)}|{\lambda_\omega(c)}\rangle}=\omega(c^*x_\alpha)\rightarrow \tilde\omega(c^*x)={\langle {\lambda_\omega(x)}|{\lambda_\omega(c)}\rangle}$, due to the definition of $\tilde\omega$. Therefore $\xi=\lambda_\omega(x)$. We recall that the weak commutant ${\mathfrak M}'_w$ of a $*-$invariant subset ${\mathfrak M}$ of ${{\cal L}}^\dagger({{\cal D}},{{\cal H}})$ is defined as $${\mathfrak M}'_w=\{C\in {{\cal B}}({{\cal H}}): {\langle {X\xi}|{C^*\eta}\rangle}={\langle {C\xi}|{X^\dagger\eta}\rangle}, \; \forall X\in {\mathfrak M},\, \xi, \eta\in {{\cal D}}\}.$$ Then we can prove the following $\pi_\omega({{\cal A}})'_w=\pi_\omega({{\cal A}_0})'_w$. The inclusion $\pi_\omega({{\cal A}})'_w\subset\pi_\omega({{\cal A}_0})'_w$ is clear. To prove the converse inclusion we take $C\in\pi_\omega({{\cal A}_0})'_w$ and $x\in{{\cal A}}$, $c_1,c_2\in{{\cal A}_0}$. Then we have, using the previous Lemma, $$\begin{aligned} {\langle {C\pi_\omega(x)\lambda_\omega(c_1)}|{\lambda_\omega(c_2)}\rangle}&=&\lim_\alpha {\langle {C\pi_\omega(x_\alpha)\lambda_\omega(c_1)}|{\lambda_\omega(c_2)}\rangle}\\ &=&\lim_\alpha {\langle {C\lambda_\omega(c_1)}|{\pi_\omega(x_\alpha^*)\lambda_\omega(c_2)}\rangle}= {\langle {C\lambda_\omega(c_1)}|{\pi_\omega(x^*)\lambda_\omega(c_2)}\rangle}.\end{aligned}$$ Let $b\in{{\cal A}_0}$. We denote by $P_\omega^b$ the projection of ${{\cal H}}_\omega$ onto ${{\cal H}}_\omega^b=\overline{\pi_\omega({{\cal A}_0})\lambda_\omega(b)}$. By Lemma \[lemma10\] we deduce the following Suppose that $\pi_\omega(a)$ is bounded for every $a\in{{\cal A}_0}$. Then $\pi_\omega({{\cal A}})_w'$ is a von Neumann algebra and $P_\omega^b\in \pi_\omega({{\cal A}})_w'$. Even if ${\pi_\omega}{\upharpoonright_{{{\cal A}_0}}}$ is bounded, $P_\omega^bD(\tilde\pi_\omega)\neq D(\tilde\pi_\omega)$ in general. Hence we introduce the following notion: Let $b \in {{\cal A}_0}$. We say that $b$ is a self-adjoint element for $\pi_\omega$ if $\tilde\pi_{\omega_b}$ is a self-adjoint \*-representation of ${{\cal A}}$. By ([@ait_book], Theorem 7.4.4) we have the following Let $b$ be a self-adjoint element for $\pi_\omega$. Then \(1) $P_\omega^b\in \pi_\omega({{\cal A}_0})'_w =\pi_\omega({{\cal A}})'_w$ \(2) $P_\omega^b {{\cal D}}(\tilde \pi_\omega) = {{\cal D}}(\tilde\pi_\omega^b)$. \(3) $\tilde\pi_\omega^b=(\tilde\pi_\omega)_{P_\omega^b}:=P_\omega^b\tilde\pi_\omega (\cdot) P_\omega^b$. By Proposition 1, $\tilde\pi_\omega^b$ is unitarily equivalent to $\tilde\pi_{\omega_b}$, and by the above Lemma we have the following result, which answer our original question Suppose that $\tilde\pi_\omega$ is self-adjoint. If there exist a family $\{b_\gamma\}_{\gamma \in \Gamma}$ of self-adjoint elements for $\pi_\omega$, such that $\{P_\omega^{b_\gamma}\}$ consists of mutually orthogonal projections and $\sum_{\gamma \in \Gamma}P_\omega^{b_\gamma}=I$, then $ \tilde\pi_\omega$ is unitarily equivalent to $\displaystyle \bigoplus_{\gamma \in \Gamma}\,\tilde\pi_{\omega_{b_\gamma}}.$ Local modifications of states ============================= We consider now the particular case in which the C\*-algebra ${{\cal A}}$ is endowed with a [*local structure*]{}. Following [@brarob] we construct the local C\*-algebra as follows. Let ${{\cal F}}$ be a set of indexes directed upward and with an orthonormality relation $\perp$ such that (i.) $\forall \alpha\in{{\cal F}}$ there exists $\beta\in{{\cal F}}$ such that $\alpha\perp\beta$; (ii.) if $\alpha\leq \beta$ and $\beta\perp \gamma$, $\alpha, \beta, \gamma\in{{\cal F}}$, then $\alpha\perp\gamma$; (iii.) if, for $\alpha, \beta, \gamma\in{{\cal F}}$, $\alpha\perp\beta$ and $\alpha\perp\gamma$, there exists $\delta\in{{\cal F}}$ such that $\alpha\perp\delta$ and $\delta\geq \beta, \gamma$. Let now $\{{{\cal A}}_\alpha(\|.\|_\alpha), \,\alpha\in{{\cal F}}\}$ be a family of C\*-algebras with C\*-norm $\|.\|_\alpha$, indexed by ${{\cal F}}$, such that (a.) if $\alpha\geq \beta$ then ${{\cal A}}_\alpha\supset{{\cal A}}_\beta$; (b.) there exists a unique identity $e$ for all ${{\cal A}}_\alpha$’s; (c.) if $\alpha\perp\beta$ then $xy=yx$ for all $x\in{{\cal A}}_\alpha$, $y\in{{\cal A}}_\beta$. Let further ${{\cal A}}_0:=\cup_\alpha {{\cal A}}_\alpha$. The uniform completion of ${{\cal A}_0}$ is, as it is well known, the quasi-local C\*-algebra[^1] with the norm $\|\cdot\|$ inherited by the $\|.\|_\alpha$’s. If we take instead the completion of ${{\cal A}_0}$ w.r.t. a locally convex topology $\tau$ which makes the involution and the multiplications continuous we get, in general, a locally convex quasi \*-algebra ${{\cal A}}$ which we call a [*quasi-local quasi \*-algebra*]{}. Given $x\in{{\cal A}}_0$, there will be some $\beta\in{{\cal F}}$ such that $x\in{{\cal A}}_\beta$. But of course, $x$ also belongs to many other ${{\cal A}}_{\beta'}$, for instance to all those algebras which contains ${{\cal A}}_\beta$ as a sub-algebra. For this reason we introduce a set $J_x$, related to $x\in{{\cal A}}_0$, which is defined as follows: $J_x=\{\alpha\in{{\cal F}}\mbox{ such that } x\in{{\cal A}}_\alpha \}$. If we now define ${{\cal A}}_\infty=\cap_{\alpha\in{{\cal F}}}{{\cal A}}_\alpha$, then we will work here under the assumption, which is verified for very general discrete and continuous models [@sew], that $\forall\, x\in{{\cal A}}_0$, $x\notin {{\cal A}}_\infty$, there exists $\alpha_x\in{{\cal F}}$ such that $\cap_{\beta\in J_x}{{\cal A}}_\beta={{\cal A}}_{\alpha_x}$. We call $\alpha_x$ the [*support of $x$*]{}. The following definition selects states on ${{\cal A}}$ with a [*reasonable asymptotic behavior*]{}. These states, indeed, factorize on regions far enough from the support of a given element. A state $\omega$ over ${{\cal A}}$ is said to be almost clustering (AC) if, $\forall \,b\in{{\cal A}_0}$ and $\forall\,\epsilon>0$, there exists $\alpha\in{{\cal F}}$, $\alpha\geq \alpha_b$, such that, $\forall\gamma\perp\alpha$ we have $\left|\omega(ab)-\omega(a)\omega(b)\right|\leq \epsilon \|a\|$, $\forall\,a\in{{\cal A}}_\gamma$. Similar definitions are given in many textbooks, like [@sew], [@brarob] and [@emch], where the physical motivations are discussed in detail. Related to the notion of factorization is also that of [*local modification*]{} of a given state. Of course, several definitions of local modifications can be introduced. The most natural one is perhaps the following: $\omega'$ is a local modification of $\omega$ if there exists $\alpha\in{{\cal F}}$ such that $\forall\,\gamma\in{{\cal F}}$, $\gamma\perp\alpha$, $\omega'(a)=\omega(a)$ for all $a\in{{\cal A}}_\gamma$. This simply implies that, outside a fixed [*region*]{} $\alpha$, the two states coincide. However this condition is rather strong and has no counterpart in the existing literature on this subject and for this reason will not be considered here. To stay in touch with the existing literature, we rather consider the following definitions. Given two states $\omega$ and $\omega'$ over ${{\cal A}}$, $\omega'$ is said to be a local modification of type 1 (1LM) of $\omega$ if, calling $\pi_{\omega'}$ and $\pi_\omega$ their associated GNS-representations, $\pi_{\omega'}$ is unitarily equivalent to a sub \*-representation of $\pi_\omega$. Also, $\omega'$ is said to be a local modification of type 2 (2LM) of $\omega$ if $\forall\,\epsilon>0$, there exists $\alpha_\epsilon\in{{\cal F}}$ such that, $\forall\gamma\in{{\cal F}}$, $\gamma\perp\alpha_\epsilon$, $\left|\omega'(x)-\omega(x)\right|\leq \epsilon \|x\|$, $\forall\,x\in{{\cal A}}_\gamma$. These definitions are physically motivated essentially from what is discussed in [@sew]. Just to clarify the situation if, for instance, $\omega'$ is a 2LM of $\omega$ then they coincide, but for an error of order $\epsilon$, outside a region whose size is, in general, proportional to $1/\epsilon$. There is an apparent difference between the conditions 1LM and 2LM: if $\omega'$ is a 2LM of $\omega$, then $\omega$ is a 2LM of $\omega'$. This symmetry is not shared by 1LM. We argue that 2LM could be used for the mathematical description of reversible local operations on a given state while 1LM seems to be more appropriate for describing the action of irreversible operations (like a quantum mechanical measurement). One immediate consequence of the results of Section II and of these definitions is that if $b\in{{\cal A}}_\alpha$ for some $\alpha\in{{\cal F}}$ then the state $\omega_b(.)$ is a 1LM of $\omega$. Less trivial is the proof of the following statement: [*let the state $\omega$ be AC and $b\in{{\cal A}_0}$ with $\omega(b^\dagger b)=1$. Then $\omega_b$ is a 2LM of $\omega$*]{}. This is not the end of the story. Indeed, let us suppose that $\omega$ is AC and that $\omega'$ is a 1LM of $\omega$. Therefore there exists a sequence $\{b_n\}$ of elements of ${{\cal A}_0}$ such that $\omega'(a)=\lim_{n\to\infty}\omega(b_n^* a b_n)$, $\forall\,a\in{{\cal A}}$, and the sequence $\{\pi_\omega(b_n)\xi_\omega\}$ converges in ${{\cal H}}_\omega$. We suppose now that there exists $n_0\in{\mathbb}{N}$ and $\lambda\in{{\cal F}}$ such that, for all $n\geq n_0$, $b_n\in{{\cal A}}_\lambda$. Then $\omega'$ is also a 2LM of $\omega$. The proof of these statements are easy and will be omitted here. We end this section, and the paper, with the following example of what a concrete local modification of a state could be. [**Discrete system:**]{} Let $V$ be a finite region of a $d$-dimensional lattice $\Lambda$ and $| V |$ the number of points in $V$. The local $C^*$-algebra ${{{\cal A}}}_V$ is generated by the Pauli operators $\vec\sigma_p = (\sigma^1_p, \sigma^2_p, \sigma^3_p)$ and by the unit $2\times 2$ matrix $e_p$ at every point $p \in V$. The $\vec\sigma_p$’s are copies of the Pauli matrices localized in $p$. If $V \subset V^{'}$ and $A_V \in {{{\cal A}}}_V$, then $A_V \rightarrow A_{V^{'}} = A_V \otimes ({{\atop \bigotimes } \atop{p \in V^{'} \setminus V}} e_p)$ defines the natural imbedding of ${{{\cal A}}}_V$ into ${{{\cal A}}}_{V^{'}}$. Let ${\vec n}=(n_1, n_2, n_3)$ be a unit vector in ${{{\mathbb}R}}^3$, and put $ (\vec\sigma\cdot {\vec n}) = n_1 \sigma^1 + n_2 \sigma^2 + n_3 \sigma^3. $ Then, denoting as $Sp(\vec\sigma \cdot {\vec n})$ the spectrum of $\vec\sigma \cdot {\vec n}$, we have $ Sp(\vec\sigma \cdot {\vec n}) = \{ 1, -1\}. $ Let $|\vec n \rangle\in {{\mathbb}C}^2$ be a unit eigenvector associated with $1$. Let now denote by ${\mathbf n}:= \{ \vec n_p \}_{p\in\Lambda}$ an infinite sequence of unit vectors in ${{{\mathbb}R}}^3$ and $| \mathbf{n} \rangle = {{\atop \bigotimes }\atop{p}} | \vec n_p\rangle$ the corresponding unit vector in the infinite tensor product ${\cal H}_\infty = {{\atop \bigotimes }\atop{p}} {{\mathbb}C}_p^2$. We put $ {{{\cal A}}}_0 = \bigcup_V {{{\cal A}}}_V $ and $ {\cal D}^0_{\mbox{\small${\mathbf{n}}$}} = {{{\cal A}}}_0 |{\mathbf{n}}\rangle $ and we denote the closure of ${\cal D}^0_{\mbox{\small${\mathbf{n}}$}}$ in ${\cal H}_\infty$ by $ {\cal H}_{\mbox{\small${\mathbf{n}}$}}$. As we saw above, to any sequence $ {\mathbf{n}}$ of three-vectors there corresponds a state $|{\mathbf{n}}\rangle$ of the system. Such a state defines a realization $\pi_{\mbox{\small${\mathbf{n}}$}}$ of ${{{\cal A}}}_0$ in the Hilbert space ${\cal H}_{\mbox{\small${\mathbf{n}}$}}$. This representation is faithful, since the norm completion ${{{\cal A}}}_S$ of ${{{\cal A}}}_0$ is a simple C\*-algebra. A special basis for $ {\cal H}_{\mbox{\small${\mathbf{n}}$}}$ is obtained from the [*ground*]{} state $|{\mathbf{n}}\rangle$ by [*flipping*]{} a finite number of spins using the following strategy:\ Let $\vec n$ be a unit vector in ${{{\mathbb}R}}^3$, as above, and $ |\vec n\rangle$ the corresponding vector of ${{\mathbb}C}^2$. Let us choose two other unit vectors ${\vec n}^1, {\vec n}^2$ so that $({\vec n}, {\vec n}^1, {\vec n}^2)$ form an orthonormal basis of ${{{\mathbb}R}}^3$. We put $ {\vec n}_{\pm} = \frac{1}{2} ({\vec n}^1 \pm i{\vec n}^2) $ and define $ |m,\vec n\rangle := (\vec\sigma \cdot {\vec n}_{-})^m |\vec n\rangle \ \ (m=0,1). $ Then we have $$(\vec\sigma \cdot {\vec n}) |m, \vec n\rangle = (-1)^m |m,\vec n\rangle \ \ (m=0,1).$$ Thus the set $ \left\{|\mathbf{m}, \mathbf{n}\rangle = {{\atop \bigotimes }\atop{p}} |m_p, \vec{n}_p\rangle ;\ m_p = 0, 1,\ \ {\displaystyle \sum_p} m_p < \infty \right\}$ forms an orthonormal basis in ${\cal H}_{\mbox{\small$\mathbf{n}$}}$, [@BCS1]. The representation $\pi_{\mbox{\small$\mathbf{n}$}}$ is defined on the basis vectors $\{ |\mathbf{m}, \mathbf{n}\rangle \}$ by $$\pi_{\mbox{\small$\mathbf{n}$}} (\sigma^i_p)|\mathbf{m}, \mathbf{n}\rangle= \sigma^i_p \mid m_p, \vec n_p\rangle \otimes ({{\atop \prod }\atop{\scriptstyle{p^{'} \neq p}}} \otimes \mid m_{p ^{'}}, \vec n_{p^{'}}\rangle)\ \ \ (i= 1, 2, 3).$$ This definition is then extended in obvious way to the whole space ${\cal H}_{\mbox{\small$\mathbf{n}$}}$. It turns out that $\pi_{\mbox{\small$\mathbf{n}$}}$ is a [*bounded*]{} representation of ${{\cal A}}_0$ into ${\cal H}_{\mbox{\small$\mathbf{n}$}}$. More details on this construction, particularly in connection with quasi \*-algebras, can be found in [@ctrev; @bagtra3]. Let now $\varphi=\otimes_{j\in\Lambda}\varphi_j$ be a fixed normalized vector in ${\cal H}_{\mbox{\small$\mathbf{n}$}}$ and $\omega$ the related vector state: if $a\in{{\cal A}}_0$ then $\omega(a)=<\varphi,\pi_{\mbox{\small$\mathbf{n}$}}(a)\varphi>$. Let now $x=\prod_{p\in\lambda}\otimes x_p$, for some bounded subset $\lambda$ in $\Lambda$. Here $x_p$ acts on ${{\mathbb}C}_p^2$ and $\lambda$ is the support of $x$. Let furthermore $\gamma$ be another bounded subset of $\Lambda$, orthogonal to $\lambda$: this means that the sets $\lambda$ and $\gamma$ have empty intersection. Then, we fix $b=\prod_{p\in\gamma}\otimes b_p$, where as before $b_p$ acts on ${{\mathbb}C}_p^2$. We further assume that $<\pi_{\mbox{\small$\mathbf{n}$}}(b)\varphi,\pi_{\mbox{\small$\mathbf{n}$}}(b)\varphi>=1$. Then we can check that $\omega(a)$ coincides with $\omega_b(a)=<\pi_{\mbox{\small$\mathbf{n}$}}(b)\varphi,\pi_{\mbox{\small$\mathbf{n}$}}(a)\pi_{\mbox{\small$\mathbf{n}$}}(b)\varphi>$, and this is true for all possible choices of $a$ and $b$ which are supported in separated regions. So $\omega_b$ is a local modification of $\omega$ in the strongest sense and, in particular, is a 2LM of $\omega$. This example shows that the definitions of local modification given here are really physically motivated. States sharing the same properties in the case of continuous physical systems, [@sew], could also be constructed with no major difficulty. To [@sew] we also refer for a more physically-minded discussion on 1LM of states. Acknowledgements {#acknowledgements .unnumbered} ================ This work was partially supported by the Japan Private School Promotion Foundation and partially by CORI, Università di Palermo. F.B. and C.T. acknowledge the warm hospitality of the Department of Applied Mathematics of the Fukuoka University. A.I. acknowledges the hospitality of the Dipartimento di Matematica ed Applicazioni, Università di Palermo, where this work was completed.\ [The authors wish to thank the referee for his suggestions and corrections]{}. [99]{} F. Bagarello, A. Inoue and C. Trapani, [*-Derivations of quasi \*-algebras*]{}, Int. Jour. Math. and Math. Sci., [**21**]{} (2004), 1077-1096. F. Bagarello, A. Inoue, C Trapani, [*Exponentiating derivations of quasi \*-algebras: possible approaches and applications*]{}, Int. Jour. Math. and Math. Sci., [**17**]{}, 2805-2820 (2005) F. Bagarello [*Algebras of unbounded operators and physical applications: a survey*]{}, Review in Math. 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[**4**]{}, 303-314 (1967). F.Bagarello, C.Trapani, [*The Heisenberg Dynamics of Spin Systems: a Quasi\*-Algebras Approach*]{}, J. Math. Phys., [**37**]{}, 4219-4234, (1996). [^1]: this terminology is due to the fact that, in concrete applications, $\alpha$ is quite often a given bounded open region in a $d-$dimensional space
--- abstract: 'We use data collected with the CLEO II detector to perform a high-statistics measurement of the resonant substructure in $D^0 \to K^-\pi^+\pi^0$ decays. We find the Dalitz Plot is well represented by a combination of seven quasi-two-body decay channels ($\ksz \piz$, $K^- \rho$, $K^{*-} \pip$, $K_0(1430)^-\pip$, $\overline{K}_0(1430)^0 \piz$, $K^- \rho^+(1700)$, and $K^*(1680)^- \pip$), plus a small non-resonant component. We see no evidence of a scalar $\kappa\rightarrow K^-\pi^+$ resonance in the mass range recently reported by other groups. Using the amplitudes and phases from this analysis, we calculate an integrated CP asymmetry of $-0.031 \pm 0.086$.' author: - CLEO Collaboration title: Dalitz Analysis of the Decay --- 6.5 in 9.0 in -0.50in 0.00in 0.00in S. Kopp,$^{1}$ M. Kostin,$^{1}$ A. H. Mahmood,$^{2}$ S. E. Csorna,$^{3}$ I. Danko,$^{3}$ K. W. McLean,$^{3}$ Z. Xu,$^{3}$ R. Godang,$^{4}$ G. Bonvicini,$^{5}$ D. Cinabro,$^{5}$ M. Dubrovin,$^{5}$ S. McGee,$^{5}$ G. J. Zhou,$^{5}$ A. Bornheim,$^{6}$ E. Lipeles,$^{6}$ S. P. Pappas,$^{6}$ M. Schmidtler,$^{6}$ A. Shapiro,$^{6}$ W. M. Sun,$^{6}$ A. J. Weinstein,$^{6}$ D. E. Jaffe,$^{7}$ G. Masek,$^{7}$ H. P. Paar,$^{7}$ D. M. Asner,$^{8}$ A. Eppich,$^{8}$ T. S. Hill,$^{8}$ R. J. Morrison,$^{8}$ R. A. Briere,$^{9}$ G. P. Chen,$^{9}$ T. Ferguson,$^{9}$ H. Vogel,$^{9}$ A. Gritsan,$^{10}$ J. P. Alexander,$^{11}$ R. Baker,$^{11}$ C. Bebek,$^{11}$ B. E. Berger,$^{11}$ K. Berkelman,$^{11}$ F. Blanc,$^{11}$ V. Boisvert,$^{11}$ D. G. Cassel,$^{11}$ P. S. Drell,$^{11}$ J. E. Duboscq,$^{11}$ K. M. Ecklund,$^{11}$ R. Ehrlich,$^{11}$ A. D. Foland,$^{11}$ P. Gaidarev,$^{11}$ L. Gibbons,$^{11}$ B. Gittelman,$^{11}$ S. W. Gray,$^{11}$ D. L. Hartill,$^{11}$ B. K. Heltsley,$^{11}$ P. I. Hopman,$^{11}$ L. Hsu,$^{11}$ C. D. Jones,$^{11}$ J. Kandaswamy,$^{11}$ D. L. Kreinick,$^{11}$ M. Lohner,$^{11}$ A. Magerkurth,$^{11}$ T. O. Meyer,$^{11}$ N. B. Mistry,$^{11}$ E. Nordberg,$^{11}$ M. Palmer,$^{11}$ J. R. Patterson,$^{11}$ D. Peterson,$^{11}$ D. Riley,$^{11}$ A. Romano,$^{11}$ J. G. Thayer,$^{11}$ D. Urner,$^{11}$ B. Valant-Spaight,$^{11}$ G. Viehhauser,$^{11}$ A. Warburton,$^{11}$ P. Avery,$^{12}$ C. Prescott,$^{12}$ A. I. Rubiera,$^{12}$ H. Stoeck,$^{12}$ J. Yelton,$^{12}$ G. Brandenburg,$^{13}$ A. Ershov,$^{13}$ D. Y.-J. Kim,$^{13}$ R. Wilson,$^{13}$ T. Bergfeld,$^{14}$ B. I. Eisenstein,$^{14}$ J. Ernst,$^{14}$ G. E. Gladding,$^{14}$ G. D. Gollin,$^{14}$ R. M. Hans,$^{14}$ E. Johnson,$^{14}$ I. Karliner,$^{14}$ M. A. Marsh,$^{14}$ C. Plager,$^{14}$ C. Sedlack,$^{14}$ M. Selen,$^{14}$ J. J. Thaler,$^{14}$ J. Williams,$^{14}$ K. W. Edwards,$^{15}$ R. Janicek,$^{16}$ P. M. Patel,$^{16}$ A. J. Sadoff,$^{17}$ R. Ammar,$^{18}$ A. Bean,$^{18}$ D. Besson,$^{18}$ X. Zhao,$^{18}$ S. Anderson,$^{19}$ V. V. Frolov,$^{19}$ Y. Kubota,$^{19}$ S. J. Lee,$^{19}$ R. Mahapatra,$^{19}$ J. J. O’Neill,$^{19}$ R. Poling,$^{19}$ T. Riehle,$^{19}$ A. Smith,$^{19}$ C. J. Stepaniak,$^{19}$ J. Urheim,$^{19}$ S. Ahmed,$^{20}$ M. S. Alam,$^{20}$ S. B. Athar,$^{20}$ L. Jian,$^{20}$ L. Ling,$^{20}$ M. Saleem,$^{20}$ S. Timm,$^{20}$ F. Wappler,$^{20}$ A. Anastassov,$^{21}$ E. Eckhart,$^{21}$ K. K. Gan,$^{21}$ C. Gwon,$^{21}$ T. Hart,$^{21}$ K. Honscheid,$^{21}$ D. Hufnagel,$^{21}$ H. Kagan,$^{21}$ R. Kass,$^{21}$ T. K. Pedlar,$^{21}$ H. Schwarthoff,$^{21}$ J. B. Thayer,$^{21}$ E. von Toerne,$^{21}$ M. M. Zoeller,$^{21}$ S. J. Richichi,$^{22}$ H. Severini,$^{22}$ P. Skubic,$^{22}$ A. Undrus,$^{22}$ V. Savinov,$^{23}$ S. Chen,$^{24}$ J. Fast,$^{24}$ J. W. Hinson,$^{24}$ J. Lee,$^{24}$ D. H. Miller,$^{24}$ E. I. Shibata,$^{24}$ I. P. J. Shipsey,$^{24}$ V. Pavlunin,$^{24}$ D. Cronin-Hennessy,$^{25}$ A.L. Lyon,$^{25}$ E. H. Thorndike,$^{25}$ T. E. Coan,$^{26}$ V. Fadeyev,$^{26}$ Y. S. Gao,$^{26}$ Y. Maravin,$^{26}$ I. Narsky,$^{26}$ R. Stroynowski,$^{26}$ J. Ye,$^{26}$ T. Wlodek,$^{26}$ M. Artuso,$^{27}$ R. Ayad,$^{27}$ C. Boulahouache,$^{27}$ K. Bukin,$^{27}$ E. Dambasuren,$^{27}$ G. Majumder,$^{27}$ G. C. Moneti,$^{27}$ R. Mountain,$^{27}$ S. Schuh,$^{27}$ T. Skwarnicki,$^{27}$ S. Stone,$^{27}$ J.C. Wang,$^{27}$ A. Wolf,$^{27}$ and J. Wu$^{27}$ $^{1}$[University of Texas, Austin, TX 78712]{}\ $^{2}$[University of Texas - Pan American, Edinburg, TX 78539]{}\ $^{3}$[Vanderbilt University, Nashville, Tennessee 37235]{}\ $^{4}$[Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061]{}\ $^{5}$[Wayne State University, Detroit, Michigan 48202]{}\ $^{6}$[California Institute of Technology, Pasadena, California 91125]{}\ $^{7}$[University of California, San Diego, La Jolla, California 92093]{}\ $^{8}$[University of California, Santa Barbara, California 93106]{}\ $^{9}$[Carnegie Mellon University, Pittsburgh, Pennsylvania 15213]{}\ $^{10}$[University of Colorado, Boulder, Colorado 80309-0390]{}\ $^{11}$[Cornell University, Ithaca, New York 14853]{}\ $^{12}$[University of Florida, Gainesville, Florida 32611]{}\ $^{13}$[Harvard University, Cambridge, Massachusetts 02138]{}\ $^{14}$[University of Illinois, Urbana-Champaign, Illinois 61801]{}\ $^{15}$[Carleton University, Ottawa, Ontario, Canada K1S 5B6\ and the Institute of Particle Physics, Canada]{}\ $^{16}$[McGill University, Montréal, Québec, Canada H3A 2T8\ and the Institute of Particle Physics, Canada]{}\ $^{17}$[Ithaca College, Ithaca, New York 14850]{}\ $^{18}$[University of Kansas, Lawrence, Kansas 66045]{}\ $^{19}$[University of Minnesota, Minneapolis, Minnesota 55455]{}\ $^{20}$[State University of New York at Albany, Albany, New York 12222]{}\ $^{21}$[Ohio State University, Columbus, Ohio 43210]{}\ $^{22}$[University of Oklahoma, Norman, Oklahoma 73019]{}\ $^{23}$[University of Pittsburgh, Pittsburgh, Pennsylvania 15260]{}\ $^{24}$[Purdue University, West Lafayette, Indiana 47907]{}\ $^{25}$[University of Rochester, Rochester, New York 14627]{}\ $^{26}$[Southern Methodist University, Dallas, Texas 75275]{}\ $^{27}$[Syracuse University, Syracuse, New York 13244]{} Introduction ============ A clearer understanding of final state interactions in exclusive weak decays is an important ingredient for our ability to model decay rates as well as for our understanding of interesting phenomena such as mixing [@ref:hepph9802291]. There are several theoretical methods  [@ref:bsw; @ref:bedaque; @ref:chau; @ref:terasaki; @ref:buccella] used to understand the dynamics of two body charmed meson decays with experimental measurements as input. Unfortunately, final-state interactions are often not well understood, and may not be included properly in many models. These long-distance strong interaction effects can cause significant changes in decay rates for specific final states, and can cause shifts in the phases of the decay amplitudes. Three-body decays provide a rich laboratory in which to study the interference between intermediate state resonances, and provide a direct probe of the final state interactions in certain decays. When a particle decays into three or more daughters, intermediate resonances dominate the decay rate. These resonances will cause a non-uniform distribution of events in phase space when analyzed using a “Dalitz Plot” technique [@ref:dalitz]. Since all events of a particular decay mode have the same final state, multiple resonances at the same location in phase space will interfere. This provides the opportunity to experimentally measure both the amplitudes and phases of the intermediate decay channels, which in turn allows us to deduce their relative branching fractions. These phase differences can even allow details about very broad resonances to be extracted by observing their interference with other intermediate states. This paper describes a study of the underlying structure in  decays [@ref:cc]. Four previous groups have made similar measurements, each with less than 1000 signal events[@ref:e687; @ref:e691; @ref:mk31; @ref:e516]. In addition to significantly increasing our statistical power, the large CLEO II dataset on which this analysis is based also permits us to tighten analysis requirements to drastically reduce the effect of backgrounds. Coupled with the superb resolution of the CLEO-II detector, this has allowed us to extract significantly more information about this decay than was possible in the past. Theoretical Models {#sec:theory} ================== Since we are studying the decay of a spin-zero particle to three daughters, only two degrees-of-freedom are required to completely describe the kinematics. To see this, consider the decay in the  rest frame. The four-momenta of the three final state particles correspond to twelve unknowns. We have one constraint for each known mass and four additional constraints from the conservation of momentum and energy in the decay. Finally, since the three degrees of freedom describing the spatial orientation of the decay are irrelevant (the  having spin zero) only two independent degrees of freedom remain. There are three invariant masses that can be formed by considering all possible pairs of final state particles: $M^2_{K^-\pi^+}$, $M^2_{K^-\pi^0}$ and $M^2_{\pi^+\pi^0}$. Only two of these are independent, however, since energy and momentum conservation results in the additional constraint $$\label{eq:epcons} M^2_{D^0} + M^2_{K^-} + M^2_{\pi^+} + M^2_{\pi^0} = M^2_{K^-\pi^+} + M^2_{K^-\pi^0} + M^2_{\pi^+\pi^0}.$$ Choosing two of the above three invariant masses as dynamic variables has two compelling advantages: i) Their relativistic invariance means the Lorentz frame in which they are evaluated is irrelevant; ii) As the expression for the partial width in Equation \[eq:dgamma\] indicates, we expect that a scatter plot of events in the $M_{12}^2$ vs $M_{23}^2$ plane (known as a Dalitz Plot) will be uniformly distributed if phase-space alone determines the decay dynamics. This allows the decay fraction at each point to be readily correlated with the decay matrix element $\cal M$ without additional corrections: $$\label{eq:dgamma} d\Gamma = { |{\cal M}|^2 \over{256 \pi^3 M_D^3}} dM_{12}^2 dM_{23}^2.$$ In this analysis we choose $M^2_{K^-\pi^+}$ and $M^2_{\pi^+\pi^0}$ as our two Dalitz Plot variables, and must next construct the relevant decay amplitudes in terms of these. The dynamics can be understood with the aid of Figure \[fig:fyn\]. We first consider the decay of the $D^0$ meson into particle C plus an AB resonance, followed by the decay of the AB resonance into particles A and B. To properly describe the structure of this decay using our Dalitz Plot variables, we need to obtain the angular dependence of the decay products. Each vertex in Figure \[fig:fyn\] contains a spin factor $\varepsilon_\lambda$ which depends on the type of the decay: scalar, vector, tensor, etc. The matrix element for a vector decay is $$\label{eq:vecspin} {\cal M} = F_D (P_{D^0} + P_C)_\mu {\sum_\lambda \varepsilon^{\mu*}_\lambda \varepsilon^\nu_\lambda \over{M^2_r - M^2_{AB} - i M_r \Gamma_{AB}}} (P_A - P_B)_\nu F_r$$ where $P$ denotes 4-momentum, and $M_r$ is the mass of the resonance. In general the form factors at each vertex, $F_D$ and $F_r$, are unknown functions, however in practice they are either set to a constant value or to the Blatt-Weisskopf penetration factors [@ref:blatt]. The spin-sum in the numerator of Equation \[eq:vecspin\] is evaluated to give $$\label{eq:spinsum} \sum_\lambda \varepsilon^{\mu*}_\lambda \varepsilon^\nu_\lambda = -g^{\mu\nu} + {{P_{AB}^\mu P_{AB}^\nu}\over{M_{AB}^2}}$$ and the “mass dependent width” $\Gamma_{AB}$ is a function of the AB invariant mass $M_{AB}$, the momentum of either daughter in the AB rest frame $p_{AB}$, the momentum of either daughter in the resonance rest frame $p_r$, the spin of the resonance $J$, the width of the resonance $\Gamma_r$, and is expressed as [@ref:pilkuhn]: $$\Gamma_{AB} = \Gamma_r \left( p_{AB} \over{p_r} \right)^{2J+1} \left( M_r \over{M_{AB}} \right) F_r^2$$ We relax the transversality requirement on the vector resonance in Equation \[eq:spinsum\] and divide by $M_r^2$ instead of $M_{AB}^2$. This substitution gives rise to a small spin zero component when the vector resonance is off mass-shell, a behavior which is observed to occur with the $W$ boson and which should also be expected in the resonance behavior we are studying here. Inserting this expression for the spin-sum into Equation \[eq:vecspin\] and summing over the repeated indices gives the Lorentz invariant expression for the matrix element of a vector particle as a function of position in the Dalitz Plot: $$\label{eq:svec} {\cal A}_1(ABC|r) = F_D F_r {M^2_{AC} - M^2_{BC} + { (M^2_D - M^2_C)(M^2_B - M^2_A)\over{M^2_{r}}} \over {M^2_r - M^2_{AB} - i M_r \Gamma_{AB}}}.$$ The procedure for calculating the vector matrix element is generalizable to intermediate particles having other spin. For example, we can easily find the amplitude for a spin zero resonance to be $$\label{eq:sscalar} {\cal A}_0(ABC|r) = F_D F_r { 1 \over {M^2_r - M^2_{AB} - i M_r \Gamma_{AB}}}.$$ The procedure for higher spin resonances involves a bit more algebra. For example, the spin two case starts with $$\label{eq:mten} {\cal A}_2(ABC|r) = F_D (P_D + P_C)_\mu (P_D + P_C)_\nu {\sum_\lambda \varepsilon^{\mu\nu*}_\lambda \varepsilon^{\alpha\beta}_\lambda \over{M^2_r - M^2_{AB} - i M_r \Gamma_{AB}}} (P_A - P_B)_\alpha (P_A - P_B)_\beta F_r.$$ In this case the spin sum has been previously calculated by Pilkuhn [@ref:pilkuhn] to be $$\sum\limits_\lambda {\varepsilon _\lambda ^{*\mu \nu } \varepsilon _\lambda ^{\alpha \beta } } = \frac{1}{2}\left( {T ^{\mu \alpha } T ^{\nu \beta } + T ^{\mu \beta } T ^{\nu \alpha } } \right) - \frac{1}{3}T ^{\mu \nu } T ^{\alpha \beta }$$ where $$T^{\mu \nu} = -g^{\mu\nu} + {{P^\mu P^\nu}\over{M^2}}$$ When this expression is inserted into Equation \[eq:mten\] and the implied sums performed we find the final form of the tensor matrix element: $${\cal A}_2(ABC|r) = {F_D F_r \over{M_r^2 - M^2_{AB} - i\Gamma_{AB} M_r}} \left[ \left( M^2_{BC} - M^2_{AC} + {(M_D^2 - M_C^2) (M_A^2 - M_B^2) \over{M_{r}^2}}\right)^2 \right .$$ $$-\frac{1}{3} \left(M^2_{AB} - 2 M_D^2 - 2 M_C^2 + {( M_D^2 - M_C^2)^2 \over{ M_{r}^2}}\right)\left. \left(M^2_{AB} - 2 M_A^2 - 2 M_B^2 + {( M^2_A - M^2_B)^2 \over{ M_{r}^2}}\right) \right].$$ Next we return to the form factors $F_D$ and $F_r$, which attempt to model the underlying quark structure of the $D^0$ meson and the intermediate resonances. We use the Blatt-Weisskopf penetration factors shown in Table \[tbl:bwff\]. These have one free parameter, R, which is the “radius” of the meson, and are dependent on the momentum $P$ of the decay particles in the parent rest frame. In all cases, we normalize the form factor to have unit value at the nominal meson mass. The fits display very little sensitivity to the meson radii; good fits are obtained when these values vary between $0\ {\rm GeV}^{-1}$ and $10\ {\rm GeV}^{-1}$ for the $D^0$ and between $0\ {\rm GeV}^{-1}$ and $3\ {\rm GeV}^{-1}$ for the intermediate resonances. To be consistent with other experiments [@ref:e687] we have chosen the $D^0$ to have $R=5\ {\rm GeV}^{-1}$ and the intermediate resonances all to have $R=1.5\ {\rm GeV}^{-1}$ Spin Form Factor $ F_r $ ------ ---------------------------------------------------------------------------------------- 0 1 1 ${\sqrt{1 + R^2 p^2_r}\over{\sqrt{1+R^2 p^2_{AB}}}} $ 2 ${\sqrt{9 + 3 R^2 p^2_r + R^4 p^4_r }\over{\sqrt{9+ 3 R^2 p^2_{AB} + R^4 p^4_{AB}}}} $ : \[tbl:bwff\] Blatt-Weisskopf Penetration Form Factors. $p_r$ is the momentum of either daughter in the meson rest frame. $p_{AB}$ is the momentum of either daughter in the candidate rest frame (same as $p_r$ except the parent mass used is the two-track invariant mass of the candidate rather than the mass of the meson). $R$ is the meson radial parameter. Before continuing, we must specify our phase conventions for the intermediate resonances. We can explicitly see the importance of specifying the ordering of particles in the decay by examining Equation \[eq:svec\]. If we were to switch the labels A and B we would generate an overall minus sign causing the phase to change by 180$^o$. In an attempt to be consistent with previous results we have chosen the phases in the same way as the E687 collaboration [@ref:e687] since they are the only group to have explicitly published their choice of phases and matrix elements. Now that we know the form of the intermediate resonance amplitudes, and have chosen a phase convention that will allow us to compare our results with previous measurements, we can write down an expression for the overall matrix element of the decay. Guided by the results of previous measurements [@ref:e687; @ref:e691; @ref:mk31], we begin with only three vector resonances $\rho(770)^+$, $K^{*-}$ and $\bar{K}^{*0}$ [@ref:kstar] as well as a flat non-resonant ([*nr*]{}) component: $$\begin{aligned} \label{eq:decm} {\cal M}(D^0 \to K^-\pi^+\pi^0) & = & a_{nr} e^{i\phi_{nr}} \nonumber \\ && + a_{\rho} e^{i\phi_{\rho}} {\cal A}_1(\pi^+\pi^0K^-|\rho^+) \nonumber \\ && + a_{\bar{K}^{*0}} e^{i\phi_{\bar{K}^{*0}}} {\cal A}_1(K^-\pi^+\pi^0|\bar{K}^{*0}) \nonumber \\ && + a_{K^{*-}} e^{i\phi_{K^{*-}}} {\cal A}_1(K^-\pi^0\pi^+|K^{*-}),\end{aligned}$$ where the $a_i$ and $\phi_i$ are the amplitude and relative phase of the $i$’th component respectively. The overall normalization is arbitrary, and is chosen to be $$\label{eq:dpint} \int | {\cal M} |^2 d{\cal DP} = 1$$ where $d{\cal DP}$ indicates that the integral is performed over the Dalitz Plot. This is equivalent to saying that we are sensitive only to relative phases and amplitudes, which in turn means that we are free to fix one phase and one amplitude in Equation \[eq:decm\]. To minimize correlated errors on the phases and amplitudes we choose the largest mode, $K^-\rho$, to have a fixed zero phase and an amplitude of one. Since the choice of normalization, phase convention, and amplitude formalism may not always be identical for different experiments, fit fractions are reported instead of amplitudes to allow for more meaningful comparisons between results. The fit fraction is defined as the integral of a single component divided by the coherent sum of all components: $$\label{eq:ddpint} {\rm Fit\ Fraction} = {\int \left| a_r e^{i\phi_r} {\cal A}(A B C |r) \right |^2 d{\cal DP} \over { \int \left| \sum_j a_j e^{i\phi_j} {\cal A}(A B C |j) \right |^2 d{\cal DP}} }.$$ The sum of the fit fractions for all components of a fit will in general not be one because of interference. One must also consider the statistical errors on the fit fractions. We have chosen to use the full covariance matrix from the fits to determine the errors on fit fractions so that the assigned errors will properly include the correlated components of the errors on the amplitudes and phases. After each fit, the covariance matrix and final parameter values are used to generate 500 sample parameter sets. For each set, the fit fractions are calculated and recorded in histograms. Each histogram is fit with a single Gaussian to extract its width, which is used as a measure of the statistical error on the fit fraction. Experimental Details {#sec:dalcuts} ==================== The CLEO II detector is described elsewhere [@ref:cleoii]. This measurement uses the entire CLEO II dataset, which represents approximately $4.7$ fb$^{-1}$ of integrated $e^+e^-$ luminosity at $\sqrt{s}\sim 10.6$ GeV. The $D^0$’s used in this analysis are required to be produced by the decay chain $D^{*+} \to D^0 \pi^+_s$, which significantly reduces the combinatorial background. To reconstruct the $D^0$’s, we take pairs of oppositely charged tracks and assign the track with the same sign as the pion from the $D^{*+}$ decay to be the pion from the $D^0$ decay. This Cabibbo-favored correlation between the signs of the pions eliminates the need for other particle identification techniques in this analysis. For tracks to be used they must be well fitted, reconstruct to within 5 cm of the interaction point along the beam pipe and within 5 mm perpendicular to the beam pipe (corresponding to about 5 standard deviations in length and more than 10 standard deviations in the width of the beam spot). We fit pairs of tracks passing these requirements to a common vertex, which is the candidate decay position of the $D^0$ meson. Each such pair of charged tracks is combined with all $\piz$ candidates in an event. The $\piz$ candidates are found by combining all pairs of electromagnetic showers which are unmatched to charged tracks. To reduce the number of fake $\piz$’s from random shower combinations and to improve their resolution, we require that each shower have energy above 100 MeV and be in the central region of the CLEO II detector. Furthermore, the invariant mass of the two photon combination is restricted to be between 128 and 140 ([*i.e.*]{} within about one standard deviation of the $\piz$ mass). The two shower combination is kinematically fit to give the known $\piz$ mass. Once we have a vertex with a $\km$, a $\pip$ and a $\piz$ candidate, we combine the momenta of the three particles to find the $D^0$ momentum. With the decay location and the momentum known, the $D^0$ is projected back to the beam spot. In CLEO, the beam spot has a ribbon like shape with a width of 700 $\mu$m, a height of 20 $\mu$m, and a length of about 2 cm. We project the $D^0$ candidate back to the vertical position of the beam, since this dimension of the beam is most precisely known. The intersection of the $D^0$ projection and the beam position defines the production point of the $D^{*+}$. We refit the slow pion track to include the $D^{*+}$ production point as an additional constraint, providing a better measurement of its true momentum. The result of this is that the width of the mass difference peak, $\Delta M =M(D^{*+}) - M(D^0)$, is reduced from 590 keV to 490 keV, providing a 15% reduction in the number of background events in our final sample. We make a requirement that $\Delta M$ is between 144.9 and 145.9 . We also require that the normalized $D^*$ momentum, $X_{D^*} ={ P_{D^*}/\sqrt{E^2_{beam} - M^2_{D^*}}} $, is greater than 0.6, which significantly reduces the combinatorial background level and kinematically excludes the possibility that a $D^*$ candidate came from a decaying $B$ meson. After obtaining the candidate $D^0$’s as described above, we can plot the mass of the  candidates as shown in Figure \[fig:md0\], where the fit to the mass distribution is also shown. When examining the Dalitz Plot, we only use the events which have $1.85\, {\rm GeV/c}^2 < M_{D^0} < 1.88\, {\rm GeV/c}^2$ ([*i.e.*]{} within about one standard deviation of the known $D^0$ mass). We have chosen quite restrictive cuts on our kinematic variables (approximately one standard deviation on each) to minimize the effect of the background on our result. Since we are studying the shape of the distribution and are not trying to extract a branching ratio, the fact that this increases the systematic uncertainty of the overall efficiency somewhat is not an issue. Applying the above requirements produces 7,070 events in the Dalitz Plot. Figure \[fig:daldata\] shows the distribution of this sample as a scatter plot in the chosen mass squared variables $M_{K^-\pip}^2$ and $M_{\pip\piz}^2$. In order to reduce the smearing effects introduced by the detector, those combinations passing the above requirements are kinematically fit such that when combined, the $\km$, $\pip$ and $\piz$ reconstruct to give the correct $D^0$ mass. This kinematic fit has two effects. First, the uncertainty of the 4-momentum of the particles is reduced, giving a more precise measurement of the mass squared variables used to define an event’s position in the Dalitz Plot. Second, the decay position in these variables is guaranteed to respect the kinematic boundaries of the Dalitz Plot. Background ---------- Turning again to Figure \[fig:md0\], we can see that the signal region contains a small but non-zero number of background events. We use the fit shown to measure the fraction of events in this region which are “true signal” by integrating the signal function (a double bifurcated Gaussian) and the background function (a line) and comparing the two. The signal fraction and its associated statistical error, $0.967 \pm 0.011$, are used in the likelihood function minimized during the fitting procedure. Knowing only the amount of background is not enough if we want to correctly extract the amplitudes and phases of the signal component; the shape of the background in the Dalitz Plot is also important. There are several sidebands [@sideband] that could be chosen to study the shape of this background using data, and a Monte Carlo study (outlined below) is used to determine which one is best. As will become apparent in the section on systematic errors, the overall low level of the background means that the final result has very little sensitivity to this choice. To determine which sideband will best represent the Dalitz Plot shape of the background in the signal region, a signal-free sample of $e^+e^-\rightarrow q\overline{q}$ Monte Carlo simulated data is used (referred to below as the “vetoed” sample). These data are generated using a full GEANT based detector simulation [@ref:geant], and are processed by the same reconstruction code that is used for real data. This Monte Carlo sample represents the background we want to measure in data, and we use it as a reference in our study. The next step is to consider a number of possible sideband samples, and see which does the best job representing the Dalitz shape of the vetoed sample. Many sideband samples can be formed in the space defined by the three mass variables, $\Delta M$, $M_{D^0}$ and $M_{\pi^0}$. To choose the best one, we fit the distribution in the Dalitz Plot using an unbinned likelihood fit to a cubic polynomial in $M^2_{K^-\pi^+}$ and $M^2_{\pi^+\pi^0}$ as well as non-interfering squared amplitudes for the $\rho(770)$, $K^*(892)^-$ and $\ksz$. A $\chi^2$ is formed between each Monte Carlo sideband sample and the reference vetoed sample to give us a measure of their relative merits. Based on this, the sideband which seems to best represent the vetoed sample consists of those events which have $\Delta M < 0.1549$ , are in the $M_{\pi^0}$ signal region, and are in the off-peak regions of $M_{D^0}$: $1.76 < M_{D^0}{\rm (GeV/c^2)} < 1.80$ or $1.91 < M_{D^0}{\rm (GeV/c^2)} < 1.95$. This choice of sidebands along with the  candidates are shown in Figure \[fig:2dside\]. The assumption is now made that the sideband method which best represents the background in the Dalitz Plot when analyzing the Monte Carlo simulated data is also the best sideband method for use in real data. Those events from the actual data which are in the selected “best” sideband are fit with the cubic polynomial plus the three non-interfering resonances. The resulting best fit parameters are shown in Table \[tbl:backeff\]. We project the fit and the background points onto the three mass squared variables and show the results in Figure \[fig:back\], along with a two dimensional Manhattan plot of the fit result. We use this parameterization of the background shape in the fit to the distribution of events in the Dalitz Plot by including both the parameters and the covariance matrix in the final likelihood function (as described in Section \[sec:fittest\]). -------------- ------------------------------------ ------------ ----------------------------------- $B_0$ $ 1.0 $ (fixed) $E_0$ $ (22.1 \pm 1.8)\times 10^{-5}$ $B_x$ $ -1.188 \pm 0.018 $ $E_x$ $ (-6.89 \pm 2.9)\times 10^{-5}$ $B_y$ $ -0.742 \pm 0.044 $ $E_y$ $ (-27.1 \pm 3.7)\times 10^{-5}$ $B_{x^2}$ $0.483 \pm 0.015 $ $E_{x^2}$ $ (10.4 \pm 1.6)\times 10^{-5}$ $B_{xy} $ $ 0.874 \pm 0.032 $ $E_{xy} $ $ (38.2 \pm 3.2)\times 10^{-5}$ $B_{y^2}$ $ 0.122 \pm 0.047 $ $E_{y^2}$ $ (12.4 \pm 2.8)\times 10^{-5}$ $B_{x^3}$ $ -0.052 \pm 0.004 $ $E_{x^3}$ $ (-3.00 \pm 0.27)\times 10^{-5}$ $B_{x^2y}$ $ -0.162 \pm 0.010 $ $E_{x^2y}$ $ (-7.97 \pm 0.73)\times 10^{-5}$ $B_{xy^2}$ $ -0.202 \pm 0.014 $ $E_{xy^2}$ $ (-12.8 \pm 0.94)\times 10^{-5}$ $B_{y^3}$ $0.061 \pm 0.016 $ $E_{y^3}$ $ (-0.53 \pm 0.73)\times 10^{-5}$ $B_{\ksz}$ $(1.65 \pm 1.70) \times 10^{-5} $ $B_{\rho^+}$ $(3.69 \pm 0.58) \times 10^{-4} $ $B_{K^{*-}}$ $(7.69 \pm 1.95) \times 10^{-5} $ -------------- ------------------------------------ ------------ ----------------------------------- : \[tbl:backeff\]Background and efficiency best fit parameters. The fitting functions are described in Section \[sec:fittest\]. Efficiency ---------- Next, we determine the efficiency for detecting signal events as a function of position in the two dimensional Dalitz Plot. After generating 4.2 million signal Monte Carlo events with a flat distribution in phase space ([*i.e.*]{}, uniform across the Dalitz Plot), these events are analyzed to find the number of observed events as a function of $M^2_{K^-\pi^+}$ and $M^2_{\pi^+\pi^0}$. The events observed are binned into regions with $50~({\rm MeV/c}^2)^2$ on a side, and we divide the number of events observed in each bin by the number generated to give a measure of the efficiency for that bin. Due to the finite number of Monte Carlo events observed in each bin, each individual efficiency measurement has about a 10% statistical error. Since we expect (and observe) that the efficiency is a slowly varying function across the Dalitz Plot, we fit the efficiency measurements with a cubic polynomial in $M^2_{K^-\pi^+}$ and $M^2_{\pi^+\pi^0}$ and use the resulting function to parameterize the efficiency. As a check that the efficiency function obtained using phase space Monte Carlo is reasonable, we repeat the procedure described above with another 2.4 million signal Monte Carlo events generated with the Dalitz distribution found by E691 [@ref:e691]. Again the efficiency in each bin is calculated and fit. Since the resulting fit agrees well with the efficiency calculated from the phase space distribution of points, we combine the two Monte Carlo samples and calculate the efficiency using the full 6.6 million events. The fit parameters for this combined fit are shown in Table \[tbl:backeff\]. Figure \[fig:eff\] shows the raw efficiency for each bin as well as the fit and the projections onto each of the three mass-squared axes. Fitting Procedure {#sec:fittest} ================= Having a parameterization for both the background and efficiency as well as knowing the fraction of events in the signal region which are in fact background, we can fit the data in the Dalitz Plot to extract the amplitudes and phases of any contributing intermediate resonances. To do this we use an unbinned maximum likelihood fit which minimizes the function ${\cal F}$ given by $${\cal F} = {\big[\sum_{events}-2\ln {\cal L}\big] + \chi^2_{\rm penalty} }$$ where $$\label{eq:likelihood} {\cal L} = \left({F{{\cal E}(M^2_{K^-\pi^+}, M^2_{\pi^+\pi^0}) \left| {\cal M} \right|^2 }\over{ {\cal N}_{signal}}} + (1 - F) {{{\cal B}(M^2_{K^-\pi^+}, M^2_{\pi^+\pi^0})}\over{{\cal N}_{background}}}\right )$$ $$\begin{aligned} \label{eq:chisquared} \chi^2_{\rm penalty} & = \left( {F - F_o \over{\sigma_F}} \right)^2 + \sum_{ij} (B_i-B_{io}) V_{ij} (B_j-B_{jo}) \nonumber \\ &+ E_{sys}\sum_{ij} (E_i-E_{io}) W_{ij} (E_j-E_{jo}) \end{aligned}$$ and $${\cal N}_{signal} = \int {{\cal E}(M^2_{K^-\pi^+}, M^2_{\pi^+\pi^0}) \left| {\cal M} \right|^2} d{\cal DP}$$ $${\cal N}_{background} = \int {\cal B}(M^2_{K^-\pi^+}, M^2_{\pi^+\pi^0}) d{\cal DP}.$$ The signal fraction $F_o$ and its error $\sigma_F$ (0.967 and 0.011 respectively) are determined from the fit to the  mass spectrum shown in Figure \[fig:md0\], and the the parameters $B_{jo}$ and $E_{jo}$ describe the nominal background and efficiency shapes (see Table \[tbl:backeff\]) via the cubic polynomial shapes $$\begin{aligned} {\cal B} & = & B_0 + B_x M^2_{K^-\pi^+} + B_y M^2_{\pi^+\pi^0} + B_{x^2} (M^2_{K^-\pi^+})^2 + B_{xy} M^2_{K^-\pi^+} M^2_{\pi^+\pi^0} + B_{y^2} (M^2_{\pi^+\pi^0})^2 + \nonumber \\ & & B_{x^3} (M^2_{K^-\pi^+})^3 + B_{x^2y} (M^2_{K^-\pi^+})^2 M^2_{\pi^+\pi^0} + B_{xy^2} M^2_{K^-\pi^+} (M^2_{\pi^+\pi^0})^2 + B_{y^3} (M^2_{\pi^+\pi^0})^3 +\nonumber \\ & & B_{\ksz} | {\cal A}_1(\km \pip \piz | \ksz )|^2 + \beta_{\rho} |{\cal A}_1(\pip \piz \km | \rho^+)|^2 + \beta_{K^{*-}} |{\cal A}_1(\km \piz \pip | K^{*-})|^2 \end{aligned}$$ and $$\begin{aligned} {\cal E} & = & E_0 + E_x M^2_{K^-\pi^+} + E_y M^2_{\pi^+\pi^0} + E_{x^2} (M^2_{K^-\pi^+})^2 + E_{xy} M^2_{K^-\pi^+} M^2_{\pi^+\pi^0} + E_{y^2} (M^2_{\pi^+\pi^0})^2 + \nonumber \\ & & E_{x^3} (M^2_{K^-\pi^+})^3 + E_{x^2y} (M^2_{K^-\pi^+})^2 M^2_{\pi^+\pi^0} + E_{xy^2} M^2_{K^-\pi^+} (M^2_{\pi^+\pi^0})^2 + E_{y^3} (M^2_{\pi^+\pi^0})^3 \end{aligned}$$. In expressing this likelihood function we have made the explicit assumption that background events and signal events are distinct, allowing us to factor the likelihood into two components which do not interfere. The $\chi^2_{\rm penalty}$ terms represent the information from the fits used to determine the signal fraction, the background parameterization, and the efficiency parameterization. $V_{ij}$ and $W_{ij}$ are the covariance matrices from the background and efficiency fits respectively. The last term is used only when evaluating the systematic errors due to the efficiency parameterization, hence $E_{sys}$ is set to zero during “normal” fitting. In addition to the likelihood, we need a measure to assess how well any given fit represents the data. A confidence level can be calculated directly from the likelihood function by utilizing the best fit parameters. This idea was described by ARGUS [@ref:goodfit] and is a direct application of the Central Limit Theorem from statistics [@ref:clt]. Assuming the candidates are truly distributed according to the likelihood function which gives the best fit, the average value is $$\mu = \frac{1}{N}\sum_{i=1}^N ( -2 \ln {\cal L}) \approx \int {\cal L} ( -2 \ln {\cal L}) d{\cal DP}$$ where $N$ is the number of candidates. The variance is given by $$\sigma^2_\mu = \frac{1}{N}\sum_{i=1}^N ( -2 \ln {\cal L} - \mu)^2 \approx \int {\cal L} ( -2 \ln {\cal L})^2 d{\cal DP} - \mu^2.$$ Because we have a large number of candidates distributed according to this function, the Central Limit Theorem tells us that the mean should follow a normal distribution. The sum of minus log likelihoods, which is the value minimized in the fit, has a mean of $N\mu$ and follows a normal distribution with a variance of $N\sigma^2_\mu$. Thus, the minimal value will come from a normal distribution with mean $$<-2 \sum \ln {\cal L}> = N \int {\cal L} ( -2 \ln {\cal L}) d{\cal DP} - n$$ and standard deviation $$\sigma_{<-2 \sum \ln {\cal L}>} = \sqrt{N \int {\cal L} ( -2 \ln {\cal L})^2 d{\cal DP} - N\mu^2 }$$ where $n$ is the number of parameters extracted from the fit. The confidence level for the fit is then just the area of a Gaussian with the above mean and width which lies above the value obtained in our fit. It is worth pointing out that this value only gives a measurement of the goodness of fit assuming the fit function correctly describes the true distribution. Having a second measure of the goodness of the fit would be extremely valuable, and an obvious choice is the $\chi^2$. This requires the data to be binned, and furthermore that there are enough events in each bin that Gaussian statistics can be assumed. As we saw in Figure \[fig:daldata\], the density of candidates in the  Dalitz Plot varies significantly as a function of position, hence to form a sensible $\chi^2$ measure we will need to have bins of varying size. To systematically choose these bins, we start by placing a grid of small regions, $50~({\rm MeV/c}^2)^2$ on a side, over the Dalitz Plot. Next, adjacent regions are combined into bins until each contains approximately 30 candidates. After completing this procedure, our Dalitz Plot is divided into 228 bins of varying size, and a $\chi^2$ variable for the multinomial distribution [@ref:lnchi2; @ref:eadie] can be calculated as $$\chi^2 = -2 \sum_{i=1}^{228} n_i \ln \left(\frac{p_i}{n_i}\right)$$ where $n_i$ is the number of events observed in bin $i$, and $p_i$ is the number predicted from the fit. For a large number of events this formulation of the $\chi^2$ becomes equivalent to the usual one [@ref:dof]. One can naively calculate the number of degrees of freedom for the fit as the number of bins (r) minus the number of fit parameters (k) minus one, as would be correct for a binned maximum likelihood fit. However, since we are minimizing the unbinned likelihood function, our “$\chi^2$” variable does not asymptotically follow a $\chi^2$ distribution [@ref:dof], but it is bounded by a $\chi^2$ variable with $(r-1)$ degrees of freedom and a $\chi^2$ variable with $(r-k-1)$ degrees of freedom. Because it is bounded by two $\chi^2$ variables, it should be a useful statistic for comparing the relative goodness of fits. In what follows, we use both the $\chi^2$ and the confidence level described above as our “goodness of fit” measures to determine which of the many possible sets of intermediate resonances are preferred. Before analyzing the  data, we performed many checks of both the fitting and fit evaluation procedures. One of these was a double-blind study in which several Monte Carlo samples containing  decays generated with “secret” mixtures of intermediate resonances were analyzed. In each case, our fitting and evaluation procedure identified the correct set of resonances, and recovered their amplitudes and phases within statistical errors. The resulting amplitudes and phases for one of the fits is shown in Table \[tbl:ajwfit\]. ----------------- ----------- ----------------- ----------------- ----------------- Resonance Amplitude Phase (degrees) Amplitude Phase (degrees) $\ksz$ 1.0 45 $1.03 \pm 0.02$ $47 \pm 1 $ $\rho^+$ 1.0 0 $1.0$ (fixed) $ 0$ (fixed) $K^{*-}$ 1.0 -115 $1.03 \pm 0.02$ $-113 \pm 2 $ $K^*_0(1430)^-$ 0.5 -115 $0.54 \pm 0.05$ $-107 \pm 6 $ Non resonant 1.0 -90 $1.08 \pm 0.05$ $273 \pm 3 $ ----------------- ----------- ----------------- ----------------- ----------------- : \[tbl:ajwfit\] A comparison between input Monte Carlo parameters and the results from a subsequent fit to the Dalitz Plot using the techniques described in Section \[sec:fittest\]. Note that the input amplitudes and phases are completely fictitious. Fitting the Data {#sec:datafits} ================ Armed with the tools described in the previous section, we are ready to fit the data distribution shown in Figure \[fig:daldata\]. Previous experiments have observed three intermediate resonances in  decays: $\rho^+$, $\ksz$ and $K^{*-}$, hence we begin by considering only these in addition to a non-resonant component. The resulting fit parameters are given in Table \[tbl:fit\]. -- ---------------------------- -- -- ---------------------- -- $a_{nr}$ $ 1.70 \pm 0.07 $ $a_{\rho^+}$ $ 1.00 $ (fixed) $a_{K^{*-}}$ $ 0.378 \pm 0.008 $ $a_{\overline{K}^{*0}}$ $0.422 \pm 0.009 $ $\phi_{nr}$ $ 59.7^o \pm 2.0^o $ $\phi_{\rho^+}$ $ 0^o $ (fixed) $\phi_{K^{*-}}$ $ 166.7 \pm 2.0^o$ $\phi_{\overline{K}^{*0}}$ $ -7.8^o \pm 2.2^o$ $-2 \ln {\cal L}$ 7070 Conf. Level. 0.0% $\chi^2$ 650 -- ---------------------------- -- -- ---------------------- -- : \[tbl:fit\] Results of the best fit to the data with only $\rho^+$, $\ksz$, $K^{*-}$, and non-resonant components included. Figure \[fig:s3fit\] shows the projections of both the fit and the data onto the three mass squared variables, as well as a two dimensional Manhattan plot of the final fit function. Even a quick glance suggests that the data is not well represented by this function, and the large value of $\chi^2$ as well as the zero confidence level confirm this observation. These parameters are useful for comparison with previous experiments, however, which reported observation of these three resonances with much less statistics. We show the comparison in Tables \[tbl:resultsff\] and \[tbl:resultsphase\] and see good agreement. Unfortunately, we can only compare the results for the phases to E687 since the other experiments do not give their choice of particle ordering or potential complex constants in their choice for ${\cal A}(ABC|r)$. Although the phases match for the three resonant components, the non-resonant phase seems to be off by 180$^o$. This observation is consistent with comments that E687 had an unreported negative sign in their vector amplitude [@ref:jew]. Decay Mode CLEO II (3 Resonance) E687 Mark III E691 -------------------------- ----------------------- -------------------- ------------------ ------------------- $K^-\rho^+$ $0.834 \pm 0.007$ $0.765 \pm 0.041 $ $0.81 \pm 0.03 $ $0.647 \pm 0.039$ $K^{*-}\pi^+$ $0.129 \pm 0.006$ $0.148 \pm 0.028 $ $0.12 \pm 0.02 $ $0.084 \pm 0.011$ $\overline{K}^{*0}\pi^0$ $0.157 \pm 0.007 $ $0.165 \pm 0.031 $ $0.13 \pm 0.02$ $0.142\pm 0.018$ Non resonant $0.074\pm 0.006 $ $0.101 \pm 0.033 $ $0.09 \pm 0.02 $ $0.036$ : \[tbl:resultsff\]A comparison of the fit fractions obtained with our “three resonance” fit and those reported by previous experiments. The errors shown are statistical only. Note that although the data is not well fit by this model, the results are consistent with those reported by previous experiments. Decay Mode [CLEO II (3 Resonance)]{} [E687]{} [Mark III]{} [E691 (Rotated)]{} -------------------------- --------------------------- --------------- -------------- -------------------- $K^-\rho^+$ $0$ (fixed) $0$ (fixed) $0$ (fixed) $0 \pm 7 $ $K^{*-}\pi^+$ $166.7 \pm 2.0$ $162 \pm 10$ $154\pm 11$ $-152 \pm 9 $ $\overline{K}^{*0}\pi^0$ $-7.8 \pm 2.2$ $-2\pm 12$ $7\pm 7$ $127 \pm 9 $ Non-resonant $59.7 \pm 2.0$ $-122 \pm 10$ $52\pm 9$ $-40 $ (fixed) : \[tbl:resultsphase\]A comparison of the phases (in degrees) obtained with our “three resonance” fit and those reported by previous experiments. The errors shown are statistical only. In the “Rotated” column we have shifted the reported phases such that the $\rho$ has a phase of $0^o$ in order to ease comparison with the other results. Note that although the data is not well fit by this model, the results are consistent with those reported by previous experiments. Since we have at least a factor of ten more statistics for this analysis, one should not be surprised that more resonances are needed to accurately represent the data. The question now becomes how best to determine which additional resonances to include. We have tried two procedures: a) adding all possible resonances and subsequently removing those which do not contribute significantly, and b) adding new resonances one at a time and choosing the best additional one at each iteration, stopping when no additional resonances contribute significantly. Both of these methods lead us to the same results, hence only the first one is described below. We begin by fitting the Dalitz Plot with all known resonances which can possibly contribute to this decay, as listed in Table \[tbl:resonances\] [@ref:pdg98]. The results of this fit are shown in the “All Resonances” column of Table \[tbl:dalsubtract\], and in Figure \[fig:s0fit\]. There are five resonances which have fit fractions that are less than one standard deviation away from zero: $\overline{K}^*_3(1780)^0$, $K^*_3(1780)^-$, $\overline{K}^*(1410)^0$, $K^*(1410)^-$ and $\overline{K}^*(1680)^0$. Two other resonances, $\overline{K}^*_2(1430)^0$ and $K^*_2(1430)^-$, have fit fractions close to zero. When the first five resonances are removed and the fit repeated, the fit fractions of these last two resonances do become consistent with zero, and hence are also removed. ----------------------------- ------- -------------------- -------------------- Resonance $J^P$ Mass (GeV/c$^2$) Width (GeV/c$^2$) $\rho(770)^+$ $1^-$ $0.770\pm 0.001$ $0.1507\pm 0.0011$ $\overline{K}^*(892)^0$ $1^-$ $0.8961\pm 0.0003$ $0.0505\pm 0.0006$ $K^*(892)^- $ $1^-$ $0.8915\pm 0.0003$ $0.050\pm 0.001$ $K^*(1410)^-$ $1^-$ $1.414\pm 0.015$ $0.232\pm 0.021$ $\overline{K}^*(1410)^0 $ $1^-$ $1.414\pm 0.015$ $0.232\pm 0.021$ $K^*_0(1430)^-$ $0^+$ $1.412\pm 0.006$ $0.294\pm 0.023$ $\overline{K}^*_0(1430)^0 $ $0^+$ $1.412\pm 0.006$ $0.294\pm 0.023$ $K^*_2(1430)^-$ $2^+$ $1.425\pm 0.002$ $0.098\pm 0.003$ $\overline{K}^*_2(1430)^0 $ $2^+$ $1.432\pm 0.001$ $0.109\pm 0.005$ $\rho(1450)^+$ $1^-$ $1.465\pm 0.025$ $0.310\pm 0.060$ $\rho(1700)^+$ $1^-$ $1.700\pm 0.020$ $0.240\pm 0.060$ $K^*(1680)^-$ $1^-$ $1.717\pm 0.027$ $0.322\pm 0.110$ $\overline{K}^*(1680)^0 $ $1^-$ $1.717\pm 0.027$ $0.322\pm 0.110$ $\overline{K}^*_3(1780)^0$ $3^-$ $1.776\pm 0.007$ $0.159\pm 0.021$ $K^*_3(1780)^- $ $3^-$ $1.776\pm 0.007$ $0.159\pm 0.021$ ----------------------------- ------- -------------------- -------------------- : \[tbl:resonances\]The resonances considered when fitting the  Dalitz Plot, along with the masses and widths used when evaluating the matrix element. --------------------------- ----------------- ------------------ ----------------- ------------------ Component Phase (degrees) Fit Fraction (%) Phase (degrees) Fit Fraction (%) $\overline{K}_3(1780)^0$ $263 \pm 16$ $0.3 \pm 7.5$ $K_3(1780)^- $ $86 \pm 12$ $0.5 \pm 2.9$ $\overline{K}^*(1680)^0 $ $175 \pm 25$ $0.4 \pm 0.5$ $K^*(1680)^-$ $67 \pm 19$ $1.0 \pm 0.5$ $103 \pm 8$ $1.3 \pm 0.3$ $\rho(1700)^+$ $149 \pm 8$ $75 \pm 18$ $171 \pm 6$ $5.7 \pm 0.8$ $\rho(1450)^+$ $-45 \pm 10$ $34 \pm 11$ Non Res. $30 \pm 5$ $9.1 \pm 1.3$ $31 \pm 4$ $7.5 \pm 0.9$ $\overline{K}^*(1410)^0 $ $279 \pm 52$ $0.1 \pm 0.2$ $\overline{K}_2(1430)^0 $ $148 \pm 13$ $0.3 \pm 0.14$ $\overline{K}_0(1430)^0 $ $168 \pm 5$ $8.0 \pm 1.3$ $166 \pm 5$ $4.1 \pm 0.6$ $K^*(1410)^-$ $152 \pm 31$ $0.2 \pm 0.2$ $K_2(1430)^-$ $339 \pm 21$ $0.12 \pm 0.08$ $K_0(1430)^-$ $42 \pm 6$ $5.6 \pm 1.1$ $55.5 \pm 5.8$ $3.3 \pm 0.6$ $K^*(892)^- $ $159 \pm 2.6$ $12.8 \pm 1.8$ $163 \pm 2.3$ $16.1 \pm 0.7 $ $\rho(770)^+$ $0$(fixed) $74 \pm 4$ $0$ (fixed) $78.7 \pm 2.0 $ $\overline{K}^*(892)^0 $ $2.8 \pm 3.2$ $11.3 \pm 1.5$ $-0.2 \pm 3.3 $ $ 12.7 \pm 0.9 $ $\chi^2$ $-2 \ln {\cal L}$ C.L. --------------------------- ----------------- ------------------ ----------------- ------------------ : \[tbl:dalsubtract\]The parameters from the fits to the  Dalitz Plot with all resonances included (“All Resonances” column), and after we remove resonances consistent with zero fit fraction (“Final Resonances” column). The $\rho(1450)^+$ and $\rho(1700)^+$ contributions are discussed in the text. Notice that in the “All Resonances” column of Table \[tbl:dalsubtract\] there are two heavy $\rho$ mesons ($\rho(1450)^+$ and $\rho(1700)^+$) which have surprisingly large fit fractions. Both have masses which place their peak outside the Dalitz Plot, but both are wide enough ($310\pm 60$ MeV/c$^2$ and $240\pm 60$ MeV/c$^2$ respectively [@ref:pdg98]) that their tails extend well into the region of interest, making it difficult to distinguish between them. Since the fitted phases of these $\rho$’s are very close to being $180^o$ apart, their large fit fractions are assumed to be an artifact of the fit’s inability to tell them apart. Supporting this is the additional fact that when both resonances are combined, their net contribution to the fit fraction is much smaller, $(9 \pm 2)\%$. Since the inclusion of both $\rho$ resonances is probably a misrepresentation of the contents of the Dalitz Plot, only one of these is included in all following fits. We choose the one which gives the best $\chi^2$ and goodness of fit, the $\rho(1700)^+$, and consider the $\rho(1450)^+$ only when evaluating our systematic errors. After the seven resonances consistent with zero fit fraction are removed along with the $\rho(1450)^+$ (as discussed above), seven resonances remain in addition to the non-resonant component: $\rho(770)^+$, $K^*(892)^- $, $\overline{K}^*(892)^0$, $\rho(1700)^+$, $\overline{K}_0(1430)^0$, $K_0(1430)^-$, and $K^*(1680)^-$. Figure \[fig:finalfit\] shows the result of fitting the Dalitz Plot with these components. The fit fractions and phases are shown in the “Final Resonances” column of Table \[tbl:dalsubtract\], and the full set of parameters extracted from this fit is shown in Table \[tbl:finalamps\]. -- ----------------------------------- -- -- ---------------------- -- -- -------------- -- -- ---------------------------------- -- $a_{nr}$ $ 1.75 \pm 0.12 $ $a_{\rho^+}$ $ 1.00 $ (fixed) $B_0$ $1.0 \pm 0.0$ $a_{K^{*-}}$ $ 0.44 \pm 0.01 $ $B_x$ $-1.206 \pm 0.001$ $a_{\overline{K}^{*0}}$ $0.39 \pm 0.01 $ $B_y$ $-0.74 \pm 0.23$ $a_{K_0(1430)^-}$ $0.77 \pm 0.08 $ $B_{x^2}$ $0.468 \pm 0.001$ $a_{\overline{K}_0(1430)^{0}}$ $0.85 \pm 0.06 $ $B_{xy}$ $0.842 \pm 0.008$ $a_{\rho(1700)^+}$ $2.50 \pm 0.19 $ $B_{y^2}$ $0.168 \pm 0.001$ $a_{K^*(1680)^-}$ $2.50 \pm 0.3 $ $B_{x^3}$ $-0.055 \pm 0.001$ $\phi_{NR}$ $ 31.2^o \pm 4.3^o $ $B_{x^2y}$ $-0.16 \pm 0.06$ $\phi_{\rho^+}$ $ 0^o $ (fixed) $B_{xy^2}$ $-0.188 \pm 0.001$ $\phi_{K^{*-}}$ $ 163 \pm 2.3^o$ $B_{y^3}$ $0.077 \pm 0.001$ $\phi_{\overline{K}^{*0}}$ $ -0.2^o \pm 3.3^o$ $B_{\ksz}$ $(3.4 \pm 0.1) \times 10^{-5}$ $\phi_{K_0(1430)^-}$ $55.5^o \pm 5.8^o $ $B_{\rho}$ $(4.27 \pm 0.05) \times 10^{-4}$ $\phi_{\overline{K}_0(1430)^{0}}$ $ 166^o\pm 5^o $ $B_{K^{*-}}$ $(9.64 \pm 0.01)\times 10^{-5}$ $\phi_{\rho(1700)^+}$ $171^o \pm 6^o $ $\phi_{K^*(1680)^-}$ $103^o \pm 8^o $ Signal Fraction $0.968 \pm 0.007$ $-2 \ln {\cal L}$ 6570 Conf. Level. 94.9% $\chi^2$ 257 -- ----------------------------------- -- -- ---------------------- -- -- -------------- -- -- ---------------------------------- -- : \[tbl:finalamps\] Summary of our best fit to the data with the final set of eight components included. As a curious side note, if a single vector ($K^-\piz$) resonance with a floating mass and width is added in place of the four new “standard resonances” discussed above, a good fit can obtained. Unfortunately, while this new resonance has a reasonable mass of 1.406 , it prefers a negative width of $\Gamma = -0.25$  which does not seem to represent the underlying dynamics we are trying to measure. It is possible that the desire for this resonance is an indication of an inaccuracy of the formalism used for the resonance shapes, or an indication that multiple resonances are needed (as we have assumed). We note that the optimum set of seven resonances used above, all of which have positive widths, provide a fit which has a lower $\chi^2$ than the inclusion of this single unphysical state. Other experiments have reported evidence of a light scalar ($\pi^+\pi^-$) resonance, the $\sigma$, in $D^+\rightarrow \pi^-\pi^+\pi^+$ decays [@ref:sigma791], as well as evidence of a scalar ($K^-\pi^+$) resonance, the $\kappa$, in $D^+\rightarrow K^-\pi^+\pi^+$ decays  [@ref:kappa791]. Since a significant fit fraction for $D^+\rightarrow \kappa\pi^+$ has been reported by these authors, we have searched for a scalar $\kappa\rightarrow K^-\pi^+$ resonance in the $D^0\rightarrow \kappa\pi^0$ channel, fixing the mass and width of the $\kappa$ to the values reported in [@ref:kappa791], (0.815   and 0.560   respectively). We find a fit fraction consistent with zero ($0.4\pm 0.3$%), and see no improvement in the confidence level of the fit with this additional resonance included. We have also allowed the mass and width of the $\kappa$ to float in the fit, and again see no significant contribution. Lastly, since this analysis considers only $D^0$ mesons produced from a decaying $D^{*+}$ in the mode $D^{*+} \to D^0 \pi^+_s$, we have the ability to divide our data into separate $D^0$ and $\overline{D^0}$ samples by simply considering the sign of the $\pi^\pm_s$ from the $D^{*\pm}$ decay. The Dalitz Plots of these samples can then be fitted separately and compared in a search for CP violation. We have fitted these samples with the same set of resonances described above, and the results are shown in Table \[tbl:acp\]. When performing these fits, the efficiency functions were found separately for the $D^0$ and $\overline{D^0}$ samples, however a common background shape was assumed. Forming a simple $\chi^2$ between the two sets of fit parameters we find $\chi^2_{cp} = 16.2$ for 14 degrees of freedom. We calculate an integrated CP asymmetry across the Dalitz Plot by evaluating $${\cal A}_{cp} = \int { |{\cal M}_{D^0}|^2 - |{\cal M}_{\overline{D^0}}|^2 \over{ |{\cal M}_{D^0}|^2 + |{\cal M}_{\overline{D^0}}|^2 }} d{\cal DP}$$ and obtain ${\cal A}_{cp} = -0.031 \pm 0.086$, consistent with zero. Note that this number is not dependent on the number of $D^0$ and $\overline{D^0}$ candidates in our data sample, but rather on the shapes of these distributions in the respective Dalitz Plots. --------------------------- ------------------ ------------------ ------------------- ------------------- Component Amplitude Phase (degrees) Amplitude Phase (degrees) $\rho(770)^+$ $1.0\pm 0.0 $ $0^o$(fixed) $1.0\pm 0.0 $ $0^o$(fixed) $K^*(892)^- $ $0.433\pm 0.034$ $168.9\pm 3.3$ $0.442\pm 0.015 $ $157.8\pm 3.4 $ $\overline{K}^*(892)^0 $ $0.391\pm 0.026$ $ 1.3\pm 3.7$ $0.410\pm 0.022 $ $ -4.9\pm 4.9 $ $\rho(1700)^+$ $2.590\pm 0.538$ $ 175.0\pm 7.5$ $2.720\pm 0.272 $ $ 163.9\pm 7.6 $ $\overline{K}_0(1430)^0 $ $0.989\pm 0.124$ $ 173.9\pm 8.2$ $0.774\pm 0.089 $ $ 159.3\pm 8.1 $ $K_0(1430)^-$ $0.701\pm 0.211$ $ 59.0\pm 10.0$ $0.917\pm 0.117 $ $ 55.0\pm 7.1 $ $K^*(1680)^-$ $2.567\pm 1.540$ $ 107.4\pm 69.2$ $2.060\pm 0.423 $ $ 106.4\pm 13.5 $ Non Res. $1.840\pm 0.146$ $ 39.9\pm 7.9$ $1.780\pm 0.160 $ $ 21.3\pm 6.0 $ $\chi^2$ $-2 \ln {\cal L}$ C.L.(%) --------------------------- ------------------ ------------------ ------------------- ------------------- : \[tbl:acp\] Fit results when the $D^0$ and $\overline{D^0}$ samples are considered separately. Systematic Uncertainties {#sec:systematics} ======================== After finding the best fit to the data, we must attempt to estimate the systematic uncertainties on the fit parameters. There are several possible sources: the background, the efficiency, biases due to experimental resolution, and the modeling of the decay. These contributions are discussed in order, and the final systematic errors are shown in Table \[tbl:sys\], where experimental and model dependent sources of systematic uncertainty are summarized in detail. The background was modeled by the choice of sideband sample that gave the best parameterization of the vetoed data sample from Monte Carlo. Furthermore, the background parameters were allowed to float in our fits to the data, constrained only by the covariance matrix from the fit that determined the nominal background function. To search for any systematic effects due the background parameterization, the fitting procedure was repeated for a number of different sideband choices. Because the background fraction our sample is a mere 3.3%, or about 230 out of the 7070 events in the Dalitz plot, these changes have a minimal effect on the fit parameters. We use the RMS spread of these results as our estimate of the systematic error due to our choice of background parameterization. These values are shown in the “Bkgnd” column of Table \[tbl:sys\]. To obtain the efficiency across the Dalitz Plot, signal Monte Carlo events were fit to a cubic polynomial. As a check, we have allowed this polynomial to float in our fit (as was done with the background) subject to a $\chi^2$ constraint from its covariance matrix in the likelihood function ([*i.e.*]{} setting $E_{sys}=1$ in Equation \[eq:chisquared\]). If the efficiency is not well modeled by a cubic polynomial, there could still be an effect that this check would fail to find. To search for this we tried a local smoothing algorithm rather than the global polynomial fit. The efficiency was smoothed by fitting either nine or twenty-five neighbors around each bin with a local plane. Each bin’s efficiency value was then replaced by the height of this plane interpolated to its center. As a final check, we used the raw measurements of the efficiency in each bin of our fits. We conclude that the effects of parameterization of the efficiency function over the Dalitz Plot is not a significant source of concern as most of the fit parameters vary by less than their one sigma error bars in the above checks. Since we make no requirement on the momentum of the charged tracks, one might worry that low momentum tracks may be poorly measured and could affect the Dalitz Plot distribution in a way not well modeled by our Monte Carlo. To search for such a momentum dependent effect, we fit the data with the additional requirement that all tracks have a momentum above 350 MeV/c. The cuts used to obtain our signal determine the structure of our efficiency. To assess how well the Monte Carlo reproduces the data distributions, we varied the cuts used in the analysis and fit the resulting Dalitz distributions. Each cut was relaxed in turn. The cuts on the masses, $M_{D^0}$, $\Delta M$ and $M_{\pi^0}$, were opened to double the size of the signal region. The minimum energy on the photons was relaxed to 90 MeV, and the requirement on $X_{D^*}$ was loosened to 0.5. The RMS variation in the fit parameters from each of the tests described above was taken as our estimate of the systematic uncertainty on the efficiency. These values are shown in the “Eff” column of Table \[tbl:sys\]. A final contribution to the experimental systematic error, presented in column “Resol” of Table \[tbl:sys\], is due to the finite resolution of the Dalitz Plot variables. As a check, we have included the effects of smearing when fitting the data. This was done by measuring the resolution as a function of position across the Dalitz plot and numerically convoluting this with the amplitude at each point when performing the fit. Again, the parameters vary by less than the statistical errors on the nominal best fit, and their variation from the nominal values is taken as an estimate of the systematic uncertainty. The above three systematic error categories (background, efficiency and resolution) are summarized in Table \[tbl:sys\]. They are combined in quadrature to give the total experimental uncertainty, which is shown in the “Total” column under “Experiment”. ------------------------------------------ -------- ---------- ------- ------ ------- ------- -------------------- ------ -------------------- Parameter Value Stat Err Bkgnd Eff Resol Total Shape Add Total $\overline{K}^*(892)^0$ Fit Frac ($\%$) 12.66 0.91 0.17 0.40 0.24 0.50 1.42 0.38 1.47 $\overline{K}^*(892)^0$ Phase (deg.) -0.20 3.28 1.06 1.62 1.04 2.20 6.99 0.67 7.02 $\rho(770)^+$ Fit Frac ($\%$) 78.76 1.93 0.52 1.10 0.53 1.33 4.40 1.33 4.60 $K^*(892)^-$ Fit Frac ($\%$) 16.11 0.69 0.47 0.53 0.18 0.73 $^{+2.58}_{-0.48}$ 0.59 $^{+2.65}_{-0.76}$ $K^*(892)^-$ Phase (deg.) 163.40 2.32 0.94 2.62 1.30 3.08 4.20 1.09 4.34 $K_0(1430)^-$ Fit Frac ($\%$) 3.32 0.64 0.13 0.60 0.40 0.73 1.16 0.40 1.23 $K_0(1430)^-$ Phase (deg.) 55.52 5.76 1.20 2.76 1.31 3.28 $^{-12.8}_{+3.1}$ 2.85 $^{+4.2}_{-13.1}$ $\overline{K}_0(1430)^0$ Fit Frac ($\%$) 4.05 0.61 0.15 0.66 0.24 0.72 $^{+3.04}_{-0.24}$ 0.39 $^{+3.06}_{-0.46}$ $\overline{K}_0(1430)^0$ Phase (deg.) 165.90 5.23 2.39 3.83 0.70 4.57 11.4 3.20 11.8 $\rho(1700)^+$ Fit Frac ($\%$) 5.65 0.76 0.20 0.43 0.50 0.68 5.71 0.59 5.74 $\rho(1700)^+$ Phase (deg.) 170.50 6.07 1.90 3.90 1.50 4.59 $^{-54.7}_{+3.3}$ 5.17 $^{+6.1}_{-54.9}$ $K^*(1680)^-$ Fit Frac ($\%$) 1.33 0.33 0.07 0.32 0.11 0.34 0.17 0.32 0.36 $K^*(1680)^-$ Phase (deg.) 103.20 7.90 3.71 5.91 2.00 7.26 9.21 9.89 13.5 Non Res Fit Frac ($\%$) 7.50 0.95 0.35 0.42 0.05 0.55 $^{+5.54}_{-0.79}$ 0.41 $^{+5.56}_{-0.89}$ Non Res Phase (deg.) 31.20 4.28 1.28 5.08 1.70 5.51 $^{-14.4}_{+3.5}$ 1.19 $^{+3.7}_{-14.4}$ ------------------------------------------ -------- ---------- ------- ------ ------- ------- -------------------- ------ -------------------- : \[tbl:sys\] A summary of the systematic errors on each fit parameter. The first two columns show the results from the best fit and the associated statistical errors. The next four (three) columns summarize the systematic uncertainties due to experimental (modeling) sources respectively. Details are provided in the text. Modeling systematic errors can arise from our choice of resonances and the uncertainty in their shapes. In Section \[sec:theory\] we motivated our choice of parameterization of the intermediate resonances; however, other groups have used different functional forms in their fits [@ref:e687; @ref:e691]. We varied these shapes to study any systematic effects resulting from our choice. We examine three variations: (i) the Zemach formalism [@ref:zemach1] which enforces the transversality of the mesons by using $M_{AB}^2$ rather than $M_{r}^2$ in the denominator of the spin sums, (ii) a simple cosine distribution for the spin sum and (iii) a non-relativistic rather than relativistic Breit-Wigner in the propagator. Further consideration was also given to the radial parameters used in the form factors, which were varied between $0\ {\rm GeV}^{-1}$ and $3\ {\rm GeV}^{-1}$ for the intermediate resonances and between $0\ {\rm GeV}^{-1}$ and $10\ {\rm GeV}^{-1}$ for the $D^0$ meson. The masses and widths of the intermediate resonances were allowed to vary within the known errors [@ref:pdg98]. The non-resonant contribution was described in our fits by a constant term, but as a check we also modeled it by a linear function or a shape given by the spin structure without the Breit-Wigner amplitudes [@ref:nr1]. The above tests were used to explore the systematic dependance of the fit parameters on the way the physics was modeled. The variations using a simple cosine distribution in place of the spin sum and using a spin structured rather than constant non-resonant component resulted in fits with significantly worsened $\chi^2$ (368 and 322 respectively), and are not considered when assigning a systematic error as the data suggests these forms could not be correct. We take the largest of the remaining variations as the systematic error due to our choice of modeling shapes, and the results are shown in the “Shape” column of Table \[tbl:sys\]. The final systematic check is on our choice of which resonances to include. For example, there is only a slight preference for the $\rho(1700)^+$ over the $\rho(1450)^+$ based on the goodness of fit. To account for this uncertainty, both fits were performed and the variation of the parameters were noted. Fits were also performed which included additional resonances from Table \[tbl:resonances\]. The RMS variation in the fit parameters from the above checks is presented in the “Add” column of Table \[tbl:sys\]. We also considered the effects of removing resonances, and two of these studies deserve further comment. The first is the removal of the $K^*(1680)^-$. We considered this because the final fit fraction for this resonance is a rather small $1.3\pm 0.3$%. When the $K^*(1680)^-$ is removed the $\chi^2$ increases from 257 to 316 indicating that this resonance should remain. The parameters for this fit are shown in the “Removed $K^*(1680)^-$” column of Table \[tbl:minus\]. For comparison, when the other “new” resonances, $\overline{K}_0(1430)^0$, $K_0(1430)^-$, and $\rho(1700)^+$, are removed, the $\chi^2$ increases to 379, 348, and 381 respectively. The second case which deserves special attention is the removal of the non-resonant component. Some theoretical models, such as chiral perturbation theory [@ref:nonr], prefer a small non-resonant component, suggesting it proceeds only by the coherent sum of two body decays. When this test is performed on our data, the resulting $\chi^2$ jumps to 411, suggesting that a non resonant component is indeed present. The parameters for this fit are shown in the “Removed Non Resonant” column of Table \[tbl:minus\]. Since removal of any of the fit components causes a significant increase in the $\chi^2$ of the fit, these variations were not included in the modeling systematic error. To obtain the final model dependent systematic error we add the “Shape” and “Add” columns of Table \[tbl:sys\] in quadrature to obtain the result shown in the “Total” column under “Model”. --------------------------- ----------------- ------------------ ----------------- ------------------ Component Phase (degrees) Fit Fraction (%) Phase (degrees) Fit Fraction (%) $\rho(770)^+$ $0$ (fixed) $80.8\pm 8.5$ $0$(fixed) $77.8\pm 1.8$ $K^*(892)^- $ $157\pm 6.7$ $13.8\pm 1.0$ $161\pm 2.2$ $18.2\pm 0.7$ $\overline{K}^*(892)^0 $ $-4.7\pm 5.7$ $14.5\pm 1.3$ $-1.5\pm 2.8$ $10.7\pm 0.8$ $\rho(1700)^+$ $161\pm 20$ $6.7\pm 0.8$ $161\pm 5$ $5.4\pm 0.8$ $\overline{K}_0(1430)^0 $ $164\pm 9$ $4.4\pm 0.5$ $194\pm 9$ $1.0\pm 0.3$ $K_0(1430)^-$ $47.8\pm 3.6$ $4.5\pm 0.7$ $11\pm 4$ $5.2\pm 0.7$ $K^*(1680)^-$ $0$ $0.0$ $90\pm 5$ $1.9\pm 0.5$ Non Res. $37\pm 6$ $7.7\pm 2.6$ $0$ $0.0$ $\chi^2$ $-2 \ln {\cal L}$ C.L.(%) --------------------------- ----------------- ------------------ ----------------- ------------------ : \[tbl:minus\] Fit results after removal of the either the $K^*(1680)^-$ resonance or the non-resonant component. See Section \[sec:systematics\] for discussion. Summary of Results ================== We have fit the distribution of data in the  Dalitz Plot obtained with the CLEO II experiment to a coherent sum of seven intermediate resonances plus a non-resonant component. All resonances are either scalar or vector; no significant tensor contribution was found. The non-resonant contribution is significant, and cannot be removed without seriously compromising the quality of the fit. We see no evidence of a scalar $\kappa\rightarrow K^-\pi^+$ resonance in the mass range recently reported by other groups. The final fit fraction and phase for each component is given in Table \[tbl:conclusion\]. These fit fractions, multiplied by the world average  branching ratio of $(13.9 \pm 0.9)$% [@ref:pdg2000], yield the partial branching fractions shown in Table \[tbl:fractions\]. The error on the world average branching ratio is incorporated by adding it in quadrature with the experimental systematic errors on the fit fractions to give the experimental systematic error on the partial branching fractions. Note that due to interference the fit fractions do not add to unity, and consequently the partial branching fractions do not sum to the world average. By separately fitting the  and  Dalitz Plots, we have calculated the integrated CP asymmetry across the Dalitz Plot to be ${\cal A}_{cp} = -0.031 \pm 0.086$. Mode Fit Fraction Phase (degrees) --------------------------------- ---------------------------------------------- ---------------------------------------- $\rho(770)^+ K^-$ $0.788\pm 0.019\pm 0.013 \pm0.046$ $0.0$ (fixed) $K^*(892)^- \pip$ $0.161\pm 0.007\pm 0.007 ^{+0.026}_{-0.008}$ $163 \pm 2.3 \pm 3.1 \pm 4.3$ $\overline{K}^*(892)^0 \piz$ $0.127\pm 0.009\pm 0.005 \pm0.015$ $-0.2 \pm 3.3 \pm 2.2 \pm 7.0$ $\rho(1700)^+ K^-$ $0.057\pm 0.008\pm 0.007\pm 0.006$ $171 \pm 6 \pm {5} ^{+6.1}_{-55}$ $\overline{K}^*_0(1430)^0 \piz$ $0.041\pm 0.006\pm 0.007^{+0.031}_{-0.005}$ $166 \pm 5 \pm {4.6} \pm 12$ $K^*_0(1430)^- \pip$ $0.033\pm 0.006\pm 0.007\pm 0.012$ $55.5 \pm 5.8 \pm {3.3} ^{+4.2}_{-13}$ $K^*(1680)^- \pip$ $0.013\pm 0.003\pm 0.003\pm 0.003$ $103 \pm 8 \pm {7} \pm 14$ Non Resonant $0.075\pm 0.009\pm 0.006 ^{+0.056}_{-0.009}$ $31 \pm 4 \pm {5.5} ^{+14}_{-3.7}$ : \[tbl:conclusion\] Final fit results. The errors shown are statistical, experimental systematic, and modeling systematic respectively, as discussed in Section \[sec:systematics\] and summarized in Table \[tbl:sys\]. Mode Partial Branching Fraction ---------------------------------------------------------------------------------------- -------------------------------------------------- $B(D^0\to\rho(770)^+ K^-)\times B(\rho(770)^+\to\pi^+\pi^0)$ $0.109 \pm 0.003 \pm 0.007 \pm 0.006$ $B(D^0\to K^*(892)^- \pip)\times B(K^*(892)^-\to K^-\pi^0)$ $0.022 \pm 0.001 \pm 0.002 ^{+0.004}_{-0.001}$ $B(D^0\to\overline{K}^*(892)^0 \piz)\times B(\overline{K}^*(892)^0\to K^-\pi^+)$ $0.018 \pm 0.001 \pm 0.001 \pm 0.002$ $B(D^0\to\rho(1700)^+ K^-)\times B(\rho(1700)^+\to\pi^+\pi^0)$ $0.008 \pm 0.001 \pm 0.001 \pm 0.001$ $B(D^0\to\overline{K}^*_0(1430)^0 \piz)\times B(\overline{K}^*_0(1430)^0\to K^-\pi^+)$ $0.006 \pm 0.001 \pm 0.001 ^{+0.004}_{-0.001}$ $B(D^0\to K^*_0(1430)^- \pip)\times B(K^*_0(1430)^-\to K^-\pi^0)$ $0.005 \pm 0.001 \pm 0.001 \pm 0.002$ $B(D^0\to K^*(1680)^- \pip)\times B(K^*(1680)^-\to K^-\pi^0)$ $0.0018\pm 0.0004 \pm 0.0004 \pm 0.0004$ $B(D^0\to K^-\pi^+\pi^0)$ Non Resonant $0.010 \pm 0.001 \pm 0.001 ^{+0.008}_{-0.001}$ : \[tbl:fractions\] Partial branching fractions calculated by combining our fit fractions with the previously measured  branching ratio as described in the text. The errors shown are statistical, experimental systematic, and modeling systematic respectively. We gratefully acknowledge the effort of the CESR staff in providing us with excellent luminosity and running conditions. M. Selen thanks the PFF program of the NSF and the Research Corporation, A.H. Mahmood thanks the Texas Advanced Research Program, F. Blanc thanks the Swiss National Science Foundation, and E. von Toerne thanks the Alexander von Humboldt Stiftung for support. This work was supported by the National Science Foundation, the U.S. Department of Energy, and the Natural Sciences and Engineering Research Council of Canada. [99]{} E. Golowich and A. Petrov, Phys.Lett. [**B 427**]{} (1998) 172. M Bauer, B. Stech and M. Wirbel, Z. Phys. C [**34**]{}, 103 (1987). P. Bedaque, A. Das and V.S. Mathur, Phys. Rev. D [**49**]{}, 269 (1994). L.-L. Chau and H.-Y. Cheng, Phys. Rev. D [**36**]{}, 137 (1987). K. Terasaki, Int. J. Mod. Phys. A [**10**]{}, 3207 (1995). F. Buccella, M. 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--- address: - | UPV/EHU\ Dpto. de Matemáticas\ Apto. 644, 48080 Bilbao, Spain. - | Department of Mathematics\ University of Chicago\ Chicago, Il. 60637\ USA. - | Department of Mathematics\ University of California\ Santa Barbara, CA 93106\ USA. - | UPV/EHU\ Dpto. de Matemáticas\ Apto. 644, 48080 Bilbao, Spain. author: - 'L. Escauriaza' - 'C. E. Kenig' - 'G. Ponce' - 'L. Vega' title: Uniqueness Properties of Solutions to Schrödinger Equations --- [^1] Introduction {#S: Introduction} ============ To place the subject of this paper in perspective, we start out with a brief discussion of unique continuation. Consider solutions to $$\label{aa1} \Delta u(x)=\sum_{j=1}^n\frac{\partial^2u}{\partial x_j^2}(x)=0,$$ (harmonic functions) in the unit ball $\,\{x\in {\mathbb R}^n\,:\,|x|<1\}$. When $n=2$, these functions are real parts of holomorphic functions, and so, if they vanish of infinite order at $x=0$, they must vanish identically. We call this the strong unique continuation property (s.u.c.p.). The same result holds for $n>2$, since harmonic functions are still real analytic in $\,\{x\in {\mathbb R}^n\,:\,|x|<1\}$. In fact, it is well-known that if $\,P(x,D)$ is a linear elliptic differential operator with real analytic coefficients, and $\,P(x,D)u=0$ in a open set $\,\Omega\subset {\mathbb R}^n$, then $u$ is real analytic in $\,\Omega$. Hence, the (s.u.c.p.) also holds for such solutions. Through the work of Hadamard [@Had] on the uniqueness of the Cauchy problem (which is closely related to the strong unique continuation property discussed earlier) it became clear (for applications in nonlinear problems) that it would be desirable to establish the strong unique continuation property for operators whose coefficients are not necessarily real analytic, or even $\, C^{\infty}$. The first results in this direction were found in the pioneering work of Carleman [@Car] (when $n=2$) and Müller [@Mu] (when $n>2$), who proved the (s.u.c.p) for $$P(x,D)=\Delta+V(x),\;\;\;\;\;\text{with}\;\;\;\;\;\;V\in L^{\infty}_{loc}({\mathbb R}^n).$$ In order to establish his result, Carleman introduced a method (the method of Carleman estimates") which has permeated the subject ever since. In this context, an example of a Carleman estimate is : .05in *For $\,f\in C^{\infty}_0(\{x\in {\mathbb R}^n\,:\,|x|<1\}-\{0\})$, $\,\alpha>0$ and $$w(r)=r\,\exp (\,\int_0^r\frac{e^{-s}-1}{s} ds),$$ one has $$\label{1aa} \alpha^3\,\int\,w^{-1-2\alpha}(|x|) f^2(x) dx\leq c\,\int w^{2-2\alpha}(|x|)\,|\Delta f(x)|^2 dx,$$ with $\,c\,$ independent of $\,\alpha$* .05in For a proof of this estimate, see [@EsVe], [@BoKe]. The (s.u.c.p.) of Carleman-Müller follows easily from (see [@Ke2] for instance). In the late 1950’s and 1960’s there was a great deal of activity on the subject of (s.u.c.p.) and the closely related uniqueness in the Cauchy problem, some highlights being [@AKS] and [@Cal] respectively, both of which use the method of Carleman estimates. These results and methods have had a multitude of applications to many areas of analysis, including to non-linear problems. (For a recent example, see [@KeMe] for an application to energy critical non-linear wave equations). In connection with the Carleman-Müller (s.u.c.p.) a natural question is : How fast is a solution $u$ allowed to vanish, before it must vanish identically? By considering $n=2$, $u(x_1,x_2)=\Re (x_1+ix_2)^N$, we see that to make sense of the question, a normalization is required, for instance $$\sup _{|x|<3/4} |u(x)|\geq 1,\;\;\;\;\;\;\;\|u\|_{L^{\infty}(|x|<1)}<\infty.$$ We refer to questions of this type as quantitative unique continuation". It is also of interest to consider unique continuation type questions around the point at infinity. For instance, a conjecture of E. M. Landis [@KoLa] was : if $$\Delta u +V u =0,\;\;\;\;x\in{\mathbb R}^n,\;\;\;\text{with}\;\;\;\|V\|_{\infty}\leq 1 ,\;\;\;\|u\|_{\infty}<\infty,$$ and for some $\,\epsilon>0$ one has $$|u(x)|\leq c_{\epsilon}\,e^{-c_{\epsilon}|x|^{1+\epsilon}},$$ then $\,u\equiv 0$. For the case of complex valued potentials $V(x)$, this conjecture was disproved by Meshkov [@Me] who constructed $V,\,u,\,u\not \equiv 0$ with $$|u(x)|\leq c\, e^{-c|x|^{4/3}},\;\;\;\;n\geq 2.$$ Meshkov also showed that if $$|u(x)|\leq c_{\epsilon}\, e^{-c_{\epsilon}|x|^{4/3+\epsilon}},\;\;\;\;\text{for some}\;\;\;\;\epsilon>0,$$ then $\,u\equiv 0$. It turns out that a quantitative" formulation of this can also be proved, as it was done in [@BoKe], and this was crucial for the resolution in [@BoKe] of a long-standing problem in disordered media, namely Anderson localization near the bottom of the spectrum, for the continuous Anderson-Bernoulli model in $\,{\mathbb R}^n,\,n\geq 1$. Next, we turn to versions of unique continuation for evolution equations. We start with parabolic equations and consider solutions of $$\partial_t u-\Delta u=W\cdot \nabla u+ Vu,\;\;\;\;\;\;\;\text{with}\;\;\;\;\;\;\|W\|_{\infty}+\|V\|_{\infty} <\infty,$$ (or equivalently $|\partial_tu-\Delta u|\leq M(|\nabla u|+|u|)$). Using a parabolic analog of the Carleman estimate described earlier, one can show that if $$|\partial_tu-\Delta u|\leq M(|\nabla u|+|u|),\;\;\;\;\;\;(x,t)\in \{x\in{\mathbb R}^n:|x|<4R\}\times [t_0,t_1],\;\;\;R>0,$$ with $\,|u(x)|\leq A$ and $$u\equiv 0,\;\;\;(x,t)\in \{x\in{\mathbb R}^n:R<|x|<4R\}\times [t_0,t_1],$$ then $$u\equiv 0,\;\;\;(x,t)\in \{x\in{\mathbb R}^n:|x|<R\}\times [t_0,t_1].$$ We call this type of result unique continuation through spatial boundaries“, (see [@EsVe], [@Ve] and references therein for this type of result and strengthenings of it). This result is closely related to the elliptic” (s.u.c.p.) discussed before. On the other hand, for parabolic equations, there is also a backward uniqueness" principle, which is very useful in applications to control theory (see [@lm60] for an early result in this direction) : Consider solutions to $$|\partial_tu-\Delta u|\leq M(|\nabla u|+|u|),\;\;\;\;\;(x,t)\in{\mathbb R}^n\times (0,1],$$ with $\,\|u\|_{\infty}\leq A$. Then, if $\,u(\cdot,1)\equiv 0$, we must have $\,u\equiv 0$. This result is also proved through Carleman estimates (see [@lm60]). Recently, a strengthening of this result has been obtained in [@EsSS], where one considers solutions only defined in $\,R^n_{+}\times (0,1]$, $\,R^n_{+}=\{(x_1,..,x_n)\in{\mathbb R}^n\,:\,x_1>0\}$, without any assumptions on $\,u\,$ at $\,x_1=0$, and still obtains the backward uniqueness" result. This strengthening had an important application to non-linear equations, allowing the authors of [@EsSS] to establish a long-standing conjecture of J. Leray on regularity and uniqueness of solutions to the Navier-Stokes equations (see also [@Se] for a recent extension). Finally, we turn to dispersive equations. Typical examples of these are the $k$-generalized KdV equation $$\label{kdv1} \partial_t u +\partial_x^3 u+ u^k\partial_xu=0,\;\;\;\;\;\; (x,t)\in {\mathbb R}\times {\mathbb R},\;\,\;k\in {\mathbb Z}^+,$$ and the non-linear Schrödinger equation $$\label{ae1} \partial_t u =i( \Delta u \pm |u|^{p-1}u),\;\;\;\;\;\; (x,t)\in \mathbb{R}^n\times {\mathbb R},\;\;\,p>1.$$ These equations model phenomena of wave propagation and have been extensively studied in the last 30 years or so. For these equations,unique continuation through spatial boundaries “ also holds, as it was shown by Saut-Scheurer [@SaSc] for the KdV-type equations and by Izakov [@Iza] for Shrödinger type equations. (All of these results were established trough Carleman estimates). These equations however are time reversible (no preferred time direction) and so backward uniqueness” is immediate, unlike in parabolic problems. Once more in connection with control theory, this time for dispersive equations, Zhang [@BZ] showed, for solutions of $$\label{zhang} \partial_t u =i( \partial_x^2 u \pm |u|^{2}u),\;\;\;\;\;\; (x,t)\in \mathbb{R}\times[0,1],$$ that if $\,u(x,t)=0$ for $(x,t)\in (-\infty,a)\times\{0, 1\}$ (or $(x,t)\in (a,\infty)\times\{0, 1\}$) for some $a\in{\mathbb R}$, the $\,u\equiv 0$. Zhang’s proof was based on the inverse scattering method which uses that this is a completely integrable model, and did not apply to other non-linearities or dimensions. This type of result was extended to the $k$-generalized KdV and the general non-linear Schrödinger equation in in all dimensions (where inverse scattering is no longer available) using suitable Carleman estimates (see [@KPV02], [@IK04], [@IK06], and references therein). For recent surveys of the results presented so far, see [@Ke1], [@Ke2]. Returning to backward uniqueness“ for parabolic equations, in analogy with Landis’ elliptic” conjecture mentioned earlier, Landis-Oleinik [@LaOl] conjectured that in the backward uniqueness" result one can replace the hypothesis $\,u(\cdot,1)\equiv 0$ with the weaker one $$|u(x,1)|\leq c_{\epsilon}\,e^{- c_{\epsilon}|x|^{2+\epsilon}},\;\;\;\text{for some }\;\;\;\epsilon>0.$$ This is indeed true and was established in [@EKPV06a] and [@Ng]. Similarly, one can conjecture (as it was done in [@EKPV08b]) that for Schrödinger equations, if $$|u(x,0)|+ |u(x,1)|\leq c_{\epsilon}\,e^{- c_{\epsilon}|x|^{2+\epsilon}},\;\;\;\text{for some }\;\;\;\epsilon>0,$$ then $\,u\equiv 0$. This was established in [@EKPV06a]. In analogy with the improvement of backward uniqueness" in [@EsSS], one can show that it suffices to deal with solutions in $\,{\mathbb R}^n_{+}\times(0,1]$ (for parabolic problems) and require $$|u(x,1)|\leq c_{\epsilon}\,e^{- c_{\epsilon}x_1^{2+\epsilon}},\;\;\;x_1>0,\;\;\;\;\text{for some }\;\;\;\epsilon>0,$$ to conclude that $\,u\equiv 0$ ([@Ng]), and that for the Schrödinger equations it suffices to have $\,u\,$ a solution in $\,{\mathbb R}^n_{+}\times[0,1]$, with $$|u(x,0)|+ |u(x,1)|\leq c_{\epsilon}\,e^{- c_{\epsilon}x_1^{2+\epsilon}},\;\;\;x_1>0,\;\;\;\;\text{for some }\;\;\;\epsilon>0,$$ to conclude that $\,u\equiv 0$, as we will prove in section 5 of this paper. In [@EKPV06] it was pointed out for the first time (see also [@Cha]) that both the results in [@EKPV06a] and in [@EKPV06], in the case of the free heat equation $$\partial_tu=\Delta u,$$ and the free Schrödinger equation $$\partial_tu=i \Delta u,$$ respectively, are in fact a corollary of the more precise Hardy uncertainty principle for the Fourier transform, which says : .05in *If $f(x)=O(e^{-|x|^2/\beta^2})$, $\widehat f(\xi)=O(e^{-4|\xi|^2/\alpha^2})$ and $1/\alpha\beta>1/4$, then $f\equiv 0$, and if $1/\alpha\beta=1/4$, $f(x)=ce^{-|x|^2/\beta^2}$* as will be discussed below. .05in Thus, in a series of papers ([@EKPV06]-[@EKPV10], [@CEKPV]) we took up the task of finding the sharp version of the Hardy uncertainty principle, in the context of evolution equations. The results obtained have already yielded new results on non-linear equations. For instance in [@EKPV08m] and [@EKPV10] we have found applications to the decay of concentration profiles of possible self-similar type blow-up solutions of non-linear Schrödnger equations and to the decay of possible solitary wave type solutions of non-linear Schrödinger equations. In the rest of this work we shall review some of our recent results concerning unique continuation properties of solutions of Schrödinger equations of the form $$\label{e1} \partial_t u =i( \Delta u + F(x,t,u,\bar u)),\;\;\;\;\;\; (x,t)\in \mathbb{R}^n\times {\mathbb R}.$$ We shall be mainly interested in the case where $$\label{F1a} F(x,t,u,\bar u)=V(x,t) u(x,t)$$ is describing the evolution of the Schrödinger flow with a time dependent potential $V(x,t)$, and in the semi-linear case $$\label{F1b} F(x,t,u,\bar u)= F(u,\bar u),$$ with $ F: {\mathbb C}\times {\mathbb C}\to {\mathbb C}$, $F(0,0)=\partial_uF(0,0)=\partial_{\bar u}F(0,0)=0$. Let us consider a familiar dispersive model, the $k$-generalized Korteweg-de Vries equation and recall a theorem established in [@EKPV07] : \[theorem1\] There exists $c_0>0$ such that for any pair $$u_1,\,u_2\in C([0,1]:H^4(R)\cap L^2(|x|^2dx))$$ of solutions of such that if $$\label{3:2} u_1(\cdot,0)-u_2(\cdot,0),\,\;\, u_1(\cdot,1)-u_2(\cdot,1)\in L^2(e^{c_0x_{+}^{3/2}}dx),$$ then $u_1\equiv u_2$. Above we have used the notation: $ x_{+}=max\{x;\,0\}$. Notice that taking $u_2\equiv 0\,$ Theorem \[theorem1\] gives a restriction on the possible decay of a non-trivial solution of at two different times. The power $3/2$ in the exponent in reflects the asymptotic behavior of the Airy function. More precisely, the solution of the initial value problem (IVP) $$\begin{aligned} \begin{cases} \partial_t v + \partial_x^3 v=0,\\ v(x,0)=v_0(x), \end{cases} \end{aligned}$$ is given by the group $\{U(t)\,:\,t\in R\}$ $$U(t)v_0(x)=\frac{1}{\root{3}\of{3t}}\,Ai\left(\frac{\cdot}{\root {3}\of{3t}}\right)\ast v_0(x),$$ where $$Ai(x)=c\,\int_{-\infty}^{\infty}\,e^{ ix\xi+i \xi^3 }\,d\xi,$$ is the Airy function which satisfies the estimate $$|Ai(x)|\leq c (1+x_{-})^{-1/4}\,e^{-c x_{+}^{3/2}}.$$ It was also shown in [@EKPV07] that Theorem 1 is optimal : \[theorem2\] There exists $ \,u_0\in S({\mathbb R}),\;u_0\not \equiv 0$ and $\Delta T>0$ such that the IVP associated to the k-gKdV equation with data $u_0$ has solution $$u\in C([0,\Delta T] : \mathbb S({\mathbb R})),$$ satisfying $$|u(x,t)|\leq \tilde d \,e^{-x^{3/2}/3}, \;\;\;\;\;\;\;\;x>1,\;\;\;\,t\in [0,\Delta T],$$ for some constant $\tilde d>0$. In the case of the free Schrödinger group $\{e^{it\Delta}\,:\,t\in{\mathbb R}\}$ $$e^{it\Delta}u_0(x)=(e^{-i|\xi|^2t} \widehat u_0)^\lor(x)=\frac{e^{i|\cdot|^2/4t}}{(4\pi i t)^{n/2}}*u_0(x),$$ the fundamental solution does not decay. However, one has the identity $$\label{formula1} \begin{aligned} &u(x,t)= e^{it\Delta}u_0(x)= \int_{{\mathbb R}^n} \frac{e^{i|x-y|^2/4t}}{(4\pi i t)^{n/2}}\, u_0(y)\,dy\\ \\ &=\frac{e^{i|x|^2/4t}}{(4\pi i t)^{n/2}} \int_{{\mathbb R}^n}e^{-2ix\cdot y/4t} e^{i|y|^2/4t} u_0(y)\,dy\\ \\ &= \frac{e^{i|x|^2/4t}}{(2 i t)^{n/2}}\; \widehat{\;(e^{i|\cdot|^2/4t}u_0)\,}\left(\frac{x}{2 t}\right), \end{aligned}$$ where $$\widehat f(\xi)=(2\pi)^{-n/2} \int_{{\mathbb R}^n} e^{-i\xi\cdot x} f(x)dx.$$ Hence, $$c_t e^{-i|x|^2/4t} \,u(x,t) = \widehat{(e^{i|\cdot|^2/4t}u_0)}\left(\frac{x}{2 t}\right),\,\,\,\,\,\,\,c_t=(2 i t)^{n/2},$$ which tells us that $e^{-i|x|^2/4t} \,u(x,t)$ is a multiple of the rescaled Fourier transform of $\;e^{i|y|^2/4t}u_0(y)$. Thus, as we pointed out earlier, the behavior of the solution of the free Schrödinger equation is closely related to uncertainty principles for the Fourier transform. We shall study these uncertainty principles and their relation with the uniqueness properties of the solution of the Schrödinger equation . In the early $1930$’s N. Wiener’s remark (see [@Hardy], [@In], and [@Mo]): .1in “a pair of transforms $f$ and $g$ ($\widehat f$) cannot both be very small”, .1in motivated the works of G. H. Hardy [@Hardy], G. W. Morgan [@Mo], and A. E. Ingham [@In] which will be considered in detail in this note. However, before that we shall return to a review of some previous results concerning uniqueness properties of solutions of the Schrödinger equation which we mentioned earlier and which were not motivated by the formula . For solutions $u(x,t)$ of the $1$-D cubic Schrödinger equation B. Y. Zhang [@BZ] showed : .05in *If $u(x,t)=0$ for $(x,t)\in (-\infty, a)\times \{0,1\}\,\, ($or $(x,t)\in (a,\infty)\times \{0,1\})\,$ for some $\,a\in {\mathbb R}$, then $u\equiv 0$.* .03in As it was mentioned before, his proof is based on the inverse scattering method, which uses the fact that the equation in is a completely integrable model. .05in In [@KPV02] it was proved under general assumptions on $F$ in that : .03in *If $u_1,\,u_2\in C([0,1]:H^s({\mathbb R}^n))$, with $\,s>\max \{n/2; \,2\}\, $ are solutions of the equation with $F$ as in such that $$u_1(x,t)=u_2(x,t),\;\;\;(x,t)\in \Gamma^c_{x_0}\times \{0,1\},$$ where $ \Gamma^c_{x_0}$ denotes the complement of a cone $\Gamma_{x_0}$ with vertex $x_0\in {\mathbb R}^n$ and opening $<180^0$, then $u_1\equiv u_2$.* (For further results in this direction see [@KPV02], [@IK04], [@IK06], and references therein). .03in A key step in the proof in [@KPV02] was the following uniform exponential decay estimate: \[ultimo\] There exists $\epsilon_0>0$ such that if $$\label{hyp2} \mathbb V:\mathbb R^n\times [0,1]\to\mathbb C,\;\;\;\;\text{with}\;\;\;\; \|\mathbb V\|_{L^1_tL^{\infty}_x}\leq \epsilon_0,$$ and $u\in C([0,1]:L^2(\mathbb R^n))$ is a strong solution of the IVP $$\begin{cases} \begin{aligned} \label{eq1} &\partial_tu=i(\Delta +\mathbb V(x,t))u+\mathbb G(x,t),\\ &u(x,0)=u_0(x), \end{aligned} \end{cases}$$ with $$\label{hyp3} u_0,\,u_1\equiv u(\,\cdot\,,1)\in L^2(e^{2\lambda\cdot x}dx),\;\mathbb G\in L^1([0,1]:L^2(e^{2\lambda\cdot x}dx)),$$ for some $\lambda\in\mathbb R^n$, then there exists $c_n$ independent of $\lambda$ such that $$\begin{aligned} \label{uno} &\sup_{0\leq t\leq 1}\| e^{\lambda\cdot x} u(\,\cdot\,,t)\|_{L^2(\mathbb {\mathbb R}^n)} \\ &\leq c_n \Big(\|e^{\lambda\cdot x} u_0\|_{L^2(\mathbb {\mathbb R}^n)} + \|e^{\lambda\cdot x} u_1\|_{L^2(\mathbb {\mathbb R}^n)} +\int_0^1 \|e^{\lambda\cdot x}\, \mathbb G(\cdot, t)\|_{L^2(\mathbb {\mathbb R}^n)} dt\Big). \end{aligned}$$ Notice that in the above result one assumes the existence of a reference $L^2$-solution $u$ of the equation and then under the hypotheses and shows that the exponential decay in the time interval $[0,1]$ is preserved. The estimate can be combined with the subordination formula $$\label{est1} e^{\gamma |x|^p/p}\simeq \int_{{\mathbb R}^n} \,e^{\gamma^{1/p}\lambda\cdot x-|\lambda|^q/q}\, |\lambda|^{n(q-2)/2}\,d\lambda,\,\,\,\forall\, x\in {\mathbb R}^n\,\,\,\text{and}\,\,\,p>1,$$ to get that for any $\alpha>0$ and $ a>1$ $$\label{dos} \begin{aligned} \sup_{0\leq t\leq 1}\| e^{\alpha|x|^a} u(\,\cdot\,,t)\|_{L^2(\mathbb {\mathbb R}^n)} &\\ \leq c_n \Big(\|e^{\alpha|x|^a} u_0\|_{L^2(\mathbb {\mathbb R}^n)} +& \|e^{\alpha|x|^a} u_1\|_{L^2(\mathbb {\mathbb R}^n)} +\int_0^1 \|e^{\alpha|x|^a}\, \mathbb G(\cdot, t)\|_{L^2(\mathbb {\mathbb R}^n)} dt\Big). \end{aligned}$$ Under appropriate assumptions on the potential $V(x,t)$ in one writes $$V(x,t)u= \chi_{R} V(x,t)u + (1-\chi_{R}) V(x,t)u = \mathbb V(x,t)u + \mathbb G(x,t),$$ with $\chi_R\in C^{\infty}_0,\,$ $\chi_R(x)=1,\,|x|<R$, supported in $|x|<2R$, and applies the estimate by fixing $\,R\,$ sufficiently large. Also under appropriate hypothesis on $F$ and $u$ a similar argument can be used for the semi-linear equation in . .03in The estimate gives a control on the decay of the solution in the whole time interval in terms of that at the end points and that of the external force”. As we shall see below a key idea will be to get improvements of this estimate based on logarithmically convex versions of it. We recall that if one considers the equation with initial data $u_0\in \mathbb S({\mathbb R}^n)$ and a smooth potential $V(x,t)$ in or smooth nonlinearity $F$ in , it follows that the corresponding solution satisfies that $u\in C([-T,T] :\mathbb S({\mathbb R}^n))$. This can be proved using the commutative property of the operators $$L=\partial_t-i\Delta,\;\;\;\;\;\;\text{and}\;\;\;\;\;\;\Gamma_j=x_j+2t\partial_{x_j},\,\,\,j=1,..,n,$$ see [@HKT1]-[@HKT2]. From the proof of this fact one also has that the persistence property of the solution $u=u(x,t)$ (i.e. if the data $u_0\in X$, a function space, then the corresponding solution $u(\cdot)$ describes a continuous curve in $X$, $u\in C([-T,T]:X)$, $\,T>0$) with data $u_0\in L^2(|x|^m)$ can only hold if $u_0\in H^s({\mathbb R}^n)$ with $s\geq 2m$. Roughly speaking, for exponential weights one has a more involved argument where the time direction plays a role. Considering the IVP for the one dimensional free Schrödinger equation $$\label{*} \begin{cases} \begin{aligned} &\partial_t u=i\partial_x^2u,\;\;\;\;\;\;\;\;\;\;\;\,x,\,t \in {\mathbb R},\\ &u(x,0)=u_0(x)\in L^2({\mathbb R}), \end{aligned} \end{cases}$$ and assuming that $ e^{\beta x}u_0 \in L^2({\mathbb R}),\,\,\beta>0$, then one formally has that $$v(x,t)=e^{\beta x}u(x,t)$$ satisfies the equation $$\partial_t v=i(\partial_x-\beta)^2v.$$ Thus, $$v(x,\pm 1)=e^{\beta x}u(x,\pm 1)\in L^2({\mathbb R})\,\;\;\;\text{if}\;\;\;\,e^{\pm 2\beta \xi}\, \widehat{e^{\beta x}u_0}\in L^2({\mathbb R}).$$ However, if we knew that $ e^{\beta x}u(x,1),\;\;e^{\beta x}u(x,-1) \in L^2({\mathbb R})$ integrating forward in time the positive frequencies of $e^{\beta x}u(x,t)$ and backward in time the negative frequencies of $e^{\beta x}u(x,t)$ one gets an estimate similar to that in with $\lambda=\beta$ and $\mathbb G=0$. This argument motivates the idea behind Lemma \[ultimo\] and its proof. .2in The rest of this paper is organized as follows: section 2 contains the results related to Hardy’s uncertainty principle including a short discussion on the version of this principle in terms of the heat flow. Section 3 those concerned with Morgan’s uncertainty principle. In section 4 we shall consider the limiting case in section 3. Also, section 4 includes the statements of some related forthcoming results. Earlier in the introduction we have discussed uniqueness results obtained under the assumption that the solution vanishes at two different time in a semi-space (see [@BZ], [@IK04], [@IK06], [@EKPV08b]). In section 2 similar uniqueness results will be established under a Gaussian decay hypothesis, in the whole space. In section 5 we shall obtain a unifying result, i.e. a uniqueness result under Gaussian decay in a semi-space of $\,{\mathbb R}^n$ at two different times. The appendix contains an abstract lemma and a corollary which will be used in the previous sections. Hardy’s Uncertainty Principle {#hardy} ============================= In [@Hardy] G. H. Hardy’s proved the following one dimensional ($n=1$) result: .07in *If $f(x)=O(e^{-|x|^2/\beta^2})$, $\widehat f(\xi)=O(e^{-4|\xi|^2/\alpha^2})$ and $1/\alpha\beta>1/4$, then $f\equiv 0$. Also, if $1/\alpha\beta=1/4$, $f(x)$ is a constant multiple of $e^{-|x|^2/\beta^2}$.* .06in To our knowledge the available proofs of this result and its variants use complex analysis, mainly appropriate versions of the Phragmén-Lindelöf principle. There has also been considerable interest in a better understanding of this result and on extensions of it to other settings: [@bonamie1], [@bonamie2], [@CoPr], [@Ho], and [@SST]. In particular, the extension of Hardy’s result to higher dimension $n\geq 2$ (via Radon transform) was given in [@SST]. The formula allows us to re-write this uncertainty principle in terms of the solution of the IVP for the free Schrödinger equation $$\begin{cases} \begin{aligned} &\partial_tu=i\triangle u, \;\;\,\,(x,t)\in{\mathbb R}^n\times (0,+\infty),\\ &u(x,0)=u_0(x), \end{aligned} \end{cases}$$ in the following manner : .05in *If $u(x,0)=O(e^{-|x|^2/\beta^2})$, $u(x,T)=O(e^{-|x|^2/\alpha^2})$ and $T/\alpha\beta> 1/4$, then $u\equiv 0$. Also, if $T/\alpha\beta=1/4$, $u$ has as initial data $u_0$ equal to a constant multiple of $e^{-\left(1/\beta^2+i/4T\right)|y|^2}$.* .05in The corresponding $L^2$-version of Hardy’s uncertainty principle was established in [@CoPr2] : .05in *If $\,e^{|x|^2/\beta^2}f$, $\,e^{4|\xi |^2/\alpha^2}\widehat f$ are in $L^2({\mathbb R}^n)$ and $1/\alpha\beta\ge 1/ 4$, then $f\equiv 0$.* .05in In terms of the solution of the Schrödinger equation it states : .05in *If $\,e^{|x|^2/\beta^2}u(x,0)$, $\,e^{|\xi |^2/\alpha^2}u(x,T)$ are in $L^2({\mathbb R}^n)$ and $T/\alpha\beta\ge 1/4$, then $u\equiv 0$.* .05in More generally, it was shown in [@CoPr2] that : .05in *If $\,e^{|x|^2/\beta^2}f\in L^p({\mathbb R}^n)$, $\,e^{4|\xi |^2/\alpha^2}\widehat f\in L^q({\mathbb R}^n)$, $p, q\in [1,\infty]\,$ with at least one of them finite and $1/\alpha\beta\ge 1/ 4$, then $f\equiv 0$.* .05in In [@EKPV08b] we proved a uniqueness result for solutions of with $F$ as in for bounded potentials $V$ verifying that either, $$V(x,t)=V_1(x)+V_2(x,t),$$ with $V_1$ real-valued and $$\sup_{[0,T]}\|e^{T^2|x|^2/\left(\alpha t+\beta\left(T-t\right)\right)^2}V_2(t)\|_{L^\infty({\mathbb R}^n)}<+\infty,$$ or $$\label{condition} \lim_{R\rightarrow +\infty}\int_0^T\|V(t)\|_{L^\infty({\mathbb R}^n\setminus B_R)}\,dt =0.$$ More precisely, it was shown that the only solution $u\in C([0,T], L^2({\mathbb R}^n))$ to with $F=V(x,t)u$, verifying $$\label{E: condicion fundamental} \|e^{|x|^2/\beta^2}u(0)\|_{L^2({\mathbb R}^n)}+\|e^{|x|^2/\alpha^2}u(T)\|_{L^2({\mathbb R}^n)}<+\infty$$ with $\,T/\alpha\beta>1/ 2$ and $V$ satisfying one of the above conditions is the zero solution. Notice that this result differs by a factor of $1/2$ from that for the solution of the free Schrödinger equation given by the $L^2$-version of the Hardy uncertainty principle described above ($T/\alpha\beta\ge 1/4$). In [@EKPV09] we showed that the optimal version of Hardy’s uncertainty principle in terms of $L^2$-norms, as established in [@CoPr2], holds for solutions of $$\label{E: 1.11} \partial_tu=i\left(\triangle u+V(x,t)u\right), \,\,\,\, (x,t)\in {\mathbb R}^n\times [0,T],$$ such that holds with $T/\alpha\beta>1/4$ and for many general bounded potentials $V(x,t)$, while it fails for some complex-valued potentials in the end-point case, $T/\alpha\beta=1/4$. .05in \[T: hardytimeindepent\] Let $u\in C([0,T]):L^2({\mathbb R}^n))$ be a solution of the equation . If there exist positive constants $\alpha$ and $\beta$ such that $T/\alpha\beta > 1/4$, and $$\|e^{|x|^2/\beta^2}u(0)\|_{L^2({\mathbb R}^n)},\,\,\,\,\|e^{|x|^2/\alpha^2}u(T)\|_{L^2({\mathbb R}^n)}<\infty,$$ and the potential $V$ is bounded and either, $V(x,t)=V_1(x)+V_2(x,t)$, with $V_1$ real-valued and $$\sup_{[0,T]}\|e^{T^2|x|^2/\left(\alpha t+\beta \left(T-t\right)\right)^2}V_2(t) \|_{L^\infty({\mathbb R}^n)} < +\infty$$ or $$\lim_{R\rightarrow +\infty}\|V\|_{L^1([0,T], L^\infty({\mathbb R}^n\setminus B_R)}=0.$$ Then, $u\equiv 0$. We remark that there are no assumptions on the size of the potential in the given class or on the dimension and that we do not assume any decay of the gradient, neither of the solutions or of the time-independent potential or any *[a priori ]{}regularity on this potential or the solution.* .03in \[T: hardytimeindepent2\] Assume that $T/\alpha\beta=1/4$. Then, there is a smooth complex-valued potential $V$ verifying $$|V(x,t)|\lesssim\frac 1{1+|x|^2},\, (x,t)\in {\mathbb R}^n\times [0,T],$$ and a nonzero smooth function $u\in C^\infty([0,T],\mathcal S({\mathbb R}^n))$ solution of such that $$\label{007} \|e^{|x|^2/\beta^2}u(0)\|_{L^2({\mathbb R}^n)},\,\,\,\,\|e^{|x|^2/\alpha^2}u(T)\|_{L^2({\mathbb R}^n)}<\infty.$$ .03in Our proof of Theorem \[T: hardytimeindepent\] does not use any complex analysis, giving, in particular, a new proof (up to the end-point) of the $L^2$-version of Hardy’s uncertainty principle for the Fourier transform. It is based on Carleman estimates for certain evolutions. More precisely, it is based on the convexity and log-convexity properties present for the solutions of these evolutions. Thus, the convexity and log-convexity of appropriate $L^2$-quantities play the role of the Phragmén-Lindelöf principle. We observe that the product of log-convex functions is log-convex which, roughly speaking, replaces the fact that the product of analytic functions is analytic. In [@CEKPV] in collaboration with M. Cowling, we gave new proofs, based only on *[real variable ]{}techniques, of both the $L^2$-version of the Hardy uncertainty principle and the original Hardy’s uncertainty principle $ (L^{\infty}$) $n$-dimensional version for the Fourier transform as stated at the beginning of this section, including the end point case $1/\alpha \,\beta=1/4$.* Returning to Theorem \[T: hardytimeindepent\] as a by product of our proof, we obtain the following optimal interior estimate for the Gaussian decay of solutions to . \[T: lamejora\] Assume that $\,u\,$ and $\,V\,$ verify the hypothesis in Theorem \[T: hardytimeindepent\] and $\,T/\alpha\beta\le 1/4$. Then, $$\label{oda} \begin{aligned} &\sup_{[0,T]}\|e^{a(t)|x|^2}u(t)\|_{L^2({\mathbb R}^n)} + \| \sqrt{t(T-t)}\nabla \left(e^{\left(a(t)+\frac{i\dot a(t)}{8a(t)}\right)|x|^2}u\right)\|_{L^2({\mathbb R}^n\times [0,T])}\\ &\le N\left[\|e^{|x|^2/\beta^2}u(0)\|_{L^2({\mathbb R}^n)}+ \|e^{|x|^2/\alpha^2}u(T)\|_{L^2({\mathbb R}^n)}\right], \end{aligned}$$ where $$a(t)=\frac {\alpha\beta RT}{2\left(\alpha t+\beta (T-t)\right)^2+2R^2\left(\alpha t - \beta (T-t)\right)^2}\ ,$$ $R$ is the smallest root of the equation $$\frac T{\alpha\beta}=\frac R{2\left(1+R^2\right)}$$ and $N$ depends on $T$, $\alpha$, $\beta$ and the conditions on the potential $V$ in Theorem \[T: hardytimeindepent\]. One has that $1/a(t)$ is convex and attains its minimum value in the interior of $[0,T]$, when $$|\alpha-\beta|<R^2\left(\alpha+\beta\right).$$ To see the optimality of Theorem \[T: lamejora\], we write $$\label{E: el enemigo} u_R(x,t)=R^{-\frac n2}\left(t-\frac iR\right)^{-\frac n2}e^{-\frac{|x|^2}{4i(t-\frac iR)}}= \left(Rt-i\right)^{-\frac n2}e^{-\frac{(R-iR^2t)}{4(1+R^2t^2)}\,|x|^2},$$ which is a free wave (i.e. $V\equiv 0$, in ) satisfying in ${\mathbb R}^n\times [-1,1]$ the corresponding time translated conditions in Theorem \[T: lamejora\] with $T=2$ and $$\frac1{\beta^2}=\frac1{\alpha^2}=\mu=\frac R{4\left(1+R^2\right)}\le\frac 18\, .$$ Moreover $$\frac R{4\left(1+R^2t^2\right)}\, ,$$ is increasing in the $R$-variable, when $0<R\le 1$ and $-1\le t\le 1$. Our improvement over the results in [@EKPV06] and [@EKPV08b] is a consequence of the possibility of extending the following argument (for the case of free waves) to prove Theorem \[T: hardytimeindepent\] (a non-free wave case). We recall the conformal or Appell transformation: If $u(y,s)$ verifies $$\label{2.1} \partial_su=i\left(\triangle u+V(y,s)u+F(y,s)\right),\;\;\;\;\;\;\;(y,s)\in {\mathbb R}^n\times [0,1],$$ and $\alpha$ and $\beta$ are positive, then $$\label{2.2} \widetilde u(x,t)=\left(\tfrac{\sqrt{\alpha\beta}}{\alpha(1-t)+\beta t}\right)^{\frac n2}u\left(\tfrac{\sqrt{\alpha\beta}\, x}{\alpha(1-t)+\beta t}, \tfrac{\beta t}{\alpha(1-t)+\beta t}\right)e^{\frac{\left(\alpha-\beta\right) |x|^2}{4i(\alpha(1-t)+\beta t)}},$$ verifies $$\label{2.3} \partial_t\widetilde u=i\left(\triangle \widetilde u+\widetilde V(x,t)\widetilde u+\widetilde F(x,t)\right),\;\; \text{in}\ {\mathbb R}^n\times [0,1],$$ with $$\label{potencial} \widetilde V(x,t)=\tfrac{\alpha\beta}{\left(\alpha(1-t)+\beta t\right)^2}\,V\left(\tfrac{\sqrt{\alpha\beta}\, x}{\alpha(1-t)+\beta t}, \tfrac{\beta t}{\alpha(1-t)+\beta t}\right),$$ and $$\label{externalforce} \widetilde F(x,t)=\left(\tfrac{\sqrt{\alpha\beta}}{\alpha(1-t)+\beta t}\right)^{\frac n2+2}F\left(\tfrac{\sqrt{\alpha\beta}\, x}{\alpha(1-t)+\beta t}, \tfrac{\beta t}{\alpha(1-t)+\beta t}\right)e^{\frac{\left(\alpha-\beta\right) |x|^2}{4i(\alpha(1-t)+\beta t)}}.$$ Thus, to prove Theorem \[T: hardytimeindepent\] for free waves, it suffices to consider $u\in C([-1,1], L^2(R^n))$ being a solution of $$\label{E: free wave} \partial_tu-=i\triangle u,\,\,\,\,(x,t)\in R\times [-1,1],$$ and $$\label{E: decaimineto} \|e^{\mu |x|^2}u(-1)\|_{L^2(R^n)}+\|e^{\mu |x|^2}u(1)\|_{L^2({\mathbb R}^n)}<+\infty,$$ for some $\mu >0$. The main idea consists of showing that either $u\equiv 0$ or there is a function $\theta_{R}: [-1,1]\longrightarrow [0,1]$ such that $$\label{E: gaussian improvement} \|e^{\frac{R|x|^2}{4\left(1+R^2t^2\right)}}u(t)\|_{L^2(R^n)}\le \|e^{\mu |x|^2}u(-1)\|_{L^2(R^n)}^{\theta_{R}(t)}\|e^{\mu |x|^2}u(1)\|_{L^2({\mathbb R}^n)}^{1-\theta_{R}(t)},$$ where $R$ is the smallest root of the equation $$\mu =\frac{R}{4\left(1+R^2\right)}\ .$$ This gives the optimal improvement of the Gaussian decay of a free wave verifying and we also see that if $\mu > 1/8$, then $u$ is zero. The proof of these facts relies on new logarithmic convexity properties of free waves verifying and on those already established in [@EKPV08b]. In [@EKPV08b Theorem 3], the positivity of the space-time commutator of the symmetric and skew-symmetric parts of the operator, $$e^{\mu |x|^2}\left(\partial_t-i\triangle\right)e^{-\mu |x|^2},$$ is used to prove that $\|e^{\mu |x|^2}u(t)\|_{L^2({\mathbb R}^n)}$ is logarithmically convex in $[-1,1]$. More precisely, defining $$f(x,t) = e^{\mu |x|^2}u(x,t)=e^{it\Delta}u_0(x),$$ it follows that $$e^{\mu |x|^2}\left(\partial_t-i\triangle\right)u=e^{\mu |x|^2}\left(\partial_t-i\triangle\right)(e^{-\mu |x|^2}f) =\partial_t f -\mathcal S f-\mathcal A f,$$ where $\mathcal S$ is symmetric and $\mathcal A$ skew-symmetric with $$\mathcal S=- i\mu(4 \,x\cdot \nabla + 2n),\,\;\,\;\;\;\;\mathcal A=i(\Delta+4\mu^2 \,|x|^2),$$ so that $$[\mathcal S;\mathcal A] = - 8 \mu (\nabla\cdot I \nabla) + 16 \mu^2\,|x|^2.$$ Formally, using the abstract Lemma \[L: freq1\] (see the appendix) and the Heisenberg inequality $$\|f\|^2_{L^2({\mathbb R}^n)}\leq \frac{2}{n} \,\|\,|x|f\|_{L^2({\mathbb R}^n)}\,\|\,\nabla f\|_{L^2({\mathbb R}^n)},$$ whose proof follows by integration by parts, one sees that $$H(t)=\|f(t)\|^2_{L^2({\mathbb R}^n)}=\|e^{\mu |x|^2}u(t)\|_{L^2({\mathbb R}^n)}$$ is logarithmically convex so $$\|e^{\mu |x|^2}u(t)\|_{L^2({\mathbb R}^n)}\le \|e^{\mu |x|^2} u(-1)\|_{L^2(R^n)}^{\frac{1-t}2} \|e^{\mu |x|^2} u(1)\|_{L^2({\mathbb R}^n)}^{\frac{1+t}2},$$ when, $-1\le t\le 1$. Setting $a_1\equiv \mu$, we begin an iterative process, where at the $k$-th step, we have $k$ smooth even functions, $a_j:[-1,1]\longrightarrow (0,+\infty)$, $1\le j\le k$, such that $$\mu\equiv a_1<a_2<\dots<a_k\in (-1,1),$$ $$F(a_i)> 0,\ a_j(1)=\mu,\ j=1,\dots,k,$$ where $$F(a)=\frac 1a\left(\ddot a-\frac{3\dot a^2}{2a\,}+32a^3\right)$$ and functions $\theta_j:[-1,1]\longrightarrow [0,1]$, $1\le j\le k$, such that for $t\in [.1,1]$ $$\label{E: algoagradable} \|e^{a_j(t) |x|^2}u(t)\|_{L^2(R^n)}\le \|e^{\mu |x|^2}u(-1)\|_{L^2(R^n)}^{\theta_j(t)}\|e^{\mu |x|^2}u(1)\|_{L^2({\mathbb R}^n)}^{1-\theta_j(t)}.$$ These estimates follow from the construction of the functions $a_i$, while the method strongly relies on the following formal convexity properties of free waves: $$\label{E: algo fundamental} \partial_t\left(\frac 1a\partial_t\log{H_b}\right)\ge -\frac{2\ddot b^2|\xi|^2}{F(a)},$$ $$\label{E: el control del gradiente} \partial_t\left(\frac 1a\partial_tH\right)\ge \epsilon_a\int_{R^n}e^{a|x|^2}\left(|\nabla u|^2+|x|^2|u|^2\right)\,dx,$$ where $$H_b(t)=\|e^{a(t)|x+ b(t)\xi|^2}u(t)\|_{L^2(R^n)}^2\ ,\ H(t)=\|e^{a(t)|x|^2}u(t)\|_{L^2({\mathbb R}^n)}^2,$$ $\xi\in {\mathbb R}^n$ and $a, b: [-1,1]\longrightarrow R$ are smooth functions with $$a> 0,\quad \;\;\;\;\;F(a)>0 \;\;\;\;\;\text{in}\;\;\;\;[-1,1].$$ Once the $k$-th step is completed, we take $a=a_k$ in with a certain choice of $b=b_k$, verifying $b(-1)=b(1)=0$ and then, a certain test is performed. When the answer to the test is positive, it follows that $u\equiv 0$. Otherwise, the logarithmic convexity associated to allows us to find a new smooth function $a_{k+1}$ in $[-1,1]$ with $$a_1<a_2<\dots<a_k<a_{k+1},\,\, (-1,1),$$ and verifying the same properties as $\,a_1,\dots,a_k$. When the process is infinite, we have for all $k\ge 1$ and there are two possibilities: $$\text{ either }\,\,\,\,\,\,\, \lim_{k\to +\infty}a_k(0)=+\infty,\,\,\,\,\,\,\, \text{or }\,\,\,\,\,\,\,\lim_{k\to +\infty}a_k(0)<+\infty.$$ In the first case and one has that $u\equiv 0$, while in the second, the sequence $a_k$ is shown to converge to an even function $a$ verifying $$\label{ole} \begin{cases} \ddot a-\frac{3\dot a^2}{2a\,\,}+32a^3=0,\,\,\,\,\, [-1,1] \\ a(1)=\mu. \end{cases}$$ Because $$a(t)=\frac{R}{4\left(1+R^2t^2\right)},\,\,\, \,\quad R\in {\mathbb R}^+,$$ are all the possible even solutions of this equation, $a$ must be one of them and $$\mu =\frac{R}{4\left(1+R^2\right)},$$ for some $R>0$. In particular, $u\equiv 0$, when $\mu >1/8$. .05in As it was already mentioned above, our proof of Theorem \[T: hardytimeindepent\] (the case of non-zero potentials $V=V(x,t)$), is based on the extension of the above convexity properties to the non-free case. .05in Theorem \[T: hardytimeindepent2\] establishes the sharpness of the result in Theorem \[T: hardytimeindepent\] by giving an example of a complex valued potential $V(x,t)$ verifying and a non-trivial solution $u\in C([0,T]:L^2({\mathbb R}^n))$ of for which holds with $T/\alpha\beta =1/4$. Thus, one may ask : Is it possible to construct a real valued potential $V(x,t)$ verifying the same properties, i.e. satisfying and having a non-trivial solution $u\in C([0,T]:L^2({\mathbb R}^n))$ of such that holds with $T/\alpha\beta =1/4\,$? The same question concerning the sharpness of the above result presents itself in the case of time independent potentials $V=V(x)$. In this regard, we consider the stationary problem $$\label{estatic} \Delta w + V(x) w =0,\,\,\,\, x\in {\mathbb R}^n,\,\,V\in L^{\infty}({\mathbb R}^n),$$ and recall V. Z. Meshkov’s result in [@Me] : .07in *If $w\in H^2_{loc}({\mathbb R}^n)$ is a solution of such that $$\label{43} \int_{{\mathbb R}^n} e^{a|x|^{4/3}}|w(x)|^2dx<\infty,\,\,\,\,\forall a>0,$$ then $\,u\equiv 0$.* .04in Moreover, it was also proved in [@Me] that for complex potentials $V$, the exponent $4/3$ in is optimal. However, it has been conjectured that for real valued potentials the optimal exponent should be 1, (see also [@BoKe] for a quantitative form of these results and applications to Anderson localization of Bernoulli models). .05in More generally, it was established in [@EKPV10], (see also [@cruz]) : .07in *If $w\in H^2_{loc}({\mathbb R}^n)$ is a solution of with a complex valued potential $V$ satisfying $$V(x)=V_1(x)+V_2(x),$$ such that $$\label{123} |V_1(x)|\leq \frac{c_1}{(1+|x|^2)^{\alpha/2}},\,\,\,\,\alpha\in [0,1/2),$$ and $V_2$ real valued supported in $\,\{x\,:\,|x|\geq 1\}$ such that $$-(\partial_r V_2(x))^- < \frac{c_2}{|x|^{2\alpha}},\,\,\,\,a^-=\min\{a;0\}.$$ Then there exists $a=a(\|V\|_{\infty};c_1;c_2;\alpha)>0$ such that if $$\label{43a} \int_{{\mathbb R}^n} e^{a|x|^{r}}|w(x)|^2dx<\infty,\,\,\,\,\,r=(4-2\alpha)/3,$$ then $\,u\equiv 0$.* .05in In addition, one can take the value $r=1$ in by assuming $\alpha>1/2$ in . .05in It was also proved in [@cruz] that for complex potentials these results for $\alpha\in [0,1/2)$ are sharp. .05in By noticing that given a solution $\phi(x) $ of the eigenvalue problem $$\label{eigen} \Delta \phi + \widetilde V(x) \phi =\lambda \phi,\,\,\,\, x\in {\mathbb R}^n,$$ with $\lambda \in{\mathbb R}, $ then $V(x)=\widetilde V(x)+\lambda$ satisfies the hypothesis of the previous result and $$u(x,t)= e^{it\lambda} \,\phi(x),$$ solves the evolution equation $$\label{evo-notime} \partial_t u=i(\Delta u + V(x)u),\,\,\,\, x\in {\mathbb R}^n,\,t\in{\mathbb R},$$ one gets a lower bound for the value of the strongest possible decay rate of non-trivial solutions $u(x,t)$ of at two different times. .05in As a direct consequence of Theorem \[T: hardytimeindepent\] we have the following application concerning the uniqueness of solutions for semi-linear equations of the form with $F$ as in . \[Theorem NL2\] Let $u_1$ and $u_2$ be strong solutions in $C([0,T],H^k({\mathbb R}^n)), \,k>n/2$ of the equation with $F$ as in such that $\,F\in C^k$ and $F(0)=\partial_uF(0)=\partial_{\bar u}F(0)=0$. If there are $\alpha$ and $\beta$ positive with $T/\alpha \beta>1/4$ such that $$e^{|x|^2/\beta^2}\left(u_1(0)-u_2(0)\right)\, ,\,\ e^{|x|^2/\alpha^2}\left(u_1(T)-u_2(T)\right) \in L^2({\mathbb R}^n),$$ then $u_1\equiv u_2$. In Theorem \[Theorem NL2\] we did not attempt to optimize the regularity assumption on the solutions $\,u_1,\,u_2$. By fixing $u_2\equiv 0$ Theorem \[Theorem NL2\] provides a restriction on the possible decay at two different times of a non-trivial solution $u_1$ of equation with $F$ as in . It is an open question to determine the optimality of this kind of result. More precisely, for the standard semi-linear Schrödinger equations $$\label{NLS} \partial_t u = i (\Delta u + |u|^{\gamma-1}u),\,\,\,\,\gamma>1,$$ one has the *[standing wave ]{}solutions $$u(x,t)=e^{\omega \,t} \varphi(x),\,\,\,\omega>0,$$ where $\varphi$ is the unique (up to translation) positive solution of the elliptic problem $$-\Delta \varphi+\omega \varphi= |\varphi|^{\gamma-1}\varphi,$$ which has a linear exponential decay, i.e. $$\varphi(x)=O(e^{-c|x|}),\,\,\,\text{as}\,\,\,|x|\to\infty,$$ for an appropriate value of $c>0$ (see [@Str], [@BLi], [@BGK], and [@Kw]). Whether or not these standing waves are the solutions of having the strongest possible decay at two different times is an open question.* .04in Hardy’s uncertainty principle also admits a formulation in terms of the heat equation $$\partial_tu=\Delta u,\;\;\;\;t>0,\;\;x\in\mathbb R^n,$$ whose solution with data $\,u(x,0)=u_0(x)$ can be written as $$u(x,t)= e^{t \Delta}u_0(x)= \int_{{\mathbb R}^n} \frac{e^{-|x-y|^2/4t}}{(4\pi t)^{n/2}}\, u_0(y)\,dy.$$ More precisely, Hardy’s uncertainty principle can restated in the following equivalent forms : .07in *(i) If $\,u_0\in L^2({\mathbb R}^n)$ and there exists $\,T>0$ such that $\,e^{|x|^2/(\delta^2T)}\,e^{T\Delta}u_0\in L^2({\mathbb R}^n)\,$ for some $\,\delta \leq 2$, then $\,u_0\equiv 0$.* .04in *(ii) If $\,u_0\in \mathcal S({\mathbb R}^n)$ (tempered distribution) and there exists $\,T>0$ such that $\,e^{|x|^2/(\delta^2T)}\,e^{T\Delta}u_0\in L^{\infty}({\mathbb R}^n)$ for some $\,\delta< 2$, then $\,u_0\equiv 0$. Moreover, if $\,\delta=2$, then $\,u_0$ is a constant multiple of the Dirac delta measure.* .03in In fact, applying Hardy’s uncertainty principle to $e^{T \triangle} u_0$ one has that $e^{\frac{|x|^2}{\delta^2 T}}e^{T \triangle} u_0$ and $e^{ T |\xi|^2}\widehat{e^{T \triangle} u_0}=\widehat u_0$ in $L^2({\mathbb R}^n)$ with $\,2\delta\le 4$ implies $e^{\triangle}u_0\equiv 0$. Then, backward uniqueness arguments (see for example [@lm60 Chapter 3, Theorem 11]) shows that $u_0\equiv 0$. In [@EKPV08b] we proved the following weaker extension of this result for parabolic operators with lower order variable coefficientes : .03in \[T: toremaparabolicco\] Let $u\in C([0,1] : L^2({\mathbb R^n}))\cap L^2([0,T]: H^1({\mathbb R^n}))$ be a solution of the IVP $$\begin{cases} \partial_tu=\triangle u+V(x,t)u,\ \text{in}\ {\mathbb R^n}\times (0,1],\\ u(x,0)=u_0(x), \end{cases}$$ where $$V\in L^{\infty}({\mathbb R^n}\times [0,1]).$$ If $$u_0\;\;\;\;\text{and}\;\;\;\;e^{\frac{|x|^2}{\delta^2}}u(1)\in L^2({\mathbb R^n}),$$ for some $\,\delta <1$, then $u_0\equiv 0$. .03in It is natural to expect that Hardy’s uncertainty principle holds in this context with bounded potentials $V$ and with the parameter $\delta$ verifing the condition of the free case, i.e. $\,\delta\leq 2$. Earlier results in this directions, addressing a question of Landis and Oleinik [@LaOl], were obtained in [@EKPV06a] and [@Ng]. .03in Uncertainty Principle of Morgan type {#morgan} ==================================== In [@Mo] G. W. Morgan proved the following uncertainty principle: .07in *If $f(x)=O(e^{-\frac{a^p |x|^p}{p}}), \,1<p\leq 2$ and $\widehat f(\xi)=O(e^{-\frac{(b+\epsilon)^q |\xi|^q}{q}}),\;1/p+1/q=1,\,\epsilon>0,$ with $$ab>\Big|\cos\left(\frac{p\,\pi}{2}\right)\Big|,$$ then $f\equiv 0$.* .07in In [@Ho] Beurling-Hörmander showed : .07in *If $f\in L^1(\mathbb R)$ and $$\label{beurling} \int_{\mathbb R} \int_{\mathbb R} |f(x)| |\widehat f(\xi)| e^{ |x\,\xi|}\,dx\,d\xi<\infty, \;\;\;\text{then} \;\;\;f\equiv 0.$$* This result was extended to higher dimensions $n\geq 2$ in [@bonamie2] and [@ray] : .07in *If $f\in L^2(\mathbb R^n), n\geq 2$ and $$\label{beurlingn} \int_{\mathbb R^n} \int_{\mathbb R^n} |f(x)| |\widehat f(\xi)| e^{ |x\,\cdot \xi|}\,dx\,d\xi<\infty, \;\;\;\text{then} \;\;\;f\equiv 0.$$* .05in We observe that from and it follows that : .05in *If $p\in(1,2]$, $\,1/p+1/q=1$, $\,a, \,b>0$, and $$\label{primera} \int_{\mathbb R^n}|f(x)|\, e^{\frac{a^p|x|^p}{p}}dx \,+ \,\int_{\mathbb R^n} |\widehat f(\xi)| \,e^{\frac{b^q|\xi|^q}{q}}d\xi<\infty,\;\;a b\geq 1\;\Rightarrow \;f\equiv 0.$$* Notice that in the case $p=q=2$ this gives us an $L^1$-version of Hardy’s uncertainty result discussed above, and for $p<2$ an $n$-dimensional $L^1$-version of Morgan’s uncertainty principle. In the one-dimensional case ($n=1$), the optimal $L^1$-version of Morgan’s result in , $$\label{primera1} \int_{\mathbb R}|f(x)|\, e^{\frac{a^p|x|^p}{p}}dx + \int_{\mathbb R} |\widehat f(\xi)| \,e^{\frac{b^q|\xi|^q}{q}}d\xi<\infty,\;\;a b>\Big|\cos\left(\frac{p\,\pi}{2}\right)\Big|\;\Rightarrow \;f\equiv 0.$$ was established in [@bonamie2] and [@monki] (for further results see [@bonamie1] and references therein). A sharp condition for $a,\,b,\,p$ in in higher dimension seems to be unknown. However, in [@bonamie2] it was shown : .07in *If $f\in L^2(\mathbb R^n)$, $1<p\leq 2\;$ and $\,1/p+1/q=1\,$ are such that for some $j=1,..,n$, $$\label{bonami11} \int_{\mathbb R^n} |f(x)|e^{\frac{a^p|x_j|^p}{p}}dx<\infty\;\;+\;\;\int_{\mathbb R^n} |\widehat f(\xi)|e^{\frac{b^q|\xi_j|^q}{q}}d\xi<\infty.$$* .05in *If $a b>\Big|\cos\left(\frac{p\,\pi}{2}\right)\Big|$, then $\;f\equiv 0$.* .05in *If $a b<\Big|\cos\left(\frac{p\,\pi}{2}\right)\Big|$, then there exist non-trivial functions satisfying* . .05in Using the above result can be stated in terms of the solution of the free Schrödinger equation. In particular, can be re-written as : .05in *If $u_0\in L^1(\mathbb R)$ or $u_0\in L^2(\mathbb R^n)$, if $n\geq 2$, and for some $\,t\neq 0$ $$\label{pq} \int_{\mathbb R^n}\;|u_0(x)|\, e^{\frac{a^p|x|^p}{p}}dx \,+ \,\int_{\mathbb R^n}\, |\,e^{it\Delta}u_0(x)| \,e^{\frac{b^q|x|^q}{q(2t)^q}}dx<\infty,$$ with $$ab>\Big|\cos\left(\frac{p\,\pi}{2}\right)\Big|\;\;\;\;\text{if}\;\;\;n=1,\;\;\;\;\text{and}\;\;\;\;ab>1\;\;\;\;\text{if}\;\;\;\;\;n\geq 2,$$ then $u_0\equiv 0$.* .07in Related with Morgan’s uncertainty principle one has the following result due to Gel’fand and Shilov. In [@GeShi] they considered the class $Z^p_p,\,p> 1$, defined as the space of all functions $\varphi(z_1,..,z_n)$ which are analytic for all values of $z_1,..,z_n\in \mathbb C$ and such that $$|\varphi(z_1,..,z_n)|\leq C_0\, e^{\sum_{j=1}^n\,\epsilon_j\,C_j\,|z_j|^p},$$ where the $C_j,\,j=0,1,..,n$ are positive constants and $\epsilon_j=1$ for $z_j$ non-real and $\epsilon_j=-1$ for $z_j$ real, $j=1,..,n$, and showed that the Fourier transform of the function space $Z_p^p$ is the space $Z_q^q$, with $\,1/p+1/q=1$. .05in Notice that the class $Z_p^p$ with $p\geq 2$ is closed with respect to multiplication by $\,e^{i c |x|^2}$. Thus, if $u_0\in Z^p_p,\,p\geq 2$, then by one has that $$|e^{it\Delta}u_0(x)|\leq d(t)\,e^{-a(t)|x|^q},$$ for some functions $\,d,\,a\,:\,\mathbb R\to(0,\infty)$. .03in In [@EKPV08m] the following results were established: \[Theorem 22\] Given $\,p\in(1,2)$ there exists $\,M_p>0$ such that for any solution $u \in C([0,1] :L^2({\mathbb R^n}))$ of $$\label{E: 1.111}\partial_tu=i\left(\triangle u+V(x,t)u\right),\;\;\;\text{in}\;\;\;\;\;{\mathbb R^n}\times [0,1],$$ with $V=V(x,t)$ complex valued, bounded (i.e. $\|V\|_{L^{\infty}({\mathbb R}^n\times[0,1])}\leq C$) and $$\label{14} \lim_{R\rightarrow +\infty}\|V\|_{L^1([0,1] : L^\infty({\mathbb R^n}\setminus B_R))}=0,$$ satisfying that for some constants $\,a_0,\,a_1,\,a_2>0$ $$\label{12} \int_{\mathbb R^n} |u(x,0)|^2\,e^{2a_0 |x|^p}dx < \infty,$$ and for any $k\in{\mathbb Z}^+$ $$\label{13} \int_{\mathbb R^n} |u(x,1)|^2\,e^{2k |x|^p}dx < a_2 e^{2 a_1 k^{q/(q-p)}},$$ $1/p+1/q=1$, if $$\label{conditionp} \,a_0\,a_1^{(p-2)} > M_p,$$ then $\,u\equiv 0$. \[Corollary 22\] Given $\,p\in(1,2)$ there exists $N_p>0$ such that if $u\in C([0,1]:L^2(\mathbb R^n))$ is a solution of $$\partial_t u=i (\Delta u +V(x,t)u),$$ with $V=V(x,t)$ complex valued, bounded (i.e. $\|V\|_{L^{\infty}({\mathbb R}^n\times[0,1])}\leq C$) and $$\lim_{R\to\infty} \,\int_0^1\,\sup_{|x|>R} |V(x,t)| dt=0,$$ and there exist $\,\alpha,\,\beta>0$ such that $$\label{con1} \int_{\mathbb R^n} |u(x,0)|^2e^{2\,\alpha^p\,|x|^p/p}dx\;\,\,+\,\,\;\int_{\mathbb R^n} |u(x,1)|^2e^{2\,\beta^q\,|x|^q/q}dx<\infty,$$ $\,1/p+1/q=1$, with $$\label{conditionp2} \;\alpha\,\beta > N_p,$$ then $\;u\equiv 0$. As a consequence of Corollary \[Corollary 22\] one obtains the following result concerning the uniqueness of solutions for the semi-linear equations with $F$ as in $$\label{E: NLS} i \partial_t u + \triangle u = F(u,\overline u).$$ \[Theorem 23\] Given $\,p\in(1,2)$ there exists $\,N_p>0$ such that if $$u_1,\,u_2 \in C([0,1] : H^k({\mathbb R}^n)),$$ are strong solutions of with $k\in {\mathbb Z}^+$, $k>n/2$, $F:{\mathbb C}^2\to {\mathbb C}$, $F\in C^{k}$ and $F(0)=\partial_uF(0)=\partial_{\bar u}F(0)=0$, and there exist $\,\alpha,\,\beta>0$ such that $$\label{con2} e^{\alpha^p\,|x|^p/p}\left(u_1(0)-u_2(0)\right),\;\;\;\ e^{\beta^q\,|x|^q/q}\left(u_1(1)-u_2(1)\right) \in L^2({\mathbb R}^n),$$ $1/p+1/q=1$, with $$\label{conditionp2b} \,\alpha\,\beta > N_p,$$ then $u_1\equiv u_2$. Notice that the conditions and are independent of the size of the potential and there is not any *[a priori ]{}regularity assumption on the potential $V(x,t)$.* The result in [@bonamie2], see , can be extended to our setting with an non-optimal constant. More precisely, \[Corollary 24\] The conclusions in Corollary \[Corollary 22\] still hold with a different constant $N_p>0$ if one replaces the hypothesis by the following one dimensional version $$\label{conn=1} \int_{\mathbb R^n} |u(x,0)|^2e^{2\,\alpha^p\,|x_j|^p/p}dx<\infty\,\;\,\,+\,\,\;\int_{\mathbb R^n} |u(x,1)|^2e^{2\,\beta^q\,|x_j|^q/q}dx<\infty,$$ for some $j=1,..,n$. Similarly, the non-linear version of Theorem \[Theorem 23\] still holds, with different constant $N_p>0$, if one replaces the hypothesis by $$e^{\alpha^p\,|x_j|^p/p}\left(u_1(0)-u_2(0)\right),\;\;\;\ e^{\beta^q\,|x_j|^q/q}\left(u_1(1)-u_2(1)\right) \in L^2({\mathbb R}^n),$$ for $j=1,..,n$. In [@EKPV08m] we did not attempt to give an estimate of the universal constant $N_p$. The limiting case $\,p=1$ will be considered in the next section. The main idea in the proof of these results is to combine an upper estimate with a lower one to obtain the desired result. The upper estimate is based on the decay hypothesis on the solution at two different times (see Lemma \[ultimo\]). In previous works we had been able to establish these estimates from assumptions that at time $t=0$ and $t=1$ involving the same weight. However, in our case (Corollary \[Corollary 22\]) we have different weights at time $t=0$ and $t=1$. To overcome this difficulty, we carry out the details with the weight $e^{a_j|x|^p},\,1<p<2$, $j=0$ at $t=0$ and $j=1$ at $t=1$, with $a_0$ fixed and $a_1=k\in\mathbb Z^+$ as in . Although the powers $\,|x|^p\,$ in the exponential are equal at time $t=0$ and $t=1$ to apply our estimate (Lemma \[ultimo\]) we also need to have the same constant in front of them. To achieve this we apply the conformal or Appell transformation described above, to get solutions and potentials, whose bounds depend on $k\in\mathbb Z^+$. Thus we have to consider a family of solutions and obtain estimates on their asymptotic value as $k\uparrow \infty$. The proof of the lower estimate is based on the positivity of the commutator operator obtained by conjugating the equation with the appropriate exponential weight, (see Lemma \[L: freq1\] in the appendix) Paley-Wiener Theorem and Uncertainty Principle of Ingham type {#ingham} ============================================================= This section is concerned with the limiting case $p=1$ in the previous section. It is easy to see that if $f\in L^1({\mathbb R}^n)$ is non-zero and has compact support, then $\,\widehat f $ cannot satisfy a condition of the type $\widehat f(y)=O(e^{-\epsilon |y|})$ for any $\epsilon>0$. However, it may be possible to have $ f\in L^1({\mathbb R}^n)$ a non-zero function with compact support, such that $\widehat f(\xi)=O(e^{-\epsilon(y) |y|})$, $\epsilon(y)$ being a positive function tending to zero as $|y|\to \infty$. In the one-dimensional case ($n=1$) soon after Hardy’s result described above, A. E. Ingham [@In] proved the following : .05in *There exists $f\in L^1({\mathbb R})$ non-zero, even, vanishing outside an interval such that $\widehat f(y)=O(e^{-\epsilon(y) |y|})$ with $\epsilon(y)$ being a positive function tending to zero at infinity if and only if $$\int^{\infty} \frac{\epsilon(y)}{y}\,dy<\infty.$$* In a similar direction the Paley-Wiener Theorem [@PW] gives a characterization of a function or distribution with compact support in term of analyticity properties of its Fourier transform. Regarding our results discussed above it would be interesting to identify a class of potentials $V(x,t)$ for which a result of the following kind holds: .05in If $u\in C([0,1]:L^2({\mathbb R}^n))$ is a non-trivial solution of the IVP $$\label{0007} \begin{cases} \begin{aligned} &\partial_tu=i(\triangle u+V(x,t)u), \;\;\,\,(x,t)\in{\mathbb R}^n\times [0,1],\\ &u(x,0)=u_0(x), \end{aligned} \end{cases}$$ with $u_0\in L^2({\mathbb R}^n)$ having compact support, then $ e^{\epsilon |x|}\,u(\cdot,t)\notin L^2({\mathbb R}^n)$ for any $\epsilon>0$ and any $t\in(0,1]$. .03in In this direction we have the following result which will appear in [@EKPV12]: \[2012\] Assume that $u\in C([0,1]:L^2(\mathbb R^n))$ is a strong solution of the IVP with $$\label{hyp1-2012} supp\,u_0\subset B_R(0)=\{x\in\mathbb R^n\,:\,|x|\leq R\},$$ $$\label{hyp2-2012} \int_{\mathbb R^n}\,e^{2a_1|x|}\,|u(x,1)|^2\,dx<\infty,\;\;\;\;\;\;a_1>0,$$ and $$\label{hyp3-2012} \|V\|_{L^{\infty}(\mathbb R^n\times [0,1])}=M_0,$$ with $$\label{hyp4-2012} \lim_{R\rightarrow +\infty}\|V\|_{L^1([0,1] : L^\infty({\mathbb R^n}\setminus B_R))}=0.$$ Then, there exists $b=b(n)>0$ (depending only on the dimension $n$) such that if $$\frac{a_1}{R\,(1+M_0)}\geq b,$$ then $\,u\equiv 0$. .05in A similar question can be raised for results of the type described above due to A. E. Ingham in [@In] and possible extensions to higher dimensions $n\geq 2$. .05in It would be interesting to obtain extensions of the above results characterizing the decay of the solution $u(x,t)$ to the equation with $F$ as in associated to data $u_0\in L^2({\mathbb R}^n)$ with compact support or with $u_0\in C_0^{\infty}({\mathbb R}^n)$. In this direction, some results can be deduced as a consequence of Theorem \[2012\], see [@EKPV12]. Hardy’s Uncertainty Principle in a half-space {#half} ============================================= In the introduction we have briefly reviewed some uniqueness results established for solutions of the Schrödinger equation vanishing at two different times in a semi-space of $\,{\mathbb R}^n$, (see [@BZ], [@ds], [@IK04], [@IK06], [@EKPV08b]). In section 2, we have studied uniqueness results gotten under the hypothesis that the solution of the Schrödinger equation at two different times has an appropriate Gaussian decay, in the whole space ${\mathbb R}^n$. In this section, we shall deduce a unified result, i.e. a uniqueness result under the hypothesis that at two different times the solution of the Schrödinger equation has Gaussian decay in just a semi-space of $\,{\mathbb R}^n$. \[hardyhalf\] Assume that $u\in C([0,1]:L^2((0,\infty)\times \mathbb R^{n-1}))$ is a strong solution of the IVP $$\begin{cases} \begin{aligned} \label{eq441} &\partial_tu=i(\Delta + V(x,t))u,\\ &u(x,0)=u_0(x), \end{aligned} \end{cases}$$ with $$\label{extrahyp} \int_0^1\,\int_{1/2}^{3/2}\,|\partial_{x_1}u(x,t)|^2\,dx\,dt<\infty,$$ $$\label{hyp442} V:\mathbb R^n\times [0,1]\to\mathbb C,\,\,\,\,\,\,\,V\in L^{\infty}(\mathbb R^n\times [0,1]),$$ and $$\label{condition44} \lim_{R\rightarrow +\infty}\int_0^1\|V(t)\|_{L^\infty(\{x_1>R\})}\,dt =0.$$ Assume that $$\begin{aligned} \label{443} &\int_{x_1>0} \,e^{c_0\,|x_1|^2}\,|u(x,0)|^2\, dx <\infty,\\ \\ &\int_{x_1>0} \,e^{c_1\,|x_1|^2}\,|u(x,1)|^2\, dx <\infty, \end{aligned}$$ with $c_0,\,c_1>\,0$ sufficiently large. Then $\,u\equiv 0$. : (a) Note that in Theorem \[hardyhalf\], the solution does not need to be defined for $\,x_1\leq 0$. In this sense, this is a stronger result that the uniqueness results in [@BZ], [@KPV02], [@IK04], [@IK06], and [@ds], which required that the solution be defined in $\,\mathbb R^n\times [0,1]$ and be $C([0,1]:L^2(\mathbb R^n))$. On the other hand, we need to assume the condition . Note that [@KPV02] also needs an extra assumption on $\,\nabla u$, stronger that , but that in [@IK04], which among other things removed any extra assumption on $\,\nabla u$, but still required the solution to be defined in $\,\mathbb R^n\times [0,1]$ and be in $C([0,1]:L^2(\mathbb R^n))$. If in the setting of Theorem \[hardyhalf\] we know that $\,u\,$ is a solution in $\,\mathbb R^n\times [0,1]$ and is in $C([0,1]:L^2(\mathbb R^n))$, then we can dispose the hypothesis as follows: First as in the first step of the proof of Theorem \[hardyhalf\], we can use the Appell transformation to reduce to the case $c_1=c_2=2\gamma$. Then, using $\,\varphi(x_1)$ a regularized" convex function which agrees with $\,x_1^+$ for $x_1>1$ , $x_1<-1$, an application of Lemma \[L: freq1\] and Corollary \[last\] in the appendix yields the estimate $$\sup_{0\leq t\leq 1}\,\int\,e^{2\gamma(x_1^+)^2}|u(x,t)|^2dx+\int_0^1\int_{x_1>2}\,t(1-t)|\nabla u(x,t)|^2e^{2\gamma(x_1^+)^2}dxdt<\infty.$$ Once this is obtained, by restricting our attention to $$(2,\infty)\times \mathbb R^{n-1}\times [\delta,1-\delta],$$ for each $\,\delta>0$, we are in the situation of Theorem \[hardyhalf\], and hence $\,u\equiv 0$ on $\{x_1>2\}\times[0,1]$. Finally, Izakov’s result in [@Iza] concludes that $\,u\equiv 0$ (more precisely, the version of Izakov’s result proved in [@IK04], which does not require $\,\nabla u$ to exist for $-1<x_1<1).$ .05in (b) We have seen that Theorem \[hardyhalf\] includes many of the uniqueness results for solutions vanishing at two different times in a semi-space. In comparison with the results in section 2, since the extra assumption can be recovered as in remark (a) when the solution is defined in $\,\mathbb R^n\times [0,1]$ and is in $C([0,1]:L^2(\mathbb R^n))$, the only weakness is that the provide an optimal estimate for the constants $\,c_1,\,c_2$, but on the other hand deals with solutions only defined in $(0,\infty)\times \mathbb R^{n-1}\times [0,1]$. .05in (c) In Theorem \[hardyhalf\] the direction $\,\vec e_1$ can be replaced by any other $\,\omega\in\mathcal S^{n-1}$. .1in : The strategy of the proof follows closely the one in [@EKPV06]. We divide the proof into three steps. .05in : Reduction to the case $c_0=c_1=2 \gamma$. .05in This follows by using the conformal or Appell transformation introduced in section 2 (see -), combined with the observation that the set $\{x_1>0\}$ remains invariant. .05in : Upper Bounds. .05in We define $$v(x,t)=\theta(x_1)\,u(x,t),$$ with $\,\theta \in C^{\infty}({\mathbb R})$, non-decreasing with $\,\theta(x_1)\equiv 1\,$ if $\,x_1>3/2$, and $\theta(x_1)\equiv 0\,$ if $\,x_1<1/2$. Therefore, $$\label{FFF} \partial_t v=i\,\Delta v + i\,V(x,t) v + i\,F(x,t),\;\;\;\,\,\,\;\;\;F(x,t)=2\,\partial_{x_1}u\,\theta'(x_1)+ u\,\theta''(x_1).$$ Using we can apply Lemma \[ultimo\] to get that $$\begin{aligned} \label{uno44} &\sup_{0\leq t\leq 1}\| e^{\lambda\cdot x_1} v(\,\cdot\,,t)\|_{L^2(\mathbb {\mathbb R}^n)} \\ &\leq c_n \Big(\|e^{\lambda\cdot x_1} v(0)\|_{L^2(\mathbb {\mathbb R}^n)} + \|e^{\lambda\cdot x_1} v(1)\|_{L^2(\mathbb {\mathbb R}^n)}\\ & +\int_0^1 \|e^{\lambda\cdot x_1}\, F(\cdot, t)\|_{L^2(\mathbb {\mathbb R}^n)} dt + \int_0^1 \|e^{\lambda\cdot x_1}\, V\,\chi_{\{x_1<R\}}v(\cdot, t)\|_{L^2(\mathbb {\mathbb R}^n)} dt\Big), \end{aligned}$$ for some fixed $R$ sufficiently large. Thus, using $$\begin{aligned} \label{uno444} &\sup_{0\leq t\leq 1}\| e^{\lambda\cdot x_1} v(\,\cdot\,,t)\|_{L^2(\mathbb {\mathbb R}^n)} \\ &\leq c_n \Big(\|e^{\lambda\cdot x_1} v(0)\|_{L^2(\mathbb {\mathbb R}^n)} + \|e^{\lambda\cdot x_1} v(1)\|_{L^2(\mathbb {\mathbb R}^n)}\\ & + c\,e^{c\,|\lambda|} + c\,\|V\|_{\infty} \,e^{c\,|\lambda|\,R}\Big). \end{aligned}$$ Thus, from the formula (with $p=2$ and $n=1$) and we obtain that $$\aligned &\sup_{0\leq t\leq 1}\| e^{\gamma\,|x_1|^2} v(\,\cdot\,,t)\|_{L^2(\mathbb {\mathbb R}^n)} \\ & \,\,\,\leq \Big(\| e^{\gamma\,|x_1|^2} v(0)\|_{L^2(\mathbb {\mathbb R}^n)} + \| e^{\gamma\,|x_1|^2} v(1)\|_{L^2(\mathbb {\mathbb R}^n)} +c +\,\|V\|_{\infty} \,e^{c\,\gamma\,R^2}\Big). \endaligned$$ Thus, $$\label{step2a} \sup_{0\leq t\leq 1}\| e^{\gamma\,|x_1|^2} v(\,\cdot\,,t)\|_{L^2(\mathbb {\mathbb R}^n)}\leq c_{\gamma}.$$ Combining this and the equation for $\,v\,$ we shall get a smoothing estimate. Using the notation $$H(t)=\|f\|^2_{L^2({\mathbb R}^n)}=\|f\|^2,$$ with $$f(x,t)= e^{\gamma|x_1|^2}\,v(x,t)$$ and the abstract Lemma \[L: freq1\] (see the appendix) one formally has that $$\label{upper-smooth} \begin{aligned} \partial_t^2H &\leq 2\partial_t\text{\it Re}\left(\partial_tf-\mathcal Sf-\mathcal Af,f\right)\\ &+ 2\left(\mathcal S_tf+\left[\mathcal S,\mathcal A\right]f,f\right) + \|\,e^{\gamma |x_1|^2}(F + V\,v)\|^2, \end{aligned}$$ with $$e^{\gamma|x_1|^2}(\partial_t-i\,\Delta) (e^{-\gamma|x_1|^2}f) = \partial_t f -\mathcal Sf-\mathcal Af= e^{\gamma |x_1|^2}(F + V v),$$ where $\, \mathcal S = - i \gamma (4x_1\,\partial_{x_1}+2)$ is symmetric, $\mathcal A=i(\Delta + 4\gamma x_1^2)$ is skew-symmetric, and $\,F\,$ as in . Since, $$[\mathcal S ; \mathcal A] = -8 \gamma \partial_{x_1}^2+ 16 \gamma^2\,x_1^2.$$ using the inequality $$\aligned &\int_{{\mathbb R}^n}\,(|\partial_{x_1}f|^2+4\gamma^2|x_1|^2|f|^2)\,dx = \int_{{\mathbb R}^n}\,e^{2\,\gamma|x_1|^2}\,(|\partial_{x_1}u|^2-2\gamma\,|u|^2)dx\\ &\geq 2\,\gamma\,\int_{{\mathbb R}^n}\,|f|^2\,dx. \endaligned$$ together with Corollary \[last\] we conclude that $$\label{009} \int_0^1\,\int \,t(1-t) \,|\partial_{x_1} v(x,t)|^2\,e^{2\,\gamma|x_1|^2}\,e^{2\,\gamma |x_1|^2}\,dx\,dt \leq c_{\gamma}.$$ Combining and and one gets that $$\label{00step2} \begin{aligned} &\sup_{0\leq t\leq 1}\| e^{\gamma\,|x_1|^2} v(\,\cdot\,,t)\|_{L^2(\mathbb {\mathbb R}^n)}\\ & + \int_0^1 \int t(1-t) |\partial_{x_1} v(x,t)|^2\,e^{2\,\gamma|x_1|^2} e^{2\,\gamma |x_1|^2}| dx dt\leq c_{\gamma}. \end{aligned}$$ We recall the following result which is a slight variation of that proven in detail in [@EKPV06] (Lemma 3.1, page 1818) : \[CPDE\] Assume that $ R>0$ and $\,\varphi : [0,1] \to {\mathbb R}$ is a smooth function. Then, there exists $\,c=c(n;\|\varphi'\|_{\infty}+\|\varphi''\|_{\infty})>0$ such that the inequality $$\label{cpde1} \frac{\alpha^{3/2}}{R^2}\,\Big\|\,e^{\alpha |\frac{x_1-x_{0_1}}{R}+\varphi(t)|^2}g \Big\|_{L^2(dxdt)} \leq c\, \Big\|\,e^{\alpha |\frac{x_1-x_{0_1}}{R}+\varphi(t)|^2}(i \partial_t+\Delta) g \Big\|_{L^2(dxdt)}$$ holds when $\,\alpha > c R^2 \,$ and $\,g\in C^{\infty}_0({\mathbb R}^{n+1})\,$ is supported in the set $$\{(x,t)=(x_1,..,x_n,t)\,\in{\mathbb R}^{n+1}\,:\, |\frac{x_1-x_{0_1}}{R}+\varphi(t)|\geq 1\}.$$ Now, we will chose $\,x_{0_1}=R/2$, $\;0\leq \varphi(t)\leq a,$ with $\,a=3/2-1/R$, $\,\varphi(t)=a,$ on $\,3/8\leq t\leq 5/8$, $\,\varphi(t)=0, $ for $\,t\in [0,1/4]\cup [3/4,1]$, and $\,\theta_R\in C^{\infty}({\mathbb R})$ with $\,\theta_R(x_1)=1\,$ on $\,1<x_1<R-1$, and $\,\theta_R(x_1)=0\,$ for $\,x_1<1/2$ or $\,x_1>R$. Also we chose $\,\eta \in C^{\infty}({\mathbb R})$ with $\,\eta(x_1)=0,\;x_1\leq 1$ and $\,\eta(x_1)=1,\;x_1\geq 1+1/2R$. We notice that up to translation we can assume that $$\label{b} \int_{3/8}^{5/8} \,\int_{2<x_1<3} \,|u(x,t)|^2dx dt=b\neq 0,$$ otherwise we would have $$u(x,t)=0\;\;\,\;\;\,\;\text{on}\,\;\;\,\;(x,t)\;\,\,s.t.\,\,\,(x_1,t)\in (0,\infty)\times (3/8,5/8),$$ and thus by Izakov’s result [@Iza] we would get that $u\equiv 0$. We let $$\label{defg} g(x,t)=\theta_R(x_1)\,\eta\Big(\frac{x_1-R/2}{R}+\varphi(t)\Big)\,u(x,t).$$ It is easy to see that $\,g$ is supported on the set $$\label{domain} \{(x,t)\in{\mathbb R}^{n+1}\,:\, 1/2<x_1<R,\,\, 1/32<t<31/32,\;\;|\frac{x_1-R/2}{R}+\varphi(t)|\geq 1\}.$$ so satisfies the hypothesis of Lemma \[CPDE\]. Also if $(x_1,t)\in (2,3)\times (3/8,5/8)$ one has $\varphi=a$, $\,\eta\Big(\frac{x_1-R/2}{R}+a\Big)=1$ and $\theta_R=1$, hence in this domain $$g(x,t)=u(x,t).$$ Thus, from it follows that $$|\frac{x_1-R/2}{R}+\varphi(t)|\geq 1 + 1/R,$$ so we have the lower bound of $$\frac{\alpha^{3/2}}{R^2} \,b\,e^{\alpha(1+1/R)^2},$$ with $\,b\,$ as in . Now we shall estimate the right hand side of . Thus, $$\label{07} \begin{aligned} &(i\partial_t-\Delta)g= - \theta_R(x_1)\eta\Big(\frac{x_1-R/2}{R}+\varphi(t)\Big) V(x,t) u(x,t)\\ &\;\;\;+\eta\Big(\frac{x_1-R/2}{R}+\varphi(t)\Big)(2\theta'(x_1)\,\partial_{x_1}u+u\,\theta_R''(x_1))\\ &\;\;\;+(i\eta'(\cdot)\,\varphi'(t)+\eta''(\cdot)\,\frac{1}{R^2}) \theta_R(x_1) u(x,t)\equiv E_1+E_2+E_3. \end{aligned}$$ Choosing $R>>\|V\|_{\infty}$, and recalling the fact that $\,\alpha>c R^2\,$ we see that the contribution of the term $E_1$ involving the potential $V$ can be absorbed by the term in the left hand side of . Next, we notice that the terms in $E_2$ involve derivatives of $\,\theta_R$ ($\theta_R'$ or $\theta_R''$) so they are supported in the $(x,t)\in {\mathbb R}^n\times [0,1]$ such that $$1/2<x_1<1,\,\,\,\,\,\text{or}\,\,\,\,\,R-1<x_1<R.$$ But, if $1/2<x_1<1$, it follows that $$\frac{x_1-R/2}{R}+\varphi(t)\leq 1/R-1/2+3/2-1/R=1,\,\,\,\,\text{so}\,\,\,\,\eta\Big(\frac{x_1-R/2}{R}+\varphi(t)\Big)=0.$$ Thus, we only get contribution from the $(x,t)\in {\mathbb R}^n\times [0,1]$ such that $\,R-1<x_1<R$, which can be bounded by $$c\,\int_{1/32}^{31/32}\,\int_{R-1<x_1<R}\,(|u|^2+|\partial_{x_1}u|^2)(x,t)\,e^{\alpha(2-1/R)^2}\,dx\,dt.$$ Finally, we look at the contribution of the term in $E_3$ in . In those the derivatives fall on $\,\eta$, thus they are supported in the region $$1\leq \frac{x_1-R/2}{R}+\varphi(t)\leq 1 + \frac{1}{2R},\;\;\,\,\,\,\,\frac{1}{2}<x_1<R,\,\,\,\,\,\;\;\frac{1}{32}<t<\frac{31}{32}.$$ Hence, their contribution in is bounded by $$c\,\int_{1/32}^{31/32}\,\int_{1/2<x_1<R}\,|u(x,t)|^2\,e^{\alpha(1+1/(2R))^2}\,dx\,dt \leq c_{\gamma}\,e^{\alpha(1+1/(2R))^2}.$$ Defining $$\label{defdelta} \delta(R)=\int_{1/32}^{31/32}\,\int_{R-1<x_1<R}\,(|u|^2+|\partial_{x_1}u|^2)(x,t)\,dx\,dt,$$ and collecting the above information using that $\,\alpha=c_n\,R^2$ we get $$c\,R\,b\,e^{\alpha(1+1/R)^2}\leq c\,\delta(R)\,e^{\alpha(2-1/R)^2}+ \,c_{\gamma}\,e^{\alpha(1+1/(2R))^2}.$$ Therefore, for $\,R\,$ sufficiently large it follows that (since $\,b\neq 0$) $$c\,R\,b\,e^{\alpha(1+1/R)^2}\leq c\,\delta(R)\,e^{\alpha(2-1/R)^2},$$ and since $\,\alpha=c_n\,R^2$ one has that $$\delta(R)\geq b\,e^{-c_n R^2}.$$ To conclude we recall that the upper bounds in gave us $$\delta(R)\leq c\,e^{-\gamma R^2},$$ hence if $\gamma>c_n/2$ we conclude that $\,b=0$, which yields the desired result $\,u\equiv 0$. Appendix {#aaa} ======== Above we have used the following abstract results established in [@EKPV08b]: \[L: freq1\] Let $\mathcal S$ be a symmetric operator, $\mathcal A$ be a skew-symmetric one, both allowed to depend on the time variable. Let $G$ be a positive function, $f(x,t)$ a reasonable function, $$\aligned &\;H(t)=\left( f, f\right)=\|f\|^2_{L^2({\mathbb R}^n)}=\|f\|^2\ ,\,\,\,\,\,\ D(t)=\left( \mathcal Sf, f\right),\\ &\; \partial_t\mathcal S=\mathcal S_t \quad \,\,\,\,\,\text{and}\,\,\,\,\,\quad N(t)=\frac{D(t)}{H(t)}\ . \endaligned$$ Then, $$\begin{gathered} \label{E: derivadasegunda} \begin{aligned} \partial_t^2H &= 2\partial_t\text{\it Re}\left(\partial_tf-\mathcal Sf-\mathcal Af,f\right)+ 2\left(\mathcal S_tf+\left[\mathcal S,\mathcal A\right]f,f\right)\\ &+\|\partial_tf-\mathcal Af+\mathcal Sf\|^2- \|\partial_tf-\mathcal Af-\mathcal Sf\|^2 \end{aligned}\end{gathered}$$ and $$\dot N(t)\ge \left(\mathcal S_tf +\left[\mathcal S,\mathcal A\right]f, f\right)/H- \|\partial_tf-\mathcal Af-\mathcal Sf\|^2/\left(2H\right).$$ Moreover, if $$\label{E: condicionesbase} |\partial_tf-\mathcal Af-\mathcal Sf|\le M_1|f| +G,\ \text{in}\ {\mathbb R^n}\times [0,1],\quad \mathcal S_t+\left[\mathcal S,\mathcal A\right]\ge -M_0,$$ and $$M_2=\sup_{[0,1]}{\|G(t)\|/\|f(t)\|}$$ is finite, then $\log H(t)$ is logarithmically convex in $[0,1]$ and there is a universal constant $N$ such that $$\label{E: convexidadlogaritmica} H(t)\le e^{N\left(M_0+M_1+M_2+M_1^2+M_2^2\right)}H(0)^{1-t}H(1)^t,\ \text{when}\ 0\le t\le 1.$$ By multiplying the formula by $\,t(1-t)$, integrating the result over $[0,1]$ and using integration by parts, one gets the following smoothing" inequality \[last\] With the same hypotheses and notation as in Lemma \[L: freq1\] $$\label{lastformula} \begin{aligned} &2\int_0^1\,t(1-t)\left(\mathcal S_tf+\left[\mathcal S,\mathcal A\right]f,f\right)\,dt+\int_0^1 \,H(t)\,dt\leq H(0)+H(1)\\ & + 2\int_0^1\,(1-2t)\text{\it Re}\left(\partial_tf-\mathcal Sf-\mathcal Af,f\right)\,dt\\ &+ \int_0^1\,t(1-t)\|\partial_tf-\mathcal Af-\mathcal Sf\|^2_2\,dt. \end{aligned}$$ [99]{} N. 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--- abstract: 'A review of the neutrino conversion and oscillations among the two neutrino species (active and sterile) induced by strong twisting magnetic field is presented and implications to neutrinos in neutron star, supernova, the Sun and interstellar galactic media are discussed. The “cross-boundary effect" (CBE) (i.e., a possible conversion of one half of neutrinos of the bunch from active into sterile specie) at the surface of neutron star is also studied for a realistic neutron star structure.' --- 1truecm 1000 [**NEUTRINO SPIN AND FLAVOUR CONVERSION AND OSCILLATIONS IN MAGNETIC FIELD**]{}\ \ [*Department of Quantum Statistic Physics, Physics Faculty, Moscow State University, 119899 Moscow, Russian Federation*]{} \ 2truemm [A.I. Studenikin]{}[^1]\ 2truemm [*Department of Theoretical Physics, Physics Faculty,\ Moscow State University, 119899 Moscow, Russian Federation*]{} Introduction ============ It is commonly believed that investigations of the neutrino properties will give a very important information for better understanding of particle interactions and for the progress in theoretical models. One of crucial problems of neutrino physics is the existance (or non existance) of the neutrino oscillations. Neutrino oscillations if observed experimentally would play an important role in explanation of different astrophysical phenomena. The subsequent investigations in this field are strongly stimulated first of all by a possible solution of the solar-neutrino puzzle on the base of the matter and magnetic field enhancement of spin and flavour neutrino conversion (see, for example, [@Wolf]–[@Lim] and [@Ber]– [@Pal] for a review). Another important motivation for consideration of neutrino conversion and oscillations is based on the common belief that these effects may be involved in the processes of supernova bursts and cooling of neutron stars (see [@Vol2]–[@Ath] and references therein). The basic idea of the neutrino conversion and oscillations between the two neutrino species in vacuum was put forward in [@Pon] and was supplied [@1; @Bil] with the time evolution analysis of a neutrino beam. Effects of neutrino interaction with matter of uniform density on neutrino conversion were considered in [@Wolf]. The flavour conversion in the case of neutrino propagation in matter with nonuniform density was studied in [@Mikh; @Mik] and the resonant amplification of neutrino oscillations was predicted (MSW effect). The neutrino spin precession in magnetic field as a possible solution for the solar-neutrino problem was studied in [@Cis]–[@Okun]. The resonant spin-flavour neutrino conversion that is analogous to the MSW effect was applied to the solar neutrino [@Akh; @Lim] and also to neutrino from supernova [@Lim]. It must be mentioned here that in the most of the performed studies of neutrino conversion and oscillations between two species in magnetized matter the considered strengths of magnetic field is of the order of $B\leq{10^5 \ G}$ that is quite adequate to the solar neutrino problem. There are also studies and discussions of the neutrino resonant conversion for the case of a supernova accounting for much stronger magnetic fields (see, for example, [@Lim; @Vol2; @Pelt; @Ath]). However, in some recent studies in this field the possible influence of strong magnetic fields on neutrino conversion and oscillations was not considered at all (see, for example, [@S-S]–[@Raf-S],[@APS]). The magnetic fields of the order of $B\sim 10^{12} - 10^{14} \ G$ are believed to exist at different stages of evolution of neutron stars. As an example we mention here new particle interaction phenomena [@ABS; @LS] that can be induced by strong magnetic fields and that can play a visible role in energetics of neutron stars (see also [@R]). The presence of strong magnetic fields may also influence the neutrino conversion and oscillations processes. In this paper supposing that neutrinos have non-vanishing magnetic or/and flavour transition moments we study the magnetic field induced effects of neutrino spin and/or spin-flavour conversion and oscillations between different neutrino species. Both the Dirac and Majorana neutrino conversion and oscillations effects induced by strong magnetic fields in the presence of matter, also accounting for mixing of neutrinos in vacuum are considered. We focus on the discussion of the case when under the influence of strong enough magnetic field numerous acts of conversion between the two neutrino specie occur (i. e., neutrino oscillations take place) for each individual neutrino of the bunch passing through different media. In Section 2 a general analysis of the problem is presented and the critical strength of magnetic field $\tilde B_{cr}$ as a function of characteristics of neutrino and matter is introduced (for magnetic fields $B\geq \tilde B_{cr}$ the magnetic field induced conversion and oscillation effects become important). The neutrino oscillations in magnetic field of a neutron star and the cross-boundary effect" (CBE) [@St]–[@prep] is discussed (Section 3). The CBE for a realistic neutron star structure accounting for variation of the matter density with distance from the centre of the star is studied in Section 4. In Section 5 the application of the magnetic field induced neutrino oscillations to the supernova reheating problem is discussed, and also neutrino oscillations in the galactic and twisting solar magnetic fields are considered. 5truemm General Analysis of Neutrino Oscillations in Magnetic Field =========================================================== The evolution of neutrinos propagating in matter and transverse twisting magnetic field $\vec B=\vec B_{\perp}e^{i\phi(t)}$, (the angle $\phi(t)$ defines the direction of the field in the plane orthogonal to the neutrino momentum) is described by the Schrödinger-type equation $$i{d\over d t}\nu(t)={H}\nu(t),$$ where the Hamiltonian $H$ can be expressed as a sum of the four terms ([@prep]–[@hadr]) $${H}={H}\sb V+{H}\sb{int}+{H}\sb F+H_{\dot\phi}.$$ Here $H_{V}$ contains a contribution from a vacuum mass matrix, $H_{int}$ contains a contribution from neutrino interactions with matter, $H_{F}$ contains a contribution from interactions with the magnetic field and the last term $H_{\dot\phi}$ accounts for the effect of rotation (twisting) of the magnetic field. If for the case of Dirac neutrinos one uses the bases in which neutrinos have a definite projection along the direction of propagation $$\nu=(\nu_{e_L}, \nu_{{\mu}_L}, \nu_{e_R}, \nu_{{\mu}_R}),$$ then the Hamiltonian is given by (see [@Pal],[@prep]– [@VW]) The Hamiltonian (4) corresponds to the case of sterile neutrinos $\nu_{e_R}$ and $\nu_{{\mu}_R}$. For the two Majorana neutrinos in the bases written as $$\nu=(\nu_{e},\nu_{\mu},\bar\nu_{e},\bar\nu_{\mu})$$ in the corresponding Hamiltonian where $\mu$ denotes the flavour transition magnetic moment. Using these Hamiltonians we can consider different neutrino conversion processes $\nu_i \rightarrow \nu_j$ and the corresponding neutrino oscillations $\nu_i \leftrightarrow \nu_j$, induced by the magnetic field such as $$\nu_{e_L} \rightarrow \nu_{e_R}, \ \nu_{e_L} \rightarrow \nu_{{\mu}_R},\ \nu_{e_L} \rightarrow \bar\nu_{{\mu}_R}.\label{proc}$$ The probabilities of neutrino conversion from the type $i$ ($\nu_i$) to the type $j$ ($\nu_j$) after passing a distance $x$ in matter and twisting magnetic field are $$P(\nu\sb i\to\nu\sb j)=\sin\sp2 2\theta\sb{eff}\sin\sp2\Big({\pi x\over L\sb{eff}} \Big),i\not=j,\label{ver}$$ while the survival probabilities are $$P(\nu\sb i\to\nu\sb i)=1-P(\nu\sb i\to\nu\sb j),$$ where the effective mixing angle $\theta_{eff}$ and effective oscillation length $L_{eff}$ are given by $$\tan2\theta\sb{eff}={2\tilde\mu B\over{{\Delta m_{\nu}^2\over2E}{\A}}-\sqrt{2} G\sb Fn\sb{eff}+{\dot\phi}},\label{tan}$$ $$L\sb{eff}=2\pi\Big[\Big({\Delta m_{\nu}^2\over2E}\A-\sqrt{2}G\sb Fn\sb{eff}+{\dot\phi}\Big) \sp2+(2\tilde\mu B)\sp2\Big]\sp{-1/2}.\label{L}$$ Note that the effective mixing angle $\theta_{eff}$ and effective oscillation length $L_{eff}$ depend on the characteristics of the magnetic field rotation $\dot\phi$ along the neutrino pass (see also [@APS; @Sm; @VW]). For different neutrino conversion processes (6) $\tilde{\mu}$, $\A$ and $n_{eff}$ are equal to $$\tilde\mu=\left\{\matrix{\mu\sb{ee}\hfill&\hfill for \ \nu\sb{e\sb L}\to \nu\sb{e\sb R}\cr \mu\sb{e\mu}\hfill&\hfill for\ \nu\sb{e\sb L}\to\nu\sb{\mu \sb R}\cr \mu\hfill &\hfill for\ \nu\sb{e\sb L}\to\bar\nu\sb{\mu\sb R}\cr} \right.,$$ $$$$ $$\A=\left\{\matrix{{1\over2}(\cos2\theta- 1)\hfill&\hfill for\ \nu\sb{e\sb L} \to\nu\sb{e\sb R}\cr {1\over2}(\cos2\theta+1)\hfill&\hfill for\ \nu \sb{e\sb L}\to\nu\sb{\mu\sb R}\cr \cos2\theta\hfill &\hfill for\ \nu\sb{e\sb L} \to\bar\nu\sb{\mu\sb R}\cr}\right.,$$ $$$$ $$n\sb{eff}=\left\{\matrix{n\sb e-n\sb n\hfill&\hfill for\ \nu\sb{e\sb L} \to\bar\nu\sb{\mu\sb R}\cr n\sb e-{1\over2}n\sb n\hfill&\hfill for\ \nu\sb{e\sb L} \to\nu\sb{e_R,\mu_R}\cr}\right..\label{n}$$ As it was in the case of non-twisting magnetic field [@prep] the probability (\[ver\]) may have a considerable value (the neutrino conversion processes and oscillations become important) if the following two conditions are valid: 1. the amplitude of oscillations" ${\sin^2}2\theta_{eff}$ is far from zero (or ${\sin^2}2\theta_{eff} \sim 1$), 2. and 3. the length $x$ of the neutrinos path in the medium must be greater than the effective oscillation length $L_{eff}$ ($x \sim $ or $>{L_{eff} \over 2}$). The condition 1) is realized if $\tan 2\theta_{eff}$ $\geq 1$, then from (9) it follows that at least one of the following two relations must be satisfied $${\Delta m_{\nu}^2\over2E}\A-\sqrt{2}G\sb Fn\sb{eff}+\dot\phi=0,\ \ (\tilde\mu B\not=0) \label{a}$$ $$2\tilde\mu B\geq\Big|{\Delta m_{\nu}^2\over2E}\A- \sqrt{2}G\sb Fn\sb{eff}+\dot\phi\Big|. \label{b}$$ Using the definitions (see, for example, in[@Bahc]) for the oscillation length in vacuum $$L_V={4\pi E\over\Delta m_{\nu}^2},$$ and the interaction oscillation length $$L_{eff}={2\pi\over\sqrt2G_Fn_{eff}},$$ one can write the equations (\[tan\]), (\[L\]) as $$L\sb{eff}=\Big[\Big({\A\over L\sb V}-{1\over L\sb{int}}+{1\over L_{\dot\phi}}\Big)\sp2+ \Big({1\over L\sb F}\Big)\sp2\Big]\sp{-1/2},\label{Leff}$$ where $$L\sb F={\pi\over\tilde\mu B},\ \ L\sb{\dot\phi}={2\pi\over\dot\phi}.$$ From the formula (\[ver\]) we can obtain the following expressions for probability in different cases $$P(\nu\sb i\to\nu\sb j)=$$ $$=\left\{\matrix{\Big({L\sb V\over L\sb F\A}\Big) \sp2\sin\sp2\Big({\pi x\A\over L\sb V}\Big),\hfill &\hfill {\rm for} \ L\sb F^{-1}\ll -L\sb{int}^{-1}+L^{-1}_{\dot\phi}\ll \A L^{-1}\sb V,\cr \Big({L\sb{int}\over L\sb F}\Big)\sp2\sin\sp2\Big({\pi x\over L\sb{int}}\Big), \hfill &\hfill {\rm for} \ L\sb F^{-1}\ll \A L^{-1}\sb V+L^{- 1}_{\dot\phi}\ll L^{-1}\sb{int},\cr \Big({L_{\dot\phi}\over L_F}\Big)^2\sin^2\Big({\pi x\over L_{\dot\phi}}\Big),\hfill &\hfill {\rm for} \ L^{-1}_F\ll \A L_V^{-1}-L_{int}^{-1}\ll L^{- 1}_{\dot\phi},\cr \sin\sp2\Big({\pi x\over L\sb F}\Big), \hfill &\hfill {\rm for} \ \A L_V^{-1}-L\sb{int}+L^{-1}_{\dot\phi}=0, \cr \to\sin\sp2\Big({\pi x\over L\sb F}\Big), \hfill &\hfill {\rm for} \ L\sb F^{-1}\gg \A L_V^{-1}-L_{int}^{-1}+L^{-1}_{\dot\phi} . \cr }\right.$$ The conditions (\[a\],) (\[b\]) can be re-written as $${\A\over L\sb V}-{1\over L\sb{int}}+{1\over L_{\dot\phi}}=0,\ \ L^{-1}_F\not=0$$ and $${1\over L\sb F}\geq\Big\vert{\A\over L\sb V}-{1\over L\sb{int}}+{1\over L_{\dot\phi}}\Big\vert.$$ Let us consider the relation (\[b\]) and suppose that the right- hand side is not equal to zero. In the case of exact equality from (\[b\]) we determine the critical strength of magnetic field [@sing]–[@hadr] $$\tilde B_{cr}= \left|{1\over2\tilde\mu}\Big({\Delta m_{\nu}^2 \A\over2E}- \sqrt{2}G_F n_{eff}+\dot\phi\Big) \right|\label{Bcr}$$ that constrain the range ($B \geq\tilde {B}_{cr}$) of field strengths for which the value of $\sin^2 2\theta_{eff}$ is not small (i.e., at least is not less than $1 \over 2$) for all possible values of the right-hand side term in (\[b\]). It is also possible to express $\tilde B_{cr}$ in a more convenient for numerical estimation form: $$\tilde B\sb{cr}\approx43\Big({\mu_B\over\tilde \mu}\Big)\Big|- \Big(2.5{n\sb{eff}\over10^{31} cm^{-3}}\Big)+ \A\Big({\Delta m\sp2\sb{\nu}\over eV^2}\Big)\Big({MeV\over E_{\nu}}\Big)+2.5\Big({1m\over L_{\dot\phi}}\Big) \Big|\ [Gauss].\label{BCR}$$ For the case of strong magnetic fields ($B > {\tilde B_{cr}}$), $\sin^2 2\theta_{eff} \approx 1$, we find that for large enough lengths of a neutrino $\nu_i$ pass given by $x\approx L_{eff}{ k\over 2}, k=1,2,...$ in the magnetized medium characterized by $n_{eff}$ the probability (\[ver\]) of conversion process $\nu_i \rightarrow \nu_j$ can reach the value of the order of $P(\nu_i \rightarrow \nu_j)\sim1$. Therefore, the initially emitted, for example, left-handed neutrino can undergo convertion to the right-handed neutrino or to the right-handed antineutrino on the path lengths $x\geq{ L_{eff}\over 2}$. It is obvious that these oscillation processes take place only in the presence of strong magnetic fields $B\gg\tilde B_{cr}$, and the oscillation length $L_{eff}$, as it follows from (\[L\]), is $L_{eff} \approx L_{F}=\pi/\tilde\mu B$. For $B\ll\tilde B_{cr}$ the influence of magnetic field is not important and oscillations (if they exist) are completely determined by the vacuum mixing angle and neutrino interaction with matter. 5truemm Neutrino Oscillations in Magnetic Field of Neutron Star (Cross-Boundary Effect) =============================================================================== Now let us consider neutrinos that are produced in the interior of a neutron star where magnetic fields of the order of $10^{13} \ G$ (or even a few orders of magnitude stronger) can exist (see, for example, [@Lan]– [@Lip]). For definiteness we suppose that initially $\nu_{eL}$’s are produced in the inner layers of the neutron star and take into account the only one of the conversion processes (\[proc\]), $\nu_{e_L} \rightarrow \nu_{e_R}$, that can be induced by the magnetic field on the neutrino pass from the centre to the surface of the neutron star. In order to determine the scale of $\tilde B_{cr}$ on the base of (\[Bcr\]) and (\[BCR\]) we use the following values for characteristics of neutrinos and matter of the neutron star: $ \mu \sim 10^{-10} \mu_B,\ \mu_B$ is the Bohr magneton, $n_{eff} \sim 10^{33}\ cm^{-3}$, $\Delta m^2_{\nu}\approx 10^{-4}\ eV^2$, $\sin 2\theta = 0.1$ and $E_{\nu} \approx 20\ MeV$. It follows that the main contribution is given by the matter" term and for this case $$B_{cr}=1.11\times10^{14}\ G,$$ (here in contrast with consideration of neutrinos from the Sun (Section 5) we do not account for a possible effect of twisting of the magnetic field). Magnetic fields just of this order may exist on the surfaces of neutron stars [@ST; @Lip]. From (\[L\]) for the effective oscillation length we get $L_{eff}\simeq 1 \ cm$, that is much less than the characteristic scales of the neutron star structures (the thickness of the crust is, for instance, $L_{crust} \sim 0.1 r_{0}\approx 1\ km$, $r_0$ is the neutron star radius). From these estimations we can conclude that for neutrinos passing from the inner layers to the surface the conversion and oscillations effects induced by the magnetic field can be important. However, if one is dealing not with a single neutrino but with a bunch of neutrinos that are emitted in different inner points of the neutron star then the average of the $x$ dependent term in formula (\[ver\]) must be taken. Therefore the probability of $\nu_{eR}$’s appearing in the initial bunch of $\nu_{e_L}$’s is given by $$\bar P(\nu_{e_R})= {1 \over 2} \sin^2 2\theta_{eff}.$$ It follows that the induced by strong magnetic field conversion and oscillations effects could yield in the approximate equal distribution of neutrinos between the two neutrino species ( $\sin^2 2\theta_{eff} \sim 1$ if $B\gg B_{cr}$); it also means that there would be a factor of two decrease in amount of initially emitted $\nu_{eL}$’s in the bunch. Now let us consider the case of not too strong magnetic field“, viz., $B<B_{cr}$ along the whole neutrinos path inside the neutron star. If we exclude the possibility for the neutrinos to pass through the resonant conversion point [@Akh; @Lim] determined by the Eq.(\[a\]) we then get that the neutrino bunch after travelling through the neutron star will still be composed only of the left-handed neutrinos. However, when the bunch of neutrinos escapes from the neutron star it passes through a sudden change of density of matter and enters into the nearly empty space where $n_{eff}\rightarrow 0$. Effectively it may result that the neutrinos enter and pass through the region of strong field ($B>B_{cr}$) determined on the base of Eq. (\[b\]). The neutrino conversion processes and oscillations may thus appear due to the cross-boundary effect” (CBE) [@sing; @prep]. To consider the CBE we suppose that the magnetic field on the surface of the neutron star is of the order of $B \sim B_0=10^{12}\ G$ and that the strength of the magnetic field decreases with the distance $r$ from the surface of the neutron star according to the law $$B(r)=B\sb 0\Big({r\sb 0\over r}\Big)\sp3,\label{Bs}$$ where $r_0$ is the radius of the neutron star. The estimation for the critical field $B^{\prime}_{cr}$ on the base of (\[Bcr\]) for the same values of $\mu$, $\Delta m^2_{\nu}$, $E_{\nu}$ and $\sin 2\theta$ (again for definiteness the conversion of the type $\nu_{eL} \leftrightarrow \nu_{eR}$ is considered) gives that $$B^{\prime}_{cr}=5.4 \times 10^3\ G.$$ From (\[Bs\]) it follows that the magnetic field exceeds $B^{\prime}_{cr}$ in regions characterized by $$r\leq r_{cr} \approx 600 r_0.$$ Therefore, along the distances of about $600 r_0$ from the neutron star the magnetic field exceeds the critical field strength $B^{\prime}_{cr}$. From the estimation for the effective oscillation length for the magnetic field at the surface of the neutron star $$L_{eff}(B \sim B_0) = {\pi \over {\tilde \mu B_0}}\simeq10^2{\mu_B\over\tilde\mu }\Big({1G \over B_0}\Big)[m] =1\ m$$ it follows that the equal distribution of neutrinos between the two neutrino species ($\nu_{e_L}$ and $\nu_{e_R}$) appears after the neutrino bunch passes through a thin layer $\Delta r \gg 1 \ m$ along which the decrease of the magnetic field is still negligible: $\Delta B(\Delta r) \ll B_0$. Thus, in the case of not too strong field“ again as it was in the case of strong field” after the neutrino bunch has passed a distance $L>L_{eff}$ from the neutron star the equal distribution of neutrinos among the two species $\nu_{e_L}$ and $\nu_{e_R}$ appears. Consider the case when the neutrinos on their path inside the neutron star pass through the resonant region [@Akh; @Lim]. In this region the condition of Eq.(\[a\]) is valid. From (\[b\]) it follows that for any fixed strength of the magnetic field there is a layer ( between the two shells with radiuses $r_1$ and $r_2$) on the neutrino path to the surface of the neutron star where effectively the strong field" case is realized. If the distance $ r_2 - r_1 $ is greater than the effective oscillation length $L_{eff} \sim L_{F}$ then after neutrinos pass through this [*resonant region*]{} again the equal neutrino distribution between the two neutrino species appears. Here it is important to note that within the discussed case of the CBE the adiabatic approximation can be used. In the most general case the adiabatic condition is $$(H_{jj}-H_{ii}){\partial\over\partial r}(H_{ij}+H_{ji})- (H_{ij}+H_{ji}){\partial\over\partial r}(H_{jj}-H_{ii}) \ll\hfill $$ $$\hfill2[(H_{jj}-H_{ii})^2+(H_{ij}+H_{ji})^2]^{3/2} \label{ADIA}$$ where $H_{ij}$ are the elements of matrixes of eqs. (4) or (5). Using expressions for $H_{ij}$ corresponding to the neutrino conversion $\nu_i\to$ $\nu_j$ we reduce the adiabatic condition (29) to the form $$\Big|\tilde B_{cr}{\partial B\over\partial r}- B{\partial\tilde B_{cr}\over \partial r}\Big|\ll4\tilde\mu(\tilde B_{cr}^2+B^2)^{3/2},\label{adia}$$ that means a slow variation of the magnetic field $B$ and the matter density $\rho$ $(\tilde B_{cr}=\tilde B_{cr}(n_{eff}))$ with distance $r$. The magnetic field $B$ is slowly varying function of $r$, whereas $\rho$ undergoes a rather abrupt change from the value $\rho_s\sim10^9\ g\times cm^{-3}$ at the surface of the neutron star to the value of nearly empty space $\rho_{vac}\to0$. For the chosen above values of $\tilde\mu$, $\Delta m_{\nu}^2$, $E_{\nu}$ and $\dot\phi=0$ we get from (\[adia\]) that the adiabatic condition is valid if matter density changes from $\rho_s$ to $\rho_{vac}$ on the distance $\Delta r\ge\Delta r_{\rho}=10\ cm\ll L_{eff}$. Therefore, even in the case of extremely abrupt change of the matter density the non-adiabatic effects can be neglected. Now in the continuation of discussions of Refs. [@St]–[@prep] we should also like to mention that the CBE can take place not only at the surface of the neutron star (when neutrinos escape the matter of the neutron star and start there path in the empty space where particle number densities $n_e,$ $ n_n,$ $ n_p \to 0$). The CBE can effectively appear for the Majorana neutrinos passing through inner layers of the neutron star composed of silicon, oxygen, nitrogen, carbon and helium. For these shells $n_{eff}=n_e-n_n \to 0$, that corresponds to isotopically neutral medium (see also [@APS]). Because of the CBE in the inner layers of the neutron star a reasonable amount of active neutrinos can be converted to the sterile (non interacting with matter) neutrinos that may cause changes in the process of neutron stars cooling. In the next Section we shell study the CBE at the surface of the neutron star using a model of the neutron star structure provided by a realistic equation of state for the matter of neutron star. 5true mm Cross-Boundary Effect for Realistic Neutron Star Structure ========================================================== Let us consider the CBE at the surface of the neutron star using a realistic model of the star structure to account for change of the matter density with distance from the centre of the star. Neutron star structure is calculated (see, for example, [@ST]) assuming that the equation of state for neutron star matter, $P=P(\rho)$ ($P$ is the pressure, $\rho$ is the mass density) at $\rho\geq 2\times10^{14}(g\times cm^{-3})$ is that of three-nucleon interaction (TNI) model [@FP]. For $\rho$ within the interval $4.3\times10^{11} (g\times cm^{-3}) \leq\rho\leq 2\times10^{14}(g\times cm^{-3})$ the Baym-Bethe-Pethieck (BBP) equation of state is used. At $\rho<4.3\times10^{11}(g\times cm^{-3})$ we use the Baym-Pethick- Sutherland (BPS) equation of state [@ST; @BPS]. For description of the non-rotating star composed of cold matter one have to integrate the general-relativistic equation of hydrostatic balance, the Tolmen-Oppenheimer-Volkoff (TOV) equation [@ST] $${dP\over dr}=-{G(\rho+P)(m(r)+4{\pi}{r^3}P) \over r^2(1-2Gm(r)/r)} , \label{P}$$ $$m(r)=\int\limits_{0}^{(r)}\rho(r^{\prime}){d^3}r^{\prime},$$ where $m(r)$ is the mass of the star, $r$ is radial coordinate ($r=0$ for the centre of the neutron star), $G$ is the gravitational constant. In equation (\[P\]) the preasure $P$ is defined as a function of the density $\rho$ by equation of state. For outside of neutron star we use the BPS equation of state. This model implay that matter is composed of free nuclei, electrons and neutrons. The Coulomb interaction energy is also accounted and the equation of state can be represented as a system (see, for example, [@ST]): $$\left\{ {\rho=\varepsilon=n_{e}M(A,Z)/Z+{\varepsilon}_{e} +\varepsilon_L ,\atop P=P_e+P_L,} \right. \label{33}$$ Here $\varepsilon$ is the total energy (per unit volume), $n_e$ is the electron density, $M(A,Z)$ is the energy of a nucleus $(A,Z)$, $\varepsilon_e$ is the energy of electrons without the energy of the Coulomb interaction, $\varepsilon_L$ is the Coulomb interection between electrons and electrons with nuclei, $P_e$ is the partial pressure of the electrons and $P_L={1\over3}\varepsilon_L.$ The values $n_e,\varepsilon_L,P_e,\varepsilon_e$ are represented as functions of parameter $X_e=p_F^e/m$ ($p_F^e$ is the electron Fermi momentum) \[36\]: $$n_{e}={1\over3{\pi}^2{\lambda}_{e}^3}{X}_{e}^3$$ (${\lambda}_{e}={1\over{m}_{e}}$ is the electron compton wavelength), $$\varepsilon_L=-1.444{Z}^{2/3}e^2{n}_e^{4/3},$$ $$P_{e}={m_e\over{\lambda}^3_e}{\Phi}(X_e) ,$$ where $\Phi(x)={1\over8\pi^2}\{x{(1+x^2)}^{1/2}(1+2x^2)- \ln[x+(1+{(1+x^2)}^{1/2}]\},$ $$\varepsilon_e={m_n\over\lambda^3}\chi(X_e),\label{38}$$ where $\chi(x)={1\over8\pi^2}\{x(1+x^2)^{1/2}(1+2x^2)- \ln[x+(1+x^2)^{1/2}].$ We assume that the density of the free neutrons is equal to zero because it is below the density of the neutron drip [@ST]. The values of A and Z used in the sistem (\[33\]) minimize the energy $\varepsilon$ that corresponds to the equilibrium nucleus. For definiteness we again consider the case of oscillations among the Majorana neutrinos ($\nu_{e_L}\leftrightarrow\overline{\nu}_{\mu_R}$). From (\[n\]) whithin the discussed model of the neutron star matter we have $$n_{eff}=n_e-\Big({A\over Z}-1\Big)n_e.\label{NEFF}$$ Using eqs. (\[33\])–(\[38\]) we perform a computer calculations of $n_{eff}$ as a function the distance from the centre of the neutron star and then determine $B_{cr}$ (see (\[Bcr\])) for the different values of $r$. It is interesting to compare the calculated value of the critical field $B_{cr}(r)$ and the neutron star magnetic field $B(r)$ for different distances $r$ from the centre of the neutron star. We suppose that the neutron star magnetic field for rather large distances from the surface ($r\geq r_0\approx10\ km$) can be approximated by eq. (\[Bs\]). However, in the close to the star surface layers the magnetic field may exhibit the more complicated behavior with variation of $r$. It is possible to suppose that the field decrease with the distance from the surface of the star ($r=r_0$) not faster than it follows from the law $B\sim\rho^{2/3}$ (the field frozen in the matter). Using these suggestions on the profile of the magnetic field of the star we plot (Fig. 1) the dependence of the critical field $B_{cr}(r)$ (solid line) and the neutron star magnetic field $B(r)$ on the distance $r$ for the close to the star surface layers ($r\sim r_0$). It is supposed that $B(r=r_0)\approx10^{14}G$. The solid line with dots corresponds to the case when the star magnetic field $B\sim\rho^{2/3 }$. $$$$ 105truemm The dashed line shows the behavior of the field for the case when the field profile is given by $B(r)\sim1/r^3$. is Fig. 2 shows the dependence of $B_{cr}(r)$ (solid line) and the neutron star magnetic field $B(r)\sim{1/r^3}$ (solid line with dots) on $r$ for remote distances ($r\geq r_0$). For large distances ($r\gg r_0$) $B_{cr}(r)$ nearly equal to its vacuum value $$B_{cr}\approx{\Delta m^2_{\nu}\A\over4\tilde\mu E}.$$ From these figures it follows that nearly for the whole space from the surface of the neutron star ($r_1=10.32\ km$) up to the distances $r_2\sim10^3\ r_0$ the magnetic field $B(r)$ exceeds the critical field $B_{cr}(r)$. Taking into account that for the field $B\sim10^{14}G$ the effective oscillation length $L_{eff}$ determined by eq. (10) is of the order of $\sim1\ cm$ we conclude that the magnetic field induced neutrino oscillation effects can be important for the space characterized by $r_1\leq r\leq10^3\ r_0$. On Fig. 3 we plot the averaged probability $P_{av}=\overline{P}(\overline{\nu}_{\mu_R})$ (similar to one of eq. (24)) that determines the amount of neutrinos $\overline{\nu}_{\mu_R}$ in the initial bunch of neutrinos $\nu_{e_L}$ as a function of $r$. For each individual neutrino (initially emited as $\nu_{e_L}$) the non-averaged probabilty $P(\overline{\nu}_{\mu_R})$ to detect the neutrino in the state $\overline{\nu}_{\mu_R}$ oscillate with the change of $r$ around the average value $\overline{P}(\overline{\nu}_{\mu_R})$ with the amplitude determined by $B(r)/B_{cr}(r)$. of the probability variation $P(\overline{\nu_{\mu_R}})$ in the narrow 5truemm Supernova Reheating Problem, Neutrino Oscillations in Galactic and Twisting Solar Magnetic Field ================================================================================================ The effects discussed above of suppression of amount of electron neutrinos (or other active neutrinos) induced by strong magnetic fields may have sufficient consequences on the reheating phase of a Type II supernova that can be used for getting constraints on the value $\tilde\mu B$. Let us suppose that the magnetic field induced neutrino oscillations do not destroy the proposed model [@FMMW] of about 60 %  increase in the supernova explosion energy. If the magnetic field $B\sim 10^{14}\ G$ exists at the radius of $r_0= 45 \ km$ from the centre of the hot proto neutron star (the matter density in this region is $\rho\sim 10^{12} \ g/cm^3$) and decreases with distance according to (\[Bs\]) then on the distances $r\sim 160 \ km$ from the centre the magnetic field is $\sim 0.6\times 10^{13} \ G$. This field is of the order of the $ B_{cr}$ determined by (\[Bcr\]) for the density $\rho \sim 6\times 10^8 \ g/cm^3$ and the magnetic moment $\tilde\mu\sim 10^{- 10}\mu_B$. For this case the probability of finding, for example, sterile $\nu_{eR}$’s among the initially emitted $\nu_{eL}$’s is $\bar P_{\nu_{eL}\rightarrow \nu_{eR}}=0.25$ (the effective length (\[L\]) for this effect is $L_{eff}\sim \ 10\ cm$). Therefore, in order to avoid the loss of a substantial amount of energy that will escape from the region behind the shock together with the sterile neutrinos, one has to constrain the magnetic moment on the level of $\tilde\mu \leq 10^{-11} \mu_B$. We should like to point out the importance of the resonance enhancement [@Akh; @Lim] of neutrino conversion and oscillations effect in magnetic fields that may substantially change the energetics of the shock and also give a stringent constraints on the value of $\tilde\mu B$. It is also interesting to consider the neutrino conversion and oscillations induced by the interstellar galactic magnetic fields that are of the order $B_G\sim 10^{- 6}\ G$. The critical field estimated on the base of eqs. (\[Bcr\]) for ultra high energy neutrinos ($E\geq 10^{17}\ eV$) are $\leq 10^{-6}\ G$. Taking into account the estimation for the effective oscillation length $L_{eff}(B\sim B_G)=10^{20}\ cm$, that is much less than the radius of the galaxy ($R_G\approx 3\times10^{22}\ cm$) we conclude that in this case the effect of neutrino conversion and oscillations in strong magnetic field" can be presented. Now let us consider the possibility of the twisting magnetic field induced neutrino conversion and oscillations (for definiteness we chose the process $\nu_{e_ L}\to\nu_{e_R}$) in the convective zone of the Sun. First of all we estimate the critical field strength, using eqs. (\[Bcr\]) with the following values for characteristics of neutrinos and matter: $\Delta m_{\nu}^2=10^{-4}\ eV^2,\ \sin2\theta=0.1,\ E_{\nu}=20\ MeV,\ n_{eff}\sim n_ e\approx10^{23}\ cm^{-3}$. For the characteristic of the variation of the field in the convective zone along the neutrinos path we suppose that $\dot\phi>0$ and use the estimation of Refs.[@APS; @Sm]: $L_{\dot\phi}\sim0.1R_{\bigodot}\approx7\times10^7\ m, \ R_{\bigodot}=7\times10^8\ m$ is the solar radius. Substituting these values to three terms of eq. (\[Bcr\]) we get $$\tilde B_{cr}\approx\Big({\mu_B\over \tilde\mu}\Big)\Big|-10^{-6}-5\times10^{-7} +1.43\times10^{-6}\Big|\ G=7\times10^{- 8}\Big({\mu_B\over\tilde\mu}\Big)\ G.$$ From this it is obvious that the account for the twisting of magnetic field reduce the value of critical field $\tilde B_{cr}$ to the order of 5% of the value $B_{cr}$ that corresponds to the case of non-twisting field. It is supposed that the typical value of magnetic fields in the convective zone is of the order of $B_{con}\sim10^5\ G$. From (\[Bs\]) it follows that $B_{con}$ exceeds the field $\tilde B_{cr}$ if the neutrino magnetic moment is greater then $\tilde\mu\geq10^{-12}\mu_B$. According to the second condition for the magnetic field induced neutrino conversion and oscillations become important the effective oscillation length $L_{eff}$ have to be of the order or less then the depth of the convective zone $L_{eff}\leq {1\over2}L_{ cz}$. This last condition holds if $\tilde\mu\sim10^{-11}\mu_B$ for the magnetic fields in the convective zone $B\sim10^5\ G$. We can conclude that the account for the variation (twisting) of the magnetic field along the neutrino path in the solar convective zone relaxes the critical field strength $\tilde B_{cr}$ to the values which can be relevant for the stimulation of a visible neutrino conversion and oscillations if the neutrino magnetic moment is of the order of $\tilde\mu\sim10^{-11}\mu_B$. 5truemm Summary ======= The Dirac and Majorana neutrino conversion and oscillations between the two neutrino species induced by the magnetic field is considered. We introduce the critical magnetic field strength $\tilde B_{cr}(\Delta m^2_{\nu}, \theta, n_{eff}, E,\tilde\mu,\dot{\phi})$ that determines the range of fields ($B\geq \tilde B_{cr}$) for which the magnetic field induced neutrino conversion and oscillations become important. This criterion is valid in the case of resonant and non-resonant amplification of neutrino conversion and oscillations in magnetic fields. 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--- abstract: 'The advancement of various research sectors such as Internet of Things (IoT), Machine Learning, Data Mining, Big Data, and Communication Technology has shed some light in transforming an urban city integrating the aforementioned techniques to a commonly known term - Smart City. With the emergence of smart city, plethora of data sources have been made available for wide variety of applications. The common technique for handling multiple data sources is data fusion, where it improves data output quality or extracts knowledge from the raw data. In order to cater evergrowing highly complicated applications, studies in smart city have to utilize data from various sources and evaluate their performance based on multiple aspects. To this end, we introduce a multi-perspectives classification of the data fusion to evaluate the smart city applications. Moreover, we applied the proposed multi-perspectives classification to evaluate selected applications in each domain of the smart city. We conclude the paper by discussing potential future direction and challenges of data fusion integration.' author: - '[Billy Pik Lik Lau, Sumudu Hasala Marakkalage, Yuren Zhou, Naveed Ul Hassan, Chau Yuen, Meng Zhang, U-Xuan Tan]{}[^1][^2][^3]' bibliography: - 'bibSpace.bib' title: A Survey of Data Fusion in Smart City Applications --- [Shell : Bare Demo of IEEEtran.cls for IEEE Journals]{} Data Fusion; Sensor Fusion;Urban Computing; Smart City; Big Data; Internet of Things; Multi-Perspectives Classification Introduction {#sec:Introduction} ============ According to UN estimates [@un2018world], $68$% of the world population would be living in cities by 2050. Hence, managing the existing resources and infrastructure to cater sustainable urban living conditions for the growing needs of the urban population has become ever more challenging. Fortunately, the advancement in Information and Communication Technologies (ICT), Internet of Things (IoT), Big Data, Data Mining, and Data Fusion is gradually paving path for the emergence of smart cities [@boulton201118; @hollands2008will; @nam2011smart]. In this paper, we adopt the following definition of smart city [@toppeta2010smart]: *“A city combining ICT and Web 2.0 technology with other organizational, design and planning efforts to de-materialize and speed up bureaucratic processes and help to identify new, innovative solutions to city management complexity, in order to improve sustainability and livability”* The integration of aforementioned technologies into various urban domains enables city managers to equip with the necessary information for better planning and resource management. Several cities around the world have already been leveraging these technologies to improve the comfort, security, mobility, health, and well-being of their citizens. To better evaluate rapid progress and to recognize the efforts of urban planners, smart city ranking systems have been established. For instance, IESE cities in motion index [@berrone2018iese] has suggested $83$ indicators to rank $165$ cities over $80$ countries. New York, London, and Paris are the top three smart cities in $2018$. Smart city projects in New York [@NewYork_:2018] aim to consistently improve the quality of residents’ life, reduce the environmental impacts, increase the street light efficiency, and enhance the water quality. Meanwhile, the focus of Smart London Projects [@GreaterLondonAuthority_:2018] is to collect city wide data to provide world class connectivity, security, and smarter streets to its residents. Digital transformation, sustainability, and urbanization for improving citizen services are at the cores of Paris Smart City Projects [@Paris_:2018]. The following up of the top smart city list includes Singapore and Tokyo, which are some other notable smart cities in the world. In Singapore, Smart Nation Project [@Nation_:2018] has been proposed, which includes e-payment systems, smart nation sensor platform, smart urban mobility, and smart community initiatives, with the aim to enhance the national digital identity of its citizens. On the other hand, Tokyo [@Government_:2016] aims to become the greenest city in Asia Pacific by improving the transportation and other sectors of their economy. Local governments in several Chinese cities [@Limited_DCPS:2018], such as, Shenzhen, Shanghai, Hangzhou, and Beijing are also shaping up their cities to facilitate economic and social development to build high income smart cities. In addition, there are several research institutes and laboratories focusing on developing smart city applications, which are currently leading the worldwide effort in smart domains. These include MIT Senseable Lab [@MIT_:2018], Future Cities Laboratory [@Zurich_:2018], SINTEF Smart Cities [@SINTEF_:2018], SMART [@SMART_:2018], etc. \[tbl:1\_reviewList\] **Surveys** **Objectives and Topics Covered** ----------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Khaledgi et al. [@Khaleghi_If:2013] Provides insights on the different types of data fusion techniques by exploring their concept, benefits, and challenges. Castanedo [@Castanedo_TSWJ:2013] Provides an overall view on the different data fusion techniques and methods. The author also reviewed common algorithms such as data association, state estimation, and decision fusion. Alam. et al. [@Alam_IA:2017] Provides a comprehensive survey on the mathematical model used in data fusion for specific IoT environments. Wang et al. [@Wang_IJoDSaTI:2016] Proposes an IoT architecture concept to survey on the different sensor data fusion techniques and also provides an overall view on their evaluation framework. Zheng [@zheng2015methodologies] Discusses about differences on fusing sources and varying techniques for cross domain data fusion. El et al. [@ElFaouzi_IF:2011] Provides a survey on the intelligent transportation systems, which use data fusion techniques. Esmaeilian et al. [@Esmaeilian_WM:2018] Provides a throughout study on waste management for smart city aspects with three categories: (1) infrastructure for the collection of product lifecycle data, (2) new adapting business model, and (3) waste upstream separation techniques. Da Xu et al. [@DaXu_IToii:2014] Provides an overall view on the current state of the industries for IoT and discusses key enabling technologies such as communication platforms, sensing technologies, and services. Chen et al. [@Chen_EaB:2018] Reviews the building occupancy estimation and detection techniques while providing a comparison between different sensor types for cost, detection and estimation accuracy, and privacy issues. Qin and Gu [@Qin_PE:2011] Introduces the data fusion algorithms in IoT domains and data acquisition characteristics. : Literature Review for Data Fusion on Smart City Nowadays, communication technology is the backbone for the smart city applications as it provides a channel for applications to transfer data effortlessly. The ongoing quest for novel, more efficient, low-latency, and cost-effective communication technologies and networks, such as, 5G [@Huang2019Iterative; @andrews2014will; @boccardi2014five], wireless sensor networks (WSN) [@Erol-Kantarci_IToSG:2011; @sreesha2011cognitive; @yue2018comprehensive], Low Power Wide Area Network (LPWAN) [@georgiou2017low; @raza2017low], and Narrow Band IoT (NB-IoT) [@chen2017narrow; @wang2017primer] and their integration in smart city projects is also relentless. These advancement has made many data sources available due to the potential of sensors collecting data with better coverage and power efficiency of the communication platform. With the large amounts of data becoming readily available in a smart city, data mining techniques [@Hashem_IJoIM:2016; @han2011data] are commonly used in the collected data. It helps in identifying the essential and important data sources in the smart city applications such as monitoring, control, resource management, anomaly detection, etc. With the availability of parallel data sources in various smart city domains, data fusion techniques that combine multiple data sources, lie at the heart of smart city platform integration. The major objectives of data fusion are to address problematic data while enhancing the data reliability and extracting knowledge from multiple data sources. [[ The existing survey papers related to smart city applications or data fusion classification are summarized in Table \[tbl:1\_reviewList\]. Majority of these review papers [@Castanedo_TSWJ:2013; @Dasarathy_PotI:1997; @DurrantWhyte_Tijorr:1988; @Steinberg_:2008] strictly focus on one particular smart city domain or one genre of classification perspective. In [@Alam_IA:2017], Alam et al. have conducted a review on data fusion technique based on mathematical model in IoT environment. Alternately in [@Wang_IJoDSaTI:2016], Wang et al. have described the frameworks of data fusion within the smart city application. Interested readers can follow these references for additional technical details.]{}]{} However, there is only a handful of limited work to provide a multi-perspectives approach for data fusion problems in smart cities and this literature gap further motivates our study. ![image](Fig01_SmartCityCat3.pdf){width="100.00000%"} Therefore, a different perspective to look at data fusion in smart city domains is necessitated by the expanding scale and scope of data sources, data collection techniques, and data processing system architectures. In order to cater evergrowing highly complicated applications, studies in smart city have to utilize data from various sources and evaluate their performance based on multiple aspects. To this end, we propose multiple generic perspectives with the ability to cover the entire depth and breadth of data fusion problems in smart city. These perspectives include data fusion objectives, data fusion techniques, data input and data output types, data sources, data fusion scales, and platform architectures for data processing. Utilizing proposed perspectives, we provide an overall view of classification techniques found in the seven domains of smart city applications such as: Smart Living, Smart Urban Area Management, Smart Environment, Smart Industry, Smart Economics, Smart Human Mobility, and Smart Infrastructure. A simple illustration of seven application domains discussed in this paper can be found in the Figure \[fig:Fig01\_SmartCityCat\]. In each domain, we only select notable papers to demonstrate the universality and effectiveness of our multi-perspective approach on evaluating the data fusion techniques. Please note that we do not provide a comprehensive review of all the smart city applications. Afterwards, we talk about emerging data fusion trends in smart cities, while outlining the best practices for deploying a smart city application. In addition, data fusion challenges in different smart city applications are also identified and discussed. To summarize, our novel contributions in this paper are three-fold as shown below: - We propose a multi-perspectives classification to evaluate common data fusion techniques in smart city applications. - We provide an overview of smart city application domains and discuss the common trend of data fusion techniques in each domain utilizing proposed multi-perspectives classification. - We list down the future challenges and the ideal scenario for deploying data fusion techniques in a smart city application. Overall, we believe that with these contributions, the readers would have a quick grasp on the current data fusion trends in smart city research without extensively going through all the details. [[ The rest of the paper is organized as follows: in Section \[sec:categoryDefinition\], we define the data fusion classification using multi-perspectives to evaluate a smart city application. This lays a foundation for evaluating the smart city applications leveraging data fusion techniques. In Section \[sec:currentSmartCityApp\], different application domains of smart city based on data fusion techniques are evaluated using the proposed multi-perspectives classification of data fusion. In addition, a brief overall view of the current research trend of respective domain is presented. Subsequently in Section \[sec:discussion\], we discuss the ideal data fusion scenario along with potential research directions/opportunities based on speculations of smart city applications from previous section. Lastly, we conclude our works in Section \[sec:conclusion\].]{}]{} Data Fusion Classification using Multi-perspectives {#sec:categoryDefinition} =================================================== In this section, we identify multiple generic perspectives with the ability to cover the entire depth and breadth of data fusion literature in smart city applications. We use smart city single perspective data fusion review papers [@Alam_IA:2017; @Qin_PE:2011] and non-smart city data fusion classification papers [@Castanedo_TSWJ:2013; @Dasarathy_PotI:1997; @DurrantWhyte_Tijorr:1988; @Steinberg_:2008] as references. In non-smart city literature, there are four well-known data fusion classification techniques, which are Dasarathy’s Classification [@Dasarathy_PotI:1997], Whyte’s Classification [@DurrantWhyte_Tijorr:1988], Fusion Architecture’s Classification [@Castanedo_TSWJ:2013], and US Joint Directories of Laboratories (JDL) data fusion classification [@Steinberg_:2008]. Dasarathy’s Classification is based on the data input and output types between data, where Whyte’s Classification focuses on the relationship between the data. JDL focuses on classifying the fusion process according to five processing levels. Meanwhile, the architecture-based classification only captures the system design level and does not consider data relationships and types. Most of the aforementioned classification of the data fusion techniques are not suitable for evaluating the applications of a smart city. Our proposed data fusion classification approach for smart city comprises of six different perspectives (also called categories): i) data fusion objectives ($O$), ii) data fusion techniques ($T$), iii) data input and output types ($D$), iv) data source types ($S$), v) system scales ($L$), and vi) platform architectures ($P$). Within each category, we further identify various sub-categories (also called classes). Overall, there are $30$ different classes. The complete list of the adopted classification indicating all the categories and their classes is shown in Table \[tbl:2\_categoryList\]. Short reference codes ($O$,$T$,$D$,$S$,$L$,$P$) for each class are also included in the table for further use in the paper. For example, $O1$ refers to the data fusion objective category and problematic data fusion class. Similarly, $S3$ refers to data source types category and participatory class. **Perspective/Category** **Code** **Classes** -------------------------- ------------- ---------------------------------------- O1 Fixing Problematic Data O2 Improving Data Reliability O3 Extracting Higher Level Information O4 Increasing Data Completeness T1 Data Association T2 State Estimation T3 Decision Fusion T4 Classification T5 Prediction / Regression T6 Unsupervised Machine Learning T7 Dimension Reduction T8 Statistical Inference and Analytics T9 Visualization D1 Data in Data Out (DAI-DAO) D2 Data In Feature Out (DAI-FEO) D3 Feature in Feature Out (FEI-FEO) D4 Feature in Decision Out (FEI-DEO) D5 Decision in Decision Out (DEI-DEO) S1 Physical Data Sources S2 Cyber Data Sources S3 Participatory Data Sources S4 Hybrid Data Sources L1 Sensor Level Fusion [[ L2]{}]{} [[ Building Wide Fusion]{}]{} [[ L3]{}]{} [[ Inter-Building Fusion]{}]{} L4 City Wide Fusion [[ L5]{}]{} [[ Inter-City Fusion (or Larger)]{}]{} P1 Edge Computation P2 Fog / Mist Computation P3 Cloud Computation P4 Hybrid Computation : Data Fusion Classifications for Smart City Applications using Multi-perspectives []{data-label="tbl:2_categoryList"} [[ Note that, there could be potentially more than one perspectives (other than data sources, fusion scales, and platform architecture) for smart city application depending on the complexity and fusion objective itself.]{}]{} Below, we provide further details of all the perspectives and classes adopted in this paper. Data Fusion Objectives (O) -------------------------- The data fusion techniques deployed in a smart city project is influenced by the objective of applications. In this paper, we have summarized the four objectives as follows: - **O1: Fixing Problematic Data**\ ‘Problematic Data’ class refers to the case when the data source is having quality issues such as, inconsistency, imperfection, disparateness, etc. Data fusion could be used as an easy approach to overcome such problems. [[ Examples of $O1$ can be found in [@Huang2019Iterative; @grime1994data; @cheng1997urban; @Huang2019Deep]. ]{}]{} - **O2: Improving Data Reliability**\ Data may suffer from reliability issues when it is collected in a less ideal (less controlled) environment with high presence of noise. In such situation, additional data sources are required to add redundancy for increasing data quality to enhance data reliability. [[ Such situations are identified as ‘Data Reliability’ class and [@hong2009evidential; @shen2016long; @kreibich2014quality; @luo2016kernel] exhibits such pattern.]{}]{} [[ In addition, security enhancement through the data fusion also belongs to this category and examples of such objectives can be found in [@li2011communication; @petit2015remote; @guo2018roboads].]{}]{} - **O3: Extracting Higher Level Information**\ Data mining advancement has contributed to many different architectures of data fusion in order to obtain knowledge from multiple data sources. For instance, the occupancy of a building can be detected using a combination of few ambient sensors with data fusion, where occupancy information cannot be directly inferred from the raw data sources. [[ We classify these approaches as ‘Higher Level Information Extraction’ class and examples can be found in [@dawar2018convolutional; @jayasinghe2019feature; @ghorpade2015integrated]. ]{}]{} - **O4: Increasing Data Completeness**\ In a situation of coverage limitations, an individual data source is insufficient to provide complete details of the desired output. [[ Therefore, in ‘Data Completeness’ class, data fusion is performed across multiple data sources to obtain a complete picture of the overall system such as [@luan2010smart; @consoli2015urban; @ricci2002travel].]{}]{} Data Fusion Techniques (T) -------------------------- In this category, we present the data fusion techniques in two different information enrichment obtained after data fusion. The $T1$ until $T3$ are the common data fusion techniques and the further details can be found in [@Dasarathy_PotI:1997; @Alam_IA:2017], where it describes the lower level information being fused to generate identical level of information. The techniques $T4$-$T8$ are associated with data mining [@han2011data; @kotsiantis2007supervised], where simple input data from multiple sources is fused to generate higher level information enrichment. Brief description of these classes is given below: - **T1: Data Association**\ Data association refers to data fusion technique that fuse data based on similarity between at least two or more data sources. [[ Common techniques for data association include Nearest Neighbors [@cover1967nearest], Probabilistic Data Association [@bar2009probabilistic], and Multiple Hypothesis Test [@shaffer1995multiple].]{}]{} - **T2: State Estimation**\ State estimation indicates the usage of multiple data sources to achieve higher sate estimation accuracy. [[ Common techniques under this category are Maximum Likelihood [@myung2003tutorial], Kalman Filter [@welch1995introduction], Particle Filter [@ristic2004beyond], and Covariance Consistency Model [@uhlmann2003covariance]]{}]{}. - **T3: Decision Fusion**\ Decision fusion is a technique that is used to fuse the decisions made by various sub-components of a system to achieve a certain overall objective. For instance, a robot can fuse different decisions from the modules to perform an actuation (direction, events, or actions). [[ General techniques include Bayesian inference [@box2011bayesian], Dempster-Shafer Inference [@wu2002sensor], and semantic approaches [@herrera2000fusion]]{}]{}. - **T4: Classification**\ Classification technique denotes methodology of grouping objects into different classes based on their unique characteristics. [[ In-depth details of generic classification techniques can be found in [@han2011data; @kotsiantis2007supervised]]{}]{}. - **T5: Prediction**\ Prediction techniques are used to forecast output based on single or multiple different data sources. Note that, this covers simple methods such as regression and as well as complicated methods such as forecast modeling. [[ Examples of such can be found in [@neter1989applied; @makhoul1975linear; @lork2017many]]{}]{} - **T6: Unsupervised Machine Learning**\ Unsupervised machine learning tries to automate the knowledge discovery without relying on the data labels. [[ Examples of such methods involves clustering [@jain1999data], anomaly detection [@liao2013intrusion] and others [@han2011data].]{}]{} [[ Note that, semi-supervised machine learning approach [@zhu2005semi] is also categorized under this class.]{}]{} - **T7: Dimension Reduction**\ Dimension reduction refers to the method of reducing data sources’ dimensions for features extraction or visualization purposes. [[ Examples of dimension reduction techniques are Principal Component Analysis (PCA) [@jolliffe2011principal], and others [@han2011data].]{}]{} The aim is to preserve the characteristic of the data sources while reducing the complexity of processing high dimensional data. - **T8: Statistical Inference and Analysis**\ Statistical inference and analysis is used for outlining certain information along with some common knowledge / hypothesis from the input data sources. [[ Examples of papers using such approaches can be found in [@Zhang_LaUP:2018; @miah2017big]]{}]{} - **T9: Visualization**\ Visualization is a technique used for the presentation of output to the end users via some platform. The end result often requires human intervention. [[ Examples of such techniques can be referred to the following papers [@nichol2005modeling; @fan2017heterogeneous; @ware2012information].]{}]{} Data Input and Output Types (D) ------------------------------- Dasarathy’s classification [@Dasarathy_PotI:1997] is based on input and output of fusion technique to determine the relation between input and output data. There are five classes in data input and output perspective. Brief details are given below: - **D1: Data In Data Out (DAI-DAO)**\ Data In Data Out (DAI-DAO) refers to the situation when multiple raw data sources are fused to increase data reliability and the output after fusion is still a raw data. - **D2: Data In Feature Out (DAI-FEO)**\ Data In Feature Out (DAI-FEO) refers to the situation when multiple raw data sources are fused to extract some unique feature of the observed system. The output feature describes certain aspect of the system and it could be further used for more feature extraction or to make certain decisions. - **D3: Feature In Feature Out (FEI-FEO)**\ Feature In Feature Out (FEI-FEO) refers to the situation when multiple unique features from different sensors are combined to generate new features. This class is commonly known as feature fusion. - **D4: Feature In Decision Out (FEI-DEO)**\ Feature In Decision Out (FEI-DEO) refers to the situation when certain features of the system are fused to make certain decisions, e.g. actuation of various system components. - **D5: Decision In Decision Out (DEI-DEO)**\ Decision In Decision Out (DEI-DEO) refers to the situation when different decision sources (maintenance status, events, etc.) are combined to obtain a final output decision. Data Source Types (S) --------------------- There are four types of generic data sources in smart city applications and [[ we categorize them based on the data sources regardless of the communication medium. Details of each category can be found as follows:]{}]{} - **S1: Physical Data Sources**\ The physical data sources are collected from sensors that are being deployed to capture information of a particular space, area, or even city wide. [[ Examples of the physical sensors include temperature [@Lau_:2016], air quality [@zheng2013u], camera [@Spinello_:2011], ultrasonic [@Lee_:2008], LiDAR [@chen2004fusion], and etc. ]{}]{} [[ Note that, we categorize smart city application based on the data sources rather than the method they are acquired. For instance, a temperature probe in a sensor nodes of a wireless sensor network (WSN) transmits data through gateway to cloud database is considered as physical data source, $S1$. ]{}]{} - **S2: Cyber Data Sources**\ [[ Cyber data sources denote datasets which are commonly obtained from the Internet domain such as social media information [@Zhang_LaUP:2018; @suma2017automatic], web access data [@breur2011data; @wang2012multi], and opinion based datasets [@balazs2016opinion]. Social media information involves major social media platforms such as Twitter, Facebook, LinkedIn, Weibo, and others. Note that, usually the data is acquired through data mining techniques.]{}]{} Meanwhile, the web access data can be obtained from web applications programming interface (API), such as transportation tickets information and online customer records. Apart from that, open datasets refer to data from third party vendors such as telecom operator or a company with readily available data. - **S3: Participatory Data Sources**\ Participatory data sources include [[ crowdsensing [@guo2015mobile; @marakkalage2019understanding] and crowdsourcing [@estelles2012towards; @howe2006rise] data contributed by the personal devices, e.g. mobile phones, wearable devices, tablets, etc. of the users in smart city.]{}]{} Users provide the data voluntarily or through some incentive mechanisms. - **S4: Hybrid Data Sources**\ The hybrid data sources include data obtained from [[ mixed data sources [@Aftab_EaB:2017; @You_IToKaDE:2018]]{}]{}, e.g. by combining the participatory and physical sensor data. As pointed in [@zheng2015methodologies], hybrid data sources can achieve more insights as compared to single data sources. Data Fusion Scales (L) ---------------------- The scale of data fusion is also an important classification perspective. Please note that data fusion scale is based on sensor coverage rather than sensor deployment. There are four different classes, which are described below: - **L1: Sensor Level Fusion**\ [[ At the sensor scale, data from various physical sensors is fused to form an output such as [@jayasinghe2019feature; @serdio2014fault].]{}]{} For instance, fusion of data collected by various smartphone sensors is an example of data fusion at sensor level. - **L2: Building Wide Fusion**\ [[ At the building wide scale, data sources collected within a premise or building is fused to form an output. For instance, fusion of building energy and building security data to develop a building management system [@Tushar_ISPM:2018; @de2006information; @park2014wireless] is an example of data fusion at building level. ]{}]{} - **[[ L3: Inter-Building Fusion]{}]{}**\ [[ In the inter-building scale, the data sources collected over several buildings are fused to form an output, where the scale of deployment normally includes small area. For example, data sources of several buildings within a university are used to generate a particular output is considered as inter-building scale. Other examples of this data fusion scale also can be found in [@zhou2018understanding; @katoch2018shading].]{}]{} - **L4: City Wide Fusion**\ [[ In the case of city wide fusion, data sources that involve whole city’s area as input for the data fusion architecture fall under this class such as [@catania2014approch; @toole2015path; @mounce2003sensor]]{}]{}. For instance, the study of citizen behavior involves fusion of data gathered in different areas of the city is considered city wide data. - **[[ L5: Inter-City Fusion (or larger)]{}]{}**\ [[ At the inter-city fusion (or larger) scale, data from large areas involving one or more cities or terrains (mountains, sea, forests, etc.) is fused to form an output. Examples of this scale involve comparing one smart city to another city or data of a city outskirts and its surrounding areas. More examples of inter-city fusion (or larger) can be referred to [@cheng1997urban; @izumi2018real; @Anjomshoaa_IIoTJ:2018]. ]{}]{} Platform Architectures (P) -------------------------- The architecture of computational platform involved in data fusion is another important classification perspective. In this category, we identify four generic classes: - **P1: Edge Computation Platform**\ In edge computation platform, data sources are processed and fused at the edge (i.e. very close to the physical location, where data is actually collected). Edge computation devices include micro-controller, computing devices (Raspberry pi), computers, etc. [[ Such architecture can be found in works such as [@serdio2014fault; @park2014wireless; @katoch2018shading].]{}]{} With this architecture, communication overheads and latency can be significantly reduced. - **P2: Fog Computation Platform**\ In fog computation platform, data sources are processed and fused at the middle layer, i.e. between the edge and the cloud. In this architecture, data is periodically or continuously sampled at the edge (without processing) and is then forwarded to a gateway (that acts as a fog device). At the gateway, computing resources are provided for data processing. [[ Both fog computing and edge computing platforms provide similar benefits of offloading computation as shown in [@catania2014approch; @tian2016agri; @mehmood2015future].]{}]{} However, fog computing architecture should be preferred when it is difficult to find stable power sources at the edge. - **P3: Cloud Computation Platform**\ In cloud computation platform, data sources are processed and fused in the cloud. This is the most common technique practiced by industry and research institutes for processing big data. [[ Examples of this architecture being used are [@consoli2015urban; @breur2011data; @ahmed2015integrated].]{}]{} The advantages of cloud computing architecture includes ready access to the data and both online and offline for further processing or fusing. The disadvantages include increased communication overheads and costs. - **P4: Hybrid Computation Platform**\ [[ In hybrid computation platform, processing is distributed among two or more layers (edge, fog and cloud) as shown in [@izumi2018real; @fleury2010svm; @hondori2012monitoring].]{}]{} In this architecture, depending on the available resources or application objectives, some low level data fusion and processing is done at the edge or fog, while high level information is extracted in the cloud. Smart City Applications Overview {#sec:currentSmartCityApp} ================================ [[ Smart city applications tend to have extremely diverse requirements, which contribute to a large variety of different techniques and requirements as stated previously in Section 2 for different domains. Thus, it is necessary to evaluate the smart city applications from a more generic perspectives rather than one specific perspective.]{}]{} [[ In this section, we select smart city applications with data fusion techniques from different domains listed in Figure \[fig:Fig01\_SmartCityCat\], and evaluate them based on multi-perspectives from the Section \[sec:categoryDefinition\].]{}]{} Note that, there exist some literatures that are cross-disciplinary, which may involve more than one domain. In order to address the cross-disciplinary smart city applications, we have grouped them into their closest relevant domain. In each application domain, we outline sub-domains and present works related to data fusion techniques. Using the proposed data fusion classification based on multi-perspectives, we discuss the common data sources and fusion techniques, along with the current research trends in each domain. **Domain** **Sources** **O** **S** **D** **T** **L** **P** **Remarks** ------------ ---------------------------------------------- ------------ -------------- -------------- -------------- -------------- ------------ ------------------------------------------ [@dawar2018convolutional] 3 1 2 4 1 4 Smart Healthcare [@hossain2017smart] 3 1 1 4 [[ 4]{}]{} 4 Voice Pathology Detection [@medjahed2011pervasive] 3 3 4 3 [[ 2]{}]{} 4 Smart Home Healthcare Monitoring [@fleury2010svm] 3 1 2 4 1 4 Daily Activity Classification [@hong2009evidential] 2 1 2 4 [[ 2]{}]{} 4 Smart Home Activity Recognition [@hondori2012monitoring] 3 1 3 4 [[ 2]{}]{} 4 Tele-Rehabilitation [@zhang2008information] 4 4 4 3 [[ 2]{}]{} 4 Smart Home Control System [@fan2017heterogeneous] 3 4 3 9 [[ 4]{}]{} 4 Intelligent Video Surveillance [@mehmood2017utilearn] [[ 3]{}]{} [[ 3,4]{}]{} [[ 4,5]{}]{} [[ 4,5]{}]{} [[ 5]{}]{} [[ 3]{}]{} [@Chan_:2008] 3 2 3 1 [[ 4]{}]{} 3 Smart Community [@Aftab_EaB:2017] 4 4 4 5 [[ 2]{}]{} 3 Building Management [@luo2007autonomous] 4 1 2 2 1 1 Fire Detection System [@Janssen_GIQ:2013] 3 3 4 3 [[ 5]{}]{} 3 Lean Government [@cheng1997urban] 1 1 1 1 [[ 5]{}]{} 1 Urban Planning with Satellite Images [@You_IToKaDE:2018; @Lau_IIoTJ:2018] 3 4 1,2 8,9 [[ 4]{}]{} 3 Urban Space Utilization Detection [@consoli2015urban] 4 3 1 9 4 3 Fault Reporting Platform [@Zhang_LaUP:2018] 3 4 3 8,9 [[ 5]{}]{} 3 Landscape Rating Systems [@Anjomshoaa_IIoTJ:2018] 3 1 1 9 [[ 5]{}]{} 3 City Environment Monitoring [@nichol2005modeling] 3 1 2 2,9 1 1 City Building Map Modeling [@Lu_RS:2017] 3 4 4 4 [[ 5]{}]{} 1 Forest Types Classification [@shen2016long] 2 1 1 1 [[ 5]{}]{} 1 Long Term Landscape Monitoring [@wolter2011multi] 4 1 4 4 [[ 5]{}]{} 1 Forest Species Classification [@chang2017developing] 3 4 2 4 1 1 Waste Water Treatment [@catania2014approch] 4 1 2,3 2,9 [[ 4]{}]{} 2 Urban Solid Waste Management [@serdio2014fault; @zhang2018engine] 4 1 2,4 2,4 1 1 Fault Detection [@jayasinghe2018temporal; @Ghosh_MSaSP:2007] 3,4 1 3,4 5 1 1 Tools Life Prediction [@de2006information] 2 4 4,5 3 [[ 2]{}]{} 1 Decision Support in Manufacturing [@guo2018roboads] [[ 2]{}]{} [[ 1]{}]{} [[ 2,4]{}]{} [[ 2,3]{}]{} [[ 1]{}]{} [[ 1]{}]{} [@huang2016data] 3 1 2,4 4,7 1 1 Seafood Freshness Classification [@moshou2005plant; @khanum2017towards] 3,4 1 2,4 2,4,5 [[ 1]{}]{} 1 Agriculture Plant Disease Classification [@breur2011data] 4 2,3 1,3 1,8 [[ 5]{}]{} 3 Customer Profiling [@sato2015design] 4 4 1,4 5 [[ 5]{}]{} 3 Consumer Awareness [@tian2016agri] 4 4 1 9 [[ 5]{}]{} 2 Blockchain and Supply Chain [@Pang2015] 3,4 4 2,4 1,5,8 [[ 5]{}]{} 3 Supply Chain Management [@miah2017big] 3 2,3 2,3 5,8 [[ 4]{}]{} 3 Tourist Behavior Analysis [@ricci2002travel] 4 4 2,3 6 [[ 4]{}]{} 3 Travel Recommendation System [@viswanath2014smart] 3 1,2,3 2 1,4 [[ 4]{}]{} 1,3 Tourist Tracking Application [@lin2010energy; @wilhelm2016wearable] 2,3 1 1 5,1 [[ 4]{}]{} 3 Outdoor Positioning [@liu2017fusing; @liu2017cooperative] 2,4 1 1 1,2 [[ 2]{}]{} 3 Indoor Positioning [@liebig2017dynamic; @teo2016bim] 4 1 4 5,1 [[ 4,2]{}]{} 3 Location-based Services [@toole2015path] 3 3 2 1 [[ 4]{}]{} 3 Obtaining Origin-Destination Matrices [@ghorpade2015integrated] 3 3 2 4 [[ 4]{}]{} 3 Identifying Transportation Modes [@yoshimura2014analysis] 3 3 2 1 [[ 2]{}]{} 3 Monitoring Visitors Inside a Building [@ahmed2015integrated] 4 1 4 3 [[ 4]{}]{} 3 Traffic Signal Controlling [@poonawala2016singapore] 3 3 2 1 [[ 4]{}]{} 3 Analyzing Public Transport Services [@li2014sensor] 4 1 4 4 1 3 Autonomous Vehicle Controlling [@luan2010smart; @wang2012multi] 3,4 1 2,4 5 [[ 4,1]{}]{} 1 Smart Grid and Power Utilities [@katoch2018shading; @huang2018data] 3 1,4 1 4,5 [[ 3]{}]{} 1 Solar Farm [@izumi2018real] 3 3 2 2 [[ 5]{}]{} 4 Smart Metering [@grime1994data; @Huang2019Iterative] 1,2 1 1 1,2 1 1,3 Communication (5G) [@kreibich2014quality; @luo2016kernel] 2 1 1 1,5 1 1 Communication (WSN) [@abeywickrama2017rf] 4 1 2,3 4 1 1 Drone Detection [@salpietro2015park] 3 4 1,2 2 [[ 4]{}]{} 3 Smart Parking System [@park2014wireless] 4 1 1 2 [[ 2]{}]{} 1 Bridge Monitoring Platform [@mounce2003sensor] 3 1 2,3 4,5 [[ 4]{}]{} 3 Water Distribution System Smart Living ------------ Smart living concerns with the life of the urban citizens and revolves around the concept of improving live-ability in urban area. [[ In the literature, the general objectives of utilizing the smart living domain involve data being used to extract higher level information or increasing the data completeness. In addition, smart city applications in this domain often leverage the cloud or hybrid platform architecture.]{}]{} In this domains, we have studied three different aspects of smart living, namely, (1) Smart Health, (2) Smart Home, and (3) Smart Community. ### Smart Health [[ Healthcare is a crucial component in everyday life concerning medical and public practices using devices as defined by Lee and Co-authors[@Lee2011SmartHC; @muhammed2018ubehealth]]{}]{}. The rapid development of technology (e.g. smartphones and their in-built sensing devices such as heart rate sensors) provides more opportunities to adopt technology in healthcare applications pervasively. For telehealth application in smart city, Hossain et al. [@hossain2017smart] have used electroencephalographic (EGG) signals and voice to monitor a specific user’s health with the support of cloud technology and doctor’s advices. In [@medjahed2011pervasive], work has shown to monitor elderly at home based on fuzzy fusion model using behavioral and acoustical environment data. Similarly, Noury [@noury2002smart] also monitors the activities and fall detection of elderly through fuzzy logic by fusing accelerometer, vibration, and orientation sensor. In [@marakkalage2019understanding], Marakkalage et al have used crowd-sensing data from a smartphone application (location, noise, light, etc.) and introduced sensor fusion based environment classification (SFEC) to profile elderly people for understanding their daily lifestyle. In addition, Dawar and Kehtarnavaz in [@dawar2018convolutional] have implemented a Convolution Neural Network (CNN) to combine both depth camera and wearable devices to detect the transition of movements to fall. Apart from that, Hondori et al. [@hondori2012monitoring] have proposed using sensor fusion between depth images and inertia to perform tele-rehab in the home. The main challenge occurs in pervasive smart healthcare data fusion is discussed in [@lee2008issues] as the need of a higher accuracy to improve sensing robustness against uncertainty and unreliable integration. ### Smart Home The concept of Smart Homes plays an important role nowadays in contemporary urban areas. According to Jiang et al. [@jiang2004smart], the definition of a smart home provides the capability of controlling, monitoring, and accessed appliances & services through implementation of ICT. There are currently many big players in developing the smart home appliances such as Amazon, Google, Apple, IBM, Intel, Microsoft, Xiaomi, and others. The challenge faced by manufacturers are related with service integration and formulating software ontology platform. These are necessary for implementing the services through different vendors and allow for a better integration. Meanwhile in [@zhang2008information], physical sensors (soil moisture) and cyber (weather, traffic) have been fused to control home appliances such as alarm clock and water sprinkle. The study of user daily activity is yet another important aspect to understand urban citizen well-being. In [@hong2009evidential], Hong et al. have combined series of life activities to understand the lifestyle pattern depends on the equally weighted sum operation and Dempster-Shafer theory. Also, similar study on the user daily activity patterns can be found in [@fleury2010svm]. Combination of house environmental sensor (infrared, door contact, temperature, hygrometry sensor, microphone) and wearable devices (kinematic sensors) using support vector machine (SVM) can be used to identify the user activity patterns. In addition, the modeling of human behavior in a smart home [@brdiczka2009learning] in order to generate learning situation models have proven the efficiency of context-aware services. [[ In addition, smart home security is yet another study field for many researchers [@fernandes2016security; @komninos2014survey; @dorri2017blockchain] due to increased usage of IoT devices in normal household.]{}]{} The research challenges is to develop the applications for the smart houses while retaining the privacy and security of the end user. ### Smart Community According to Smart Communities Guidebook [@san1997smart], a smart community is described as “a geographical area ranging in size from neighborhood to a multi-county region whose residents, organizations, and governing institutions are using information technology to transform their region in significant ways”. There is only a handful of cities focus on this aspect as majority are still in the stage of transforming from facility to community welfare. First world countries such as USA, Canada, Australia, European Union, and Singapore shown in [@lindskog2004smart] have started up initiatives to create smart communities. Information fusion for smart community video surveillance system is performed in [@fan2017heterogeneous] to aid neighborhood in terms of security. The combination of the different modal surveillance camera provides a vast amount of visual information extraction such as video summarization for highlighting certain events. [[ A distance learning framework is proposed in [@mehmood2017utilearn], which enables personalized learning to cater what is best for each individual user. It uses data fusion to understand user environment and their activities by means of hybrid data sources.]{}]{} Real-time community monitoring also helps to prevent emergency situations and it ensures the safety of community citizens. A good example for a smart community application in large-scale is the Social Credit System in China [@liang2018constructing]. It is a state-owned system to collect data from both public (traffic cameras, transit data etc.) and private (online shopping, fitness trackers etc.) data sources to monitor and analyze user behaviour to generate a single “credit score” for each person, which helps in community well-being. The techniques fuse these data sources and remains a back box to the general public. However, the effect on user privacy with the rise of “data state” remains a debate for some [@cheung2018rise]. A mature citizen should be on alert and always responds to any potential threat, while spreading the awareness to build a safer community in the urban city. Smart Urban Area Management --------------------------- Smart urban area management denotes the managing of urban area using ICT. Sub-domains in this regime composed of urban planning, governance, and smart buildings. For an application to fit into this definition, the minimum scale would be at the building level (e.g. a building management system). [[ The main trend of data fusion techniques being applied in this domain mostly consists of objectives of extracting higher level information or increasing the data completeness. The end product of data fusion include visualization of information for respective authorities.]{}]{} ### Smart Governance In smart governance, managing a city is considered as a complex task as the integration of different domains and services is proven to be challenging. Transparent services integration is an example of why many governance authorities are having difficulties to sort it out. It is hard to strike a balance in developing a transparent governance policy with consideration of sensitive information. Therefore, there is only limited study materials available to the best of our knowledge. Janssen and Estevez [@Janssen_GIQ:2013] have proposed a centralized platform for cutting down government staff by shifting existing organization to rely on integration of platforms. [[ The disaster response management is also considered as another vital element for a smart city to carry out any potential counter measurements towards disaster as shown in [@alazawi2014smart]. ]{}]{} Apart from that, urban reporting system [@consoli2015urban] has collected report from the city wide region on the faulty infrastructure so that immediate actions can be taken to remedy the situation. It uses cloud technology and focuses on the display of fused data report, which it also describes the location and types of infrastructure. Example of research challenges is to remove any potential fake report to prevent misuse of the reporting platform. [[ Another example of smart governance that involves city safety can be found in [@jin2016smart], where it can act as an emergency aid application (light pulse on emergency through mesh network) while providing energy efficient lighting to urban area.]{}]{} Moreover, there are cities also working on governance platform such as New York [@NewYork_:2018], Singapore [@Nation_:2018], Tokyo [@Government_:2016], Oslo [@Oslo_:2018], and others. The potential research opportunity is to propose consensus protocols within the city for better integration of services. ### Smart Urban Planning Urban planning plays an important role in developing the city economy by taking account of well-being of the urban residents. Traditionally in urban planning, aerial photography and statistical data sources (building size, population number, public amenities, etc.) are combined to understand the current development state of the city. The downside of such method is data sources frequently lacks of fine details, which resulting the output result is not representative. To address such issue, Cheng and Toutin [@cheng1997urban] have combined various satellite and aerial images to generate details for the exiting urban structures. Alternately, low power sensors are capable to provide a larger coverage with lower deployment cost, which give researchers the opportunity to study different points of interest in the urban area. In [@You_IToKaDE:2018; @Lau_IIoTJ:2018; @Lau_:2016], a bottom up urban planning method is implemented, where sensors are installed in a designated region to capture space utilization. From the collected data, urban planners can study public space utilization pattern using an integrated portal. Here, a hybrid processing method is proposed, where the data processing and fusion occur in different stages of data pipeline. In addition, a large variety of data sources can be used for urban planning such as physical sensors [@sohn2007data], photography [@Zhang_LaUP:2018; @xu2016multimodal], or hybrid data sources [@chen2004fusion]. Despite wide variety of data sources, human interpretation is required when it comes to make decision on a proposed urban design. The need of full automated planning system would further benefit the urban planners to combine different data sources in order to achieve a more ideal city planning. ### Smart Building Urban building management provides building owner a platform to understand building’s energy consumption rate while automating building resources management. It has been extensively studied in [@Chen_EaB:2018; @raza2015review; @baetens2010properties; @li2017optimizing] and the current trend is to optimize the building resources such as hot water systems, electrical consumption, and heating ventilation & air conditioning (HVAC). In [@Aftab_EaB:2017], Aftab et al. have combined four different parameters to predict building occupancy to control HVAC using low-cost embedded systems. Some other works such as [@Tushar_ISPM:2018; @McKenna_EaB:2015; @Chen_:2009] also have the same objectives but using different types of data sources. The potential solution for better building management system is to rely on fusing weather, human feedback, and electricity price to fine tune the building resources in order to maximize human comfort, while minimizing the energy consumption. Apart from that, fire alarm system is considered another important features of the smart building management system. Luo and Su [@luo2007autonomous] have fused three different data sources (flame, smoke, and temperature sensor) to detect any potential fire outbreak and reduce false alarms. In addition, a notification-based system is implemented to notify the property owner and manager in case of emergency. In future, potential building safety features may include a group of robots to deal with fire hazards and double duty as building security patrols. Smart Environment ----------------- Smart environment studies the surrounding of a given area of interest, which covers the internal and external surrounding of a city. [[ From the literature, we observed that majority of the data sources consist of physical and hybrid data sources, while the data scale often represent a large spatial coverage.]{}]{} Nowadays, the most common surrounding effects studied in the smart city include urban heat island (UHI), green house effect, and global warming. In addition, we have grouped urban waste management under this domain because it also has an environmental impact. ### Landscape Monitoring The main challenge of landscape monitoring in smart city is the sensing coverage of the data sources. To address such issue, two different sensing approaches have been used such as relying on mobile sensing or satellite-based data. Mobile sensing [@Anjomshoaa_IIoTJ:2018; @cardone2014participact] offers greater sensing capability by leveraging the mobility of moving objects (vehicles or humans). The mobile sensing technique provides a large spatial coverage, but it is not suitable for real-time applications unless there are multiple data sources to compensate the lack of spatial resolution concurrently. The output type of this mobile sensing includes combination of different spatial data in order to complete the data sources before proceed to data processing stage. Mobile sensing works such as [@zheng2013u; @antonic2014urban] utilized different data sources to complete spatial resolution and visualized the ambient changes across the city. The common characteristic of aforementioned works is feature extraction, which they visualize the processed features from the raw data sources. Majority of data input and output types in this domain are DAI-DAO and DAI-FEO since physical sensors are the common data sources. Using the satellite-based data sources, Shen et al. [@shen2016long] have studied the UHI effect in a city using data sources collected over $26$ years. The UHI index changes are measured through the combination of Landsat and MODIS images data. Mobile sensing offers a lower deployment cost, where it sacrifice the spatial resolution given there is limited number of sensors. Also, it has a lower coverage compared to satellite data sources. In contrast, satellite data has a wider coverage of spatial resolution but it frequently needs data enhancement and lacks of finer details. ### Urban City Modeling The surrounding natural resources of an urban city such as mountains and forests are considered as important assets of a city. The most common data sources in modeling the city area are satellite images, which as stated before, it requires data enhancement such as [@tu2001new; @wu2016improved] before using it. Therefore, prior work of data fusion  [@Zeng_:2010] was focused on improving the satellite images quality. Only until recently, the emergence of machine learning algorithms and faster computers have created new ways to extract large variety of satellite image features. For instance in [@Lu_RS:2017] and [@wolter2011multi], forest types classification have been conducted in order to understand the variety of tree species in a specific region of interest. Both methods involve region-wide data sources and classification techniques, which are used to identify the tree species based on the forest types. With a lower deployment cost, small satellite (smallsat) and nano satellite (nanosat) could improve spatial coverage to generate a better data sources. Smart city applications leveraging satellite data will also beneficial from these deployment. ### Waste Management With astonishing rate of garbage being generated daily, waste management for an urban city can be rather challenging. Thus, it is essential to handle the waste efficiently to improve on sustainability of a city. An example of such effort could be found in [@Esmaeilian_WM:2018], where they have proposed three new aspects of a smart waste management system such as: (1) infrastructure to overlook the overall life cycle of the product, (2) new business models revolving the product life cycle for preventing any waste generation, and (3) intelligent sensor networks for waste management facilities. In [@catania2014approch], Catania and Ventura have combined the proximity reading and weight sensor from garbage bin to estimate the garbage capacity of a typical household. Afterwards, rubbish categories collected from user mobile devices and garbage trucks are combined to keep track of residential participation in recycling scheme. On the other hand, waste water treatment helps to manage liquid waste of urban city before discharging to river or reuse. Chang et al. [@chang2017developing] have combined landsat and MODIS dataset in order to trace the water pollution level of a lake. On top of that, a web portal has been deployed to visualize and monitor the water pollution region over the time. Currently, many researchers are working together to develop an efficient waste management system since there is only limited resources available on earth. The goal is to adopt the 3R (Reduce, Reuse, and Recycle) concept with the help of ICT to improve city resource sustainability. Smart Industry -------------- With the upcoming Industry $4.0$ standards [@Wang_CN:2016] touted as the gold standard of the future, various industries have been experiencing transformation with automation and data driven approaches. [[ The majority of smart industry applications often leverage data collected from physical sensors while data fusion techniques are often performed at sensor or building level.]{}]{} Here, smart industry can be divided into three sub-domains, which are Smart Manufacturing, Smart Maintenance, and Smart Agriculture. ### Smart Manufacturing Smart manufacturing denotes the factory that depends on ICT to optimize the manufacturing process by increasing the production throughput. In [@de2006information], De Vin et al. have proposed a simulation tool to test out the management decision support by fusing undisclosed data entries and manufacturing process events. Similar to the aforementioned approach, decision based fusion can also be seen in [@groger2012data; @lee2003manufacturing], which combines different machinery sensors data and data warehouse entries. The data fusion integration also considers supply chain demand in order to further optimize the manufacturing process. The challenge in this domain is to develop a self-optimizing manufacturing process while delivering the products to meet the demand of supply chain. Therefore, smart manufacturing frequently has a high correlation with the supply chain and attempts to deliver the market needs. [[ In addition, the robotics usage in the smart manufacturing domain is nothing new. Guo et al. [@guo2018roboads] have proposed an anomaly detection to combat potential security aspects in the robots using sensor fusion technique such as state estimation.]{}]{} ### Smart Maintenance The reliability and stability of the equipment and machinery is vital to all the industries to ensure smooth operation in production. Without the guarantee of smooth operation, any downtime can cost damages to reputation and also loses profit. Thus, preventive maintenance has been studied in [@jayasinghe2018temporal; @Ghosh_MSaSP:2007; @Niu_RESS:2010; @schmidt2018cloud] and attempts to predict the remaining useful life (RUL) of a machine accurately. By accurately predicting the RUL, maintenance can be carried out on time to save cost only when needed. The common data fusion techniques for predicting RUL are neural network (NN) based model such as CNN and Deep NN (DNN). Please note that, common data source in this sub-domain is physical data source such as machine states, sensors readings, and related parameters. Nonetheless on the fault detection domain, machine fault detection can be found in [@serdio2014fault; @zhang2018engine], where they describe the problem of fault diagnosis and apply data fusion techniques to overcome. State estimation and classification have been used to detect the current state of the machinery. The data sources share some similarity with the preventive maintenance, where lower level of data information is preferred. This yields a faster fault detection when compared to a complex data pipeline. The research challenge here is to develop a generic and a flexible maintenance system for different scale of applications adhering to the goal of accurate fault detection. ### Smart Agriculture In order to produce sustainable food resources in smart city, smart farming [@wolfert2017big; @walter2017opinion] has become a trend to meet the food supply demand in a smart city. There are two different sub-domains in smart farming such as land and sea agriculture. In the land agriculture aspect, planting crops using controlled environment has shed some light in fulfilling the city needs of fresh supplies. However, plant disease remains a potential threat to a highly-dense plantation crop framing. In [@moshou2005plant], Moshou et al. have classified the plant disease infection through Self Organizing Map (SOM) by fusing the spectral reflection and fluorescence imaging data. This helps to isolate infected crops while it focuses on the production of healthy plants. Apart from that, electromagnetic induction sensors, vegetarian index, water stress level, and radiance data are combined in [@de2013field] to better determine the partition of the crop field. [[ Similar work also can be found in [@khanum2017towards], where Khanum et al. propose an ontology-based fuzzy logic to classify plant disease.]{}]{} The research gaps in this domain involve improving live stock management as well as optimizing smart farm. On the other hand, sea agriculture is responsible for supplying the seafood supplies in a city. Obtaining fresh seafood supplies in an urban city sometimes can be rather difficult due to various factors such as delivery, city location, weather, seasonal pricing, etc. Therefore, a fresh seafood supply in a city is often not guaranteed. In order to address such issue, Huang et al. [@huang2016data] have provided a solution by integrating two types of cameras for seafood freshness inspection. Camera and near infrared spectroscopy are fused through PCA and use NN to classify the freshness index. The research gaps in this domain involve developing large scale fish breeding and also wide varieties of seafood product such as calm, mussels, abalone, etc. A potential solution such as smart fish breeding with IoT has been proposed in [@Atlas_:2018], where it suggests using a moving pod to breed fishes while transporting them to destination in a particular destination simultaneously. Smart Economics --------------- Smart economics can be defined as the generic commercial activities in an urban city ranging from supply chain, logistic, finance center, to tourism. All these activities yield potential commercial value to a city, which it depends on the unilateral or bilateral trading relationship. In this subsection, we discuss smart economics in three major sub-domains, namely, (1) Smart Commerce, (2) Smart Supply Chain, and (3) Smart Tourism. ### Smart Commerce Today, modern e-commerce platforms use multi modal data sources to reach and better understand their customers. This helps e-commerce vendors to give better product recommendations for their customers and it helps customers to make their decisions easily. Fusing customer data such as mobility, credit card purchases, and social media interactions is commonly used in modern recommender systems. In [@breur2011data], Breur introduced the fusion of customer behavior data and market research data to obtain a holistic picture of the customer. Investors can leverage financial data to make investment decisions, as Hassan et al. [@hassan2007fusion] have introduced a fusion model of Hidden Markov Model (HMM), NN, and Genetic Algorithm (GA) for stock market prediction. Improving the consumer awareness is conducted in [@sato2015design], by fusing real world (weather, geographical) and cyber world (Twitter, Facebook) data. The proposed system has two levels of fusion, which relies on hierarchical-based processing architecture. The data combined bottom level input and fed it into upper level for further processing to achieve its objectives. ### Smart Supply Chain In a smart supply chain, it often involves sources and destination tracking in order to understand the flow / processing of the objects. As discussed in [@christopher2016logistics], supply chain management and logistic are the fundamental of modern supplies on fulfilling the needs of an urban city. For instance in food supply chain, three tiers information fusion framework is proposed in [@Pang2015] such as: (1) to accelerate data processing, (2) shelf life prediction, and (3) real-time supply chain planning. The proposed hierarchical information fusion architecture (HIFA) includes a process that is intelligently transforming the sensor’s data sources into usable decision-making information. Recently, combination of blockchain technology has paved a new way for revolutionizing the existing supply chain. In [@tian2016agri], Tian has shown the integration of blockchain and supply chain in the agri-food supply application. It aids consumers to trace the origin of food using Radio Frequency Identification (RFID) along with database or WSN. The information also includes food origin to help consumers to identify the brand authenticity and avoids consuming counterfeit products. The research gap in this sub domain concerns with the implementation of smart supply and it needs the involvement from various commercial organizations. The consensus and national regulations are also parts of the critical factors of smart supply implementation. ### Smart Tourism The advancement of transportation technology has granted accessibility for the humans to move around the globe with ease. This phenomenon has caused rapid expansion of the tourism commercial values contributed to a city side income. Since then, Internet resources such as travel blogs and recommendation systems have influenced public to venture different locations. For instance, recommendation system [@ricci2002travel] has been developed to recommend the place to travel based on user’s information such as socioeconomic (e.g. age, education, and income) and psychological and cognitive (experience, personality, involvement, and so forth) groups. User choices are used as feedback to further fine-tune the recommendation system using Rocchio’s method. Apart from that, Miah et al. [@miah2017big] have combined social media-generated big data (geo-tagged photos of tourist attraction places) to predict tourist behavioral patterns. Alternately, Viswanath et al. [@viswanath2014smart] used a smartphone based mobile application to passively track tourist location data and obtain user ratings for tourist attraction places to better understand the preferences of tourists when they visit tourist attractions. The potential research development for smart travel is to focus on using a smartphone application for improving travel experience by relying on real-time translation and augmented reality (AR) navigation. [[ Smart Human Mobility]{}]{} ----------------------------- [[ Human mobility has been an important research area as commuting and traveling play big roles in modern life. With the help of advanced ICT, plentiful data sources related to human mobility have been collected and accessible to researchers, which yields deeper insights into the nature of human mobility as well as better improvement strategies for transportation systems. [[ Smart human mobility]{}]{}, therefore, means collecting, managing, and analyzing (fusing) various data sources related to different aspects of residents’ movement in order to better understand and improve the way people move. Depending on the purpose of different applications, smart human mobility domain can be further divided into three sub-domains:(1) Smart Location-Based Services, (2) Human Mobility Understanding, and (3) Smart Transportation Systems.]{}]{} ### [[ Smart Location-Based Services]{}]{} [[ This sub-domain aims to get the accurate position of individuals and further to provide services, such as route planning and navigation, to help them travel efficiently and comfortably, in both outdoor and indoor environments.]{}]{} For outdoor positioning, Global Positioning System (GPS) has been the most accurate, reliable and dominant technology since it was allowed for civilian use in 1980s [@parkinson1996global; @misra2006global]. Less-accurate non-GPS positioning approaches, such as wifi-based localization and cell-tower triangulation, are sometimes used instead of (or together with) GPS, because they consume less energy [@lin2010energy; @wilhelm2016wearable]. For indoor positioning, since GPS does not work well indoors, other positioning approaches have been proposed. The data collection technologies used for these approaches mainly include Wi-Fi (WLAN), inertial measurement unit (IMU), RFID tags, Bluetooth, global system for mobile communications (GSM), frequency modulation (FM), and ultra-wide band (UWB) [@yassin2016recent; @tariq2017non]. Meanwhile, multiple data sources are often fused to achieve more accurate localization results [@liu2017fusing; @liu2017cooperative]. Once accurate locations are obtained, either indoors or outdoors, location-based services (e.g. route planning and navigation) can be provided to end users by fusing the location sequences with other information sources such as geographic information system (GIS) data, real-time traffic data, and [[ user preference data [@liebig2017dynamic; @teo2016bim; @schlingensiepen2016autonomic; @delling2017controlling; @han2014building; @arfat2017parallel].]{}]{} Since the outdoor positioning and location-based services have been well developed and commercialized, the current research trend in this field is mainly focused on improving the performance (accuracy, deployment cost, and energy cost) of indoor systems and services. ### [[ Human Mobility Understanding]{}]{} Positioning systems not only enable the location-based services for individuals but also provide data sources for further monitoring and understanding human mobility in a larger and more comprehensive scale. By aggregating and analyzing (fusing) the location data of residents along with GIS data of the environment, various aspects of human mobility can be monitored and the hidden patterns can be obtained. As summarized in [@zhou2018understanding], the most common subjects of monitoring and understanding human mobility include distance and duration distributions [@gonzalez2008understanding], origin-destination matrices [@toole2015path], individual activity-based mobility patterns [@jiang2017activity], transportation mode identification [@ghorpade2015integrated], and densities and flows within a building (or a cluster of buildings) [@yoshimura2014analysis; @prentow2015spatio]. Results obtained from these subjects provide clues for improving transportation system [@demissie2016inferring], urban planning [@horner2001embedding], and communication network [@karamshuk2011human]. Typical studies in this sub-domain usually fuse one data source of people’s movement trajectories with the environment information, such as GIS data of the city or floor plan of a building. Although this type of approach has produced much deeper insights compared with traditional approach relied on survey data, there is a trend to fuse multiple data sources related to people’s movement and obtain a more comprehensive picture of human mobility [@zhang2014exploring; @zhang2015comobile]. [[ Moreover, social media data sources. such as Tweets, also bring in more information regarding the mobility status in cities due to the combination of spatio-temporal data and descriptive text [@suma2017automatic; @alomari2017analysis].]{}]{} ### Smart Transportation Systems Another large part of smart mobility is the improvement of transportation systems, which mainly comes from three aspects: relieving traffic congestion, improving public transportation, and introducing new transport systems. To relieve traffic congestion, effective light control plays an important role. While existing light control systems are usually based on hand-crafted rules and do not adjust to the rapid dynamics of traffic flows, intelligent light control approaches have been proposed using different data sources, data fusion techniques, and decision making (optimization and control) algorithms [@ahmed2015integrated; @wei2018intellilight]. Challenges in this aspect mainly come from the implementation of such intelligent light control approaches. Improvement of the public transportation system is mainly conducted through the network and schedule optimization [@yao2014transit]. Although these two topics have been thoroughly discussed in the literature, new insights related to the public transit system [[ (e.g. origin-destination matrices and service level obtained from big data) [@poonawala2016singapore; @munizaga2012estimation] and more advanced transport modeling tools enabled by big data [@mehmood2017exploring]]{}]{} have brought new opportunities. Even if the existing transportation manner has been optimized, there are still problems that cannot be solved, such as last mile issue and driving accidents. Therefore, new transport systems, such as bike sharing systems and autonomous vehicle systems, are introduced. Advanced ICT and data fusion techniques are the core of the realization of these systems. For a bike sharing system, data fusion and analysis helps to understand how the system works and evaluate different operational strategies [@shaheen2011china; @schuijbroek2017inventory]. As for the autonomous vehicle system, the control of an autonomous vehicle itself is a complex data fusion process, fusing various data sources about the vehicle and the road by advanced machine learning and control algorithms [@li2014sensor; @falcone2007predictive]. [[ Security plays an important role in the autonomous vehicles deployment to ensure reliability of the autonomous driving. Examples of such techniques can be found in [@petit2015remote; @ferdowsi2018robust].]{}]{} Smart Infrastructure -------------------- In a smart city, infrastructure aims to provide convenience for the public by supplying resources (electricity, gas, and water) or providing services (public facility or communication systems). Here, we outline four different sub-domains for discussion, which are (1) Smart Grid, (2) Smart Energy, (3) Smart Facility, and (4) Smart Communication. ### Smart Grid The electrical grid provides an intermediate platform for relaying the electricity from the power plant to residential and industrial area. The common goal in this sub-domain is to provide reliable and stable electricity supply with the integration of ICT, which is commonly known as smart grid. Smart grid has been extensively studied in [@luan2010smart; @gao2012survey; @kordestani2017data; @thirugnanam2018energy] and the goal is to address on load and demand balancing of electricity in a particular area, building, or even household. Common technique applied in this sub-domain is forecasting, and example of such application can be found in [@luan2010smart], which it combines the information received from residential meters and predicts the electricity consumption load. Wang et al. [@wang2012multi] have proposed a different approach, where the concept of multi agent systems (MAS) is used to predict building energy consumption by denoting each meter as an agent. The common goal is to use a higher information extraction technique such as prediction, where it allows grid operators to forecast the grid demand to ensure sufficient electricity load. Test bed currently is the common method for testing out the smart grid use case and has been studied in [@tushar2016smart]. [[ Another common research topic is security and reliability of the smart grid system. Li et al. [@li2011communication] have proposed a secure state estimation, which it can be used to address single sensor or multi-sensor scenarios. Similar works addressing smart grid security also can be found in [@huang2012state; @liu2015abnormal].]{}]{} On the other hand, advanced metering infrastructure (AMI) has been studied along with the smart grid to ensure the electrical metering is tamper-proof while able to accurately measure energy consumption. For instance, work in [@mclaughlin2013multi] uses the clustering algorithm to identify energy theft accurately while reducing potential false positives. Meanwhile, work in [@izumi2018real] has presented a real-time price estimation by fusing local power and global power consumption to understand real-time electric load of the grid. In future, prosumers (producer and consumer) will emerge in the smart grid market and sole distributor paradigm will be no longer valid. This scenario greatly increase the difficulty of the energy demand and load when accounting the energy as a live market ### Smart Energy The search for clean energy resources has been an ongoing effort for many researchers in order to cut down the dependency on the fossil fuels. Therefore, the clean energy research direction mostly focuses on renewable energy, which propose to go for a green and less carbon footprint energy producing approach. Nowadays, the most common renewable energy sources emerged in the market are solar farm [@katoch2018shading; @huang2018data] and wind power [@sideratos2007advanced; @foley2012current]. Solar energy is generated based on the conversion of the sunlight into electricity, but the energy harvesting technique suffers from limited energy harvesting time. Thus, solar irradiance prediction is crucial to ensure maximum energy throughput in the solar farm within the limited time. Huang et al. [@huang2018data] have proposed to use data driven algorithms such as ABB, SVM, BRT, and Lasso, in which the information from neighboring solar plants are combined to accurately predict the solar irradiance. Meanwhile in [@jung2014current], Jung and Broadwater have implemented a statistical model to fuse wind speed, direction, temperature from forecast station and online measurement to determine the total power output of the wind farm. Most of the aforementioned methods focus on improving efficiency of the existing energy harvesting methodology. Future research on the clean energy relies on various data and energy sources in order to construct a high efficiency energy harvesting model. ### Smart Facility Smart facility denotes access of physical facility that provides services to the public such as parking facility, water supply, etc. The most vital facility in a smart city would be water treatment center as clean water is an important necessity for the urban citizens. Any potential leakage or downtime of water supply in a city would be proven troublesome. Mounce et al. [@mounce2003sensor] propose a water leakage detection using classification technique, which combines all the district water meter data. Similar concept can be applied on other resources such as gas pipe leakage detection or electricity theft in smart grid. In the public facility, the wear and tear of structures can be a major issue due to the frequent rate of public usage. Hence in [@park2014wireless], Park et al. have combined multi-metric sensors to estimate the bridge displacement. Through this, a rough estimation of the structural health can be determined. Alternately, Khoa et al. [@khoa2017smart] have proposed a tensor decomposition approach using the facility data sources in order to understand the facility usage details. [[ In addition, the emergence of data centers providing various functionalities to the smart city applications such as [@cioara2019exploiting; @li2018ultra; @kong2015survey] also one of the focuses for the ongoing efforts of smart city.]{}]{} There is also a few domains that is highly correlated with smart facility such as Smart Maintenance and Governance [@consoli2015urban], where integration of a web portal is used to report potential damages. ### Smart Communication Communication in an urban city remains an essential infrastructure for various application platforms to communicate with each other. Not all communication platforms and standards are designed equally as each of them serve different purposes. Therefore, different standards and protocols to meet varying requirements have been established. Currently, the upcoming 5G technology [@andrews2014will; @rappaport2013millimeter] has promised to bring integration of 5G interface with support for older generation spectrum such as LTE and Wi-Fi in order to provide seamless user experience. The common data source in 5G standards is raw signal, and that is the reason why data fusion only happens at the edge level. For example, Huang et al. [@Huang2019Iterative] and Rappaport et al. [@rappaport2013millimeter] have fused raw signals that are divided through multiple antenna during transmission. The receivers will receive multiple signal sources and reconstruct the original information being transferred. Further discussion of the energy efficient trade-off in wireless communication technology can be found in [@mahapatra2016energy; @wang2017survey]. In the IoT domain, wireless sensor network (WSN) is considered a common communication platform because of its wide coverage and low power consumption. WSN is built on top of nodes’ network, which is smaller than a wireless ad hoc network. Hence, multiple nodes can be combined for encoding and decoding the packets received. Kreibich et al. [@kreibich2014quality] and Luo et al. [@luo2016kernel] have proposed approaches to improve communication between WSN focusing on the communication mechanism between nodes. The main objective is to focus on the reliability of communication channel while maintaining the coverage (from relay to sink nodes) and also low power consumption. The research significance of communication is undoubtedly a necessity in smart city as it benefits all domains leveraging communication technology. The main goal is to design efficient and reliable communication protocols to meet different requirements of applications. Alternately, low power communication is yet another goal for IoT in order to achieve long sensing operation. Challenges and Open Research Directions {#sec:discussion} ======================================= [[ After outlining the applications of the smart city that use data fusion, we discuss the potential aspects to improve the data fusion in the smart city applications observed from previous section. These aspects include potential categories or perspectives that are not discussed in Section \[sec:categoryDefinition\] and \[sec:currentSmartCityApp\].]{}]{} As shown in Figure \[fig:Fig02\_open\], we identify four major research directions, which are (1) data quality, (2) data representation, (3) data privacy and security, and (4) data fusion technique. ![image](Fig02_researchDirections2){width="100.00000%"} Data Quality ------------ Quality of the data sources directly determine the quality of output results since processing module follows the “garbage in and garbage out" theorem in fusing data sources. Thus, we discuss two aspects to improve the data sources in the smart city applications, which are sensing coverage and sensing longevity. ### Sensing Coverage Sensing coverage is one of the important factors to determine the quality of data sources. Insufficient data coverage will generate a result that is not representative, and often it implies more sensors need to be installed to increase the sensing coverage. This indirectly affects the deployment cost since more physical hardware is required to compensate the sensing coverage. Apart from that, it also affects the design of communication architecture because more physical sensors are required to transmit data, and thus potentially congests the communication platform. These factors are common obstacles for a large-scale deployment in smart city applications and getting worse when increasing the deployment scale. There are two commonly used approaches to address the aforementioned issues, which are crowdsensing and mobile sensing platform. As shown in [@ma2014opportunities], crowdsensing is one of the most cost-efficient method as personal mobile devices such as smartphones. Smartphones offer wide variety of sensors such as vibration, magnetic field, IMU, GPS, and others. The problems with crowdsensing are related to user privacy intrusion and high battery consumption when actively collecting data. User privacy is a challenge in collecting data as regulations in many countries have been facilitated to prevent applications to collect any sensitive information. This issue will be further discussed in user privacy and security sub-section. Another problems with crowdsensing are the unavailability of geolocations information or random distribution of geographical located data. These scenarios lead to inconsistent data quality. Potential way to resolve this limitation is to collect data at a fixed time and location only when needed, where incentive is provided for valid participants. Also, the trade-off problem of the mobile sensing can be further found in [@wang2018distributed]. Through this method, only qualified data will be included as data sources, while invalid information will be automatically filtered. Using similar concept as crowdsensing, mobile sensing has offered the same data sensing approach but only follows designated route to collect data. The idea is to leverage the mobility of the transportation (normally public transports, cabs, and garbage trucks) to conduct data collection, where the vehicles are traveling across the city. Example of mobile sensing platform can be found in [@Anjomshoaa_IIoTJ:2018], where garbage trucks on duty will collect the ambient data across different parts of the city weekly. An identical concept can also be implemented with the public transport systems, since majority of them follow fixed schedules. The challenge with the mobile sensing is that spatial resolution of the data may not have a finer detail when compared to crowdsensing due to fixed data collection schedule. The main cause is due to the limited accessibility of the vehicles in certain areas (pedestrian path and residential area). Potential workaround of this limitation would be combining the mobile sensing and crowdsensing data sources to generate data that covers large area within the urban city. Services integration also plays an important role in supplying platforms alternate data sources to perform data enrichment. By simulating the different IoT services in smart city as shown in [@jha2018holistic], potential limitation or bottlenecks of smart services can be avoided in order to design a better smart city application. ### Data Sensing Longevity Long term data collection offers different aspects of knowledge discovery as data is able to cover more detail in a larger temporal resolution. The advancement of miniaturization has greatly reduced the power consumption of the sensors and IoT devices while maintaining the same sensing performance. As a result, combining both energy harvesting techniques and low energy devices are able to create a long self-sustaining sensing approach. This breakthrough allows physical sensors to run independently without the need of external power sources. In order to preserve the longevity of physical sensors’ sensing capability, energy harvesting is one of the common approach in large area networks. It allows sensors to draw energy from solar energy, vibration, or temperature difference. The most widely available energy harvesting technique is solar panels and it can be easily obtained. Solar panel is affected by the presence of solar irradiance, where the energy harvested varies throughout the different time of the day. Contrast to solar farm, the goal here is to conserve as much energy, while maintaining the sensing capability of the physical sensors. The most notable influence would be the energy management architecture as well as the battery capacity and the solar panel efficiency. Apart from that, although temperature difference and sensor vibration are capable of harvesting energy but it is limited to certain use case and not suitable for general usage. Alternately, potential replacement of the traditional energy harvesting technique is wireless power transfer. As shown in [@bi2015wireless; @sekitani2007large], this method offers power to be transferred wirelessly without battery and energy harvesting module. Currently, there are different types of wireless power transfer technologies such as inductive coupling, capacitive coupling, magnetodynamics coupling, microwaves, and light-waves. Each of them has their limitation such as inductive coupling only has limited range of transferring energy. That being said, this technology is still relatively new, and it requires further investigation in order to guarantee its minimum working efficiency for smart city applications. Other than using external power sources, low power sensing for carrying out the sensing tasks. In order to drive different smart city applications, various standards have been proposed for LPWAN, such as LoRaWAN by LoRa Alliance and NB-IoT Release 13 by 3GPP. LoRaWAN focuses on the long range IoT connectivity for industrial applications while the NB-IoT focuses on the indoor coverage, low cost, long battery life, and stable communication in high density communication channel. The main reason to use low power sensing approach is due to the high compatibility with large scale deployment relying on the low bit rate communication channel usage. However, standardization of these protocols remains a challenge in LPWAN due to the possibly of using unlicensed spectrum, where organizations may choose not to follow the agreed spectrum. In future, low power communication will ensure the long term sensing capability of physical sensors in the smart city applications and therefore will improve data sources quality. Data Representation ------------------- A high speed Internet connection provides easy access to many genres of data sources and creates opportunity to study wide variety of different data sources. However, large variety of data sources frequently indicate the incompatibility of data formats. The problem becomes more obvious when there is no standardization on the data format. To tackle such problem, data ontology is the building block to represent the data sources to connect different sources of data for seamless services integration. If the format of the data source cannot be interpreted, it will be marked as useless for the platform integrator. Therefore, semantic web has been proposed as an extension to WWW web services utilizing Resource Description Framework (RDF) to provide standard data exchange formats. It opens the path to create different solutions for the IoT applications and it supports the Open Government Data (OGD) principles [@ubaldi2013open]. To date, there are few common ontology languages have been developed such as Delivery Context (DCN) [@cantera2010delivery], Web Ontology Language (OWL) [@antoniou2004web], Resource Description Framework Schema (RDFS) [@brickley2000resource], Semantic Sensor Network (SSN) [@neuhaus2009semantic], and others. Majority of the ontology languages only focus on one application domain because they are not suitable for representing the metadata from other domains. This causes data segmentation in the smart city applications, where further increases the gap between different smart city domains. Thus, DBpedia [@auer2007dbpedia] is designed to address the aforementioned issue using public and private stocks of semantic web. DBpedia has provided solutions for the ontology software as it offers different classes and types that are available on the Wikipedia. That being said, not all applications adopt the idea of DBpedia and there is a fraction of applications remain conservative using proprietary data representation. Apart from that, Message Queuing Telemetry Transport (MQTT) [@Organization_:2018] v3.1 protocol has been introduced as one of the protocols to address ontology problems between brokers. It offers machine to machine (M2M) communication by providing lightweight publish and subscribe messaging services, where network bandwidth limitation is one of the main constraint. It is possible to combine the aforementioned technologies in order to generate a better data integration for data fusion purposes across different domains. Therefore, the future agenda for the ontology language is to encourage integration of different levels of data sources using different system architecture such as edge, fog, and cloud computing. Privacy and Security -------------------- ### [[ Privacy]{}]{} Collecting urban residents’ data in a smart city application can be challenging due the nature of sensitive data that can be misused if poorly managed. As privacy issue has been discussed extensively by the authors in [@martinez2013pursuit; @li2016privacy; @Chan_:2008], misuse of private information may lead to catastrophic events such as information theft, or identity fraud. Currently in Europe, General Data Protection Regulation (GDPR) as discussed in [@tankard2016gdpr; @albrecht2016gdpr] has been proposed to better address the data privacy concern of the Internet. In other countries, there are also similar efforts to enforce data privacy protection such as Canada’s Personal Information Protection and Electronic Documents Act (PIPEDA), China’s China Data Protection Regulations (CDPR), Singapore’s Personal Data Protection Act (PDPA), Japan’s Personal Information Protection Commission (PIPC), etc. Meanwhile in USA, Health Insurance Portability and Accountability Act of 1996 (HIPAA), the Children’s Online Privacy Protection Act of 1998 (COPPA), and the Fair and Accurate Credit Transactions Act of 2003 (FACTA) have been introduced to improve with the information flow efficiency across agencies. This is also a part of the efforts to prevent sensitive information being available for unauthorized parties. Majority of the policies and regulations emphasize on the users’ consent for collecting personal data and this can be problematic as not all platforms provide ample security for data storage. With insufficient security measurements, the data collected may be compromised, which may lead to tainted reputation and loss of public faith. For instance, Facebook and Cambridge Analytica scandal [@cadwalladr2018revealed] has shown potential misuse of user data collected. With that in mind, potential right of accessing data sources could be revoked if the data source is not handled properly by the right person. Hence, privacy and security should be the responsibility for both platforms and users. A thorough review has been conducted in [@ding2018survey], which works on the IoT requirements to address privacy issues. Potential solution for the aforementioned problem is to use hybrid data fusion technique in a smart city application. The idea here is to locally fuse the sensitive information (user identity, phone number, bank account number) into generic information, before uploading to the cloud for further processing. The benefits of such approach are two-folds, which are the ability to offload computational cost and to preserve sensitive information at the physical sensor only. In addition, we can leverage machine learning approaches such as [@beaulieu2017privacy; @esteban2017real] to generate synthetic datasets with identical data characteristic for study purpose. This eliminates the chances of private data been leaked out and encourage the openness of datasets to be studied by different researchers and data scientists. To draw a clear line between generic and sensitive data remains a debate among researchers. In future, the data fusion can be applied at the lower level to remove any potential sensitive data. ### [[ Security]{}]{} According to Kitchin [@kitchin2016getting], there are two general security concerns in the smart city applications, which are security of technology/infrastructure (data center, services, and system architecture) and data security (data generation, storage, and communication). The security of the technology and infrastructure highly relies on the design architecture of the system being deployed. Depending on the application requirements, it varies from traditional client server architecture to decentralized architecture. The main objective is to deploy a hack-proof/exploit-less system architecture. Alternately, there are also ways of improving security of system architecture such as incentive/bounty for reporting flaws, simulating injection attacks, security assessment from third party, etc. Nowadays, the security enhancement focuses towards continuous effort as the technology has been changing rapidly. For instance, security works in [@chakrabarty2016secure; @mocanu2019data; @talacs2017elastic] have proposed different strategies to enhance the security of the smart city application’s architecture by focusing on the common security standards/practices/protocols. This shows that as the number of smart city applications increase rapidly, system architectures implemented with the security design in mind become apparent with good practices and standard architecture design. Subsequently, regular security assessment and auditing also pave way for a safer smart city applications deployment. Meanwhile, data security also contributes to the significant part of smart city applications ecosystem from generation, storage, and communication. The common method to combat such issue is leveraging encryption techniques, where it encodes the data so that only the authorized parties have access to it. For instance in [@wang2014performance], Wang et al. have introduced an attribute based encryption scheme, which it allows fine-grained access control, scalable key management, and flexible data distribution. In addition, encryption also can be used in the communication platform between IoT devices in smart city application as shown in [@singh2015secure; @elhoseny2018secure] to prevent information hijacking. Despite constant effort of cyber security researchers developing new security schemes, the numbers of data breaches and cyber threats increase every year according to David et al. [@DavidMcCandless_:2019]. The main culprit of such occurrence is due to negligence of data security practices/implementation. Security often appears to be an afterthought in deployment of a smart city application. Thus, in order to combat such threat, the smart city application should comply with security standards as shown in [@bartoli2011security] to mitigate the chances of becoming a victim. Data Fusion Techniques ---------------------- Extracting knowledge from a smart city application frequently involves data mining techniques in order to fuse different data sources. Lower tier data fusion techniques have been well explored in [@Dasarathy_PotI:1997] and the current research trend focuses more on the machine learning approach. The main reason why machine learning approach has gained so much attention is due to its capability of handling high dimensional data. The problem of high dimensional data is also known as curse of dimensionality as described by Bellman [@bellman2013dynamic]. In this context, we discuss two research trends on applying machine learning techniques in data fusion as follows: ### Explainable Deep Neural Network Lately, supervised machine learning techniques focus on the DNN, where the in-depth reviews of the recent development can be found in [@zhang2018survey; @miikkulainen2019evolving; @liu2017survey]. Major research efforts aim to increase the explainability of the model such as NN, CNN, and DNN rather than using them as black box models. To this end, explainable AI (XAI) [@Gunning_DARPADnW:2017] is the new motivation for data scientists to explore the interpretable learning paradigm of the modeling in order to provide a semantic meaning behind modeling logic. This new learning process has driven three big fields in the deep learning domains, which are (1) Deep Explanation, (2) Interpretable Model, and (3) Model Induction. To develop a deep explanation on the model interpretation, the cognitive layers will act as an intermediate layer between learning and explanation layer in order to cast the learned abstractions, policies, and clusters information into an explainable format. Subsequently, the interpretable model such as Bayesian learning [@kendall2017uncertainties] can be built to explain the uncertainties required when developing the deep learning models to learn the choices of a learning process. Alternate approach has proposed to use subspace approximation with an adjusted bias technique [@kuo2018interpretable] to build interpretable CNN, which uses feed forward design to better explain the model’s choice in allocating certain hyper-parameters. Meanwhile, model induction refers to the technique used for inferring the model’s decision and learning progress. Through a thorough understanding of the model, parameters can be fine-tuned to increase the learning optimization rate in a long-term application deployment. Hence, the search of XAI is an important milestone for the data scientists, which can be used to explain the learning process and the decision machine learning made. An example of potential use case would be trying to understand the reason behind (also known as reasoning in some literatures) the predictive maintenance decision machine learning rather than performing maintenance due to the result of predictive algorithm. ### Unsupervised Data Fusion In the smart city applications, collecting the ground truth could be proven challenging due to the uncertainties and errors in the collected data sources. Hence, obtaining labels or data annotation are another problems with certain data sources. Despite the rapid development of advanced modeling tools like DNN, it still requires labels and data annotation in order to achieve objectives of extracting higher information. There are a few approaches that address the lack of labels such as manual annotation, crowd labeling, software annotation, and pattern labeling. However, manual annotation only works well with a small dataset while other approaches do not guarantee the correctness of end result. This shows a big research gap to seek a better way to label data sources accurately. Research works such as Zhou et al. [@da2014learning; @li2015towards] have attempted to fix unlabeled data by transforming them into useful features to achieve certain objectives. Traditionally, raw data is required to be preprocessed into something meaningful, but it still suffers from the need of data cleansing and amputation. The simplest method would be to solely depend on the filtering technique. However, aggressive filtering may remove large amount of raw data resulting potential loss of knowledge. Another simple solution is to increase the number of reliable data sources to be fused to create potential annotation. Increasing data sources often indicates an increment of the overall deployment cost. Alternative solution to the increased deployment cost is to use transfer learning [@hoo2016deep], where the knowledge from existing domain can be transferred to other domain to learn from it. ### Emergence of Hybrid Model The emergence of the hybrid models has become common due to wide variety of data sources available. It allows different levels of data sources (high, low, or both) to combine in order to create potential insights in a particular domain. It also helps to solve the data privacy problem along with machine learning technique, which has opened up many opportunities for researchers and data scientist to study on these big data collected. One example of the hybrid model is shown as follows: an urban planning system has different data sources as input such as human comfort factor index (environmental ambient sensors), positive urban city factor (feedback data on urban area such as greenery, surrounding amenities, recreational parks, and others), and cyber data (social media input) to design a fully automated urban planning system by fulfilling predefined criteria. The result from the data fusion needs to be explainable as discussed in the previous XAI for understanding choices made by the automation software. In this example, different tiers of data sources are fused using data sources types ($D1$, $D2$) and the result is some features. Eventually, these features will be combined to generate a potential plan for city through computation modeling ($D3$, $D4$). By joining different data sources, simulation can be used concurrently to verify the performance of urban planning system before deploying to the city. In future, implementation of the hybrid model will become a general trend due to wide availability of the data sources and processing platforms. As mentioned in the discussion, data ontology is another key factor to allow data sources to be connected from different platforms to provide knowledge for the smart city applications. Conclusion {#sec:conclusion} ========== This paper presents an overall view of the data fusion techniques found in the smart city applications. Easy accessibility of the data sources has paved way for data fusion in different smart city applications in various forms. The increasing trends of data fusion in the smart city applications create the need for a new evaluation method. Therefore, we propose a multi-perspectives classification for the smart city applications that involve data fusion techniques. The data fusion classification based on multi-perspectives introduced in this paper are: (1) Fusion Objectives, (2) Fusion Techniques, (3) Data Input and Output Types, (4) Data Source Types, (5) Data Fusion Scales, and (6) System Architecture. Using the proposed multi-perspectives, we evaluated some selected works in the smart city applications and we also discussed the research trend for each domain respectively. [[ Next, we also discuss four open research directions of data fusion in a smart city application such as data quality, data representation, data privacy & security, and data fusion technique. Overall, we are certain that generic nature of the multi-perspectives classification is able to perform well with various smart city applications for different domains that leverage the data fusion techniques.]{}]{} [[ In addition, an in-depth analysis can be further extended onto individual domain to study the common requirements and techniques applied, which we do not include in this paper due to limited paper length. ]{}]{} A successful smart city application is built on top of the data (also known as data-driven architecture) and data fusion has provided a wide variety of techniques to improve the input data for an application. Therefore, data fusion has opened the path for various applications to gain insights about the city. This also holds the key for a smart city to further understand and improve the domains that it is lacking. Acknowledgment {#acknowledgment .unnumbered} ============== The research work was supported in part by the National Research Foundation (NRF) of Singapore via the Green Buildings Innovation Cluster (GBIC) administered by the Building and Construction Authority (BCA)–Green Building Innovation Cluster (GBIC) Program Office; in part, by the SUTD-MIT International Design Center (IDC; idc@sutd.edu.sg); in part by Natural Science Foundation of China (NSFC) through Project No. 61750110529,61850410535 and Higher Education Commission (HEC) Pakistan through grant number NRPU P\#5913. We thank our colleagues and reviewers, who have provided insight and expertise that greatly assisted with improving the context of this survey paper. [^1]: B. P. L. Lau, S. H. Marakkalage, Y. Zhou, N. Ul Hassan, C. Yuen, U-X. Tan are with the Engineering Product Development, Singapore University of Technology and Design, 8 Somapah Rd, Singapore 487372. [^2]: N. Ul Hassan is also with Lahore University of Management Sciences (LUMS), Lahore, Pakistan 54792 [^3]: Meng Zhang is with Southeast University, Nanjing, China 210096
--- abstract: 'We study the role of a possible nonet of light scalar mesons in the still interesting $\eta \rightarrow 3\pi$ decay process, with the primary motivation of learning more about the scalars themselves. The framework is a conventional non-linear chiral Lagrangian of pseudoscalars and vectors, extended to include the scalars. The parameters involving the scalars were previously obtained to fit the s-wave $\pi\pi$ and $\pi$K scatterings in the region up to about 1 GeV as well as the strong decay $\eta'' \rightarrow \eta \pi\pi$. At first, one might expect a large enhancement from diagrams including a light $\sigma(560)$. However there is an amusing cancellation mechanism which prevents this from occurring. In the simplest model there is an enhancement of about 13 per cent in the 3p decay rate due to the scalars. In a more complicated model which includes derivative type symmetry breakers, the cancellation is modified and the scalars contribute about 30 percent of the total decay rate (although the total is not significantly changed). The vectors do not contribute much. Our model produces a reasonable estimate for the related $a_0(980)-f_0(980)$ mixing strength, which has been a topic of current debate. Promising directions for future work along the present line are suggested.' author: - 'Abdou Abdel-Rehim$^{\it \bf a}$ ' - 'Deirdre Black$^{\it \bf b}$ ' - 'Amir H. Fariborz$^{\it \bf c}$ ' - 'Joseph Schechter$^{\it \bf a}$ ' title: Effects of light scalar mesons in 3p decay --- , , , Introduction ============ There has been a revival of interest recently [@kyotoconf] in the possible existence of a broad scalar meson (sigma) with a mass in the 560 MeV region and its corresponding nonet partners. A large number of workers [@vanBev]-[@Deirdreetal] have found evidence for the sigma in models of $\pi \pi$ scattering even though it is partially obscured by background. Generally this state is considered to be of exotic nature (more complicated than $q \bar q$) and hence an important clue to an understanding of QCD in its low energy non-perturbative regime. Similarly, analyses of $\pi\pi$ , $\pi$K and $\pi\eta$ scattering have provided evidence for the existence of the remaining members of a possible light scalar nonet: the $\kappa$, the $a_0(980)$ and the $f_0(980)$. In fact, the latter two states have been well established experimentally for some time. Of course, the treatment of such strongly interacting processes is inevitably model dependent and there are a number of different opinions as to the correct approach [@kyotoconf]. Thus it is of great interest to see whether treatments of the role of scalars in other processes using the same models employed in the scattering processes above give consistency with experiment. From this point of view we will study the role of possible light scalars in the interesting 3p decay. Typically this process has been treated by chiral perturbation theory [@Gasser85], in which the possible effects of scalars have been amalgamated into effective contact interactions among the pseudoscalars. This is probably the most effective way to study the 3p decay. However, our goal here is to learn more about the scalars so it is natural to keep them rather than integrating them out. Also there is a possibility that a light scalar \[like the $\sigma(560)$\] might give an enhancement due to closeness of its propagator to the pole \[see for instance Feynman diagrams like (a) and (b) of Fig.\[scdiagrams\]\]. Another reason for including light scalars explicitly is to become more familiar with the isospin violating $a_0(980)-f_0(980)$ transition which should play a role in the 3p decay and has also recently been postulated [@Close01] to provide an explanation for observations of anomalously strong $a_0(980)$ central production and the large $\Gamma(\phi \rightarrow f_0 \gamma) / \Gamma(\phi \rightarrow a_0 \gamma)$ ratio. It is important to know whether the value consistent with the eta decay determination is consistent with these proposed new effects. Doubts about whether an unreasonably large value was assumed in [@Close01] were expressed in [@Achasov02]. These doubts were confirmed [@Black02] using the work of the present paper. Still another reason for the interest in the effects of the scalars in 3p is to provide an orientation for the discussion of the apparently puzzling $\eta' \rightarrow 3\pi$ decays in which light scalar mesons can be reasonably expected to have very large effects. We will give only a preliminary discussion of this process here. In section II we give a brief historical outline of treatments of 3p decay based on chiral symmetry. A number of well known ambiguities in the analysis are briefly described. Our calculation is based on the tree level treatment of a chiral Lagrangian containing pseudoscalars, vectors and a postulated nonet of light scalars. Since the calculation is somewhat complicated, it seems to us helpful to present the results in a series of steps. First, in section III we give the results of using a Lagrangian containing only pseudoscalars with minimal symmetry breaking terms. To this Lagrangian we add, in section IV, the scalar mesons. It will be seen that the individual scalar diagrams are quite large but there is a lot of cancellation so that the net effect is not at all dominant. However the scalars do, as desired, increase the predicted decay rate in a noticeable way. Next, the effect of adding some derivative type symmetry breakers for the pseudoscalars is described in section V. This doesn’t much change the overall rate but modifies the somewhat delicate cancellations so that the scalars end up making a larger percentage contribution than before. In low energy calculations of this sort one always may expect some contributions from the vector mesons. This is discussed in section VI where it is shown that, although there is a new type of diagram the vectors do not produce a big change in the previous results. Section VII contains a discussion of the results and directions for further work. For the convenience of readers, material describing the chiral Lagrangian used is brought together in Appendix A. Similarly the detailed expression for the decay amplitude is given in Appendix B. Historical Background on the 3p decay ===================================== The study of 3p has turned out to be surprisingly complicated and correspondingly important for understanding the non-perturbative (low energy) structure of QCD. Chiral dynamics in various forms has been the basic tool. Since the process violates G-parity it was initially assumed to be of electromagnetic nature, mediated by an effective photon exchange operator proportional to the product of two electromagnetic currents. The old “current algebra” approach had previously predicted the $K_L \rightarrow \pi^+ \pi^- \pi^0$ spectrum shape [@haranambu] to be $$1 - \frac{2E_0}{m}, \label{shape}$$ where $m$ is the $K_L$ mass and $E_0$ the energy of the $\pi ^0$ in the $K_L$ rest frame. This shape, which is in reasonable agreement with experiment, resulted from the vanishing commutator of the axial charge transforming like a $\pi^+$ with the appropriate product of two weak currents. When Sutherland [@Sutherland66] repeated this type of calculation for $\eta \rightarrow \pi^+ \pi^- \pi^0$ with the product of two electromagnetic currents he found that the decay amplitude was actually zero (to this leading order). Thus the 3p decay did not seem to be mediated by a virtual photon emission and reabsorption. In fact, it was found [@bose] that a quark scalar density operator with the $\Delta I = 1$ property proportional to $$\bar u u - \bar d d \label{scalardensity}$$ would give a non-zero result for the decay rate. A more detailed treatment [@Chiu67] showed that the quark density operator gave the same spectrum for $\eta \rightarrow \pi^+ \pi^- \pi^0$ as in Eq.(\[shape\]) with $m$ the $\eta$ mass in this case. Such a result is in fairly good agreement with experiment. The scalar density interaction in Eq.(\[scalardensity\]) was recognized [@su] to be the fundamental up-down quark mass difference generated by the Higgs boson in the electroweak theory. However, the predicted rates of the $\eta \rightarrow \pi^+ \pi^-\pi^0$ and $\eta \rightarrow 3 \pi^0$ modes (both the ratios and the absolute values) did not agree well with experiment at that time. Some years later, after more precise experiments, the ratio of the rates for $\pi^+ \pi^- \pi^0$ to $3\pi^0$ modes stabilized around the value expected from isospin invariance. On the other hand the absolute rate has only recently stabilized to a value notably larger than that predicted by theory. The theory behind the current algebra results could be economically presented in the framework of an effective chiral Lagrangian. For most low energy processes where the scheme could be expected to work, the tree level computation did produce results within 25 $\%$ or so of experiment. Thus the relatively poor prediction for 3p at tree level is somewhat surprising. An improvement was obtained by Gasser and Leutwyler [@Gasser85] who carried the computation of the chiral Lagrangian amplitude to one loop level. Since the non-linear chiral Lagrangian is non-renormalizable, this required the addition of new counterterms. Their finite parts were new parameters which could be mostly determined from other processes. They obtained the result $\Gamma(\eta \rightarrow \pi^+ \pi^- \pi^0) = 160 \pm 50$ eV which may be compared with the present experimental value [@pdg] ${\Gamma \left( \eta \rightarrow \pi^+ \pi^- \pi^0 \right)}_{\rm expt} = 267 \pm 25$ eV. The extra effects included involve both the implicitly integrated-out heavier meson exchanges and partial unitarization to one loop order. One might expect a two loop calculation in the chiral perturbation scheme to be valuable but this may involve too many unknown parameters at the present stage. A dispersion approach using the Gasser-Leutwyler result as a subtraction gave an improved estimate [@Kambor96] $\Gamma \left( \eta \rightarrow \pi^+ \pi^- \pi^0 \right) = 209 \pm 20$ eV, which still seems too small. A possible source of ambiguity arises from the determination of the coefficient of the driving scalar density interaction in Eq.(\[scalardensity\]). This is determined from the $K^0 - K^+$ mass difference, which in turn has two components $$\begin{aligned} m^2 (K^0) - m^2 (K^+) &=& { \left[ m^2 (K^0) - m^2 (K^+) \right] }_{\rm quark \, mass} + { \left[ m^2 (K^0) - m^2 (K^+) \right] }_{\gamma}, \label{deltak}\end{aligned}$$ corresponding to the quark mass differences and the virtual photon emission and reabsorption diagrams respectively. The latter is given in the chiral limit by $m^2 (\pi^0) - m^2 (\pi^+)$ according to Dashen’s theorem [@dashen] and the reasonable assumption that the photon contribution saturates the pion mass difference. A number of authors [@Donoghue92] have argued that there are important corrections to Dashen’s theorem which have the effect of boosting the 3p decay rate. If one questions Dashen’s theorem it is natural to also question Sutherland’s result, which deals with the direct electromagnetic contribution to 3p . An investigation of this point yielded [@Baur96] the estimate that there was only about a 2$\%$ change arising from this, although it decreased rather than raised the rate. Still another point which may repay further investigation concerns the possible subtleties arising from $\eta - \eta^\prime$ mixing. An understanding of the $\eta^\prime \rightarrow 3 \pi$ process, for example, might clarify this point. This process has been treated by some authors [@hudnall], [@paula] in the literature but has received only a fraction of the attention given to 3p . In the present paper we will focus on learning more about the putative nonet of light scalar mesons by studying their contribution to 3p. Chiral symmetry results to lowest order ======================================= For comparison, we first present the well-known results when only the terms present in the lowest order chiral Lagrangian of pseudoscalars are kept. $$\begin{aligned} {\cal L}_{LO} &=& \frac { {F_\pi}^2}{8} {\rm Tr} \left( \partial_\mu U \partial _\mu U^\dagger \right) + \delta^\prime {\rm Tr} \left[ {\cal M} \left( U + U^\dagger \right) \right] + \frac{\kappa}{576} {\rm ln}^2 \left( \frac { {\rm det} U}{{\rm det} U^\dagger} \right), \label{lowestorderLag}\end{aligned}$$ where the last term \[see Eq.(\[extraetaprime\]) and comments there\] supplies mass to the SU(3) singlet state and $\cal M$ is defined in (\[spurion\]). Fitting ${\cal L}_{LO}$ to the experimental masses determines $\delta^\prime = F_\pi^2m_\pi^2/8$. The $\eta \rightarrow \pi^+ \pi^- \pi^0 $ amplitude receives, in this approximation, contributions from diagrams (a), (b) and (c) of Fig.\[psdiagrams\], which are given in Eq.(\[psampls\]) (with the non leading corrections deleted). To a reasonable approximation which displays the key dependences these sum up to the lowest order result for the $\eta \rightarrow \pi^+\pi^- \pi^0 $ amplitude $$M_{0+-}(E_1,E_2,E_3) \approx \frac{16 i \delta^\prime y}{F_\pi^4} {\rm cos} \theta_p (1 - \frac {2E_1}{m_\eta} ) . \label{lowestorderAmp}$$ Here $E_1$ is the $\pi^0$ energy in the $\eta$ restframe and $y$ is the dimensionless parameter in Eq.(\[spurion\]) which measures the isospin violation in the quark mass matrix. Assuming Dashen’s theorem, Eq.(\[lowestorderLag\]) yields $$8\delta^\prime y = F_\pi^2(m_{K^0}^2-m_{K^+}^2-m_{\pi^0}^2+m_{\pi^+}^2) ,$$ which allows us to solve for $y$. Furthermore $\theta_p$ is the “nonstrange-strange" pseudoscalar mixing angle defined in Eq.(\[eta\_etap\]); it is generally taken to be about $37^o$. It is related to the “octet-singlet" angle, $\theta$ by $${\rm cos}\theta_p = \frac{{\rm cos}\theta -\sqrt 2{\rm sin}\theta}{\sqrt 3} .$$ Then Eq.(\[lowestorderAmp\]) agrees with Eq. (1.14) of [@Gasser85] except that they neglected $\eta-\eta^\prime$ mixing by replacing ${\rm cos}\theta_p \rightarrow 1/\sqrt 3$ in what was denoted the current algebra formula. The matrix element for $\eta \rightarrow 3\pi^0$ is given in general by $$M_{000} = M_{0+-}(E_1,E_2,E_3) + M_{0+-}(E_2,E_1,E_3) + M_{0+-}(E_3,E_2,E_1).$$ The widths are then, defining $\Gamma_{0+-}= \Gamma(\eta \rightarrow \pi^+ \pi^- \pi^0)$ and $\Gamma_{000}=\Gamma (\eta \rightarrow 3\pi^0)$: $$\begin{aligned} \Gamma_{0+-}&=& \frac{1}{64 \pi^3 m_\eta} \int {\rm d}E_1 {\rm d}E_2 {\left| M_{0+-} \right|}^2, \nonumber \\ \Gamma_{000}&=& \frac{1}{384 \pi^3 m_\eta} \int {\rm d}E_1 {\rm d}E_2 {\left| M_{000} \right|}^2.\end{aligned}$$ Using ${\cal L}_{LO}$, with parameters determined as described above, we get the tree-level results \[from the first terms in each of $M_{contact}^{a,b,c}$ in Eqs.(\[psampls\])\]: $$\begin{aligned} \Gamma_{0+-} &=& 106 \, {\rm eV} ,\nonumber \\ \frac{\Gamma_{000}}{\Gamma_{0+-}} &=& 1.40 . \label{eta_LO_result}\end{aligned}$$ These may be compared with the experimental results [@pdg] $$\begin{aligned} {(\Gamma_{0+-})}_{\rm expt} &=& 267 \pm 25 \hskip.2cm {\rm eV} ,\nonumber \\ {(\frac{\Gamma_{000}}{\Gamma_{0+-}})}_{\rm expt} &=& 1.40 \pm 0.01 , \label{eta_experimental_rate}\end{aligned}$$ which demonstrate the disagreement with experiment for the overall rates in the simplest model. However the width ratio has about the correct magnitude. The related energy spectrum is also about the correct magnitude. The squared matrix element is usually described by quantities $a$, $b$ and $c$ defined from $${\left| M_{0+-} \right|}^2 \propto ( 1 + aY + bY^2 + cX^2 \ldots ) , \label{spectrum}$$ with $X = \frac{\sqrt{3}}{m_\eta - 3m_\pi} (E_2 - E_3)$ and $Y=\frac{3}{m_\eta - 3 m_\pi} ( E_1 - m_\pi) - 1$. In the present paper we shall not take into account the (not completely negligible) kinematic $\pi^0-\pi^+$ mass difference. See [@Gasser85] for a discussion of this point. The predictions from this simple model, $a\approx-1$ and $b\approx0.25$ are similar to the experimental results [@Abele] $a_{\rm exp} = -1.19 \pm 0.07$ and $b_{\rm exp} = 0.19 \pm 0.11$ with $c$=0. It is of some interest to also give the predictions for the $\eta^\prime \rightarrow 3 \pi$ decay process at tree level using the simple Lagrangian Eq.(\[lowestorderLag\]). It just is necessary (see Appendix B) to replace ${\rm cos} \theta_p$ by ${\rm sin} \theta_p$ and $m_\eta$ by $m_{\eta^\prime}$ in Eq.(\[lowestorderAmp\]) to get for the $\eta' \rightarrow \pi^0 \pi^+ \pi^-$ matrix element: $$M_{0+-}^\prime(E_1,E_2,E_3) \approx \frac{16 i \delta^\prime y}{F_\pi^4} {\rm sin} \theta_p (1 - \frac {2E_1}{m_{\eta^\prime}} ).$$ This leads to the predictions for the $\eta^\prime$ modes: $$\begin{aligned} \Gamma^\prime_{0+-} &= 497 \, {\rm eV} , \nonumber \\ \Gamma^\prime_{000} &= 562 \, {\rm eV} . \label{etap_LO_result}\end{aligned}$$ The experimental results are given as [@pdg] $$\begin{aligned} {(\Gamma^\prime_{0+-})_{\rm exp}} & < {10}^4 \, {\rm eV} , \nonumber \\ {(\Gamma^\prime_{000})}_{\rm exp} & = 315 \pm 56 \, {\rm eV}. \label{etap_experimental_rate}\end{aligned}$$ Only the $3\pi^0$ mode has really been measured; its width is smaller than predicted in the simple model just presented. One would, of course, expect better agreement for the low energy process $\eta \rightarrow 3 \pi$ for which chiral perturbation theory should be more clearly reliable. In the present paper we shall just make a few remarks on this more complicated process. It may also be worthwhile to give a rough estimate of the corrections to the rates corresponding to violations of Dashen’s Theorem mentioned earlier. If we parameterize the electromagnetic contribution to the $K^+ - K^0$ mass difference as $${(m_{K^0}^2 - m_{K^+}^2 )}_\gamma = f (m_{\pi^0}^2 - m_{\pi^+}^2 ), \label{definef}$$ where $f = 1$ corresponds to Dashen’s Theorem, we would find by using Eq.(\[deltak\]) that the $\eta, \eta^\prime \rightarrow 3 \pi$ rates predicted for ${\cal L}_{LO}$ should be multiplied by $${ \left[ \frac { (m_{K^0}^2 - m_{K^+}^2 ) - f (m_{\pi^0}^2 - m_{\pi^+}^2 )} { (m_{K^0}^2 - m_{K^+}^2 ) - (m_{\pi^0}^2 - m_{\pi^+}^2) } \right] }^2 . \label{Dashen_violation_factor}$$ For $f \approx 2$, which was actually found many years ago [@Socolow65] the correction factor is about $1.54$ and would give $\Gamma_{0+-} \approx 163$ eV. This corresponds to the overall factor, $y$ taking the value $-0.33$ while the Dashen’s theorem value used to obtain Eqs.(\[eta\_LO\_result\]) and (\[etap\_LO\_result\]) is $y=-0.277$. = 8.5cm   =8.5cm   = 8.5cm   Including light scalar meson interactions ========================================= Now we will study what effects the inclusion of a nonet of light scalar mesons will have on the 3p calculation. We designate the scalar nonet by the $3 \times 3$ matrix $N_a^b$ whose interactions are also listed in Appendix A.$N$ is assumed to contain the well-established $f_0(980)$ isoscalar and the $a_0(980)$ isovector as well as the $\sigma(560)$ and the strange $\kappa(900)$. Of these only the $\kappa(900)$ will not contribute to 3p at tree level. The quark structure of such a nonet has been the subject of much discussion [@kyotoconf]-[@Deirdreetal]. If this were an ideal nonet like $\rho - \omega - K^*- \phi$ one would expect the roughly degenerate $a_0(980)$ and $f_0(980)$ to be lowest rather than highest in mass. Actually the masses are better understood intuitively [@jaffe] if $N_a^b$ is an “ideal dual nonet” constructed as $Q_a {\bar Q}^b$ with $Q_a \sim \epsilon_{abc} {\bar q}^b {\bar q}^c$; $q_a$ being the ordinary quark. Then the observed inverted mass ordering is easily seen to follow just from counting the number of strange quarks in $N^a_b$. It is important to note that the form of the couplings of $N_a^b$ to the particles of the non-linear chiral Lagrangian being used depend only on the flavor transformation properties of $N_a^b$. This does not distinguish different quark substructures. What is sensitive to the quark substructure is the scalar mixing angle, $\theta_s$, defined from $$\left( \begin{array}{c} \sigma\\ f_0 \end{array} \right) = \left( \begin{array}{c c} {\rm cos} \theta_s & -{\rm sin} \theta_s \\ {\rm sin} \theta_s & {\rm cos} \theta_s \end{array} \right) \left( \begin{array}{c} N_3^3 \\ \frac {N_1^1 + N_2^2}{\sqrt 2} \end{array} \right). \label{scalar-mixing-convention}$$ Small values of $\theta_s$ would typify a dual ideal nonet while $\left| \theta_s \right|$ about $\frac{\pi}{2}$ would typify a conventional nonet. Fitting the $\pi \pi$ and $\pi$K scattering amplitudes, including the effects of these scalar resonances, selects [@BFSS2] the small value $\theta_s = -20.3^o$. The scalar nonet mass terms in Appendix A are specified by the four parameters $(a,b,c,d)$. The needed chiral invariant scalar -pseudoscalar-pseudoscalar $S \phi \phi$-type couplings are specified by the parameters $(A,B,C,D)$ and the mixing angle $\theta_s$. These were all determined from fitting to $\pi \pi$ scattering, $\pi$K scattering and the strong decay $\eta^\prime \rightarrow \eta \pi \pi$. Actually there is reason to believe [@2nonetmixing] that the scalars may be best understood as mixtures of a lighter dual nonet and a heavier ordinary nonet. From that point of view, which will be explored more fully in the future, the present single nonet, $N_b^a$ should be regarded as an approximation to the situation where the heavier (after mixing) particles have been integrated out. The Feynman diagrams for the scalar contributions to $\eta \rightarrow \pi^+ \pi^- \pi^0$ are shown in Fig.\[scdiagrams\]. Notice that the diagram in (d) involves new $a_0 - \sigma$ and $a_0 - f_0$ isospin violating transitions rather than the $\pi^0 - \eta$ and $\pi^0 - \eta^\prime$ transitions which play an important role in the other diagrams. Their strengths $A_{a\sigma}$ and $A_{af}$ (see Appendix B) were determined simply by including the effects of isospin violation contained in the spurion matrix $\cal M$ in the $b$ and $d$ scalar mass terms. Therefore, this does not introduce any new parameters. Actually, the possibility of such contributions was suggested a long time ago [@hudnall] as a possible solution of the 3p width problem. Recently a relatively large $a_0 - f_0$ mixing has been suggested [@Close01] as a way of understanding both anomalously large $a_0$ central production and the large $\Gamma(\phi \rightarrow f_0 \gamma) / \Gamma (\phi \rightarrow a_0 \gamma)$ ratio. However criticisms of this explanation have also been presented [@Achasov02]. Clearly it may be useful for studies of processes other than $\eta \rightarrow 3\pi$ to give the coefficients of the scalar isospin violating two point Lagrangian, $${\cal L} = A_{a\sigma}a_0^0 \sigma + A_{af}a_0^0f_0, \label{scalar_isospin_violating_transitions}$$ determined consistently with the $\eta \rightarrow 3\pi$ calculation. Using the parameters from Eq.(\[abcd-parameters\]) in Eq.(\[two-point-vertices\]) we find $A_{a\sigma}=0.0170 y$ ${\rm GeV}^2$, $A_{af}=0.0234 y$ ${\rm GeV}^2$, where $y$ is the quark mass ratio $\frac{m_u-m_d}{m_u+m_d}$. Notice that $y$ (which is negative) is an overall factor for the $\eta \rightarrow 3\pi$ amplitude in the present model. We would like to discuss the effects of the scalars when added to the more realistic Lagrangian presented in Appendix A which contains both pseudoscalars and vectors. This Lagrangian contains additional symmetry breaking terms ($\alpha_p$ and $\lambda^{\prime}$) to account for the ratio of pseudoscalar decay constants, $F_K/F_\pi$ being different from unity as well as a number of terms describing the properties of the vector mesons. Of course, the vector mesons play an important role in low energy processes. Since there are many terms it seems useful to add these new features one at a time. Thus in the present section we will consider just the minimal pseudoscalar Lagrangian, ${\cal L}_{LO}$ \[Eq.(\[lowestorderLag\])\], to be present in addition to the scalars. Furthermore, it is instructive to look at the contributions to the amplitude from different diagrams in order to see how they combine to give the predicted total $\eta \rightarrow \pi^+ \pi^- \pi^0$ width. In section III we reviewed the leading order calculation of the 3p amplitude which gives the result $\Gamma (\eta \rightarrow \pi^+ \pi^- \pi^0) = 106$ eV in the Dashen’s theorem limit. In Fig.\[pseudoscalar01fig\] we show how the individual contributions of the diagrams in Fig.\[psdiagrams\] combine to give the leading order amplitude. The magnitude of the $\eta - \pi^0$ and $\eta^\prime - \pi^0$ transition coefficiencts \[Eq.(\[two-point-vertices\]) with $\alpha_p = \lambda^\prime = 0$, so independent of which state is on-shell\] are (in ${\rm GeV}^2$): $$C_{\pi \eta} \approx 0.0042, \quad \quad C_{\pi \eta^\prime} \approx 0.0031. \label{etapitrans}$$ = 6.5cm   = 6.5cm   Three of the twelve scalar diagrams in Fig.\[scdiagrams\] can be seen to be larger (by at least an order of magnitude) than all the rest. These are the three diagrams involving the lightest of the scalar mesons, $\sigma$, and are contained in Figs.\[scdiagrams\]a, \[scdiagrams\]b and \[scdiagrams\]d. For the case of Fig.\[scdiagrams\]b the graph with an $\eta - \pi^0$ transition dominates that with an $\eta^\prime$ transition because the latter is suppressed by the (square of) the $\eta^\prime$ mass in the denominator of the propagator and also the smallness of the associated coupling constants/transition coefficients. In Fig.\[scalar\_pseudoscalar01fig\] we present the $\eta \rightarrow \pi^+ \pi^- \pi^0$ amplitudes arising from the three diagrams just mentioned and notice that they cancel almost completely. In particular the $\sigma$ exchange diagrams in Figs.\[scdiagrams\]a and \[scdiagrams\]b have opposite signs, as expected – their structure is roughly similar except the propagators have a relative minus sign. We note also that the new isospin violating diagram, involving an $a_0 - \sigma$ transition, turns out not to lead to dramatically larger contributions than the other diagrams. The cancellation between different diagrams involving the sigma means that the total scalar contribution to the $\eta \rightarrow \pi^+ \pi^- \pi^0 $ width is smaller than might be expected and in fact arises mainly from the $a_0^{\pm}$ exchanges in Fig.\[scdiagrams\]c. Comparing Fig.\[scalar\_pseudoscalar01fig\]b with Fig.\[pseudoscalar01fig\]b shows that the net scalar contribution does enhance the overall $\eta \rightarrow \pi^+ \pi^- \pi^0$ rate. Specifically, including the scalar contributions with the pseudoscalar Lagrangian $\cal L_{LO}$ has increased $\Gamma_{0+-}$ by 16 per cent, from 106 eV to 124 eV. The ratio $\Gamma_{000}/\Gamma_{0+-}$ is essentially unchanged. Actually, the calculations above have neglected the finite widths of the $\sigma$, $f_0$ and $a_0$ particles. We take these into account by making the replacements in the corresponding propagators \[see Eq.(\[scampls\])\]: $$\frac{1}{m_X^2 +q^2} \rightarrow \frac{1}{m_X^2 + q^2 -im_X\Gamma_X},$$ where X stands for $\sigma$, $f_0$ or $a_0$. The $\Gamma_X$ are given in Appendix A. These replacements modify the result to $\Gamma_{0+-}= 120$ eV, a 13 per cent increase relative to the leading order result. The width effect is mainly due to the $\sigma$ propagator. We may note that the improvement due to the scalars is consistent with the lower values of the prediction, $160 \pm 50$ eV obtained from the next order of chiral perturbation theory in ref.[@Gasser85]. The numerical amount of suppression of the scalar contribution to the decay rate from cancellation of Figs. (a) and (b) of Fig.\[scdiagrams\] is due to the fitted values of $\gamma_{\sigma \pi \pi}$ and $\gamma_{\sigma \eta \eta}$ given in Eq.(\[scalar\_vertices\]). If we wanted to raise the predicted rate to about 150 eV (still keeping Dashen’s theorem in the evaluation of $y$) it would be necessary to raise $\gamma_{\sigma \eta \eta}$ to about 10. = 6.5cm   = 6.5cm   Effects of Higher Order Pseudoscalar Symmetry Breakers ====================================================== So far we have worked only with the leading order chiral Lagrangian of pseudoscalars and scalars and obtained (to linear order in $y = - \frac{m_d - m_u}{m_d + m_u}$) isospin-breaking amplitudes proportional to $\delta^\prime, b$ and $d$ in Eq. (\[LagSB\]) of Appendix A. In order to better fit the properties of the pseudoscalar mesons we consider, as mentioned above, the higher-order symmetry breaking terms in Eq.(\[LagSB\]) with coefficients $\alpha_p$ and $\lambda^\prime$. The numerical values of these parameters are obtained in section 4 of Appendix A, based on an overall fitting of pseudoscalar meson properties. Next we examine the effects of these two new symmetry breaking terms on our previous calculation. It will be seen that these effects include an interesting redistribution of the contributions from the scalar and pseudoscalar diagrams to the total amplitude. First, the contact diagram Fig.\[psdiagrams\]a will receive corrections due to the $\alpha_p$ and $\lambda^\prime$ terms \[see Eq.(\[psampls\])\]. This results in some energy dependence since $\alpha_p$ gives a four-point derivative coupling. More importantly, the $\eta - \pi^0$ and $\eta^\prime -\pi^0$ transition coefficients relevant for 3p now depend on which particle is on-shell and are numerically (in ${\rm GeV}^2$): $$\begin{aligned} C_{\pi\eta}^\eta \approx -0.00583y, & \quad & C_{\pi\eta}^\pi \approx -0.0151y \nonumber \\ C_{\pi\eta^\prime}^\pi & \approx & -0.0113y. \label{newetapi}\end{aligned}$$ Since $C_{\pi\eta}^\eta$ is now considerably suppressed in magnitude, the Feynman amplitude for Fig.\[psdiagrams\]b is now suppressed, while $C_{\pi\eta}^\pi$ and the amplitude for Fig.\[psdiagrams\]c remain about the same. These results, due to only pseudoscalars, are summarized in Fig.\[scalar\_pseudoscalar\_alphap01fig\] which may be compared with Fig.\[pseudoscalar01fig\]. The net result is that the total $\eta \rightarrow \pi^+ \pi^- \pi^0$ amplitude (shown in the second of Fig.\[scalar\_pseudoscalar\_alphap01fig\]) due to pseudoscalar mesons is reduced compared with the leading order result. The pseudoscalars themselves now give $\Gamma_{0+-}=$ 81 eV rather than 106 eV, as in section IV. = 6.5cm   = 6.5cm   However, for the diagrams involving scalar mesons the effect of higher order symmetry breaking is even more important. As we noted above, the scalar meson contribution to $\eta \rightarrow \pi^+ \pi^- \pi^0 $ was rather small as the main diagrams tended to cancel. When we include the $\alpha_p$ and $\lambda^\prime$ corrections to the $\eta \pi^0$ transition this cancellation will not be so complete. Specifically, comparing Eqs.(\[etapitrans\]) and (\[newetapi\]) we see that the magnitude of the amplitude for Fig.\[scdiagrams\]a, where the $\eta - \pi^0$ transition occurs with an on-shell $\eta$, will be reduced by a factor of approximately four, while that of Fig.\[scdiagrams\]b, where the $\eta - \pi^0$ transition has an on-shell pion, will remain about the same relative to our result in Section IV. Fig.\[scdiagrams\]d will be unchanged. There will now be a non-negligible contribution from the scalar mesons. It will be more negative in sign (the contribution from Fig.\[scdiagrams\]a is positive, but now smaller in magnitude) and will add “constructively” to the pseudoscalar diagrams in Fig.\[psdiagrams\] and so increase our prediction for $\Gamma(\eta \rightarrow \pi^0 \pi^+ \pi^-)$. This is shown in Fig. \[scalar\_pseudoscalar\_alphap03fig\]. Adding all of the pseudoscalar and scalar diagrams in Fig.\[psdiagrams\] and Fig.\[scdiagrams\] with the inclusion of the symmetry breaking effects due to $\alpha_p$ and $\lambda^\prime$ we get: $ \Gamma(\eta \rightarrow \pi^0 \pi^+ \pi^-)= 119.6 \, {\rm eV}$. This is essentially the same as the result in section IV but now a larger portion is due to the scalar meson diagrams. Since this is the case the damping effect of the sigma width will be more prominent; in fact it reduces the predicted rate to 103.6 eV in the Dashen’s theorem limit. = 6.5cm   = 6.5cm   It may be of interest to see how the detailed pattern just described depends on the precise value of the quark mass ratio $x$ and, as explained in section 4 of appendix A, correspondingly on the crucial isospin violating quark mass ratio $y$. This is shown for the predicted value of $\Gamma_{0+-}$ in Table \[firsttable\]. $x,y$ ps. only ps. + sc. (zero scalar widths) ps.+sc. (non-zero scalar widths) -------------- ---------- -------------------------------- ---------------------------------- 20.5, -0.202 63.7 eV 95.9 eV 82.2 eV 23, -0.241 70.7 eV 106.0 eV 91.4 eV 25.1, -0.277 80.2 eV 119.6 eV 103.6 eV : $\Gamma(\eta \rightarrow \pi^+\pi^- \pi^0 )$ for different values of $x$ and $y$ defined after Eq.(\[spurion\]). The second column applies to the case of only pseudoscalars present while the third includes scalars too. The effect of taking non zero scalar widths into account is shown in the last column. Dashen’s theorem is assumed in order to extract $y$.[]{data-label="firsttable"} Including Vectors in the calculation ==================================== It is well known that the inclusion of vector mesons is important for a realistic discussion of low energy chiral dynamics. For example, in the chiral pertubation scheme, most of the finite pieces of the counterterms can be explained by integrating out various vector contributions [@DRG]. In our present approach, of course, we are keeping the resonances, rather than integrating them out, in order to learn more about the scalars. First, the vector mesons contribute to the $\eta \rightarrow \pi^+ \pi^- \pi^0 $ amplitude corresponding to Fig.\[vmdiagrams\]a, which is just a correction to $\pi \pi$ scattering. Its value given by the amplitude, $M_\rho^a$ in (\[vmampls\]) is easily seen to be comparatively large. However the fourth term of the $U(3)_L \times U(3)_R$ invariant Lagrangian Eq.(\[LagLO\]) gives, in addition to the $\rho \pi \pi$ vertex a four pion contact term \[included in $M^a_{contact}$ in (\[psampls\])\]. Actually this contact term cancels most of the contribution from the $\rho$-exchange diagram in Fig.\[vmdiagrams\]a. This is well known from chiral treatments of $\pi \pi$ scattering: when the $\rho$ is added to the Lagrangian, chiral symmetry requires a contact term which cancels most of the $\rho$ contribution near threshold, thereby maintaining the current algebra threshold result. However, the situation is actually a bit more complicated since the process of obtaining an adequate fit to the properties of both the vectors and pseudoscalars [@Harada96] requires a number of additional symmetry breaking terms shown in Eq.(\[LagSB\]) of Appendix A. As well as the symmetry breaking terms we have already discussed involving the pseudoscalars alone, there are, in particular, two new terms, measured by the coefficients $\alpha_+$ and $\alpha_-$. It turns out that their effects are very minor. They include an additional contribution to the 4-point isospin violating $\eta \pi^+ \pi^- \pi^0$ vertex due to the $\alpha_-$ term, corrections to the 4-pion vertex in Fig.\[psdiagrams\]b due to both $\alpha_+$ and $\alpha_-$ and an additional diagram, shown in Fig.\[vmdiagrams\]b, which contains a new G-parity (and isospin) violating $\rho \pi \eta$ vertex \[the amplitude for this is given as $M_\rho^b$ in (\[vmampls\])\]. Note that there there exists a $\rho^0 - \omega$ mixing transition, which is the analog of the $\pi^0-\eta$ and $a_0^0-f_0$ mixing transitions, but it does not contribute to 3p at tree level. The decay widths with inclusion of vectors are tabulated in Table \[secondtable\] for the same values of $x,y$ used in the last section. In this table the neutral modes are also included. Furthermore, the effect of both the scalar and (actually negligible) vector widths are included too. We see that, as expected from our discussion above, the vectors do not change the overall predictions compared to the last column of Table \[firsttable\] very much but they do give a little enhancement. This is also clear by comparing the pseudoscalars + vectors column of table \[secondtable\] with the pseudoscalars only column of Table \[firsttable\]. It is seen again that the scalars make a non negligible contribution to the total amplitude. $x,y$ decay mode ps. +vec. ps.+sc.+vec.(no width) ps. +sc. + vec. (width included) -------------- ------------ ----------- ------------------------ ---------------------------------- 20.5, -0.202 0+- 64.4 eV 96.6 eV 82.8 eV 000 92.9 eV 139.4 eV 118.9 eV 23, -0.241 0+- 71.9 eV 107.4 eV 92.5 eV 000 101.9 eV 152.9 eV 131.1 eV 25.1, -0.277 0+- 82 eV 121.7 eV 105.4 eV 000 114.5 eV 171.3 eV 147.7 eV : $\Gamma (\eta \rightarrow \pi^0 \pi^+ \pi^-)$ and $\Gamma (\eta \rightarrow 3\pi^0)$ for different values of $x,y$. In the third column pseudoscalars and vectors are both present. In the fourth and fifth column pseudoscalars, vectors and scalars all present (without and with the effect of the scalar meson widths).[]{data-label="secondtable"} Finally it is interesting to display the energy spectrum parameters $a,b$ and $c$ defined in Eq.(\[spectrum\]) for the various models we have examined. These are given in Table \[energy\_table\] and are seen to be reasonable. The $\chi^2$ measures the fit of our model to the spectrum shape assumed in Eq.(\[spectrum\]) and seems to be small. $a$ $b$ $c$ $\chi^2$ ------------------------------- ------- ------ ------- --------------------- pseudoscalars(LO) -1.11 0.31 0 $5.6\times 10^{-5}$ pseudoscalars -0.96 0.23 0 $1.5\times 10^{-4}$ pseudoscalars+scalars -0.93 0.22 -0.01 $3.3\times 10^{-4}$ pseudoscalars+scalars+vectors -1.09 0.26 0.033 $5.8\times 10^{-3}$ : Fits of the energy dependence of the normalized (charged) decay amplitude for $\eta \rightarrow \pi^0 \pi^+ \pi^-$ to the form $|M_{0+-}|^2=1 +a Y + b Y^2 + c X^2 $. The first line corresponds to result at leading order with pseudoscalar mesons only. The second with inclusion of higher order symmetry breakers, the third when scalar mesons are added and the final line when vector mesons are included as well.[]{data-label="energy_table"} Discussion ========== We studied the role of a possible nonet of light scalar mesons in the still interesting $\eta \rightarrow 3\pi$ decay process. Our motivation was primarily to learn more about the scalars themselves. The framework is a conventional non-linear chiral Lagrangian of pseudoscalars and vectors, extended to include scalars (the Lagrangian is described in Appendix A). The parameters involving the scalars were previously obtained to fit the s-wave $\pi \pi$ and $\pi K$ scattering in the region up to about 1 GeV as well as the strong decay $\eta' \rightarrow \eta \pi \pi$. An initial concern is whether the model as it stands, containing essentially no undetermined main parameters (up to possible uncertainties in the quark mass ratios $ x=2m_s/(m_u+m_d)$ and $y=(m_u-m_d)/(m_u+m_d)$), does not make the $\eta \rightarrow 3 \pi$ amplitude too large. In particular, the $\sigma(560)$ exchange diagrams (a) and (b) of Fig.\[scdiagrams\] might lead to a great deal of enhancement due to the possibility of the $\sigma(560)$’s momentum being close to mass shell. However this turns out not to be the case. In our initial calculation where the scalars are added to the minimal model of pseudoscalars given in Eq.(\[lowestorderLag\]), the left part of Fig.\[scalar\_pseudoscalar01fig\] shows that these two diagrams, though not individually small, tend to cancel each other. This partial cancellation occurs because the $\eta-\pi^0$ transition leads to opposite signs when the $\eta$ is on mass shell and when the $\pi^0$ is on mass shell (In the first case we have a $\pi^0$ propagator carrying the momentum squared of an on-shell $\eta$ while in the second case, the reverse holds). In addition, the enhancement due to the sigma propagator is further suppressed by the inclusion of an imaginary piece, needed to satisfy unitarity in the scattering calculation. The net result is that the effect of including scalars in the minimal pseudoscalar Lagrangian, Eq.(\[lowestorderLag\]) is to increase the width for $\eta \rightarrow \pi^+ \pi^- \pi^0$ decay by about 13 per cent. This relatively small, due to cancellation, increase illustrates the difficulty of finding dramatic “smoking gun" evidence for the existence of a light sigma. In the scattering calculation a light sigma appears (see for example [@HSS1]) obscured by a large background and does not have a simple Breit Wigner shape. It is amusing to note the effect of higher derivative terms in the Lagrangian of pseudoscalars (see section V for details). The higher derivative terms allow one to conveniently implement at tree level the fact that the ratio of the pseudoscalar decay constants $F_{Kp}/F_{\pi p}$ is somewhat greater than unity. With these terms the important $\pi^0 \eta$ transition vertex has a momentum dependent piece. Together with a needed modification in the parameter fitting (see section 4 of Appendix A) this reduces the contribution of the pseudoscalars to the $\eta \rightarrow 3\pi$ decay width. However the modification of the $\pi^0-\eta$ transition noticeably upsets the cancellation between the two sigma exchange diagrams in (a) and (b) of Fig.\[scdiagrams\]. The net result is that, while the total prediction for the $\eta \rightarrow 3\pi$ decay width remains about the same, now about thirty percent of the value is contributed by the scalars. The vector meson contribution, discussed in section VI, actually does not change things much. This is because the $\rho$ exchange diagrams for $\pi\pi$ s-wave scattering are essentially canceled at very low energies by an extra four pion contact term which automatically arises due to the chiral symmetric formulation. Experimentalists fit the Dalitz plot describing the $\eta \rightarrow \pi^+ \pi^-\pi^0$ spectrum to the form given in Eq.(\[spectrum\]). A fit of this type to the predicted spectrum from the Lagrangian of pseudoscalars, scalars and vectors was seen to be close to the experimental one. The basic spectrum shape is already reasonable with the very simplest model discussed in section III. As both the theory and experiment get more precise, the importance of the spectrum shape toward a deeper understanding of the underlying physics increases. A particularly interesting scalar contribution to $\eta \rightarrow 3\pi$ arises from the $a_0-\sigma$ transition shown in (d) of Fig.\[scdiagrams\] (The $a_0-f_0$ transition contribution to $\eta \rightarrow 3\pi$ is suppressed due to the propagator of the heavier $f_0(980)$). This is the analog of the important $\pi^0-\eta$ transition and, in a sense, is a new mechanism for $\eta \rightarrow 3\pi$ (although it was investigated a long time ago [@hudnall] as a possible way to increase the $\eta \rightarrow 3\pi$ width). The formula in raw form for this transition is given in Eq.(\[two-point-vertices\]). We evaluated its strength from the knowledge of the isospin violating piece of the dimensionless quark mass matrix ${\cal M}$ in Eq.(\[spurion\]), determined from the pseudoscalar sector and the coefficients: $a,b,c$ and $d$ of the scalar meson mass terms \[see Eqs.(\[LagLO\]\] and (\[LagSB\])) determined from the isospin conserving sector of the scalars. However as one can see from the left sides of Figs.\[scalar\_pseudoscalar01fig\] and \[scalar\_pseudoscalar\_alphap03fig\], the contribution to $\eta \rightarrow 3\pi$ due to the $a_0-\sigma$ transition is not very large, although it has the right sign to boost the decay rate. The method just described also evaluates the strength of the $a_0(980)-f_0(980)$ transition. For convenience this is given after Eq.(\[scalar\_isospin\_violating\_transitions\]), where the overall factor, $y$ is displayed. This transition has been very much “in the news" recently as a proposed [@Close01] explanation for the large observed $\Gamma(\phi \rightarrow f_0\gamma)/\Gamma(\phi\rightarrow a_0\gamma)$ ratio and the anomalously strong $a_0$ central production. However criticisms of this explanation have been given [@Achasov02], [@Black02], pointing out that the $a_0-f_0$ mixing expected from a transition strength like the one determined above is insufficient to give a large effect. Intuitively, because of the near degeneracy of the $a_0(980)$ and $f_0(980)$ as well as the similarity of their widths, one might expect the mixing to be very large. But the mixing amplitude is governed by a dimensionless factor $iA_{af}/ (m_a\Gamma_a)$ \[see for example Eq.(12) of [@Black02]\] which is suppressed by the scalar meson width, $\Gamma_a$. In section II we discussed the current comparison between theory and experiment for the $\eta \rightarrow \pi^0 \pi^+ \pi^-$ width. The experimental width [@pdg] is $\Gamma_{0+-}=267 \pm 25$ eV. This may be compared with the one loop chiral perturbation theory result [@Gasser85] of $160 \pm 50$ eV. More recent attempts [@Kambor96] to estimate final state interaction effect outside of the chiral perturbation theory approach have increased this somewhat to $ 209 \pm 20$ eV. It seems to us that the thirty per cent contribution of the scalars compared to the pseudoscalars we have found should probably not be considered on top of this latter figure. That is because a good portion of the increase due to scalars we have found may be considered as resulting from final state interactions. Many attempts to close the gap between theory and experiment have focused [@Donoghue92] on a reanalysis of electromagnetic corrections to the $K^+-K^0$ mass difference. This is argued to increase the quark mass ratio, $y$ which is an overall factor for the $\eta \rightarrow 3\pi$ amplitude. From the standpoint of learning more about the properties of the scalar mesons it is clear that the $\eta' \rightarrow 3\pi$ decays represent a potentially important source of information. In this case there is sufficient energy available for the $a_0(980)$ and $f_0(980)$ propagators to be close enough to their mass shells to avoid suppressing the contributions of these resonances. On the other hand, the theoretical analysis is more difficult since large non-perturbative unitarity corrections are expected. In addition, other more massive particles may also contribute. The experimental information \[see Eq.(\[etap\_experimental\_rate\])\] is more preliminary than in the $\eta \rightarrow 3\pi$ case. While a number with reasonably small errors has been presented for $\Gamma_{000}^\prime$, there is only a weak upper bound for $\Gamma_{+-0}^\prime$ and also no information on its Dalitz plot. In the model employed in the present papper the $\eta' \rightarrow 3\pi$ amplitudes are simply obtained from the $\eta \rightarrow 3\pi$ amplitudes by the simple substitution given in Eq.(\[etap-amplitude\]). Notice that this substitution rule would get modified if a more complicated $\eta-\eta'$ mixing scheme \[e.g. the one mentioned after Eq.(\[eta\_etap\])\] is adopted. As shown in Eq.(\[etap\_LO\_result\]) the prediction of the minimal model of only pseudoscalars is somewhat too high, but at least of the correct order of magnitude. Adding the scalars without any readjustment of parameters does not improve the prediction for $\Gamma'_{000}$ but makes it considerably larger (about 2300 eV). A similar large value was recently found in [@paula]. Since the phase space is fairly large it is perhaps to be expected that large values are typically obtained. Presumably it is a sign for including more detailed unitarity corrections or other physical effects which result in cancellations. We are particularly hopeful that a careful study of mixing between a lower mass exotic scalar nonet and a more conventional higher mass scalar nonet [@BFS3; @Deirdreetal; @2nonetmixing] may solve this problem and perhaps contribute to an improved understanding of the $\eta \rightarrow 3\pi$ decays also. We are grateful to Masayasu Harada, Paula Herrera-Siklody and Francesco Sannino for very helpful discussions related to this problem. The work of A. A-R. and J.S. has been supported in part by the US DOE under contract DE-FG-02-85ER 40231. D.B. wishes to acknowledge support from the Thomas Jefferson National Accelerator Facility operated by the Southeastern Universities Research Association (SURA) under DOE Contract No. DE-AC05-84ER40150. The work of A.H.F. has been supported by grants from the State of New York/UUP Professional Development Committee, and the 2002 Faculty Grant from the School of Arts and Sciences, SUNY Institute of Technology. The chiral Lagrangian ===================== For convenience we collect here needed terms from the pseudoscalar-vector chiral Lagrangian presented in [@Harada96] and from the scalar addition presented in [@BFSS2]. Transformation Properties ------------------------- These are constructed to mock up the symmetry properties of the fundamental quark Lagrangian, under which left and right projected light quark fields tranfsorm as $$q_{L,R} \rightarrow U_{L,R} q_{L,R},$$ $U_L$ and $U_R$ being $3 \times 3$ constant unitary matrices. The pseudoscalar nonet $\phi (x)$ is a $3 \times 3$ matrix which fits into the unitary chiral matrix $$U = {\rm exp} ( \frac {2 i \phi (x)}{F_\pi} )$$ where $F_\pi$ is the (bare) pion decay constant. Under a chiral transformation $$U \rightarrow U_L U U_R^\dagger.$$ It is convenient to define the $3 \times 3$ unitary matrix $\xi$ by $U= \xi^2$. Then $\xi$ transforms as $$\xi \rightarrow U_L \xi K^\dagger(\phi,x) = K(\phi,x) \xi U_R^\dagger,$$ which implicitly defines the unitary matrix $K$. The intuitive significance of $K$ is that the objects $Kq$ behave like bare quarks surrounded by a pseudoscalar meson cloud, or “constituent quarks”. The objects $$v_\mu \, p_\mu = \frac{i}{2} (\xi \partial_\mu \xi^\dagger \pm \xi^\dagger \partial_\mu \xi ),$$ transform as $$\begin{aligned} p_\mu &\rightarrow& K p_\mu K^\dagger \nonumber \\ v_\mu &\rightarrow& K v_\mu K^\dagger + i K \partial_\mu K^\dagger.\end{aligned}$$ A putative scalar nonet matrix $N(x)$ is taken to transform as $$N \rightarrow K N K^\dagger,$$ The vector meson nonet $\rho_\mu$ transforms as $$\rho_\mu \rightarrow K\rho_\mu K^\dagger + \frac{i}{\tilde g} K\partial_\mu K^\dagger,$$ where we have included the dimensionless coupling constant, $\tilde g$. The “field-strength tensor” $$F_{\mu \nu} (\rho) = \partial_\mu \rho_\nu - \partial_\nu \rho_\mu - i {\tilde g} \left[ \rho_\mu , \rho_\nu \right] \rightarrow K F_{\mu \nu} K^\dagger.$$ $U(3)_L \times U(3)_R$ Invariant Terms -------------------------------------- These comprise the kinetic terms for the three multiplets, mass terms for the scalars and vectors and appropriate interaction terms: $$\begin{aligned} {\cal L}_0 = &-& \frac{F_\pi^2}{8} {\rm Tr} (\partial_\mu U \partial_\mu U^\dagger) - \frac{1}{4} {\rm Tr} ( F_{\mu \nu}(\rho) F_{\mu \nu}(\rho) ) \nonumber \\ & - & \frac{1}{2} {\rm Tr} ( {\cal D}_\mu N {\cal D}_\mu N ) - \frac{m_v^2}{2 {\tilde g}^2} {\rm Tr} \left[ { ( \tilde g \rho_\mu - v_\mu)}^2 \right] \nonumber \\ &-& a {\rm Tr} (NN) - c {\rm Tr} (N) {\rm Tr}(N) \nonumber \\ &+& F_\pi^2 \left[ A \epsilon^{abc} \epsilon_{def} N_a^d {(p_\mu)}_b^e {(p_\mu)}_c^f + B {\rm Tr} (N) {\rm Tr}{(p_\mu p_\mu)} + C {\rm Tr}(Np_\mu){\rm Tr}(p_\mu) + D {\rm Tr} (N) {\rm Tr} (p_\mu) {\rm Tr} (p_\mu) \right], \label{LagLO}\end{aligned}$$ where ${\cal D}_\mu N = \partial_\mu N - i v_\mu N + i Nv_\mu$. These include the parameters $m_v^2, a, c, A, B, C$ and $D$. Note that the pseudoscalars are still massless at this level. Further note that for the interactions and mass terms of the scalars we do not restrict ourselves to a single trace. For $q \bar q$ mesons the single trace is suggested by the OZI rule while for an ideal dual nonet the A term is in fact expected to be dominant. We made a fit for $m_v^2, a, c, A, B, C$ and $D$ assuming only SU(3) invariance. Symmetry Breaking Terms ----------------------- The fundamental QCD Lagrangian contains the quark mass term $ - \frac{ (m_u + m_d)}{2} \bar q {\cal M} q$, with the dimensionless matrix $${\cal M} = \left[ \begin{array}{c c c} 1+y & 0 & 0 \\ 0 & 1-y & 0 \\ 0 & 0 & x \end{array} \right] \label{spurion}$$ where $x = \frac {2m_s}{m_u + m_d}$ and $y= - \frac{m_d - m_u}{m_d + m_u}$. It is convenient to define $${\hat {\cal M}}_{\pm} = \frac{1}{2} \left( \xi {\cal M} \xi \pm \xi^\dagger {\cal M} \xi^\dagger \right).$$ Then, the symmetry breaking Lagrangian is taken as $$\begin{aligned} {\cal L}_{SB} &=& \delta^\prime {\rm Tr} \left[ {\cal M} ( U + U^\dagger ) \right] + {\lambda^\prime}^2 {\rm Tr} \left[ {\cal M} U^\dagger {\cal M} U^\dagger + {\cal M} U {\cal M} U \right] \nonumber \\ &-& \frac{2\alpha_p}{{\tilde g}^2} {\rm Tr} \left( {\hat {\cal M}}_+ p_\mu p_\mu \right) + 2 \alpha_+ {\rm Tr} \left[ {\hat {\cal M}}_+ \left( \rho_\mu - \frac {v_\mu}{\tilde g} \right) \left( \rho_\mu - \frac {v_\mu}{\tilde g} \right) \right] \nonumber \\ &-& \frac{2 \alpha_-} {\rm Tr} \left({\hat {\cal M}}_- \left[ \left( \rho_\mu - \frac {v_\mu}{\tilde g} \right), p_\mu \right]\right) \nonumber \\ &+& 2 \gamma^\prime {\rm Tr} \left[ {\hat {\cal M}}_+ F_{\mu \nu} (\rho) F_{\mu \nu}(\rho) \right] \nonumber \\ &-& b {\rm Tr}(NN{\cal M}) - d {\rm Tr}(N) {\rm Tr}(N{\cal M}). \label{LagSB}\end{aligned}$$ Only the parameters $\delta^\prime$, ${\lambda^\prime}^2$, $\alpha_p$, $b$ and $d$ here contribute to the isospin violating vertices of interest in the present paper. The parameter $\gamma^\prime$ was included in the overall parameter fit obtained in [@Harada96] but its small effect on the isospin-conserving vertices will be neglected. In addition to the quark mass induced symmetry breaking terms there is an important term induced by instanton effects which breaks just the ${U(1)}_A$ piece of ${\rm SU}(3)_L \times {\rm SU}(3)_R \times {\rm U}(1)_V \times {\rm U}(1)_A$. It may be summarized as $${\cal L}_{\eta^\prime} = \frac{\kappa}{576} {\rm ln}^2 \left( \frac { {\rm det} U}{ {\rm det} U^\dagger}\right) + \ldots \label{extraetaprime}$$ where $\kappa$ is a constant essentially proportional to the squared mass of the $\eta^\prime$ meson. The three dots stand for other terms which will be neglected here but are listed in Eq. (2.12) of [@Schechter93]. Effectively this term gives an important contribution to the $\eta^\prime$ mass and an $\eta - \eta^\prime$ mixing angle defined by $$\left( \begin{array}{c} \eta\\ \eta' \end{array} \right) = \left( \begin{array}{c c} {\rm cos} \theta_p & -{\rm sin} \theta_p \\ {\rm sin} \theta_p & {\rm cos} \theta_p \end{array} \right) \left( \begin{array}{c} (\phi^1_1+\phi^2_2)/ \sqrt{2} \\ \phi^3_3 \end{array} \right). \label{eta_etap}$$ When the extra terms in Eq. \[extraetaprime\] are included they will not only give rise to an additional isospin violating transition but will also modify the $\eta-\eta'$ mixing transformation above to be the non-orthogonal one given in Eq. (4.9) of [@Schechter93]. We will not include these effects in the present paper, however. Numerical values of parameters used ----------------------------------- For the averaged pseudoscalar masses we used, $$m_\pi = 0.137 \hskip.2cm {\rm GeV}, \quad\quad m_K = 0.4957 \hskip.2cm {\rm GeV}.$$ In section III we gave the fitted parameters for the lowest order Lagrangian containing only pseudoscalars. This also yields the isospin conserving quark mass ratio $x=25.1$ (assuming that $f$, defined in Eq. (\[definef\]) is unity). A refitting of these parameters is necessary when the $\alpha_p/{\tilde g}^2$ and $\lambda'^2$ symmetry breaking terms are included. This can be conveniently done following the method used in preparing Table III of [@Harada96]. There, a value of $x$ is assumed and the four quantities $F_\pi$ (unrenormalized pion decay constant), $\delta'$, $|\lambda'|^2$ and $\alpha_p/{\tilde g}^2$ are calculated in terms of the four physical quantities $m_\pi$, $m_K$, $F_{\pi p}=0.1307$ GeV and $F_{Kp}=0.1598$ GeV, using: $$\begin{aligned} \lambda'^2 &=& \frac{(1+x)F_{\pi p}^2m_\pi^2/16-F_{Kp}^2m_K^2/8}{1-x^2}, \nonumber \\ \delta'&=& F_{\pi p}^2m_\pi^2/8-4\lambda'^2, \nonumber \\ \frac{\alpha_p}{{\tilde g}^2F_\pi^2} &=& \frac{(F_{Kp}/F_{\pi p})^2-1}{ 2(1+x)-4(F_{Kp}/F_{\pi p})^2}, \nonumber \\ F_\pi &=& \frac{F_{\pi p}}{(1+4\alpha_p/({\tilde g}^2F_\pi^2))^{1/2}}, \nonumber \\ \frac{\alpha_p}{{\tilde g}^2} &=& F_\pi^2(\frac{\alpha_p}{{\tilde g}^2F_\pi^2}). \label{paramfitting}\end{aligned}$$ In addition, the isospin violating quark mass ratio $y$ is obtained from $$(m_{K^0}^2-m_{K^+}^2)-f(m_{\pi^0}^2-m_{\pi^+}^2)= (4y/F_{Kp}^2)(-2\delta'-8(1+x)\lambda'^2+m_K^2\alpha_p/{\tilde g}^2), \label{gety}$$ for a particular value of $f$. To isolate the effects of the scalars we may choose an $x$ such that, with the value $f=1$ corresponding to Dashen theorem, we recover the value $y=-0.277$ found in sections III and IV. That gives $x=25.1$ and $$\begin{aligned} F_\pi = 0.128 \hskip.2cm {\rm GeV}, \quad\quad \delta'= 0.0386 \times 10^{-3} \hskip.2cm {\rm GeV}^4, \nonumber \\ \alpha_p/{\tilde g}^2= 0.176 \times 10^{-3} \hskip.2cm {\rm GeV}^2, \quad\quad |\lambda'|=0.643 \times 10^{-3} \hskip.2cm {\rm GeV}^2. \label{symbreakingparameters}\end{aligned}$$ The needed dependences on the quark mass ratio ,$x$ of the parameters involving vector mesons ($\gamma'$ , $\alpha_+$, $\alpha_-$, $m_v$ and $\tilde g$) are given in Table 3 of [@Harada96]; the additional point $x=25.1$ used in Table \[secondtable\] above was treated by interpolation. The masses and widths of the scalars are taken to be (in MeV) $$\begin{aligned} m_\sigma = 550, \quad m_\kappa = 897, \quad m_{a_0} = 983.5, \quad m_{f_0} = 980 \nonumber \\ \Gamma_\sigma = 370 , \quad \Gamma_{a_0} = 70.0, \quad \Gamma_{f_0} = 64.6 .\end{aligned}$$ Note that the values of $\Gamma_\sigma$ and $\Gamma_\kappa$ are not “Breit-Wigner” widths but are chosen to unitarize the $\pi \pi$ and $\pi K$ scattering amplitudes. The masses above fix the parameters (in ${\rm GeV}^2)$ in (\[LagLO\]) and (\[LagSB\]) $$a = 0.492, \, b = -0.00834, \, c = -0.0160, \, d=-0.00557 \label{abcd-parameters}$$ and the mixing angle $\theta_s = -20.3^0$. The parameters $A,B,C,D$ define all the trilinear $S\phi \phi$ coupling constants according to the formulas given in Appendix C of [@BFSS2]. The needed coupling constants are (in ${\rm GeV}^{-1}$) $$\begin{aligned} \gamma_{\sigma \pi \pi} = 7.27, \, \gamma_{\sigma \eta \eta} = 3.90, \, \gamma_{\sigma \eta \eta^\prime} = 1.25, \, \gamma_{\sigma \eta^\prime \eta^\prime} = -3.82, \nonumber \\ \gamma_{f\pi \pi} = 1.47, \, \gamma_{f \eta \eta} = 1.50, \, \gamma_{f\eta \eta^\prime} = -10.19, \, \gamma_{f\eta^\prime \eta^\prime} = 1.04, \nonumber \\ \gamma_{a\pi \eta} = -6.87, \, \gamma_{a \pi \eta^\prime} = -8.02. \label{scalar_vertices}\end{aligned}$$ Decay amplitude =============== The Feynman diagrams representing the $\eta(p) \rightarrow \pi^0 (p_1) \pi^+ (p_2) \pi^- (p_3)$ decay are shown in Figs.1-3. The contact diagrams (1a, 1b and 1c) receive contributions from the pseudoscalar and vector part of the Lagrangian $$\begin{aligned} M_{contact}^a &=& i { { 16 y \delta' {\rm cos}\theta_p }\over { 3 F_{\pi p}^4} } + i { { 8 y \alpha_p {\rm cos}\theta_p }\over{3 {\tilde g}^2 F_{\pi p}^4} } \left( - 3 p_2.p_3 + p.p_1 + p.p_2 + p.p_3 \right) + i { {512 y \lambda'^2 {\rm cos}\theta_p }\over { 3 F_{\pi p}^4} } \nonumber \\ M_{contact}^b &=& + i \left( 1 - { {3 m_v^2} \over {4 {\tilde g}^2 F_{\pi p}^2} } \right) { {C_{\pi \eta}^\eta}\over {m_\pi^2 - m_\eta^2} } {2\over {3 F_{\pi p}^2}} \left( - 2 p_2.p_3 + p_1.p_3 - p.p_3 + p_1.p_2 - p.p_2 + 2p.p_1 \right) \nonumber \\ &+&i\frac{2\alpha_+C_{\pi\eta}^\eta}{{\tilde g}^2F_{\pi p}^4(({m_\pi}^2- {m_\eta}^2)}(-2p.p_1+2p_2.p_3+p_3.p-p_3.p_1+p_3.p-p_2.p_1) \nonumber \\ &+& i { {8 \alpha_p\, C_{\pi\eta}^\eta}\over { 3 {\tilde g}^2 F_{\pi p}^4 (m_\pi^2 - m_\eta^2) } } \left( 5 p.p_1 - 5 p_2.p_3 - p.p_3 + p_1.p_3 - p.p_2 + p_1.p_2 \right) \nonumber \\ &+& i { {256 \lambda'^2 C_{\pi \eta}^\eta }\over { 3 F_{\pi p}^4 (m_\pi^2 - m_\eta^2 ) } } + i { {16 \delta'\, C_{\pi\eta}^\eta}\over { 3 F_{\pi p}^4 (m_\pi^2 - m_\eta^2) } } \nonumber \\ M_{contact}^c &=& i { {16 \delta' } \over {F_{\pi p}^4} } \left( { { C_{\pi\eta}^\pi{\rm cos}^2 \theta_p}\over {m_\eta^2 - m_\pi^2 } } + { { C_{\pi\eta'}^\pi{\rm sin}\theta_p {\rm cos} \theta_p} \over {m_{\eta'}^2 - m_\pi^2 } } \right) \nonumber \\ &+& i { {8 \alpha_p}\over { {\tilde g}^2 F_{\pi p}^4} } \left( { { C_{\pi\eta}^\pi{\rm cos}^2 \theta_p}\over {m_\eta^2 - m_\pi^2 } } + { { C_{\pi\eta'}^\pi{\rm sin}\theta_p {\rm cos} \theta_p} \over {m_{\eta'}^2 - m_\pi^2 } } \right) \left( p.p_1 - p_2.p_3 + p.p_3 + p.p_2 -p_1.p_3 - p_1.p_2 \right) \nonumber \\ &+& i { {256 \lambda'^2 }\over { F_{\pi p}^4 } } \left( { { C_{\pi\eta}^\pi{\rm cos}^2 \theta_p}\over {m_\eta^2 - m_\pi^2 } } + { { C_{\pi\eta'}^\pi{\rm sin}\theta_p {\rm cos} \theta_p} \over {m_{\eta'}^2 - m_\pi^2 } } \right) \label{psampls}\end{aligned}$$ The scalar contributions (Figs.\[scdiagrams\]a, b, c, and d) are: $$\begin{aligned} M_{scalar}^a &=& -i { {2 C_{\pi\eta}^\eta\gamma_{\sigma\pi\pi}^2}\over {m_\pi^2 - m_\eta^2}} { {(p.p_1)(p_2.p_3)} \over {m_\sigma^2 + (p-p_1)^2 }} + (\sigma \leftrightarrow f_0) \nonumber \\ M_{scalar}^b &=& -i \sqrt{2} \left( { {2 C_{\pi\eta}^\pi \gamma_{\sigma\pi\pi} \gamma_{\sigma\eta\eta}} \over {m_\eta^2 - m_\pi^2} } + { { C_{\pi\eta'}^\pi \gamma_{\sigma\pi\pi} \gamma_{\sigma\eta\eta'}} \over {m_{\eta'}^2 - m_\pi^2} } \right) { {(p.p_1)(p_2.p_3)} \over {m_\sigma^2 + (p-p_1)^2 }} + (\sigma \leftrightarrow f_0) \nonumber \\ M_{scalar}^c &=& -i \left( { {C_{\pi\eta}^\pi \gamma_{a\pi\eta}^2} \over {m_\eta^2 - m_\pi^2} } + { { C_{\pi\eta'}^\pi \gamma_{a\pi\eta} \gamma_{a\pi\eta'}} \over {m_{\eta'}^2 - m_\pi^2} } \right) { {(p.p_3)(p_1.p_2)} \over {m_{a_0}^2 + (p-p_3)^2 }} + (p_2 \leftrightarrow p_3) \nonumber \\ M_{scalar}^d &=& -i \sqrt{2} A_{a\sigma}\gamma_{a\pi\eta} \gamma_{\sigma\pi\pi} { {(p.p_1)(p_2.p_3)} \over { \left[{m_{a_0}^2 + (p-p_3)^2}\right]\left[{m_\sigma^2 + (p-p_1)^2 }\right] } } + (\sigma \leftrightarrow f_0) \label{scampls}\end{aligned}$$ The $\rho$ contributions are: $$\begin{aligned} M_{\rho}^a &=& i { {m_\rho^4 C_{\pi\eta}^\eta } \over { 2 {\tilde g}^2 F_{\pi p}^4 (m_\pi^2 - m_\eta^2) } } { {p.p_1 + p_1.p_3 - p.p_2 - p_2.p_3 } \over { m_\rho^2 + (p - p_3)^2 } } + (p_2 \leftrightarrow p_3) \nonumber \\ M_\rho^b &=& i \frac {4 \alpha_- y {\rm cos} \theta_p g_{\rho \pi \pi}}{\tilde g F_{\pi p}^2} \frac{p_2 \cdot (p_3 - p_1)}{m_\rho^2 + {(p - p_2)}^2} + ( p_2 \rightarrow p_3) \label{vmampls}\end{aligned}$$ . The two point vertices are: $$\begin{aligned} C_{\pi\eta}^\pi & = & - { {8 y \, {\rm cos} \theta_p \, \delta'}\over F_{\pi p}^2} - { { 64 y \,\lambda'^2 \,{\rm cos} \theta_p }\over {F_{\pi p}^2 } } + { {4 y \,\alpha_p \,{\rm cos}\theta_p \,m_\pi^2} \over {{\tilde g}^2 F_{\pi p}^2 }} \nonumber \\ C_{\pi\eta}^\eta & = & - { {8 y \,{\rm cos} \theta_p \,\delta'}\over F_{\pi p}^2} - { { 64 y \,\lambda'^2 \,{\rm cos} \theta_p }\over {F_{\pi p}^2 } } + { {4 y \alpha_p \,{\rm cos}\theta_p \,m_\eta^2} \over {{\tilde g}^2 F_{\pi p}^2}} \nonumber \\ C_{\pi\eta'}^\pi & = & - { {8 y \,{\rm sin} \theta_p \,\delta'}\over F_{\pi p}^2} - { { 64 y \,\lambda'^2 \,{\rm sin} \theta_p }\over {F_{\pi p}^2 } } + { {4 y \,\alpha_p {\rm sin}\theta_p \,m_\pi^2} \over {{\tilde g}^2 F_{\pi p}^2}} \nonumber \\ C_{\pi\eta'}^{\eta^\prime} & = & - { {8 y \,{\rm sin} \theta_p \,\delta'}\over F_{\pi p}^2} - { { 64 y \,\lambda'^2 \,{\rm sin} \theta_p }\over {F_{\pi p}^2 } } + { {4 y \,\alpha_p {\rm sin}\theta_p \,m_{\eta^\prime}^2} \over {{\tilde g}^2 F_{\pi p}^2 }} \nonumber \\ A_{a\sigma} & = & 2 y\, ( b + d)\, {\rm sin} \theta_s - \sqrt{2} y \, d \,{\rm cos}\theta_s \nonumber \\ A_{af} & = & - 2 y\, ( b + d)\, {\rm cos} \theta_s - \sqrt{2} y\, d \, {\rm sin}\theta_s \label{two-point-vertices}\end{aligned}$$ Notice that the superscript on $C$ indicates which of the two particles involved in the $\Delta I = 1$ transition is on-shell; this only affects the $\alpha_p$ term which has derivative coupling. 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--- address: | $^{1}$ DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK\ $^{2}$ School of Physics and Astronomy & Institute for Gravitational Wave Astronomy, University of Birmingham, Birmingham, B15 2TT, UK\ $^{3}$ TAPIR 350-17, California Institute of Technology, 1200 E. California Blvd., Pasadena, California 91125, USA\ $^{4}$ Kavli Institute for Cosmology Cambridge, Madingley Road CB3 0HA, Cambridge, UK bibliography: - 'regen.bib' --- Introduction {#sec:intro} ============ Ever since its formulation in 1915, Einstein’s general relativity (GR) has been a tremendously successful theory of gravity, combining mathematical elegance with enormous predictive power. Phenomena ranging from Mercury’s perihelion precession to the formation of black holes (BHs), the generation of gravitational waves and the big bang, find a mathematical description within this single theory. A wide range of lab-based experiments, solar-system tests and observations of astronomical phenomena have systematically scrutinized the accuracy of the theory’s predictions and unanimously seen GR passing these tests with flying colors [@Will:2014kxa]. With the advent of gravitational-wave (GW) astronomy, marked by the detection of GW150914 by LIGO [@Abbott:2016blz], new tests of GR have become possible, in spacetime regions with strong and dynamical gravitational fields and sources moving at relativistic velocities. Once again, all GW observations so far are compatible with GR [@TheLIGOScientific:2016src; @Monitor:2017mdv; @Abbott:2018lct; @LIGOScientific:2019fpa; @Yunes:2016jcc]. Notwithstanding GR’s success, the search for possible alternative theories of gravity has for many decades been a highly active area of research [@Will:2014kxa; @Berti:2015itd; @Barack:2018yly; @Olmo:2019flu], motivated by important theoretical considerations, such as the incompatibility of GR with quantum theory at a fundamental level, as well as open questions in observational astronomy and cosmology. Astronomical observations of galactic rotation curves, micro-lensing, primordial nucleosynthesis or the accelerated expansion of the universe cannot be explained in GR without evoking [*dark matter*]{} and [*dark energy*]{}, enigmatic entities beyond the standard model of particles; see e.g. [@Freese:2008cz; @Novosyadlyj:2015zpa]. Alternatively to either the dark-matter or dark-energy hypotheses, we may consider modifications in the laws of gravity; just in the same way GR explained Mercury’s anamolous perihelion precession in terms of modifications of the then prevailing Newtonian laws of gravity. Modifications of GR may also overcome one of the most important theoretical concerns about Einstein’s theory, its nonrenormalizability in quantum theory terms [@Hamber:2009zz]. For a theory as well established as GR, however, the quest for modifications faces an obvious difficulty; the longstanding success of the old theory suggests that modifications either be extremely weak or become measurable only under new, in some sense extreme, conditions. Quite remarkably, however, this conclusion is not quite correct: nonperturbative effects of an alternative theory of gravity may lead to order-of-unity deviations from GR even if departures at [*linearized*]{} level are small. The prototypical example of this phenomenon is the [*spontaneous scalarization*]{} of neutron stars (NSs) in scalar-tensor (ST) theory of gravity discovered by Damour and Esposito-Far[è]{}se in 1993 [@Damour:1993hw]. Here, the additional degree of freedom – in the form of the scalar field – allows for additional families of solutions describing stars in equilibrium. Moreover, these new families of solutions may appear “abruptly”, in a manner akin to phase-transitions, as one varies certain parameters of the theory or the star’s density profile. In the case of compact stars in ST gravity, the new solutions consist of stars with strong scalar-field profiles, as opposed to the GR-like models with negligible or zero scalar field. Often, the new scalarized configurations are energetically favored over their GR-like counterparts (assuming equal baryon mass or number), so that they represent the expected endpoints of dynamical scenarios. Spontaenous scalarization bears a qualitative resemblence to other effects known in physics; Damour and Esposito-Far[è]{}se have highlighted its analogy to the spontaneous magnetization of ferromagnets [@Damour:1996ke] and later studies have interpreted its onset in terms of catastrophe theory [@Harada:1998ge] or a tachyonic instability [@Ramazanoglu:2016kul]. Originally, spontaneous scalarization has been identified for spherically symmetric NS models in a class of massless ST theories sometimes refered to as Bergmann-Wagoner [@Bergmann:1968ve; @Wagoner:1970vr] theories; these complement the metric sector of GR with a single scalar field, are governed by second-order covariant field equations at most linear in second and quadratic in first derivatives, and obey the weak equivalence principle. The phenomenon has by now been demonstrated to occur over a wide range of configurations and also in other theoretical frameworks [@Andreou:2019ikc; @Ventagli:2020rnx]. Analogous phenomena occur in many scenarios involving fields non-minimally coupled to a spacetime metric. Examples include scalarized BHs in theories with Gauss-Bonnet coupling [@Silva:2017uqg; @Doneva:2017bvd], universal horizons in Lorentz violating gravity theories [@Barausse:2011pu] and the spotaneous [*vectorization*]{} or [*tensorization*]{} of compact stars in modified gravity [@Ramazanoglu:2017xbl; @Annulli:2019fzq; @Ramazanoglu:2019gbz]. Numerous studies have demonstrated that spontaneous scalarization features as prominently in rotating NS models, either in the slow-rotation limit [@Sotani:2012eb; @Pani:2014jra; @Silva:2014fca; @Yazadjiev:2016pcb; @Motahar:2017blm; @Staykov:2018hhc], for fast rotation [@Doneva:2013qva; @Doneva:2014faa], or with differential rotation [@Doneva:2018ouu]. Spontaneous scalarization has also been found a robust phenomenon under variations of the equation of state (EOS) [@Ramazanoglu:2016kul; @Motahar:2017blm; @Sotani:2017pfj; @Rezaei:2018jur; @Anderson:2019hio]. While quantitative differences occur, the phenomenon as such appears to be ubiquitous and also preserve the approximate EOS universality of the relation between the moment of inertia $I$ and the quadrupole moment $Q$ known from GR [@Doneva:2014faa; @Pani:2014jra] and the inertia vs. compactness universality [@Motahar:2017blm]. Numerical calculations find that the families of scalarized NSs can have larger maximal masses than the corresponding GR solutions [@Damour:1993hw; @Ramazanoglu:2016kul; @Morisaki:2017nit; @Doneva:2018ouu]. Often this is accompanied with an even stronger increase in the NS radius, so that the maximum compactness of NSs in ST gravity is smaller than in GR [@Sotani:2018aiz]. These findings suggest that the scalar field may effectively stiffen the equation of state and thus counteract the normal gravitational pull. This effect, does not appear to be generic, however, but rather depends on details of the matter sources. A generalization of the Buchdahl limit [@Tsuchida:1998jw] has found that compactness above the Buchdahl limit is possible in ST theory, but only if the energy density $\rho$ and pressure $p$ satisfy $\rho < 3p$. Sotani and Kokkotas [@Sotani:2017pfj] find that the maximum NS mass is larger in ST theory than in GR for sufficiently small sound speeds in the core, but that the reverse holds if this velocity exceeds 0.79 of the speed of light. In light of the recent discovery by LIGO and Virgo of the compact binary GW190814, whose light component’s mass likely falls in the so-called mass gap between NS and BHs [@Abbott:2020khf], it is worth noting that MST gravity allows for the possibility of such objects being strongly scalarized NSs. The presence or absence of the spontaneous scalarization phenomenon is largely determined by the quadratic coupling parameter $\beta_0$ between the scalar and tensor sectors of the theory; cf. Eq.(\[eq:conformalfactor\]) below. Strongly scalarized NS models are found for $\beta_0 \lesssim -4.35$ and this threshold has been found remarkably robust against variations of other parameters such as the EOS; see e.g. [@Novak:1998rk; @Pani:2014jra]. In a series of studies, however, Mendes and collaborators [@Mendes:2014vna; @Mendes:2014ufa; @Mendes:2016fby] have demonstrated that strongly scalarized solutions can also be obtained for positive values of $\beta_0$ [*provided*]{} there exist stable equilibrium solutions for matter fields in the GR limit where the trace $T$ of the energy-momentum tensor acquires positive values. This can be understood, for instance, in terms of the tachyonic instability by noticing that the scalar field is sourced by a term $\propto \beta_0 T$; cf. Eq. (3) in Ref. [@Ramazanoglu:2016kul]. This $\beta_0 > 0$ scenario has been explored in time evolutions of NSs close to the upper NS mass limit in Ref. [@Palenzuela:2015ima]. These simulations demonstrate an instability of the star to collapse for large $\beta_0$ of $\mathcal{O}(10^2)$, suggesting an upper bound on the parameter $\beta_0$. Massless ST theories of gravity have by now been significantly constrained – not least of all because of the large magnitude of the spontaneous scalarization effect – by the Cassini mission [@Bertotti:2003rm], Lunar Laser Ranging [@Williams:2005rv], binary pulsar observations [@Wex:2014nva] and gravitational wave (GW) observations with LIGO-Virgo [@Yunes:2016jcc]. While spontaneous scalarization has been seen to occur in dynamical evolutions in massless ST theory, either for the gravitational collapse of single stars [@Novak:1997hw; @Novak:1998rk; @Novak:1999jg; @Gerosa:2016fri] or the merger of binary NSs [@Barausse:2012da; @Palenzuela:2013hsa; @Shibata:2013pra], the most recent constraints on $\beta_0$ severely limit the magnitude of the resulting GW signals and, thus, make it difficult to constrain this theory further with GW observations. In the context of this work, the most important extension of the scalarization phenomenon is the inclusion of a non-zero scalar mass. This is because the above constraints only apply to ST theories with a scalar mass parameter $\mu \lesssim 10^{-16}\,{\rm eV}$. Otherwise, the Compton wavelength $\lambda_c=(2\pi \hbar)/(\mu c)$ is smaller than the distance between the objects involved in the systems under consideration and, hence, the scalar contribution to the objects’ interaction is suppressed. GW observations, on the other hand, provide exquisite constraints on dispersion which in turn can be interpreted as a constraint on the graviton mass, but this does not apply to radiation in the scalar sector. In consequence, massive ST theory remains compatible with present observations over much of its parameter space. Motivated by this realization, many recent studies have explored spontaneous scalarization in massive ST gravity. Computations of stationary models have confirmed that the spontaneous scalarization phenomenon persists under the inclusion of a mass term in the scalar potential [@Chen:2015zmx; @Doneva:2016xmf; @Yazadjiev:2016pcb; @Ramazanoglu:2017xbl; @Morisaki:2017nit]. A non-zero scalar mass $\mu > 0$ does, however, dramatically affect the GW signals generated in stellar collapse in ST gravity through dispersion. A Fourier mode with frequency $\omega$ propagates at group velocity $v_g = \sqrt{1-\omega_*^2/\omega^2}$, $\omega_*=c^2 \mu/\hbar$, so that high-frequencies reach a detector first with lower frequencies arriving later, so the signal acquires an [*inverse-chirp*]{} or [*howl*]{} character. Furthermore these signals get extremely stretched out and become approximately monochromatic (in the sense that the frequency changes by very little over one period; $\mathrm{d}f/\mathrm{dt}\ll f^2$), can reach considerable amplitude for sufficiently negative $\beta_0$ and may last for years or even centuries for scalar masses $\mu \lesssim 10^{-12}~{\rm eV}$ [@Sperhake:2017itk; @Rosca-Mead:2020ehn; @Geng:2020slq]. While the inclusion of self-interaction terms may reduce the degree of scalarization and the amplitude of the scalar GWs [@Cheong:2018gzn; @Staykov:2018hhc], this requires considerable finetuning of the scalar potential parameters [@Rosca-Mead:2019seq]. In this work, we focus on spherically symmetric, static NS models in the framework of Bergmann-Wagoner ST theory with massive scalar fields. The main purpose of our study is two-fold. First, to introduce a numerical scheme that enables us to compute these stellar models over the entire spatial domain, all the way out to infinity, while maintaining complete control over exponentially diverging solutions. Second, to present an in-depth analysis of the structure of the different solution branches and their dependence on the parameters of the ST theory. We begin this discussion in Sec. \[sec:formalism\] with a review of the field equations governing the stars. In Sec. \[sec:numerics\], we describe the numerical framework used for our computations. Our results on the structure of NS solutions in massive ST gravity are presented in Sec. \[results\] and we conclude in Sec. \[sec:conclusions\]. Formalism {#sec:formalism} ========= The Bergmann-Wagoner class of ST theories, i.e. theories involving a single scalar field that are governed by two-derivative, covariant field equations and obey the weak equivalence principle, can be described in terms of the action [@Fujii:2003pa] $$S_J = \int dx^4 \sqrt{-g} \left[ \frac{F(\phi)}{16\pi G} R - \frac{1}{2}Z(\phi) g^{\mu \nu} (\partial_{\mu} \phi)(\partial_{\nu}\phi) - W(\phi) \right] + S_m(\psi_m,g_{\mu \nu})\,. \label{eq:Jordan_action}$$ Here, $\psi_m$ collectively denotes the matter fields and $S_m$ represents their coupling to the spacetime geometry of the [*physical*]{} or [*Jordan*]{} metric $g_{\mu\nu}$ with determinant $g$ and Ricci scalar $R$. The functions $F(\phi)$ and $Z(\phi)$ encapsulate the nonminimal coupling of the scalar field $\phi$ to the metric sector, and $V(\phi)$ is the potential function. As we shall see shortly, the function $Z$ can be eliminated through an appropriate redefinition and is therefore often set to unity in the literature; see e.g. [@Salgado:2005hx]. This class of theories is conveniently described in the so-called Einstein frame, obtained from the physical or Jordan frame through a conformal transformation of the metric and a redefinition of the scalar degree of freedom and its potential, $$g_{\alpha \beta}=\frac{1}{F}\bar{g}_{\alpha \beta},~~~~~~~~~~ \frac{\partial \varphi}{\partial \phi} = \sqrt{\frac{3}{4} \frac{F_{,\phi}(\phi)^2}{F(\phi)^2} + \frac{4\pi Z(\phi)}{F(\phi)}} \,,~~~~~~~~~~ V(\varphi) = \frac{4\pi W(\phi)}{F(\phi)^2}\,, \label{eq:EJtrafo}$$ where $F_{,\phi}=dF/d\phi$. In terms of these new functions, the action (\[eq:Jordan\_action\]) becomes $$S_E = \frac{1}{16\pi G} \int dx^4 \sqrt{-\bar{g}} \left[ \bar{R} - 2\bar{g}^{\mu \nu} (\partial_{\mu} \varphi) (\partial_{\nu} \varphi) - 4V(\varphi) \right] +S_m\left[ \psi_m,\frac{\bar{g}_{\mu \nu}}{F(\varphi)}\right]\,, \label{eq: einsteinaction}$$ where an overbar distinguishes tensors in the Einstein frame from their Jordan counterparts. Henceforth, we use natural units where $c=G=1$, unless stated otherwise. By transforming to the Einstein frame, we have eliminated the function $Z$. The equivalence (or lack thereof) of the Einstein and Jordan frame formulations has been the subject of a long standing debate (see e.g. [@Faraoni:1999hp; @Salgado:2008xh; @Geng:2020ftu] and references therein). Without entering this debate here, we merely note the extra freedom that the function $Z$ introduces to the transformation (\[eq:EJtrafo\]) between the frames and henceforth follow the recommendation of Ref. [@Faraoni:1999hp] and work in the Einstein frame. To complete the description of the gravitational theory we must specify the remaining free functions $F$ and $V$. Following most previous studies in the literature, we write the conformal factor as[^1] $$F(\varphi) = e^{-2\alpha_0 \varphi-\beta_0 \varphi^2} \,, \label{eq:conformalfactor}$$ and take as our potential function the quadratic function $$V(\varphi) = \frac{\mu^2\varphi^2}{2\hbar^2}\,,$$ which describes a non-self-interacting scalar field of mass $\mu$. The field equations obtained by varying the Einstein action (\[eq: einsteinaction\]) with respect to $\bar{g}_{\alpha\beta}$, $\varphi$ are given by $$\begin{aligned} \bar{R}_{\alpha\beta}-\frac{1}{2}\bar{g}_{\alpha\beta} \bar{R} &=& 2\partial_{\alpha}\varphi \partial_{\beta}\varphi -\bar{g}_{\alpha\beta}\bar{g}^{\mu\nu}\partial_{\mu}\varphi \partial_{\nu}\varphi+8\pi \bar{T}_{\alpha\beta} \label{eq:barG}, \\ \bar{\Box}\varphi &=& 2\pi \frac{F_{,\varphi}}{F} \bar{T}+V_{,\varphi}\,, \label{eq:boxvarphi}\end{aligned}$$ with the energy momentum tensor $$\begin{aligned} &&\bar{T}^{\alpha\beta} = \frac{2}{\sqrt{-\bar{g}}} \frac{\delta S_m}{\delta \bar{g}_{\alpha\beta}} = \frac{1}{{F(\varphi)}^3}\frac{2}{\sqrt{-g}} \frac{\delta S_m}{\delta g_{\alpha\beta}} = \frac{1}{{F(\varphi)}^3}T^{\alpha\beta}\,,\end{aligned}$$ for which the Bianchi identity now implies the following conservation law $$\bar{\nabla}_{\mu}\bar{T}^{\mu\alpha} = -\frac{1}{2} \frac{F_{,\varphi}}{F}\bar{T}\bar{g}^{\alpha\mu} \bar{\nabla}_{\mu}\varphi\,. \label{eq:barT}$$ From now on, we restrict our attention to spherically symmetric, time independent stellar models. More specifically, we employ polar slicing and radial gauge in the Einstein frame, so that the line element is of the form $$\mathrm{d}\bar{s}^2 = \bar{g}_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu} = -F\,\alpha^2 \mathrm{d}t^2+F\,X^2 \mathrm{d}r^2 + r^2\mathrm{d} \Omega^2\,, \label{eq:lineelement}$$ where $\alpha$ and $X$ as well as the scalar field $\varphi$ are functions of the radius $r$, and $\mathrm{d}\Omega^2$ denotes the standard line element on the unit 2-sphere. It is furthermore common practice to introduce the gravitational potential $\Phi$ and mass function $m$ according to $$F\,\alpha^2 = e^{2\Phi},~~~~~F\,X^2 = \left(1-\frac{2m}{r}\right)^{-1}\,. \label{eq_nu_m}$$ In this work we explore the behaviour of NSs in equilibrium and model their matter as a perfect fluid at zero temperature; the temperature of NS interiors in equilibrium, despite being of order $10^{6}\,\mathrm{K}$, is well below the Fermi temperature $\mathcal{O}(10^{11})\,\mathrm{K}$ of matter at nuclear densities. The energy momentum tensor is then given in terms of the baryon density $\rho(r)$, the specific enthalphy $h(r)$ and the pressure $P(r)$ by $$\begin{aligned} &&\bar{T}_{\alpha \beta}=\frac{1}{F} T_{\alpha \beta} = \frac{1}{F} \left( \rho h u_{\alpha} u_{\beta} + P g_{\alpha \beta} \right)\,, \nonumber \\ && \text{with}~~~~~~~~ u^{\alpha} = \left[ \alpha^{-1},~0,~0,~0 \right]\,, ~~~~~~~~h = 1+\epsilon + \frac{P}{\rho}\,, \label{eq:mom}\end{aligned}$$ and where $\epsilon$ is the specific internal energy. Inserting the Einstein frame metric (\[eq:lineelement\]) and the energy momentum tensor (\[eq:mom\]) into the field equations (\[eq:barG\])-(\[eq:barT\]), we obtain the set of differential equations $$\begin{aligned} \partial_r \Phi &=& \frac{F X^2-1}{2r}+\frac{4\pi rP X^2}{F} +\frac{r X^2 \eta^2}{2}-Wr X^2 F\,, \label{eq:Phi}\\ \partial_r X &=& \frac{4\pi r X^3}{F}(\rho h -P)+\frac{r X^3 \eta^2}{2}\,, -\frac{X^3 F}{2 r}+\frac{X}{2 r} -\frac{F_{,\varphi} X^2 \eta}{2 F}+X^3FWr\label{eq:X}\,,\\ \partial_r P &=& -\rho hF X^2\left(\frac{m}{r^2}+4\pi r \frac{P}{F^2} +\frac{r}{2F}{\eta}^2-rW\right)+\rho h \frac{F_{,\varphi}}{2F}X \eta\,, \label{eq:P} \\ \partial_r \varphi &=& X\eta \label{eq:varphi}\,, \\ \partial_r \eta &=& -\frac{3 \eta}{2 r} -\frac{2 \pi X F_{,\varphi}}{F^2}(\rho h-4P) -\frac{X^2 \eta F}{2 r}-\frac{4 X^2 \eta \pi r P}{F} -\frac{X^2 \eta^3 r}{2} \nonumber\\ && +\frac{X \eta^2 F_{,\varphi}}{2 F}+X^2\eta FWr+XF\partial_\varphi W\,. \label{eq:eta}\,\end{aligned}$$ In order to close this set of equations, we need to relate the pressure and internal energy to the baryon density. In this work, we use cold polytropic EOSs with exponent $\Gamma$, $$P = K \rho^{\Gamma}\,, \label{eq:EOS}$$ The internal energy is then determined by the first law of thermodynamics $dE=\dbar Q-pdV$ for adiabatic processes with $\dbar Q=0$. For a total baryon number $N$, the specific internal energy and baryon density are given by $\epsilon = E/N$ and $\rho = N/V$, respectively, and the first law results in $$\epsilon = \frac{P}{(\Gamma-1)\rho}\,. \label{eq:epsilon}$$ The set of equations (\[eq:Phi\])-(\[eq:eta\]) can then be solved subject to the boundary conditions $$\begin{aligned} \text{at}~~r=0:~~~~\,&&\eta = 0\,,~~~~~~~~~~~FX^2 = 1\,, \nonumber \\ \text{at}~~r\rightarrow \infty:~~&&\Phi = 0\,,~~~~~~~~~~ \rho = 0\,,~~~~~~~~~~\varphi = 0\,. \label{eq:BCs}\end{aligned}$$ The computation of solutions to this problem is complicated by three issues, which we list in increasing order of difficulty. - The boundary conditions are specified at different locations of the domain, so that we have a [*two-point-boundary-value problem*]{}. - For realistic values of the polytropic exponent $\Gamma$, the pressure will reach zero at a finite radius $R_S$; at this point, we need to match to an exterior solution with vanishing baryon density $\rho$. - The asymptotic behaviour of the scalar field near infinity is determined by the scalar mass $\mu$ and is given by $$\lim_{r\rightarrow\infty} \varphi \sim A_1 \frac{e^{-(\mu/\hbar) r}}{r}+A_2 \frac{e^{(\mu/\hbar) r}}{r}\,, \label{eq:scalarasymptotics}$$ for constants $A_1$, $A_2$. We are only interested in bounded solutions with $\varphi \propto e^{-(\mu/\hbar) r}/r$. This exponential fall-off is responsible for the suppressed scalar contribution in the interaction of pulsar binaries in massive ST gravity and forms the key motivation for our study. From a purely numerical point of view, however, Eq. (\[eq:scalarasymptotics\]) creates a significant challenge. Numerical algorithms will pick up all possible modes of a solution – even if only through roundoff error. We therefore seek an algorithm that provides us with explicit control over the asymptotic behaviour of our numerical solutions. In the next section, we will discuss how this can be achieved inside the more standard frameworks employed to address items (i) and (ii) of our above list. Numerical framework {#sec:numerics} =================== Numerical methods for solving two-point-boundary-value problems are well developed and fall into two main classes, [*shooting algorithms*]{} and [*relaxation schemes*]{} (including collocation methods) [@Press1992] To the best of our knowledge, all literature on static NS models in ST gravity has employed shooting algorithms; see e.g. [@Ramazanoglu:2016kul; @Yazadjiev:2016pcb]. This process integrates the differential equations from one end of the domain by supplementing the known boundary conditions at this point with appropriate trial values for the remaining variables. The resulting integration will typically not match the boundary conditions at the other end of the domain, but the degree of violation can be used, e.g. through a Newton-Raphson or a bisection method, to iteratively improve the trial values until all boundary conditions are satisfied within a user-specified threshold accuracy. For the case of our system of differential equations (\[eq:Phi\])-(\[eq:eta\]) with boundary conditions (\[eq:BCs\]), this would work as follows. We first note that the function $\Phi$ appears only in Eq. (\[eq:Phi\]) and in the form of its spatial derivative. We can therefore set $\Phi(0)=0$ and add an arbitrary constant to match its boundary condition [*after*]{} having solved for all variables. Bearing in mind this freedom, we start the integration at the origin $r=0$ by selecting values for the central baryon density $\rho(0)$, the central scalar field amplitude $\varphi(0)$ and the metric function $\Phi(0)$, additionally to the known $\eta(0)=0$ and $X(0)=1/\sqrt{F(\varphi(0))}$. The integration will reach zero pressure at a finite $r$ which represents the NS radius. Beyond this radius, the integration continues setting $\rho=P=0$ in Eqs. (\[eq:Phi\])-(\[eq:eta\]). In principle this surmounts the issue (ii) mentioned in the previous section. We note, though, that $P=0$ is, in general, not realized on a grid point which adds a small discontinuity to the solution; the data on the outermost grid point inside the star and on the first point outside the star do not satisfy Eqs. (\[eq:Phi\])-(\[eq:eta\]). This discontinuity is typically not problematic, but we will see below how it can be simply eliminated in a relaxation approach. For a massless scalar field, it is even possible to analytically match the spacetime to an exterior vacuum[^2] metric; cf. Eqs. (8), (9) in [@Damour:1993hw]. Integration beyond the neutron star radius is not required and the trial value $\varphi(0)$ can be improved in accordance with the selected shooting algorithm. Such an analytic matching is not known, however, for [*massive*]{} scalar fields. And now a more problematic issue arises as the integration is continued beyond the NS radius; no matter how accurate the central value $\varphi(0)$ has been chosen, the numerical solution will contain an exponentially growing contribution from the asymptotic behaviour (\[eq:scalarasymptotics\]) and eventually blow up exponentially. Worse, this blowup prevents us from improving our trial value $\varphi(0)$ through measuring the departure from the correct boundary condition at infinity; this departure is infinite and, hence, useless for numerical purposes. In shooting algorithms, this problem is circumvented by imposing the outer boundary conditions at a finite radius rather than infinity; cf. Sec. III A in [@Ramazanoglu:2016kul]. While this method is still capable of generating accurate stellar models, a scheme covering the complete exterior and imposing the boundary conditions at infinity provides practical advantages besides the more rigorous boundary treatment. By extending all the way to infinity, our scheme can provide initial data for time evolutions on arbitrarily large computational domains (including compactified evolution schemes that incorporate spatial or null infinity) without resorting to adhoc procedures to extend results beyond the inevitably finite blow-up radius of shooting methods. We will also obtain a vacuum exterior that is matched to the NS interior on a grid point; in fact, the value of the NS surface, rather than the central density, will select the specific stellar model. Furthermore, the relaxation scheme provides an exceptionally elegant and simple way to implement the matching between the interior and exterior domain that we expect to be applicable to a wider range of problems, including extension to time evolutions. For our method, we first introduce the NS radius $R_S$ as a free parameter. On the domain $r\in[0,R_S]$, we use the differential equations (\[eq:Phi\])-(\[eq:eta\]) with boundary conditions (\[eq:BCs\]) for $\eta$ and $X$ at $r=0$; the condition $FX^2=1$ is formulated as an equation involving the unknown $\varphi(0)$. In the exterior, we introduce a compactified radial coordinate $$y = \frac{1}{r}\,,$$ set $\rho=P=0$, and introduce rescaled variables $$\sigma = \varphi e^{(\mu/\hbar)r}\,,~~~~~~~~~~ \kappa = -\eta e^{(\mu/\hbar)r}\,.$$ By factoring the exponential dependence into our scalar field variables, we ensure that regular solutions $\sigma$ and $\kappa$ asymptote towards a $\propto y$ dependence at $y=0$. We find this step crucial in achieving convergence of our relaxation scheme which struggles with the exponential fall-off of $\varphi$ and $\eta$ but copes smoothly with the benign linear behaviour of the rescaled $\sigma$ and $\kappa$. We also notice a minor (but not crucial) improvement in the speed of convergence when switching from $X$ to the mass function $m$ of Eq. (\[eq\_nu\_m\]) and hence use the set of variables $\Phi,~m,~\sigma,~\kappa$ in the exterior. The differential equations in the exterior domain $y\in [0,y_{\rm S}]$, $y_{\rm S}=1/R_{\rm S}$, thus become $$\begin{aligned} \partial_y \Phi &=& -\frac{m}{1-2my}-\frac{1}{2\left(1-2my\right) y^3e^{2\mu/y}} \left(\frac{\kappa^2}{F}-\mu^2\sigma^2\right)\,, \label{eq:Phiy} \\ \partial_y m &=& -\frac{1}{2y^4e^{2\mu/y}}\left( \frac{\kappa^2}{F}-\mu^2\sigma^2\right)\,, \label{eq:my} \\ \partial_y \sigma &=& \frac{X\kappa-\mu\sigma}{y^2}\,,\\ \partial_y \kappa &=& -\kappa\partial_y\Phi+\frac{FX\mu^2\sigma+\kappa \left(2y-\mu\right)}{y^2}+\frac{X\kappa^2}{2y^2} \frac{F_{,\varphi}}{F}\,, \label{eq:kappay}\end{aligned}$$ and the matching conditions imposed at the surface of the NS are given by $$\begin{aligned} \Phi(y_{\rm S}) &=&\Phi(R_{\rm S})\,, \nonumber \\ m(y_{\rm S}) &=& \left[ \frac{r}{2}\left(1-\frac{1}{FX^2}\right) \right]_{r=R_{\rm S}}\,, \nonumber \\ \sigma(y_{\rm S}) &=& \varphi(R_{\rm S}) e^{(\mu/\hbar) R_{\rm S}} \,, \nonumber \\ \kappa(y_{\rm S}) &=& -\eta(R_{\rm S}) e^{(\mu/\hbar) R_{\rm S}}\,. \label{eq:matching}\end{aligned}$$ We formally also use the trivial equation $\partial_y P=0$ in the exterior which allows us to use a constant number of five variables over the entire grid. This grid consists of $N$ grid points in the interior and $M$ points in the exterior. We discretize the differential equations using cell-centered second-order stencils which provides us with $5(N-1)$ algebraic equations in the interior and $5(M-1)$ equations in the exterior. The boundary conditions provide two further equations at $r=0$ and three further equations at $y=0$. The surface radius is represented twice on our grid, the outermost point $r=R_{\rm S}$ of the interior and the innermost point $y_{\rm S}$ of the exterior grid. The variables used on these points are related by the matching conditions (\[eq:matching\]) as well as the trivial $P(R_{\rm S})=P(y_{\rm S})=0$. In total, we thus have $5(N+M)$ non-linear algebraic equations for the $5(N+M)$ unknown values of the variables on the grid points. Given an initial guess, we can linearize the equations around this trial solution which leads to a matrix equation with block-diagonal structure that is readily inverted to improve the guess iteratively; we use the algorithm of Ref. [@Press1992] and typically obtain convergence after about ten iterations. The initial guess is obtained by integrating Eqs. (\[eq:Phi\])-(\[eq:eta\]) up to $R_{\rm S}$, fixing $\sigma$ and $\kappa$ as linear functions $\propto y$ in the exterior and integrating Eqs. (\[eq:Phiy\]), (\[eq:my\]) with these specified scalar sources. Note that in this calculation we set $\rho=P=0$ in the exterior irrespective of whether or not they have reached zero at $R_{\rm S}$; the discontinuity that may result at the matching point is removed in the ensuing relaxation process. Even for modest resolutions such as $N=M=401$, this approach provides an accuracy of $O(10^{-4})$. All models discussed in the remainder of this paper have been computed with this code. Results ======= Overall phenomenology --------------------- We start this section by defining the terminology and diagnostic quantities as well as providing a qualitative review of the different branches of static NS models encountered in massive ST gravity. We then explore in the following subsections in more detail the impact of the ST parameters $\alpha_0$, $\beta_0$ and $\mu$ on the structure of these branches. In the following, we will use the term “family” for the set of all NS models obtained for fixed EOS and ST parameters $\alpha_0$, $\beta_0$ and $\mu$. We will use the term “branch” to denote a subset of solutions of a family that share some specific property, for example strong scalarization. A family thus consists of one or more branches. In some cases, we find a branch to have the shape of a closed loop disconnected from other branches, and we also refer to such a branch as a “loop”. For reference, we note that the scalar mass $\mu$ introduces a Compton wavelength and characteristic frequency given by $$\lambda_C = 1.24\times 10^6\,\mathrm{km}~\left( \frac{\mu}{10^{-15}\,\mathrm{eV}}\right)^{-1}\,,~~~~~~~~~~ f_* = \frac{\omega_*}{2\pi} = 24.2\,\mathrm{Hz}~\left( \frac{\mu}{10^{-15}\,\mathrm{eV}}\right)^{-1}\,.$$ Unless stated otherwise, our numerical NS models in massive ST theory are computed with the polytropic EOS labelled “EOS1” in Ref. [@Novak:1997hw]. Translated into our notation, we therefore compute the pressure and specific internal energy from the baryon density $\rho$ through Eqs. (\[eq:EOS\]) and (\[eq:epsilon\]) with $$K =1.543~\frac{\mathrm{cm}^{3\Gamma-1}}{\mathrm{g}^{\Gamma-1}\mathrm{s}} ,~~~~~~~~~~ \Gamma=2.34\,. \label{eq:EOS1}$$ The families of solutions thus obtained are conveniently represented in a mass-radius diagram. For this purpose, we define the total baryon mass $M_b$ as the volume integral of the baryon number density multiplied by the mass per baryon $m_b$. Translated into our baryon density $\rho = m_b n_b$, the expression becomes $$M_b=m_b\int d^3 x\sqrt{-g}n_bu^t=4\pi\int_{0}^{R_S}dr \left(r^2\frac{\rho}{F^{3/2}\sqrt{1-2m/r}}\right)\,.$$ The motivation for using the baryon mass, rather than the gravitational mass of Eq. (\[eq\_nu\_m\]), arises from the conservation of the baryon number; if we consider the possibility that a NS might migrate dynamically from one branch to another, we expect it to do so at constant $M_b$, whereas the binding energy and, hence, gravitational mass will, in general, change. As in massless ST theories, all NS solutions can be classified as either [*weakly scalarized*]{} with scalar field profiles reaching a magnitude $\varphi\sim O(\alpha_0)$ or [*strongly scalarized*]{} solutions where the scalar field reaches values $\varphi\sim O(1)$ [@Damour:1993hw]. In this work, we call these branches W (for weak) and S (for strong scalarization); see e. g. Fig. \[fig:various\]. The distinction between the two regimes naturally blurs for large values $\alpha_0 = \mathcal{O}(1)$; in this work we consider only $\alpha_0 \ll 1$ and thus retain a clear division between weakly and strongly scalarized NSs. ![Branches of NS models are shown in the form of baryon-mass vs. radius ($M_b-R_S$) diagrams. [*Left:*]{} For fixed values $\alpha_0=10^{-4}$ and $\mu=4.8\times 10^{-13}\,\mathrm{eV}$, we plot the strongly scalarized branches obtained for selected values of $\beta_0$. For reference, the dashed black curve displays the solutions obtained in GR with $\alpha_0=\beta_0=0$. [*Right:*]{} Here we fix $\alpha_0=10^{-4}$ and a more extreme value of $\beta_0=-15$ and vary the scalar mass $\mu$; larger deviations from the GR structure are clearly visible in this case. In both panels the color scale measures the central value of $|\varphi|$ and the “S” and “W” label the strongly and weakly scalarized branches described in the text. []{data-label="fig:various"}](various_beta-eps-converted-to.pdf "fig:"){width="48.00000%"} ![Branches of NS models are shown in the form of baryon-mass vs. radius ($M_b-R_S$) diagrams. [*Left:*]{} For fixed values $\alpha_0=10^{-4}$ and $\mu=4.8\times 10^{-13}\,\mathrm{eV}$, we plot the strongly scalarized branches obtained for selected values of $\beta_0$. For reference, the dashed black curve displays the solutions obtained in GR with $\alpha_0=\beta_0=0$. [*Right:*]{} Here we fix $\alpha_0=10^{-4}$ and a more extreme value of $\beta_0=-15$ and vary the scalar mass $\mu$; larger deviations from the GR structure are clearly visible in this case. In both panels the color scale measures the central value of $|\varphi|$ and the “S” and “W” label the strongly and weakly scalarized branches described in the text. []{data-label="fig:various"}](various_mphi-eps-converted-to.pdf "fig:"){width="48.00000%"} As is well known, the NS solutions in GR[^3] form a one-parameter family whose members can be characterized by its central density $\rho_c$ [@Shapiro1983]. For the EOS (\[eq:EOS1\]), this leads to the black branch in the left panel[^4] of Fig. \[fig:various\]. In ST gravity, the introduction of a scalar field leads to additional branches of NS solutions with a non-vanishing scalar-field profile $\varphi(r)$. The shape of the extra branches depends on the values of the parameters $\alpha_0$, $\beta_0$ and $\mu$. In agreement with the literature [@Novak:1998rk], we observe strongly scalarized solutions if $\beta_0 \lesssim -4.35$. These new branches are displayed in the left panel of Fig. \[fig:various\] in terms of a color code that denotes the central scalar field amplitude. In contrast, the weakly scalarized solutions we obtain for $\beta_0 \gtrsim -4.35$ have macroscopic properties that barely differ from those of the GR solutions and their branch would be indistinguishable from the GR family in the figure. For these cases, we have set $\alpha_0=10^{-4}$ and $\mu = 4.8\times 10^{-13}\,\mathrm{eV}$. We now discuss in more detail how the solutions and their branches vary when the parameter values are changed. Dependence on mu {#sec:mu} ---------------- The variation of scalarized NS branches in massive ST theory has already been studied in Ref. [@Ramazanoglu:2016kul] who generally observe that an increase in the scalar mass $\mu$ results in a weakening of the scalarization. By computing a sequence of models with equal gravitational or “Arnowitt-Deser-Misner” (ADM) [@Arnowitt:1962hi] mass, they observe a monotonic decrease in the scalarization as the scalar mass is increased. Around $10^{-12}\,\mathrm{eV}$, their scalar profile drops to negligible levels when $\beta_0=-4.5$; cf. their Fig. 2. Our results exhibit a similar drop in scalarization. We illustrate this general behaviour by comparing the cases $\mu=10^{-15}\,\mathrm{eV}$ and $\mu=4.8\times 10^{-13}\,\mathrm{eV}$ in the right panel of Fig. \[fig:various\]; the color code of the branches represents the central scalar field amplitude and displays lower values for the larger $\mu$. We furthermore notice that the strongly scalarized NS branches reach out to smaller baryon mass and radii as we increase the scalar mass $\mu$. As we shall discuss in more detail in Sec. \[sec:stability\], the stable NS model for a given baryon mass is that with the largest radius. For fixed $M_b$, a larger scalar mass $\mu$ results in smaller and more compact stable NS models. Dependence on alpha0 {#sec:alpha0} -------------------- ![$M_b-R_S$ diagrams are shown for $\mu=4.8\times 10^{-13}\,\mathrm{eV}$ and $\beta_0=-4.5$, as well as $\alpha_0=0$ (top left), $\alpha_0=10^{-4}$ (top right) and $\alpha_0=10^{-3}$ (bottom). The color scale measures the central value of $|\varphi|$. Whereas the S and W branches connect at two points when $\alpha_0=0$, the S branch splits in two for nonzero $\alpha_0$ with each part connecting to GR-like models in such a way that we obtain a “loop” of models separate from the main branch of solutions. We refer to the main branch as branch I and to the loop as branch $II$. []{data-label="split_starting"}](new_24_plot_within_plot-eps-converted-to.pdf "fig:"){width="48.00000%"} ![$M_b-R_S$ diagrams are shown for $\mu=4.8\times 10^{-13}\,\mathrm{eV}$ and $\beta_0=-4.5$, as well as $\alpha_0=0$ (top left), $\alpha_0=10^{-4}$ (top right) and $\alpha_0=10^{-3}$ (bottom). The color scale measures the central value of $|\varphi|$. Whereas the S and W branches connect at two points when $\alpha_0=0$, the S branch splits in two for nonzero $\alpha_0$ with each part connecting to GR-like models in such a way that we obtain a “loop” of models separate from the main branch of solutions. We refer to the main branch as branch I and to the loop as branch $II$. []{data-label="split_starting"}](new_4_plot_within_plot-eps-converted-to.pdf "fig:"){width="48.00000%"} ![$M_b-R_S$ diagrams are shown for $\mu=4.8\times 10^{-13}\,\mathrm{eV}$ and $\beta_0=-4.5$, as well as $\alpha_0=0$ (top left), $\alpha_0=10^{-4}$ (top right) and $\alpha_0=10^{-3}$ (bottom). The color scale measures the central value of $|\varphi|$. Whereas the S and W branches connect at two points when $\alpha_0=0$, the S branch splits in two for nonzero $\alpha_0$ with each part connecting to GR-like models in such a way that we obtain a “loop” of models separate from the main branch of solutions. We refer to the main branch as branch I and to the loop as branch $II$. []{data-label="split_starting"}](new_25_plot_within_plot-eps-converted-to.pdf "fig:"){width="48.00000%"} ![Distribution of scalarized NS models based on the sign of $\varphi_c$ in the $M_b-R_S$ plane for $\mu=4.8\times 10^{-13}\,\mathrm{eV}$ and, from top-left to bottom-right, $(\alpha_0,\beta_0)= (10^{-1},-4.5)$, $(10^{-2},-5.5)$,    $(10^{-2},-5)$, $(3\times 10^{-2},-5)$. The orange points represent models with $\varphi_c<0$ whereas the black ones have $\varphi_c>0$. The type $II$ models on the loop differ in the sign of $\varphi_c$ from the nearby main branch $I$ models. Furthermore, we always observe a sign flip at the high-density end of branch $I$ (around $R_S\approx 8\,\mathrm{km}$ in the figure) but these NS models are unstable; cf. Sec. \[sec:stability\]. []{data-label="positive_negative_1"}](various_alpha_4_5-eps-converted-to.pdf "fig:"){width="48.00000%"} ![Distribution of scalarized NS models based on the sign of $\varphi_c$ in the $M_b-R_S$ plane for $\mu=4.8\times 10^{-13}\,\mathrm{eV}$ and, from top-left to bottom-right, $(\alpha_0,\beta_0)= (10^{-1},-4.5)$, $(10^{-2},-5.5)$,    $(10^{-2},-5)$, $(3\times 10^{-2},-5)$. The orange points represent models with $\varphi_c<0$ whereas the black ones have $\varphi_c>0$. The type $II$ models on the loop differ in the sign of $\varphi_c$ from the nearby main branch $I$ models. Furthermore, we always observe a sign flip at the high-density end of branch $I$ (around $R_S\approx 8\,\mathrm{km}$ in the figure) but these NS models are unstable; cf. Sec. \[sec:stability\]. []{data-label="positive_negative_1"}](various_alpha_5_1-eps-converted-to.pdf "fig:"){width="48.00000%"} ![Distribution of scalarized NS models based on the sign of $\varphi_c$ in the $M_b-R_S$ plane for $\mu=4.8\times 10^{-13}\,\mathrm{eV}$ and, from top-left to bottom-right, $(\alpha_0,\beta_0)= (10^{-1},-4.5)$, $(10^{-2},-5.5)$,    $(10^{-2},-5)$, $(3\times 10^{-2},-5)$. The orange points represent models with $\varphi_c<0$ whereas the black ones have $\varphi_c>0$. The type $II$ models on the loop differ in the sign of $\varphi_c$ from the nearby main branch $I$ models. Furthermore, we always observe a sign flip at the high-density end of branch $I$ (around $R_S\approx 8\,\mathrm{km}$ in the figure) but these NS models are unstable; cf. Sec. \[sec:stability\]. []{data-label="positive_negative_1"}](new_27_mass_positive_negative_closeup-eps-converted-to.pdf "fig:"){width=".49\linewidth"} ![Distribution of scalarized NS models based on the sign of $\varphi_c$ in the $M_b-R_S$ plane for $\mu=4.8\times 10^{-13}\,\mathrm{eV}$ and, from top-left to bottom-right, $(\alpha_0,\beta_0)= (10^{-1},-4.5)$, $(10^{-2},-5.5)$,    $(10^{-2},-5)$, $(3\times 10^{-2},-5)$. The orange points represent models with $\varphi_c<0$ whereas the black ones have $\varphi_c>0$. The type $II$ models on the loop differ in the sign of $\varphi_c$ from the nearby main branch $I$ models. Furthermore, we always observe a sign flip at the high-density end of branch $I$ (around $R_S\approx 8\,\mathrm{km}$ in the figure) but these NS models are unstable; cf. Sec. \[sec:stability\]. []{data-label="positive_negative_1"}](new_32_mass_positive_negative_closeup-eps-converted-to.pdf "fig:"){width=".49\linewidth"}\ ![Distribution of scalarized NS models based on the sign of $\varphi_c$ in the $M_b-R_S$ plane for $\mu=4.8\times 10^{-13}\,\mathrm{eV}$ and, from top-left to bottom-right, $(\alpha_0,\beta_0)= (10^{-1},-4.5)$, $(10^{-2},-5.5)$,    $(10^{-2},-5)$, $(3\times 10^{-2},-5)$. The orange points represent models with $\varphi_c<0$ whereas the black ones have $\varphi_c>0$. The type $II$ models on the loop differ in the sign of $\varphi_c$ from the nearby main branch $I$ models. Furthermore, we always observe a sign flip at the high-density end of branch $I$ (around $R_S\approx 8\,\mathrm{km}$ in the figure) but these NS models are unstable; cf. Sec. \[sec:stability\]. []{data-label="positive_negative_1"}](new_29_mass_positive_negative_closeup-eps-converted-to.pdf "fig:"){width=".49\linewidth"} ![Distribution of scalarized NS models based on the sign of $\varphi_c$ in the $M_b-R_S$ plane for $\mu=4.8\times 10^{-13}\,\mathrm{eV}$ and, from top-left to bottom-right, $(\alpha_0,\beta_0)= (10^{-1},-4.5)$, $(10^{-2},-5.5)$,    $(10^{-2},-5)$, $(3\times 10^{-2},-5)$. The orange points represent models with $\varphi_c<0$ whereas the black ones have $\varphi_c>0$. The type $II$ models on the loop differ in the sign of $\varphi_c$ from the nearby main branch $I$ models. Furthermore, we always observe a sign flip at the high-density end of branch $I$ (around $R_S\approx 8\,\mathrm{km}$ in the figure) but these NS models are unstable; cf. Sec. \[sec:stability\]. []{data-label="positive_negative_1"}](new_35_mass_positive_negative_closeup-eps-converted-to.pdf "fig:"){width=".49\linewidth"} When $\alpha_0=0$, our system of equations (\[eq:Phi\])-(\[eq:eta\]) is invariant under the transformation[^5] $\varphi \to - \varphi$. In this case, each strongly scalarized model consists of two solutions that only differ by a minus sign in the scalar-field profile. Additionally to this degenerate scalarized branch, there exists a branch with zero scalar field, i.e. the set of models we also obtain in GR. A nonzero $\alpha_0$ breaks the degeneracy of the strongly scalarized branch which now splits into two branches with unequal macroscopic properties and whose scalar field magnitudes differ at a level $\mathcal{O}(\alpha_0)$. This split is illustrated in Fig. \[split\_starting\] where we consider NS models in the $M_b-R_S$ plane for $\mu = 4.8\times 10^{-13}\,\mathrm{eV}$, $\beta_0=-4.5$ and different value of $\alpha_0$. For $\alpha_0=0$, we see that the scalarized branch directly connects to the GR branch. For $\alpha_0\ne 0$, we obtain a weakly scalarized branch W with $\varphi = \mathcal{O}(\alpha_0)$ in place of the GR models. In terms of their mass $M_b$ and radius $R_S$, however, these models are barely distinguishable from their GR counterparts, and we refer to them as “GR like” models. The strongly scalarized branch S, on the other hand, splits into two, each of them connecting to separate parts of the GR-like branch in such a way that we obtain one loop of models that is separated from the single, large branch; cf. the insets in the top right and bottom panels of Fig. \[split\_starting\]. We find the gap between the loop and the main branch to be proportional to $\alpha_0$ and independent of $\beta_0$. For small but nonzero $\alpha_0$, we thus find two sets of solutions: A branch $I$ that approximately follows the $M_b-R_S$ curve of GR for small and for large central density $\rho_c$, and a separate branch $II$ on a closed loop located in the region where strong scalarization occurs. The central baryon density strictly increases as we move along branch $I$, starting at $(M_b,R_S)=(0,0)$. We note that branches $I$ and $II$ both contain weakly as well as strongly scalarized models. Instead, their distinction arises from the separation of the loop from the main branch of models. As we increase $\alpha_0$, however, the loop of branch $II$ solutions shrinks and eventually disappears, leaving branch $I$ as the only class of solutions. This remaining branch approximately overlaps with the GR family in the very high and low $\rho_c$ regime but shows a strong bulge of strongly scalarized solutions when $\rho_c$ has values comparable to nuclear density. In agreement with the literature, we observe that these scalarized models can reach significantly larger masses and radii than their GR counterparts. We illustrate these features in Fig. \[closed\_contour\] for $\beta_0=-4.5$ and $-5$, but note that this behaviour occurs universally in all cases we have studied. We conclude the discussion of the $\alpha_0$ dependence with a subtle observation we make throughout our computations: For all NS models of branch $II$ the central scalar field value has the same sign; $\varphi_c>0$ for our convention. For the vast majority (though not all) branch $I$ solutions, $\varphi_c$ has the opposite sign; $\varphi_c<0$ in our case. We display this observation graphically in Fig. \[positive\_negative\_1\] for several combinations of $\beta_0$ and $\alpha_0$. We note, however, that branch $I$ always contains a swap in $\mathrm{sign}(\varphi_c)$ at very large central baryon density: we always observe $\varphi_c>0$ as $\rho_c\rightarrow\infty$. ![$M_b-R_S$ diagrams are shown for several values of $\beta_0$ in the regime of spontaneous scalarization $\beta_0<-4.35$. The other scalar field parameters are $\mu=-4.8\times 10^{-13}$ eV, $\alpha_0=10^{-2}$. For increasingly negative values of $\beta_0$, the S branch extends to larger values of the NS radius and baryon mass. []{data-label="various_alpha_0_01"}](various_alpha_0_01-eps-converted-to.pdf){width="0.60\linewidth"} Dependence on beta0 {#sec:beta_dependence} ------------------- ![$M_b-R_S$ diagrams are shown for $\mu=4.8\times 10^{-13}\,\mathrm{eV}$ and $\alpha_0=10^{-4}$, as well as $\beta_0=-15$ (top left), $\beta_0=-17$ (top right) and $\beta_0=-20$ (bottom panel). The color scale measures the central value of $|\varphi|$. This sequence of plots (from top left to bottom right) shows the upper end of the S branch disconnecting and separating from the W branch as $\beta_0$ becomes more negative. []{data-label="plot_within_plot"}](new_1_plot_within_plot-eps-converted-to.pdf "fig:"){width="48.00000%"} ![$M_b-R_S$ diagrams are shown for $\mu=4.8\times 10^{-13}\,\mathrm{eV}$ and $\alpha_0=10^{-4}$, as well as $\beta_0=-15$ (top left), $\beta_0=-17$ (top right) and $\beta_0=-20$ (bottom panel). The color scale measures the central value of $|\varphi|$. This sequence of plots (from top left to bottom right) shows the upper end of the S branch disconnecting and separating from the W branch as $\beta_0$ becomes more negative. []{data-label="plot_within_plot"}](new_13_plot_within_plot-eps-converted-to.pdf "fig:"){width="48.00000%"} ![$M_b-R_S$ diagrams are shown for $\mu=4.8\times 10^{-13}\,\mathrm{eV}$ and $\alpha_0=10^{-4}$, as well as $\beta_0=-15$ (top left), $\beta_0=-17$ (top right) and $\beta_0=-20$ (bottom panel). The color scale measures the central value of $|\varphi|$. This sequence of plots (from top left to bottom right) shows the upper end of the S branch disconnecting and separating from the W branch as $\beta_0$ becomes more negative. []{data-label="plot_within_plot"}](new_12_plot_within_plot-eps-converted-to.pdf "fig:"){width="48.00000%"} ![$M_b-R_S$ diagrams are shown for $\mu=4.8\times 10^{-13}\,\mathrm{eV}$ and $\alpha_0=10^{-4}$, as well as $\beta_0=-15$ (top left), $\beta_0=-17$ (top right) and $\beta_0=-20$ (bottom panel). The color scale measures the central value of $|\varphi|$. This sequence of plots (from top left to bottom right) shows the upper end of the S branch disconnecting and separating from the W branch as $\beta_0$ becomes more negative. []{data-label="plot_within_plot"}](new_22_plot_within_plot-eps-converted-to.pdf "fig:"){width="48.00000%"} Spontaneous scalarization is a non-linear phenomenon and driven by the quadratic coupling parameter when $\beta_0 \lesssim -4.35$. It has already been remarked in [@Ramazanoglu:2016kul], that this threshold value for strong scalarization is barely affected by the introduction of a non-zero scalar mass. Our results confirm this observation as is illustrated by the W and S branches of solutions shown in the left panel of Fig. \[fig:various\] for $\beta_0=-4.5$, $-5$ and $-6$ with fixed $\alpha_0=10^{-4}$ and $\mu=4.8\times 10^{-13}\,\mathrm{eV}$. Note that the transition from weak to strong scalarization along a sequence of NS models, while strictly speaking continuous, is sufficiently abrupt to allow for a clear distinction between models belonging to branch W or S. The three branches displayed in the left panel of Fig. \[fig:various\] for $\beta_0=-4.5$, $-5$ and $-6$ also demonstrate the increasing deviation in terms of mass and radius of branch S models from their GR-like counterparts. Increasingly negative values of $\beta_0$ allow for larger maximum mass and radius; cf. also Sec. IV A in [@Ramazanoglu:2016kul]. This rather strong effect may provide opportunities for constraining $\beta_0$ through mass and radius measurement of NSs, although a reevaluation of the measurements in the framework of ST gravity (rather than assuming GR) will be required for this purpose. In Fig. \[fig:various\], the strongly scalarized branch S models appear as an arc splitting off from the GR-like branch W. We always find branch S to have this qualitative shape and the size of the arc grows monotonically as $\beta_0$ takes on increasingly negative values. This is illustrated in Fig. \[various\_alpha\_0\_01\], which displays branches W and S for several values of $\beta_0 \le -4.5$. This figure also demonstrates that branch S has a shape resembling an inverted ’S’; it initially splits off from branch W towards [*smaller*]{} radii (around $R_S\approx 13.5\,\mathrm{km}$ and $M_b \approx 1\,M_{\odot}$ in the figure) before turning around and crossing branch W towards larger $R_S$. Note also that for each choice of $\beta_0$, branch S consists of two nearby but distinct curves; this splitting results from the relatively large value $\alpha_0=10^{-2}$ as we have already seen in the last section; cf. the bottom right panel in Fig. \[positive\_negative\_1\]. For highly negative $\beta_0$, we obtain NS models with yet larger radius and baryon mass as illustrated in Fig. \[plot\_within\_plot\], where we plot branches W and S for $\beta_0=-15$, $-17$, $-20$, $-25$ and fixed $\alpha=10^{-4}$, $\mu=4.8\times 10^{-13}\,\mathrm{eV}$. In this figure, we also notice a new effect: the upper (in the sense of larger $\rho_c$) end of branch S exhibits a more complex structure. Instead of connecting to branch W, branch S appears to remain separate and curl around; cf. the insets for $\beta_0=-15$ and $\beta_0=-17$. This behaviour becomes clearer for yet more negative $\beta_0$: between $\beta_0=-20$ and $\beta_0=-25$, the intersection of branch S with branch W is lost and instead, branch S forms its own tail of NS models with very small central values of the scalarfield $|\varphi_c|$; note the magenta color of this end of branch S. Contrary to what one might guess, the NS models on this tail of branch S are still [*strongly*]{} scalarized; the profile $\varphi(r)$ merely reaches its maximum away from the center $r=0$. We explore this behaviour in more detail by comparing in Fig. \[fig:frames\] the sequence of models obtained for $\beta_0=-6$ with that for $\beta_0=-17$, keeping $\alpha_0=10^{-4}$ and $\mu=4.8\times10^{-13}\,\mathrm{eV}$ fixed. The bottom panels show the respective families in the $M_b-R_S$ diagram analogous to Fig. \[plot\_within\_plot\]. Along the branches, we have marked several NSs by colored circles, and for these models we plot the baryon-density and scalar-field profiles $\rho(r)$, $\varphi(r)$ in the upper panels[^6]. For $\beta_0=-6$, we observe a simple pattern: At the lower branch point, we obtain a weakly scalarized model (“1”) with comparatively small central baryon density. As we continue along branch S, the scalar field increases in strength, reaching a maximum at maximal radius (model “3”). Beyond that point, the central baryon density $\rho_c$ keeps increasing, but the scalar profile weakens. Eventually (model “6”), a weakly scalarized but highly compact NS marks the smooth reconnection with branch W; note that this weakly scalarized model is located on the unstable branch of the GR-like models. ![The branches of NS models are shown in the $M_b-R_S$ plane in the bottom panels for $\mu=4.8\times 10^{-13}\,\mathrm{eV}$, $\alpha_0=10^{-4}$ as well as $\beta_0=-6$ (left) and $\beta_0=-17$ (right). Several NS models are marked along the branches as colored circles. The top panels show the radial profiles of the baryon density $\rho(r)$ and the scalar field $\varphi(r)$ for these NSs using their respective color. The density profile always reaches a maximum at the origin; however, the scalar field profile in some cases reaches a peak at a non-zero radius. []{data-label="fig:frames"}](_Frames_case_7-eps-converted-to.pdf "fig:"){width="49.00000%"} ![The branches of NS models are shown in the $M_b-R_S$ plane in the bottom panels for $\mu=4.8\times 10^{-13}\,\mathrm{eV}$, $\alpha_0=10^{-4}$ as well as $\beta_0=-6$ (left) and $\beta_0=-17$ (right). Several NS models are marked along the branches as colored circles. The top panels show the radial profiles of the baryon density $\rho(r)$ and the scalar field $\varphi(r)$ for these NSs using their respective color. The density profile always reaches a maximum at the origin; however, the scalar field profile in some cases reaches a peak at a non-zero radius. []{data-label="fig:frames"}](_Frames_case_13-eps-converted-to.pdf "fig:"){width="49.00000%"} The analogous results for $\beta_0=-17$ in the right column of Fig. \[fig:frames\] display a qualitatively similar behaviour near the lower branch point (model “1”); the central baryon-density and scalar-field values increase as we move along branch S (model “2”). Eventually, however, the scalar field profile changes its qualitative behaviour and peaks away from $r=0$ while the central value $\varphi_c$ decreases. In consequence, the upper tail of branch S now consists of models with $\varphi_c \approx 0$ but strong scalarization at $r>0$ and does not directly connect to branch W; compare model “8” for $\beta_0=-17$ with model “6” for $\beta_0=-6$. As branch S curls around, the scalar profile strengthens once again and we encounter models with a steep density cusp (model “9”). We cannot rigorously rule out that after further curling around, branch S might eventually connect with branch W, but our numerical results do not show any signs of this happening. We finally note the remarkable structure of these upper tail stars: A highly compact star of baryonic matter is surrounded by a shell of scalar-field (i.e. bosonic) matter. This structure is reminiscent of the atom like shape noticed for stars in massless ST gravity in [@Brito:2015yfh] and scalarized black holes in modified gravity  [@Baumann:2019eav]. Stability of models {#sec:stability} ------------------- ![Plots showing the distribution of stable (green) and unstable (black) NS configurations in the $M_b-R_S$ plane. When two solutions with the same baryon mass $M_b$ exist, the one with the lower ADM mass is energetically favored. The scalar parameters are $\mu=4.8\times 10^{-13}$ eV, $\alpha_0=-10^{-4}$ and, from top left to bottom right, $\beta_0=-5$, $-5.5$, $-6$ and $-10$. []{data-label="stable models"}](new_10_stable_branch-eps-converted-to.pdf "fig:"){width=".49\linewidth"} ![Plots showing the distribution of stable (green) and unstable (black) NS configurations in the $M_b-R_S$ plane. When two solutions with the same baryon mass $M_b$ exist, the one with the lower ADM mass is energetically favored. The scalar parameters are $\mu=4.8\times 10^{-13}$ eV, $\alpha_0=-10^{-4}$ and, from top left to bottom right, $\beta_0=-5$, $-5.5$, $-6$ and $-10$. []{data-label="stable models"}](new_23_stable_branch-eps-converted-to.pdf "fig:"){width=".49\linewidth"} ![Plots showing the distribution of stable (green) and unstable (black) NS configurations in the $M_b-R_S$ plane. When two solutions with the same baryon mass $M_b$ exist, the one with the lower ADM mass is energetically favored. The scalar parameters are $\mu=4.8\times 10^{-13}$ eV, $\alpha_0=-10^{-4}$ and, from top left to bottom right, $\beta_0=-5$, $-5.5$, $-6$ and $-10$. []{data-label="stable models"}](new_7_stable_branch-eps-converted-to.pdf "fig:"){width=".49\linewidth"} ![Plots showing the distribution of stable (green) and unstable (black) NS configurations in the $M_b-R_S$ plane. When two solutions with the same baryon mass $M_b$ exist, the one with the lower ADM mass is energetically favored. The scalar parameters are $\mu=4.8\times 10^{-13}$ eV, $\alpha_0=-10^{-4}$ and, from top left to bottom right, $\beta_0=-5$, $-5.5$, $-6$ and $-10$. []{data-label="stable models"}](new_6_stable_branch-eps-converted-to.pdf "fig:"){width=".49\linewidth"} ![Same as Fig. \[stable models\] using scalar mass $\mu=4.8\times 10^{-13}$ eV, and coupling parameters from top-left to bottom, $(\alpha_0,\beta_0)=(10^{-3},-4.5)$, $(10^{-2},-5.5)$, $(3\times10^{-2},-5)$. []{data-label="stable models_split"}](new_25_stable_branch-eps-converted-to.pdf "fig:"){width=".49\linewidth"} ![Same as Fig. \[stable models\] using scalar mass $\mu=4.8\times 10^{-13}$ eV, and coupling parameters from top-left to bottom, $(\alpha_0,\beta_0)=(10^{-3},-4.5)$, $(10^{-2},-5.5)$, $(3\times10^{-2},-5)$. []{data-label="stable models_split"}](new_32_stable_branch-eps-converted-to.pdf "fig:"){width=".49\linewidth"} ![Same as Fig. \[stable models\] using scalar mass $\mu=4.8\times 10^{-13}$ eV, and coupling parameters from top-left to bottom, $(\alpha_0,\beta_0)=(10^{-3},-4.5)$, $(10^{-2},-5.5)$, $(3\times10^{-2},-5)$. []{data-label="stable models_split"}](new_35_stable_branch-eps-converted-to.pdf "fig:"){width=".49\linewidth"} In the previous sections, we have seen many cases where for fixed ST parameters $\alpha_0$, $\beta_0$, $\mu$ several equilibrium NS models with equal baryon mass exist; see e.g. $M_b = 2\,M_{\odot}$ in the left panel of Fig. \[fig:various\]. We can analyze the stability of these models by comparing their binding energy. The model with the lowest ADM mass, i.e. strongest binding energy, represents the stable configuration and other models with equal baryon mass are expected to migrate to this configuration under perturbations. We note, however, that the physical relevance of this instability depends on the instability timescale as compared to other dynamical timescales under consideration. Using this method, we classify in Fig. \[stable models\] stable and unstable NS models for several values of $\beta_0$ at fixed $\alpha=10^{-4}$ and $\mu=4.8\times 10^{-13}\,\mathrm{eV}$. The results confirm the theoretical prediction that the weakly scalarized branch W becomes unstable when strongly scalarized counterparts with equal baryon mass exist [@Novak:1998rk]. Note also that the scalarized branch S exhibits a stability structure analogous to the well known GR case: The maximum mass model separates stable from unstable stars and the stable models are those with larger radius. In Fig. \[stable models\_split\], we analyze how the stable and unstable models spread among our “loop” branches $I$ and $II$ of Sec. \[sec:alpha0\] for different values of $\alpha_0$. The stable NSs are the strongly scalarized models with the largest radius, whereas the NSs on branch $II$ (i.e. on the closed loop) are always unstable. As a general pattern in all our computations, we find the models with the strongest central scalar field value $|\varphi_c|$ to be the stable configurations. For our convention, these always turn out to be models with $\varphi_c<0$; for example compare Fig. \[stable models\_split\] with Fig. \[positive\_negative\_1\]. Conclusions {#sec:conclusions} =========== In this study, we have numerically computed solutions of spherically symmetric NSs in massive ST theory using a numerical scheme that enables us to eliminate the exponentially growing modes from the scalar field. For this purpose, we split the domain into the NS interior and the exterior from the stellar surface to infinity and discretize the resulting equations with a second-order relaxation scheme. This method enables us to compute NS spacetimes extending all the way to infinity where we can prescribe the boundary conditions in simple Dirichlet form. This formalism also provides a trivially simple implementation of the matching conditions without the need to perform interpolation. We have used the resulting code to compute solutions of static, spherically symmetric NSs in massive ST theory and explored in detail the structure of the resulting branches of solutions in the (baryon) mass-radius plane for combinations of the linear and quadratic coupling parameters $\alpha_0$, $\beta_0$ of the ST theory and the scalar mass $\mu$. We summarize the main findings of our analysis as follows. - In agreement with previous literature studies of NS equilibrium models in massive and massless ST gravity, we find larger values of $\alpha_0$ and $\beta_0$ to result in larger deviations from the NS solutions in GR, whereas larger values of the scalar mass tend to reduce these deviations; cf. Figs. \[fig:various\] and \[closed\_contour\]. - For $\alpha_0=0$, the NS models of GR are also solutions of the field equations of massive ST gravity. For $\beta\lesssim -4.35$, we find, additionally to the GR branch, the spontaneously scalarized class of NS solutions that Damour & Esposito-Far[è]{}se discovered in their original exploration of massless ST theory [@Damour:1993hw] and that were also identified in massive ST theory in [@Ramazanoglu:2016kul]. These solutions are invariant under the scalar field transformation $\varphi \rightarrow -\varphi$. - A non-zero $\alpha_0$ breaks this degeneracy and results in a dissection of the branches around the branch points; instead of the two connected branches of scalarized and non-scalarized solutions for $\alpha_0=0$, we now find a main branch $I$ and a smaller loop of branch $II$ solutions; cf. Fig. \[split\_starting\]. The solutions on branches $I$ and $II$ are characterized by different signs of the central scalar-field value $\varphi_c$; cf. Fig. \[positive\_negative\_1\]. - For sufficiently negative $\beta_0$, roughly $\beta_0 \lesssim -15$, we observe a qualitative change in the strongly scalarized branch S of solutions. Instead of smoothly approaching the weakly scalarized branch W as happens for milder $\beta_0$, its upper (in the sense of increasing central baryon density) tail now either crosses or completely detaches from the W branch. - For highly negative values of $\beta_0$, we furthermore encounter a new type of strongly scalarized solutions at this upper end of the S branch: the maximum of the scalar field is located away from the stellar center; cf. Figs. \[plot\_within\_plot\], \[fig:frames\]. In its most extreme form, these solutions are composed of highly compact NS models surrounded by a scalar shell; see e.g. [@Brito:2015yfh; @Baumann:2019eav] for similar “gravitational atom” like configurations in other theories of gravity. - Whenever multiple NS models with equal baryon mass exist, we find the scalarized model to be the stable configurations in the sense of minimal binding energy. Typically, though not always, this is the model with the largest radius; cf. Figs. \[stable models\], \[stable models\_split\]. We also observe that the stable configurations agree in the sign of the central scalar field value, $\varphi_c<0$ in our convention. The behavior with respect to the scalar parameters seems to be universal as we have encountered the same $M_b-R_S$ profile deviations with respect to GR for all other equations of state that we have studied. We have explored in a similar manner, though less exhaustively, the cold hybrid EOS1, EOS3 and EOSa [@Rosca-Mead:2020ehn], APR4 [@Akmal:1998cf], 2H and HB [@Read:2009yp] and observe qualitatively similar behaviour. [^1]: We note that alternative, equivalent formulations use instead the function $A=F^{-2}$ and/or replace $\alpha_0$ in terms of a non-zero asymptotic value $\varphi_0$; cf. the discussion in Sec. 3.2 of Ref. [@Gerosa:2016fri]. [^2]: The term “vacuum” here refers to the baryonic matter; the scalar field is nonzero exterior to the star. [^3]: We produce GR solutions with our code by simply setting $\alpha_0=\beta_0=0$. [^4]: The smooth curves in all our mass-radius plots are in fact made up of a large number of crosses which in most cases are not individually visible. We opt [*not*]{} to connect these with lines to avoid spurious cross-branch connections. In consequence, some curves appear to have breaks when the gradient becomes nearly vertical; these breaks are not physical. [^5]: Recall that $\varphi \to -\varphi$ implies $\eta\to -\eta$ and that $F_{,\varphi}$ and $V_{,\varphi}$ are linear in $\varphi$ when $\alpha_0=0$. [^6]: We have selected here exclusively models with $\varphi_c>0$. The small $\alpha_0=10^{-4}$ leads to such a small splitting that the corresponding figure using the models with $\varphi_c$ would be indistinguishable besides the sign reversal in $\varphi(r)$.
--- author: - 'D. J. Mowbray^,^, P. Ayala^^, V. Despoja^^, T. Pichler^^, and A. Rubio^^' title: Theoretical spectroscopy techniques applied to graphene EELS and optics --- Introduction ============ ![Schematic of the orthorhombic graphene unit cell repeated twice in the surface plane. The $x$-direction corresponds to the zigzag direction or circumference of a zigzag SWNT, while the $y$-direction corresponds to the armchair direction or circumference of an armchair SWNT. The $z$-direction is normal to the graphene surface. []{data-label="Graphene_Schematic"}](Fig1){width="0.35\linewidth"} Understanding of the momentum dependent response function is one of the major challenges in solid state spectroscopy. One typical example is the analysis of the two particle excitation spectra as determined by the loss function of inelastic electron scattering. In contrast to direct optical excitations, such electron energy loss spectroscopy gives a direct probe of both collective longitudinal excitations (inter and intra-band plasmons), as well as single particle excitations. The momentum dependence of these plasmon excitations as well as single particle excitation gives a measure of the combined dispersion of the valence and conduction bands. Additionally, one can extract information on the screening and dispersion of free charge carriers in low dimensional systems. To address these questions, we have performed density functional theory (DFT) calculations within the projector augmented wavefunction (PAW) method, and employed the Casida method to calculate the non-interacting density-density response function within the random-phase approximation (TDDFT-RPA), to obtain the loss function $-\Im\{\varepsilon^{-1}(\textbf{q},\omega)$ [@Angel; @Oni2002RMP]. Using this methodology, we have calculated the loss function for graphene along the armchair or $y$-direction, as shown schematically in Fig. \[Graphene\_Schematic\]. Methodology {#Methodology} =========== All DFT calculations were performed using the real-space projector augmented wavefunction (PAW) method code <span style="font-variant:small-caps;">gpaw</span> [@GPAW], with a grid spacing of 0.2 Å, and the PBE exchange correlation (xc)-functional [@PBE]. An electronic temperature of $k_B T \approx$ 0.05 eV was used to obtain the occupation of the Kohn-Sham orbitals, with all energies extrapolated to $T = 0$ K, and one unoccupied band per C atom included to improve convergence. Structural minimization was performed within the Atomic Simulation Environment (ASE)[@ASE], until a maximum force below 0.05 eV/Å was obtained. An orthorhombic 2.46 Å $\times$ 4.26 Å $\times$ 8.00 Å supercell was employed, consisting of four C atoms, as depicted in Fig. \[Graphene\_Schematic\]. Non-periodic boundary conditions were enforced in the $z$-direction normal to the graphene surface, so that both the electron density and Kohn-Sham wavefunctions $\rightarrow $ 0 as $z \rightarrow$ 0 or $z\rightarrow L$, where $L$ is the length of the unit cell in the $z$-direction. A Monkhorst-Pack $k$-point sampling of 25 $k$-points along the zigzag direction, and 15 $k$-points along the armchair direction of the graphene surface was employed, yielding a longitudinal momentum transfer resolution $\Delta q$ of 0.102 Å$^{-1}$ and 0.098 Å$^{-1}$ respectively. 14 unoccupied bands per C atom were converged in the self-consistent calculation, which was ultimately found to be more than sufficient to converge the loss function for energies up to 50 eV, as shown in Fig. \[Fig1\]. Calculations of the loss function have been performed within the Casida methodology [@Angel], employing time dependent density functional theory within the random phase approximation (TDDFT-RPA), as recently implemented within <span style="font-variant:small-caps;">gpaw</span> [@TDDFT]. Within this framework the non-interacting density-density function $\chi_{\textbf{GG}'}^0(\textbf{q},\omega)$ for momentum transfer $\textbf{q}$ at energy $\omega$ is given by $$\begin{aligned} \chi_{\textbf{G}\textbf{G}'}^0(\textbf{q},\omega) &=& \frac{1}{\Omega}\sum_{\textbf{k}}\sum_{n,n'}\frac{f_{n\textbf{k}} - f_{n'\textbf{k}+\textbf{q}}}{\omega + \varepsilon_{n\textbf{k}} - \varepsilon_{n'\textbf{k}+\textbf{q}} + i\gamma}\nonumber\\ &&\times\int_{\Omega_{\textrm{Cell}}}\!\!\!d\textbf{r}\psi^*_{n\textbf{k}}(\textbf{r})e^{-i(\textbf{q}+\textbf{G})\cdot\textbf{r}}\psi_{n'\textbf{k}+\textbf{q}}(\textbf{r})\nonumber\\ &&\times\int_{\Omega_{\textrm{Cell}}}\!\!\!d\textbf{r}'\psi^*_{n\textbf{k}}(\textbf{r}')e^{i(\textbf{q}+\textbf{G}')\cdot\textbf{r}'}\psi_{n'\textbf{k}+\textbf{q}}(\textbf{r}').\label{chi0}\end{aligned}$$ Here the sum is over reciprocal lattice vectors $\textbf{k}$ and band numbers $n$ and $n'$, with $\varepsilon_{n\textbf{k}}$ the eigenenergy of the $n^{\textrm{th}}$ band at $\textbf{k}$, $f_{n\textbf{k}}$ the Fermi-Dirac occupation of the $n^{\textrm{th}}$ band at $\textbf{k}$, $\gamma$ the peak broadening, $\Omega$ the volume of the supercell, $\textbf{G}$ and $\textbf{G}'$ the reciprocal unit cell vectors, and $\psi_{n\textbf{k}}(\textbf{r})$ the real-space Kohn-Sham wavefunctions for the $n^{\textrm{th}}$ band with reciprocal lattice-vector $\textbf{k}$. Including local field effects, we may write the inverse macroscopic dielectric function $\varepsilon^{-1}(\textbf{q},\omega)$ within the random phase approximation (RPA) in terms of the non-interacting density-density response function $\chi_{\textbf{GG}'}^0(\textbf{q},\omega)$ as $$\begin{aligned} \varepsilon^{-1}(\textbf{q},\omega) &\approx& \left.\left[\delta_{\textbf{G}\textbf{G}'} - v_{\textbf{G}}(\textbf{q})\chi_{\textbf{GG}'}^0(\textbf{q},\omega)\right]^{-1}\right|_{\textbf{G} = \textbf{G}' = 0},\label{3}\end{aligned}$$ where $\delta_{\textbf{GG}'}$ is the Kronecker delta, and $v_{\textbf{G}}(\textbf{q})$ is the Fourier transform of the Coulomb kernel. It should be noted that the inclusion of exchange and correlation effects in $v_{\textbf{G}}(\textbf{q})$ at the LDA level adds a minor correction to the present results, as already shown for the case of graphite [@Pichler; @Kramberger08PRL]. As discussed in Ref. [@RadialCutoff], for a 3D periodic system with translational invariance, the Coulomb kernel is $$\begin{aligned} v^{3\textrm{D}}_{\textbf{G}}(\textbf{q}) &=& \iiint d\textbf{r}\frac{e^{i(\textbf{q}+\textbf{G})\cdot\textbf{r}}}{\|\textbf{r}\|} = \frac{4\pi}{\|\textbf{q}+\textbf{G}\|^2}.\end{aligned}$$ However, for a system which is periodic in only two dimensions, such as a bulk slab or graphene, interactions between periodic images in a TDDFT-RPA calculation may be significant due to the long-range behaviour of $v^{3\textrm{D}}$. This will be the case even for systems with sufficient vacuum to converge the electron density at the DFT level. On the other hand, image—image interactions are included at the TDDFT-RPA level only through $v^{3\textrm{D}}$. This motivates us to introduce a 2D periodic Coulomb kernel, $v^{2\textrm{D}}$, which is both translationally invariant and zero for $|z| > R$, where $R$ is the “radial cutoff” for the Coulomb kernel. In this case $$\begin{aligned} v^{2\textrm{D}}_{\textbf{G}}(\textbf{q}) &=& \int_{-R}^R dz\iint dx dy \frac{e^{i (\textbf{q}+\textbf{G})\cdot(\textbf{x}+\textbf{y}+\textbf{z})}}{\sqrt{x^2+y^2+z^2}}\nonumber\\ &=& \frac{4\pi}{\|\textbf{q}+\textbf{G}_\|\|} \int_0^R \cos(G_z z) e^{-\|\textbf{q}+\textbf{G}_\|\|z} dz\nonumber\\ &=& \frac{4\pi\left[1 + e^{-\|\textbf{q}+\textbf{G}_\|\|R}\left[\frac{G_z \sin G_z R}{\|\textbf{q}+\textbf{G}_\|\|} - \cos G_zR\right]\right]}{\|\textbf{q}+\textbf{G}\|^2}\nonumber.\end{aligned}$$ Employing the suggested choice of $R = L/2$ [@RadialCutoff], since $G_z = n \frac{2\pi}{L}$, where $n \in \mathbb{Z}$, we find $$\begin{aligned} v^{2\textrm{D}}_{\textbf{G}}(\textbf{q}) &=& \frac{4\pi}{\|\textbf{q}+\textbf{G}\|^2}\left[1 - e^{-\|\textbf{q}+\textbf{G}_\|\|L/2}\cos n\pi\right].\label{v2D}\end{aligned}$$ From Eqn.  we clearly see that for $L \gg 2/q$ or $q \gtrsim$ 1 Å$^{-1}$, $v^{2\textrm{D}} \rightarrow v^{3\textrm{D}}$. Alternatively, we may introduce further regions of vacuum separating the images directly at the TDDFT-RPA level. Since in the added vacuum regions the matrix elements for the occupied Kohn-Sham wavefunctions are zero, the inclusion of extra vacuum in Eqn.  only enters into the non-interacting density-density response function through the unit cell volume $\Omega$ and the reciprocal unit cell vectors $\textbf{G}$. We may thus introduce extra unit cells of vacuum, or “zero padding” in the non-periodic direction, by doubling or tripling $L$ when computing the set of $\textbf{G}$ vectors to include at the TDDFT-RPA level. In this way, increasing the length of the unit cell in the non-periodic direction through the inclusion of vacuum effectively increases the density of sampling of the reciprocal unit cell. Finally, the quantities of fundamental interest are the loss function $-\Im\{\varepsilon^{-1}(\textbf{q},\omega)\}$, the adsorption or imaginary part of the dielectric function $\Im\{\varepsilon(\textbf{q},\omega)\}$, and the real part of the dielectric function $\Re\{\varepsilon(\textbf{q},\omega)\}$, which may be obtained from Eqn. . Results & Discussion {#Results} ==================== ![Convergence with respect to the number of unoccupied bands $n_\textrm{unocc}$ of the calculated loss function $-\Im\{\varepsilon^{-1}(\textbf{q},\omega)\}$, imaginary part of the dielectric function $\Im\{\varepsilon(\textbf{q},\omega)\}$, and real part of the dielectric function $\Re\{\varepsilon(\textbf{q},\omega)\}$ versus energy $\omega$ in eV for momentum transfer $\|\textbf{q}\| \approx$ 0.1 Å$^{-1}$ along the armchair direction of graphene. Contributions from including $n_{\textrm{unocc}}$ = 1, 2, 3, 4, 5, 6, 7, 8, 10, and 14 unoccupied bands per C atom are shown. A plane-wave cutoff energy $\varepsilon_{\textrm{cut}}$ = 60 eV, corresponding to 89 $\textbf{G}$ vectors was used.[]{data-label="Fig1"}](Fig2){height="4.5in"} ![Influence of neighbouring image interactions on the calculated loss function $-\Im\{\varepsilon^{-1}(\textbf{q},\omega)\}$, imaginary part of the dielectric function $\Im\{\varepsilon(\textbf{q},\omega)\}$, and real part of the dielectric function $\Re\{\varepsilon(\textbf{q},\omega)\}$ versus energy $\omega$ in eV for momentum transfer $\|\textbf{q}\| \approx$ 0.1 Å$^{-1}$ along the armchair direction of graphene, and including $n_{\textrm{unocc}}$ = 8 bands/atom. Results are shown for $\varepsilon_{\textrm{cut}}$ = 15 eV with 8 Å of vacuum (——), 16 Å of vacuum (), and 24 Å of vacuum (), corresponding to 11, 25, and 37 $\textbf{G}$ vectors respectively, and for $\varepsilon_{\textrm{cut}}$ = 60 eV with 8 Å of vacuum, corresponding to 89 $\textbf{G}$ vectors, both with ([****]{}) and without ([**——**]{}) applying a radial cutoff $R$ = 4 Å to the Coulomb kernel.[]{data-label="Fig2D"}](Fig3){height="4.5in"} ---------------------- ------------ ---------------------- ------------ $n_{\textrm{unocc}}$ $\Delta f$ $n_{\textrm{unocc}}$ $\Delta f$ (bands/atom) (%) (bands/atom) (%) 1 66% 6 18% 2 40% 7 16% 3 28% 8 15% 4 23% 10 14% 5 20% 14 12% ---------------------- ------------ ---------------------- ------------ : Percentage error for the $f$-sum rule $\Delta f$, and convergence with the number of unoccupied bands $n_{\textrm{unocc}}$ per C atom.[]{data-label="f-sum-rule"} To test the convergence of the TDDFT-RPA calculated dielectric function with respect to the number of unoccupied bands included within the calculation $n_{\textrm{unocc}}$, we make use of the $f$-sum rule, $$\begin{aligned} \int_0^\infty d\omega \omega \Im\{\varepsilon(\omega)\} &=& \frac{2\pi^2 N_e}{\Omega},\end{aligned}$$ where $N_e$ is the number of electrons in the unit cell of volume $\Omega$. The percentage error in the adherence of the imaginary part of the calculated dielectric function to the sum rule, $\Delta f$, for $n_{\textrm{unocc}} =$ 1 to 14 unoccupied bands per C atom is shown in Table \[f-sum-rule\]. For $n_{\textrm{unocc}} =$ 8 bands/atom, the $f$-sum rule is quite well satisfied, considering that the integral has been truncated at $\omega_{\textrm{max}} = $ 60 eV, and the approximation $d\omega \approx \Delta\omega = $ 0.02 eV. It should also be noted that the $f$-sum rule is not satisfied exactly when using non-local pseudopotentials, although the correction is within the errors given in Table \[f-sum-rule\] [@Oni2002RMP]. ![Convergence with respect to the plane-wave cutoff energy $\varepsilon_\textrm{cut}$ for the calculated loss function $-\Im\{\varepsilon^{-1}(\textbf{q},\omega)\}$ versus energy $\omega$ in eV and momentum transfer $\|\textbf{q}\|$ in Å$^{-1}$ along the armchair direction of graphene, including $n_{\textrm{unocc}}$ = 8 bands/atom, and a radial cutoff $R =$ 4 Å. Results for $\varepsilon_\textrm{cut}$ = 15 (), 30 (), 60 eV (——), and 90 eV ([**——**]{}), corresponding to 11, 27, 89, and 149 $\textbf{G}$ vectors respectively, are shown.[]{data-label="Fig5"}](Fig4){width="\linewidth"} ![Intensity of the calculated loss function $-\Im\{\varepsilon^{-1}(\textbf{q},\omega)\}$ as a function of energy $\omega$ in eV and momentum transfer $\|\textbf{q}\|$ in Å$^{-1}$ along the armchair direction of graphene, with $n_{\textrm{unocc}}$ = 8 bands/atom, $\varepsilon_{\textrm{cut}}$ = 60 eV, corresponding to 89 $\textbf{G}$ vectors, and using a radial cutoff $R$ = 4 Å. []{data-label="Fig3"}](Fig5){width="\linewidth"} As a more direct test, we plot in Fig. \[Fig1\] the loss function and both the real and imaginary parts of the dielectric function for $n_{\textrm{unocc}}$ between 1 and 14 bands per C atom. We consider the limit of low momentum ($\|\textbf{q}\| \approx$ 0.1 Å$^{-1}$ in the armchair direction) since in this limit $\chi_{\textbf{G}\textbf{G}'}^0$ will be rather insensitive to number of $\textbf{G}$ vectors included. Figure \[Fig1\] clearly shows that (1) the dielectric function converges faster than the loss function with respect to $n_{\textrm{unocc}}$, (2) for $n_{\textrm{unocc}} =$ 8 bands/atom we have convergence up to about 50 eV, and (3) with only 2–3 unoccupied bands per C atom the loss function is converged semi-quantitatively up to 10 eV, as shown in the inset. Combined with the values for the $f$-sum rule error in Table \[f-sum-rule\], this suggests that $n_{\textrm{unocc}} = $ 8 bands/atom is sufficient to describe the dielectric response of graphene-like materials up to 50 eV. Further, a near quantitative description of the behaviour of the low energy $\pi$ plasmons is obtained by including only 2–3 unoccupied bands per C atom. However, although both the electron density and Kohn-Sham wavefunction have been set to zero at $z = 0$ and $z=L$ through the use of non-periodic boundary conditions at the DFT level, interactions between periodic images in the $z$-direction are included through the Coulomb kernel in Eqn. . Figure \[Fig2D\] shows how much neighbouring image interactions contribute to the loss and dielectric functions for low momentum transfers. By either applying a radial cutoff of $R=$ 4 Å using $v^{2\textrm{D}}$ or increasing the unit cell length $L$ to 24 Å at the TDDFT-RPA level, we obtain the same converged loss and dielectric functions, with image—image interactions removed. However, as the addition of extra unit cells of vacuum, even at only the TDDFT-RPA level, increases the size of the $\chi_{\textbf{G}\textbf{G}'}^0$ matrix, we shall employ a radial cutoff of $R = $ 4 Å from hereon. Figure \[Fig5\] shows the convergence of the loss function dispersion with respect to the plane-wave energy cutoff $\varepsilon_{\textrm{cut}}$, and the corresponding number of $\textbf{G}$ vectors included in the calculation. As mentioned above, for low to medium momentum transfers ($\|\textbf{q}\| \lesssim$ 0.8 Å$^{-1}$) only a few $\textbf{G}$ vectors (10—30) are required to converge the loss function. Further, even at high momentum transfers ($\|\textbf{q}\| \gtrsim$ 1 Å$^{-1}$), the differences between $\varepsilon_{\textrm{cut}} =$ 15 and 90 eV lie mainly in the intensities rather than the peak positions. This is especially true if we restrict consideration to the perhaps most interesting low energy $\pi$ plasmon regime. The intensity of the converged loss function for graphene is plotted in Fig. \[Fig3\] for $n_{\textrm{unocc}} = $ 8 bands/atom, 89 **G** vectors, and a radial cutoff $R = $ 4 Å. We clearly see the near linearly dispersive $\pi$ plasmon at 5—10 eV, the non-dispersive $\pi$ plasmon at about 5.5 eV, and the dispersing $\sigma+\pi$ plasmon between 15–30 eV, from experiment [@Kramberger08PRL]. We also find that the optical limit $\|\textbf{q}\|\rightarrow 0^+$ for graphene is well reproduced [@PhysRevLett.106.046401]. Conclusions {#Conclusions} =========== In conclusion, we have extended the TDDFT-RPA implementation within <span style="font-variant:small-caps;">gpaw</span> to employ both a radial cutoff of the Coulomb kernel $v^{2\textrm{D}}$ for 2D periodic systems, and include extra unit cells of vacuum at the TDDFT-RPA level. We find these spurious image—image interactions have a significant impact on the calculated loss function for isolated systems at low momentum transfer ($\|\textbf{q}\| \lesssim$ 0.5 Å$^{-1}$), as demonstrated for graphene. We also find for carbon systems, including about 3 unoccupied bands per atom is sufficient to obtain a near quantitative description of the loss function up to 10 eV. These results are particularly important in the area of nanoplasmonics, for the description of the low energy free-charge carrier plasmons induced by electrostatic or potassium doping. Such systems can be well described within a simple RPA model by shifting the Fermi level rigidly, as we will show in future work. D.J.M. acknowledges funding through the Spanish “Juan de la Cierva” program (JCI-2010-08156). We acknowledge funding by Spanish MEC (FIS2010-21282-C02-01), “Grupos Consolidados UPV/EHU del Gobierno Vasco” (IT-319-07), ACI-Promociona (ACI2009-1036), and the Ikerbasque Foundation. The European Theoretical Spectroscopy Facility is funded through ETSF-I3 (Contract Number 211956). 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--- abstract: 'Probabilistic forecasts are becoming more and more available. How should they be used and communicated? What are the obstacles to their use in practice? I review experience with five problems where probabilistic forecasting played an important role. This leads me to identify five types of potential users: Low Stakes Users, who don’t need probabilistic forecasts; General Assessors, who need an overall idea of the uncertainty in the forecast; Change Assessors, who need to know if a change is out of line with expectatations; Risk Avoiders, who wish to limit the risk of an adverse outcome; and Decision Theorists, who quantify their loss function and perform the decision-theoretic calculations. This suggests that it is important to interact with users and to consider their goals. The cognitive research tells us that calibration is important for trust in probability forecasts, and that it is important to match the verbal expression with the task. The cognitive load should be minimized, reducing the probabilistic forecast to a single percentile if appropriate. Probabilities of adverse events and percentiles of the predictive distribution of quantities of interest seem often to be the best way to summarize probabilistic forecasts. Formal decision theory has an important role, but in a limited range of applications.' author: - | Adrian E. Raftery\ University of Washington bibliography: - 'probforecasting.bib' title: Use and Communication of Probabilistic Forecasts --- Introduction ============ Much progress has been made over the past few decades in the development of methods for probabilistic forecasting, and probabilistic forecasts are now routinely used in several disciplines. These include finance, where trading decisions are made based on predictive distributions of assets, often using automated computer trading programs. In marketing, predictive distributions of future sales and inventory are commonly made using statistical models such as ARIMA models [@BoxJenkins2008], and used as the basis for stocking and other decisions. However, in other areas the development of probabilistic forecasting methods is more recent, and use of these methods in practice is at an earlier stage. How should probabilistic forecasts be used and communicated? What are the obstacles to their use in practice? Can these be overcome? Can they be presented in ways that make them more useful to possibly sceptical users? Communicating uncertainty is inherently a challenging problem. @Kahneman2011 identified people’s resistance to uncertainty as > “a puzzling limitation of our mind: our excessive confidence in what we believe we know, and our apparent inability to acknowledge the full extent of our ignorance and the uncertainty of the world we live in. We are prone to overestimate how much we understand the world and to underestimate the role of chance in events. Overconfidence is fed by the illusory certainty of hindsight.” There are various possible explanations for this. One is that people’s cognitive bandwidth is limited, and uncertainty information increases cognitive load. For example, adding a range to a point or “best” forecast triples the cognitive load. A more fundamental explanation is proposed, again by @Kahneman2011: > “An unbiased appreciation of uncertainty is a cornerstone of rationality, but it is not what people and organizations want. Extreme uncertainty is paralyzing under dangerous circumstances, and the admission that one is merely guessing is especially unacceptable when the stakes are high. Acting on pretended knowledge is often the preferred solution.” A related possible explanation arises when forecasters and decision-makers are different people, as is often the case in policy-making contexts. Then the decision-maker may wish to push the responsibility for the decision onto the forecaster, and when the forecasters provides a range or a probabilistic forecast, this is harder to do than when a single number is given. If things go wrong, it’s easier to blame the forecaster who gave an incorrect forecast. In this article, I will describe experience with probabilistic forecasting in five different contexts and try to draw some conclusions. These will lead me to identify five types of potential users of probabilistic forecasts: Low Stakes Users, General Assessors, Change Assessors, Risk Avoiders, and Decision Theorists. Each may have different needs. Some suggestions are that it is important to interact with users and consider their goals; ways of doing this include meetings and web surveys. This is a cognitive problem as well as a statistical one. The cognitive research tells us that calibration is important for trust in probability forecasts, and that it is important to match the verbal expression with the goal. The cognitive load should be minimized to the extent possible, even reducing the probabilistic forecast to a single number if appropriate. Probabilities of adverse events and percentiles of the predictive distribution of quantities of interest seem often to be the best way to summarize probabilistic forecasts. Formal decision theory has an important role in a limited range of applications, particularly when users are aware of their loss functions, and when there is agreement on the loss function to use. This arises most clearly when costs and losses are measured in monetary terms. Decision theory is also useful in research on the use of probabilistic forecasts, to analyze different possible decision rules. This article is organized as follows. In the following sections I will describe experience with five problems where probabilistic forecasting played an important role: setting aboriginal whaling quotas, probabilistic weather forecasting, projecting the worldwide HIV/AIDS epidemic, probabilistic population projections for the UN, and deciding on the number of funded graduate students to admit. I will then discuss what conclusions can be drawn from this experience. Setting Aboriginal Bowhead Whaling Quotas {#sect-iwc} ========================================= For centuries, the Western Arctic stock of bowhead whales, [*Balaena mysticetus*]{}, off the coasts of Alaska and Siberia, has been the object of small-scale subsistence hunting by the Inuit, or Eskimo, peoples of the area, for whom it is vital both nutritionally and culturally; see Figure \[fig-bowhead\]. The stock was severely depleted by commercial whaling by Yankee and European whalers in the late 19th and early 20th centuries. Commercial whaling of bowhead whales (although not other whale species) effectively ended around 1915, and the species was first protected legally from commercial whaling from 1931 by the League of Nations Convention, and then by the International Whaling Commission (IWC), founded in 1946. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![\[fig-bowhead\] Left: Bowhead whale, [*Balaena mysticetus*]{}. Right: Community celebration after Inuit bowhead whale hunt.](Figs/BowheadWhale.jpg "fig:"){height="0.22\textheight"} ![\[fig-bowhead\] Left: Bowhead whale, [*Balaena mysticetus*]{}. Right: Community celebration after Inuit bowhead whale hunt.](Figs/WhaleHunt.jpg "fig:"){height="0.22\textheight"} ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- This left the question of whether and how to regulate aboriginal whaling by the Inuit. It was generally recognized that it would be unfair to penalize the Inuit people for a problem that was not of their making, since they had been whaling sustainably for centuries, and that to ban aboriginal whaling would damage their livelihood and culture. This led to a tension between two conflicting goals: on the one hand, to protect the whale stock and allow it to recover to its pre-commercial whaling levels, and on the other hand to satisfy the subsistence and cultural needs of the Inuit people. The IWC’s solution was to allow continued limited aboriginal subsistence whaling, but with a quota to be set at a level low enough to allow the stock to recover. A key quantity for setting the quota in a given future year is the replacement yield (RY) in that year, namely the greatest number of whales that could be taken without the population decreasing. This is unknown and is subject to considerable uncertainty. Because it is important that the quota not exceed this unknown value, a conservative value or “lower bound” is sought, which should take account of all nonnegligible sources of uncertainty. The future RY has traditionally been forecast using a deterministic population dynamics model, in which births are added and natural deaths and kills are subtracted. This requires as inputs age-specific fertility and natural mortality rates, and outputs the population for each future year, broken down by age and sex. The inputs are unknown and subject to considerable uncertainty. Until 1991, the lower bound was set by doing several runs of the model with different scenarios or variants, consisting of combinations of “central,” “high,” and “low” values of the inputs. The range of values of RY output was then treated as a rough prediction interval. In 1991, however, the IWC Scientific Committee rejected this approach as statistically invalid, noting that it had no probabilistic interpretation and could lead to, for example, decisions that were riskier than they seemed. They recommended that statistically principled methods be developed. In response to this, we developed the Bayesian melding method for making inference about outputs from the population dynamics model, taking account of all known substantial uncertainties about the inputs [@rgz1992; @rgz1995; @PooleRaftery2000]. This yielded a posterior predictive distribution for RY for future years; an example is shown in Figure \[fig-RY\]. ![\[fig-RY\] Posterior Predictive Distribution of the 1990 Replacement Yield of Bowhead Whales, obtained by Bayesian Melding. [*Source:*]{} Raftery, Givens and Zeh (1995).](Figs/RY.pdf){height="0.25\textheight"} Once this was available, the IWC Scientific Committee recommended that the 5th percentile of this distribution be taken as a precautionary lower bound on RY, and thus as an upper bound on the allowed hunting quota. The recommendation was accepted by the Commission itself (consisting mostly of politicians and senior civil servants, such as Fisheries Ministers and officials from the then 40 IWC member countries). Taking account of this lower bound, the desire to allow a margin for future recovery of the stock, and the Inuit subsistence and cultural needs, the Commission set a quota slightly below the lower bound. This approach was used successfully for the following ten years. Over that period, the bowhead whale stock prospered, indeed increasing substantially, while the Inuit whale hunt continued and the related Inuit culture was preserved. The basic statistical ideas have since been used for other wildlife management problems . The 5th percentile of the posterior predictive distribution effectively became the “point forecast” for this problem. To calculate it, it was necessary to compute the full posterior distribution. But once the 5th percentile had been calculated and agreed as valid by the IWC Scientific Committee, most of the policy-making attention focused on it, and the rest of the distribution (including measures of its center such as the median or mode) was largely ignored. Thus the cognitive load was no larger than for a single “best” forecast. Also, the responsibility for making a single best forecast had been met by the forecasters (in this case the IWC Scientific Committee) — only in this case it was a lower bound rather than a predictive median or mode, or a deterministic point forecast. Probabilistic forecasts were important in this application because the first priority was to limit the risk of an adverse outcome, namely a decrease of the whale stock. Note that formal decision theory was not used in this problem. The IWC Scientific Committee has considered using formal decision theory for such problems, but in general has not done so, because they considered that reaching agreement on the relative costs involved was not feasible. For example, what is the ratio of the cost to the stock of killing a whale to the benefit to the Inuit community? Consensus on the answers to questions like these would be hard to achieve . Instead, the preferred approach was to set the quota so that the risk of the stock decreasing as a result would be no more than 5%, and this eventually commanded broad agreement, even in a body where debates have often been contentious because of the environmental sensitivities associated with whaling. Probabilistic Weather Forecasting {#sect-pwf} ================================= Methods and Probcast Website {#sect-pwf.probcast} ---------------------------- Probabilistic weather forecasts consist of predictive probability distributions of future weather quantities. In particular they yield probabilities of future adverse weather events, such as freezing temperatures, high rainfall or wind storms. Since 1992, probabilistic weather forecasts have been produced by major weather forecasting agencies using ensembles of deterministic numerical weather predictions [@GneitingRaftery2005]. However, these have been little used as the basis for public forecasts, because they are typically poorly calibrated. In response to this situation, methods for postprocessing ensembles to produce calibrated probabilistic weather forecasts have been developed, based on statistical methods, including ensemble Bayesian model averaging and ensemble model output statistics (EMOS) . In addition to temperature, methods were developed for precipitation , wind speeds , wind directions , wind vectors , and visibility [@ChmieleckiRaftery2011]. Based on these forecasts, we set up a prototype real-time probabilistic weather forecasting website for the general public in the North American Pacific Northwest, at [www.probcast.com]{} ; see Figure \[fig-probcast\]. Its design and content were based on extensive cognitive experiments and ethnographic studies of forecasters and end-users . ![\[fig-probcast\] Screenshot of the Probabilistic Weather Forecasting Website at [www.probcast.com]{}.](Figs/Probcast20140613zoom2.pdf){width="\textwidth"} The website contains three kinds of information. First are percentiles of decision-critical weather quantities, namely temperature and the amount of precipitation. The 10th, 50th and 90th percentiles of future temperature are given. For precipitation the (upper) 90th percentile is given. The second kind of information consists of probabilities of adverse weather events of common interest, namely freezing temperatures, precipitation (defined as more than 0.01 inches in the 12-hour period of the forecast), heavy precipitation (defined as more than 0.25 inches), and very heavy precipitation (defined as more than 1 inch). When the probabilities are below 5%, these fields are left blank. The third kind of information consists of maps of any of the percentiles or probabilities in the upper part of the web page, showing how they vary over the spatial domain. The kinds of display used were chosen on the basis of cognitive experiments. For example, to choose the icon representing probability of precipitation seen in Figure \[fig-probcast\], cognitive experiments were carried out to compare the relative effectiveness of several kinds of icon [@JoslynNadav-Greenberg2009]. Three of the icons are shown in Figure \[fig-icons\]: a question mark icon, a pie icon, and a bar icon. In the question mark icon, higher probability of precipitation is represented by darker colors. The pie icon produced the fewest misunderstandings among study participants and so was used on the Probcast website. ![\[fig-icons\] Icons used in cognitive experiments to compare the relative effectiveness of different icons for probability of precipitation: question mark icon (where the icon is darker when the probability is higher), pie icon and bar icon. [*Source:*]{} @JoslynNadav-Greenberg2009.](Figs/icons.pdf){width="\textwidth"} On the Probcast website we gave the 10th and 90th percentiles of temperature, corresponding to an 80% prediction interval. There is a trade-off in choosing the default probability levels to display: larger intervals (e.g. the 95% interval) contain a higher proportion of actual outcomes, but they are also much wider, and hence may be judged less useful. In the event, we received almost no requests for higher probability level intervals, and so we stuck with the 80% intervals. It would of course be possible to display multiple probability levels, but this would add to the cognitive load and so make the website harder to use. Cognitive Findings {#sect-pwf.cognitive} ------------------ An important part of the probabilistic weather forecasting project consisted of carrying out cognitive experiments to determine how best to convey the uncertainty information. There is a long tradition in psychology of assessing people’s understanding of probability and uncertainty by offering them simple gambles [@KahnemanTversky1984], but less research on how best to communicate uncertainty about complex real-life outcomes. Calibration of the probability forecast (e.g. 80% prediction intervals contain the truth 80% of the time on average) is an important requirement for probabilistic forecasts . One series of experiments showed that providing calibrated probability forecasts improve weather-related decision-making and increases trust in the forecast . This is good news for probabilistic forecasting, showing that ordinary people can understand and use probabilities to improve their decision-making. @JoslynSavelli2010 found that users of standard (deterministic) weather forecasts have well-formed uncertainty expectations, suggesting that they are prepared to understand explicit uncertainty forecasts for a wide range of parameters. Indeed, explicit uncertainty estimates may be necessary to overcome some of the anticipated forecast biases that may be affecting the usefulness of existing weather forecasts. Despite the fact that these bias expectations are largely unjustified, they could lead to adjustment of forecasts that could in turn have serious negative consequences for users, especially with respect to extreme weather warnings. reported on a series of experiments to investigate the effects of various aspects of how probability forecasts are presented: framing (positive versus negative), format (frequency versus probability), probability (low versus high), and compatibility between the presentation and the decision task. They showed that the key factor is the match between the verbal expression and the task goal. The other three factors (framing, format and probability) made a much smaller difference. In one experiment, people were asked to decide whether or not to post a wind advisory for winds above 20 knots, and were given probability information. When people were told the probability that wind speed would be above 20 knots, they made few errors. However, when they were told the probability that wind speed would be below 20 knots, they made far more errors, even though the information is mathematically equivalent. This indicates that when the verbal expression and the task were mismatched, more errors were made. Another series of experiments was carried out to assess whether it was better to present probability forecasts in terms of probability (e.g. 10% chance) or frequency (e.g. 1 time in 10). It has been argued that uncertainty presented as frequency is easier for people to understand [@Fiedler1988; @HertwigGigerenzer1999]. However, @JoslynNichols2009 found that people better understood the forecast when it was presented in probability format rather than a frequency format, in contrast with the earlier research. This is more good news for probabilistic forecasting, indicating again that ordinary people can understand probabilities. Assessment ---------- Overall, the Probcast website has been reasonably successful, attracting about two million unique visits since it was set up in 2005 [@Jones2011]. Public probabilistic weather forecasting (beyond probability of precipitation, which has been issued by the U.S. National Weather Service for about 40 years) is now being considered and evaluated by several national and other weather agencies, and Probcast provides both a methodology for producing calibrated probabilistic forecasts and a model of how they might be communicated to the public. It has also been cited by @NRC2006 as a possible model for communication of uncertainty in weather forecasting. While specialists sometimes argue that the public doesn’t understand probabilities and so that there’s little point in issuing probabilistic forecasts, the research results from the Probcast project suggest otherwise. The cognitive results indicate that users are ready for explicit uncertainty statements in forecasts, and that including them can improve decision-making and increase trust in the forecast. The fairly wide public use of the Probcast website, in spite of its lack of substantial institutional backing and its narrow geographic range (the North American Pacific Northwest), suggest that the public is ready for probabilistic forecasts on a broader scale, although of course only a portion of the public would actively use them (notably those with higher-stakes weather-related decisions to make). The cognitive experiments carried out as part of our project by Susan Joslyn’s research group at the University of Washington suggest that probabilities of particular adverse weather events (e.g. freezing temperatures, precipitation, heavy precipitation, high winds), and percentiles (10th, 50th, 90th) of the predictive distribution of continuous weather quantities of interest (e.g. temperature, amount of precipitation, wind speed) are useful quantities to provide to users [@SavelliJoslyn2013]. The work suggests that both are understandable to people and that they make better decisions when they have this information. A common prescription is that probabilities should be used in decision-making using decision theory [@vonNeumannMorgenstern1947]. This says that each possible outcome imposes a loss on the decision-maker, and that the decision made should minimize the expected overall loss. In this case the expectation would be taken over possible future weather outcomes, and the losses might relate, for example, to the costs of issuing a high wind warning if no high winds occur, and to the damage that high winds would cause in the absence of a warning. This seems to be a very useful framework when the utilities associated with different outcomes can be quantified on the same scale, typically money. The clearest weather example that I know of is decision-making by wind energy companies that have to bid for contracts to provide specified amounts of energy at given prices and with specified penalties for failing to fulfil the contract, in the presence of great uncertainty about future wind speeds . However, we did not incorporate decision-theoretic concepts explicitly into the Probcast website. It seems that most people are unaware of their utility functions, and may even be unwilling to specify them when the losses involved are on different scales (e.g. money versus possible loss of life). Thus people may find it easier to use probabilistic forecasts to make decisions that limit the risk of adverse outcomes to acceptable levels, rather than carrying out a full decision-theoretic analysis. Nevertheless, [@JoslynLeclerc2012] showed that when costs and benefits are on the same scale (e.g. money), while people do not match the optimal decision-making standard, they are closer to it when they have probabilistic information. [@JoslynLeclerc2012] also found that if people were given decision advice based on optimal decision-theoretic calculations, they followed the advice only if they were also given the probabilities. Projecting the HIV/AIDS Epidemic {#sect-unaids} ================================ The Joint United Nations Programme on HIV/AIDS (UNAIDS) publishes updated estimates and projections of the number of people living with HIV/AIDS in the countries with generalized epidemics every two years. Generalized epidemics are defined by overall prevalence being above 1% and the epidemic not being confined to particular subgroups; there are about 38 such countries . UNAIDS projections are typically provided for no more than five years into the future. As part of this, statements of uncertainty are also provided. This exercise has two main goals. The first is to develop estimation and projection methods and software for use by country health officials for planning, for example to meet future medication needs. There, statements of uncertainty may be used, for example, for determining the amount of medication needed to be reasonably sure of having enough to meet the need; this would correspond to an upper percentile of the predictive distribution. The second goal is to contribute to the basis for the UNAIDS annual reports [@UNAIDS2013]. Uncertainty statements about estimates are routine in the UNAIDS reports, perhaps because UNAIDS is a newer agency, established in 1996, by which time it had become the norm to include uncertainty measures of some kind with estimates of uncertain quantities. See Figure \[fig-unaids\](a) for an example. While the uncertainty statements do not feature prominently in the published report for the broad public, they underlie assessments in the report such as the following: > “The annual number of new HIV infections among adults and adolescents decreased by 50% or more in 26 countries between 2001 and 2012. However, other countries are not on track to halve sexual HIV transmission, which underscores the importance of intensifying prevention efforts.” The phrase “not on track” reflects conclusions drawn in part from probabilistic projections. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![\[fig-unaids\] Uncertainty Statements in Estimates and Projections of the HIV/AIDS Epidemic. Left: Estimates of global number of new HIV infections, 2001–2012, with uncertainty bounds, from UNAIDS’ main annual public statement. [*Source:*]{} @UNAIDS2013. Right: Estimates and projections of HIV prevalence in Gabon, 1970–2015, based on antenatal clinic data up to 2009; the bands for 2010–2015 summarize probabilistic projections. The results are shown for two different models, EPP and R-flex, and the observed prevalence at individual clinics is shown by unfilled circles. [*Source:*]{} . ](Figs/HIVinfections.pdf "fig:"){width="45.00000%"} ![\[fig-unaids\] Uncertainty Statements in Estimates and Projections of the HIV/AIDS Epidemic. Left: Estimates of global number of new HIV infections, 2001–2012, with uncertainty bounds, from UNAIDS’ main annual public statement. [*Source:*]{} @UNAIDS2013. Right: Estimates and projections of HIV prevalence in Gabon, 1970–2015, based on antenatal clinic data up to 2009; the bands for 2010–2015 summarize probabilistic projections. The results are shown for two different models, EPP and R-flex, and the observed prevalence at individual clinics is shown by unfilled circles. [*Source:*]{} . ](Figs/GabonUrbanPrevalence.pdf "fig:"){width="45.00000%"} ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- We developed methods for assessing uncertainty about estimates and projections using Bayesian melding . One example of the output is shown in Figure \[fig-unaids\](b). Figures such as these typically do not make their way into the most visible public reports; instead they provide background support for the conclusions presented in these reports. There seem to be two main kinds of use for the probabilistic estimates and projections developed by UNAIDS. The first is to provide a general assessment of the estimates and projections and how accurate they are likely to be. The second kind of use is to assess changes. For example, reported HIV prevalence might increase in a given year, but the question then arises whether the increase is out of the range of normal expections, perhaps warranting some new policy intervention. Probabilistic forecasts such as those summarized by the uncertainty bands in Figure \[fig-unaids\](b) can be useful in this context. For example, if the new estimated prevalence is inside the range of the projection (even if there is an increase), then there is little evidence that what is happening is out of what could be expected in the normal run of things, and the chosen policy could be to continue as before while monitoring the situation. On the other hand, if the new estimate is outside the projected range, there may be grounds for concern and for an intervention. Probabilistic Population Projections for the United Nations {#sect-ppp} =========================================================== The United Nations (UN) publishes projections of the populations of all countries broken down by age and sex, updated every two years in a publication called the [*World Population Prospects*]{} (WPP). It is the only organization to do so. These projections are used by researchers, international organizations and governments, particularly those with less developed statistical systems, and researchers. They are used for planning, social and health research, monitoring development goals, and as inputs to other forecasting models such as those used for predicting climate change and its impacts . They are the de facto standard [@LutzSamir2010]. Like almost all other population projections, the UN’s projections are produced using the standard cohort-component projection method . This is a deterministic method based on an age-structured version of the basic demographic identity that the number of people in a country at time $t+1$ is equal to the number at time $t$ plus the number of births, minus the number of deaths, plus the number of immigrants, minus the number of emigrants. The UN projections are based on assumptions about future fertility, mortality and international migration rates; given these rates the UN produces the “Medium” projection, a single value of each future population number with no statement of uncertainty. The UN also produces “Low” and “High” projections using total fertility rates (the average number of children per woman) that are, respectively, half a child lower and half a child higher than the Medium projections. These are alternative scenarios that have no probabilistic interpretation. Like the UN up to 2008, most national statistical offices, including the U.S. Census Bureau and the U.K. Office of National Statistics, use assumptions about future fertility, mortality and migration rates from experts: either internal experts or panels of outside experts. Expert knowledge is an essential part of the population projection process, and experts are generally agreed to be good at assembling and reviewing the underlying science, as well as assessing the actual forecasts. However, evidence has been mounting over the past 60 years that experts in several domains are less good at producing forecasts themselves from scratch. @Meehl1954 found that very simple statistical models beat expert human forecasters overall in a range of clinical disciplines, and this finding has been replicated in many subsequent studies [@Meehl1986]. @OeppenVaupel2002 showed that expert forecasts of life expectancy at birth, both by leading demographers and forecasting organizations, had performed poorly over the previous 70 years. Forecasters generally tended to project that the future would be like the present, and in particular that a limit to life expectancy would be reached soon, whereas in fact life expectancy continued to increase throughout the period. @Tetlock2005 evaluated the quality of about 3,000 forecasts of political events and outcomes by experts, many highly distinguished, and found their performance to be startlingly poor. He memorably concluded that many of the experts would have been beaten by a “dart-throwing chimpanzee.” In a rare counterexample, @MandelBarnes2014 found that analysts in a Canadian intelligence agency provided calibrated forecasts of good quality. In collaboration with the UN Population Division, we developed new statistical methods for projecting future fertility and mortality rates probabilistically, and translating these into probabilistic population projections for all countries (Alkema et al., 2011; Raftery et al., 2012, 2013; Fosdick and Raftery 2014; Raftery, Lalic et al., 2014; Raftery, Alkema and Gerland, 2014). An experimental version of the new probabilistic projections was issued by the UN in November 2012, at http://esa.un.org/unpd/ppp. This release was accompanied by no fanfare, but the experimental probabilistic projections have still had about 10,000 downloads per month. Official UN probabilistic population projections for all countries were issued for the first time on the same website on July 11, 2014 (World Population Day). There are other indications of the beginning of a paradigm shift from deterministic population projections based on expert assumptions to probabilistic population projections based on statistical models. Statistics New Zealand changed its official population projection method to probabilistic projections in 2012 [@Bryant2014]. But these releases are recent, and it remains to be seen how and to what extent ultimate users, such as policy-makers and planners, make use of them. One possible use is in setting future international goals, similar to the Millenium Development Goals for 2015 for things like child and maternal mortality. It is desirable to set goals that are ambitious but also realistic, and probabilistic projections could be useful in indicating what is realistic, suggesting setting goals that are towards the “good” end of the probability distribution [@Gerland2014]. A possible use of probabilistic population projections is in making decisions about policy issues that depend directly on future population numbers, such as school and hospital infrastructure. One such decision is whether or not to close schools. These decisions are often based on deterministic population projections, which can have a spurious air of certainty. It is not desirable to close a school unless the probability of having to reopen it or find other premises in the future is small [@Louis2012]. Even if a deterministic population projection points to school enrollments declining, there can still be a substantial probability of them staying essentially constant or even increasing, in which case closing the school would typically not be a good idea. Basing such decisions on reasonable upper percentiles of future school enrollments (such as the 90th percentile), rather than on a deterministic projection or a predictive mean or median, could be a reasonable approach. Conditional Probabilistic Forecasts: How Many Graduate Students to Admit? {#sect-gradadm} ========================================================================= Like most U.S. academic departments with graduate programs, the Department of Statistics at the University of Washington, of which I am a faculty member, faces the problem of deciding how many potential entering graduate students to make funded offers to for the next academic year. Offers are made in December for entry in the following September, nine months later, and are binding on the department. Entering graduate students are funded by a mix of teaching assistantships, fellowships and research assistantships. There are several major uncertainties to deal with in making this decision. The number of research assistantships available depends on the outcome of faculty research grant applications, which are often unknown nine months ahead of time. Not all students accept our offers, and we do not know ahead of time how many will. We also do not know exactly how many current students will leave in the next nine months through graduation or dropout. Up to 2009, departmental practice was to make a number of offers based on expected numbers of students graduating and grants, and on an assumed acceptance rate. However, these calculations were based only on expectations and were not probabilistic, and also did not incorporate past data in a systematic way. This often led in practice to too few acceptances relative to the number of positions available, with the result that teaching assistants for Statistics courses had to be recruited from among non-Statistics graduate students. This was undesirable in that statistics teaching was not being done by optimally qualified people, departmental teaching assistantships were “lost” to other departments, and the pool of future potential research assistants was depleted. Also, there are currently more jobs available for Ph.D. statisticians that graduates, so increasing the number of entering graduate students is desirable from the labor market point of view as well. In the five years up to 2009, about four teaching assistantships were being “lost” to the department every year, compared with a typical incoming class size of about ten graduate students. The downside is that if students accept and there is no identified funding for them, the deparment has to scramble to find funding. This is difficult but possible within the university, because many non-Statistics departments have research and teaching needs for statistically qualified people that they find it hard to meet from within their own pool of students. In 2010, the departmental faculty decided to base the decision about the number of students to admit on a probabilistic calculation instead of the then current expectation-based approach, and I took on the task of developing the appropriate method. For each possible number of offers, I computed the predictive probability distribution of the number of TA positions lost to the department, as this seemed to be the key quantity for decision-making. Ideally this would be equal to zero. With perfect knowledge, the number of TA positions lost conditional on a given number of offers is equal to $$Y = T + R_1 + R_2 + G + L + D - C - A, \label{eq-lostTA}$$ where $$\begin{aligned} Y &=& \mbox{ Number of TA positions lost to department } \\ T &=& \mbox{ Number of TA positions available} \\ R_1 &=& \mbox{ Number of RA positions available within the department} \\ R_2 &=& \mbox{ Number of RA positions available outside the department} \\ G &=& \mbox{ Number of students graduating by September} \\ L &=& \mbox{ Number of students dropping program by September} \\ C &=& \mbox{ Number of current students} \\ A &=& \mbox{ Number of acceptances}.\end{aligned}$$ $T$ and $C$ are taken as known exactly, but the other quantities in equation (\[eq-lostTA\]) are uncertain at the time when the decision has to be made. The predictive distributions of $R_1$, $R_2$, $G$, $L$, and $A$ are derived from past data and elicited information. They are treated as independent in order to derive a joint distribution. The predictive distribution of $A$ depends on the number of offers, $O$, and is modeled as Binomial $(O,\pi)$, where $\pi$ is estimated from historical data. The predictive distribution of $R_1$ is obtained by polling departmental faculty to elicit from each of them a predictive distribution of the number of research assistantships they will have available in the next academic year. The distribution of $R_1$ is then the distribution of the sum of the numbers from faculty, obtained by convolving the elicited distributions. The predictive distributions of $R_2$ and $L$ are based on historical data on these quantities; empirical rather than model-based distributions are used. The predictive distribution of $G$ is based on current information about student progress and is typically quite tight. The predictive distribution of $Y$, the number of lost TA positions, which is the primary quantity for decision-making, is then obtained by Monte Carlo. A large number of values of each of $R_1$, $R_2$, $G$, $L$, and $A$ are simulated from their predictive distributions, and the corresponding simulated values of $Y$ are found from equation (\[eq-lostTA\]). Figure \[fig-GradAdm12\] shows conditional predictive distributions of the number of lost TA positions given several possible number of offers, and Table \[tbl-GradAdm12\] shows percentiles of these distributions. Note that negative numbers correspond to the number of students that would not be funded with current funding sources. In these cases, alternative funding sources would be sought, such as research or teaching assistantships in departments that currently fund few or no statistics graduate students. ![\[fig-GradAdm12\] Conditional Probabilistic Forecasts of the Number of Lost Teaching Assistant Positions given Different Numbers of Graduate Student Offers with Funding. Negative values indicate the number of students that would not be funded from current funding sources.](Figs/GradAdm12.pdf){width="\textwidth"} \#Offers 10% 33% 50% 67% 90% Description ---------- ----- ----- ----- ----- ----- ------------------- 12 0 2 3 5 7 Very conservative 17 -3 0 1 2 5 Conservative 20 -4 -2 0 1 4 Break-even 23 -6 -3 -2 0 3 Bold 30 -9 -6 -5 -3 0 Very bold : \[tbl-GradAdm12\] Percentiles of the Predictive Distributions of the Conditional Probabilistic Forecasts of the Number of Lost Teaching Assistant Positions given Different Numbers of Graduate Student Offers with Funding. Negative values indicate the number of students that would not be funded from current funding sources. The verbal descriptions in Table \[tbl-GradAdm12\] characterize how aggressive a decision is relative to the uncertainty. For example, 20 offers is the break-even point, because with that number the department is equally likely to lose TA positions as to have to seek additional funding sources. Similarly, 23 offers is described as “bold” because there is only one chance in three of losing TA positions, but a larger chance of having to seek additional positions. Given these numbers, the then department chair decided to take a “bold” stance and make 23 offers. Under the previous system fewer offers would likely have been made. In the event, the department was able to fund the students who accepted the offers quite comfortably, so that the bold stance turned out well. The previous expectation-based more conservative approach could have led to several TA positions being lost to the department, as in the preceding years. The probabilistic approach made it possible to go beyond the break-even point, and to quantify the risk in so doing, thus helping the decision-maker to decide how far beyond the break-even point to go. Given this successful outcome, the department decided to continue to use this approach, which has now been used in four successive years. The decision to be made in this case involves trading off losses of different kinds (lost TA positions against the possible need to seek additional funding sources outside the department, which could be difficult and stressful). Thus it would seem like a possible candidate for formal decision analysis, especially given that the decision-makers are trained statisticians. Nevertheless, a loss function was not assessed at any point, and decision theory was not used; the predictive distributions by themselves provided enough information to the decision-maker. After the fact, it seems possible to argue that the decision-maker was using a loss function under which losing a TA position was twice as bad as having to find funding for an additional student outside current sources, but if so this was never explicitly articulated. It would be possible to improve the statistical model used for generating the probabilistic forecasts. For example, the students the department ranks most highly for funding typically are less likely to accept the offer, because they often have more options. However, the model assumes that all students with an offer are equally likely to accept it; it would be possible to relax this assumption. Also, a second round of offers is sometimes made, depending on initial responses to the first round of offers. It would be possible to extend the model to include the second round, about which decisions are currently made without similar quantitative analysis. But overall, the method seems developed enough to provide useful guidance to the decision-makers, and there has not yet been a strong demand for further methodological refinement. Discussion ========== I have described five cases in which probabilistic forecasts have been used with a certain degree of success. These lead me to identify five types of potential user of probabilistic forecasts (where the five cases don’t map exactly onto the five types of user): 1. Low Stakes User: This is a user for whom the stakes are low and/or the losses from over- and underpredicting are similar. An example might be someone deciding whether to wear a sweater or a short-sleeved shirt based on temperature; a single “best” temperature forecast will often be enough in this case. 2. General Assessor : This is a user for whom the probabilistic forecast provides a general assessment of the likely quality of the forecast. The UNAIDS annual report is a possible example. This is important also for the process of forecast improvement. The goal of forecast development should be to improve forecast accuracy, and hence to reduce the uncertainty around the forecast [@SonejiKing2012]. It is hard to guide this process without an accurate assessment of forecast uncertainty. 3. Change Assessor: For this kind of user, the probabilistic forecast provides a way of assessing whether a change in some measurement is in line with expectations, or instead is a source of concern warranting action. An example might be the probabilistic forecasts of HIV prevalence produced by UNAIDS, where some changes (including increases) are to be expected, but larger increases that are “significant” would sound an alarm. One-number forecasts provide no way of making this kind of assessment. 4. Risk Avoider: Here the goal includes keeping the risk of an adverse outcome to an acceptable level. The IWC bowhead whale quota is a good example of this, in which the risk of possible damage to the stock from aboriginal whaling was to be kept to a low level. Note that this did lead to a “one number” forecast, but the forecast was not the “best” or “central” forecast, but rather a lower percentile of the predictive distribution, in this case the 5th percentile. 5. Decision Theorist: This user has an explicit loss function and is able to quantify it. He or she uses the probabilistic forecast to explicitly minimize expected loss, as advocated by formal decision theory. This did not arise in any of the cases I described, and seems most likely when the different kinds of possible loss being traded off are on the same scale, typically money. One example would a wind energy company, which needs to bid on a contract to supply a given amount of energy, with specified penalties if the contract is not fulfilled . The fact that there are different types of user and use of probabilistic forecasts suggests that it is important for developers of probabilistic forecasts to interact with users and consider their goals. While this may seem obvious, it is often not done. Interaction can take the form of direct contact (meetings, phone, email and so on) between developers and users. This can be in the context of an established scientific advisory committee with regular meetings and an official membership (as used by the IWC), or a small less formal reference group with rotating members (as used by UNAIDS), or expert group meetings, which are effectively workshops lasting several days (as used by the UN Population Division). If the probabilistic forecasts are delivered to the general public using a website (as in the case of probabilistic weather forecasting), the interaction can take the form of a web survey [@JoslynSavelli2010]. It is important for trust in the forecast that the probabilistic statements be at least approximately calibrated, so that, for example, events given predictive probability 80% happen about 80% of the time on average. For the forecast to be useful, it is also important that forecast intervals be narrow, or sharp, enough to provide a basis for action. Indeed, defined the key design principle of probabilistic forecasting as being to maximize sharpness subject to calibration, and this has been widely accepted. The experience I have described suggests that formal decision theory, much advocated in theory by statisticians and economists, may have less practical application than sometimes claimed. One reason may be that people are often not aware of their loss functions. Another may be that using formal decision theory greatly increases the cognitive load, in that one’s loss function has to be assessed and then the decision theoretic calculations performed. One also needs to be careful because in practice people tend to attribute different values to equivalent losses and gains, contrary to decision theory, a finding referred to as “prospect theory” [@KahnemanTversky1979; @TverskyKahneman1992]. Nevertheless, a recent result suggests that the scope of decision theory may be wider than I have conceded. @Gneiting2011 showed that if the loss function is generalized piecewise linear as a function of the quantity being predicted probabilistically, then the optimal point forecast is a quantile of the predictive distribution. An important special case of this is when the cost of an overestimate is a fixed multiple of the cost of an underestimate. This will often be at least roughly true, and it may be much easier to elicit that multiple than the full loss function. People may be able to say, at least approximately, how much worse an overestimate is than an underestimate, or vice versa. This also greatly simplifies the practical use of decision theory, reducing it to the calculation of a predictive quantile, so that the cognitive load is little greater than that of probabilistic forecasting by itself. One overarching conclusion is that people can use and understand probabilities and probabilistic forecasts, even if they do not have advanced training in statistics. The cognitive research shows that probabilistic forecasts lead to better decision-making than deterministic ones, and also to increased trust in the forecast by users. Experience with probabilistic weather forecasting and probabilistic population projection websites has shown that there is considerable public interest in probabilistic forecasts, even in the absence of much publicity. This suggests that once probabilistic forecasts become available in a domain, they will be used: “Build it and they will come.” #### Acknowledgements: This work was supported by the Eunice Kennedy Shriver National Institute of Child Health and Development through grants nos. R01 HD054511 and R01 HD070936, and by a Science Foundation Ireland E. T. S. Walton visitor award, grant reference 11/W.1/I2079. The author is grateful to Geof Givens, Susan Joslyn, Giampaolo Lanzieri and Elizabeth Thompson for helpful comments and discussions, and to Nial Friel and the School of Mathematical Sciences at University College Dublin for hospitality during the preparation of this paper.
--- author: - 'J.L. Raath, M.S. Potgieter, R.D. Strauss, A. Kopp' title: The effect of magnetic field modifications on the modulation of cosmic rays in the heliosphere --- Abstract {#abstract .unnumbered} ======== A numerical model for the solar modulation of cosmic rays, based on the solution of a set of stochastic differential equations, is used to illustrate the effects of modifying the heliospheric magnetic field, particularly in the polar regions of the heliosphere. To this end, the differences in the modulation brought about by each of three choices for the heliospheric magnetic field, i.e. the unmodified Parker field, the Smith-Bieber modified field, and the Jokipii-Kóta modified field, are studied. It is illustrated that both the Jokipii-Kóta and Smith-Bieber modifications are effective in modifying the Parker field in the polar regions. In addition, it is argued that the modification of Smith and Bieber is based on observational evidence and has a firm physical basis, while these motivations are lacking in the case of the Jokipii-Kóta modification. From a cosmic ray modulation point of view, we found the Smith-Bieber modification to be the most suitable choice for modifying the heliospheric magnetic field. The features and effects of these three modifications are illustrated both qualitatively and quantitatively. It is also shown how the Smith-Bieber modified field can be applied in cosmic ray modulation models to reproduce observational cosmic ray proton spectra from the PAMELA mission during the solar minimum of 2006 - 2009. These results are compared with those obtained in previous studies of this unusual solar minimum activity period and found to be in good qualitative agreement.\ \ **Keywords** Solar modulation $\cdot$ Cosmic rays $\cdot$ Stochastic differential equations $\cdot$ Heliospheric magnetic field $\cdot$ PAMELA $\cdot$ Solar minimum Introduction ============ The intensities of Galactic cosmic rays (CRs) entering the heliosphere are modulated by a number of physical mechanisms, including convection, particle drifts and diffusion, as well as adiabatic energy changes. Modeling these mechanisms and the associated modulation of CRs is done primarily by employing numerical modulation models, since analytical models are limited to only a few simplified cases. The numerical modulation model used in this study is based on the solution of an appropriate set of stochastic differential equations (SDEs). Studies in the past have mainly employed numerical models based on finite difference approaches, the alternating direction implicit (ADI) being the most extensively utilised scheme. However, after Zhang (1999) successfully applied SDEs to study CR modulation in the heliosphere, the use of such models has become popular. SDE-based models have been used to study the transport of pickup ions (Fichtner *et al.*, 1996), solar energetic particles (Dröge *et al.*, 2010), the propagation of Galactic and Jovian electrons (Strauss *et al.*, 2011a), and other aspects of CR modulation (e.g. Pei *et al.* 2010; Strauss *et al.*, 2011b, 2012; Luo *et al.*, 2011, 2013a, 2013b). Some historical and contemporary models are discussed by e.g. Yamada, Yanagita and Yoshida (1998), Alanko-Huotari *et al.* (2007), Bobik *et al.* (2012), and Kopp *et al.* (2012) who also discussed the mathematical and practical implementation of SDEs for a variety of general transport equations. The SDE-approach leads to numerical models that are ideal for implementation on multiple processors, allowing for parts of the numerical code to be executed simultaneously and independently, saving a great deal of computation time. Besides the parallel use of multiple central processing units (CPUs) and computer clusters, high performance calculations on graphics processing units (GPUs) have also become available during the last years; in such a GPU-accelerated implementation to study the solar modulation of CRs, presented by Dunzlaff, Strauss and Potgieter (2015), a performance increase of a factor of about 10 - 60 was found when compared to the CPU-version of the same algorithm. ADI-based models, on the other hand, only allow for a linear execution of the numerical code so that improvement in computation time cannot be achieved by these means. Furthermore, SDE-based models, being independent of a spatial grid, have also displayed remarkable numerical stability. This makes it possible to incorporate detailed descriptions of the heliosphere and heliospheric structures into these models. Examples of such implementations include the works of Strauss *et al.* (2011b), where the relative motion of Jupiter $-$ as the source of Jovian electrons $-$ was accounted for; Strauss *et al.* (2012), where a fully three-dimensional wavy heliospheric current sheet (HCS) was implemented; Luo *et al.* (2013a), where acceleration of the anomalous component and the re-acceleration of Galactic CRs by the solar wind termination shock (TS) were included; and Luo *et al.* (2011, 2013b), where time-varying modulation parameters were implemented. ADI-based models, on the other hand, are notoriously unstable when solving differential equations in higher dimensions and the complexity of these models are thus restricted. In addition to these considerations, SDE-based models are especially powerful in their ability to visualise the modulation process, being able to calculate pseudo-particle trajectories or $-$ equivalently stated $-$ follow the evolution of individual phase space (density) elements; it is important to note that these trajectories are not the trajectories of actual particles, but can at most be interpreted to obtain an indication of how the modulation of actual particles would proceed. The ability to calculate these trajectories leads to the possibility of calculating additional modulation features which were previously not possible, e.g. the propagation times and energy losses of particles in the heliosphere. The heliospheric magnetic field (HMF) plays a central role in the modulation mechanisms pointed out above and, as such, the choice of the HMF profile is a very important question in modulation studies. In this paper, we apply the SDE-based model to investigate the effect of different choices for the HMF. We consider the unmodified Parker HMF (Parker, 1958) and two modifications to this field, namely that of Jokipii and Kóta (1989) and that of Smith and Bieber (1991). The focus is on the modification of Smith and Bieber (SBM), and it is argued that this modification should be preferred when modifying Parker like HMFs. To this end the numerical model is applied, employing the SBM, to reproduce the Galactic proton spectra observed at Earth by the PAMELA mission during the peculiar solar minimum of 2006 to 2009. The results are then compared to those of other authors who have studied this minimum. The transport model for Galactic protons ======================================== An equation including most of the relevant heliospheric modulation mechanisms was derived by Parker (1965) and is known as the Parker transport equation (TPE), given by $$\label{eq:TPE} \dfrac{\partial f}{\partial t} = -\left( \vec{V}_{\mathrm{sw}}+\langle\vec{v}_{\mathrm{d}}\rangle \right)\cdot\nabla f +\nabla\cdot\left( \mathrm{\bf{K}}_{\mathrm{s}} \cdot \nabla f \right) +\dfrac{1}{3}\left(\nabla\cdot \vec{V}_{\mathrm{sw}}\right)\dfrac{\partial f}{\partial \ln p}+Q,$$ in terms of the omnidirectional distribution function $f\left(\vec{r},p,t\right)$, where $\vec{r}$ is the position, $p$ the particle momentum, and $t$ the time. The first term on the right hand side describes the process of convection via the expanding solar wind (SW), with velocity $\vec{V}_{\mathrm{sw}}$; the second term represents the effects of gradient and curvature drifts, where $\langle\vec{v}_{\mathrm{d}}\rangle$ is the pitch-angle averaged guiding centre drift velocity; the third term describes particle diffusion through the symmetric diffusion tensor $\textbf{K}_{\mathrm{s}}$; the fourth term accounts for adiabatic energy changes; the final term on the right hand side, $Q$, includes any source terms, if required. For a detailed description of these modulation processes, see e.g. Potgieter (2013). The geometry, magnitude and turbulence of the HMF play a key role in most of these modulation processes, in particular in determining particle drifts, as will be shown and emphasised below. In most CR modulation models the straight-forward Parker HMF (Parker, 1958) is used. This HMF profile is valid for $r\geq r_\odot$, where $r_\odot=0.005$ AU is the solar radius, and is expressed in spherical coordinates $\left(r,\theta,\phi\right)$ as $$\vec{B}=B_0\left[\dfrac{r_0}{r}\right]^2\left(\textbf{e}_r-\tan\psi\textbf{e}_\phi\right),$$ where $\textbf{e}_r$ and $\textbf{e}_\phi$ are unit vectors in the radial and azimuthal directions respectively, $B_0$ is the magnitude of the HMF at $r_0=1$ AU (i.e. at Earth), and the spiral angle $\psi$ is defined by $$\label{eq:PHMFspiral} \tan\psi=\dfrac{\Omega\left(r-r_\odot\right)}{V_{\mathrm{sw}}}\sin\theta = \Psi.$$ In the above expression, $\Omega=2.66\times10^{-6}$ $\mathrm{rad.s^{-1}}$ is the average angular rotation speed of the Sun. The magnitude of the Parker HMF is then given by $$\label{eq:PHMF} B=B_0\left[\dfrac{r_0}{r}\right]^2\sqrt{1+\Psi^2}.$$ Figure \[fig:PHMF\] shows the magnetic field lines in the case of the Parker HMF, converted to Cartesian coordinates, and its spiral structure is clearly illustrated over the first 10 AU from the Sun, which is located at $\left(X,Y,Z\right)=\left(0,0,0\right)$; these lines originate at consecutive azimuthal angles and a latitude of $\theta=45^\circ$. The Parker HMF, however, leads to very large gradients in particle intensities $j=fP^2$, with $P$ the particle rigidity, in the polar regions; this makes for a gross over-estimation of drift effects in these regions, as was originally pointed out by Potgieter, le Roux and Burger (1989); see also Potgieter (2013). An example of such drift dominated modulation is found in Jokipii and Kopriva (1979). The Parker HMF has therefore been modified in several studies to scale down these effects in the polar regions; this forms the topic of Section \[sec:HMFmod\]. The SW profile selected is rather simple, accelerating from zero to a constant and supersonic velocity within the first 0.3 AU from the Sun. The only other radial structure occurs at the position of the TS, i.e. at $r=r_{\mathrm{TS}}$, where $V_{\mathrm{sw}}$ drops off to subsonic levels; The SW also has a latitudinal profile, which accounts for a slow SW of $\sim$400 $\mathrm{km.s^{-1}}$ in the equatorial regions and a fast SW of $\sim$800 $\mathrm{km.s^{-1}}$ in the polar regions, in accordance with Ulysses observations during solar minimum (Phillips *et al.*, 1995). Particle diffusion is realised through the diffusion tensor $\textbf{K}_{\mathrm{s}}$ in Eq. (\[eq:TPE\]), of which each element is known as a diffusion coefficient. An approach similar to that of Potgieter *et al.* (2014) is followed; this will also allow for a direct comparison with their modulation modeling, which did not use the SBM. The parallel diffusion coefficient $\kappa_{||}$, i.e. parallel to the mean HMF, is assumed to have a power-law dependence on particle rigidity, while its spatial dependence is assumed to be inversely proportional to the magnitude of the magnetic field, so that $$\label{eq:kpar} \kappa_{||}=\kappa_{||,0}\beta\dfrac{B_0}{B}\left[ \dfrac{\left(\dfrac{P}{P_0^{'}}\right)^{a_3}+\left(\dfrac{P_{\mathrm{k}}}{P_0^{'}}\right)^{a_3}}{1+\left(\dfrac{P_{\mathrm{k}}}{P_0^{'}}\right)^{a_3}} \right]^{\dfrac{a_2-a_1}{a_3}}\left(\dfrac{P}{P_0^{'}}\right)^{a_1},$$ where $\kappa_{||,0}$ is a constant in units of $6\times10^{20}$ $\mathrm{cm^2.s^{-1}}$, with $P_0^{'}=1$ GV and $B_0=1$ nT added to obtain the correct dimensions; $\beta=v/c$ is the ratio of particle speed $v$ to the speed of light $c$. Here $a_1$ and $a_2$ are dimensionless constants and determine the slope of the rigidity dependence below and above a rigidity $P_{\mathrm{k}}$ respectively; in this study, we set $a_2=1.95$. The quantity $a_3=3.0$ is another dimensionless constant and determines the smoothness of the transition between the two slopes $P^{a_1}$ and $P^{a_2}$ at $P_{\mathrm{k}}$. The rigidity dependence is therefore essentially a double power-law. The perpendicular diffusion coefficient $\kappa_{\bot}$ consists of perpendicular diffusion in the $\textbf{e}_r$ and $\textbf{e}_\theta$ directions respectively, i.e. $\kappa_{\bot r}$ and $\kappa_{\bot\theta}$. Here, $\kappa_{\bot}$ is scaled as $\kappa_{||}$. Giacalone and Jokipii (1999) found that the ratio $\kappa_{\bot}/\kappa_{||}$ has a value between 0.02 and 0.04. In addition, observations from the Ulysses spacecraft have revealed that the latitude dependence of CR protons is significantly less than predicted by classical drift models (Potgieter and Haasbroek, 1993), which led Kóta and Jokipii (1995) to propose the concept of an anisotropic perpendicular diffusion, where $\kappa_{\bot\theta}>\kappa_{\bot r}$ in the off-equatorial regions (e.g. Burger, Potgieter and Heber, 2000; Ferreira *et al.*, 2000; Potgieter, 2000). This anisotropy of $\kappa_{\bot}$ is accounted for in this work, as was done by e.g. Ngobeni and Potgieter (2008) as well as Potgieter *et al.* (2014), so that $$\kappa_{\bot r}=\kappa_{\bot r}^0\kappa_{||}$$ and $$\kappa_{\bot\theta}=f(\theta)\kappa_{\bot\theta}^{0}\kappa_{||},$$ where $\kappa_{\bot r}^0=\kappa_{\bot\theta}^0=0.02$ are dimensionless constants, and $$f(\theta)=A^{+}+A^{-}\tanh\left[\dfrac{1}{\Delta\theta}\left(\overset{\sim}{\theta}-\dfrac{\pi}{2}+\theta_{\mathrm{F}}\right)\right].$$ Here $A^{\pm}=\dfrac{d\pm1}{2}$, $\Delta\theta=1/8$, with $$\label{eq:Apm} \overset{\sim}{\theta}=\left\{ \begin{array}{lcl} \theta & \mathrm{for} & \theta\geq\dfrac{\pi}{2} \\ \\ \pi-\theta & \mathrm{for} & \theta < \dfrac{\pi}{2}, \end{array} \right.$$ and $$\theta_{\mathrm{F}}=\left\{\begin{array}{lcl}\dfrac{-35^{\circ}\pi}{180^{\circ}} & \mathrm{for} & \theta\geq\dfrac{\pi}{2} \\ \\ \dfrac{35^{\circ}\pi}{180^{\circ}} & \mathrm{for} & \theta < \dfrac{\pi}{2},\end{array} \right.$$ where $d=3.0$ is a dimensionless constant that determines the enhancement factor of $\kappa_{\bot\theta}$ from its value in the equatorial plane towards the poles, with respect to $\kappa_{||}$; see also Potgieter (2000). Cosmic rays will undergo a combination of gradient and curvature drifts $\langle\vec{v}_{\mathrm{d}}\rangle_{\mathrm{gc}}$ caused by the large scale HMF, as well as current sheet drift $\left(\vec{v}_{\mathrm{d}}\right)_{\mathrm{ns}}$ because of a switch in magnetic polarity over the HCS. These drifts are both calculated from the general equation for the pitch-angle averaged guiding center drift velocity, written as $$\begin{aligned} \label{eq:drift_gen} \langle\vec{v}_{\mathrm{d}}\rangle & = & \nabla\times\kappa_{\mathrm{A}}\textbf{e}_{B} \nonumber \\ & = & \left[\nabla\times\left(\kappa_{\mathrm{A}}\textbf{e}_B^{'}\right)\right]\left[1-2H\left(\theta-\theta^{'}\right)\right]+2\delta_{\mathrm{Dirac}}\left(\theta-\theta^{'}\right) \kappa_{\mathrm{A}}\textbf{e}_B^{'}\times\nabla\left(\theta-\theta^{'}\right) \nonumber \\ & = & \langle\vec{v}_{\mathrm{d}}\rangle_{\mathrm{gc}}\left[1-2H\left(\theta-\theta^{'}\right)\right]+\left(\vec{v}_{\mathrm{d}}\right)_{\mathrm{ns}}\delta_{\mathrm{Dirac}}\left(\theta-\theta^{'}\right),\end{aligned}$$ where $$\textbf{e}_B = \left[1-2H\left(\theta-\theta^{'}\right)\right]\textbf{e}_B^{'},$$ and $\textbf{e}_B^{'}=\vec{B}/B$; $H$ is the Heaviside function, $\delta_{\mathrm{Dirac}}$ is the Dirac delta function, $\theta^{'}$ is the HCS latitudinal extent defined below, and $\kappa_{\mathrm{A}}$ the global drift coefficient which is given by $$\label{eq:ka} \kappa_{\mathrm{A}}=\kappa_{\mathrm{A}}^0qA\dfrac{P\beta}{3B}\dfrac{\left(P/P_0\right)^2}{1+\left(P/P_0\right)^2}.$$ In this equation $q$ is the particle charge sign so that in this study, which only considers results pertaining to CR proton modulation, $q=+1$. The quantity $P_0\in\lbrace1/\sqrt{10},1/\sqrt{40}\rbrace$ GV is added on dimensional grounds. The value $A=\pm1$, i.e. either $A>0$ or $A<0$; this represents the magnetic polarity cycle with $A>0$ ($A<0$) defined as the polarity cycle in which the HMF points outward (inward) in the Northern hemisphere of the Sun; during an $A>0$ ($A<0$) polarity cycle positively charged particles will drift inward (outward) along the polar regions and outward (inward) along the HCS in the equatorial regions, while the reverse is true for negatively charged particles. The quantity $\kappa_{\mathrm{A}}^0$ is a constant which can be used as a scaling factor, having values between 0.0 (zero drift) and 1.0 (full drift); in this work, $\kappa_{\mathrm{A}}^0=1.0$, which is known as the so-called weak scattering maximal value for the drift coefficient. Remembering the change in sign that occurs when crossing the HCS, the gradient and curvature drifts can thus be calculated by means of $$\begin{aligned} \langle\vec{v}_{\mathrm{d}}\rangle_{\mathrm{gc}} & = &\nabla\times\left(\kappa_{\mathrm{A}}\dfrac{\vec{B}}{B}\right) \nonumber \\ & = & \kappa_{\mathrm{A}}^0qA\dfrac{ P \beta}{3}\dfrac{\left(P/P_0\right)^2}{1+\left(P/P_0\right)^2}\nabla\times\left(\dfrac{\vec{B}}{B^2}\right)\\ \nonumber & =& \kappa_{\mathrm{A}}^{'}\nabla\times\left(\dfrac{\vec{B}}{B^2}\right),\end{aligned}$$ where $\kappa_{\mathrm{A}}^{'}$ was defined as $$\kappa_{\mathrm{A}}^{'}=\kappa_{\mathrm{A}}^0qA\dfrac{P\beta}{3}\dfrac{\left(P/P_0\right)^2}{1+\left(P/P_0\right)^2}.$$ For the Parker HMF, the averaged gradient and curvature drift velocities in the radial, latitudinal, and azimuthal directions respectively are given by $$\begin{aligned} \label{eq:PHMFdrift} v_{\mathrm{dr}} & = & -v_{\mathrm{d}0}\cot\theta \nonumber \\ v_{\mathrm{d\theta}} & = & v_{\mathrm{d}0}\left(2+\Psi^2\right)\nonumber \\ v_{\mathrm{d\phi}} & = & v_{\mathrm{d}0}\Psi\cot\theta,\end{aligned}$$ where $$v_{\mathrm{d}0}=2\dfrac{\kappa_{\mathrm{A}}^{'}\Psi}{rB\sqrt{1+\Psi^2}^3}$$ Figure \[fig:PHMFdrift\] shows the drift velocity streamlines for this case, i.e. solving for $$\label{eq:streamlines} \dfrac{d\vec{r}(l)}{dl}=\langle\vec{v}_{\mathrm{d}}\rangle_{\mathrm{gc}},$$ where $\vec{r}=\left(r,\theta,\phi\right)$ is the position vector in spherical coordinates and $l$ is the arc length along the field lines of $\langle\vec{v}_{\mathrm{d}}\rangle_{\mathrm{gc}}$. These lines are calculated in the polar regions and it is clear from their narrow, cone-like configuration that drift in these regions occurs very effectively, towards the Sun. Note that Figure \[fig:PHMFdrift\] again makes use of Cartesian coordinates so that the Sun is located at $\left(X,Y,Z\right)=\left(0,0,0\right)$; the polar regions are centered around the $Z$-axis. When accounting for current sheet drift, which is directed parallel to the HCS and perpendicular to the HMF (e.g. Burger, Moraal and Webb, 1985; Burger and Potgieter, 1989), the model does not make use of the $\delta_{\mathrm{Delta}}$ function in Eq. (\[eq:drift\_gen\]); rather, it is assumed that a particle will experience current sheet drift upon the condition that its distance $d_{\mathrm{ns}}$ to the HCS is less than two gyro radii, i.e. $$\label{eq:HCScondition} d_{\mathrm{ns}}\leq2r_{\mathrm{L}}=2\left[\dfrac{mv}{q^{'}B}\right],$$ with $q^{'}$ the particle charge (not to be confused with the particle charge sign $q$ as in Eq. (\[eq:ka\])), and $m$ is the relativistic mass of the particle. Assuming that the latitudinal extent of the HCS is given by the expression from Kóta and Jokipii (1983) $$\theta^{'}=\dfrac{\pi}{2}-\arctan\left[\tan\alpha\sin\left(\phi+\dfrac{\Omega(r-r_\odot)}{V_{\mathrm{sw}}}\right)\right],$$ Eq. (\[eq:PHMFdrift\]) is replaced by $$\begin{aligned} \label{eq:HCSdrift} v_{\mathrm{d}r} & = & v_{\mathrm{ns},0}\sin\psi\cos\left(\pm\beta_{\mathrm{rot}}\right) \nonumber \\ v_{\mathrm{d}\theta} & = & v_{\mathrm{ns},0}\sin\left(\pm\beta_{\mathrm{rot}}\right) \nonumber \\ v_{\mathrm{d}\phi} & = & v_{\mathrm{ns},0}\cos\psi\cos\left(\pm\beta_{\mathrm{rot}}\right),\end{aligned}$$ when the condition of Eq. (\[eq:HCScondition\]) is met. Here $v_{\mathrm{ns},0}$ is given by the approximation of Burger, Moraal and Webb (1985) $$v_{\mathrm{ns},0}=vqA\left[0.457-0.412\dfrac{d_{\mathrm{ns}}}{r_\mathrm{L}}+0.0915\left(\dfrac{d_{\mathrm{ns}}}{r_{\mathrm{L}}}\right)^2\right],$$ and $\beta_{\mathrm{rot}}$ is the angle between the radial direction and a vector parallel to $\left(\vec{v}_{\mathrm{d}}\right)_{\mathrm{ns}}$, defined by $$\tan\beta_{\mathrm{rot}}=r\dfrac{\partial\theta^{'}}{\partial r},$$ as in Strauss *et al.* (2012) and discussed by Burger (2012). The correct sign of $\beta$ in Eq. (\[eq:HCSdrift\]) is determined by $$\begin{array}{lcl} \dfrac{\partial\theta^{'}}{\partial r}<0&\Rightarrow&\beta_{\mathrm{rot}} <0 \\ \\ \dfrac{\partial\theta^{'}}{\partial r}\geq0&\Rightarrow&\beta_{\mathrm{rot}} >0. \end{array}$$ The proton LIS assumed in this study has its origin in that of Langner and Potgieter (2004). Potgieter *et al.* (2014) improved on this LIS by taking into account PAMELA measurements at energies between 30 and 50 GeV. However, this LIS did not provide for the heliopause (HP) crossing of Voyager 1 and was therefore modified to take into account these new Voyager 1 observations in the lower energy range; see Potgieter (2014) in this regard. The LIS resulting from these modifications, expressed in units of $\mathrm{particles.m^{-2}.s^{-1}.sr^{-1}.MeV^{-1}}$, is employed in this study and is given by the expressions $j=0.6978\exp\left\{ 4.64 - 0.023(\ln E)^2 - 2.91E^{-0.5}\right\} $ for energies $E<1.4$ GeV, and $$j=0.6847\exp\left\{3.22 - 2.78(\ln E) - 1.5E^{-1}\right\}$$ for $E>1.4$ GeV. The HP position is assumed at 120 AU, where this LIS is specified as an initial condition for cosmic ray modulation. The Numerical Modulation Model ============================== Since analytical solutions to the TPE in Eq. (\[eq:TPE\]) are only possible to a very limited extent, modulation models solve this equation numerically to various degrees of complexity. The numerical model of this work makes use of an SDE approach and is based on the well-benchmarked model of Strauss **et al.** (2011a, 2011b, 2012). It is only briefly explained here; an extensive treatment can be found in Kopp *et al.* (2012). The model is solved in a steady state and assuming no sources so that $Q=0$. It is solved in spherical coordinates $(r,\theta,\phi)$ and particle kinetic energy $E$. Each Fokker-Planck type equation has a corresponding set of independent SDEs, which is calculated in either a time forward or time backward fashion. The TPE, written in spherical coordinates, is associated with such a time backward set of SDEs, given by $$\label{eq:SDEs} \begin{array}{lcl} dr & = & \left[\dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2\kappa_{rr}\right)+\dfrac{1}{r\sin\theta}\dfrac{\partial\kappa_{r\phi}}{\partial\phi}-V_{\mathrm{sw}}-v_{dr}\right]ds + \sqrt{2\kappa_{rr}-\dfrac{2\kappa^2_{r\phi}}{\kappa_{\phi\phi}}}dW_r+\dfrac{\sqrt{2}\kappa_{r\phi}}{\sqrt{\kappa_{\phi\phi}}}dW_{\phi} \\ \\ d\theta & = & \left[\dfrac{1}{r^2\sin\theta}\dfrac{\partial}{\partial \theta}\left(\sin\theta\kappa_{\theta\theta}\right)-\dfrac{v_{d\theta}}{r}\right]ds + \dfrac{\sqrt{2\kappa_{\theta\theta}}}{r}dW_{\theta} \\ \\ d\phi & = & \left[\dfrac{1}{r^2\sin^2\theta}\dfrac{\partial\kappa_{\phi\phi}}{\partial\phi}+\dfrac{1}{r^2\sin\theta}\dfrac{\partial}{\partial r}\left(r\kappa_{r\phi}\right)-\dfrac{v_{d\phi}}{r\sin\theta}\right]ds + \dfrac{\sqrt{2\kappa_{\phi\phi}}}{r\sin\theta}dW_{\phi} \\ \\ dE & = & \left[\dfrac{1}{3r^2}\dfrac{\partial}{\partial r}\left(r^2V_{\mathrm{sw}}\right)\Gamma E\right] ds,\end{array}$$ for each of the spherical coordinates $r$, $\theta$, and $\phi$, as well as for the particle energy $E$; $d\vec{W}=\left[dW_r,dW_{\theta},dW_{\phi}\right]$ is the multi-dimensional Wiener process with each of its elements containing a Gaussian distributed random number. The quantity $\Gamma$ is defined by $$\Gamma=\dfrac{E+2E_0}{E+E_0}$$ where $E_0$ is the particle rest energy; $ds$ is the infinitesimal backward time increment. The diffusion tensor has here been converted to spherical coordinates and also includes the drift coefficient $\kappa_{\mathrm{A}}$. Its elements are related to the coefficients in HMF aligned coordinates through $$\label{eq:Kspher} \begin{array}{lcl} \left[{\begin{array}{ccc}\kappa_{rr}&\kappa_{r\theta}&\kappa_{r\phi}\\ \kappa_{\theta r} & \kappa_{\theta\theta} & \kappa_{\theta\phi} \\ \kappa_{\phi r}&\kappa_{\phi\theta}&\kappa_{\phi\phi}\end{array}}\right] &=&\left[{\begin{array}{ccc} \kappa_{||}\cos^2\psi+\kappa_{\bot r}\sin^2\psi & -\kappa_{\mathrm{A}}\sin\psi & (\kappa_{\bot r}-\kappa_{||})\cos\psi\sin\psi \\ \kappa_{\mathrm{A}}\sin\psi & \kappa_{\bot\theta} & \kappa_{\mathrm{A}}\cos\psi \\ (\kappa_{\bot r}-\kappa_{||})\cos\psi\sin\psi & -\kappa_{\mathrm{A}}\cos\psi & \kappa_{||}\sin^2\psi+\kappa_{\bot r}\cos^2\psi \end{array}}\right].\\ \\ \end{array}$$ This tensor transformation is only valid for Parker-like HMF profiles, i.e. for an HMF that does not contain a $\theta$ component; more general transformations are discussed in e.g. Effenberger *et al.* (2012). Solving the TPE in a time backward fashion implies that a number of pseudo-particles, $N$, are traced from Earth outward to the boundary of the heliosphere, stepping in $(r,\theta,\phi)$ and calculating energy losses according to Eq. (\[eq:SDEs\]). At the outer modulation boundary, $r_{\mathrm{out}}=120$ AU, particle relative intensities are convolved with the LIS and an energy spectrum at Earth is obtained. For the purposes of this study, no modulation is assumed beyond the HP, so that the heliopause spectrum and the very LIS is synonymous (see Potgieter *et al.*, 2013). A reflective inner modulation boundary is used at $r_{\mathrm{in}}=0.1$ AU, so that any pseudo-particle for which $r<r_{\mathrm{in}}$ is reflected back into the modulation volume, i.e. $r_{\mathrm{in}}<r<r_{\mathrm{out}}$. Modified Magnetic Field Profiles {#sec:HMFmod} ================================ The modification of Jokipii and Kóta ------------------------------------ Jokipii and Kóta (1989) showed that the effect of small perpendicular magnetic field components near the solar surface leads to a larger magnetic field at greater radial distances in the polar regions. These authors argued that the average direction of these small components would cancel out so that only the magnitude of the magnetic field is modified. They put forward the following modified expression for the Parker HMF magnitude $$\label{eq:JKM} B=B_0\left[\dfrac{r_0}{r}\right]^2\sqrt{1 + \Psi^2 + \left( \dfrac{r\delta }{r_{\odot}} \right)^2},$$ introducing the quantity $\delta$ to signify the magnitude of the extra, superimposed, transverse magnetic field. The Jokipii-Kóta modification is henceforth referred to as the JKM, while the term containing $\delta$ will be referred to as the JKM modification term. In this study we use $\delta=0.002$ (see e.g. Potgieter, 2000). Geometrically speaking, a cone centered around the polar regions is induced in which the HMF is altered so that it decreases as $r^{-1}$ instead of $r^{-2}$. The effect of this modification, therefore, is to bring about the changes which are required to reduce the large drift velocities in the polar regions, without noticeably altering the field in the equatorial plane. However, upon this modification, the requirement that the magnetic field remains divergence free, i.e. $\nabla\cdot\vec{B} = 0 $, is no longer complied with and therefore the JKM is technically incorrect. Through inspection, the HMF will remain divergence free if $\delta=\delta(\theta)\propto\left(\sin\theta\right)^{-1}$, so that $$\label{eq:JKMmod} \delta (\theta) = \dfrac{\delta_{\mathrm{m}}}{\sin\theta},$$ with $\delta_{\mathrm{m}}=8.7\times 10^{-5}$ (Langner, 2004). This value for $\delta_{\mathrm{m}}$ implies $\delta (\theta) = 0.002$ near the poles and $\delta (\theta) \approx 0 $ in the ecliptic plane. This will again bring about the intended changes to the HMF. The modification of Smith and Bieber ------------------------------------ This modification was introduced by Smith and Bieber (1991), who analysed data collected by various satellites and found magnetic field spirals at Earth more tightly wound than that predicted by the Parker theory. Consequently, they suggested a modification to the Parker spiral by reasoning that the difference in rotational speed of the equatorial and polar regions of the Sun would cause small azimuthal magnetic field components to develop. This would lead to larger spiral angles at larger radial distances, and would provide an explanation for the larger spiral angles at Earth. At the same time, this modification would have a significant effect on the HMF structure at the high latitude regions. The expression for the spiral angle, i.e. Eq. (\[eq:PHMFspiral\]), was modified to yield $$\label{eq:SBM} \Psi^{'}=\dfrac{\Omega(r-b)\sin\theta}{V_{\mathrm{sw}}(r,\theta)}-\dfrac{r}{b}\dfrac{V_{\mathrm{sw}}(b,\theta)}{V_{\mathrm{sw}}(r,\theta)}\left(\dfrac{B_{\mathrm{T}}(b)}{B_{\mathrm{R}}(b)}\right),$$ where $B_{\mathrm{T}}(b)/B_{\mathrm{R}}(b)$ is the ratio of the azimuthal to the radial magnetic field components at a position $b$ near the solar surface, here taken as $b=20r_{\odot}$. Smith and Bieber (1991) showed that the value of $B_{\mathrm{T}}(b)/B_{\mathrm{R}}(b)$ is approximately -0.02 and although the value of $b$ assumed here is larger than the original $b=5r_{\odot}$ assumed by these authors, the value of -0.02 is retained for the purposes of this paper, except where explicitly indicated otherwise. This modification, based on sound physical arguments, has been mostly ignored in the numerical modeling of the solar modulation of cosmic rays, which is rather surprising. It will be shown next that this modification is indeed appropriate and as such an improvement. Figure \[fig:SBM\] is an illustration of HMF lines that results from applying the SBM; these lines once again originate at consecutive azimuthal angles and at a latitude of $\theta=45^{\circ}$. According to Eq. (\[eq:SBM\]), and clearly evident from comparing Figures \[fig:SBM\] and \[fig:PHMF\], the SBM alters the direction of the Parker HMF by winding up the magnetic field spirals more tightly; as a consequence, in contrast to the straight line case of Eqs. (\[eq:PHMFspiral\]) and (\[eq:PHMF\]), the HMF still has an azimuthal component in the polar regions, even at $\theta=0^{\circ}$. It is also clear from this comparison that the field lines in the case of the SBM are closer together than in the case of the Parker HMF, in accordance with the larger HMF magnitudes of the modified field. Note that, in Figure \[fig:SBM\], $B_{\mathrm{T}}(b)/B_{\mathrm{R}}(b)=-0.5$ was chosen to be very small, corresponding to a very large modification, in order to illustrate these effects more clearly. It should be noted that the ultimate, very complex, HMF modification was introduced by Fisk (1996). However, it has been seen as quite controversial and without convincing observational support so that it is not pursued any further in this work; see Sternal *et al.* (2011) for an appreciation of this modification. The SDE-approach does however allow for the implementation of a HMF of this complexity whereas traditional ADI based numerical schemes are quite unsuitable. Comparison of the features of HMF modifications ----------------------------------------------- The top panel of Figure \[fig:Brad\] compares the radial profiles of the Parker HMF and HMFs which have been modified according to respectively the SBM and JKM; this is shown in the equatorial plane with $\theta=90^\circ$, assuming a magnetic field magnitude of 5.05 nT at Earth. All three of these HMF profiles start out with an $r^{-1.95}$ dependence at the smallest radial distances and end with an $r^{-1.00}$ dependence just in front of the TS at $r=r_{\mathrm{TS}}$, where $V_{\mathrm{sw}}$ drops by a factor of 2.5 and brings about a step-like increase; beyond the TS, the $r^{-1.00}$ dependent decrease is continued. In the case of the Parker HMF, the $r^{-1.95}$ dependence continues up to a radial distance of about 0.5 AU, where it begins to transition into a weaker dependence to eventually reach $r^{-1.00}$ at $\sim$ 3.0 AU. Although not clearly discernible on the scale of this graph, the SBM and JKM profiles were determined to enter the $r^{-1.00}$ dependence by a distance of up to 1 AU earlier than the Parker HMF. The bottom panel of Figure \[fig:Brad\] compares the radial profiles of the Parker HMF, SBM, and JKM in the polar regions and necessarily illustrates the effects of the modifications more clearly. The Parker HMF and both the SBM and the JKM start out with an $r^{-1.96}$ dependence at the smallest radial distances. At a radial distance between 6 AU and 7 AU the Parker HMF starts transitioning into a weaker radial dependence and keeps up this transition to reach its weakest dependence of $r^{-1.10}$ at a radial distance of between 40 AU and 50 AU; this is then maintained up to the TS. The SBM begins its transition into the weaker radial dependence at a smaller radial distance between 1 AU and 2 AU and reaches $r^{-1.00}$ at about 20 AU, continuing up to the TS. The JKM begins the transition into the weaker radial dependence even earlier than the SBM at $\sim$ 1 AU and reaches $r^{-1.00}$ between 8 and 9 AU, once again continuing up to the TS. In all three cases, the dependence in front of the TS is regained beyond the TS. Notice that, in the polar regions, the JKM profile does not undergo the upward step at the TS to the extent shown by the Parker HMF and SBM; in fact, the JKM profile barely shows any visible increase. This suppression of the TS in the polar regions is not an intended effect and is a consequence of the fact that the TS is simulated entirely by the drop in $V_{\mathrm{sw}}$ at $r=r_{\mathrm{TS}}$. In the instance of the JKM, the modification term in Eq. (\[eq:JKM\]) is much larger than the term containing $V_{\mathrm{sw}}$ and hence the effect of the drop in $V_{\mathrm{sw}}$ at $r=r_{\mathrm{TS}}$ is markedly less pronounced. It is clear from Figure \[fig:Brad\] that both the SBM and JKM have the desired effect of drastically reducing the radial dependence of the HMF in the polar regions: the transition into the weaker radial dependence is initiated at much smaller $r$ than in the case of the Parker HMF, ensuring significantly larger magnetic field magnitudes $B$. It is also noted that because the SBM starts its transition into the weaker radial dependence at larger radial distances than does the JKM, its effect on $B$ is less pronounced than that of the JKM. Although not illustrated in Figure \[fig:Brad\], it is of importance to note that this less effectual status of the SBM when compared to the JKM is only *locally* true, i.e. it is only valid at or *very close* to the poles as in this case with $\theta=5^{\circ}$. The four panels of Figure \[fig:Blat\] show the unfolding of the Parker HMF, SBM, and JKM latitudinal profiles over successive radial distances of 1 AU, 10 AU, 50 AU, and 100 AU. From the first panel it is clear that, at 1 AU, the effects of both the SBM and JKM are negligible in the equatorial regions, but become more pronounced towards the poles. Taking the equator at $\theta=90^{\circ}$ as reference and moving towards the poles, the deviation of the SBM from the Parker HMF is seen to start at about $\theta=90^\circ\pm22^\circ$, continuing towards the respective poles. The SBM is larger than the Parker HMF up to about $\theta=90^\circ\pm76^\circ$, where it drops below and eventually reaches a maximum deviation from the Parker HMF of $\sim$ 0.08 nT at the poles. The difference between the Parker HMF and JKM is seen to consistently increase towards the polar regions $\theta\in \lbrace 0^\circ,180^\circ\rbrace$, where the JKM eventually reaches levels of $\sim$ 0.12 nT higher than the Parker HMF. The picture at a radial distance of 10 AU has unfolded to be significantly different from the profiles at 1 AU. The SBM is seen to have already modified the Parker HMF in the equatorial regions, amounting to a value of about 0.04 nT higher than the Parker HMF at $\theta = 90^\circ$. Towards the poles the difference between the SBM and Parker HMF increases to about 0.06 nT at intermediate latitudes, declining again to about 0.03 nT at $\theta\in\lbrace 0^\circ, 180^\circ \rbrace$. The SBM, at the equator, is initially higher than the JKM, but drops below it at about $\theta=90^\circ\pm42^\circ$, and continues to do so up to the polar regions. Smaller deviations from the Parker HMF in the equatorial regions are also seen to be induced by the JKM, amounting to no more than 0.02 nT at $\theta = 90^\circ$. As was the case at 1 AU, the effect of the JKM increases towards the poles where it reaches levels of about 0.10 nT higher than the Parker HMF. The picture at 50 AU looks qualitatively the same as that at 10 AU, only the magnitude of the HMF being smaller. Comparing with Figure \[fig:Brad\], it is clear that this qualitative picture is continued up to the TS. The fourth panel of Figure \[fig:Blat\] shows the latitudinal profiles at 100 AU, a distance comfortably beyond the TS. The SBM is still higher than the Parker HMF at all latitudes, now being higher than the Parker HMF by about 0.01 nT in the equatorial regions and by about 0.02 nT in the polar regions. The JKM is, curiously enough, at more or less the same level as the SBM in the polar regions; it lies just below the Parker HMF at $\theta = 90^\circ$, but climbs above it at about $\theta=90^\circ\pm45^\circ$, eventually reaching the level of the SBM at the poles. This unfolding of the HMF latitudinal profiles over successive radial distances effectively illustrates the previous statement about the more effective status of the JKM being only locally true: the SBM is effected over a larger part of the modulation volume so that its overall effect can indeed be greater than that of the JKM. From Eqs.(\[eq:kpar\]) to (\[eq:Apm\]), defining the diffusion coefficients straight-forwardly as used in this study, it is clear that the choice of HMF profile directly influences the spatial dependence of these coefficients. Figure \[fig:lambda\] shows the radial profiles of the parallel mean free paths $\lambda_{||}$ of 1 GV protons, which is related to the parallel diffusion coefficient via $$\lambda_{||}=\dfrac{3}{v}\kappa_{||}.$$ This is shown for each of the Parker HMF (solid red lines), the SBM (dashed green lines), and the JKM (dashed blue lines). The upper three lines represent the values of $\lambda_{||}$ in the polar ($\theta=5^\circ$) regions, while the lower three, almost coinciding, lines do so in the equatorial plane at $\theta=90^\circ$. As expected, the effects of the modifications are minimal in the equatorial regions, while being far more significant in the polar regions. In the equatorial regions, the $\lambda_{||}$s for each of the Parker HMF, SBM and JKM are identical at small $r$ and then start diverging from one another at about 1 AU. This divergence is not significant however, and both the $\lambda_{||}$s for the SBM and JKM end up being only marginally below the Parker HMF, the SBM being the lowest. In the polar regions, the $\lambda_{||}$s for each of the Parker HMF, SBM and JKM are again seen to start out together at small $r$ and then to diverge at greater $r$. As should be the case, the $\lambda_{||}$s for the SBM and JKM are observed to start diverging from the Parker HMF at more or less the same radial distances at which the HMF profiles in Figure \[fig:Brad\] started diverging from one another. In the polar regions the $\lambda_{||}$s for both the SBM and JKM then keep significantly lower than that for the Parker HMF, the JKM being the lowest up to $r=r_{\mathrm{TS}}$ where the SBM drops below it due to the failure of the JKM to decrease over the TS. The drift coefficient also has a $1/B$ dependence according to Eq. (\[eq:ka\]), and a modification to the HMF will therefore affect the particle drift velocity, so that in each of the cases for the JKM and SBM a set of expressions analogous to Eq. (\[eq:PHMFdrift\]) can be obtained. For the modified JKM, i.e. using Eq. (\[eq:JKMmod\]), the drift velocities of Eq. (\[eq:PHMFdrift\]) are replaced by $$\begin{aligned} \label{eq:JKMdrift} v_{\mathrm{d}r} & = & -v_{\mathrm{d}0}\left(1+\left(\dfrac{\delta r}{r_\odot}\right)^2\right)\Psi\cot\theta \nonumber \\ v_{\mathrm{d}\theta} & = & v_{\mathrm{d}0}\left(2+\left(\dfrac{\delta r}{r_\odot}\right)^2+\Psi^2\right)\Psi\nonumber \\ v_{\mathrm{d}\phi} & = & v_{\mathrm{d}0}\left(\Psi^2\cot\theta+\dfrac{\delta r}{r_\odot}\left(2+\left(\dfrac{\delta r}{r_\odot}\right)^2+\Psi^2\right)\right),\end{aligned}$$ with $$\label{eq:JKMkappa} v_{\mathrm{d}0}=2\dfrac{\kappa_{\mathrm{A}}^{'}}{rB\sqrt{1+\left(\dfrac{\delta r}{r_\odot}\right)^2+\Psi^2}^3}.$$ To obtain the analogous set of equations in the case of an HMF modified according to the SBM, we write $$\Psi^{'}\approx\Psi-\delta^{'}r,$$ where the SBM modification term $\delta^{'}$ is defined as $$\delta^{'}=-\dfrac{0.02}{b}\dfrac{V_{\mathrm{sw}}\left(b,\theta\right)}{V_{\mathrm{sw}}\left(r,\theta\right)},$$ since we have set $B_{\mathrm{T}}(b)/B_{\mathrm{R}}(b)=-0.02$ for our choice of $b$; finally, substituting $$\gamma = r\delta^{'} - \Psi,$$ Eq. (\[eq:PHMFdrift\]) is replaced by $$\begin{aligned} v_{\mathrm{d}r} & = & v_{\mathrm{d}0}\left(\gamma^3+\gamma^2\Psi + \gamma-\Psi\right)\cot\theta \nonumber \\ v_{\mathrm{d}\theta} & = & -2v_{\mathrm{d}0}\left(2+\gamma^2\right)\gamma\nonumber \\ v_{\mathrm{d}\phi} & = & -2v_{\mathrm{d}0}\gamma\Psi\cot\theta,\end{aligned}$$ with $$v_{\mathrm{d}0}=\dfrac{\kappa_{\mathrm{A}}^{'}}{rB\sqrt{1+\gamma^2}^3}.$$ Figure \[fig:SBMdrift\] shows the drift velocity streamlines in the case of the SBM. Compared to Figure \[fig:PHMFdrift\], it is clear that the inward drift from the poles is not as effective when employing the SBM as it was when using the Parker HMF: the lines in the case of the SBM fan out significantly more than in the case of the Parker HMF. The value of $B_{\mathrm{T}}(b)/B_{\mathrm{R}}(b)=-0.1$, and has once again been set to a value low enough so that these differences can be clearly illustrated. The Effect on Modulation ------------------------ Very effective use can be made of the properties of the SDE-based model to qualitatively illustrate the effect that these modifications to the HMF can have. Figure \[fig:traj\] presents a meridional cut of the heliosphere and shows trajectories for pseudo-particles (1.5 GeV protons) entering in the polar regions during an $A>0$ polarity cycle, and propagating down towards the Sun, which is located at $\left(Y,Z\right)=\left(0,0\right)$; the solid black line shows the warped HCS with tilt angle $\alpha=10^\circ$, for illustrative purposes. The position of the TS, $r_{\mathrm{TS}}=88$ AU, is indicated by the vertical red lines. The red, blue and green trajectories respectively shows the case for the Parker HMF, JKM and SBM, and the differences are significant: in the case of the Parker HMF, the drift is obviously more effective, followed by consecutively reduced drift effects, i.e. reduced in terms of the ratio to diffusive processes, in the case of the JKM and SBM. This is not only evident from the longer trajectories in the latter two cases, but also by the reduced occurrence of the long straight and jump-like segments in the trajectories, which indicate cases where $v_{\mathrm{d}}$ reaches unphysically large speeds. From this figure, therefore, it is evident that both the JKM and SBM are effective in reducing drift effects over the poles and are thus successful in their original purpose. Next, we look at the effect that these HMF modifications have on CR proton spectra; in this way a quantitative measure of the effect on the modulation is obtained. The model is used to reproduce the December 2009 CR proton spectrum, which was observed by the PAMELA experiment and averaged over 1 month (e.g. Adriani *et al.*, 2013; Potgieter *et al.*, 2014). This is shown in Figure \[fig:PAM2009\], where the observations from PAMELA are indicated by the circles and the model reproduction, assuming the SBM, is indicated by the green line; the LIS at 120 AU is represented by the black line. The values of all the relevant quantities for this model reproduction are indicated in the last column of Table 1. To obtain a comparison with the SBM, these same values are used while assuming the Parker HMF (red line) and JKM (blue line) respectively; the difference between these three HMF profiles is clearly depicted: the JKM and SBM are consecutively lower than the Parker HMF; these spectra peak in the vicinity of 200 MeV and, at this energy, the SBM intensities are reduced relative to the LIS by about 86$\%$; the JKM intensities are reduced by about 82$\%$ while, in the case of the Parker HMF, intensities show a decrease of only about 71$\%$. The fact that the spectrum resulting from the SBM is lower than that resulting from the JKM once again illustrates the fact that, although the effects of the JKM may be locally greater than that of the SBM, the overall effect of the SBM is larger than that of the JKM when integrated over the whole of modulation space. Further motivation for the applicability of the SBM comes from applying this modification to model the peculiar solar minimum stretching from 2006 to 2009. Monthly averaged proton spectra measured by PAMELA at Earth, during November 2006 and December of 2007 and 2008 are now also reproduced using our model. This is depicted in Figure \[fig:PAM2006-9\], the observations from PAMELA once again indicated by the circles, while the model reproductions for 2006, 2007, 2008, and 2009 are indicated by the red, orange, green, and purple lines respectively; the LIS at 120 AU is represented by the black line. The parameters to obtain these results are summarised in Table 1 and it is clear that: 1) the diffusion was required to continually increase from 2006 to 2009, with the value of $\kappa_{||,0}$ increasing from 16.0 in 2006 to 17.1, 17.5, and 19.4 in 2007, 2008, and 2009 respectively; 2) the spectra became progressively softer towards 2009, so that the rigidity dependence of the diffusion coefficient below $\sim$ 3 GeV was required to change from $P^{0.85}$ in 2006 to $P^{0.81}$, $P^{0.78}$, and $P^{0.73}$ 2007, 2008, and 2009. These type of changes are in agreement with those found by various other authors, including Bazilevskaya *et al.* (2012), Ndiitwani *et al.* (2013), Potgieter *et al.* (2014), and Pacini and Usoskin (2015). Except for providing evidence for the applicability of the SBM in particular, this qualitative correspondence in modulation results over the entire solar minimum period of 2006 to 2009 also illustrates the already well established credibility of the SDE-based model. Figure \[fig:lambdaPAM\] illustrates the results of Table 1 by depicting the mean free paths for each of 2006, 2007, 2008, and 2009. The grey lines show the mean free paths obtained by Potgieter *et al.* (2014) and, when compared to the results obtained in this study, the qualitative agreement is clearly illustrated. Quantitative differences must exist between these two sets of results for three reasons. Firstly, these two studies did not select the same HMF profile, i.e. this study employs the SBM, while the study of Potgieter *et al.* (2014) utilised the JKM; referring to Figure \[fig:PAM2009\], this will clearly have a quantitative influence on the results. Secondly, our model uses an updated LIS that was adapted according to observations by Voyager 1 after its crossing of the HP, as explained earlier. Thirdly, there were some differences in the treatment of current sheet drift, the details of which are not relevant to the topic of this paper. **Parameter** **2006** **2007** **2008** **2009** ----------------------------------------------------- --------------- --------------- --------------- --------------- $\alpha$ \[deg\] 15.7 14.0 14.3 10.0 $B_{\mathrm{e}}$ \[nT\] 5.05 4.50 4.25 3.94 $r_{\mathrm{TS}}$ 88.0 86.0 84.0 80.0 $\kappa_{||,0}$ $[6\times 10^{20}$ cm$^2$.s$^{-1}]$ 16.0 17.1 17.5 19.4 $a_1$ 0.85 0.81 0.78 0.73 $P_0$ \[GV\] $1/\sqrt{10}$ $1/\sqrt{10}$ $1/\sqrt{10}$ $1/\sqrt{40}$ $P_{\mathrm{k}}$ \[GV\] 4.0 4.0 4.0 4.2 : Summary of the parameters used in this study to reproduce the 2006 to 2009 PAMELA proton spectra, using the SBM; see Eqs. (\[eq:PHMFspiral\]), (\[eq:PHMF\]) and (\[eq:SBM\]). Conclusions =========== An investigation into the effects of the SBM and JKM modifications were undertaken, making use of an SDE-based numerical modulation model. These two modified HMF profiles were compared to that of the unmodified Parker HMF and it was illustrated that both these modifications change the radial dependence of the HMF in the polar regions, such as to bring about a larger HMF magnitude as function of radial distance. This larger HMF magnitude in the polar regions then reduces both the diffusion and drift coefficients in these regions. It was noted that, although the effects of the JKM may be locally larger than those of the SBM, the effects of the SBM are exercised over a greater part of modulation space and hence its overall effect on CR intensities is greater than that of the JKM. The eventual effect on modulation of these two modifications were also illustrated and it was concluded that both are effective in reducing drift effects over the poles; this was illustrated by plotting drift velocity streamlines, as well as utilising the unique abilities of the SDE-based numerical model, which enables the construction of pseudo-particle trajectories. It was then shown that the SBM can be used to reproduce the Galactic proton spectra observed by the PAMELA experiment during the peculiar solar minimum period of 2006 to 2009, and that the results thus obtained correspond qualitatively to that found by various other authors who have previously investigated this solar minimum. This agreement was presented as veritable evidence for the applicability of the SBM in CR modulation studies; it was also argued that the SBM is grounded in observational evidence and that, for these reasons, it should be preferred as modification to Parker type HMFs. 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--- author: - | Tom Schmiedlechner\ CSAIL, MIT, USA\ `schmied@mit.edu` Ignavier Ng Zhi Yong\ CSAIL, MIT, USA\ `ignavier@mit.edu` Abdullah Al-Dujaili\ CSAIL, MIT, USA\ `aldujail@mit.edu` Erik Hemberg\ CSAIL, MIT, USA\ `hembergerik@csail.mit.edu` Una-May O’Reilly\ CSAIL, MIT, USA\ `unamay@csail.mit.edu`\ title: 'Lipizzaner: A System That Scales Robust Generative Adversarial Network Training' ---
--- abstract: | Let $G$ be a matroid on ground set [$\mathcal A$]{}. The Orlik-Solomon algebra $A(G)$ is the quotient of the exterior algebra [$\mathcal E$]{} on [$\mathcal A$]{} by the ideal [$\mathcal I$]{}  generated by circuit boundaries. The quadratic closure ${\ensuremath{\overline{A}}}(G)$ of $A(G)$ is the quotient of [$\mathcal E$]{} by the ideal generated by the degree-two component of [$\mathcal I$]{}. We introduce the notion of [[*nbb*]{}]{} set in $G$, determined by a linear order on [$\mathcal A$]{}, and show that the corresponding monomials are linearly independent in the quadratic closure ${\ensuremath{\overline{A}}}(G)$. As a consequence, $A(G)$ is a quadratic algebra only if $G$ is line-closed. An example of S. Yuzvinsky proves the converse false. These results generalize to the degree $r$ closure of ${\ensuremath{\mathcal A}}(G)$. The motivation for studying line-closed matroids grew out of the study of formal arrangements. This is a geometric condition necessary for [$\mathcal A$]{} to be free and for the complement $M$ of [$\mathcal A$]{} to be a $K(\pi,1)$ space. Formality of [$\mathcal A$]{} is also necessary for $A(G)$ to be a quadratic algebra. We clarify the relationship between formality, line-closure, and other matroidal conditions related to formality. We give examples to show that line-closure of $G$ is not necessary or sufficient for $M$ to be a $K(\pi,1)$, or for [$\mathcal A$]{} to be free. author: - Michael Falk title: 'Line-closed matroids, quadratic algebras, and formal arrangments' --- [Introduction]{} \[intro\] Let [$\mathbb{K}$]{} be a field. An [*arrangement*]{} is a finite set ${\ensuremath{\mathcal A}}=\{H_1, \ldots, H_n\}$ of linear hyperplanes in $V={\ensuremath{\mathbb{K}}}^\ell$. Each $H_i$ is the kernel of a linear form ${\ensuremath{\alpha}}_i : V {\ensuremath{\longrightarrow}}{\ensuremath{\mathbb{K}}}$, unique up to nonzero scalar multiple. Let $[n]$ denote the set $\{1,\ldots,n\}$ and $2^{[n]}$ the set of subsets of $[n]$. A coordinate-free combinatorial model of the arrangement [$\mathcal A$]{} is provided by the [*underlying matroid*]{} of [$\mathcal A$]{}, which we denote by $G({\ensuremath{\mathcal A}})$, or simply $G$. This matroid contains the same information as the intersection lattice $L({\ensuremath{\mathcal A}})$ – see [@OT]. By definition the matroid $G$ is the collection of dependent subsets of the set of defining forms $\{{\ensuremath{\alpha}}_1, \ldots, {\ensuremath{\alpha}}_n\}$. We identify these subsets with the corresponding sets of labels. Then it is easy to see that $$G=\{ \, S \subseteq [n] \ | \ {\operatorname{codim}}(\bigcap_{i\in S} H_i)<|S| \, \}.$$ Elements of $G$ are called [*dependent sets*]{}, and elements of $2^{[n]} - G$ are [*independent sets*]{}. The projective point configuration ${\ensuremath{\mathcal A}}^* \subseteq {\ensuremath{\mathbb{P}}}(V^*)$ determined by $\{{\ensuremath{\alpha}}_1, \ldots, {\ensuremath{\alpha}}_n\}$ is called a [ *projective realization*]{} of $G$. There are several other data besides the dependent sets which suffice to determine $G$ uniquely. Among these are the [ *circuits*]{} of $G$, which are the minimal dependent sets, and the [*bases*]{} of $G$, which are the maximal independent sets. Besides these, we single out two functions which also uniquely determine $G$. The [*rank function*]{} ${\operatorname{rk}}: 2^{[n]} {\ensuremath{\longrightarrow}}{\ensuremath{\mathbb{Z}}}$ is given by ${\operatorname{rk}}(X)={\operatorname{codim}}(\bigcap_{i\in X} H_i)$. In the abstract setting, ${\operatorname{rk}}(X)$ is the (unique) size of a maximal independent subset of $X$. The rank ${\operatorname{rk}}(G)$ of $G$ is ${\operatorname{rk}}([n])$. The [*closure operator*]{} ${\ensuremath{c\ell}}: 2^{[n]} {\ensuremath{\longrightarrow}}2^{[n]}$ , given by $${\ensuremath{c\ell}}(X)=\{ i\in [n] \ | \ {\operatorname{rk}}(X\cup\{i\})={\operatorname{rk}}(X)\},$$ also uniquely determines $G$. We refer the reader to the recent survey [@FR4] for more discussion of the role of matroid theory in the study of complex hyperplane arrangements. A set $S$ is [*closed*]{} if ${\ensuremath{c\ell}}(S)=S$. Closed sets are also called [*flats*]{}. A flat corresponds to the collection of hyperplanes in [$\mathcal A$]{}  containing a fixed subspace of ${\ensuremath{\mathbb{K}}}^\ell$, or equivalently, the intersection of the point configuration ${\ensuremath{\mathcal A}}^*$ with a fixed projective subspace of ${\ensuremath{\mathbb{P}}}(V^*)$. The set of flats, ordered by inclusion, forms a geometric lattice $L(G)$ isomorphic to the intersection lattice $L({\ensuremath{\mathcal A}})$. The flats of rank one are the singletons, called [*points*]{}. Flats of rank two are called [*lines*]{}. This terminology is natural with regard to the dual projective point configuration ${\ensuremath{\mathcal A}}^*.$ Let ${\ensuremath{\mathbb{K}}}={\ensuremath{\mathbb{C}}}$. The [*complement*]{} of [$\mathcal A$]{} is $V - \bigcup_{i=1}^n H_i,$ denoted by $M$. The cohomology $H^*(M)$ is isomorphic to the [*Orlik-Solomon algebra*]{} $A(G)$ of the underlying matroid $G$, defined in the next section. Study of the lower central series of $\pi_1(M)$ [@F4; @Y5] leads to the consideration of arrangements for which the cohomology algebra $H^*(M)$, or equivalently, the Orlik-Solomon algebra $A(G)$, is quadratic. Here $A(G)$ is [*quadratic*]{} if it has a presentation in which all relations have degree two. While this condition depends only on $G$, the underlying combinatorial meaning has never been understood. The best results in this direction involve the notion of formality. An arrangement is [*formal*]{} if it is uniquely determined up to linear isomorphism by the dependence [*relations*]{} yielding dependent sets of rank two in $G$. This is a geometric, non-matroidal condition, and is a necessary condition for $A(G)$ to be quadratic. In looking for a matroidal analogue of formality we were led naturally to the study of line-closed matroids. A subset $S\subseteq [n]$ is [*line-closed*]{} if ${\ensuremath{c\ell}}(\{i,j\}) \subseteq S$ for every $i,j \in S$. The matroid $G$ is [*line-closed*]{} if every line-closed set is closed. In attempting to sort out how this property fits in with other properties related to formality, we were led to the following. $G$ is line-closed if and only if $A(G)$ is quadratic. \[boston\] In this paper we prove half of this conjecture, that $A(G)$ quadratic implies $G$ line-closed, for arbitrary coefficient fields [$\mathbb{K}$]{}. The author sketched this proof and stated Conjecture \[boston\] in a lecture at the workshop “Arrangements in Boston" in 1999 [@F13]. Subsequently S. Yuzvinsky found a counter-example for the full conjecture (at least for ${\ensuremath{\mathbb{K}}}={\ensuremath{\mathbb{C}}}$), a line-closed matroid whose Orlik-Solomon algebra is not quadratic. We exhibit Yuzvinsky’s example, and refer the reader to the companion paper [@DY] for details on Yuzvinsky’s approach, and for a stronger condition, also necessary but not sufficient for quadraticity of $A(G)$. One can define a quadratic algebra ${\ensuremath{\overline{A}}}(G)$ and a surjection ${\ensuremath{\overline{A}}}(G) {\ensuremath{\longrightarrow}}A(G)$ which is an isomorphism if and only if $A(G)$ is quadratic; ${\ensuremath{\overline{A}}}(G)$ is called the [*quadratic closure*]{} of $A(G)$. Our main theorem follows from a more general construction, a partial generalization of the well-known [[*nbc*]{}]{} (“no broken circuits”) basis for $A(G)$. We generalize one of the characterizations of [[*nbc*]{}]{} sets to the lattice of line-closed sets of $G$. The result is the notion of [[*nbb*]{}]{} [*set.*]{} We show that the monomials corresponding to [[*nbb*]{}]{} sets are linearly independent in the quadratic closure ${\ensuremath{\overline{A}}}(G)$. In contrast to the situation for [[*nbc*]{}]{} sets, the number of [[*nbb*]{}]{} sets is not independent of the linear ordering of the atoms. But the collection of [[*nbb*]{}]{} sets will include all of the [[*nbc*]{}]{} sets, for any given linear ordering. The two collections coincide, for every linear ordering, if and only if $G$ is line-closed. Thus, if $G$ is not line-closed, ${\ensuremath{\overline{A}}}(G)$ must be strictly bigger than $A(G)$, so $A(G)$ is not quadratic. The entire development generalizes to any degree, with the line-closed sets and quadratic closure replaced by $r$-closed sets and degree $r$ closure. The main theorem and its generalization are developed and proved in Section \[main\]. The problem of finding a (monomial, or combinatorial) basis for ${\ensuremath{\overline{A}}}(G)$ is an interesting problem with some applications to lower central series calculations. Yuzvinsky’s example shows that there may be no linear ordering for which the [[*nbb*]{}]{} monomials form a basis of ${\ensuremath{\overline{A}}}(G)$. Our definition of [[*nbb*]{}]{} set is a special case of the [[*NBB*]{}]{} (“no bounded below”) sets of A. Blass and B. Sagan [@BSag], for the lattice of line-closed sets of $G$, with a linear ordering on the set of atoms. Blass and Sagan define [[*NBB*]{}]{} sets for finite atomic lattices with an arbitrary partial order on the set of atoms. Although general [[*NBB*]{}]{}  monomials for the lattice of line-closed sets are not linearly independent in ${\ensuremath{\overline{A}}}(G)$, we present a partial generalization of our main result to non-linear orderings, possibly yielding better lower bounds on $\dim {\ensuremath{\overline{A}}}(G)$ for $G$ of rank four or greater. Much of the research in complex hyperplane arrangements focuses on the extent to which properties of the complement $M$ as a topological space or algebraic variety are determined by the combinatorial structure of $G$. In particular, two important open problems are whether asphericity of $M$ [@FR1] or freeness of [$\mathcal A$]{} [@OT] are dependent only on $G$ – see [@FR1] and [@OT]. Formality of [$\mathcal A$]{} is also a necessary condition for each of these two properties. Thus attempts were made to replace the definition of formality with some stronger purely combinatorial notion – line-closure is one example. In the last section we give several other natural candidates for combinatorial analogues of formality. each of them is stronger than formality. We establish the relationships among these various notions and show by example that in fact none of them have true topological implications. The discussion leads to an interesting conjecture concerning matroids which are determined by their points and lines. [Quadratic closure and [[*nbb*]{}]{} sets]{} \[main\] We will use the matroid-theoretic terminology developed in the introduction without further comment. The reader is referred to [@Wh1; @Ox] for further background. We begin with the definition of the Orlik-Solomon algebra $A(G)$ of a matroid $G$ on ground set $[n]$. For the remainder of the paper, let [$\mathbb{K}$]{} be any field, or indeed any commutative ring. Assume $G$ has no loops or multiple points. Let ${\ensuremath{\mathcal E}}=\Lambda(V)$, the exterior algebra generated by $1$ and $\{e_i \ | \ 1\leq i \leq n\}$, with the usual grading by degree. If $S=(i_1,\ldots,i_p)$ is an ordered $p$-tuple we denote the product $e_{i_1}\cdots e_{i_p}$ by $e_S$. We occasionally use the same notation when $S$ is an unordered set – in this case $e_S$ is well-defined up to sign. Define the linear mapping $\partial : {\ensuremath{\mathcal E}}^p {\ensuremath{\longrightarrow}}{\ensuremath{\mathcal E}}^{p-1}$ by $$\partial(e_{i_1} \cdots e_{i_p})= \sum_{k=1}^p (-1)^{k-1}e_{i_1} \cdots \widehat{e}_{i_k}\cdots e_{i_p},$$ where $\widehat{\hspace{.5em}}$ indicates an omitted factor. Then $\partial$M is a graded derivation, that is, $$\partial (x{\ensuremath{\wedge}}y)=\partial x{\ensuremath{\wedge}}y+(-1)^{\deg(x)}x{\ensuremath{\wedge}}\partial y$$ for homogeneous $x,y\in {\ensuremath{\mathcal E}}$. Let ${\ensuremath{\mathcal I}}$ denote the ideal of [$\mathcal E$]{} generated by $\{\partial e_S \ | \ S \ \text{is dependent}\}$. The [*Orlik-Solomon algebra*]{} $A=A(G)$ of $G$ is the quotient ${\ensuremath{\mathcal E}}/{\ensuremath{\mathcal I}}$. Since [$\mathcal I$]{} is generated by homogeneous elements, both [$\mathcal I$]{} and $A$ inherit gradings from [$\mathcal E$]{}. We denote the image of $e_S$ in $A$ by $a_S$. The topological significance of $A$ is given in the following. If [$\mathcal A$]{} is an arrangement in ${\ensuremath{\mathbb{C}}}^\ell$ with complement $M$ and underlying matroid $G$, then $A(G)\cong H^*(M,{\ensuremath{\mathbb{K}}})$. \[orsol\] [The quadratic closure of $A(G)$]{} A graded algebra $U$ is [ *quadratic*]{} if $U$ has a presentation with generators of degree one and relations of degree at most two. Let [$\mathcal J$]{} denote the ideal of [$\mathcal E$]{} generated by ${\ensuremath{\mathcal I}}^2$, the degree two part of the relation ideal [$\mathcal I$]{}. Because [$\mathcal E$]{} itself is quadratic, the Orlik-Solomon algebra $A$ will be quadratic if and only if ${\ensuremath{\mathcal J}}={\ensuremath{\mathcal I}}$. More generally the quotient ${\ensuremath{\mathcal E}}/{\ensuremath{\mathcal J}}$ is called the [*quadratic closure*]{} of $A(G)$, denoted ${\ensuremath{\overline{A}}}(G)$, or sometimes ${\ensuremath{\overline{A}}}$. Quadratic Orlik-Solomon algebras appear in the study of complex arrangements, in the rational homotopy theory of the complement $M$ [@F4] and the Koszul property of $A(G)$ [@Y5]. Rational homotopy theory provides a connection with the lower central series of the fundamental group. In that vein an invariant $\phi_3$ of $A(G)$ was introduced in [@F6], defined as follows: $$\phi_3(G)=\mbox{nullity}\left(\delta: {\ensuremath{\mathcal E}}^1\times {\ensuremath{\mathcal I}}^2 {\ensuremath{\longrightarrow}}{\ensuremath{\mathcal E}}^3\right),$$ where $\delta$ is multiplication in ${\ensuremath{\mathcal E}}$. When ${\ensuremath{\mathbb{K}}}={\ensuremath{\mathbb{C}}}$ and $G$ is the matroid of an arrangement with complement $M$, $\phi_3(G)$ is the rank of the third factor in lower central series of $\pi_1(M)$. Because the image of $\delta$ is precisely ${\ensuremath{\mathcal J}}^3$, the cokernel of $\delta$ is ${\ensuremath{\overline{A}}}\,^3$, and we find a simple relationship between $\phi_3(G)$ and $\dim({\ensuremath{\overline{A}}}\,^3)$. The proof of the identity is left as an exercise. $$\begin{split} \phi_3(G)&=\dim({\ensuremath{\overline{A}}}\,^3) + n\dim({\ensuremath{\mathcal I}}^2)-{n\choose 3}\\ &=2{{n+1}\choose 3} - n\dim(A^2) +\dim({\ensuremath{\overline{A}}}\,^3) \end{split}$$ The preceding identity can be stated in a simpler way, indicating that $\dim({\ensuremath{\overline{A}}}\,^3/A^3)$ measures of the failure of the LCS (lower central series) formula relating the ranks of lower central series factors of $\pi_1(M)$ to the betti numbers $\dim(A^p)$, $p\geq 0$. Let $\gamma_3=\dim(A^3) +n\dim({\ensuremath{\mathcal I}}^2)-{n\choose 3}$. Then, according to [@F8], $\gamma_3$ is the value of $\phi_3$ predicted by the LCS formula for the given values of $\dim(A^p)$, and $\phi_3\geq \gamma_3$ with equality if and only if ${\ensuremath{\mathcal I}}^3={\ensuremath{\mathcal J}}^3$. $\dim({\ensuremath{\overline{A}}}\,^3) - \dim(A^3)=\phi_3-\gamma_3$ \[count\] [The line-closure of a matroid]{} Next we refine further the material on line closure from Section \[intro\]. Let us define an idempotent, order-preserving closure operator on subsets of $S\subseteq [n]$ using line-closed sets: the [*line-closure*]{} ${\ensuremath{\ell c}}(S)$ is by definition the intersection of the line-closed sets containing $S$. Since closed sets are automatically line-closed, ${\ensuremath{\ell c}}(S)\subseteq {\ensuremath{c\ell}}(S)$. We will consider the combinatorial structure consisting of the set $[n]$ equipped with the closure operator ${\ensuremath{\ell c}}: 2^{[n]} {\ensuremath{\longrightarrow}}2^{[n]}$ to be the [*line-closure*]{} of the matroid $G$, and denote it by ${\ensuremath{\overline{G}}}$. This set system ${\ensuremath{\overline{G}}}$ will not be a matroid in general, because the operator ${\ensuremath{\ell c}}$ fails to satisfy the Steinitz exchange axiom. The arguments and constructions in this section are seriously affected by this defect. Clearly $G$ is line-closed if and only if ${\ensuremath{\overline{G}}}=G$. The collection of line-closed sets, partially-ordered by inclusion, will be denoted by ${\ensuremath{\overline{L}}}(G)$. We call ${\ensuremath{\overline{L}}}(G)$ the [ *line-closure*]{} of $L(G)$. The poset ${\ensuremath{\overline{L}}}(G)$ is a lattice [@Rota Section 2] in which every element is a join of atoms. But ${\ensuremath{\overline{L}}}(G)$ is not a graded lattice, as the example below shows. This is a reflection of the failure of the exchange axiom. Again, $G$ is line-closed if and only if ${\ensuremath{\overline{L}}}(G)=L(G)$. Let $G$ be the rank-three matroid on $[6]$ with rank-two circuits $\{1,2,3\}, \{3,4,5\}$, and $\{1,5,6\}$. This example, pictured in Figure , is the “rank-three wheel" [@Ox]. Then there are two maximal chains in ${\ensuremath{\overline{L}}}(G)$ of different lengths, namely $$\emptyset<1<123<123456, \ \text{and}$$ $$\emptyset<2<24<246<123456.$$ \[wheel\] [[[*nbc*]{}]{} and [[*nbb*]{}]{} sets]{} Fix a linear order of the ground set $[n]$. A broken circuit of $G$ is a set of the form $C-\min(C)$, where $C$ is a circuit of $G$. An [[*nbc*]{}]{} set of $G$ is a subset of $[n]$ which contains no broken circuits. The collection of [[*nbc*]{}]{} sets of $G$ will be denoted ${\operatorname{\it nbc}}(G)$. The set of elements of ${\operatorname{\it nbc}}(G)$ of cardinality $p$ is denoted ${\operatorname{\it nbc}}^p(G)$. The dependence on the linear order of $[n]$ is suppressed in the notation. For a flat $X$ of $G$, let ${\operatorname{\it nbc}}_X(G)$ denote the set of [[*nbc*]{}]{} sets with closure equal to $X$. Among the properties of ${\operatorname{\it nbc}}(G)$ we highlight the following. For proofs and a more complete discussion see [@Bj3]. Let $\mu: L {\ensuremath{\longrightarrow}}{\ensuremath{\mathbb{Z}}}$ be the Möbius function of $L$. For any linear order on $[n]$, 1. ${\operatorname{\it nbc}}(G)$ is a pure simplicial complex of dimension ${\operatorname{rk}}(G)-1$. 2. The cardinality of ${\operatorname{\it nbc}}^p(G)$ is equal to $w_p(L)$, the $p^{\rm th}$ Whitney number of $L$. 3. For every flat $X$ of $G$, the cardinality of the set ${\operatorname{\it nbc}}_X(G)$ is equal to $(-1)^{{\operatorname{rk}}(X)}\mu(X)$ \[purity\] The relevance of [[*nbc*]{}]{} sets to Orlik-Solomon algebras was established by several authors independently – see [@Bj3 Section 7.11, 7.10]. The set $\{a_S \ | \ S\in {\operatorname{\it nbc}}(G)\}$ forms a basis for $A(G)$. \[nbcbase\] There are several natural ways in which one might attempt to relate ${\ensuremath{\overline{A}}}(G)$ directly to ${\ensuremath{\overline{G}}}$, motivated by the various connections between $A(G)$ and $G$. (For instance, independent sets in $G$ correspond to nonzero monomials in $A(G)$.) None of these seem to work; the difficulties can all be traced back to the failure of the exchange axiom. There is at least an indirect connection between ${\ensuremath{\overline{A}}}(G)$ and ${\ensuremath{\overline{G}}}$ obtained by generalizing the following well-known property of [[*nbc*]{}]{} sets. Let us impose the natural linear order on $[n]$, unless otherwise noted. An increasing subset $S=\{i_1,\ldots,i_p\}\subseteq [n]$ is nbc if and only if $i_k=\min {\ensuremath{c\ell}}(\{i_k,\ldots,i_p\})$ for each $1\leq k\leq p$. \[bjorner\] Replacing matroid closure with line-closure, we propose the following. An increasing subset $S=\{i_1,\ldots,i_p\}\subseteq [n]$ is [[*nbb*]{}]{} if and only if $$i_k=\min {\ensuremath{\ell c}}(\{i_k,\ldots,i_p\})$$ or each $1\leq k\leq p$. \[nbb\] The collection of [[*nbb*]{}]{} sets of $G$ will be denoted by ${\operatorname{\it nbb}}(G)$. Of course ${\operatorname{\it nbb}}(G)$ is dependent only on ${\ensuremath{\overline{G}}}$, rather than $G$. ${\operatorname{\it nbb}}(G)$ is a simplicial complex, containing ${\operatorname{\it nbc}}(G)$ as a subcomplex. \[ncinnb\] The first assertion follows from the monotonicity of the line-closure operator. The second is a consequence of the fact that ${\ensuremath{\ell c}}(S)\subseteq {\ensuremath{c\ell}}(S)$ for any subset $S$ of $[n]$. We will customarily specify ${\operatorname{\it nbb}}(G)$ and ${\operatorname{\it nbc}}(G)$ by listing the facets, or maximal simplices. Because of the lack of exchange, ${\operatorname{\it nbb}}(G)$ depends heavily on the linear ordering of the points. Let $G$ be the matroid of Example \[wheel\]. Then, with the natural linear order on $[6]$, the facets of ${\operatorname{\it nbb}}(G)$ are $$1246,136,135,125,134, \ \text{and} \ 124.$$ If we adopt the linear ordering $2<1<3<4<5<6$, the new [[*nbb*]{}]{} complex has facets $$246,236,216,235,215,234,\ \text{and} \ 214.$$ In fact, for this second linear ordering. ${\operatorname{\it nbb}}(G)={\operatorname{\it nbc}}(G)$. We see from this example that the number of [[*nbb*]{}]{} sets of a fixed size $p$ is not independent of the linear ordering, and the complex ${\operatorname{\it nbb}}(G)$ may fail to be pure. Compare with Theorem \[purity\](i) and (ii). We also see that ${\operatorname{\it nbb}}(G)$ may agree with ${\operatorname{\it nbc}}(G)$ even when $G$ is not line-closed. However, these [[*nbb*]{}]{} sets do capture the lack of line-closure, in the following sense. The matroid $G$ is line-closed if and only if ${\operatorname{\it nbb}}(G)={\operatorname{\it nbc}}(G)$ for every linear ordering of $[n]$ \[nb=nc\] Suppose $G$ is not line-closed. Then there exists a line-closed set $X$ which is not closed. Let $i\in{\ensuremath{c\ell}}(X)-X$, and choose a linear order on $[n]$ such that $i$ precedes $\min(X)$. Now, let $S=(i_1,\ldots,i_p)$ be the lexicographically first ordered basis for the flat ${\ensuremath{c\ell}}(X)$ which is contained in $X$. Then, by the choice of ordering, $S\not \in {\operatorname{\it nbc}}(G)$, by Theorem \[bjorner\]. We claim $S\in {\operatorname{\it nbb}}(G)$. Suppose not. Then, for some $k$, $i:= \min {\ensuremath{\ell c}}\{i_k, \ldots,i_p\}$ is less than $i_k$. Since $X$ is line-closed, $i\in X$. Also, by the exchange axiom in $G$, $S-\{i_k\}\cup\{i\}$ is a basis for ${\ensuremath{c\ell}}(X)$, and is lexicographically smaller than $S$. This contradicts the choice of $S$. Thus $S\in{\operatorname{\it nbb}}(G)-{\operatorname{\it nbc}}(G)$, so ${\operatorname{\it nbc}}(G)\not = {\operatorname{\it nbb}}(G)$. Conversely, if $G$ is line-closed, then ${\operatorname{\it nbb}}(G)={\operatorname{\it nbc}}(G)$ by Theorem \[bjorner\]. [Independence of [[*nbb*]{}]{} monomials in ${\ensuremath{\overline{A}}}(G)$]{} We turn now to the analysis of the quadratic closure ${\ensuremath{\overline{A}}}(G)$ of the Orlik-Solomon algebra. For each line-closed set $X\in {\ensuremath{\overline{L}}}(G)$, let ${\ensuremath{\mathcal E}}_X$ be the subspace of ${\ensuremath{\mathcal E}}$ spanned by monomials $e_S$ for which ${\ensuremath{\ell c}}(S)=X$. Then we have a grading of ${\ensuremath{\mathcal E}}$ by ${\ensuremath{\overline{L}}}(G)$: $${\ensuremath{\mathcal E}}=\oplus_{X\in {\ensuremath{\overline{L}}}(G)} {\ensuremath{\mathcal E}}_X.$$ Let ${\ensuremath{\mathcal J}}_X={\ensuremath{\mathcal J}}\cap {\ensuremath{\mathcal E}}_X$ and ${\ensuremath{\overline{A}}}_X(G)={\ensuremath{\mathcal E}}_X/{\ensuremath{\mathcal J}}_X$. Then we have the following analogue of [@OT Theorem 3.26]. ${\ensuremath{\overline{A}}}(G)=\oplus_{X\in{\ensuremath{\overline{L}}}(G)} {\ensuremath{\overline{A}}}_X(G)$. \[sum\] The ideal [$\mathcal J$]{} is generated by elements $\partial e_{ijk}$ where $\{i,j,k\}$ is dependent. Since $G$ has no multiple points, $\{i,j,k\}$ is a circuit. Then ${\ensuremath{\ell c}}(\{i,j\})={\ensuremath{\ell c}}(\{i,k\})={\ensuremath{\ell c}}(\{j,k\})$, each being equal to ${\ensuremath{c\ell}}(\{i,j,k\})$. This shows that $\partial e_{ijk}$ is homogeneous in the grading above. Thus ${\ensuremath{\mathcal J}}=\oplus_{X\in{\ensuremath{\overline{L}}}(G)} {\ensuremath{\mathcal J}}_X$, and the result follows. We will also use the following elementary observation, proof left to the reader. The graded derivation $\partial: {\ensuremath{\mathcal E}}{\ensuremath{\longrightarrow}}{\ensuremath{\mathcal E}}$ induces a graded derivation ${\ensuremath{\overline{\partial}}}: {\ensuremath{\overline{A}}}(G) {\ensuremath{\longrightarrow}}{\ensuremath{\overline{A}}}(G)$. We are now prepared to prove the main result. For $S\subset [n]$ we denote by ${\ensuremath{\overline{a}}}_S$ the image of $e_S$ in the quadratic closure ${\ensuremath{\overline{A}}}(G)$. The set $\{{\ensuremath{\overline{a}}}_S \ | \ S\in {\operatorname{\it nbb}}(G)\}$ is linearly independent in ${\ensuremath{\overline{A}}}(G)$. \[indep\] With Lemma \[sum\] in hand the proof is identical to the argument in the proof of Theorem \[nbcbase\]. It is enough to prove the result for [[*nbb*]{}]{} sets of a fixed size $p$. Then we induct on $p$. Suppose $$\sum_{S\in {\operatorname{\it nbb}}^p(G)} \lambda_S {\ensuremath{\overline{a}}}_S=0.$$ By Lemma \[sum\] we may assume that ${\ensuremath{\ell c}}(S)=X$ for a fixed element $X\in {\ensuremath{\overline{L}}}(G)$ and all $S$ in the sum. Setting $i_0=\min(X)$, we have $\min(S)=i_0$ for all $S$, by definition of ${\operatorname{\it nbb}}(G)$. Write $S'=S-\{i_0\}$. Then we have $${\ensuremath{\overline{a}}}_{i_0}\wedge \biggl(\underset{{\ensuremath{\ell c}}(S)=X}{\sum_{S\in{\operatorname{\it nbb}}^p(G)}} \lambda_S {\ensuremath{\overline{a}}}_{S'}\biggr)=0.$$ Applying the derivation ${\ensuremath{\overline{\partial}}}$ we obtain $$\underset{{\ensuremath{\ell c}}(S)=X}{\sum_{S\in{\operatorname{\it nbb}}^p(G)}} \lambda_S {\ensuremath{\overline{a}}}_{S'} + \underset{{\ensuremath{\ell c}}(S)=X}{\sum_{S\in{\operatorname{\it nbb}}^p(G)}} {\ensuremath{\overline{a}}}_{i_0} \wedge {\ensuremath{\overline{\partial}}}{\ensuremath{\overline{a}}}_{S'}=0.$$ Using again the definition of ${\operatorname{\it nbb}}(G)$, we have that $i_0\not\in {\ensuremath{\ell c}}(S')$ for ${\ensuremath{\ell c}}(S)=X$. Then, applying Lemma \[sum\] once more, we have $$\underset{{\ensuremath{\ell c}}(S)=X}{\sum_{S\in{\operatorname{\it nbb}}^p(G)}} \lambda_S {\ensuremath{\overline{a}}}_{S'} =0.$$ Since $S\in {\operatorname{\it nbb}}^p(G)$ implies $S'\in {\operatorname{\it nbb}}^{p-1}(G)$, we conclude $\lambda_S=0$ for all $S$ by the inductive hypothesis. As a consequence we obtain half of Conjecture \[boston\]. Suppose $A(G)$ is quadratic. Then $G$ is line-closed. \[oneway\] If $G$ is not line-closed, then by Theorems \[ncinnb\] and \[nb=nc\] there is a linear ordering of $[n]$ such that the cardinality of ${\operatorname{\it nbb}}(G)$ is strictly greater than that of ${\operatorname{\it nbc}}(G)$. Then, by Theorems \[indep\] and \[nbcbase\], we have $\dim {\ensuremath{\overline{A}}}(G)>\dim A(G)$, so $A(G)$ is not quadratic. Because of Example \[wheel\], it is not the case that $\{{\ensuremath{\overline{a}}}_S \ | \ S\in {\operatorname{\it nbb}}(G)\}$ spans ${\ensuremath{\overline{A}}}(G)$ for every linear order. When we announced Theorem \[indep\] in the Boston lecture[@F13] we expressed some hope that one could show the existence of some linear order for which the [[*nbb*]{}]{}  monomials span ${\ensuremath{\overline{A}}}(G)$, yielding a proof of the converse of Corollary \[oneway\] as well as a combinatorial calculation of $\phi_3(G)$ via Corollary \[count\]. Subsequently S. Yuzvinsky found a counterexample. Let $G$ be the rank-three matroid on $[8]$ with nontrivial lines $$123,\ 148,\ 257,\ 3678,\ \text{and}\ 456,$$ pictured in Figure . One can use Theorem \[baseclosure\] of Section \[formal\] below to check fairly quickly by hand that $G$ is line-closed. On the other hand, one computes $\phi_3(G)=16$. (We use a [ *Mathematica*]{} script available from the author.) Then, by Theorem \[count\], we have $\dim {\ensuremath{\overline{A}}}\,^3 = 16$. But $\dim A^3(G) = 14$. Thus $A(G)$ is not quadratic. \[yuzex\] For us, Example \[yuzex\] shows that there may be no “good” linear order, for which ${\operatorname{\it nbb}}(G)$ spans ${\ensuremath{\overline{A}}}(G)$, for some matroids $G$. See [@DY] for further discussion of the converse of Corollary \[oneway\]. Using Theorem \[oneway\] we have an alternate proof of [@F4 Prop. 5.1]. If $n>{\operatorname{rk}}(G)$ and $G$ has a basis each of whose two-point subsets is closed in $G$, then $A(G)$ is not quadratic. \[doublepoint\] Such a basis $B$ would form a line-closed set by hypothesis, but it cannot be closed in $G$ since $|{\ensuremath{c\ell}}(B)|=n>{\operatorname{rk}}(G)=|B|$. We may also use the work on quadratic algebras together with Corollary \[oneway\] to give a nice sufficient condition for line-closure of matroids. The following assertion is a generalization of [@F8 Proposition 3.2], with essentially the same proof. Suppose, for every circuit $S$ of $G$ with $|S|\geq 4$, the closures in $G$ of two disjoint two-point subsets of $S$ meet. Then $G$ is line-closed. \[parallel\] We show that $A(G)$ is quadratic by verifying directly that ${\ensuremath{\mathcal I}}^p\subseteq {\ensuremath{\mathcal J}}^p$ for all $p\geq 3$. Let $S=\{i_1,\ldots, i_p\}$ be a circuit, $p\geq 4$, and suppose $i_0\in {\ensuremath{c\ell}}(\{i_1,i_2\})\cap {\ensuremath{c\ell}}(\{i_3,i_4\})$. Then $$\partial e_{i_1i_2i_3i_4}=(e_{i_3}-e_{i_4})\partial e_{i_0i_1i_2} + (e_{i_1}-e_{i_2})\partial e_{i_0i_3i_4}.$$ Thus $\partial e_{i_1i_2i_3i_4}$ lies in ${\ensuremath{\mathcal J}}$. Then $e_{i_1i_2i_3i_4} \in {\ensuremath{\mathcal J}}$, and it follows from the Leibniz rule that $\partial e_S$ lies in ${\ensuremath{\mathcal J}}$. If $G$ has rank three, the hypothesis of Theorem \[parallel\] can be weakened, for in this case it suffices to show that $\partial e_S\in {\ensuremath{\mathcal J}}$ for those circuits $S$ with $|S|\geq 4$ and $1\in S$. Arrangements of rank three whose matroids satisfy the weaker hypothesis are called [*parallel arrangements.*]{} See [@FR1]. [Generalization to high rank/degree]{} All of the results of this section on line-closure and quadraticity can be generalized, with essentially identical proofs. A subset of $[n]$ is [*$r$-closed*]{} if it contains the closures of all of its $p$-subsets for all $p\leq r$. The matroid $G$ is [*$r$-closed*]{} if every $r$-closed set is closed. For arbitrary $G$ the collection $L_r(G)$ of $r$-closed sets forms a lattice, and we have a sequence of surjective order-preserving maps $$B_n=L_1(G){\ensuremath{\longrightarrow}}{\ensuremath{\overline{L}}}(G)=L_2(G) {\ensuremath{\longrightarrow}}\cdots {\ensuremath{\longrightarrow}}L_r(G) {\ensuremath{\longrightarrow}}\cdots {\ensuremath{\longrightarrow}}L_n(G)=L(G),$$ where $B_n$ is the boolean lattice. The [*degree $r$ closure*]{} ${\ensuremath{\overline{A}}}_r(G)$ of $A(G)$ is ${\ensuremath{\mathcal E}}/{\ensuremath{\mathcal J}}_r$, where ${\ensuremath{\mathcal J}}_r$ is the ideal generated by the elements of [$\mathcal I$]{} of degree less than or equal to $r$. Thus we have a sequence of surjective homomorphisms $${\ensuremath{\mathcal E}}=A_1(G) {\ensuremath{\longrightarrow}}{\ensuremath{\overline{A}}}(G)=A_2(G) {\ensuremath{\longrightarrow}}\cdots {\ensuremath{\longrightarrow}}A_r(G) {\ensuremath{\longrightarrow}}\cdots A_n(G)=A(G).$$ An ordered subset $S=\{i_1,\ldots, i_p\}$ is $r$-[[*nbb*]{}]{} if for each $k$, $i_k$ is the first element in the $r$-closure of $\{i_k\ldots, i_p\}$. $A_r(G)=\oplus_{X\in{\ensuremath{\overline{L_r}}}(G)} {\ensuremath{\overline{A}}}_X(G)$. The crucial points are (i) that ${\ensuremath{\mathcal J}}_r$ is generated by boundaries of [*circuits*]{} of size at most $r+1$, and (ii) that $r$-closure agrees with matroid closure on sets of size at most $r$. Using these observations the proof of Lemma \[sum\] is easy to adapt to the more general setting. The proof of the following generalization is now identical to the proof of Theorem \[indep\]. The set of monomials in ${\ensuremath{\mathcal A}}_r(G)$ corresponding to the $r$-[[*nbb*]{}]{} sets of $G$ forms a linearly independent set. If $A_r(G)=A(G)$, then $G$ is $r$-closed. [The [[*nbb*]{}]{} complex and a generalization to nonlinear orders]{} After formulating Definition \[nbb\] and proving Theorem \[indep\], we found that our notion of [[*nbb*]{}]{} set coincides with a special case of the more general notion of [[*NBB*]{}]{} set in a finite lattice with a partial ordering of the atoms, introduced by Blass and Sagan in [@BSag]. These results are stated only for the line-closure of $G$, but again analogous results will hold for $r$-closure. ([@BSag]) Suppose $({\ensuremath{\overline{L}}},\leq)$ is a finite lattice, and $\preceq$ is a partial ordering of the atoms of ${\ensuremath{\overline{L}}}$. A set $T$ of atoms is [*bounded below*]{} if there exists an atom $a$ such that $a< \bigvee T$ and $a\prec t$ for all $t\in T$. A set $S$ is [[*NBB*]{}]{} if $S$ does not contain any set $T$ which is bounded below. A set $S$ is [[*nbb*]{}]{} if and only if $S$ is an [[*NBB*]{}]{} set in the lattice ${\ensuremath{\overline{L}}}(G)$ for the given linear ordering of the atoms. \[sagan\] In our setting the atom ordering $\preceq$ is the natural linear ordering on $[n]$. In this context a set $T\subseteq [n]$ is bounded below if and only if there exists $i\in {\ensuremath{\ell c}}(T)$ with $i<\min(T)$. Suppose $S=\{i_1,\ldots,i_p\}$ is [[*nbb*]{}]{} and $T\subseteq S$. Let $i_k=\min(T)$. Then ${\ensuremath{\ell c}}(T)\subseteq {\ensuremath{\ell c}}(\{i_k,\ldots,i_p\}$, so $\min({\ensuremath{\ell c}}(T))\geq \min({\ensuremath{\ell c}}(\{i_k,\ldots,i_p\}))$. Since $S$ is [[*nbb*]{}]{} we conclude $\min({\ensuremath{\ell c}}(T))=i_k$, so $T$ is not bounded below. Conversely, if $S$ is not [[*nbb*]{}]{} then for some $k$, $T=\{i_k,\ldots,i_p\}$ is bounded below by $\min({\ensuremath{\ell c}}(T))<i_k,$ and thus $S$ is not [[*NBB*]{}]{}. This observation yields a numerical result on the number of [[*nbb*]{}]{}  sets, by one of the main results of [@BSag]. Let ${\ensuremath{\overline{\mu}}}: {\ensuremath{\overline{L}}}(G) {\ensuremath{\longrightarrow}}{\ensuremath{\mathbb{Z}}}$ denote the Möbius function of ${\ensuremath{\overline{L}}}(G)$. The sum $\underset{{\ensuremath{\ell c}}(S)=X}{\underset{S\in {\operatorname{\it nbb}}(G)}{\sum}} (-1)^{|S|}$ is equal to ${\ensuremath{\overline{\mu}}}(X)$. Let ${\operatorname{\it NBB}}(G,\preceq)$ denote the collection of [[*NBB*]{}]{}  sets of ${\ensuremath{\overline{L}}}(G)$ under the atom-order $\preceq$. Suppose $\preceq$ is a partial order on $[n]$ with the property that each line-closed set $X$ has a unique smallest element relative to $\preceq$. Then $$\{{\ensuremath{\overline{a}}}_S \ | \ S\in {\operatorname{\it NBB}}(G,\preceq)\}$$ is linearly independent in ${\ensuremath{\overline{A}}}(G)$. \[nonlinear\] Assume without loss that the natural order on $[n]$ is a linear extension of $\preceq$. Then the proof of \[indep\] goes through without change. We also have the following analogue of Theorem \[nb=nc\]. Recall that ${\operatorname{\it nbc}}(G)$ is determined by a linear order on $[n]$. ${\operatorname{\it nbc}}(G)\subseteq{\operatorname{\it NBB}}(G,\preceq)$ for any linear order which extends $\preceq$. Suppose $T$ is a bounded below set. Let $a\leq \bigvee T$ with $a\prec t$ for all $t\in T$. Then $a$ precedes $\min(T)$ in any linear extension of $\preceq$. Since $\bigvee T ={\ensuremath{\ell c}}(T)\subseteq {\ensuremath{c\ell}}(T)$, it follows that $T$ contains a broken circuit. Theorem \[nonlinear\] raises the possibility of finding more than $|{\operatorname{\it nbc}}(G)|$ linearly independent monomials in ${\ensuremath{\overline{A}}}(G)$, even when $G$ is line-closed. At this point we have no examples of this phenomenon. [Combinatorial notions of formality]{} \[comb\] Our research on line-closed matroids was motivated by the study of formal arrangements, and specifically by attempts to describe combinatorially the property of an arrangement being “generic with given codimension two structure." We start this section by recalling the definition of formal arrangement, and outlining some of the motivation and main results. We then turn to various combinatorial versions of formality. These have been studied to some extent before, but the definitions have never been published. We present some newly rediscovered results and examples, which appeared long ago in the combinatorial literature but not in the context of formal arrangements. Line-closure turns out to be the strongest among the properties we study here. The remaining notions form a hierarchy descending to formality of a realization, which in itself is not a combinatorial notion. We show by example that none of the combinatorial notions have the nice topological or algebraic consequences that formality affords. [Formal arrangements]{} The notion of formality was introduced in [@FR1] and studied further in [@Y1; @BT; @BB]. The terminology is unfortunate; there is a notion of formality of spaces that is important in rational homotopy theory, and has implications for arrangements, but the definition of formal arrangement is completely unrelated. We adopt the definition from [@BT]. Henceforth assume [$\mathcal A$]{} is an [*essential*]{} arrangement, that is, ${\operatorname{rk}}(G)=\dim(V)$. Let $e_i$ denote the $i^{\rm th}$ standard basis vector in ${\ensuremath{\mathbb{K}}}^n$. The [ *weight*]{} of a vector in ${\ensuremath{\mathbb{K}}}^n$ is the number of nonzero entries. Let $\rho: {\ensuremath{\mathbb{K}}}^n {\ensuremath{\longrightarrow}}V^*$ be the linear mapping defined by $\rho(e_i)={\ensuremath{\alpha}}_i$. Then [$\mathcal A$]{} is [*formal*]{} if the kernel of $\rho$ is spanned by elements of weight at most three. \[formaldef\] An element of $\ker(\rho)$ of weight three corresponds to a dependent set of $G$ of size three, and thus of rank two. A vector in $\ker(\rho)$ gives the coefficients in a dependence relation among the linear forms. Thus [$\mathcal A$]{} is formal if all dependence relations among the forms ${\ensuremath{\alpha}}_i$ are consequences of “rank two dependence relations." There is a natural way to associate a subspace $W$ of ${\ensuremath{\mathbb{K}}}^n$ of dimension $r$ with a (possibly degenerate) arrangement of $n$ hyperplanes in ${\ensuremath{\mathbb{K}}}^r$, by considering the set of intersections of $W$ with the $n$ coordinate hyperplanes as an arrangement in $W$. If $K\subseteq {\ensuremath{\mathbb{K}}}^n$ denotes the kernel of $\rho$, then its orthogonal complement $K^\perp$ returns the original arrangement [$\mathcal A$]{}  under this construction. Indeed, the linear mapping $$\Phi=({\ensuremath{\alpha}}_1,\ldots,{\ensuremath{\alpha}}_n): V {\ensuremath{\longrightarrow}}{\ensuremath{\mathbb{K}}}^n$$ carries $V$ isomorphically to $K^\perp$ and $H_i$ to the intersection of $K^\perp$ with $\{x_i=0\}$. Let $F\subseteq K$ denote the subspace spanned by elements of weight three. The arrangement corresponding to the subspace $F^\perp \subseteq {\ensuremath{\mathbb{K}}}^n$ is called the [*formalization*]{} of [$\mathcal A$]{}, denoted ${\ensuremath{\mathcal A}}_F$. This construction first appears in [@Y1]. A [*section*]{} ${\ensuremath{\overline{{\ensuremath{\mathcal B}}}}}$ of an arrangement [$\mathcal B$]{} in $V$ is formed by intersecting the hyperplanes of [$\mathcal B$]{} with a linear subspace $W$ of $V$. The section ${\ensuremath{\overline{{\ensuremath{\mathcal B}}}}}$ is [*generic*]{} if $W$ is transverse to every intersection of hyperplanes of [$\mathcal B$]{}. In this case the combinatorics and topology of ${\ensuremath{\overline{{\ensuremath{\mathcal B}}}}}$ depend only on [$\mathcal B$]{} and $\dim(W)$. A section of [$\mathcal B$]{} by a 3-dimensional subspace is called a [*planar section*]{}. Let [$\mathcal A$]{} be an essential arrangement. Then > 1. ${\ensuremath{\mathcal A}}_F$ is formal. > > 2. ${\ensuremath{\mathcal A}}$ is a section of ${\ensuremath{\mathcal A}}_F$. > > 3. ${\ensuremath{\mathcal A}}$ and ${\ensuremath{\mathcal A}}_F$ have identical generic planar sections. > > 4. ${\ensuremath{\mathcal A}}$ is formal if and only if ${\operatorname{rk}}({\ensuremath{\mathcal A}})={\operatorname{rk}}({\ensuremath{\mathcal A}}_F)$. > \[formalprop\] We remark that [$\mathcal A$]{} need not be a generic section of ${\ensuremath{\mathcal A}}_F$, nor of any other arrangement. Indeed, if there are no nontrivial lines in $G$, then ${\ensuremath{\mathcal A}}_F$ is the boolean arrangement. Then, if [$\mathcal A$]{} has some nontrivial plane, and has more than four elements, [$\mathcal A$]{} will not be a generic section of ${\ensuremath{\mathcal A}}_F.$ If [$\mathcal A$]{} is inerectible, it will not be a generic section of any arrangement. Such arrangements are easy to construct. The interest in formal arrangements is due to the following theorem. Let [$\mathcal A$]{} be an essential arrangement. Then > 1. If [$\mathcal A$]{} is a $K(\pi,1)$ arrangement, then [$\mathcal A$]{} is formal. > > 2. If [$\mathcal A$]{} is a rational $K(\pi,1)$ arrangement, then [$\mathcal A$]{}  is formal. > > 3. If [$\mathcal A$]{} is a free arrangement, then [$\mathcal A$]{} is formal. > > 4. If [$\mathcal A$]{} has quadratic Orlik-Solomon algebra, then [$\mathcal A$]{} is formal. > \[goodstuff\] Assertions (i) and (iv) above are easy consequences of Theorem \[formalprop\], and (ii) is a consequence of (iv), because rational $K(\pi,1)$ arrangements have quadratic Orlik-Solomon algebras. See [@FR2]. Assertion (iii) was proved in [@Y1]. In [@Y1], Yuzvinsky presented examples of two arrangements, one formal and the other not, with the same underlying matroid. See [@FR4] for diagrams of the dual point configurations. The underlying matroid is the dual of the matroid of complete bipartite graph $K_{3,3}$. This observation yields different realizations and a geometric explanation of this phenomenon. \[k33\] The diagram on the left in Figure  is a representation of the rank-four matroid $G^*$ dual to the graphic matroid of $K_{3,3}$. By considering intersecting planes (in ${\ensuremath{\mathbb{C}}}^3$), one can see that the three dotted lines in this, or in any [$\mathbb{C}$]{}-representation of $G^*$, must be concurrent. The diagram on the right is a representation of the truncation $T(G^*)$ in which the corresponding lines are not concurrent. And indeed, one can show that the configuration on the right is formal. In fact, it cannot be lifted to a rank four configuration with the same points and lines. [Combinatorial formality]{} \[formal\] Example \[k33\] shows that formality is not a combinatorial property. Since the discovery of these examples, efforts have been made to strengthen Theorem \[goodstuff\] by replacing the formality assumption with some purely combinatorial property. In this subsection we present several reasonable candidates. Theorem \[formalprop\](ii) suggests a natural combinatorial formulation of formality. We recall for the reader the notion of strong map (or quotient) of matroids. See [@Wh1 Section 7.4 and Chaps. 8-9] for more details. Suppose $G'$ and $G$ are two matroids on ground set $[n]$. We say $G$ is a [*quotient*]{} of $G'$ if every closed set in $G$ is closed in $G'$. This is the case precisely when the identity map on $[n]$ is a strong map from $G'$ to $G$. If $G'$ is the matroid of an arrangement [$\mathcal A$]{}, then the matroid of any section of [$\mathcal A$]{} is a quotient of $G$. The matroid of a generic $r$-dimensional section of [$\mathcal A$]{} coincides with the [*truncation*]{} $G^{[r]}$ of the matroid $G$ to rank $r$, the matroid whose dependent subsets are those of $G$ together with every subset of size greater than $r$. A matroid $G$ is [*taut*]{} if $G$ is not a quotient of any matroid $G'\not=G$ satisfying $(G')^{[3]}=G^{[3]}$. The condition $(G')^{[3]}=G^{[3]}$ says merely that $G$ and $G'$ have the same points and lines. The following assertion is a consequence of Theorem \[formalprop\]. If $G$ is a taut matroid, then every arrangement realizing $G$ is formal. \[cformal\] In a lecture in 1992 [@Y6], Yuzvinsky formulated a definition of “dimension" of a geometric lattice, or matroid, based on line-closure: the [*dimension*]{} of $G$ is the size of the smallest set of points whose line-closure is $[n]$. Corollary \[look\] below was presented in [@Y6], but was never published. The result was already known to matroid theorists [@Cra]. Suppose $G$ has a basis (of ${\operatorname{rk}}(G)$ points) whose line-closure is $[n]$. Then $G$ is taut. \[tautness\] Suppose $G$ is a quotient of a matroid $G'$ with the same points and lines as $G$. Since the closure of $B$ in $G'$ contains the line-closure of $B$ in $G'$, which agrees with the line-closure of $B$ in $G$, we conclude that $B$ is a basis for $G'$. Thus ${\operatorname{rk}}(G')={\operatorname{rk}}(G)$. It follows that $G'=G$. The following criterion is the standard method to prove an arrangement is formal, although it has never appeared in the literature. Suppose $G$ has a basis whose line-closure is $[n]$. Then every realization of $G$ is formal. \[look\] We now have four notions which might capture the combinatorics of formality, at least in spirit: 1. $G$ is line-closed, 2. $G$ has a basis whose line-closure is $[n]$, 3. $G$ is taut, that is, $G$ is not a proper quotient preserving points and lines, and 4. every realization of $G$ is formal. We have the string of implications $$(i)\implies (ii) \implies (iii) \implies (iv).$$ The first of these is trivial, and the others were proved in the preceding paragraphs. We now give counter-examples for the first two of the reverse implications. We note that Example \[k33\] provides a formal arrangement whose matroid fails to satisfy (iv). In [@Cra] there appears an example of a matroid $G$ of rank three on nine points, with the property that no set of three points line-closes to the entire ground set $[9]$. See Figure . Thus (ii) fails. But it is easy to see that $G$ is not a quotient of a rank-four matroid with the same points and lines, so (iii) is satisfied. \[threenottwo\] The rank-three wheel, illustrated in Figure , provides an example of a matroid which satifies (ii) but is not line-closed. \[twonotone\] Finding a counter-example for the implication “$(iv) \implies (iii)$” presents a delicate problem. One needs a [$\mathbb{C}$]{}-representable matroid which is a proper quotient, preserving points and lines, but such that the quotient “mapping” is not representable over [$\mathbb{C}$]{}, either because the larger matroid is not [$\mathbb{C}$]{}-representable, or because the larger matroid is not realizable in such a way that the original matroid is obtained from it via projection. [Local formality]{} In [@Y1] an arrangement is defined to be [*locally formal*]{} if, for every flat $X$, the arrangement ${\ensuremath{\mathcal A}}_X:=\{H_i \ | \ i \in X\}$ is formal. This idea can be applied to the combinatorial notions of formality discussed above. We will focus on the local version of tautness, for reasons that will become clear. A matroid $G$ is [*locally taut*]{} if the restriction of $G$ to any flat is taut. By Corollary \[cformal\], any realization of a locally taut matroid is locally formal. Theorem \[tautness\] can be used to compare local tautness to line-closure, in the next pair of results. Suppose $G$ is a matroid in which every flat $X$ has a basis whose line-closure is equal to $X$. Then $G$ is locally taut. \[localtaut\] Thus is an immediate consequence of Theorem \[tautness\]. By way of contrast, the definition of line-closed matroid may be restated as follows. A matroid $G$ is line-closed if and only if, for every flat $X$, [*every*]{} basis of $X$ has line-closure equal to $X$. \[baseclosure\] Thus every line-closed matroid is even locally taut, strengthening Theorem \[tautness\]. The rank-three wheel $W_3$ (Examples \[wheel\] and \[twonotone\]) serves as an example of a locally taut matroid which is not line-closed. We make one more observation concerning locally taut matroids. Suppose $G$ is locally taut. Then $G$ is determined by its points and lines. \[pointline\] This is a consequence of part (5) of Theorem 7.5.4 in [@Bry3], which asserts that $G$ is determined by its essential flats, along with their ranks. A flat of $G$ is essential if it is a truncation of a matroid of higher rank. Truncation is in particular a quotient map, and preserves points and lines, so long as the image has rank at least two. Thus, if $G$ is locally taut, the only essential flats of $G$ are the nontrivial lines. So $G$ is determined by the number of points and the list of nontrivial lines. The converse of this result also holds. For if $G$ is not locally taut, then $G$ is not taut, so $G$ is a quotient of a distinct matroid with the same points and lines. A natural question is evident: what axioms govern the point-line incidence structures of locally taut matroids? [Topology vs. combinatorics]{} One motivation of the present discussion is to find combinatorial conditions with the same consequences as formality, as in Theorem \[goodstuff\]. Unfortunately, we have no positive results of this sort, aside from the implication “quadratic Orlik-Solomon algebra implies line-closed matroid” strengthening assertions (ii) and (iv) of that theorem. First of all, it is not the case that $K(\pi,1)$ or free arrangements must have line-closed matroids. Consider the arrangement [$\mathcal A$]{} with defining equation $Q(x,y,z)=z(x+y)(x-y)(x+z)(x-z)(y+z)(y-z)$, denoted $J_2$ in [@FR1 section 2.6]. The underlying matroid is the non-Fano plane, pictured in Figure . Then [$\mathcal A$]{} is not line-closed: apply Corollary \[doublepoint\] to the three “edge-midpoints.” But this arrangement is well-known to be both free and $K(\pi,1)$ - see [@FR1]. \[jay2\] Line-closure is not sufficient for $K(\pi,1)$-ness or freeness either. Let [$\mathcal A$]{} be the parallel arrangement with defining equation $Q(x,y,z)= z(x+z)(x-z)(y+z)(y-z)(x+y+2z)(x+y-2z)$, which appears as $X_2$ in [@FR1 Section 2.6]. The underlying matroid $G$ of [$\mathcal A$]{} is pictured in Figure . Since the hypothesis of Theorem \[parallel\] is satisfied, $G$ is line-closed. But [$\mathcal A$]{} is not a free arrangement, nor is [$\mathcal A$]{} a $K(\pi,1)$ arrangement, nor a rational $K(\pi,1)$ arrangement (though it has quadratic Orlik-Solomon algebra). The first of these assertions holds because the characteristic polynomial of $G$ has non-integer roots. The second assertion was proved by L. Paris (unpublished), and the last was proved in [@Y5]. See [@FR4]. We close with a fascinating conjecture about the combinatorial structure of free or $K(\pi,1)$ arrangements inspired by the preceding discussion. First we note the stronger version of Theorem \[goodstuff\]: a free or $K(\pi,1)$ arrangement must in fact be [*locally*]{} formal [@F1; @Y1]. Now, let us consider the hierarchy “$(i)\implies (ii)\implies (iii)\implies (iv)$” among the combinatorial notions we have introduced. We have seen that that the matroids of $K(\pi,1)$ or free arrangements need not satisfy (i). The question whether such matroids satisfy (iv) is very close to two famous and important problems in the theory of arrangements. Indeed, a matroid which underlies both a free arrangement and also a non-formal arrangement would be a counter-example to Terao’s conjecture, that freeness is a matroidal property. A matroid which underlies both a $K(\pi,1)$ arrangement and a non-formal arrangement would improve upon Rybnikov’s (non-$K(\pi,1)$) counter-examples to the homotopy-type conjecture, that the homotopy type of the complement is matroidal. See [@FR4] for a discussion of the latter problem. Finally, we note that the matroid appearing in Example \[jay2\], while not line-closed, is taut (i.e., satisfies (iii)), in fact locally taut, by Theorem \[look\], and is therefore determined by its points and lines. The underlying matroid of a free or $K(\pi,1)$ arrangement is determined by its points and lines. These ideas have developed through conversations with several people over the years. We are especially grateful to Sergey Yuzvinsky and Joseph Kung for helpful discussions along the way. In particular, Kung pointed out Theorem \[pointline\] to us. Samantha Melcher helped with examples related to the [[*nbb*]{}]{} complex during a summer REU project in 1998. We also thank Henry Crapo for explaining to us his interpretation of Yuzvinsky’s examples, which we reproduced in Example \[k33\]. [Fal89b]{} M. Bayer and K. Brandt. Discriminantal arrangements, fiber polytopes, and formality. , 6:229–246, 1997. A. Bj[ö]{}rner. Homology and shellability of matroids and geometric lattices. 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--- abstract: 'Given an irrational number $\alpha\in(0,1)$, the Sudler product is defined by $P_N(\alpha) = \prod_{r=1}^{N}2|\sin\pi r\alpha|$. Answering a question of Grepstad, Kaltenböck and Neumüller we prove an asymptotic formula for distorted Sudler products when $\alpha$ is the golden ratio $(\sqrt{5}+1)/2$ and establish that in this case $\limsup_{N \to \infty} P_N(\alpha)/N < \infty$. We obtain similar results for quadratic irrationals $\alpha$ with continued fraction expansion $\alpha = [a,a,a,\dots]$ for some integer $a \geq 1$, and give a full characterization of the values of $a$ for which $\liminf_{N \to \infty} P_N(\alpha)>0$ and $\limsup_{N \to \infty} P_N(\alpha) / N < \infty$ hold, respectively. We establish that there is a (sharp) transition point at $a=6$, and resolve as a by-product a problem of the first named author, Larcher, Pillichshammer, Saad Eddin, and Tichy.' address: - 'Institute of Analysis and Number Theory, TU Graz, Steyrergasse 30, 8010 Graz, Austria; E-mail:aistleitner@math.tugraz.at' - 'School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel; E-mail address: niclast@mail.tau.ac.il' - 'Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway; Email address: agamemnon.zafeiropoulos@ntnu.no' author: - Christoph Aistleitner - Niclas Technau - Agamemnon Zafeiropoulos title: on the order of magnitude of Sudler products --- [^1] Introduction and statement of results – the case of the golden ratio ==================================================================== Definition and overview ----------------------- Given a real number $\alpha$ and an integer $N\geq 1,$ the [*Sudler product*]{} at stage $N$ with parameter $\alpha$ is defined as $$\label{defn} P_N(\alpha) = \prod_{r=1}^{N}2|\sin \pi r\alpha| .$$ When $\alpha= m/n$ is a rational number we can see that $P_N(\alpha) = 0$ for all $N\geq n$, so in this case the asymptotic behavior of $P_N$ as $N \to \infty$ is trivial. However, we note that for $\alpha=1/n$ and $N = n-1$ we have the important trigonometric identity $\prod_{r=1}^{n-1} 2\sin(\pi r/n)= n$, which in terms of the notation we introduced above can be written as $P_{n-1}(1/n)=n$; this identity will play a role throughout the paper. Since the periodicity of the sine function implies that $P_{N}(\alpha) = P_{N}(\{\alpha\})$, where $\{\alpha\}$ is the fractional part of $\alpha$, in order to study the asymptotic behavior of products $P_{N}(\alpha)$ as $N \to \infty$ we can restrict ourselves to irrational numbers $0\leq \alpha < 1$.\ Sudler products have been studied in various contexts, as they appear to have connections with many different areas of research. The first known appearance of products of the form is in a paper of Erdős and Szekeres [@erdos]. There it is proved that $$\liminf_{N\to\infty}P_N(\alpha)=0 \text{ and } \limsup_{N \to \infty}P_N(\alpha)=\infty \hspace{2mm} \text{ for almost all } \alpha,$$ and it is conjectured that $$\label{erdosconj} \liminf_{N\to\infty}P_N(\alpha) = 0 \hspace{3mm} \text{ for all $\alpha$.}$$ In [@erdos] it is also claimed without proof that the limit $E := \lim_{N\to \infty}\|P_N\|_{\infty}^{1/N}$ exists and $1<E<2.$ (Here $\|\cdot \|_{\infty}$ denotes the supremum norm on the interval $[0,1]$.) A formal proof of this claim was given by Sudler [@sudler], who also gave a precise formula for the limit $E$ and provided asymptotic estimates for the points $\alpha_N\in (0,1)$ for which $\|P_N\|_{\infty} = P_{N}(\alpha_N) $.\ The size of the $L_{\infty}$- as well as the $L_p$-norms of Sudler products and of certain subproducts has been studied extensively, and for more results we refer to [@atkinson; @bell; @jordanbell; @bourgain; @freiman; @kol1; @kol2; @wright]. In the present paper we focus our attention on results concerning the pointwise growth of Sudler products. Towards this direction, the asymptotic estimate $|\sin\pi x | \asymp \|x\|, \, x\to 0 ,$ shows the intimate connection of the size of $P_N(\alpha)$ with the Diophantine properties of $\alpha$. (We write $\|x\|$ for the distance of $x$ to its nearest integer).\ Pointwise estimates for Sudler products play a key role in the proof of the Ten Martini Problem by Avila and Jitomirskaya [@aj1]. Pointwise lower bounds for Sudler products (in the case of general irrational $\alpha$) also play a crucial role in [@aj2] by Avila, Jitomirskaya and Marx. Furthermore, we want to point to very recent work by Bettin and Drappeau [@bd1; @bd2; @bd3]. Following work of Zagier [@zagier], they study the order of magnitude of the Kashaev invariant of certain hyperbolic knots, which in terms of the present paper could be described as an average of Sudler products with fixed rational parameter. Some aspects of their work also play an explicit (continued fractions, Diophantine approximation) or implicit (cotangent sums) role in the present paper, but some aspects (modularity, reciprocity formulas) do not play a visible role in our paper at all. In all the papers mentioned in this paragraph, the approach taken is somewhat different from the one taken in the present paper, but it seems very interesting to compare all these approaches and to find a unified picture.\ Although the exponential growth of $\|P_N\|_{\infty}$ proved in [@sudler] could lead one to believe that the sequence $( P_N(\alpha) )_{N=1}^{\infty}$ also exhibits the same behaviour for most values of $\alpha$ (from the metrical point of view), it has been shown that this is not the case. Lubinsky and Saff [@lubinsky2] proved that $\lim_{N\to\infty}P_N(\alpha)^{1/N}=1 $ for almost all $\alpha$. Subsequently, Lubinsky [@lubinsky] proved several results which explicitly exhibit the underlying relation between the asymptotic order of magnitude of $P_N(\alpha)$ and the Diophantine properties of $\alpha$ as encoded in its continued fraction expansion $\alpha=[a_1,a_2,a_3,\ldots]$. To be more specific, it is proved in [@lubinsky] that for any ${\varepsilon}>0$ we have $$\log P_N(\alpha) \, \ll \, \log N (\log\log N)^{1+{\varepsilon}}\quad \text{ as }\, N\to \infty, \qquad \text{ for almost all } \alpha$$ (in this statement and throughout our paper, “$\ll$” is the usual Vinogradov symbol). We refer the reader to [@lubinsky Theorem $1.1$] for the more precise metrical statement. This result gives an upper bound on the order of magnitude of $P_N(\alpha)$ for typical $\alpha$ which is almost polynomial. On the other hand, Lubinsky showed that $$\label{limsuplowerbound} \limsup_{N \to \infty} \frac{\log P_N(\alpha)}{\log N} \geq 1 \hspace{4mm} \text{ for all irrational } \alpha,$$ which means heuristically that the higher order of magnitude of $P_N(\alpha)$ is at least linear. Since then it has been conjectured by the first named author, Larcher, Saad Eddin, and Tichy [@products]) that equality is true in , and additionally that the even stronger statement $$\label{folkloreconj} \limsup_{N \to \infty} \frac{P_{N}(\alpha)}{N} <\infty$$ holds for all irrational $\alpha$. Green [@green] points out in his mathematical review of [@products] that not even the existence of a particular $\alpha$ satisfying is known, and mentions the specific case of $\alpha = \sqrt{2}$ as an example. Later in the paper we show that is indeed true for $\alpha=\sqrt{2}$ but fails for many quadratic irrationals: the maximal order of magnitude of $P_{N}(\alpha)$ depends sensitively on the size of the partial quotients of $\alpha$, see Theorem \[lineargrowththeorem\] and Corollary \[height corr\].\ Lubinsky also showed in [@lubinsky] that $$\label{liminf} \liminf_{N \to \infty}P_N(\alpha) = 0$$ for any irrational $\alpha$ with unbounded partial quotients in its continued fraction expansion, while for irrational numbers $\alpha$ with bounded partial quotients (such numbers are called [*badly approximable*]{} in the context of Diophantine approximation) he showed that $$\log P_N(\alpha) \ll \log N, \hspace{4mm} \text{ as } N\to \infty,$$ where the implicit constant depends on $\alpha$. Thus for badly approximable $\alpha$ the product $P_N(\alpha)$ has polynomial upper order of magnitude. In addition to all results stated explicitly in [@lubinsky], there is a statement which is mentioned as a byproduct of the proof of for irrationals with unbounded partial quotients; it is mentioned that relation is actually also true for $\alpha$ with bounded partial quotients, provided that the partial quotients are infinitely often large enough. For a more detailed explanation of this phenomenon, we refer to Theorem $5.1$ of the survey paper [@sigrid3].\ More recently, Mestel and Verschueren [@mv] examined the behaviour of the sequence of Sudler products evaluated on the golden ratio $\frac{\sqrt{5}+1}{2}$. By periodicity the Sudler product for the golden ratio is the same as the Sudler product for $\phi := \frac{\sqrt{5}-1}{2} = \frac{\sqrt{5}+1}{2} -1 \approx 0.618\dots$, which is the conjugate of the golden ratio. Throughout our paper, $\phi$ will always denote this number (and by a very slight abuse of notation, we will call $P_N(\phi)$ the Sudler product for the golden ratio). Mestel and Verschueren established the convergence of the subsequence $P_{F_n}(\phi)$ to some positive and finite limit, where $F_n$ is the $n$-th Fibonacci number. Note that the Fibonacci numbers are the denominators of continued fraction approximations to $\phi$, which emphasizes the connection between the Sudler product and Diophantine properties of the parameter. The methods developed in [@mv] form the basis for many subsequent works as well as for the analysis in the current paper, and we will return to them with more details later in the course of the proofs.\ Expanding the techniques of [@mv], Grepstad and Neumüller [@sigrid2] showed the convergence of specific subsequences of $P_N(\alpha)$ when $\alpha$ is a [*quadratic irrational*]{}, while Grepstad, Kaltenböck and Neumüller [@gkn] proved that for the specific case of the golden ratio we actually have $$\label{liminf_pos} \liminf_{N \to \infty}P_N(\phi) > 0,$$ thus disproving the conjecture of Erdős and Szekeres in [@erdos]. This last result is particularly striking, since it shows that whether $\alpha$ satisfies or not depends on the actual size of the partial quotients of $\alpha$, and not only on whether they are bounded or not. As previously mentioned, this dependence on the size of partial quotients is a phenomenon which we also encounter in one of the main results of this paper.\ Main results, Part 1: The case of the golden ratio $\phi$ {#golden_case} --------------------------------------------------------- As noted in the introduction, throughout this paper we write $\phi=\frac{\sqrt{5}-1}{2}$ for the conjugate of the golden ratio. The continued fraction expansion of this number is $[1,1,1,\dots]$. Throughout the paper $(F_n)_{n=0}^{\infty}$ denotes the Fibonacci sequence defined by $F_0=0, F_1=1,$ and $F_{n+1}= F_n + F_{n-1}$ for all $n=1,2,\ldots$ The sequence of Fibonacci numbers is closely associated with $\phi $, since each $F_n$ is a denominator of a convergent of $\phi$.\ The precise result obtained by Mestel and Verschueren in [@mv] is the following. \[th\_mv\]For the sequence $P_{F_n}(\phi)$, there exists a constant $C_1>0$ such that $$\label{c} C_1 = \lim_{n\to\infty} P_{F_n}(\phi).$$ Moreover, for the same constant $C_1$ we have $$\label{mv2} \lim_{n\to \infty} \frac{P_{F_n-1}(\phi)}{F_{n}} = \frac{C_1\sqrt{5}}{2\pi} \cdot$$ Regarding the value of the constant $C_1$, Mestel and Verschueren in [@mv Theorem $1$] give the approximate value $C_1\approx 2.407 .$ However, this value seems to be purely based on experimental observation, and what they actually prove is only that $C_1$ exists and that $C_1>0$.\ Refining the arguments of [@mv], Grepstad, Kaltenböck and Neumüller [@gkn] showed that $\liminf\limits_{N \to \infty}P_N(\phi)>0$, based on an analysis of perturbed Sudler products of the form $$\prod_{r=1}^{N}2| \sin \pi (r \phi + {\varepsilon})|,\vspace{-2mm}$$ where ${\varepsilon}$ are some specific small parameters coming from the so-called Zeckendorff representation of positive integers (see below). The significance of this perturbed version of the standard Sudler products defined in will become apparent in the heuristic analysis after the statement of the main results of this paper.\ Following the approach of Grepstad, Kaltenböck and Neumüller in [@gkn], we define perturbed Sudler products of the form $$\label{sudlereps} P_{F_n}(\phi,{\varepsilon}) = \prod_{r=1}^{F_n} 2 \left| \sin \pi\left( r \phi + (-1)^{n+1} \frac{{\varepsilon}}{F_n} \right) \right|.$$ Thus we consider the perturbed Sudler products as functions of a real variable ${\varepsilon}$, rather than as quantities that arise in an ad-hoc way from the Zeckendorff expansion of specific positive integers. We point out that in our text the definition of $P_{F_n}(\phi,{\varepsilon})$ is different from the one given in [@gkn]. The role of the factor $(-1)^{n+1}$ within the argument of the sine function will become clear during the proofs; this alternating factor reflects the fact that the error terms in successive continued fraction convergents also have alternating signs.\ As our first main result, we establish the convergence of $P_{F_n}(\phi,{\varepsilon})$ as a sequence of functions in the variable ${\varepsilon}$. Note that we already know that $\lim\limits_{n \to \infty} P_{F_n}(\phi,0) = C_1$, where $C_1>0$ is the constant from .\ For the sake of convenience, throughout the text we use the notation $$\label{ur} u(r) = 2\sqrt{5}\left( r-\frac{1}{\sqrt{5}} \left(\{r\phi\}-\frac{1}{2} \right) \right) ,\hspace{5mm} r=1,2,\ldots$$ \[th1\] For every ${\varepsilon}\in \mathbb{R}$, the limit $ \lim\limits_{n \to \infty} P_{F_n}(\phi,{\varepsilon})$ exists and is equal to $$\label{limit} G({\varepsilon}) = K |{\varepsilon}\sqrt{5}+1|\cdot \prod_{r=1}^{\infty} \left|1 - \frac{(2{\varepsilon}\sqrt{5}+1)^2}{u(r)^2} \right|,$$ where $K>0$ is some absolute constant and the sequence $u(r), r\geq 1,$ is as in . The convergence is uniform on any compact interval where $G$ is nonzero.\ The recursive structure of the Fibonacci sequence allows us to calculate the exact value of $G({\varepsilon})$ at a certain value of ${\varepsilon}$. As a consequence, we are able to determine the precise value of the constant $C_1$ in Theorem A $1$.\ \[th2\] Let $ C_1 = \lim\limits_{n\to\infty} P_{F_n}(\phi) $ be the constant in , and let the sequence $(u(r))_{r=1}^{\infty}$ be defined by . Then $$C_1 = (1+\phi) \cdot \prod_{r=1}^{\infty} \left( 1 - \frac{1}{u(r)^2}\right)\left( 1- \frac{(1+2\phi)^2}{u(r)^2}\right)^{-1} .$$ Since $G(0)=C_1$, we can calculate the constant $K$ in Theorem \[th1\] and obtain a completely explicit formula for the function $G$. \[co\_golden\] The limiting function $G$ satisfies $$\label{gformula} G({\varepsilon}) = (1+\phi) |{\varepsilon}\sqrt{5}+1|\cdot \prod_{r=1}^{\infty}\left| 1 - \frac{(2{\varepsilon}\sqrt{5}+1)^2}{u(r)^2}\right|\left|1- \frac{(2\phi+1)^2}{u(r)^2}\right|^{-1} .$$ Some of the key properties of the function $G({\varepsilon})$ are captured by the proposition below. \[prop1\] We have $$\label{eq: preim} G \left(-\frac{\phi}{\sqrt{5}} \right) = 1.$$ Furthermore, we have $G({\varepsilon})>1.01$ for all ${\varepsilon}$ in the range $$\label{eq: varepsilon range} \varepsilon \in (-0.26, 0.58).$$ The function $G$ is continuous on $\mathbb{R}$. Finally, $G$ is a $C^{\infty}$ function and it is strictly log-concave (i.e. the logarithm of $G$ is strictly concave) in any interval whose endpoints are two consecutive roots of $G$. \[fig1\] ![Plot of $G({\varepsilon})$ in the range $-1 \leq {\varepsilon}\leq 1$. The function has zeros at ${\varepsilon}\approx -0.45$ and ${\varepsilon}\approx 0.72$, and it equals 1 at ${\varepsilon}\approx -0.28$ and ${\varepsilon}\approx 0.60$. The fact that $\liminf P_N(\phi) > 0$, first proved in [@gkn], can be reduced to the fact that $G({\varepsilon})>1$ throughout the shaded range ${\varepsilon}\in (-0.17, 0.27)$, since this turns out to be the range of possible perturbations ${\varepsilon}$ coming from the Zeckendorff representation of positive integers. A very similar reasoning applies to the problem of establishing an upper bound for $\limsup P_N(\phi)/N$ by using a “backward” Zeckendorff expansion, see Theorem \[th3\] below.](G1.eps "fig:") To explain the significance of Theorems \[th1\] and \[th2\] and Proposition \[prop1\], we briefly indicate how (upper and lower) bounds for Sudler products can be calculated using the Zeckendorff representation of an integer. Let $N\geq 1$ be a positive integer. Then $N$ can be written in a unique manner in the form $$\label{zeck} N = F_{n_k} + F_{n_{k-1}} + \ldots + F_{n_1},$$ where the integers $1\leq n_1 \leq \ldots \leq n_{k}$ are such that $n_{i+1} - n_i \geq 2$ for all $i=1,2,\ldots, k-1$. The representation of $N$ as in is referred to as the Zeckendorff representation of $N$. We note that the Zeckendorff representation of integers is a special case of the Ostrowski expansion, to which we refer later in the text.\ In order to calculate the Sudler product $P_N(\phi)$, we expand $N$ into its Zeckendorff representation as above, split the full Sudler product into factors corresponding to the components of the Zeckendorff representation, and accordingly obtain $$\label{zeckexpansion} P_N(\phi)= \left( \prod_{r=1}^{F_{n_k}} 2|\sin\pi r\phi| \right) \left( \prod_{r=F_{n_k}+1}^{F_{n_k} + F_{n_{k-1}}} \!\!\!\!\!2|\sin\pi r\phi| \right) \dots \left(\prod_{r=F_{n_k}+\dots+F_{n_2} + 1}^{F_{n_k}+\dots+F_{n_1}}\!\!\!\!\!\!2|\sin\pi r\phi| \right).$$ The first product on the right-hand side is equal to $P_{F_{n_k}}(\phi)$, and by Mestel and Verschueren’s result it converges to $C_1$ as $N\to\infty$. The second product equals $$\prod_{r=F_{n_k}+1}^{F_{n_k} + F_{n_k-1}}\hspace{-3mm} 2 |\sin \pi r \phi| = \prod_{r=1}^{F_{n_{k-1}}} 2 |\sin\pi (r \phi + F_{n_k} \phi)|,$$ so it is of the form $P_{F_n}(\phi,{\varepsilon}_k)$, as defined in , with ${\varepsilon}_k = (-1)^{1+n_{k-1}} F_{n_{k-1}}\{F_{n_k} \phi\}$. Consequently, by Theorem \[th1\], this factor is roughly $G({\varepsilon}_k)$, assuming that $n_{k-1}$ is “large”. Using the recursive structure of the Fibonacci sequence — see and below — we have $|{\varepsilon}_k| = F_{n_{k-1}}\phi^{n_k}$ and one can show that, for sufficiently large $N$, this implies that ${\varepsilon}$ lies within the range $(-\phi^3/ \sqrt{5}, \phi^2/\sqrt{5})$. The other products on the right-hand side of are estimated in a similar way, and give contributions of size roughly $G({\varepsilon}_j)$ for some appropriate values of ${\varepsilon}_j$ depending on the Zeckendorff expansion of $N$. Thus the problem of estimating the whole Sudler product $P_N(\phi)$ essentially boils down to estimating a product of values of the limit function $G$, evaluated at positions (“perturbations”) which depend on the Zeckendorff representation of $N$ in a relatively simple way.\ Using continued fractions, it turns out that in such a product we can only encounter values of the perturbation variable which are within the range $(-\phi^2/\sqrt{5}, \phi/\sqrt{5})$, which is roughly $(-0.17, 0.27)$, with the possible exception of finitely many indices at the final segment of the representation, which correspond to sub-products of short length. Note that throughout the range $(-0.17, 0.27)$ we have $G({\varepsilon})>1$ (cf. Proposition \[prop1\] and Figure 1). Thus, heuristically speaking, $P_N(\phi)$ decomposes into a product of factors which are all at least $1$ (with the possible exception of finitely many indices at the final segment), and thus $\liminf\limits_{N\to\infty}P_N(\phi)$ cannot be equal to 0.\ The analysis carried out in [@gkn] to show involves precisely these arguments, but the authors directly provide estimates for the perturbed products without having first established the convergence of the sequence $P_{F_n}(\phi,{\varepsilon})$ in . A crucial advantage of the “functional” approach is that once we have established the existence of a limit function $G({\varepsilon})$, together with a concavity property, it is not necessary anymore to estimate $G({\varepsilon})$ for all perturbations ${\varepsilon}$ that we might encounter during the proof, but it is rather sufficient that we show that the value of $G$ exceeds $1$ at two appropriate left and right endpoints; concavity of $\log G$ then tells us that we must also have $G({\varepsilon})>1$ everywhere in between.\ In the sequel we show that Theorem \[th1\] also allows us to calculate *upper* bounds for the size of the Sudler product: more precisely, $P_N(\phi)$ grows at most linearly in $N$. \[th3\] We have $$\limsup_{N \to \infty} \frac{P_N(\phi)}{N} < \infty. \vspace{4mm}$$ For the proof of Theorem \[th3\] we will utilize the following decomposition. Given $N\geq 1$, when $n$ is such that $F_{n-1} \leq N + 1 < F_{n}$ we write $$\label{reflectionprinciple} P_N(\phi) \,= \, \dfrac{ P_{F_{n}-1} (\phi)}{ \prod\limits_{r=N+1}^{F_{n}-1} 2 |\sin \pi r \phi|} \, = \, \frac{P_{F_{n}-1}(\phi) }{ \prod\limits_{r=1}^{F_{n}-N-1} 2 | \sin \pi (r-F_n) \phi | } \, \cdot$$ The motivation for this decomposition is that by , the quotients $P_{F_n-1}(\phi) /F_n $ are bounded from above, and since $F_{n-1}\leq N+1 < F_n$ we can show that $P_{F_n-1}(\phi)/N$ will be also bounded from above. It remains to control the product $\prod_{r=1}^{F_{n}-N-1} 2 | \sin \pi (r-F_n) \phi |$. This can be factorised as in , but with an additional perturbation coming from the term $- F_n \phi$. Since $\|F_n\phi\|$ is sufficiently small, it will turn out that this additional perturbation does not cause any particular problems.\ In other words, relation shows that the lower asymptotic order of magnitude of $P_N(\phi)$ reflects on the upper asymptotic order of magnitude of $P_N(\phi)/N$; for this reason, throughout the text we refer to as the reflection principle.\ Thus the problem of finding a finite upper bound for $\limsup P_N(\phi )/N$ is, in a very natural sense, the dual problem of finding a positive lower bound for $\liminf P_N(\phi)$, and as sketched above this can be done by showing that $G({\varepsilon})>1$ in an appropriate range of values of ${\varepsilon}$. (In this particular case the range turns out to be roughly $(-0.10,0.17)$, which is even smaller than the range of possible perturbations in the $\liminf$ problem). See Section \[sec:proofs\_phi\] below for details. Main results, Part 2: The case of quadratic irrationals $\beta=[b,b,b,\ldots]$ ------------------------------------------------------------------------------ As already mentioned, it has been conjectured that $\limsup_{N \to \infty} P_N(\alpha)/N$ is finite for *all* irrationals $\alpha$. However, we will show that this is false even if $\alpha$ is restricted to the class of quadratic irrationals whose continued fraction expansion is of the simplest possible form.\ For any positive integer $b\geq 1$, let $$\beta = \beta(b) = [b,b,b,\ldots]$$ be the quadratic irrational with all partial quotients in its continued fraction expansion equal to $b$. It is well-known that $$\label{eq: explicit form of beta} \beta=\frac{1}{2}(-b+\sqrt{b^2+4}),$$ and if $(q_n)_{n=0}^{\infty}$ are the denominators of the convergents of $\beta$, then by induction we have that $$\label{b1} \beta = \frac{q_{n-1}}{q_n} + (-1)^{n}\frac{\beta^{n+1}}{q_n}, \,\hspace{5mm} n=1,2,\ldots$$ and $$\label{b2} q_n = \frac{1}{\sqrt{b^2+4}} \left(\beta^{-(n+1)}-(-\beta)^{n+1} \right), \,\hspace{5mm} n=1,2,\ldots$$ The generalisation of the theorem of Mestel and Verschueren (Theorem A $1$ above) to the case of quadratic irrationals of the form ${\beta}= {\beta}(b)$ looks as follows. \[generalthm\] Let ${\beta}=[b,b,b,\ldots]$, and let $(q_n)_{n=1}^{\infty}$ be the sequence of denominators associated with its continued fraction expansion. There exists a constant $C_b>0$ such that $$\label{cb} C_b = \lim_{n\to\infty} P_{q_n}({\beta}).$$ Moreover, for the constant $C_b$ we have $$\label{cb2} \lim_{n\to\infty} \frac{P_{q_n-1}({\beta})}{q_n} = \frac{C_b \sqrt{b^2+4}}{2\pi} \cdot$$ Relation is a special case of [@sigrid2 Theorem 1.2] by Grepstad and Neumüller, which covers the case of arbitrary quadratic irrationals. The limit in is not stated explicitly in [@sigrid2], but it can be deduced easily from arguing as in [@mv Corollary 8.1].\ We prove an analogue of Theorem \[th1\] for perturbed Sudler products in the case of the irrational ${\beta}$. The objects of our study are now the functions [^2] $$\label{pqndefinition} P_{q_n}({\beta},{\varepsilon}) = \prod_{r=1}^{q_n}2\left|\sin \pi \left(r{\beta}+ (-1)^n \frac{{\varepsilon}}{q_n} \right) \right|.$$ We also define $$\label{ubr} u_b(r)=2\sqrt{b^2+4}\left(r-\frac{1}{\sqrt{b^2+4}}\left(\{r\beta\}-\frac{1}{2}\right)\right), \hspace{2mm} r=1,2,\ldots$$ Note that these formulas are straightforward generalisations of the ones for the case of the golden mean, which corresponds to the case $b=1$ and which we considered in Section \[golden\_case\] above.[^3] The following result is a generalisation of Theorem \[th1\].\ \[th6\] For every ${\varepsilon}\in \mathbb{R}$, the limit $ \lim\limits_{n \to \infty} P_{q_n}({\beta},{\varepsilon})$ exists and is equal to $$\label{limit_pert} G_{\beta}({\varepsilon}) = K_b \cdot |{\varepsilon}\sqrt{b^2+4}+1|\cdot \prod_{r=1}^{\infty} \left|1 - \frac{(2{\varepsilon}\sqrt{b^2+4}+1)^2}{u_b(r)^2} \right|,$$ where $K_b>0$ is some absolute constant and $u_b(r)$ is defined in . The convergence is uniform on any compact interval where $G_{\beta}$ is nonzero. We can also prove generalised versions of Theorem \[th2\] and of Corollary \[co\_golden\]. \[cbvaluethm\] Let $C_b>0$ be the constant as in . Then $$\label{cbvalue} C_b^b = \frac{1}{\beta(\beta+1)\cdots(\beta+b-1)} \prod_{r=1}^{\infty}\prod_{j=1}^{b}\left(1- \frac{1}{u_b(r)^2}\right) \left( 1 - \frac{(2b-2j+2\beta+1)^2}{u_b(r)^2} \right)^{-1}.$$ The limiting function $G_{\beta}({\varepsilon})$ in satisfies $$\label{pinftyformula} G_{\beta}({\varepsilon})^b = \frac{|{\varepsilon}\sqrt{b^2+4}+1|^b }{ \beta \cdots(\beta+b-1)} \cdot \prod_{r=1}^{\infty}\prod_{j=1}^{b}\dfrac{ \left|1 - \dfrac{(2{\varepsilon}\sqrt{b^2+4}+1)^2}{u_b(r)^2} \right|}{ \left|1 - \dfrac{(2b-2j+2\beta+1)^2}{u_b(r)^2} \right|} \cdot$$ The following proposition states the basic properties of the function $G_{{\beta}}$ that we use later in the paper. We omit the proof since it involves precisely the same convergence arguments as the proof of Proposition \[prop1\]. \[gbetaproperties\] For any $b\geq 2$, the function $G_{{\beta}}$ defined in Theorem \[th6\] is continuous on $\mathbb{R}$. Furthermore, it is a $C^{\infty}$-function and strictly log-concave in any interval with endpoints two of its consecutive roots. Interestingly, the $\liminf$ result of Grepstad, Kaltenböck and Neumüller and the $\limsup$ result of Theorem \[th3\] cannot be extended to the general case of $b \geq 2$. Instead, it turns out that both results fail when $b$ is sufficiently large, and the proofs rely on the particular structure of the function $G_{\beta}$ in an extremely delicate way. Theorem \[lineargrowththeorem\] gives a full characterisation for this problem.\ \[lineargrowththeorem\] Let ${\beta}= [b,b,b,\ldots]$, where $b$ is a positive integer. Then the following holds. - If $b \leq 5$, then $\liminf \limits_{N \to \infty} P_N({\beta}) > 0$ and $\limsup \limits_{N\to\infty}\dfrac{P_N({\beta})}{N} < \infty.$ - If $b \geq 6$, then $\liminf \limits_{N \to \infty} P_N({\beta}) = 0$ and $\limsup \limits_{N\to\infty}\dfrac{P_N({\beta})}{N} = \infty.$ The above theorem implies that is not true for every irrational $\alpha$. In fact, we have a quantitative lower bound on the number of quadratic irrationals whose Sudler products have super-linear growth in $N$; here we are ordering the quadratic irrationals in a natural way — by their naive height. For not interrupting the flow of presentation, we have recorded the relevant corollary, that is Corollary \[height corr\], right before the bibliography.\ Sudler products for quadratic irrationals ${\beta}=[b,b,b,\ldots ]$ were examined in [@sigrid3], where it was proved that there exists some (finite) value $B_0$ such that $\liminf_{N \to \infty}P_N({\beta})=0$ provided that $b \geq B_0$. The authors of [@sigrid3] showed that one can take $B_0 = e^{803}$, and, based on numerical calculations, they conjectured that actually $b=6$ should be the transition point where the $\liminf$ behaviour of $P_N({\beta})$ changes. This conjecture is established in our theorem above. We note that the theorems cited and proved in this paper imply that $$\liminf_{N \to \infty} \frac{\log P_N({\beta})}{\log N} = 0, \qquad \limsup_{N \to \infty} \frac{\log P_N({\beta})}{\log N} = 1 \qquad \text{when \, $b \leq 5$},$$ and that modifying the proof of Theorem \[lineargrowththeorem\] we could also establish the slightly stronger conclusion that $$\liminf_{N \to \infty} \frac{\log P_N({\beta})}{\log N} < 0, \qquad \limsup_{N \to \infty} \frac{\log P_N({\beta})}{\log N} > 1 \qquad \text{when\, $b \geq 6$,}$$ instead of part (ii) of Theorem \[lineargrowththeorem\].\ It turns out that for all $b \geq 7$ we have $C_b = G_{\beta}(0) <1$ (see Corollary \[cor: when M-V const greater one\] below). By continuity of $G_\beta$ this implies that $G_\beta({\varepsilon})<1$ for all sufficiently small perturbations ${\varepsilon}$, and we will see that this implies that $\liminf P_N({\beta})=0$ as well as $\limsup P_N({\beta})/N = \infty$ by a rather straightforward argument (see Lemma \[lemma&gt;1\] below). The case $b=6$ is special, since this is the only case where we have $C_b > 1$ but where $$\liminf \limits_{N \to \infty} P_N({\beta}) = 0\qquad \text{and} \qquad \limsup \limits_{N\to\infty}\dfrac{P_N({\beta})}{N} = \infty.$$ We will need a separate proof for this case (Lemma \[lemma\_b6\] below). The case $b=1$ is the golden ratio case, for which the desired conclusions have been already established in Section \[golden\_case\] above. It remains to deal with the case when $b \in \{2, 3, 4, 5\}$. The analysis in these cases is quite involved, and significantly more complex than in the case $b=1$. Roughly speaking, in the golden ratio case we could show that from the Ostrowski (Zeckendorff) representation there can arise no perturbations ${\varepsilon}$ which lead to values of $G$ being smaller than $1$. This continues to be true when $b=2$, but in the case $3 \leq b \leq 5$ it can indeed happen that the Ostrowski representation leads to particular (negative) perturbations ${\varepsilon}$ for which $G_\beta({\varepsilon})<1$. However, we show that such problematic perturbations can only arise from very particular configurations of the Ostrowski representation, in such a way that contributions from an earlier stage compensate for the small factors, and the overall product still exceeds $1$. For a heuristic explanation of these effects see Figures $2-4$ below. \ \[fig3\] ![Plot of $G_\beta({\varepsilon})$ for $b=35$ (solid line). Compare this with the plots in Figure 2. The peak has “wandered off” so far to the right that $G_{\beta}$ only takes a very small value at ${\varepsilon}=0$. The graph of $G_{\beta}({\varepsilon})$ has lost much of its similarity with the corresponding graphs for small values of $b$, and now rather resembles $2 |\sin \pi {\varepsilon}|$ (dashed line).](G6.eps "fig:") Proofs of the theorems for the golden ratio {#sec:proofs_phi} =========================================== Throughout this section, we will make use of the following relations: $$\label{phi1} \phi = \frac{F_{n-1}}{F_n} + (-1)^{n+1}\frac{\phi^n}{F_n}, \,\hspace{5mm} n=1,2,\ldots$$ and $$\label{phi2} F_n = \frac{1}{\sqrt{5}} \left(\phi^{-n}-(-\phi)^n \right), \,\hspace{5mm} n=1,2,\ldots$$ In order to show the convergence result of Theorem \[th1\], we follow the same factorisation argument as in [@mv] and [@gkn]. Here we present the basic steps of the proof and refer to [@gkn; @mv] for the remaining details which are nearly identical.\ In what follows, we assume that an integer $n\geq 1$ is given, and we write $[k]$ for the residue of the integer $k\in\mathbb{N}$ modulo $F_n$. Following [@mv] we define $$\begin{aligned} s_n(0,{\varepsilon}) &=& 2\sin\pi\left(\frac{{\varepsilon}}{F_n}+\frac{\phi^{n}}{2} \right), \\ s_n(r) &=& 2\sin\pi\left( \frac{r}{F_n}-\phi^{n}\left(\frac{[F_{n-1}r]}{F_n} -\frac{1}{2}\right)\right), \qquad r=1,2,\ldots, F_n-1.\end{aligned}$$ The product $P_{F_n}(\phi,{\varepsilon})$ may be further factorized as $$\label{factors} P_{F_n}(\phi,{\varepsilon}) = A_n(\phi,{\varepsilon})\cdot B_n(\phi)\cdot C_n(\phi,{\varepsilon}),$$ where $$\begin{aligned} A_n(\phi,{\varepsilon}) &=& 2 F_n\left| \sin \pi\left(F_n\phi + (-1)^{n+1}\frac{{\varepsilon}}{F_n} \right)\right|,\\ B_n(\phi) &=& \prod_{r=1}^{F_n-1} \frac{s_n(r)}{2\left| \sin(\pi r/F_n)\right|}, \hspace{3mm} \text{ and} \\ C_n(\phi, {\varepsilon}) &=& \prod_{r=1}^{F_n-1}\left( 1-\frac{s_n(0,{\varepsilon})^2}{s_n(r)^2} \right)^{\frac{1}{2}} .\end{aligned}$$ The proof of is based on elementary trigonometric identities, see [@mv Lemma 5.1] or [@gkn Lemma 3.1]. Using properties and together with the asymptotic estimate $\sin x \sim x, \, x\to 0,$ we deduce that $$\begin{aligned} A_n(\phi,{\varepsilon}) & = & 2F_n \left|\sin\pi \left((-1)^{n+1}\phi^{n} +(-1)^{n+1}\frac{{\varepsilon}}{F_n} \right) \right| \\ & = & 2F_n\left|\sin\pi\left( \phi^{n} + \frac{{\varepsilon}}{F_n}\right)\right| \\ & \sim & 2\pi |F_n\phi^n +{\varepsilon}| \\ &\sim & \frac{2\pi}{\sqrt{5}}|{\varepsilon}\sqrt{5}+1|, \qquad n\to \infty .\end{aligned}$$ Regarding the products $B_n(\phi)$, it is shown in [@mv] that there exists a constant $B>0$ such that $B_n(\phi)\to B$ as $n\to \infty$. This is the most difficult part of the proof of Theorem A $1$ in [@mv], but since the factor $B_n(\phi)$ does not depend on the perturbation ${\varepsilon}$ at all we can just use this fact without any further work.\ Finally, regarding the factor $C_n(\phi,{\varepsilon})$ we can show, arguing as in [@mv Section 6], that $$\lim_{n\to\infty}C_n(\phi,{\varepsilon})^2 = \prod_{r=1}^{\infty}\left(1-\frac{(2{\varepsilon}\sqrt{5}+1)^2}{u(r)^2} \right)^2, \hspace{3mm} \text{ for all } {\varepsilon}\in\mathbb{R},$$ where $u(r)$ is as in . The only difference in comparison with [@mv] is that the numerators $s_n(0,{\varepsilon})$ now depend on the perturbation ${\varepsilon}$ and satisfy $$\label{snestimate} s_n(0,{\varepsilon}) = \pi\phi^n (2{\varepsilon}\sqrt{5}+1) + O(\phi^{3n}) \quad \text{ as } n\to \infty.$$ Combining all the previous formulas, we obtain the requested convergence result for $P_{F_n}(\phi,{\varepsilon})$.\ Now let $I$ be a compact interval. In order to show that the convergence of $P_{F_n}(\phi,{\varepsilon})$ is uniform on $I$, it suffices to show that all three factors appearing in converge uniformly. This is trivial for $B_n(\phi)$, while for $A_n(\phi,{\varepsilon})$ it can be done using the estimate $\sin x = x + O(x^3),\, x\to 0$. Finally regarding $C_n(\phi,{\varepsilon})$, the estimates as well as $(6.2)$ of [@mv] hold uniformly on $I$, and this allows us to deduce that $C_n(\phi,{\varepsilon})$ is uniformly Cauchy on $I$; the details are left to the interested reader. Since by the Mestel–Vershueren Theorem the sequence $(P_{F_n}(\phi))_{n=1}^{\infty}$ has a limit $0<C_1 <\infty$, we get $$\begin{aligned} 1 &=& \lim_{n\to\infty} \frac{P_{F_{n+1}}(\phi)}{P_{F_{n-1}}(\phi)} \\ & = & \lim_{n\to\infty} \prod_{r=F_{n-1}+1}^{F_{n+1}}\!\!\!2|\sin \pi r\phi | \\ &=& \lim_{n\to\infty}\prod_{r=1}^{F_{n+1}-F_{n-1}}\hspace{-3mm}2| \sin\pi(F_{n-1}+r)\phi| \\ & = & \lim_{n\to\infty}\prod_{r=1}^{F_{n}}2| \sin\pi(r\phi + F_{n-1}\phi)| \\ & \stackrel{\eqref{phi1}}{=}& \lim_{n\to\infty}\prod_{r=1}^{F_{n}}2| \sin\pi \left(r\phi + (-1)^{n}\phi^{n-1} \right)| \\ & = & \lim_{n\to\infty}P_{F_n}(\phi, -F_n\phi^{n-1}) .\end{aligned}$$ Here implies that $$\label{asymptotic} {\varepsilon}_n := -F_{n}\phi^{n-1} \sim -\frac{1}{\phi \sqrt{5}}, \qquad n\to \infty.$$ At this point we can write $$\label{factorisation} P_{F_n}(\phi, {\varepsilon}_n) = A_n(\phi,{\varepsilon}_n)\cdot B_n(\phi) \cdot C_n(\phi, {\varepsilon}_n),$$ where the factors appearing are as in . Now $$\begin{aligned} A_n(\phi, {\varepsilon}_n) &=& 2F_n \left| \sin \pi\left(\phi^{n} -\phi^{n-1}\right)\right| \\ &\sim & 2\pi F_n \phi^{n}|1-\phi^{-1}| \\ &\sim & \frac{2\pi\phi }{\sqrt{5}} , \qquad n\to \infty.\end{aligned}$$ Regarding $B_n(\phi)$, we have already mentioned in the proof of Theorem \[th1\] that there exists $B>0$ such that $$\lim_{n\to\infty} B_n(\phi) = B.$$ Finally, for $C_n(\phi, {\varepsilon}_n)$ one can employ the arguments of [@mv p.10–12] to prove that $$\lim_{n\to\infty} C_n(\phi, {\varepsilon}_n) = \prod_{r=1}^{\infty} \left( 1 - \frac{(1+2\phi)^2}{u(r)^2} \right).$$ Thus it follows by taking limits in that $$\label{onelimit} \frac{2\pi\phi}{\sqrt{5}} \cdot B \cdot \prod_{r=1}^{\infty} \left( 1 - \frac{(1+2\phi)^2}{u(r)^2} \right) = 1 .$$ Regarding the constant $C_1>0$, in the proof of [@mv Theorem 3.1] it is actually shown that $$\label{constantlimit} C_1 = \frac{2\pi}{\sqrt{5}}\cdot B \cdot \prod_{r=1}^{\infty} \left( 1 - \frac{1}{u(r)^2} \right)$$ Hence combining with we obtain the requested relation. To deduce Theorem \[th3\] from Theorem \[th1\], let $N$ be given, and let $n=n(N)\geq 1$ be such that $F_{n-1} \leq N + 1 < F_n$. Factorizing $P_N(\phi)$ as in , we obtain $$\begin{aligned} P_N(\phi) & = & \frac{P_{F_{n}-1}(\phi)}{\prod\limits_{r=N+1}^{F_{n}-1} 2 | \sin \pi r \phi |} \, = \, \frac{P_{F_n-1}(\phi)}{\prod\limits_{r=1}^{F_n-N-1} 2 | \sin \pi (F_n - r) \phi |} \nonumber\\ & \stackrel{\eqref{phi1}}{=} & \frac{P_{F_{n}-1}(\phi)}{\prod\limits_{r=1}^{F_n-N-1} 2|\sin\pi ( r \phi -(-\phi)^{n})|} \cdot \label{line_y} $$ Now let $$F_n - (N+1) = F_{n_k} + F_{n_{k-1}} + \ldots + F_{n_1}$$ be the Zeckendorff representation of $F_n-(N+1)$ as in . Since $F_{n-1} \leq N +1 < F_n,$ we have $n_k \leq n-2$. Set $N_k=0$ and $N_i = F_{n_k} + \ldots + F_{n_{i+1}}$ for $i=1,2,\ldots, k-1$. The product in the denominator in line can then be written as $$\begin{aligned} \prod\limits_{r=1}^{F_n-N-1} 2|\sin\pi ( r \phi -(-\phi)^{n})| &=& \prod_{i=1}^{k} \prod_{r=N_i+1}^{N_i + F_{n_i}}2|\sin \pi(r\phi - (-\phi)^{n})| \nonumber \\ &=& \prod_{i=1}^{k} \prod_{r=1}^{F_{n_i}} 2|\sin\pi(r\phi + N_i\phi -(-\phi)^{n}) | \label{finitely}\\ &=& \prod_{i=1}^{k} \prod_{r=1}^{F_{n_i}} 2\left|\sin\pi\left(r\phi + \frac{(-1)^{n_i+1} {\varepsilon}_i}{F_{n_i}}\right)\right| \nonumber ,\end{aligned}$$ where for $ i=1,2,\ldots, k$ we define ${\varepsilon}_i$ so that $$\frac{ (-1)^{n_i+1} {\varepsilon}_i}{F_{n_i}} = (N_i\phi - (-\phi)^n) \underbrace{- F_{n_{i+1}-1} - F_{n_{i+2}-1} - \dots - F_{n_{k-1} - 1} - F_{n_k - 1}}_{\in \mathbb{Z}}.$$ The integer which is subtracted in the formula above is chosen in such a way that ${\varepsilon}_i$ is small (see below). Note that subtracting this integer is possible without problems by the periodicity of the sine-function. Comparing with the definition of the perturbed Sudler products in we note that we have $$\label{prodl} \prod\limits_{r=1}^{F_n-N-1} 2|\sin\pi ( r \phi -(-\phi)^{n})| = \prod_{i=1}^{k}P_{F_{n_i}}(\phi, {\varepsilon}_i).$$ Furthermore, according to we have $$\begin{aligned} \frac{{\varepsilon}_i}{(-1)^{n_i+1}F_{n_i}} & = & \left(N_i\phi + (-\phi)^{n+1}\right) - F_{n_{i+1}-1} - \ldots - F_{n_k - 1} \\ & = & (F_{n_{i+1}}+\ldots + F_{n_{k}})\phi - (-\phi)^{n} - F_{n_{i+1}-1} - \ldots -F_{n_k - 1} \\ & = & - \big((-\phi)^{n_{i+1}} + \ldots + (-\phi)^{n_{k}} + (-\phi)^n \big).\end{aligned}$$ Thus $$\begin{aligned} \frac{{\varepsilon}_i}{F_{n_i} \phi^{n_i}} & = & \phi^{-n_i} (-1)^{-n_i} \left( (-\phi)^{n_{i+1}} + \ldots + (-\phi)^{n_{k}} + (-\phi)^n \right) \label{as_in_line}\\ & = & (-1)^{n_{i+1}-n_i} \phi^{n_{i+1}-n_i} + \ldots + (-1)^{n_k-n_i} \phi^{n_{k}-n_i} + (-1)^{n - n_i} \phi^{n-n_i} \label{pos_to}\end{aligned}$$ for all $i$. In the last line, we (in general) have positive as well as negative summands. The positive terms are those coming from indices having the same parity mod $2$ as $n_i$, while the other terms give negative contributions. Note, however, that by the properties of the Zeckendorff representation (which cannot have two consecutive digits equal to $1$) we necessarily have $n_{i+1} - n_i \geq 2$; consequently, the maximal possible negative contribution is smaller than the maximal possible positive contribution, since $n_{i+1}$ and $n_i$ can only have different parity mod 2 if $n_{i+1} - n_i \geq 3$. This implies that the maximal possible positive contribution to is bounded above by $$\label{range_1} \phi^2 + \phi^4 + \phi^6 + \dots = \phi,$$ while the maximal possible negative contribution is bounded by $$\label{range_2} -\phi^3 - \phi^5 - \phi^7 - \dots = - \phi^2.$$ By we have $F_{n_i}\phi^{n_i} \sim \frac{1}{\sqrt{5}}$ as $n_i\to \infty$. Thus we have $$- \phi^2/\sqrt{5} - 0.001 \leq {\varepsilon}_i \leq \phi / \sqrt{5} + 0.001,$$ whenever $n_i \geq i_0$ (here and in the sequel we write $i_0$ for generic absolute lower bounds for elements of the index set, not necessarily the same at different occurrences). By $\phi^2/\sqrt{5} \approx 0.171$ and $\phi / \sqrt{5} \approx 0.276$ this implies that for $n_i \geq i_0$ we have $$- 0.18 \leq {\varepsilon}_i \leq 0.28.$$ Consequently, by Proposition \[prop1\] we have $G({\varepsilon}_i)>1.01$ for all $n_i \geq i_0$, which implies that $P_{F_{n_i}}(\phi,{\varepsilon}_i) \geq 1$ for $n_i \geq i_0$ (recall here that the convergence towards $G$ is uniform in ${\varepsilon}$). Thus in the product on the right-hand side of all factors are at least $1$, except for the contribution of a finite number of small indices $n_i \leq i_0$.\ The contribution of the finitely many indices with $n_i \leq i_0$ to the full product, in the decomposition in line , is $$\begin{aligned} & & \prod_{\substack{1 \leq i \leq k,\\n_i \leq i_0}} \prod_{r=1}^{F_{n_i}} 2|\sin\pi(r\phi + N_i\phi -(-\phi)^{n})|. \label{the_term}\end{aligned}$$ Let $j = \max\{n_i:~n_i \leq i_0\}$, and assume that $j$ is odd (the even case is perfectly analogous). Also define $(\!(x)\!) = x$ for $x\in(-1/2,1/2]$ and extend to all $x\in\mathbb{R}$ with period $1$. We can write the number $N_i + r$ as $y + z$, where the Zeckendorff representation of $y$ contains only Fibonacci numbers of size at least $F_{j+2}$, and where that of $z$ only contains Fibonacci numbers of size at most $F_j$ (note that $F_{j+1}$ cannot occur at all, since we know that $F_j$ does occur). By the best approximation properties of continued fraction convergents, we can easily deduce that $(\!( z \phi)\!) \not\in (-\phi^{j},\phi^{j+1})$, with the left and right endpoints of this interval corresponding to $z=F_j$ and $z=F_{j-1}$, respectively. Recall here that we assumed that $j$ is odd, so that $(\!(F_j z)\!)$ is positive and $(\!(F_{j-1} z)\!)$ is negative by . On the other hand, using and arguing as in the lines leading to and , we have $$(\!(y \phi - (-\phi)^n )\!) \in \left( - \sum_{\substack{\ell \geq j+2,\\ \ell \text{~even}}} \phi^{-\ell},~ \sum_{\substack{\ell \geq j+2,\\ \ell \text{~odd}}} \phi^{-\ell} \right) = \left( - \phi^{j+2}, \phi^{j+1} \right).$$ Thus we have $$(\!( r\phi + N_i\phi -(-\phi)^{n} )\!) = (\!( (y+z) \phi - (-\phi)^{n} )\!) \not\in \left(-\phi^{j} + \phi^{j+1}, \phi^{j+1} - \phi^{j+2} \right).$$ Noting that trivially $j \leq i_0$, we deduce that for every individual factor appearing in one of the products in , the term $|\sin\pi(r\phi + N_i\phi -(-\phi)^{n})|$ is bounded below by an absolute constant (depending only on $i_0$). Since the product contains a bounded number of factors, we can deduce that this product is bounded below by an absolute constant.\ Thus we can conclude that there exists an absolute constant $K>0$ such that $$\liminf_{N\to\infty}\hspace{-1mm}\prod_{r=1}^{F_n-N-1}\hspace{-3mm}2|\sin\pi ( r \phi - (-\phi)^{n})| \geq K.$$ Finally we deduce $$\limsup_{N\to\infty}\frac{P_N(\phi)}{N} = \limsup_{N\to\infty} \frac{F_n}{N}\cdot \frac{P_{F_n-1}(\phi)}{F_n}\cdot \left( \prod_{r=1}^{F_n-N-1}\hspace{-3mm}2|\sin\pi ( r \phi - (-\phi)^{n})| \right)^{-1} < \infty.$$ For illustration, we show that a proof for the fact that $\liminf P_N(\phi)>0$ can be obtained in a way which is completely analogous to the one above, just without the “reflection” at $F_n$. Let $N$ be given. We expand $N$ into its Zeckendorff representation $$N = F_{n_k} + F_{n_{k-1}} + \ldots + F_{n_1}.$$ With $N_i$ defined as in the proof of Theorem \[th3\], we have $$\begin{aligned} P_N(\phi) & = & \prod_{r=1}^N 2 |\sin \pi r \phi| \\ & = & \prod_{i=1}^k \prod_{r=1}^{F_{n_i}} 2 |\sin\pi (r \phi + N_i \phi)|. \end{aligned}$$ This is very similar to line , except that the term $\phi^n$ is missing. We can write $P_N(\beta) = \prod_{i=1}^k P_{F_{n_i}} (\phi,{\varepsilon}_i)$ for some appropriate perturbations ${\varepsilon}_i$. Since the term $\phi^n$ is now missing, the perturbations ${\varepsilon}_i$ are slightly different. However, the possible *range* for these perturbations is exactly the same as above, since $\phi^n$ is just one term of a geometric progression, and we have estimated the possible range of perturbations by considering the whole infinite geometric progressions – cf.  and . So we obtain the same range of perturbations as previously, i.e. ${\varepsilon}_i \in [-0.18,0.28]$ for all $i$, and throughout this range of possible perturbations the function $G$ is uniformly bounded below by 1.01 (see above). This allows us to deduce that $\liminf\limits_{N \to \infty} P_N(\phi) > 0$. As in the proof of Theorem \[th2\], we write $$\begin{aligned} 1 &=& \lim_{n\to\infty} \frac{P_{F_{n+2}}(\phi)}{P_{F_{n+1}}(\phi)} = \lim_{n\to\infty} \prod_{r=F_{n+1}+1}^{F_{n+2}}\!\!\!2|\sin \pi r\phi | = \lim_{n\to\infty}\hspace{-3mm}\prod_{r=1}^{F_{n+2}-F_{n+1}}\hspace{-3mm}2|\sin\pi(r+ F_{n+1})\phi| \\ &\stackrel{\eqref{phi1}}{=}& \lim_{n\to\infty}\prod_{r=1}^{F_{n}}2| \sin\pi(r\phi + (-1)^{n}\phi^{n+1})| = \lim_{n\to\infty}P_{F_n}(\phi, -F_n\phi^{n+1}) . \end{aligned}$$ Here for the sequence $\zeta_n := -F_n\phi^{n+1}, n=1,2,\ldots$ we have $\lim \zeta_n = \zeta_0 := -\phi/\sqrt{5}$ by , and since the functions $P_{F_n}(\phi,{\varepsilon})$ converge uniformly to $G({\varepsilon})$ we deduce that $$1 = \lim_{n\to \infty}P_{F_n}(\phi,\zeta_n) = G\left( -\phi/\sqrt{5} \right) \, .$$ To prove the remaining assertions of the theorem, we let $I \subseteq \mathbb{R}$ be a closed interval such that $G({\varepsilon}) \neq 0$ for all ${\varepsilon}\in I$. Note that $\log G$ is well defined in the set of ${\varepsilon}$ for which $G({\varepsilon})\neq 0$. In view of , to show that $G$ is a $C^{\infty}$ function on $I$, it is enough to establish the uniform convergence of $$H_{R}\left(\varepsilon\right)=\log\prod_{r\leq R} \vert 1-\delta_{r}^{2}({\varepsilon}) \vert, \qquad \mathrm{where}\quad \delta_{r}({\varepsilon})= \frac{2\sqrt{5}\varepsilon+1}{u(r)} \,\, \text{ for any } \varepsilon\in I$$ and its derivatives of any order as $R\to \infty$. First, we note that since $I$ is compact, the distance $ \min\{ \left|1- \delta_{r}({\varepsilon}) \right| : {\varepsilon}\in I, r\geq 1\}$ is strictly positive. Furthermore, the estimate $$0\geq\log\left|1-\delta\right| = -\int_{1-\delta}^{1}\frac{\mathrm{d}x}{x}\geq -\int_{1-\delta}^{1}\frac{\mathrm{d}x}{1-\delta} =-\frac{\delta}{1-\delta} \, , \hspace{5mm} 0< \delta < 1$$ implies that for $r$ large enough, $$\vert \log\vert 1-\delta_{r}^{2}({\varepsilon}) \vert \vert \leq \frac{\delta_r^2({\varepsilon})}{1-\delta_r^2({\varepsilon})} \ll_{I} \frac{(2\sqrt{5}\varepsilon+1)^{2}}{ u(r)^{2}} \ll_{\, I} \, \frac{1}{r^{2}} \cdot \label{eq: supnorm bound G}$$ Thus, the partial sums $ H_R({\varepsilon})=\sum\limits_{r\leq R} \log\vert 1-\delta_{r}^{2}({\varepsilon}) \vert $ converge uniformly to $$H({\varepsilon}) = \sum_{r=1}^{\infty} \log\vert 1-\delta_{r}^{2}({\varepsilon}) \vert, \hspace{5mm} {\varepsilon}\in I .$$ We proceed to show that the derivatives $H_R^{(k)}({\varepsilon})$ of any order $k\geq 1$ converge uniformly, too. For $k\geq 1$, the $k$-th derivative of each summand of $H_R({\varepsilon})$ is $$\begin{aligned} \frac{\mathrm{d}^{k}}{\mathrm{d}\varepsilon^{k}} \log\vert 1- \delta_{r}^{2}({\varepsilon})\vert & = \frac{(-1)^{k-1} (2\sqrt{5})^k (k-1)! }{u(r)^k} \Bigg( \frac{1}{\left( 1+ \delta_{r}({\varepsilon}) \right)^k} + \frac{(-1)^k}{\left( 1- \delta_{r}({\varepsilon}) \right)^k} \Bigg). \end{aligned}$$ For the case $k=1$, we infer that $$\frac{\mathrm{d}}{\mathrm{d}\varepsilon} \log \vert 1 - \delta_{r}^{2}({\varepsilon}) \vert \, =\, \frac{4\sqrt{5}(2\sqrt{5}\varepsilon+1)}{u\left(r\right)^2 (1-\delta_r^2({\varepsilon})) } \, \ll_I \, \frac{1 }{u\left(r\right)^{2}}\, \ll \, \frac{1}{r^{2}} \, ,$$ which is summable over $r$. Now we observe that for each $k\geq 2$ fixed, $$\left| u(r)^{k}\left(1\pm \delta_{r}({\varepsilon}) \right)^{k}\right|\, \gg_I \, u(r)^{k} \, \gg \, r^{k} ,$$ and this implies that the $k$-th derivative of $H_R$ converges uniformly on $I$. The upshot is that, for any $k\geq1$, the partial sums of $$\sum_{r=1}^{\infty}\frac{\mathrm{d}^{k}} {\mathrm{d}\varepsilon^{k}}\log \vert 1 - \delta_{r}^{2}({\varepsilon}) \vert$$ converge uniformly on $I$ to $H^{(k)}$. It follows that $G$ is $C^{\infty}$ in any interval where it is nonzero, and by its definition we deduce that it is also continuous on $\mathbb{R}$. Having established that $G\left(\varepsilon\right)$ is $C^{\infty}$, except in the discrete set of its roots, makes demonstrating the concavity assertion now easier. Indeed, it suffices to argue that the second derivative of $\log G\left(\varepsilon\right)$ is strictly negative. In the previous part of the proof, we have actually shown that $$\label{logG} \log G(\varepsilon) = \log |\sqrt{5}\varepsilon+1| + H(\varepsilon) + A,$$ where $A$ is an absolute constant. Hence, the second derivative of $\log G(\varepsilon)$ is $$- 5 (\sqrt{5}{\varepsilon}+1)^{-2} -20 \sum_{r=1}^{\infty}(\vert u(r) -2\sqrt{5}\varepsilon-1 \vert ^{-2} + \vert u(r)+2\sqrt{5}\varepsilon+1 \vert ^{-2})\, <\, 0,$$ which implies the log-concavity of $G$ on the intervals under consideration. Recall that $G(-\phi/\sqrt{5})=1$, and note that $-\phi/\sqrt{5} < -0.26$. For the values ${\varepsilon}=-0.26$ and ${\varepsilon}= 0.58$ we can prove that $G({\varepsilon})>1$ (or, equivalently, that $\log G({\varepsilon})>0$). Indeed, we can explicitly estimate the error we make when we approximate the infinite series $H({\varepsilon})$ in by a finite series - the tail behavior of this infinite series is essentially the same as that of $\sum_{r=1}^{\infty} 1/r^2$. Accordingly, we can calculate that $G(-0.26) \in [1.09, 1.11]$ and that $G(0.58) \in [1.10, 1.12]$, and in particular that $G(-0.26)>1.01$ and $G(0.58)>1.01$. The log-concavity of $G$ implies that actually $G({\varepsilon})>1.01$ holds throughout the whole range ${\varepsilon}\in (-0.26,0.58)$. Proofs of Theorems for the quadratic irrationals ${\beta}=[b,b,\ldots]$ ======================================================================== As one might expect, the proof goes entirely along the lines of the proof of Theorem \[th1\]. Given a fixed integer $n\geq 1$, we write $[k]$ for the residue of the integer $k\in\mathbb{N}$ modulo $q_n$. Define $$\begin{aligned} s_n(0,{\varepsilon}) &=& 2\sin\pi\left(\frac{{\varepsilon}}{q_n}+\frac{{\beta}^{n+1}}{2} \right), \\ s_n(r) &=& 2\sin\pi\left( \frac{r}{q_n}+{\beta}^{n+1}\left(\frac{[q_{n-1}r]}{q_n} -\frac{1}{2}\right)\right), \, r=1,2,\ldots, F_n-1. \end{aligned}$$ The product $P_{q_n}({\beta},{\varepsilon})$ is further factorized as $$\label{bfactorisation} P_{q_n}({\beta},{\varepsilon}) = A_n({\beta},{\varepsilon})\cdot B_n({\beta})\cdot C_n({\beta},{\varepsilon}),$$ where $$\begin{aligned} A_n({\beta},{\varepsilon}) &=& 2 q_n\left| \sin \pi\left(q_n{\beta}+ (-1)^n\frac{{\varepsilon}}{q_n} \right)\right|,\\ B_n({\beta}) &=& \prod_{r=1}^{q_n-1} \frac{s_n(r)}{2\left| \sin(\pi r/q_n)\right|}, \hspace{3mm} \text{ and} \\ C_n({\beta}, {\varepsilon}) &=& \prod_{r=1}^{q_n-1}\left( 1-\frac{s_n(0)^2}{s_n(r)^2} \right)^{\frac{1}{2}} . \end{aligned}$$ Using , together with the asymptotic estimate $\sin x \sim x, x\to 0$ we get $$\begin{aligned} A_n({\beta},{\varepsilon}) &=& 2q_n|\sin\pi((-1)^n{\beta}^{n+1} +(-1)^n\frac{{\varepsilon}}{q_n} )| \\ &=& 2q_n\left|\sin\pi\left( {\beta}^{n+1} + \frac{{\varepsilon}}{q_n}\right)\right| \\ &\sim & 2\pi q_n \left|{\beta}^{n+1} + \frac{{\varepsilon}}{q_n} \right| \\ &\sim & 2\pi\left| {\varepsilon}+ \frac{1}{\sqrt{b^2+4}}\right|, \qquad n\to\infty . \end{aligned}$$ Regarding the products $B_n({\beta})$, it is shown in [@sigrid2] that there exists a constant $B_b>0$ such that $B_n({\beta})\to B_b$ as $n\to \infty$ and finally for the factor $C_n({\beta},{\varepsilon})$ we can show arguing as in [@mv] that $$\lim_{n\to\infty}C_n(\phi,{\varepsilon}) = \prod_{r=1}^{\infty}\left(1-\frac{(2{\varepsilon}\sqrt{b^2+4}+1)^2}{u_b(r)^2} \right) \hspace{3mm} \text{ for all } {\varepsilon}\in\mathbb{R},$$ where $u_b(r)$ was defined in . Combining these facts for the asymptotic behavior of the factors in we obtain the requested convergence result for $P_{q_n}({\beta},{\varepsilon})$. The fact that the convergence is uniform can be established as in the proof of Theorem \[th1\]. We calculate $$\begin{aligned} \frac{P_{q_{n+1}}({\beta})}{P_{q_{n-1}}({\beta})} &=&\! \prod_{r=q_{n-1}+1}^{q_{n+1}}\!\!\!2|\sin \pi r {\beta}| = \prod_{r=1}^{bq_n}2| \sin \pi(r+q_{n-1})\beta | \\ &\stackrel{\eqref{b1}}{=}& \prod_{r=1}^{bq_n} 2| \sin \pi(r\beta + (-1)^{n-1}{\beta}^n)| \\ &=& \prod_{j=1}^{b}\prod_{r=(j-1)q_n+1}^{jq_n}\!\!2|\sin\pi(r{\beta}-(-{\beta})^n)| \\ &=& \prod_{j=1}^{b}\prod_{r=1}^{q_n}2|\sin\pi(r\beta + (j-1)q_n{\beta}-(-{\beta})^n)| \\ &\stackrel{\eqref{b1}}{=}& \prod_{j=1}^{b}\prod_{r=1}^{q_n}2|\sin \pi(r{\beta}+ (-1)^{n}(j-1){\beta}^{n+1} -(-{\beta})^n) | \\ &=& \prod_{j=1}^{b}P_{q_n}({\beta}, {\varepsilon_{n}^{(j)}}) , \end{aligned}$$ where for $j=1,2,\ldots,b$ we have set $$\begin{aligned} {\varepsilon_{n}^{(j)}}& :=& q_n{\beta}^n [(j-1){\beta}-1] \\ &=& q_n{\beta}^{n+1} [(j-1){\beta}-1]{\beta}^{-1} \\ & \sim & - \frac{b-j + {\beta}+1}{\sqrt{b^2+4}},\hspace{4mm} n\to \infty. \end{aligned}$$ At this point we may once again factorise $$P_{q_n}({\beta}, {\varepsilon_{n}^{(j)}}) = A_n({\beta},{\varepsilon_{n}^{(j)}}) \cdot B_n({\beta}) \cdot C_n({\beta},{\varepsilon_{n}^{(j)}}), \qquad j=1,2,\ldots,b$$ where the factors are as in . Now $$\begin{aligned} A_n({\beta}, {\varepsilon_{n}^{(j)}}) &=& 2q_n| \sin \pi(q_n\beta + (-1)^n\frac{{\varepsilon_{n}^{(j)}}}{q_n} )| \\ &=& 2q_n |\sin \pi({\beta}^{n+1} + {\beta}^n [(j-1){\beta}-1] ) | \\ & \sim & 2\pi q_n {\beta}^{n+1} ({\beta}+ b - j) \\ &\sim & \frac{2\pi ({\beta}+ b - j)}{\sqrt{b^2+4}}, \hspace{4mm} n\to \infty . \end{aligned}$$ As mentioned before, it is shown in [@sigrid2] that there is $B_b>0$ such that $\lim\limits_{n\to\infty}B_b(\phi)=B_{b}$. Finally regarding the third factor we notice that $$\begin{aligned} s_n(0,{\varepsilon_{n}^{(j)}}) &=& 2|\sin \pi\left(\frac{{\varepsilon_{n}^{(j)}}}{q_n}+ \frac{{\beta}^{n+1}}{2} \right)| \\ &=& 2| \sin \pi ({\beta}^{n+1} [j-1-{\beta}^{-1}] + \frac{{\beta}^{n+1}}{2}) | \\ &\sim & \pi {\beta}^{n+1} ( 2b- 2j +2{\beta}+1), \hspace{4mm} n \to \infty \end{aligned}$$ and $$s_n(r) \sim 2\pi {\beta}^{n+1}\sqrt{b^2+4} \left( r - \frac{\{r{\beta}\}-\frac{1}{2}}{\sqrt{b^2+4}}\right) = \pi \beta^{n+1}u_b(r) , \hspace{3mm} \, n\to\infty$$ hence repeating once again the arguments utilised in [@mv; @sigrid2] we deduce that $$\lim_{n\to\infty} C_n({\beta}, {\varepsilon_{n}^{(j)}}) = \prod_{r=1}^{\infty}\left(1 - \frac{(2b-2j+2{\beta}+1)^2}{u_b(r)^2} \right)$$ Combining, $$\lim_{n\to\infty}P_{q_n}({\beta}, {\varepsilon_{n}^{(j)}}) = \frac{2\pi({\beta}+b-j)}{\sqrt{b^2+4}} \cdot B_{(b)} \cdot \prod_{r=1}^{\infty}\left(1 - \frac{(2b-2j+2{\beta}+1)^2}{u_b(r)^2} \right) .$$ Now by Theorem A $2$ we may deduce $$1 = \lim_{n\to\infty} \frac{P_{q_{n+1}}({\beta})}{P_{q_{n-1}}({\beta})} = \lim_{n\to\infty}\prod_{j=1}^{b}P_{q_n}({\beta}, {\varepsilon_{n}^{(j)}}),$$ whence $$\label{cbeq1} 1 = \left( \frac{2\pi}{\sqrt{b^2+4}} \right)^b \cdot \beta(\beta+1)\cdots (\beta + b-1) \cdot B_{b}^b \cdot \prod_{r=1}^{\infty} \prod_{j=1}^{b} \left(1 - \frac{(2b-2j +2{\beta}+1)^2}{u_b(r)^2} \right) \, .$$ At this point we observe that in the proof of Theorem A $2$ in [@sigrid2] it is actually shown that the constant $C_b>0$ satisfies $$\label{cbeq2} C_b = \frac{2\pi}{\sqrt{b^2+4}} \cdot B_{b} \cdot \prod_{r=1}^{\infty} \left( 1 - \frac{1}{u_b(r)^2}\right) .$$ Combining with we deduce the value of $C_b$. We now formulate and prove some complementary results that will be used later in the proof of Theorem \[lineargrowththeorem\]. \[prop: MV constant for b at least 15\] If $b\geq 11$, then $C_{b}<1$. We have $ {\beta}\geq (b + b^{-1})^{-1}$ and $({\beta}+ 1)\cdots ({\beta}+ b-1) \geq (b-1)!$. Thus by ,$$\begin{aligned} C_b^b & \leq& \frac{b + b^{-1}}{ (b-1)!} \prod_{r=1}^{\infty}\prod_{j=1}^{b} \left( 1- \frac{1}{u_b(r)^2}\right)\left( 1 - \frac{(2j +2{\beta}-1)^2}{u_b(r)^2} \right)^{-1} \\ & = & \frac{b+b^{-1}}{ (b-1)!} \prod_{r=1}^{\infty}\prod_{j=1}^{b} \left( 1 + \frac{(2j+2{\beta}-1)^2 -1 }{u_b(r)^2 - (2j + 2{\beta}-1)^2 } \right) . \end{aligned}$$ Now using the estimates $1+x \leq e^x, x\in\mathbb{R}$ and $u_b(r)^2-(2b+2{\beta}-1)^2 \geq 2(b^2+4)r^2, r\geq 1$ we find $$\begin{aligned} C_b^b & \leq & \frac{b+ b^{-1}}{ (b-1)!} \prod_{r=1}^{\infty}\prod_{j=1}^{b} \exp\left( \frac{(2j+2{\beta}-1)^2 -1 }{u_b(r)^2 - (2j + 2{\beta}-1)^2 } \right) \\ & \leq & \frac{b+ b^{-1}}{ (b-1)!} \exp\left( \sum_{r=1}^{\infty}\sum_{j=1}^{b} \frac{(2j+2{\beta}-1)^2 -1 }{u_b(r)^2 - (2b + 2{\beta}-1)^2 } \right) \\ & \leq & \frac{b+ b^{-1}}{ (b-1)!} \exp\left( \sum_{r=1}^{\infty} \frac{ \frac{4}{3}b^3 + 4{\beta}b^2 - (\frac{4}{3}-4{\beta}^2 )b }{2(b^2+4)r^2 } \right) \\ &\leq & \frac{b+ b^{-1}}{ (b-1)!} \exp\left( \frac{(2b^3 + b + 6b^{-1} )\pi^2 }{18(b^2 + 4)} \right) . \end{aligned}$$ The right hand side is decreasing as a function of $b$ for $b\geq 11$, and we can verify that it is less than $1$ when $b=11$. This proves that $C_b<1$ for all $b\geq 11$. \[cor: when M-V const greater one\] The constant $C_{b}$ exceeds $1$ if and only if $b\in \{1, 2, 3, 4, 5, 6\}$. By , $\log C_b$ is given by an infinite series, and we can provide explicit estimates for the error of approximation of the series by a finite sum. By Proposition \[prop: MV constant for b at least 15\], in order to prove the corollary it remains to compute the value of $C_{b}$ for $1\leq b \leq 10$. These values are shown in the following table, with precision of $6$ decimal digits.\ $b$ $1$ $2$ $3$ $4$ $5$ --------- ------------ ------------ ------------ ------------ ------------ $C_{b}$ $2.406152$ $2.159658$ $1.800517$ $1.499350$ $1.267273$ $b$ $6$ $7$ $8$ $9$ $10$ $C_{b}$ $1.089429$ $0.951175$ $0.841663$ $0.753296$ $0.680773$ \  \ The corollary is now proved. The proof of Theorem \[lineargrowththeorem\] is given in several lemmas, which deal with different cases for the value of the integer $b$. We make use of the *Ostrowski expansion* of integers with respect to some given irrational number, which we now briefly present. Fix some irrational number $\alpha\in (0,1)$ with continued fraction expansion $\alpha=[a_1, a_2, \ldots]$ and let $(q_n)_{n=0}^{\infty}$ be the corresponding sequence of denominators. Every positive integer $N\geq 1$ can be written uniquely in the form $$\label{ostrowski} N = c_{n+1}q_n + c_{n}q_{n-1} + \ldots + c_2q_1 + c_1q_0$$ where the integers $(c_i)_{i=1}^{n+1}$ are such that 1. $0\leq c_1 < a_1$ 2. $0\leq c_{i+1} \leq a_{i+1}$ $i=1,\ldots, n$ 3. $c_i=0$ $c_{i+1}=a_{i+1}$. The expansion of $N$ as in is called the [*Ostrowski expansion*]{} of $N$ with respect to $\alpha$. For more details we refer to [@rocket Chapter II].\ \[lemma&gt;1\] Let ${\beta}$ be defined as in Theorem \[lineargrowththeorem\]. If $b \geq 7$, then $$\liminf \limits_{N \to \infty} P_N({\beta}) = 0 \qquad \text{and} \qquad \limsup \limits_{N\to\infty}\dfrac{P_N({\beta})}{N} = \infty.$$ Let $b \geq 7$ be fixed, and let ${\beta}$ be defined as in Theorem \[lineargrowththeorem\]. The key fact in the proof will be that $C_b <1$. By the upper bound for $C_b$ given in the proof of Proposition \[prop: MV constant for b at least 15\] and the table in Corollary \[cor: when M-V const greater one\] we see that we actually have $C_b \leq 0.96$ for $b\geq 7$. By Proposition \[gbetaproperties\], $G_\beta $ is continuous near ${\varepsilon}=0$, so there exists $\eta>0$ small enough such that $G_\beta({\varepsilon}) \leq 0.98$ for $|{\varepsilon}| \leq \eta$. Set $m_1=1$. We construct a sequence $m_1 < m_2 < m_3 < \dots$ of positive integers, such that: 1. every $m_k$ is the denominator of a convergent to $\beta$, 2. $m_k \geq 2 m_{k-1}$ for all $k \geq 2$, and 3. $\|m_k \beta\| < \eta / (4 m_{k-1})$ for $k \geq 2$. This construction is always possible by choosing $m_k$ sufficiently large compared to $m_{k-1}$. Let $N_k = m_k + m_{k-1} + \dots + m_1, \, k\geq 1$. Then $$\begin{aligned} P_{N_k}(\beta)\, & = \,& \prod_{r=1}^{N_k} 2|\sin \pi r\beta| \,\, = \,\, \prod_{j=1}^{k}\prod_{r=1}^{m_j}2|\sin \pi (M_j + r) \beta| \nonumber\\ & = \, & \prod_{j=1}^{k} \prod_{r=1}^{m_j}2 \left|\sin \pi\left(r \beta + \frac{{\varepsilon}_j}{m_j}\right)\right| \,\, = \,\, \prod_{j=1}^{k}P_{m_j}(\beta,(-1)^j{\varepsilon}_j) , \label{prod_nk} \end{aligned}$$ where $$\label{mj} M_k = 0 \qquad \text{and} \qquad M_j = m_k + m_{k-1} + \dots + m_{j+1}, \quad j=1,2,\ldots, k-1$$ and where either $${\varepsilon}_j =m_j \| M_j \beta\| \qquad \text{or} \qquad {\varepsilon}_j = -m_j \| M_j \beta\|.$$ In any case we have $$\begin{aligned} |{\varepsilon}_j| & = & m_j \|(m_{j+1} + m_{j+2} \dots + m_{k}) \beta\| \nonumber\\ & \leq & m_j (\|m_{j+1} \beta\| + \| m_{j+2} \beta \| + \dots + \|m_{k} \beta\|) \label{anycase}\\ & \leq & \frac{\eta}{4} \left( \frac{m_j}{m_j} + \frac{m_j}{m_{j+1}} + \dots + \frac{m_j}{m_{k}} \right) \nonumber\\ & \leq & \frac{\eta}{2}. \label{anycase2} \end{aligned}$$ Thus we have $G_\beta({\varepsilon}_j) \leq 0.98$ for all $j$, and consequently $P_{m_j}(\beta,{\varepsilon}_j) \leq 0.99$ for all sufficiently large $j$. Accordingly, in all factors (except for finitely many) are smaller than 0.99 and the product tends to 0 as $k \to \infty$. This proves the first assertion of the lemma.\ For the second part, we set $N_k = m_{k+1} - 1 - m_k - m_{k-1} - \ldots - m_1,\, k\geq 1$. Then $$\label{pnkbeta} P_{N_k}(\beta) = \frac{P_{m_{k+1}-1}(\beta)}{\prod\limits_{r=N_k+1}^{m_{k+1}-1}\!\!2|\sin \pi r \beta|} = \frac{P_{m_{k+1}-1}(\beta)}{\prod\limits_{j=1}^{k}\prod\limits_{r=1}^{m_j} 2|\sin \pi (-m_{k+1} + M_j + r) \beta|},$$ with $M_j$ defined as in . By Theorem A 2 the numerator in satisfies $$P_{m_{k+1}-1}(\beta) \, \, \asymp \,\, m_{k+1} \, \asymp \, \, N_k\, , \qquad k\to \infty,$$ and it remains to show that the denominator in tends to zero as $k \to \infty$. This can be done in exactly the same way as above, the only difference being that in we now have an additional term $m_j \|m_{k+1} \beta\|$, which does not affect the validity of . Thus $\prod_{r=N_k+1}^{m_{k+1}-1} 2|\sin \pi r \beta| \to 0$ as $k \to \infty$, and $P_{N_k}(\beta) / N_k \to \infty$ as $k \to \infty$, which proves the second part of the lemma. The following lemma settles the case $b=6$ in Theorem \[lineargrowththeorem\]. \[lemma\_b6\] Let ${\beta}$ be defined as in Theorem \[lineargrowththeorem\], for $b=6$; that is, $\beta = \sqrt{10}-3$. Then $$\liminf \limits_{N \to \infty} P_N({\beta}) = 0 \qquad \text{and} \qquad \limsup \limits_{N\to\infty}\dfrac{P_N({\beta})}{N} = \infty.$$ It was already noted in [@gkn] that $P_N(\beta)$ seems to be decreasing along a subsequence of indices $N$. In [@gkn] the following table concerning the evolution of minima of $P_N(\beta)$ is given:\ $N$ 1 7 44 272 1677 10335 -------------- ------- ------- ------- ------- ------- ------- $P_N(\beta)$ 0.977 0.907 0.849 0.794 0.742 0.693  \ The denominators of continued fraction convergents to $\beta$ are $1,6,37,228,1405,8658, \dots$, and one can see that the indices $N$ above arise as sums of such denominators. For example, we have $272 = 228 + 37 + 6 + 1$, or $1677 = 1405 + 228 + 37 + 6$. We will exploit this structure to construct a subsequence of indices along which the Sudler product at $\beta$ tends to zero.\ A similar table shows values of $N$ for which the ratio $P_N(\beta)/N$ is large.\ $N$ 30 184 1133 6981 -------------- ------- ------- ------- ------- $P_N(\beta)$ 1.061 1.213 1.286 1.378  \ From the analogue of the reflection principle for $\beta$, it is not surprising that these indices $N$ also arise from denominators of convergents to $\beta$, but in a “reflected” way. Indeed, we can see that $30 = 37 - 6 - 1$, that $184 = 228 - 37 - 6 - 1$, and so on. Thus we can imitate this structure to construct a subsequence of indices along which the Sudler product at $\beta$ shows the desired growth behaviour.\ Let $\beta =[6,6,\ldots]= \sqrt{10}-3$, and let $(q_n)_{n=0}^{\infty}$ be the sequence of denominators of convergents to $\beta$. Set $N_k = q_k + q_{k-1} + \dots + q_1+q_0$ for $k \geq 1$. Furthermore, we set $M_j^{(k)} = q_k + q_{k-1} + \dots + q_{j+2} + q_{j+1}$, for $j=0, \dots, k-1$, and $M_k^{(k)}=0$. Then $$\begin{aligned} P_{N_k} (\beta) & = & \prod_{r=1}^{N_k} 2 |\sin \pi r \beta | \, =\, \prod_{j=1}^{k} \prod_{r=M_j^{(k)}+1}^{M_j^{(k)} + q_j}\!\!\!2 |\sin \pi r \beta | \nonumber\\ & = & \prod_{j=1}^{k} \prod_{r=1}^{q_j} 2 |\sin \pi (r \beta + M_j^{(k)} \beta) | \nonumber\\ & = & \prod_{j=1}^{k} \prod_{r=1}^{q_j} 2 \left|\sin \pi \left(r \beta + (-1)^{j} \frac{{\varepsilon}_j^{(k)}}{q_j} \right) \right| \nonumber\\ & = & \prod_{j=1}^{k} P_{q_j}(\beta,{\varepsilon}_j^{(k)}), \label{p-fact} \end{aligned}$$ where in view of , ${\varepsilon}_j^{(k)}$ is defined so that $$\begin{aligned} \frac{(-1)^{j} {\varepsilon}_j^{(k)}}{q_j} & = & M_j^{(k)} \beta - q_{k-1} - q_{k_2} - \dots - q_{j+1} - q_j \\ & = & (q_k + q_{k-1} + \dots + q_{j+2} + q_{j+1}) \beta - q_{k-1} - q_{k_2} - \dots - q_{j+1} - q_j \\ [1ex] & = & (-1)^{k} \beta^{k+1} + (-1)^{k-1} \beta^{k} + \dots + (-1)^{j+2} \beta^{j+3} + (-1)^{j+1} \beta^{j+2}. \end{aligned}$$ Accordingly $$\frac{{\varepsilon}_j^{(k)}}{q_j \beta^{j+1}} =(-1)^{k-j} \beta^{k-j} + (-1)^{k-j-1} \beta^{k-j-1} + \dots + (-1)^{2} \beta^{2} + (-1)^{1} \beta^{1}.$$ Since $q_1\beta^2 \leq q_j \beta^{j+1} \leq q_2{\beta}^3$ for all $j=1,2,\ldots$ and $$-\frac{{\beta}}{\sqrt{40}}\, \leq \, \sum_{r=1}^n \frac{ (-{\beta})^{r} }{\sqrt{40}}\, \leq \, -\frac{{\beta}}{\sqrt{40}} + \frac{{\beta}^2}{\sqrt{40}} , \quad n=1,2,\ldots$$ we deduce that ${\varepsilon}_j^{(k)} \in [-0.0257,-0.02]$ for all $k\geq 1$ and $ j< k$. Furthermore we can verify that $G_{{\beta}}(-0.257) < G_\beta(-0.020) \approx 0.949$, up to an error of $\pm 0.01$ – again this can be formally proved with an appropriate estimate for the approximation errors in .\ *Claim:* We have $G_\beta({\varepsilon}) \leq 0.96$ for any ${\varepsilon}\in [-0.0257,-0.02]$.\ *Proof of Claim:* We know that $\log G_{{\beta}}$ is strictly concave in the interval $[-1/\sqrt{40},(6+{\beta})/\sqrt{40} ]$ that is formed by two consecutive roots of $G_{{\beta}}$. Consequently, $\log G_{{\beta}}$ –and hence also $G_{{\beta}}$– is increasing in some interval $[-1/\sqrt{40}, s_0]$ and decreasing in $[s_0, (6+{\beta})/\sqrt{40}]$. We saw that $G_{{\beta}}(-0.025)< G_{{\beta}}(-0.02)< 0.96$ and also we can show that $G_{{\beta}}(0)> 1.05 $, so $s_0>0$ and $G_{\beta}$ is increasing in $[-1/\sqrt{40}, 0]$. The Claim now follows.\ Since the functions $P_{q_j}$ converge uniformly to $G_{\beta},$ we may bound all factors $P_{q_j}(\beta,{\varepsilon}_j^{(k)})$ in from above by $0.97$ whenever $j \geq j_0$ for some appropriate $j_0$ (independent of $k$). Furthermore we can show that the remaining initial factors are bounded from below arguing as in the proof of Theorem \[th3\]. This proves that $P_{N_k} (\beta) \ll 0.97^{k - j_0}$ and thus $P_{N_k}(\beta) \to 0$ as $k \to \infty$. Consequently, $\liminf\limits_{N \to \infty} P_N(\beta)=0$.\ Proving that $\limsup \limits_{N\to\infty}\dfrac{P_N({\beta})}{N} = \infty$ can be done in a perfectly analogous way. We define $N_k = q_{k+1} - q_k - q_{k-1} - \ldots - q_1$, and use the reflection principle adjusted to the case of the irrational ${\beta}$. Then we are led to estimating products similar to the ones above, except that there is a additional perturbation $- q_{k+1} \beta$. However, the influence of this additional perturbation is very small (just one additional term in a geometric series – cf. the remarks after the proof of Theorem \[th3\]), so we can use exactly the same estimates as above in order to obtain the desired conclusion. It remains to deal with the case when $b \in \{2,3,4,5\}$. \[lemma\_b2-5\] Let ${\beta}$ be defined as in Theorem \[lineargrowththeorem\], for $b \in \{2,3,4,5\}$. Then $$\liminf \limits_{N \to \infty} P_N({\beta}) > 0 \qquad \text{and} \qquad \limsup \limits_{N\to\infty}\dfrac{P_N({\beta})}{N} < \infty.$$ In principle we could settle all cases $b \in \{2,3,4,5\}$ simultaneously, keeping track of all possible structures of the perturbations ${\varepsilon}$ coming from the Ostrowski expansions of the corresponding integers with respect to $\beta$. However, for the reader’s convenience we decided to give a proof only for the particular case $b=5$, which is the most delicate one. The other cases $b \in \{2,3,4\}$ can be treated in a perfectly analogous way.\ So we fix $b=5$, which means that $\beta = \frac{\sqrt{29}-5}{2}$. Before we start with the proof, let us give a heuristic description of what will happen. The first few denominators of convergents to $\beta$ are given by $1,5,26,135,701,\dots$. We have $G_\beta(0) = C_5 > 1$ by Corollary \[cor: when M-V const greater one\], and (as we will show) we have $G_\beta({\varepsilon})>1$ for all possible *positive* perturbations ${\varepsilon}$ that could come from the Ostrowski expansion. Thus $\liminf P_N(\beta)=0$ could only happen as the effect of *negative* perturbations ${\varepsilon}$ for which $G_\beta({\varepsilon})<1$. Since the differences $\beta - p_n/q_n$ have alternating signs, we know which structure in the Ostrowski expansion can give negative perturbations; essentially, there are the ones coming from an *odd* difference in the index of the convergent denominator. Indeed, consider for example the case when $N = 31 = 26+5$. According to the Ostrowski expansion, we split the product $P_N(\beta)$ in the form $$\begin{aligned} P_N(\beta) & = & \prod_{r=1}^N 2 |\sin \pi r \beta | \\ & = & P_{26}(\beta) \cdot \prod_{r=1}^5 2 |\sin \pi (26 + r) \beta| \\ & = & P_{26}(\beta) \cdot \prod_{r=1}^5 2 |\sin \pi (\underbrace{26 \beta}_{\approx 5.007} + r \beta)| \\ & = & P_{26}(\beta) \cdot P_5 (\beta, {\varepsilon}),\end{aligned}$$ where ${\varepsilon}$ is the deviation of $26 \beta \cdot 5$ from the nearest integer (which is 25), and thus ${\varepsilon}\approx -0.035$. Note that ${\varepsilon}$ has negative sign. This essentially is a consequence of the fact that $26=q_3$ and $5 = q_2$, and that the difference of the indices is $3-2$, which is *odd*.\ To give a few other examples, when $N = 135 + 5$ where $135 = q_4$ and $5 = q_2$, then we get $P_{140}(\beta) = P_{135}(\beta) \cdot P_5 (\beta, {\varepsilon})$ with ${\varepsilon}\approx 0.0069$, which is positive (and thus is good, since it implies $G_\beta({\varepsilon})>1$). When we have $N = 701 + 5$ where $701=q_5$ and $5 = q_2$ then we get $P_{706}(\beta) = P_{701}(\beta) \cdot P_5 (\beta, {\varepsilon})$ with ${\varepsilon}\approx -0.0013$, which again is negative (since $5-2=3$ is *odd*).\ So we saw that negative perturbations can only come from *odd* differences of indices in the Ostrowski expansion. However, it turns out that the perturbation above, which was roughly $-0.035$, still does not cause us problems, since there $G_\beta$ still exceeds one (we have $G_\beta(-0.035) \approx 1.02$).\ To reach a region where indeed $G_\beta<1$, we need the Ostrowski expansion of $N$ to have a specific structure. Consider now the number $N = 83 = 26 + 26 + 26 + 5$. According to the Ostrowski expansion, we decompose the product $P_N(\beta)$ into $$\begin{aligned} P_N(\beta) & = & \left(\prod_{r=1}^{26} 2 |\sin \pi r \beta|\right) \cdot \left(\prod_{r=1}^{26} 2 |\sin \pi (26 + r) \beta|\right) \cdot \\ & & \cdot \left(\prod_{r=1}^{26} 2 |\sin\pi (26 + 26 + r) \beta|\right) \cdot \left(\prod_{r=1}^5 2 |\sin\pi (26 + 26 + 26 + r) \beta|\right).\end{aligned}$$ The last product is $P_5(\beta,{\varepsilon})$ with a perturbation ${\varepsilon}$ coming from the difference between $3 \cdot 26 \beta \cdot 5$ and the nearest integer (which is 75), which gives the large negative perturbation ${\varepsilon}\approx - 0.107$. We have $G_\beta(-0.107) \approx 0.54$, which is significantly smaller than 1, so in the decomposition of $P_N$ there is a factor which is much smaller than 1. However, note that we could only reach such a large negative perturbation by constructing $N$ in a way such that in the Ostrowski representation the same number (in our case 26) occurs more than once. Thus in the decomposition of $P_N$ we also see the products $$\prod_{r=1}^{26} 2 |\sin\pi (26 + r) \beta| = P_{26}(\beta,{\varepsilon}),$$ with ${\varepsilon}\approx 0.19$, and $$\prod_{r=1}^{26} 2 |\sin\pi (26 + 26 + r)\beta| = P_{26}(\beta,{\varepsilon})$$ with ${\varepsilon}\approx 0.37$, which means that we see two perturbed Sudler products $P_{26}$ with large *positive* perturbations. This gives us two additional large factors of size roughly $G_\beta(0.19) \approx 2.27$ and $G_\beta(0.37) \approx 2.67$ in the decomposition of $P_N(\beta)$, which actually even overcompensate the influence of the single small factor of size roughly $0.54$.\ \ ![An analogue of the picture on the left-hand side of Figure 4, but for the case $b=6$ instead of $b=5$. Here $N=43=37+6$, which again is the sum of two consecutive convergent denominators. The product $P_{37}(\beta)$ is roughly 1.08. The perturbed product is roughly $P_6(\beta,-0.026) \approx G_\beta(-0.026) \approx 0.92$, which in contrast to the case $b=5$ now is smaller than 1. Choosing a number $N$ which is the sum of more convergent denominators to $\beta$, such as $N=1676=1405+228+37+6$, would give more contributions near the dot in the picture, all of them producing factors which are smaller than 1. This is the principle behind the proof of Lemma \[lemma&gt;1\].](Gcase6.eps) This outlines the basic strategy for the proof of the theorem. We will make a case distinction, depending on whether a “digit” in the Ostrowski expansion of $N$ exceeds one or not. Whenever a digit is one or zero, it cannot lead to a large negative perturbation for the subsequent partial product, and everything is okay. If a digit exceeds one, then this gives large additional factors in our product decomposition, which overcompensate the potential small factor coming from the subsequent partial product which may have a large negative shift.\ For the arbitrary $N\geq 1$ let $n=n(N)\geq 1$ be such that $q_{n}\leq N+1 < q_{n+1}$. We can write $N$ in its Ostrowski representation in the form $$N = c_{n+1} q_n + c_n q_{n-1} + \dots + c_2 q_1 + c_1 q_0,$$ where we have $0 \leq c_1 < 5$ and $$0 \leq c_i \leq 5 \quad \text{for all $i$} \qquad \text{and } \qquad \text{$c_i=0$ \, whenever \, $c_{i+1}=5$.}$$ For $0 \leq i \leq n$ and $0 \leq a_{i} < c_{n+1-i}$ we set $$M_{i,a_i} = c_{n+1} q_n + c_n q_{n-1} + \dots + c_{n+2-i} q_{n+1-i} + a_{i} q_{n-i},$$ so that we can split the whole product $P_N(\beta)$ in the form $$\begin{aligned} P_N({\beta}) & = & \prod_{i=0}^n ~\prod_{a_i=0}^{c_{n+1-i} - 1}~ \prod_{r=M_{i,a_{i}}+1}^{M_{i,a_{i}}+q_{n-i}} 2 |\sin \pi r \beta| \\ & = & \prod_{i=0}^n ~\prod_{a_i=0}^{c_{n+1-i} - 1}~ \prod_{r=1}^{q_{n-i}} 2 |\sin\pi (M_{i,a_{i}}+r) \beta|.\end{aligned}$$ To exploit the phenomenon addressed above, we will now split the product in such a way that we combine the contribution of “large” digits of $q_i$ with the contribution of the first (zero) digit of $q_{i-1}$. In formulas, we re-organize the product above in the form $$\begin{aligned} P_N({\beta}) & = & \prod_{i=0}^n ~\prod_{a_i=0}^{c_{n+1-i} - 1}~ \prod_{r=1}^{q_{n-i}} 2 |\sin\pi (M_{i,a_{i}}+r) \beta| \nonumber\\ & = & \left( \prod_{r=1}^{q_n} 2 |\sin\pi (M_{0,0}+r) \beta| \right) \cdot \label{prod_l} \\ & & \cdot \prod_{i=0}^n \left(\left(\prod_{a_i=1}^{c_{n+1-i} - 1}~ \prod_{r=1}^{q_{n-i}} 2 |\sin \pi (M_{i,a_{i}}+r) \beta | \right) \left(\prod_{r=1}^{q_{n-i-1}} 2 |\sin \pi (M_{i+1,0}+r) \beta | \right)\right) \label{prod_l2}\end{aligned}$$ We have $M_{0,0}=0$, so for the product in line we simply have $\prod_{r=1}^{q_n} 2 |\sin \pi (M_{0,0}+r) \beta | \to C_b$ as $n\to\infty$. To emphasize the decomposition again, we have split the product in such a way that in the contribution of the “digits” greater than one attached to some $q_j$ is combined with the contribution of the digit one attached to the next-smallest convergent denominator $q_{j-1}$.\ Note that the product over $a_i$ in is empty when $c_{n+1-i} \in \{0,1\}$. Similarly the final product in can be empty, which happens when $c_{n-i} = 0$. We remind the reader that products over empty index sets are understood to equal 1.\ So let us assume that we have chosen some value of $i$ in . We wish to show that there is a $i_0$ such that $$\label{prod_ex} \left(\prod_{a_i=1}^{c_{n+1-i} - 1}~ \prod_{r=1}^{q_{n-i}} 2 |\sin \pi (M_{i,a_{i}}+r) \beta | \right) \left(\prod_{r=1}^{q_{n-i-1}} 2 |\sin \pi (M_{i+1,0}+r) \beta | \right)$$ is large enough whenever $n-i \geq i_0$. Ideally we would wish that all factors in are $>1$, since that would directly imply our desired result. It will turn out that it is not true that all factors in exceed 1. While sometimes it may happen that such a factor is a bit smaller than 1, this will be compensated by other factors in which exceed 1.\ We distinguish the following cases depending on the values of $c_{n+1-i}, c_{n-i}$ and also, at times, depending on the values of $c_{n+2-i}$, and $c_{n+3-i}$.\ $ \bullet $ Case 1: $c_{n-i} \neq 0$ and $c_{n+1-i}=0$.\ In this case the product $\prod\limits_{a_i=1}^{c_{n+1-i} - 1}\prod\limits_{r=1}^{q_{n-i}} 2 |\sin \pi (M_{i,a_{i}}+r) \beta | $ in is empty, and we will\ show that $$\label{to_show} \prod_{r=1}^{q_{n-i-1}} 2 |\sin \pi (M_{i+1,0}+r) \beta | > 1.$$ Recall that by definition $$M_{i+1,0} = c_{n+1} q_n + c_n q_{n-1} + \dots + c_{n+1-i+1} q_{n-i+1} + c_{n+1-i} q_{n-i} + 0 \cdot q_{n-i-1},$$ where the last term is zero because $a_{i+1}=0$. Since we assumed that $c_{n+1-i}=0$ the penultimate term also vanishes. Hence $$M_{i+1,0} = c_{n+1} q_n + c_n q_{n-1} + \dots + c_{n+2-i} q_{n-i+1}.$$ We write $$\begin{aligned} \prod_{r=1}^{q_{n-i-1}} 2 |\sin \pi (M_{i+1,0}+r) \beta | & = & \prod_{r=1}^{q_{n-i-1}} 2 \left|\sin \pi \left(r \beta + \frac{(-1)^{n-i-1} {\varepsilon}}{q_{n-i-1}} \right) \right| = P_{q_{n-i-1}} (\beta, {\varepsilon}) , \end{aligned}$$ where by and $\varepsilon$ can be chosen such that $$\begin{aligned} \frac{(-1)^{n-i-1} {\varepsilon}}{q_{n-i-1}} &=& M_{i+1,0} \beta - (c_{n+1} q_{n-1} + \dots + c_{n+1-i+1} q_{n-i} ) \\ &=& - c_{n+1} (-\beta)^{n+1} - c_n (-\beta)^{n} - \dots - c_{n+2-i} (- \beta)^{n+2-i}.\end{aligned}$$ Thus we have $$\begin{aligned} \label{final_t} \frac{{\varepsilon}}{q_{n-i-1} \beta^{n-i}} & = & c_{n+1} (-1)^{i+1} \beta^{i+1} + c_n (-1)^{i} \beta^{i} + \dots + c_{n+2-i} (-1)^2 \beta^{2}.\end{aligned}$$ In the sum on the right of there are both negative and positive contributions. Note that the last (largest) term is positive. Since the coefficients $c_{n+1}, ~c_n, \dots, c_{n+2-i}$ are all bounded above by 5, the total contribution of the positive terms is at most $$\sum_{j=1}^\infty 5 \beta^{2j} = \beta,$$ and the total contribution of the negative terms is at most $$- \sum_{j=1}^\infty 5 \beta^{2j+1} = -\beta^2.$$ By we have $q_{n-i-1} \beta^{n-i} \to 1/\sqrt{29}$. Note that $\beta/\sqrt{29} \approx 0.036 < 0.04$ and $- \beta^2 / \sqrt{29} \approx -0.007 > -0.01$. So we have that $${\varepsilon}\in [-0.01,0.04],$$ provided that $i$ is sufficiently large. We have $G_\beta(0.04) \approx 1.50$, and in particular we can formally prove (again using formula for $G_\beta$ and estimates for approximation errors for the infinite product) that $G_\beta(0.04) > 1.1$ . Similarly, we have $G_\beta(-0.01) \approx 1.19$, and we can formally prove that $G_\beta(-0.01) > 1.1$ . Thus by Proposition \[gbetaproperties\], the log-concavity of $G_\beta$ implies that $G_\beta({\varepsilon}) > 1.1$ throughout the whole range $[-0.01,0.04]$ of possible perturbations ${\varepsilon}$. By the uniform convergence in Theorem \[th6\] this means that $P_{q_{n-i-1}} (\beta, {\varepsilon})> 1.05$ whenever $i$ is sufficiently large. Thus we have established for all $i \geq i_0$. The upshot is, all factors appearing in which belong to Case $1$ are $>1$, except for finitely many of them; the overall contribution of the Case $1$ factors to the product in can be bounded below by an absolute constant as in the proof of Theorem \[th3\].\ $\bullet$ Case $2$: $c_{n-i} \neq 0$ and $c_{n+1-i}=1$.\ In this case again the product $\prod\limits_{a_i=1}^{c_{n+1-i} - 1}\prod\limits_{r=1}^{q_{n-i}} 2 |\sin \pi (M_{i,a_{i}}+r) \beta | $ in is empty, and we\ have to control the size of $$\label{to_show_2} \prod_{r=1}^{q_{n-i-1}} 2 |\sin \pi (M_{i+1,0}+r) \beta |.$$ Now we have $$M_{i+1,0} = c_{n+1} q_n + c_n q_{n-1} + \dots + c_{n+1-i+1} q_{n-i+1} + \underbrace{c_{n+1-i}}_{=1} q_{n-i}.$$ Similar to Case 1, we can write the product in in the form $$\prod_{r=1}^{q_{n-i-1}} 2 |\sin \pi (M_{i+1,0}+r) \beta| = P_{q_{n-i-1}} (\beta, {\varepsilon}),$$ where instead of we now obtain $$\begin{aligned} \frac{{\varepsilon}}{q_{n-i-1} \beta^{n-i}} \label{final_t_2} &=& c_{n+1} (-1)^{i+1} \beta^{i+1} + c_n (-1)^{i} \beta^{i} + \dots + c_{n+1-i+1} (-1)^2 \beta^{2} + (-1)^1 \beta^1. \nonumber\end{aligned}$$ Here the last term “$+(-1)^1 \beta^1$” comes from the assumption that $c_{n+1-i}=1$. Again, we find the maximal positive and negative contributions to determine the range of all possible perturbations ${\varepsilon}$. We have $$\frac{{\varepsilon}}{q_{n-i-1} \beta^{n-i}} \leq - \beta + \sum_{j=2}^\infty 5 \beta^{2j} < 0$$ and $$\begin{aligned} \frac{{\varepsilon}}{q_{n-i-1} \beta^{n-i}} & \geq & - \beta + c_{n+2-i} \beta^2 - c_{n+3-i} \beta^3 - \sum_{j=2}^\infty 5 \beta^{2j+1} \nonumber\\ & = & -\beta + c_{n+2-i} \beta^2 - c_{n+3-i} \beta^3 - \beta^4.\label{detail}\end{aligned}$$ We know that $G_\beta(0) = C_5 \approx 1.25$, and we can formally verify that $G_\beta(0) > 1.1$, so $P_{q_{n-i-1}} (\beta, {\varepsilon}) > 1.1 $ for all $i\geq i_0$. However, the situation is more delicate regarding the lower bound of possible perturbations: suppose that in we ignore the influence of the digits $c_{n+2-i}$ and $c_{n+3-i}$, and just use the estimates $c_{n+2-i} \geq 0$ and $c_{n+3-i} \leq 5$. This would lead to ${\varepsilon}\geq (- \beta - \beta^2)/\sqrt{29}$. Now $(-\beta - \beta^2)/\sqrt{29} \approx -0.04$, but $G_\beta(-0.04<1) .$ So we need to provide a sharper estimate, and we distinguish further sub-cases based on the values of $c_{n+2-i}$ and $c_{n+3-i}$.\ - Case 2a: $c_{n+3-i} \leq 2$ or $c_{n+2-i} \neq 0$.\ In this case from we can deduce that $$\frac{{\varepsilon}}{q_{n-i-1} \beta^{n-i}} \,\geq \, \min \{- \beta - 2 \beta^3 - \beta^4,\, - \beta + \beta^2 - 5 \beta^3 - \beta^4\}.$$ Since $q_{n-i-1} \beta^{n-i} \to 1/\sqrt{29},$ this implies that ${\varepsilon}> -0.0387$ for sufficiently large $n$. We have $G_\beta(-0.0387) \approx 1.002$, and we can formally verify that $G_\beta(-0.0387) > 1.001$. We noted above that $G_\beta(0)>1.1$. Thus we have $G_\beta({\varepsilon}) > 1.001$ for all possible perturbations in Case 2a, and thus $P_{q_{n-i-1}} (\beta, {\varepsilon}) > 1.0001$ for all sufficiently large $i$. Thus all factors in Case 2a exceed 1, except for finitely many factors coming from indices $i < i_0$ that can be treated similarly as in the proof of Theorem \[th3\]. Thus the overall contribution of the Case 2a factors is bounded below by a positive absolute constant.\ - Case 2b: $c_{n+3-i} \geq 3$ and $c_{n+2-i} = 0$.\ In this case from we have ${\varepsilon}/(q_{n-i-1} \beta^{n-i}) \geq - \beta - 5 \beta^3 - \beta^4$, and thus by $q_{n-i-1} \beta^{n-i} \to 1/\sqrt{29}$ we have ${\varepsilon}\geq - 0.043$. We have $G_\beta(-0.043) \approx 0.973$, and we can formally prove that $G(-0.043) > 0.97$ . Note that in Case 2b we can thus *not* guarantee that we have a factor which is 1 or less. Instead we can only deduce that $P_{q_{n-i-1}} (\beta, {\varepsilon}) > 0.96$, say, for all sufficiently large $i$. Note that Case 2b only occurs when the Ostrowski expansion has a “digit” of at least 3, followed by a zero “digit”. Thus the joint overall contribution of the Case 2b factors is not less that an absolute constant multiplied with $$\label{small_c} 0.96^{A}, \qquad \textrm{where $A = \# \{2 \leq i \leq n:~c_i \geq 3,~ \text{and} ~ c_{i-1} \neq 0 \}$}$$ (as always, the absolute constant comes from the constribution of finitely many indices for which $i<i_0$). We will need to show that the “small” contribution of to the product is compensated by an overshoot in the contribution of Case 6.\ $\bullet$ Case 3: $c_{n-i} \neq 0$ and $c_{n+1-i}=2$.\ In this case the product $\prod\limits_{a_i=1}^{c_{n+1-i} - 1}\prod\limits_{r=1}^{q_{n-i}} 2 |\sin \pi (M_{i,a_{i}}+r) \beta | $ in is not empty, and we\ need to show that $$\label{to_show_3} \left(\prod_{r=1}^{q_{n-i}} 2 |\sin \pi (M_{i,1}+r) \beta| \right) \left(\prod_{r=1}^{q_{n-i-1}} 2 |\sin \pi (M_{i+1,0}+r) \beta |\right) > 1.$$ We analyse the two products separately, and show that their product exceeds 1. It is crucial to combine these products, since the second product in alone does *not* necessarily exceed 1, and we need the first product for compensation. We have $$M_{i,1} = c_{n+1} q_n + c_n q_{n-1} + \dots + c_{n+2-i} q_{n-i+1} + q_{n-i}$$ and $$M_{i+1,0} = c_{n+1} q_n + c_n q_{n-1} + \dots + c_{n+2-i} q_{n-i+1} + 2 q_{n-i} .$$ The product equals $$\label{equals_what} P_{q_{n-i}} (\beta, {\varepsilon}_1) P_{q_{n-i-1}} (\beta, {\varepsilon}_2),$$ where $$\begin{aligned} \frac{{\varepsilon}_1}{q_{n-i} \beta^{n-i+1}} & = & c_{n+1} (-1)^{i} \beta^{i} + c_n (-1)^{i-1} \beta^{i-1} + \dots + c_{n+2-i} (-1)^1 \beta^{1} + \underbrace{1 (-1)^0 \beta^0}_{=1} \nonumber \end{aligned}$$ and $$\begin{aligned} \frac{{\varepsilon}_2}{q_{n-i-1} \beta^{n-i}} & = & c_{n+1} (-1)^{i+1} \beta^{i+1} + c_n (-1)^{i} \beta^{i} + \dots + c_{n+2-i} (-1)^2 \beta^{2} + 2 (-1)^1 \beta^1. \end{aligned}$$ Note that $c_{n+1-i}=2$ implies that $c_{n+2-i} \leq 4$. Thus we have $$\frac{{\varepsilon}_1}{q_{n-i} \beta^{n-i+1}} \leq 1 + 5 \sum_{j=1}^\infty \beta^{2j} = 1 + \beta,$$ and $$\frac{{\varepsilon}_1}{q_{n-i} \beta^{n-i+1}} \geq 1 - 4 \beta - 5 \sum_{j=1}^\infty \beta^{2j+1} \geq 1 - 4 \beta - \beta^2.$$ Again using that $q_{n-i-1} \beta^{n-i} \to 1/\sqrt{29}$, this implies that ${\varepsilon}_1 \in [0.03,0.23]$ for sufficiently large $i$. Similarly, we estimate the range for ${\varepsilon}_2$ and obtain $$\frac{{\varepsilon}_2}{q_{n-i-1} \beta^{n-i}} \leq -2 \beta + 5 \sum_{j=1}^\infty \beta^{2j} \leq - \beta,$$ as well as $$\frac{{\varepsilon}_2}{q_{n-i-1} \beta^{n-i}} \geq - 2 \beta - 5 \sum_{j=1}^\infty \beta^{2j+1} = -2 \beta - \beta^2.$$ Thus we have ${\varepsilon}_2 \in [-0.079,-0.036]$ for sufficiently large $i$. We can establish that $G_\beta({\varepsilon}) > 1.44$ throughout the possible range for ${\varepsilon}_1$, and that $G_\beta({\varepsilon}) > 0.72$ throughout the possible range for ${\varepsilon}_2$. Thus $P_{q_{n-i}} (\beta, {\varepsilon}_1) > 1.43$ and $P_{q_{n-i-1}} (\beta, {\varepsilon}_2) > 0.71$ whenever $i \geq i_0$, which implies that $P_{q_{n-i}} (\beta, {\varepsilon}_1) P_{q_{n-i-1}} (\beta, {\varepsilon}_2) > 1.43 \cdot 0.71 > 1.01$ whenever $i \geq i_0$. Thus the overall joint contribution of the Case 3 factors is bounded below by a positive constant (coming from the factors for which $i \geq i_0$ is not satisfied).\ $\bullet$ Case 4: $c_{n-i} \neq 0$ and $c_{n+1-i}=3$.\ In this case we wish to obtain a lower bound for the product $$\label{to_show_4} \left(\prod_{r=1}^{q_{n-i}} 2 |\sin \pi (M_{i,2}+r) \beta | \right)\!\left(\prod_{r=1}^{q_{n-i}} 2 |\sin\pi (M_{i,1}+r) \beta| \right)\!\left(\prod_{r=1}^{q_{n-i-1}}\!2|\sin\pi (M_{i+1,0}+r) \beta |\right)\!\!.$$ This product is equal to $$\label{this_prod} P_{q_{n-i}} (\beta, {\varepsilon}_1) P_{q_{n-i}} (\beta, {\varepsilon}_2) P_{q_{n-i-1}} (\beta, {\varepsilon}_3),$$ where for $j=1,2$ $$\begin{aligned} \frac{{\varepsilon}_j}{q_{n-i} \beta^{n-i+1}} &=& c_{n+1} (-1)^{i} \beta^{i} + c_n (-1)^{i-1} \beta^{i-1} + \dots + c_{n+2-i} (-1)^1 \beta^{1} + j\end{aligned}$$ and $$\begin{aligned} \frac{{\varepsilon}_3}{q_{n-i-1} \beta^{n-i}} &=& c_{n+1}(-1)^{i+1}\beta^{i+1} + c_n(-1)^{i} \beta^{i} + \dots + c_{n+2-i} (-1)^2 \beta^{2} + 3(-1)^1 \beta^1.\end{aligned}$$ Using a similar analysis as in Case 3, we obtain the restrictions ${\varepsilon}_1 \in [0.03,0.23],~{\varepsilon}_2 \in [0.21,0.42]$, and ${\varepsilon}_3 \in [-0.115,-0.07]$. In these respective ranges the function $G_\beta$ is uniformly bounded below by the values $1.44$, $2.34$ and $0.48$. Note that $1.44 \cdot 2.34 \cdot 0.48 \approx 1.62$. Thus we have $P_{q_{n-i}} (\beta, {\varepsilon}_1) P_{q_{n-i}} (\beta, {\varepsilon}_2) P_{q_{n-i-1}} (\beta, {\varepsilon}_3) > 1.61$ whenever $i \geq i_0$ for appropriate $i_0$. Consequently the joint overall contribution of the Case 4 factors is bounded below by an absolute constant.\ $\bullet$ Case 5: $c_{n-i} \neq 0$ and $c_{n+1-i}=4$.\ Now we need to control the product $$\label{this_prod_2} P_{q_{n-i}} (\beta, {\varepsilon}_1) P_{q_{n-i}} (\beta, {\varepsilon}_2) P_{q_{n-i}} (\beta, {\varepsilon}_3) P_{q_{n-i-1}} (\beta, {\varepsilon}_4),$$ where $$\begin{aligned} \frac{{\varepsilon}_j}{q_{n-i} \beta^{n-i+1}} & = & c_{n+1} (-1)^{i} \beta^{i} + c_n (-1)^{i-1} \beta^{i-1} + \dots + c_{n+3-i} (-1)^2 \beta^{2} + c_{n+2-i } (-1)^1 \beta^{1} + j\end{aligned}$$ for $j= 1,2,3$, and where $$\begin{aligned} \frac{{\varepsilon}_4}{q_{n-i-1} \beta^{n-i}} &= & c_{n+1} (-1)^{i+1} \beta^{i+1} + c_n (-1)^{i} \beta^{i} + \dots + c_{n+2-i} (-1)^2 \beta^{2} + 4 (-1)^1 \beta^1.\end{aligned}$$ This gives us the restrictions ${\varepsilon}_1 \in [0.03,0.23],~{\varepsilon}_2 \in [0.21,0.42], ~{\varepsilon}_3 \in [0.39,0.61]$ and ${\varepsilon}_4 \in [-0.151,-0.10]$. Throughout these ranges the function $G_\beta$ is uniformly bounded below by $1.44, ~2.34, ~2.18$ and $0.23$, respectively. We have $1.44 \cdot 2.34 \cdot 2.18 \cdot 0.23 \approx 1.69$. Thus the overall contribution of the Case 5 factors is bounded below by an absolute constant, since the finitely many factors corresponding to indices $n-i\leq i_0$ can be bounded from below arguing as in the proof of Theorem \[th3\].\ $\bullet$ Case 6: $c_{n-i} = 0$.\ Note that this case includes the case $c_{n+1-i}=5$, since necessarily we always have $c_{n-i} = 0$ when $c_{n+1-i}=5$. We distinguish five subcases:\ - Case 6a: $c_{n+1-i} \in \{0,1\}$. In this case all products in are empty, and the value of an empty product is 1.\ - Case 6b: $c_{n+1-i} = 2$. We can estimate similar to Case 3 above, but since $c_{n-i} = 0$ we only have $P_{q_{n-i}} (\beta, {\varepsilon}_1)$ instead of . As we showed in Case 3 above, we have $P_{q_{n-i}} (\beta, {\varepsilon}_1) > 1.43$, except for finitely many indices. Thus the joint overall contribution of Case 6b factors is bounded below by an absolute constant.\ - Case 6c: $c_{n+1-i} = 3$. In the same way that Case 6b was reduced to Case 3, we can reduce this case to Case 4 from above. We have $P_{q_{n-i}} (\beta, {\varepsilon}_1) P_{q_{n-i}} (\beta, {\varepsilon}_2)$ instead of , and by the Case 4 analysis this can be bounded below by a constant below the product $1.44 \cdot 2.34$, such as $3$. Thus the joint overall contribution of Case 6c factors is bounded below by an absolute constant, multiplied with $$3^{A^{(3)}}, \qquad \textrm{where $A^{(3)} = \# \{2 \leq i \leq n:~c_i = 3 ~ \text{and} ~ c_{i-1} = 0 \}$}.$$ - Case 6d: $c_{n+1-i} = 4$. This can be reduced to Case 5 from above. We have $P_{q_{n-i}} (\beta, {\varepsilon}_1) P_{q_{n-i}} (\beta, {\varepsilon}_2) P_{q_{n-i}} (\beta, {\varepsilon}_3)$ instead of , and according to the Case 5 analysis this is bounded below by any constant below $1.44 \cdot 2.34 \cdot 2.18$, such as $7$ (except for the contribution of finitely many indices). Thus the joint overall contribution of Case 6d factors is bounded below by an absolute constant, multiplied with $$7^{A^{(4)}}, \qquad \textrm{where $A^{(4)} = \# \{2 \leq i \leq n:~c_i = 4 ~ \text{and} ~ c_{i-1} = 0 \}$}.$$ - Case 6e: $c_{n+1-i} = 5$. In this case we have to work a bit again, since we encounter a large *positive* perturbation whose influence has to be controlled. In this case we have to control a product that can be written in the form $$\label{product_in} P_{q_{n-i}} (\beta, {\varepsilon}_1) P_{q_{n-i}} (\beta, {\varepsilon}_2) P_{q_{n-i}} (\beta, {\varepsilon}_3) P_{q_{n-i}} (\beta, {\varepsilon}_4),$$ where $$\begin{aligned} \frac{{\varepsilon}_j}{q_{n-i} \beta^{n-i+1}} & = & c_{n+1} (-1)^{i} \beta^{i} + \ldots + c_{n+3-i} (-1)^2 \beta^{2} + c_{n+2-i} (-1)^1 \beta^{1} + j,\end{aligned}$$ for $j=1,2,3,4$. This gives us the restrictions $${\varepsilon}_1 \in [0.03,0.23],~{\varepsilon}_2 \in [0.21,0.42], ~{\varepsilon}_3 \in [0.39,0.61],~{\varepsilon}_4 \in [0.57,0.80].$$ In these ranges $G_\beta$ is uniformly bounded below by the values $1.44,~2.34,~2.18$ and $1.12$, respectively. We have $1.44 \cdot 2.34 \cdot 2.18 \cdot 1.12 \approx 8.23$, so the product in is always bounded below by $8$ (provided that $n-i \geq i_0$). So the total contribution of the Case 6e factors is bounded below by an absolute constant (coming from the terms with $n-i <i_0$ ; see also the proof of Theorem \[th3\]), multiplied with $$8^{A^{(5)}}, \qquad \textrm{where $A^{(5)} = \# \{2 \leq i \leq n:~c_i = 5 ~ \text{and} ~ c_{i-1} = 0 \}$}.$$ Finally, we collect the contribution of all cases. The overall contribution of each of the Case 1, 2a, 3, 4, 5, 6a, 6b and 6e factors to the product in is bounded below by an absolute constant. The joint overall contribution of the Case 2b factors in bounded below by $$0.9^{A}, \qquad \textrm{where $A = \# \{2 \leq i \leq n:~c_i \geq 3 ~ \text{and} ~ c_{i-1} = 0 \}$},$$ while the joint overall contribution of the Case 6c, Case 6d and Case 6e factors was bounded below by $$3^{A^{(3)}}, \qquad \textrm{where $A^{(3)} = \# \{2 \leq i \leq n:~c_i = 3 ~ \text{and} ~ c_{i-1} = 0 \}$},$$ by $$7^{A^{(4)}}, \qquad \textrm{where $A^{(4)} = \# \{2 \leq i \leq n:~c_i = 4 ~ \text{and} ~ c_{i-1} = 0 \}$},$$ and by $$8^{A^{(5)}}, \qquad \textrm{where $A^{(5)} = \# \{2 \leq i \leq n:~c_i = 5 ~ \text{and} ~ c_{i-1} = 0 \}$},$$ respectively (up to multiplication with an absolute constant). Now note that $A = A^{(3)} + A^{(4)} + A^{(5)}$. Thus the joint overall contribution of the Case 2b and the Case 6c, 6d, 6e factors together is bounded below by an absolute constant multiplied by the product $0.9^A \cdot 3^{A^{(3)}} \cdot 7^{A^{(4)}} \cdot 8^{A^{(5)}} \geq 0.9^A \cdot 3^{A^{(3)}+A^{(4)}+A^{(5)}} = 0.9^A \cdot 3^A > 1$.\ Combining all these estimates proves that is bounded below by an absolute constant. Consequently $P_N(\beta)$ is bounded below by an absolute constant, as desired.\ We can establish that $\limsup_{N \to \infty} P_N(\beta)/N < \infty$ in a completely analogous way, using the reflection principle for the irrational $\beta$ as in the proof of Theorem \[th3\]. This “reflection” only generates an additional very small perturbation (coming from the reflection at the endpoint), but this does not actually change the range for the permissible perturbations ${\varepsilon}$ in all our estimates above, since this additional small perturbation only appears as one further term within a geometric progression, and we have always used the estimated coming from summation of the whole infinite geometric progression. So the proof of $\limsup_{N \to \infty} P_N(\beta)/N < \infty$ can be carried out in exactly the same way as the proof of $\liminf\limits_{N \to \infty} P_N(\beta)>0$ above. We conclude the manuscript with the corollary announced after Theorem \[lineargrowththeorem\]. Loosely speaking, its purpose is to record that the class of quadratic irrationals that we investigated is not overly exceptional. In this context, it would clearly be interesting to know the answer to the following problem: \[question1\] For which $\beta\in [0,1]$ does the associated Sudler product $P_N(\beta)$ grow at most linearly, i.e. is satisfied? A first step towards a solution of this question would be to completely settle the case of Sudler products of numbers whose continued fraction expansion is two-periodic: \[question2\] For which numbers $\beta$ of the form $\beta=[a,b,a,b,a,b,\ldots]$, with $a \neq b$, is satisfied? To formulate our corollary, which is a first partial result towards and answer of Question \[question1\], we let $H(\alpha)= \max_{0\leq i\leq d} \vert a_i\vert$ denote the (naive) height of an algebraic number $\alpha \in \mathbb{C}$ with minimal polynomial $$\mu_{\alpha}(X) =\sum_{0\leq i\leq d} a_i X^i.$$ \[height corr\] There are $\gg X$ many quadratic irrationals[^4] $\beta\in \mathbb{R}$ of height at most $X$ with $$\liminf \limits_{N \to \infty} P_N({\beta}) = 0, \quad \mathrm{and} \quad \limsup \limits_{N\to\infty}\dfrac{P_N({\beta})}{N} = \infty.$$ Let $\beta$ be as in where $b$ is an integer such that $b^2+4$ is not a perfect square. We note that, for any integer $c$, the number $\gamma(b,c) = \beta + c$ is a quadratic irrationality whose minimal polynomial is seen, by using , to be $$\mu(X)=(2(X-c)+b)^2-(b^2+4)= 4X^2 + 2(b-2c)X+4c^2-4bc-4.$$ Choosing $b,c \in (\sqrt{X}/100, \sqrt{X}/200)$ as above, the coefficients of the aforementioned polynomial are $\leq X$. A well-known theorem of Besicovitch [@besicovitch], about the $\mathbb{Q}$-linear independence of square-roots, implies that all these $\gg X$ many numbers $\gamma(b,c)$ are pairwise distinct. Acknowledgements {#acknowledgements .unnumbered} ---------------- We thank Martin Widmer for bringing Green’s comment [@green] to our attention, which was one of the starting points of the present manuscript.\ [8]{} C. Aistleitner, G. Larcher, F. Pillichshammer, S. Saad Eddin, R.F. Tichy, [*On Weyl products and uniform distribution modulo one*]{}. Monatsh. Math. 185 (2018), no. 3, 365–395. F.V. Atkinson, [*On a problem of Erdős and Szekeres.*]{} Canad. Math. Bull., 4 (1961), 7–12. A. Avila, S. Jitomirskaya, *The Ten Martini Problem*. Ann. of Math. (2) 170 (2009), no. 1, 303–342. A. Avila, S. Jitomirskaya, C.A. Marx, *Spectral theory of extended Harper’s model and a question by Erdős and Szekeres.* Invent. Math. 210 (2017), no. 1, 283–339. A. 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[^1]: CA is supported by the Austrian Science Fund (FWF), projects F-5512, I-3466 and Y-901.\ NT is supported by the European Research Council (ERC) under the European Union’s Horizon 2020\ research and innovation programme (Grant agreement No. 786758).\ AZ is supported by a postdoctoral fellowship funded by Grant 275113 of the Research Council of Norway [^2]: When the irrational $\beta$ is fixed, $q_n = q_n({\beta})$ always denotes the denominator of the $n$-th convergent of ${\beta}$. Furthermore, for convenience we write $P_{q_n}({\beta},{\varepsilon})$ instead of $P_{q_n({\beta})}({\beta},{\varepsilon})$, which would be more accurate. [^3]: The difference of the factors $(-1)^n$ and $(-1)^{n+1}$ in and corresponds to the fact that the $n$–th convergent of $\phi$ has denominator $F_{n+1}$. We have adopted this notation in our results relevant to the golden ratio in order to remain consistent with [@mv]. [^4]: It is well-known that there are $\asymp X^3$ many (real) quadratic irrationals of height at most $X$.
--- abstract: 'We show that the difference between the Ar and Si relative abundance ratio derived from [*FUSE*]{} absorption spectra and from the regions of I Zw 18 is a consequence of the microturbulent analysis applied to the absorption spectra. [*FUSE*]{} observations were performed with a large entrance aperture which fully covered the galaxy. This means that the observed profiles are averaged over the full body of I Zw 18, implying that large-scale velocity fields influence the absorption-line profiles. Taking this into account, we show that the absorption spectra are consistent with the same metal abundances as those derived from the regions. It follows that no significant ionization correction as suggested by Izotov and collaborators to describe metal contents in damped Ly$\alpha$ systems (DLA) is required to model abundances in the neutral gas of I Zw 18 (a local DLA system). Using a mesoturbulent approach and applying the generalized radiative transfer equation to the $\lambda1048$ and $\lambda1020$ lines observed by Vidal-Madjar et al., we found that the profiles may be reproduced with log (Ar/Si) $\simeq - 0.8$ and $N(\ion{Si}{ii}) \simeq 4\times10^{15}$ cm$^{-2}$.' author: - 'Sergei A. Levshakov' - 'Wilhelm H. Kegel' - 'Irina I. Agafonova' date: 'Received 00 December 2000 / Accepted 00 December 2000' title: ' Argon and Silicon abundances in the damped Ly$\alpha$ system I Zw 18 ' --- Introduction ============ The blue compact galaxy (BCG) I Zw 18 (Mrk 116) has been an intensively studied object for the last three decades since the first spectroscopic observations by Zwicky (1966). I Zw 18 shows an intense and recent burst of star formation which makes this galaxy an attractive target for studies of star formation history. Recent observations revealed, for example, two stellar populations in I Zw 18: 10–20 Myr red supergiants and 0.1–5 Gyr asymptotic giant branch stars (Östlin 2000). Amongst BCGs, I Zw 18 shows the lowest oxygen abundance : the northwest (NW) and the southeast (SE) bright regions yield log (O/H) $= -4.83 \pm 0.03$ and $-4.82 \pm 0.03$, respectively (Izotov et al. 1999). Relative to solar values, one finds \[O/H\][^1] $ = -1.70 \pm 0.08$, i.e. $Z/Z_\odot \simeq 1/50$. For argon abundances, Izotov et al. obtained log (Ar/H) $= -6.90 \pm 0.05$ (NW) and a slightly lower ratio for the SE region, log (Ar/H) = $-7.23 \pm 0.05$. Silicon abundances measured by Izotov & Thuan (1999) give log (Si/H) $= -6.29 \pm 0.22$ (NW) and log (Si/H) $= -6.30 \pm 0.22$ (SE). Thus, these values imply for the NW emission patch \[Ar/H\] = $-1.42 \pm 0.06$ and \[Si/H\] = $-1.85 \pm 0.22$. The neutral gas properties in I Zw 18 have been probed in both the radio (see van Zee et al. 1998 and references cited therein) and the UV range (Kunth et al. 1994; Pettini & Lipman 1995; Vidal-Madjar et al. 2000). High velocity and high spatial resolution radio observations have shown that the overall kinematics of the gas associated with I Zw 18 is very complex and the neutral gas velocity dispersion $\sigma$ equals $13 - 14$ km s$^{-1}$, or $b \equiv \sqrt{2}\sigma = 18 - 20$ km s$^{-1}$ (van Zee et al. 1998). The column density in front of I Zw 18 deduced from the Ly$\alpha$ absorption profile by Kunth et al., $N(\ion{H}{i}) = (3.5 \pm 0.5)\times10^{21}$ cm$^{-2}$, is comparable to the peak surface density found by van Zee et al., $N(\ion{H}{i}) \simeq 3.0\times10^{21}$ cm$^{-2}$. This means that the neutral gas in I Zw 18 can be considered as a local damped Ly$\alpha$ system (DLA) which is similar to high redshift DLAs observed in the light of background quasars. First measurement of the O/H abundance in the region have indicated a possible discrepancy between the metal content in the neutral gas and in the regions (Kunth et al. 1994). This, however, was not confirmed in later studies by Pettini & Lipman (1995) and by van Zee et al. (1998) who have shown that both the neutral and ionized gas in I Zw 18 may have the same oxygen abundance. New observations carried out with the [*Far Ultraviolet Spectroscopic Explorer*]{} ([*FUSE*]{}) by Vidal-Madjar et al. (2000) produce a similar puzzle : the column density ratio deduced from the $\lambda1048$ and $\lambda1020$ lines in the neutral gas, log (Ar/Si) $= -1.32$, differs significantly from that observed in the regions, log (Ar/Si)$_{\rm NW} = -0.61 \pm 0.22$ and log (Ar/Si)$_{\rm SE} = -0.74 \pm 0.22$ (Izotov et al. 2000). Izotov et al. suggested that ionization in DLAs may affect abundance ratios. To interpret the observations, they suggested a model consisting of two regions with [*substantially different*]{} metal contents, i.e. implicitly recalling an idea of the existence of two media in BCGs (a pristine gas, unprocessed since the big bang and a gas polluted by nucleosynthetic products – Kunth & Sargent 1986). In general, the discovery of such a primordial gas would be of great importance for cosmology, as noted by Kunth et al. (1994). In this Letter, we report on the study of the $\lambda1048$ and $\lambda1020$ profiles of I Zw 18 published by Vidal-Madjar et al. (2000). We show that these lines can be modeled under the assumption that the metal content in the neutral gas is the same as in the regions without refering to an ionization correction. Data analysis and results ========================= Spectroscopic observations of I Zw 18 with [*FUSE*]{} in the range $\sim 980 - 1187$ Å are described in detail by Vidal-Madjar et al. (2000). The spectrum was obtained with a resolution of about $\lambda/\Delta\lambda \sim 10,000$ and a signal-to-noise ratio of S/N $\sim 10$ per resolution element. The large entrance aperture ($30'' \times 30''$) fully covers the galactic surface ($\sim 10'' \times 4''$). As noted by Vidal-Madjar et al., [*FUSE*]{} produces the average absorption over the full body of the galaxy. In this case the analysis of saturated absorption lines is not an easy and unambiguous task. The main difficulty is connected with the line broadening by large-scale irregular (stochastic) velocity fields. The influence of the finite correlation length on the line profile depends on the details of the observation. If one considers the line formation process in the light of a point source, then the observed spectrum reflects only one realization of the velocity field and, hence, large deviations from the expectation value of the intensity $\langle I_\lambda \rangle$ may occur if the correlation length of the velocity field $\ell$ is not very small compared to the size of the absorbing region (Levshakov & Kegel 1997). If, however, the spectrograph aperture covers an essential part of the galactic surface, then $\langle I_\lambda \rangle$ should reasonably well correspond to the observations. But also in this case the standard Voigt-fitting analysis (based on the assumption of microturbulence) may yield misleading results (Levshakov & Kegel 1994). Below we demonstrate this effect using the published [*FUSE*]{} data. A microturbulent approach ------------------------- We begin with the standard Voigt-fitting analysis which assumes a completely uncorrelated velocity field, i.e. $\ell = 0$ ([*microturbulence*]{}). In our example we use the published data from Fig. 1 of Vidal-Madjar et al. to illustrate how the column density ratio deduced from two lines, and , is affected by the underlying assumptions. From these data given in arbitrary units we have calculated normalized intensities using the plotted synthetic profiles, and then added to the normalized data $1\sigma$ error bars corresponding to S/N = 10 (dots and error bars in Figs. 1[**a**]{}, 1[**b**]{}). Both profiles were centered at $v = 0$ km s$^{-1}$ according to the synthetic spectra of Vidal-Madjar et al., and the internal uncertainty of our velocity scale calibration was estimated to be about $\pm 5$ km s$^{-1}$. The oscillator strengths of the $\lambda1048.2119$ Å and $\lambda1020.6989$ Å lines, $f_{\ion{Ar}{i}} = 0.257$ and $f_{\ion{Si}{ii}} = 0.01391$, were taken from Federman et al. (1992) and from Charro & Martín (2000), respectively. The absorption lines of and are partly blended with the Galactic H$_2$ L(4-0)P(1) and L(7-0)P(4) lines, respectively. To estimate the and column densities and the $b$-parameter, we firstly fitted two component Voigt profiles to the observed intensities via $\chi^2$ minimization. In this procedure the theoretical profiles were convolved with the instrumental point-spread function which is supposed to be a Gaussian with the width of 26 km s$^{-1}$. The best fit with $\chi^2_{\rm min} = 34.25$ ($M = 35$ data points, and $\nu = 28$ degrees of freedom) is shown in Fig. 1[**a**]{} by grey solid curves which simultaneously mark the data points involved in the optimization procedure. Dotted curves in Fig. 1[**a**]{} show the corresponding unconvolved profiles. As seen from this figure, the H$_2$ blends do not affect the cores of the and lines much. This is why in the further analysis we used only the blue wings and the central parts of these lines. Our main results are shown in Figs. 1[**b**]{} and 1[**c**]{}. Fig. 1[**b**]{} presents the best fit with $\chi^2_{\rm min} = 18.36$ ($M = 19$, $\nu = 16$), $N_{\ion{Ar}{i}} = 6.83\times10^{13}$ cm$^{-2}$, $N_{\ion{Si}{ii}} = 1.11\times10^{15}$ cm$^{-2}$, and $b = 17.4$ km s$^{-1}$. Fig. 1[**c**]{} shows the calculated $\chi^2_{\rm min}$ values as a function of $b$ with the $1\sigma$ and $2\sigma$ confidence levels. Here, for each point we fixed the value of $b$ and minimize $\chi^2$ by varying all other parameters. The estimated ratio of log (Ar/Si) $= -1.21$ is consistent within the uncertainty interval of 0.11 dex with the value $-1.32$ cited by Izotov et al. (2000). The most likely value for $b$, $17.4 \pm 3$ km s$^{-1}$, is in an excellent agreement with the radio observations of the gas in I Zw 18 carried out by van Zee et al. (1998). The fact that $b_{\ion{H}{i}} \simeq b_{\ion{Ar}{i}}$ implies that the kinetic temperature of the neutral gas in I Zw 18 is less than 600 K and that the line broadening is caused mainly by turbulent motions. To summarize this section we note that the microturbulent results lead to the puzzle mentioned above which encouraged us to try a more general model. [lccccc]{} $L/\ell$ & $N_{\ion{Si}{ii}}$, cm$^{-2}$ & Ar/Si$^\ddagger$ & $\frac{1}{\nu}\chi^2_{\rm min}$ & \[Ar/H\]$^\dagger$ & \[Si/H\]$^\dagger$\ 0 & 1.0(19) & $-1.93$ & 1.33 & $\;\;\;1.03$&$\;\;\;1.91$\ 0.50& 4.4(18) & $-1.82$ & 1.13 & $\;\;\;0.77$&$\;\;\;1.11$\ 0.75& 1.6(18) & $-1.57$ &1.08 & $\;\;\;0.58$&$\;\;\;1.11$\ 0.90& 4.1(17) & $-1.12$ &1.06 & $\;\;\;0.43$&$\;\;\;0.51$\ 1.00& 4.9(16) & $-0.32$ &1.06 & $\;\;\;0.31$&$-0.41$\ 1.10& 1.3(16) & $\;\;\;0.07$ & 1.07 & $\;\;\;0.14$ &$-0.98$\ 1.15& 9.5(15) & $\;\;\;0.10$ & 1.07 & $\;\;\;0.02$ &$-1.12$\ 1.20& 7.5(15) & $\;\;\;0.02$ & 1.07 & $-0.17$ &$-1.22$\ 1.25& 6.3(15) & $-0.05$& 1.08 & $-0.32$ &$-1.30$\ 1.30& 5.4(15) & $-0.34$& 1.08 & $-0.66$ &$-1.36$\ 1.35& 5.0(15) & $-0.71$& 1.08 & $-1.08$&$-1.40$\ 1.40& 4.4(15) & $-0.75$& 1.08 & $-1.17$&$-1.46$\ 1.45& 4.0(15) & $-0.84$& 1.08 & $-1.30$&$-1.50$\ 1.50& 3.7(15) & $-0.90$& 1.08 & $-1.39$&$-1.53$\ 2.00& 2.5(15) & $-1.09$ & 1.08 & $-1.76$&$-1.71$\ 5.00& 1.5(15) & $-1.19$ & 1.08 & $-2.09$&$-1.94$\ 10.0 & 1.3(15) & $-1.20$ & 1.08 & $-2.17$&$-2.00$\ $\infty$ &1.1(15)&$-1.21$& 1.09 & $-2.23$&$-2.06$\ \ A mesoturbulent approach ------------------------ 21 cm observations with high spatial resolution show that the line profile varies with position (van Zee et al. 1998). This is a clear indication that large-scale motions determine the line profiles and that the microturbulent approach is not well founded. We therefore generalize our analysis to include the effects of a finite correlation length, i.e. $\ell \neq 0$ ([*mesoturbulence*]{}) and to study the limiting case $L/\ell \rightarrow 0$ ([*macroturbulence*]{}). The simulation of the mesoturbulent and profiles has been carried out using the simplified model of a plane-parallel slab of geometrical size $L$ with homogeneous turbulence and uniform kinetic temperature, $T_{\rm kin}$. Specifying the and column densities, the $L/\ell$ ratio, the velocity dispersion $\sigma$, and the thermal widths for each line, we can calculated their average absorption-line profiles employing the generalized radiative transfer equation (Levshakov & Kegel 1994, 1997). The same $\chi^2$-minimization procedure as before was applied to the same data set, but now with two fitting parameters $\{ N_{\ion{Si}{ii}}$, $\log ({\rm Ar}/{\rm Si})\}$, i.e. $\nu = 17$. We fixed $\sigma = 13$ km s$^{-1}$ and $T_{\rm kin} = 200$ K (i.e. $b \simeq 18$ km s$^{-1}$) and calculated $\chi^2_{\rm min}$ for a given $L/\ell$. The results are presented in Table 1. It should be emphasized that all and profiles corresponding to the listed – very different – solutions are [*identical*]{} to those shown in Fig. 1[**b**]{}. Our numerical results show that accounting for correlation effects strongly affects the derived column densities as well as the abundance ratio Ar/Si. These effects are closely related to the changes in the curves of growths caused by the finite $L/\ell$ value. Fig. 2 shows three curves of growth for each of the two lines. They correspond to the two limiting cases of micro- and macroturbulence (Gail et al. 1974), as well as to an intermediate – mesoturbulent – case ($L/\ell = 1.45$). In general, one may say that the column density derived from a measured equivalent width increases with increasing correlation length $\ell$. Fig. 2 also allows a qualitative interpretation of the Ar/Si ratio in dependence of $L/\ell$. In the microturbulent limit both lines are still close to the linear part of the curve of growth, while in the mesoturbulent regime they lie on the flat part. Since the slope here is smaller than on the linear part, the Ar/Si ratio is larger. In the macroturbulent limit, both lines lie on the square root part. In this case the slope is again steeper and, more importantly, the two curves of growth are well separated from each other (due to the different damping constants), implying a lower Ar/Si ratio than in the mesoturbulent case. Thus, going from micro- to macroturbulence we expect the Ar/Si ratio at first to rise and, after going through a maximum, to decline again. From Fig. 2 we also see that in the macroturbulent limit the Ar/Si ratio is lower than in the microturbulent one. Conclusions =========== We have shown that metal abundances derived from the absorption profiles of $\lambda1048$ and $\lambda1020$, observed in the spectrum of the damped Ly$\alpha$ system I Zw 18 by Vidal-Madjar et al. (2000), become fully consistent with those derived from emission-line spectra of the regions by Izotov & Thuan (1999) and by Izotov et al. (1999), if correlations in the large-scale velocity field are accounted for. The adequacy of the obtained mesoturbulent solutions is supported by the fact that the derived velocity dispersion of the neutral gas, $\sigma \simeq 13$ km s$^{-1}$, is in an excellent agreement with the 21 cm observations, $\sigma = 12 - 14$ km s$^{-1}$ (van Zee et al. 1998). This implies that the gas in I Zw 18 is efficiently mixed. To estimate more accurately metal abundances in the gas within the framework of our model, one needs to know the $L/\ell$ ratio. This parameter can be found from the analysis of saturated and optically thin lines of the same ion, since the latter are less affected by the correlation effects. For our case, observations of the $\lambda1066$ line would be of particular interest since its oscillator strength $f_{1066} = 1/4\,f_{1048}$. The $\lambda1066$ line was observed in the $z = 3.4$ DLA system toward the quasar Q0000-2620 and its analysis showed a remarkably similar abundance to the other $\alpha$-chain elements O, S, and Si, $Z/Z_\odot \simeq 1/80$ (Molaro et al. 2000). The similarity of both DLA systems is also supported by the very low molecular hydrogen contents found in I Zw 18, $f({\rm H}_2) \equiv 2N_{{\rm H}_2}/N_{\ion{H}{i}} \ll 10^{-6}$ (Vidal-Madjar et al. 2000), and at $z=3.4$, $f({\rm H}_2) \simeq 4\times10^{-8}$ (Levshakov et al. 2000). Since Ar can hardly be depleted onto dust grains, but can be partially ionized by nearby UV stellar radiation with energies $h\nu > 15.76$ eV, the relative abundance of is a good indicator of the intensity of the local photoionizing flux (Sofia & Jenkins 1998). Our measurements rule out the presence of a significant amount of partially ionized gas in the damped L$\alpha$ system I Zw 18 and, hence, metal abundances in the neutral gas do not require ionization corrections as suggested by Izotov et al. (2000). Thus, it is extremely important in future observations to investigate low-ion lines to undestand better the kinematic characteristics of the neutral gas bulk motion which will enable us to obtain more reliable estimations of metallicities. The work of S.A.L. and I.I.A. is supported in part by the Deutsche Forschungsgemeinschaft and by the RFBR grant No. 00-02-16007. Charro E., Martín I. 2000, ApJS 126, 551 Federman S. R., Beideck D. J. et al. 1992, ApJ 401, 367 Gail H. P., Hundt E., Kegel W. H., Schmid-Burgk J., Traving G. 1974, A&A 32, 65 Grevesse N., Noels A., Sauval A. J. 1996, in Cosmic Abundances ASP Conf. Series, Vol. 99, p. 117 Izotov Yu. I., Thuan T. X. 1999, ApJ 511, 639 Izotov Yu. I., Chaffee F. H. et al. 1999, ApJ 527, 757 Izotov Yu. I., Schaerer D., Charbonnel C. 2000, ApJ, in press (astro-ph/0010643) Kunth D., Lequeux J., Sargent W. L. W., Viallefond F. 1994, A&A 282, 709 Kunth D., Sargent W. L. W. 1986, ApJ 300, 496 Levshakov S. A., Kegel W. H. 1994, MNRAS 271, 161 Levshakov S. A., Kegel W. H. 1997, MNRAS 288, 787 Levshakov S. A., Molaro P. et al. 2000, A&A 361, 803 Molaro P., Levshakov S. A. et al. 2000, ApJ, in press (astro-ph/0010434) Östlin, G. 2000, ApJ 535, L99 Pettini M, Lipman K. 1995, A&A 297, L63 Sofia U. J., Jenkins E. B. 1998, ApJ 499, 951 van Zee L., Westpfahl D. et al. 1998, AJ 115, 1000 Vidal-Madjar A., Kunth D. et al. 2000, ApJ 538, L77 Zwicky F. 1966, ApJ 143, 192 [^1]: Using the customary definition \[X/H\] = $\log [N({\rm X})/N({\rm H})] - \log [N({\rm X})/N({\rm H})]_\odot$, and solar abundances from Grevesse et al. (1996) except for Ar for which the weighted average value $-5.48 \pm 0.04$ from Sofia & Jenkins (1998) is adopted.
--- abstract: 'We completely classify all quotient bundles of a given vector bundle on the Fargues-Fontaine curve. As consequences, we have two additional classification results: a complete classification of all vector bundles that are generated by a fixed number of global sections and a nearly complete classification of subbundles of a given vector bundle. For the proof, we combine the dimension counting argument for moduli of bundle maps developed in [@Arizonaext] with a series of reduction arguments based on some reinterpretation of the classifying conditions.' address: 'Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor MI 48109' author: - Serin Hong bibliography: - 'Bibliography.bib' title: 'Classification of quotient bundles on the Fargues-Fontaine curve' --- Introduction ============ In [@FF08], Fargues and Fontaine constructed a remarkable scheme, now commonly referred to as the *Fargues-Fontaine curve*, which serves as the “fundamental curve of $p$-adic Hodge theory". In fact, many constructions in $p$-adic Hodge theory and related fields have geometric interpretations in terms of vector bundles on the Fargues-Fontaine curve. As an example, Fargues in [@Far16] formulates the conjectural geometrization of the local Langlands correspondence in terms of certain sheaves on the stack of vector bundles on the Fargues-Fontaine curve. In this paper we obtain several classification results regarding vector bundles on the Fargues-Fontaine curve. Our main result is a complete classification of all quotient bundles of a given vector bundle. As a special case, we obtain a complete classification of all vector bundles that are generated by a fixed number of global sections. In addition, a dual statement of our main result gives a nearly complete classification of subbundles of a given vector bundle. Statement of results -------------------- $ $ For a precise statement of our results, we briefly recall the classification of vector bundles on the Fargues-Fontaine curve. \[classification of vector bundles on FF curve, intro\] Fix a prime number $p$. Let ${E}$ be a finite extension of ${\mathbb{Q}}_p$, and let ${F}$ be an algebraically closed perfectoid field of characteristic $p$. Denote by ${X}= {X}_{{E}, {F}}$ the Fargues-Fontaine curve associated to the pair $({E}, {F})$. 1. The scheme ${X}$ is complete in the sense that the divisor of an arbitrary nonzero rational function on ${X}$ has degree zero. As a consequence, there is a well-defined notion of the slope of a vector bundle on ${X}$. 2. For every rational number $\lambda$, there is a unique stable bundle of slope $\lambda$ on ${X}$, denoted by ${{\mathcal{O}}}(\lambda)$. 3. Every semistable bundle of slope $\lambda$ is of the form ${{\mathcal{O}}}(\lambda)^{\oplus m}$. 4. \[HN decomp of vector bundles, intro\] Every vector bundle ${\mathcal{V}}$ on ${X}$ admits a splitting and canonical Harder-Narasimhan filtration. As a result, it admits a direct sum decomposition $${\mathcal{V}}\simeq \bigoplus_i {{\mathcal{O}}}(\lambda_i)^{\oplus m_i}$$ where $\lambda_i$’s run over the Harder-Narasimhan slopes of ${\mathcal{V}}$; in other words, the isomorphism class of ${\mathcal{V}}$ is determined by the Harder-Narasimhan polygon ${\mathrm{HN}}({\mathcal{V}})$ of ${\mathcal{V}}$. We retain the notation from Theorem \[classification of vector bundles on FF curve, intro\]. In addition, for a vector bundle ${\mathcal{V}}$ with a direct sum decomposition as in \[HN decomp of vector bundles, intro\] of Theorem \[classification of vector bundles on FF curve, intro\], we define $${\mathcal{V}}^{\leq \mu}:= \bigoplus_{\lambda_i \leq \mu} {{\mathcal{O}}}(\lambda_i)^{\oplus m_i}\quad\quad \text{ and } \quad\quad {\mathcal{V}}^{\geq \mu}:= \bigoplus_{\lambda_i \geq \mu} {{\mathcal{O}}}(\lambda_i)^{\oplus m_i} \quad\quad\text{ for every } \mu \in {\mathbb{Q}}.$$ Now we can state our main result as follows: \[classification of quotient bundles, intro\] Let ${\mathcal{E}}$ be a vector bundle on ${X}$. Then a vector bundle ${\mathcal{F}}$ on ${X}$ is a quotient bundle of ${\mathcal{E}}$ if and only if the following conditions are satisfied: 1. \[rank inequalities for quotients, intro\] $\operatorname{rank}({\mathcal{E}}^{\leq \mu}) \geq \operatorname{rank}({\mathcal{F}}^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$. 2. \[equal rank condition for quotient bundles, intro\] ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{F}}^{\leq \mu}$ whenever equality holds in \[rank inequalities for quotients\]. Moreover, if we align the Harder-Narasimhan polygons ${\mathrm{HN}}({\mathcal{E}})$ and ${\mathrm{HN}}({\mathcal{F}})$ so that their right endpoints lie at the origin, the conditions \[rank inequalities for quotients, intro\] and \[equal rank condition for quotient bundles, intro\] are equivalent to the following conditions: 1. \[dual slopewise dominance for quotients, intro\] For each $i = 1, \cdots, \operatorname{rank}({\mathcal{F}})$, the slope of ${\mathrm{HN}}({\mathcal{F}})$ on the interval $[-i, -i+1]$ is greater than or equal to the slope of ${\mathrm{HN}}({\mathcal{E}})$ on this interval. 2. \[dual slopewise dominance equality condition for quotient bundles, intro\] If both ${\mathrm{HN}}({\mathcal{E}})$ and ${\mathrm{HN}}({\mathcal{F}})$ have vertices at some integer $-j$, then the slope of ${\mathrm{HN}}({\mathcal{F}})$ on $[-j, -j+1]$ is greater than or equal to the slope of ${\mathrm{HN}}({\mathcal{E}})$ on $[-j-1, j]$ unless ${\mathrm{HN}}({\mathcal{E}})$ and ${\mathrm{HN}}({\mathcal{F}})$ agree on $[-j, 0]$. (right) at (0, 0); (q0) at (-1,2); (q1) at (-2.5, 3.4); (q2) at (-6, 4.8); (q3) at (-9, 4); (p0) at (-2, 1.5); (p1) at (-4.5, 2); (p2) at (-6, 1.3); (p3) at (-7, 0.1); (right) – (q0) – (q1) – (q2) – (q3); (right) – (p0) – (p1) – (p2) – (p3); (q0) circle \[radius=0.05\]; (q1) circle \[radius=0.05\]; (q2) circle \[radius=0.05\]; (q3) circle \[radius=0.05\]; (right) circle \[radius=0.05\]; (p0) circle \[radius=0.05\]; (p1) circle \[radius=0.05\]; (p2) circle \[radius=0.05\]; (p3) circle \[radius=0.05\]; (-3, -0.4) – (-3, 5); (-3.5, -0.4) – (-3.5, 5); (-6, -0.4) – (-6, 5); at (-2.8,-0.8) [$-i+1$]{}; at (-3.7,-0.8) [$-i$]{}; at (-6,-0.8) [$-j$]{}; (q3) ++(-0.8, 0.05) node [${\mathrm{HN}}({\mathcal{E}})$]{}; (p3) ++(-0.8, 0.05) node [${\mathrm{HN}}({\mathcal{F}})$]{}; (right) ++(0.3, -0.05) node [$O$]{}; The two characterizations of quotient bundles given in Theorem \[classification of quotient bundles, intro\] have their own pros and cons. In practice, the characterization by the conditions \[dual slopewise dominance for quotients, intro\] and \[dual slopewise dominance equality condition for quotient bundles, intro\] is preferred as a classification criterion since it is easy to check for any given bundles ${\mathcal{E}}$ and ${\mathcal{F}}$. On the other hand, the characterization by the conditions \[rank inequalities for quotients, intro\] and \[equal rank condition for quotient bundles, intro\] is simple to state and preferable for studying consequences of Theorem \[classification of quotient bundles, intro\]. If we take ${\mathcal{E}}= {{\mathcal{O}}}_{X}^{\oplus n}$ for some positive integer $n$ in Theorem \[classification of quotient bundles, intro\], we obtain the following classification of finitely globally generated vector bundles on ${X}$. \[classification of globally generated bundles, intro\] A vector bundle ${\mathcal{F}}$ on ${X}$ is generated by $n$ global sections if and only if the following conditions are satisfied: 1. \[nonpositivity of slopes for globally generated bundles, intro\] All Harder-Narasimhan slopes of ${\mathcal{F}}$ are nonnegative. 2. \[rank bound for globally generated bundles, intro\] $\operatorname{rank}({\mathcal{F}}) \leq n$ with equality if and only if ${\mathcal{F}}\simeq {{\mathcal{O}}}_{X}^{\oplus n}$. In addition, dualizing the statement of Theorem \[classification of quotient bundles, intro\] yields a classification of a majority of subbundles of a given vector bundle on ${X}$. \[almost classification of subbundles, intro\] Let ${\mathcal{E}}$ be a vector bundle on ${X}$. Then a vector bundle ${\mathcal{D}}$ on ${X}$ is (isomorphic to) a subbundle of ${\mathcal{E}}$ if the following conditions are satisfied: 1. \[rank inequalities for subbundles, intro\] $\operatorname{rank}({\mathcal{E}}^{\geq \mu}) \geq \operatorname{rank}({\mathcal{D}}^{\geq \mu})$ for every $\mu \in {\mathbb{Q}}$. 2. \[equal rank condition for subbundles, intro\] ${\mathcal{E}}^{\geq \mu} \simeq {\mathcal{D}}^{\geq \mu}$ whenever equality holds in \[rank inequalities for subbundles, intro\]. Moreover, if we align the Harder-Narasimhan polygons ${\mathrm{HN}}({\mathcal{D}})$ and ${\mathrm{HN}}({\mathcal{E}})$ so that their left endpoints lie at the origin, the conditions \[rank inequalities for subbundles, intro\] and \[equal rank condition for subbundles, intro\] are equivalent to the following conditions: 1. \[slopewise dominance for subbundles, intro\] For each $i = 1, \cdots, \operatorname{rank}({\mathcal{D}})$, the slope of ${\mathrm{HN}}({\mathcal{D}})$ on the interval $[i, i+1]$ is less than or equal to the slope of ${\mathrm{HN}}({\mathcal{E}})$ on this interval. 2. \[slopewise dominance equality condition for subbundles, intro\] If both ${\mathrm{HN}}({\mathcal{D}})$ and ${\mathrm{HN}}({\mathcal{E}})$ have vertices at some integer $j$, then the slope of ${\mathrm{HN}}({\mathcal{D}})$ on $[j-1, j]$ is less than or equal to the slope of ${\mathrm{HN}}({\mathcal{E}})$ on $[j, j+1]$ unless ${\mathrm{HN}}({\mathcal{D}})$ and ${\mathrm{HN}}({\mathcal{E}})$ agree on $[0, j]$. (left) at (0, 0); (q0) at (1,2); (q1) at (2.5, 3.4); (q2) at (6, 4.8); (q3) at (9, 4); (p0) at (2, 1.5); (p1) at (4.5, 2); (p2) at (6, 1.3); (p3) at (7, 0.1); (left) – (q0) – (q1) – (q2) – (q3); (left) – (p0) – (p1) – (p2) – (p3); (q0) circle \[radius=0.05\]; (q1) circle \[radius=0.05\]; (q2) circle \[radius=0.05\]; (q3) circle \[radius=0.05\]; (left) circle \[radius=0.05\]; (p0) circle \[radius=0.05\]; (p1) circle \[radius=0.05\]; (p2) circle \[radius=0.05\]; (p3) circle \[radius=0.05\]; (3, -0.4) – (3, 5); (3.5, -0.4) – (3.5, 5); (6, -0.4) – (6, 5); at (2.8,-0.8) [$i-1$]{}; at (3.5,-0.8) [$i$]{}; at (6,-0.8) [$j$]{}; (q3) ++(0.8, 0.05) node [${\mathrm{HN}}({\mathcal{E}})$]{}; (p3) ++(0.8, 0.05) node [${\mathrm{HN}}({\mathcal{D}})$]{}; (right) ++(-0.3, -0.05) node [$O$]{}; We remark that Corollary \[almost classification of subbundles, intro\] does not give a complete classification of all subbundles since the condition \[equal rank condition for subbundles, intro\] is not necessary. In fact, we conjecture that the condition \[rank inequalities for subbundles, intro\] alone should give a complete classification of all subbundles. \[conjecture classification of subbundles, intro\] Let ${\mathcal{E}}$ be a vector bundle on ${X}$. Then a vector bundle ${\mathcal{D}}$ is (isomorphic to) a subbundle of ${\mathcal{E}}$ if and only if it satisfies the condition \[rank inequalities for subbundles, intro\] (or its equivalent condition \[slopewise dominance for subbundles, intro\]) in Corollary \[almost classification of subbundles, intro\]. Outline of the strategy {#introstrategy} ----------------------- $ $ It is relatively easy to see that the conditions \[rank inequalities for quotients, intro\] and \[equal rank condition for quotient bundles, intro\] in Theorem \[classification of quotient bundles, intro\] are indeed necessary and that they are equivalent to the conditions \[dual slopewise dominance for quotients, intro\] and \[dual slopewise dominance equality condition for quotient bundles, intro\]. Therefore the main part of our proof will concern sufficiency of the conditions \[rank inequalities for quotients, intro\] and \[equal rank condition for quotient bundles, intro\] in Theorem \[classification of quotient bundles, intro\]. Our argument will be based on the dimension counting method for certain moduli spaces of bundle maps developed in [@Arizonaext]. Our goal is to show that the moduli space ${\mathcal{S}\mathrm{urj}}({\mathcal{E}}, {\mathcal{F}})$ parametrizing surjective bundle maps ${\mathcal{E}}{\twoheadrightarrow}{\mathcal{F}}$ is not empty if the conditions \[rank inequalities for quotients, intro\] and \[equal rank condition for quotient bundles, intro\] in Theorem \[classification of quotient bundles, intro\] are satisfied. For this, we consider auxiliary spaces ${\mathcal{H}\mathrm{om}}({\mathcal{E}}, {\mathcal{F}})_{\mathcal{Q}}$ which parametrizes bundle maps ${\mathcal{E}}\to {\mathcal{F}}$ with image isomorphic to a specified subbundle ${\mathcal{Q}}$ of ${\mathcal{F}}$. Then showing nonemptiness of ${\mathcal{S}\mathrm{urj}}({\mathcal{E}}, {\mathcal{F}})$ boils down to establishing the following inequality on dimensions of the topological spaces: $$\label{key inequality intro} \dim |{\mathcal{H}\mathrm{om}}({\mathcal{E}}, {\mathcal{F}})_{\mathcal{Q}}| < \dim |{\mathcal{S}\mathrm{urj}}({\mathcal{E}}, {\mathcal{F}})| \quad\quad \text{ if } {\mathcal{Q}}\neq {\mathcal{F}}.$$ The dimension counting method developed in [@Arizonaext] allows us to rewrite this inequality in terms of degrees of certain vector bundles related to ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$. However, the details of our arguments are completely different from those in [@Arizonaext]. The main reason is that, unlike the quantities considered in [@Arizonaext], the quantities we need to study in this paper do not generally have good interpretations in terms of areas of polygons related to the Harder-Narasimhan slopes. In fact, our proof of the inequality will consist of a series of reduction steps as follows: 1. \[reduction to integer slopes, intro\] We reduce the proof of to the case when all slopes of ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ are integers. 2. \[reduction to equal ranks, intro\] We further reduce the proof of to the case $\operatorname{rank}({\mathcal{Q}}) = \operatorname{rank}({\mathcal{F}})$. 3. \[reduction to equal slopes, intro\] When $\operatorname{rank}({\mathcal{Q}}) = \operatorname{rank}({\mathcal{F}})$, we complete the proof of by gradually “reducing" the slopes of ${\mathcal{F}}$ to the slopes of ${\mathcal{Q}}$. As a key ingredient of our reduction argument, we introduce and study the notion of *slopewise dominance* for vector bundles on the Fargues-Fontaine curve. This notion is motivated by the condition \[dual slopewise dominance for quotients, intro\] in Theorem \[classification of quotient bundles, intro\] (and the condition \[slopewise dominance for subbundles, intro\] in Corollary \[almost classification of subbundles, intro\]); indeed, we can state the condition \[dual slopewise dominance for quotients, intro\] in Theorem \[classification of quotient bundles, intro\] as slopewise dominance between the dual bundles ${\mathcal{E}}^\vee$ and ${\mathcal{F}}^\vee$ (and the condition \[slopewise dominance for subbundles, intro\] in Corollary \[almost classification of subbundles, intro\] as slopewise dominance between ${\mathcal{D}}$ and ${\mathcal{E}}$). The notion of slopewise dominance allows us to use the equivalence of the conditions \[rank inequalities for quotients, intro\] and \[dual slopewise dominance for quotients, intro\] in Theorem \[classification of quotient bundles, intro\] to its full capacity. In particular, we will use this notion to obtain a number of implications of the condition \[dual slopewise dominance for quotients, intro\] which are difficult to directly deduce from the condition \[rank inequalities for quotients, intro\]. Slopewise dominance is also crucial for formulating our process of “reducing" the slopes of ${\mathcal{F}}$ to the slopes of ${\mathcal{Q}}$ in \[reduction to equal slopes, intro\] Acknowledgments {#acknowledgments .unnumbered} --------------- The major part of this study was done at the Oberwolfach Workshop on Arithmetic of Shimura varieties. The author would like to thank the organizers of the workshop for creating such a wonderful academic environment. Preliminaries on the Fargues-Fontaine curve {#background} =========================================== The construction ---------------- $ $ Throughout this paper, we fix the following data: - $p$ is a prime number; - ${E}$ is a finite extension of ${\mathbb{Q}}_p$ with residue field ${\mathbb{F}}_q$; - ${F}$ is an algebraically closed perfectoid field of characteristic $p$. The Fargues-Fontaine curve can be constructed in two different flavors, namely as a scheme and as an adic space. We first present the construction as an adic space since it is simpler to describe than the construction as a scheme is. \[adicFFC\] Denote by ${{{E}}^\circ}$ and ${{{F}}^\circ}$ the rings of integers of ${E}$ and ${F}$, respectively. Let ${\pi}$ be a uniformizer of ${E}$, and let ${\varpi}$ be a pseudouniformizer of ${F}$. We write ${W}_{{E}^\circ}({{{F}}^\circ}):={W}({{{F}}^\circ}) \otimes_{{W}({\mathbb{F}}_q)} {{{E}}^\circ}$ for the ring of ramified Witt vectors of ${{{F}}^\circ}$ with coefficients in ${{{E}}^\circ}$, and $[{\varpi}]$ for the Teichmuller lift of ${\varpi}$. Define $${\mathcal{Y}}_{{E},{F}}:={\mathrm{Spa}}({W}_{{{{E}}^\circ}}({{{F}}^\circ}))\setminus\{|p[{\varpi}]|=0\},$$ and let $\phi:{\mathcal{Y}}_{{E},{F}}\to {\mathcal{Y}}_{{E},{F}}$ be the Frobenius automorphism of ${\mathcal{Y}}_{{E},{F}}$ induced by the natural $q$-Frobenius $\varphi_q$ on ${W}_{{{{E}}^\circ}}({{{F}}^\circ})$. The (mixed-characteristic) *adic Fargues-Fontaine curve* associated to the pair $({E}, {F})$ is defined by $${\mathcal{X}}_{{E},{F}}:={\mathcal{Y}}_{{E},{F}}/\phi^{\mathbb{Z}}.$$ ${\mathcal{X}}_{{E},{F}}$ is a Noetherian adic space over ${\mathrm{Spa}}({E})$. When ${E}$ is replaced by a finite extension of ${\mathbb{F}}_p((t))$, there is a related construction of the equal-characteristic Fargues-Fontaine curve. Our main results are equally valid for vector bundles on the equal-characteristic Fargues-Fontaine curve. Moreover, the proofs for the equal-characteristic setting are strictly easier than the proofs for the mixed-characteristic setting. Therefore in this paper we will focus on vector bundles on the mixed-characteristic Fargues-Fontaine curve. Let us relate the above construction of ${\mathcal{X}}_{{E},{F}}$ to the schematic construction of the Fargues-Fontaine curve. For this, we need to define some vector bundles on ${\mathcal{X}}_{{E},{F}}$. Note that, by descent, a vector bundle ${\mathcal{V}}$ over ${\mathcal{X}}_{{E},{F}}$ is the same as a $\phi$-equivariant vector bundle ${\mathring}{{\mathcal{V}}}$ on ${\mathcal{Y}}_{{E},{F}}$, that is, a vector bundle ${\mathring}{{\mathcal{V}}}$ on ${\mathcal{Y}}_{{E},{F}}$ together with an isomorphism $\phi^*{\mathring}{{\mathcal{V}}}\stackrel{\sim}{\to}{\mathring}{{\mathcal{V}}}$. \[o-r-over-s\] Let $\lambda = r/s$ be a rational number written in lowest terms with $r>0$. Let $v_1, v_2, \cdots, v_s$ be a trivializing basis of ${{\mathcal{O}}}_{{\mathcal{Y}}_{{E},{F}}}^{\oplus s}$. Define an isomorphism $\phi^* {{\mathcal{O}}}_{{\mathcal{Y}}_{{E},{F}}}^{\oplus s} \stackrel{\sim}{\to} {{\mathcal{O}}}_{{\mathcal{Y}}_{{E},{F}}}^{\oplus s}$ by $$v_1 \mapsto v_2, \quad v_2 \mapsto v_3, \quad\cdots, \quad v_{s-1} \mapsto v_s, \quad v_s \mapsto {\pi}^{-r} v_1,$$ where we abuse notation to view $v_1, v_2, \cdots, v_s$ as a trivializing basis for $\phi^* {{\mathcal{O}}}_{{\mathcal{Y}}_{{E},{F}}}^{\oplus s}$ as well. We write ${{\mathcal{O}}}(\lambda)$ for the vector bundle on ${\mathcal{X}}_{{E},{F}}$ corresponding to the vector bundle ${{\mathcal{O}}}_{{\mathcal{Y}}_{{E},{F}}}^{\oplus s}$ with the isomorphism $\phi^* {{\mathcal{O}}}_{{\mathcal{Y}}_{{E},{F}}}^{\oplus s} \stackrel{\sim}{\to} {{\mathcal{O}}}_{{\mathcal{Y}}_{{E},{F}}}^{\oplus s}$ as defined above. The following fact suggests that we can regard ${{\mathcal{O}}}(1)$ as an “ample" line bundle on ${\mathcal{X}}_{{E},{F}}$. Let ${\mathcal{F}}$ be a coherent sheaf on ${\mathcal{X}}_{{E},{F}}$. Then for all sufficiently large $n \in {\mathbb{Z}}$, the twisted sheaf ${\mathcal{F}}(n):= {\mathcal{F}}\otimes {{\mathcal{O}}}(1)^{\otimes n}$ satisfies the following properties: 1. $H^1({\mathcal{X}}_{{E},{F}}, {\mathcal{F}}(n)) = 0$. 2. The sheaf ${\mathcal{F}}(n)$ is generated by finitely many global sections. We now recover the schematic construction of the Fargues-Fontaine curve as follows: \[schematic FF curve\] We define the *schematic Fargues-Fontaine curve* associated to the pair $({E},{F})$ to be $${X}_{{E},{F}} := \operatorname{Proj\,}\left( \bigoplus_{n \geq 0 } H^0({\mathcal{X}}_{{E},{F}}, {{\mathcal{O}}}(n) ) \right).$$ The original construction of the schematic Fargues-Fontaine curve in [@FF08] was given in terms of certain period rings of $p$-adic Hodge theory (see also [@FF14], 4.1): $${X}_{{E},{F}} = \operatorname{Proj\,}\left( \bigoplus_{n \geq 0 } B^{\varphi_q = {\pi}^n}\right).$$ This definition agrees with Definition \[schematic FF curve\] via the identification $H^0({\mathcal{X}}_{{E},{F}}, {{\mathcal{O}}}(n) ) \simeq B^{\varphi_q = {\pi}^n}$. The scheme ${X}_{{E},{F}} $ is Noetherian, connected, and regular of (absolute) dimension one. For our purpose, the two constructions are essentially equivalent in the following sense: \[GAGA for FF curve\] There is a natural map $${\mathcal{X}}_{{E},{F}} \rightarrow {X}_{{E},{F}}$$ which induces by pullback an equivalence of categories of vector bundles. Following Kedlaya-Liu [@KL15], we can extend the construction of the adic Fargues-Fontaine curve to relative settings. \[relative FF curve\] Let $S = {\mathrm{Spa}}({R}, {R}^+)$ be an affinoid perfectoid space over ${\mathrm{Spa}}({F})$, and let ${\varpi}_{R}$ be a pseudouniformizer of ${R}$. Denote by ${{{E}}^\circ}$ the ring of integers of ${E}$, and by ${{{R}}^\circ}$ the ring of power bounded elements of ${R}$. We take the rings of ramified Witt vectors $$\begin{aligned} {W}_{{E}^\circ}({{{R}}^\circ})&:={W}({{{R}}^\circ}) \otimes_{{W}({\mathbb{F}}_q)} {{{E}}^\circ}, \\ {W}_{{E}^\circ}({R}^+)&:={W}({R}^+) \otimes_{{W}({\mathbb{F}}_q)} {{{E}}^\circ}\end{aligned}$$ and write $[{\varpi}_{R}]$ for the Teichmuller lift of ${\varpi}_{R}$. Define $${\mathcal{Y}}_{{E},S}:={\mathrm{Spa}}({W}_{{{{E}}^\circ}}({{{R}}^\circ}), {W}_{{{{E}}^\circ}}({R}^+))\setminus\{|p[{\varpi}_{R}]|=0\},$$ and let $\phi:{\mathcal{Y}}_{{E},S}\to {\mathcal{Y}}_{{E},S}$ be the Frobenius automorphism of ${\mathcal{Y}}_{{E},S}$ induced by the natural $q$-Frobenius $\varphi_q$ on ${W}_{{{{E}}^\circ}}({{{R}}^\circ})$. The *relative adic Fargues-Fontaine curve* associated to the pair $({E}, S)$ is defined by $${\mathcal{X}}_{{E},S}:={\mathcal{Y}}_{{E},S}/\phi^{\mathbb{Z}}.$$ More generally, for an arbitrary perfectoid space $S$ over ${\mathrm{Spa}}({F})$, we choose an affinoid cover $S = \bigcup S_i = \bigcup {\mathrm{Spa}}({R}_i, {R}_i^+)$ and define the relative adic Fargues-Fontaine curve ${\mathcal{X}}_{{E},S}$ by gluing the adic spaces ${\mathcal{X}}_{{E}, S_i}$. By construction, the relative curve ${\mathcal{X}}_{{E}, S}$ comes with a map ${\mathcal{X}}_{{E}, S} \to {\mathcal{X}}_{{E}, {F}}$. However, the relative curve ${\mathcal{X}}_{{E}, S}$ cannot be obtained from ${\mathcal{X}}_{{E}, {F}}$ by base change. In fact, neither ${\mathcal{X}}_{{E}, S}$ nor ${\mathcal{X}}_{{E}, S}$ is defined over ${\mathrm{Spa}}({F})$. Classification of vector bundles and Harder-Narasimhan polygons --------------------------------------------------------------- $ $ For the rest of this paper, we will simply write ${\mathcal{X}}:= {\mathcal{X}}_{{E},{F}}$ and ${X}:= {X}_{{E},{F}}$. Moreover, we will henceforth speak interchangeably about vector bundles on ${\mathcal{X}}$ and ${X}$ in light of Proposition \[GAGA for FF curve\]. In this section we review the main classification theorem for vector bundles on the Fargues-Fontaine curve and discuss some of its immediate consequences. \[completeness of FF curve\] The curve ${X}$ is complete in the sense that for an arbitrary nonzero rational function $f$ on ${X}$, the divisor of $f$ has degree zero. This yields a well-defined notion of degree for line bundles as follows: Given a line bundle ${\mathcal{L}}$ on ${X}$, we define the *degree* of ${\mathcal{L}}$ by $$\deg({\mathcal{L}}) := \deg (\mathrm{div}(s))$$ where $s$ is an arbitrary nonzero meromorphic section of ${\mathcal{L}}$. The above notion of degree readily extends to vector bundles, thereby yielding a notion of slope for vector bundles. Let ${\mathcal{V}}$ be a vector bundle on ${\mathcal{X}}$. 1. We write ${\mathrm{rk}}({\mathcal{V}})$ for the rank of ${\mathcal{V}}$, and ${\mathcal{V}}^\vee$ for the dual bundle of ${\mathcal{V}}$. 2. We define the *degree* and the *slope* of ${\mathcal{V}}$ respectively by $$\deg ({\mathcal{V}}) := \deg (\wedge^{{\mathrm{rk}}({\mathcal{V}})} {\mathcal{V}}) \quad\quad\text{ and }\quad\quad \mu({\mathcal{V}}) := \dfrac{\deg({\mathcal{V}})}{{\mathrm{rk}}({\mathcal{V}})}.$$ Let us now recall the usual notions of stability and semistability. Let ${\mathcal{V}}$ be a vector bundle on ${\mathcal{X}}$. 1. We say that ${\mathcal{V}}$ is *stable* if $\mu({\mathcal{W}}) < \mu({\mathcal{V}})$ for all nonzero proper subbundles ${\mathcal{W}}\subset {\mathcal{V}}$. 2. We say that ${\mathcal{V}}$ is *semistable* if $\mu({\mathcal{W}}) \leq \mu({\mathcal{V}})$ for all nonzero proper subbundles ${\mathcal{W}}\subset {\mathcal{V}}$. We have the following classification of stable and semistable vector bundles on ${\mathcal{X}}$: \[classification of semistable bundles\] Let $\lambda$ be a rational number. 1. The bundle ${{\mathcal{O}}}(\lambda)$ represents a unique isomorphism class of stable bundles of slope $\lambda$. 2. Every semistable bundle of slope $\lambda$ is isomorphic to ${{\mathcal{O}}}(\lambda)^{\oplus n}$ for some $n$. This is indeed the technical crux of the proof of the main classification theorem for vector bundles on ${\mathcal{X}}$. We will soon see that it is not hard to deduce the main classification theorem from this fact combined with some cohomological computations. We collect some fundamental properties of the stable bundles on ${\mathcal{X}}$. \[basic properties of stable bundles\] Let $r$ and $s$ be relatively prime integers with $s>0$. 1. \[rank and degree of stable bundles\] The bundle ${{\mathcal{O}}}(r/s)$ has rank $s$, degree $r$, and slope $r/s$. 2. \[tensor product of stable bundles\] For any relatively prime integers $r'$ and $s'$ with $s'>0$, we have $${{\mathcal{O}}}\left(\frac rs\right)\otimes{{\mathcal{O}}}\left(\frac{r'}{s'}\right)\simeq{{\mathcal{O}}}\left(\frac rs+\frac{r'}{s'}\right)^{{\oplus}\gcd(s s',rs'+r's)}.$$ In particular, the bundle ${{\mathcal{O}}}(r/s)\otimes{{\mathcal{O}}}(r'/s')$ has rank $ss'$, degree $rs'+r's$, and slope $r/s+r'/s'$. 3. \[dual of stable bundles\] ${{\mathcal{O}}}(r/s)^\vee \simeq {{\mathcal{O}}}(-r/s)$. All statements are straightforward to check using Definition \[o-r-over-s\]. \[cohomological vanishing of stable bundles\] We have the following cohomological computations: 1. $H^0({\mathcal{X}}, {{\mathcal{O}}}(\lambda)) = 0$ if and only if $\lambda < 0$. 2. $H^1({\mathcal{X}}, {{\mathcal{O}}}(\lambda)) = 0$ if $\lambda \geq 0$. It turns out that every vector bundle on ${\mathcal{X}}$ admits a direct sum decomposition into stable bundles, as stated in the following theorem: \[existence of HN decomp\] Every vector bundle ${\mathcal{V}}$ on ${\mathcal{X}}$ admits a unique filtration $$\label{HN filtration} 0 = {\mathcal{V}}_0 \subset {\mathcal{V}}_1 \subset \dotsb \subset {\mathcal{V}}_l = {\mathcal{V}}$$ such that the successive quotients ${\mathcal{V}}_i/{\mathcal{V}}_{i-1}$ are semistable vector bundles with $$\mu({\mathcal{V}}_1/{\mathcal{V}}_0) > \mu({\mathcal{V}}_2/{\mathcal{V}}_1) > \cdots > \mu({\mathcal{V}}_l/{\mathcal{V}}_{l-1}).$$ Moreover, the filtration splits into a direct sum decomposition $$\label{HN decomposition} {\mathcal{V}}\simeq \bigoplus_{i=1}^l {{\mathcal{O}}}(\lambda_i)^{\oplus m_i}$$ where $\lambda_i = \mu({\mathcal{V}}_i/{\mathcal{V}}_{i-1})$. Existence and uniqueness of the filtration is a standard consequence of slope formalism. We refer the readers to [@Ked17 §3.4] for a detailed discussion. For the direct sum decomposition , we proceed by induction on $l$. Since the base case $l=0$ is trivial, we only need to consider the induction step. By the induction hypothesis, the filtration $$0 = {\mathcal{V}}_0 \subset {\mathcal{V}}_1 \subset \dotsb \subset {\mathcal{V}}_{l-1}$$ splits into a direct sum decomposition $$\label{HN decomposition induction step} {\mathcal{V}}_{l-1} \simeq \bigoplus_{i=1}^{l-1} {{\mathcal{O}}}(\lambda_i)^{\oplus m_i}.$$ Moreover, since the quotient ${\mathcal{V}}/{\mathcal{V}}_{l-1} = {\mathcal{V}}_l/{\mathcal{V}}_{l-1}$ is a semistable bundle with slope $\lambda_l$, Proposition \[classification of semistable bundles\] yields a decomposition $$\label{final quotient decomposition} {\mathcal{V}}/{\mathcal{V}}_{l-1} \simeq {{\mathcal{O}}}(\lambda_l)^{\oplus m_l}.$$ Hence it remains to establish the identity $$\label{splitting of last step in HN filtration} \operatorname{Ext}^1({\mathcal{V}}/ {\mathcal{V}}_{l-1}, {\mathcal{V}}_{l-1}) = 0.$$ For each $i = 1, 2, \cdots, l-1$, Lemma \[basic properties of stable bundles\] yields an identification $$\operatorname{Ext}^1({{\mathcal{O}}}(\lambda_l), {{\mathcal{O}}}(\lambda_i)) \simeq H^1({\mathcal{X}}, {{\mathcal{O}}}(\lambda_i) \otimes {{\mathcal{O}}}(\lambda_l)^\vee) \simeq H^1({\mathcal{X}}, {{\mathcal{O}}}(\lambda_i - \lambda_l)^{\oplus n_i}).$$ Since $\lambda_i > \lambda_l$ for each $i = 1, 2, \cdots, l-1$, Theorem \[cohomological vanishing of stable bundles\] now yields $$\operatorname{Ext}^1({{\mathcal{O}}}(\lambda_l), {{\mathcal{O}}}(\lambda_i)) = 0 \quad\quad \text{ for each } i = 1, 2, \cdots, l-1.$$ We thus deduce the desired identity by the decompositions and . \[def of HN filtration, decomposition and polygon\] Let ${\mathcal{V}}$ be a vector bundle on ${\mathcal{X}}$. 1. We refer to the filtration in Theorem \[existence of HN decomp\] as the *Harder-Narasimhan (HN) filtration* of ${\mathcal{V}}$. 2. We refer to the decomposition in Theorem \[existence of HN decomp\] as the *Harder-Narasimhan (HN) decomposition* of ${\mathcal{V}}$. 3. We define the *Harder-Narasimhan (HN) polygon* of ${\mathcal{V}}$ as the upper convex hull of the points $({\mathrm{rk}}({\mathcal{V}}_i), \deg({\mathcal{V}}_i))$ where ${\mathcal{V}}_i$’s are subbundles in the HN filtration of ${\mathcal{V}}$. 4. We refer to the slopes of ${\mathrm{HN}}({\mathcal{V}})$ as the *Harder-Narasimhan (HN) slopes* of ${\mathcal{V}}$, or simply the *slopes* of ${\mathcal{V}}$. These are precisely the numbers $\lambda_i = \mu({\mathcal{V}}_i/{\mathcal{V}}_{i-1})$ in Theorem \[existence of HN decomp\]. We can restate Theorem \[existence of HN decomp\] in terms of HN polygons as follows: \[classification by HN polygon\] Every vector bundle ${\mathcal{V}}$ on ${\mathcal{X}}$ is determined up to isomorphism by its HN polygon ${\mathrm{HN}}({\mathcal{V}})$. The HN polygon ${\mathrm{HN}}({\mathcal{V}})$ determines the HN filtration of ${\mathcal{V}}$ by definition. Hence by Theorem \[existence of HN decomp\] it determines the isomorphism class of ${\mathcal{V}}$ via the HN decomposition. Let us now introduce some notations that we will frequently use. \[slope\_leq\_part\] Let ${\mathcal{V}}$ be a vector bundle on ${\mathcal{X}}$ with Harder-Narasimhan filtration $$0 = {\mathcal{V}}_0 \subset {\mathcal{V}}_1 \subset \ldots \subset {\mathcal{V}}_m = {\mathcal{V}}.$$ 1. We write ${\mu_\text{max}}({\mathcal{V}})$ (resp. ${\mu_\text{min}}({\mathcal{V}})$) for the maximum (resp. minimum) slope of ${\mathcal{V}}$. 2. For every $\mu \in {\mathbb{Q}}$, we define ${\mathcal{V}}^{\geq \mu}$ (resp. ${\mathcal{V}}^{> \mu}$) to be the subbundle of ${\mathcal{V}}$ given by ${\mathcal{V}}_i$ for the largest value of $i$ such that $\mu({\mathcal{V}}_i/{\mathcal{V}}_{i-1}) \geq \mu$ (resp. such that $\mu({\mathcal{V}}_i/{\mathcal{V}}_{i-1}) > \mu$). We also define ${\mathcal{V}}^{<\mu} := {\mathcal{V}}/ {\mathcal{V}}^{\geq \mu}$ and ${\mathcal{V}}^{\leq \mu} := {\mathcal{V}}/ {\mathcal{V}}^{> \mu}$. \[slope\_leq\_part via HN decomp\] Let ${\mathcal{V}}$ be a vector bundle on ${\mathcal{X}}$ with Harder-Narasimhan decomposition $${\mathcal{V}}\simeq \bigoplus_{i=1}^l {{\mathcal{O}}}(\lambda_i)^{\oplus m_i}.$$ Then we have the following identifications: $${\mathcal{V}}^{\geq \mu} \simeq \bigoplus_{\lambda_i \geq \mu}{{\mathcal{O}}}(\lambda_i)^{\oplus m_i} \quad\quad \text{ and } \quad\quad {\mathcal{V}}^{> \mu} \simeq \bigoplus_{\lambda_i>\mu}{{\mathcal{O}}}(\lambda_i)^{\oplus m_i},$$ $${\mathcal{V}}^{\leq \mu} \simeq \bigoplus_{\lambda_i \leq \mu}{{\mathcal{O}}}(\lambda_i)^{\oplus m_i} \quad\quad \text{ and } \quad\quad {\mathcal{V}}^{< \mu} \simeq \bigoplus_{\lambda_i<\mu}{{\mathcal{O}}}(\lambda_i)^{\oplus m_i}.$$ This is an immediate consequence of Definition \[slope\_leq\_part\]. \[rank and degree of dual bundle\] Given a vector bundle ${\mathcal{V}}$ on ${\mathcal{X}}$, we have identities $${\mathrm{rk}}({\mathcal{V}}) = {\mathrm{rk}}({\mathcal{V}}^\vee) \quad\text{ and }\quad \deg({\mathcal{V}}) = -\deg({\mathcal{V}}^\vee).$$ More generally, for every $\mu \in {\mathbb{Q}}$ we have identities $${\mathrm{rk}}({\mathcal{V}}^{\geq \mu}) = {\mathrm{rk}}(({\mathcal{V}}^\vee)^{\leq-\mu}) \quad\text{ and }\quad \deg({\mathcal{V}}^{\geq \mu}) = -\deg(({\mathcal{V}}^\vee)^{\leq-\mu}).$$ When ${\mathcal{V}}$ is stable, the first statement is an immediate consequence of Lemma \[basic properties of stable bundles\]. From this, we deduce the first statement for the general case using HN decomposition of ${\mathcal{V}}$. The second statement then follows from the first statement since we have $({\mathcal{V}}^{\geq \mu})^\vee \simeq ({\mathcal{V}}^\vee)^{\leq -\mu}$ by Lemma \[basic properties of stable bundles\] and Lemma \[slope\_leq\_part via HN decomp\]. \[zero hom for dominating slopes\] Given two vector bundles ${\mathcal{V}}$ and ${\mathcal{W}}$ on ${\mathcal{X}}$, we have $$\label{zero hom for dominating slopes equation} \mathrm{Hom}({\mathcal{V}}, {\mathcal{W}}) = 0 \quad\quad \text{ if and only if } \quad\quad {\mu_\text{min}}({\mathcal{V}})>{\mu_\text{max}}({\mathcal{W}}).$$ It suffices to consider the case when both ${\mathcal{V}}$ and ${\mathcal{W}}$ are stable; indeed, the general case will follow from this special case using the HN decompositions of ${\mathcal{V}}$ and ${\mathcal{W}}$. Let us now write ${\mathcal{V}}= {{\mathcal{O}}}(\lambda)$ and ${\mathcal{W}}= {{\mathcal{O}}}(\mu)$ for some $\lambda, \mu \in {\mathbb{Q}}$. Then using Lemma \[basic properties of stable bundles\] we obtain an identification $$\mathrm{Hom}({\mathcal{V}}, {\mathcal{W}}) \simeq H^0({\mathcal{X}}, {\mathcal{V}}^\vee \otimes {\mathcal{W}}) \simeq H^0({\mathcal{X}}, {{\mathcal{O}}}(\lambda)^\vee \otimes {{\mathcal{O}}}(\mu)) \simeq H^0({\mathcal{X}}, {{\mathcal{O}}}(\mu - \lambda)^{\oplus n}).$$ Since the condition ${\mu_\text{min}}({\mathcal{V}})>{\mu_\text{max}}({\mathcal{W}})$ is equivalent to $\lambda > \mu$, the assertion follows from Theorem \[cohomological vanishing of stable bundles\]. Moduli of bundle maps {#diamond_dim} ===================== Definitions and key properties {#bundlemaps} ------------------------------ $ $ In this section we define certain moduli spaces of bundle maps and collect some of their key properties. We refer the readers to [@Arizonaext §3.3] for a thorough discussion about these moduli spaces. Recall from Definition \[relative FF curve\] that for any perfectoid space $S$ over ${\mathrm{Spa}}({F})$ we have a relative Fargues-Fontaine curve ${\mathcal{X}}_S$ that comes with a map ${\mathcal{X}}_S \to {\mathcal{X}}$. Let ${\mathcal{V}}$ be a vector bundle on ${\mathcal{X}}$. For any perfectoid space $S$ over ${\mathrm{Spa}}({F})$, we denote by ${\mathcal{V}}_S$ the vector bundle obtained from ${\mathcal{V}}$ via pullback along the map ${\mathcal{X}}_S \to {\mathcal{X}}$. Let us define some moduli functors parametrizing bundle maps over ${\mathcal{X}}$ with various specified properties. Denote by ${\mathrm{Perf}}_{/{\mathrm{Spa}}({F})}$ the category of perfectoid spaces over ${\mathrm{Spa}}({F})$. Given vector bundles ${\mathcal{E}}$ and ${\mathcal{F}}$ on ${\mathcal{X}}$, we define the following Set-valued functors on ${\mathrm{Perf}}_{/{\mathrm{Spa}}({F})}$: 1. ${\mathcal{H}}^{0}$ is the functor associating $S \in {\mathrm{Perf}}_{/{\mathrm{Spa}}({F})}$ to the set $H^{0}({\mathcal{X}}_S,{\mathcal{E}}_S)$. 2. ${\mathcal{H}\mathrm{om}}(\mathcal{E},{\mathcal{F}})$ is the functor associating $S \in {\mathrm{Perf}}_{/{\mathrm{Spa}}({F})}$ to the set of ${{\mathcal{O}}}_{{\mathcal{X}}_S}$-module maps $m:{\mathcal{E}}_S\to{\mathcal{F}}_S$. Note that ${\mathcal{H}\mathrm{om}}({\mathcal{E}}, {\mathcal{F}})\cong{\mathcal{H}}^{0}({\mathcal{E}}^{\vee}\otimes{\mathcal{F}})$. 3. Let ${\mathcal{S}\mathrm{urj}}({\mathcal{E}}, {\mathcal{F}}) \subset{\mathcal{H}\mathrm{om}}({\mathcal{E}}, {\mathcal{F}})$ be the subfunctor of ${\mathcal{H}\mathrm{om}}({\mathcal{E}},{\mathcal{F}})$ whose $S$-points parametrize surjective ${{\mathcal{O}}}_{{\mathcal{X}}_S}$-module maps $m: {\mathcal{E}}_S {\twoheadrightarrow}{\mathcal{F}}_S$. 4. Let ${\mathcal{I}\mathrm{nj}}({\mathcal{E}}, {\mathcal{F}})\subset{\mathcal{H}\mathrm{om}}({\mathcal{E}},{\mathcal{F}})$ be the subfunctor of ${\mathcal{H}\mathrm{om}}({\mathcal{E}},{\mathcal{F}})$ whose $S$-points parametrize “fiberwise-injective” ${{\mathcal{O}}}_{{\mathcal{X}}_S}$-module maps. Precisely, this functor parametrizes ${{\mathcal{O}}}_{{\mathcal{X}}_S}$-module maps $m:{\mathcal{E}}_{S}\to{\mathcal{F}}_{S}$ whose pullback along the map ${\mathcal{X}}_{\overline{x}} \to{\mathcal{X}}_S$ for any geometric point $\overline{x} \to S$ gives an injective ${{\mathcal{O}}}_{{\mathcal{X}}_{\overline{x}}}$-module map. The condition defining ${\mathcal{I}\mathrm{nj}}({\mathcal{E}}, {\mathcal{F}})$ is much stronger than the condition that $m:{\mathcal{E}}_{S}\to{\mathcal{F}}_{S}$ is injective. Scholze’s theory of diamonds in [@Sch] provides a framework for making sense of these functors as moduli spaces, thereby allowing us to study their geometric properties. \[moduli of bundle maps dimension formulas\] Let ${\mathcal{E}}$ and ${\mathcal{F}}$ be vector bundles on ${\mathcal{X}}$. The functors ${\mathcal{H}}^0({\mathcal{E}})$, ${\mathcal{H}\mathrm{om}}({\mathcal{E}}, {\mathcal{F}})$, ${\mathcal{S}\mathrm{urj}}({\mathcal{E}}, {\mathcal{F}})$ and ${\mathcal{I}\mathrm{nj}}({\mathcal{E}}, {\mathcal{F}})$ are all locally spatial and partially proper diamonds in the sense of Scholze [@Sch]. Moreover, their dimensions are given as follows: 1. The diamond ${\mathcal{H}}^0({\mathcal{E}})$ is equidimensional of dimension $\deg({\mathcal{E}})^{{\geq 0}}$. 2. The diamond ${\mathcal{H}\mathrm{om}}({\mathcal{E}}, {\mathcal{F}})$ is equidimensional of dimension $\deg({\mathcal{E}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}$. 3. The diamonds ${\mathcal{S}\mathrm{urj}}({\mathcal{E}}, {\mathcal{F}})$ and ${\mathcal{I}\mathrm{nj}}({\mathcal{E}}, {\mathcal{F}})$ are both either empty or equidimensional of dimension $\deg({\mathcal{E}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}$. One can show that the natural map $${\mathcal{S}\mathrm{urj}}({\mathcal{F}}^\vee, {\mathcal{E}}^\vee) \to {\mathcal{I}\mathrm{nj}}({\mathcal{E}}, {\mathcal{F}})$$ is an open immersion. The following fact is crucial for our proof of the main result. \[dimension inequality for surj maps\] Let ${\mathcal{E}}$ and ${\mathcal{F}}$ be vector bundles on ${\mathcal{X}}$ satisfying the following properties: 1. \[existence of nonzero bundle map from E to F\] There exists a nonzero bundle map ${\mathcal{E}}\to {\mathcal{F}}$. 2. \[positive codim for Hom minus surj\] For any ${\mathcal{Q}}\subsetneq {\mathcal{F}}$ which also occurs as a quotient of ${\mathcal{E}}$ we have an inequality $$\deg({\mathcal{E}}^\vee \otimes {\mathcal{Q}})^{{\geq 0}}+ \deg({\mathcal{Q}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}< \deg({\mathcal{E}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}+ \deg({\mathcal{Q}}^\vee \otimes {\mathcal{Q}})^{{\geq 0}}.$$ Then there exists a surjective bundle map ${\mathcal{E}}{\twoheadrightarrow}{\mathcal{F}}$. We give a sketch of the proof here. Interested readers can find a complete proof in [@Arizonaext §3.3]. We wish to show that the topological space $|{\mathcal{S}\mathrm{urj}}({\mathcal{E}}, {\mathcal{F}})|$ is nonempty. Let $S$ be the set of isomorphism classes of subbundles ${\mathcal{Q}}\subsetneq {\mathcal{F}}$ which also occur as a quotient of ${\mathcal{E}}$. For each ${\mathcal{Q}}\in S$, composition of bundle maps induces a natural map of diamonds $${\mathcal{S}\mathrm{urj}}({\mathcal{E}},{\mathcal{Q}}) \times_{{\mathrm{Spd}}\,{F}} {\mathcal{I}\mathrm{nj}}({\mathcal{Q}},{\mathcal{F}}) \to {\mathcal{H}\mathrm{om}}({\mathcal{E}},{\mathcal{F}}).$$ Let us define $|{\mathcal{H}\mathrm{om}}({\mathcal{E}}, {\mathcal{F}})_{{\mathcal{Q}}}| \subset |{\mathcal{H}\mathrm{om}}({\mathcal{E}}, {\mathcal{F}})|$ to be the image of the induced map on topological spaces. Then we have the following facts ([@Arizonaext Proposition 3.3.9 and Lemma 3.3.10]): 1. \[Hom with specified image behave nicely\] For every ${\mathcal{Q}}\in S$, the set $|{\mathcal{H}\mathrm{om}}({\mathcal{E}}, {\mathcal{F}})_{{\mathcal{Q}}}|$ is stable under generalization and specialization inside $|{\mathcal{H}\mathrm{om}}({\mathcal{E}}, {\mathcal{F}})|$. 2. \[dim formula for Hom with specified image\] For every ${\mathcal{Q}}\in S$ with $|{\mathcal{H}\mathrm{om}}({\mathcal{E}}, {\mathcal{F}})_{{\mathcal{Q}}}|$ nonempty, we have $$\dim |{\mathcal{H}\mathrm{om}}({\mathcal{E}}, {\mathcal{F}})_{{\mathcal{Q}}}| = \deg({\mathcal{E}}^\vee \otimes {\mathcal{Q}})^{{\geq 0}}+ \deg({\mathcal{Q}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}- \deg({\mathcal{Q}}^\vee \otimes {\mathcal{Q}})^{{\geq 0}}.$$ Moreover, by definition we have $$|{\mathcal{H}\mathrm{om}}({\mathcal{E}}, {\mathcal{F}})| \backslash |{\mathcal{S}\mathrm{urj}}({\mathcal{E}}, {\mathcal{F}})| = \coprod_{{\mathcal{Q}}\in S}|{\mathcal{H}\mathrm{om}}({\mathcal{E}}, {\mathcal{F}})_{\mathcal{Q}}|.$$ Now we use the facts \[Hom with specified image behave nicely\] and \[dim formula for Hom with specified image\], the assumption \[positive codim for Hom minus surj\], and Proposition \[moduli of bundle maps dimension formulas\] to find $$\begin{aligned} \dim \big(|{\mathcal{H}\mathrm{om}}({\mathcal{E}}, {\mathcal{F}})| \backslash |{\mathcal{S}\mathrm{urj}}({\mathcal{E}}, {\mathcal{F}})| \big) &= \sup_{{\mathcal{Q}}\in S} \dim |{\mathcal{H}\mathrm{om}}({\mathcal{E}}, {\mathcal{F}})_{\mathcal{Q}}|\\ &\leq \sup_{{\mathcal{Q}}\in S} \big( \deg({\mathcal{E}}^\vee \otimes {\mathcal{Q}})^{{\geq 0}}+ \deg({\mathcal{Q}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}- \deg({\mathcal{Q}}^\vee \otimes {\mathcal{Q}})^{{\geq 0}}\big)\\ &< \deg({\mathcal{E}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}\\ &= \dim |{\mathcal{H}\mathrm{om}}({\mathcal{E}}, {\mathcal{F}})|.\end{aligned}$$ Hence we deduce that $|{\mathcal{S}\mathrm{urj}}({\mathcal{E}}, {\mathcal{F}})|$ is nonempty as desired. As the notation suggests, $|{\mathcal{H}\mathrm{om}}({\mathcal{E}}, {\mathcal{F}})_{{\mathcal{Q}}}|$ is the underlying topological space of a subdiamond ${\mathcal{H}\mathrm{om}}({\mathcal{E}}, {\mathcal{F}})_{{\mathcal{Q}}}$ of ${\mathcal{H}\mathrm{om}}({\mathcal{E}}, {\mathcal{F}})$, which (more or less) parametrizes bundle maps ${\mathcal{E}}\to {\mathcal{F}}$ with image isomorphic to ${\mathcal{Q}}$ at all geometric points. Dimension counting by Harder-Narasimhan polygons {#Geometric interpretation of degrees} ------------------------------------------------ $ $ As our discussion in §\[bundlemaps\] suggests, we will have to understand quantities of the form $\deg({\mathcal{V}}^\vee \otimes {\mathcal{W}})^{{\geq 0}}$ for some fairly arbitrary vector bundles ${\mathcal{V}}$ and ${\mathcal{W}}$ on ${\mathcal{X}}$. Following the strategy developed in [@Arizonaext §2.3], we prove some useful lemmas for this. \[order\_relation\] Let $v$ and $w$ be arbitrary vectors in ${\mathbb{R}}^2$. 1. We denote by $v_x$ (resp. $v_y$) the $x$-coordinate (resp. $y$-coordinate) of $v$. 2. If $v_x \neq 0$, we write $\mu(v):= v_y/v_x$ for the slope of $v$. 3. If both $v$ and $w$ have nonzero $x$-coordinates, we will often write $v \prec w$ (resp. $v \preceq w$) in lieu of $\mu(v) < \mu(w)$ (resp. $\mu(v) \leq \mu(w)$). 4. We write $v \times w$ for the (two-dimensional) cross product of $v$ and $w$, regarded as a scalar by the formula $$v \times w = v_x w_y - v_y w_x.$$ We can characterize the relations $\preceq$ and $\prec$ in terms of cross products as follows: \[relation\_area\] Let $v$ and $w$ be vectors in ${\mathbb{R}}^2$. 1. If $v_x$ and $v_y$ have the same sign, we have $v \prec w$ (resp. $v \preceq w$) if and only if $v \times w >0$ (resp. $v \times w \geq 0$). 2. If $v_x$ and $v_y$ have opposite signs, we have $v \prec w$ (resp. $v \preceq w$) if and only if $v \times w <0$ (resp. $v \times w \leq 0$). This is straightforward to check using Definition \[order\_relation\]. We will make use of Lemma \[relation\_area\] by expressing the HN polygons in terms of two-dimensional vectors. \[HN vector notation\] Let ${\mathcal{V}}$ be a vector bundle on ${\mathcal{X}}$ with Harder-Narasimhan decomposition $${\mathcal{V}}= \bigoplus_{i=1}^l {{\mathcal{O}}}(\lambda_i)^{m_i}$$ where $\lambda_1 > \lambda_2 > \cdots > \lambda_l$. We define the *HN vectors of ${\mathcal{V}}$* by $${\overrightarrow{\mathrm{HN}}}({\mathcal{V}}) := (v_i)_{1\leq i \leq l}$$ where $v_i$ is the vector representing the $i$-th edge in ${\mathrm{HN}}({\mathcal{V}})$. More precisely, writing $\lambda_i = r_i/s_i$ in lowest terms with $s_i>0$, we set $v_i :=(m_i s_i, m_i r_i)$. The following simple lemma is pivotal to our discussion in this section. \[diamond lemma for nonnegative degree\]\[degree in terms of HN vectors\] Let ${\mathcal{V}}$ and ${\mathcal{W}}$ be vector bundles on ${\mathcal{X}}$ with ${\overrightarrow{\mathrm{HN}}}({\mathcal{V}}) = (v_i)$ and ${\overrightarrow{\mathrm{HN}}}({\mathcal{W}}) = (w_j)$. Then we have $$\label{degree formula in terms of HN vectors} \deg({\mathcal{V}}^\vee \otimes {\mathcal{W}}) = \sum_{i, j} v_i \times w_j \quad\quad \text{ and } \quad\quad \deg({\mathcal{V}}^\vee \otimes {\mathcal{W}})^{{\geq 0}}= \sum_{v_i \preceq w_j} v_i \times w_j$$ When ${\mathcal{V}}$ and ${\mathcal{W}}$ are both semistable, we quickly verify both identities in using Lemma \[basic properties of stable bundles\] and Lemma \[relation\_area\]. Then we deduce the general case using the HN decompositions of ${\mathcal{V}}$ and ${\mathcal{W}}$. \[zero degree for completely dominating slopes\] For arbitrary vector bundles ${\mathcal{V}}$ and ${\mathcal{W}}$ on ${\mathcal{X}}$, we have $$\dim {\mathcal{H}\mathrm{om}}({\mathcal{V}}, {\mathcal{W}}) = 0 \quad\quad \text{ if and only if } \quad\quad {\mu_\text{min}}({\mathcal{V}}) \geq {\mu_\text{max}}({\mathcal{W}}).$$ This is an immediate consequence of Proposition \[moduli of bundle maps dimension formulas\] and Lemma \[degree in terms of HN vectors\] This is not a consequence of Lemma \[zero hom for dominating slopes\]; in fact, When ${\mu_\text{min}}({\mathcal{V}}) = {\mu_\text{max}}({\mathcal{W}})$, Lemma \[zero hom for dominating slopes\] and Corollary \[zero degree for completely dominating slopes\] respectively yield $\mathrm{Hom}({\mathcal{V}}, {\mathcal{W}}) \neq 0$ and $\dim {\mathcal{H}\mathrm{om}}({\mathcal{V}}, {\mathcal{W}}) = 0$. Given a vector bundle ${\mathcal{V}}$ on ${\mathcal{X}}$, we write ${\mathcal{V}}(\lambda) := {\mathcal{V}}\otimes {{\mathcal{O}}}(\lambda)$ for any $\lambda \in {\mathbb{Q}}$. \[degree after shear\] Let ${\mathcal{V}}$ and ${\mathcal{W}}$ be vector bundles on ${\mathcal{X}}$. For any $\lambda \in {\mathbb{Q}}$ we have $$\deg({\mathcal{V}}(\lambda)^\vee \otimes {\mathcal{W}}(\lambda))^{{\geq 0}}= {\mathrm{rk}}({{\mathcal{O}}}(\lambda))^2 \cdot \deg({\mathcal{V}}^\vee \otimes {\mathcal{W}})^{{\geq 0}}.$$ Let us write $\lambda = r/s$ in lowest term with $s>0$, and consider the HN vectors $${\overrightarrow{\mathrm{HN}}}({\mathcal{V}}) = (v_i), \quad\quad {\overrightarrow{\mathrm{HN}}}({\mathcal{V}}(\lambda)) = (v_i'), \quad\quad {\overrightarrow{\mathrm{HN}}}({\mathcal{W}}) = (w_j), \quad\quad {\overrightarrow{\mathrm{HN}}}({\mathcal{W}}(\lambda)) = (w_j').$$ By Lemma \[basic properties of stable bundles\], each $v_i$ (resp. $w_j$) is related to $v_i'$ (resp. $w_j$’) as follows: 1. \[change in slope after shear\] $\mu(v_i') = \mu(v_i) + \lambda$ and $\mu(w_j') = \mu(w_j) + \lambda$. 2. $v'_{i, x} = s v_{i, x}$ and $w'_{j, x} = s w_{j, x}$. 3. $v'_{i, y} = s v_{i, y} + r v_{i, x}$ and $w'_{j, y} = s w_{j, y} + r w_{j, x}$. Now for each $i$ and $j$ we find $$\label{cross product of HN vectors after shear} \begin{aligned} v'_i \times w'_j &= v'_{i, x} w'_{j, y} - v'_{i, y} w'_{j, x} \\ &= s v_{i, x} (s w_{j, y} + r w_{j, x}) - (s v_{i, y} + r v_{i, x}) \cdot s w_{j, x}\\ &= s^2 (v_{i, x} w_{j, y} - v_{i, y} w_{j, x}) \\ &= {\mathrm{rk}}({{\mathcal{O}}}(\lambda))^2 \cdot (v_i \times w_j) \end{aligned}$$ Moreover, by \[change in slope after shear\] we have $v'_i \preceq w'_j$ if and only if $v_i \preceq w_j$. Thus we use Lemma \[degree in terms of HN vectors\] and to obtain $$\begin{aligned} \deg({\mathcal{V}}(\lambda)^\vee \otimes {\mathcal{W}}(\lambda))^{{\geq 0}}&= \sum_{v'_i \preceq w'_j} v'_i \times w'_j = \sum_{v_i \preceq w_j} \Big( {\mathrm{rk}}({{\mathcal{O}}}(\lambda))^2 \cdot (v_i \times w_j) \Big)\\ &= {\mathrm{rk}}({{\mathcal{O}}}(\lambda))^2 \sum_{v_i \preceq w_j} v_i \times w_j \\ &= {\mathrm{rk}}({{\mathcal{O}}}(\lambda))^2 \cdot \deg({\mathcal{V}}^\vee \otimes {\mathcal{W}})^{{\geq 0}}.\end{aligned}$$ Let us give a geometric intuition behind the proof of Lemma \[degree after shear\]. In terms of HN vectors, tensoring with the bundle ${{\mathcal{O}}}(\lambda)$ is the same as the composition of a shear transformation (that makes every slope increase by $\lambda$) and a dilation by ${\mathrm{rk}}({{\mathcal{O}}}(\lambda))$. Then represents a geometric fact that the (signed) area of a parallelogram remains the same after the shear transformation and gets multiplied by ${\mathrm{rk}}({{\mathcal{O}}}(\lambda))^2$ after the dilation. \[degree after stretch\] Let ${\mathcal{V}}$ and ${\mathcal{W}}$ be vector bundles on ${\mathcal{X}}$. Take ${\tilde}{{\mathcal{V}}}$ and ${\tilde}{{\mathcal{W}}}$ to be vector bundles on ${\mathcal{X}}$ whose HN polygons are obtained by vertically stretching ${\mathrm{HN}}({\mathcal{V}})$ and ${\mathrm{HN}}({\mathcal{W}})$ by a positive integer factor $C$. Then we have $$\deg({\tilde}{{\mathcal{V}}}^\vee \otimes {\tilde}{{\mathcal{W}}})^{{\geq 0}}= C \cdot \deg({\mathcal{V}}^\vee \otimes {\mathcal{W}})^{{\geq 0}}.$$ Let us consider the HN vectors $${\overrightarrow{\mathrm{HN}}}({\mathcal{V}}) = (v_i), \quad\quad {\overrightarrow{\mathrm{HN}}}({\mathcal{W}}) = (w_j), \quad\quad {\overrightarrow{\mathrm{HN}}}({\tilde}{{\mathcal{V}}}) = ({\tilde}{v}_i), \quad\quad {\overrightarrow{\mathrm{HN}}}({\tilde}{{\mathcal{W}}}) = ({\tilde}{w}_j).$$ By construction, we have the following relations between these HN vectors. 1. \[change in slope after stretch\] $\mu({\tilde}{v}_i) = C\mu(v_i)$ and $\mu({\tilde}{w}_j) = C\mu(w'_j)$. 2. ${\tilde}{v}_{i, x} = v_{i, x}$ and ${\tilde}{w}_{j, x} = w_{j, x}$. 3. ${\tilde}{v}_{i, y} = C v_{i, y}$ and ${\tilde}{w}_{j, y} = C w_{j, y}$. Now for each $i$ and $j$ we have $$\label{cross product of HN vectors after stretch} \begin{aligned} {\tilde}{v}_i \times {\tilde}{w}_j &= {\tilde}{v}_{i, x} {\tilde}{w}_{j, y} - {\tilde}{v}_{i, y} {\tilde}{w}_{j, x} \\ &= v_{i, x} \cdot C w_{j, y} - C v_{i, y} \cdot w'_{j, x}\\ &= C (v_{i, x} w_{j, y} - v_{i, y} w_{j, x}) \\ &= C \cdot (v_i \times w_j) \end{aligned}$$ Furthermore, \[change in slope after stretch\] implies that ${\tilde}{v}_i \preceq {\tilde}{w}_j$ if and only if $v_i \preceq w_j$. We thus use Lemma \[degree in terms of HN vectors\] and to find $$\begin{aligned} \deg({\tilde}{{\mathcal{V}}}^\vee \otimes {\tilde}{{\mathcal{W}}})^{{\geq 0}}&= \sum_{{\tilde}{v}_i \preceq {\tilde}{w}_j} {\tilde}{v}_i \times {\tilde}{w}_j = \sum_{v_i \preceq w_j} C (v_i \times w_j) \\ &= C \sum_{v_i \preceq w_j} v_i \times w_j \\ &= C \cdot \deg({\mathcal{V}}^\vee \otimes {\mathcal{W}})^{{\geq 0}}.\end{aligned}$$ As in the case of Lemma \[degree after shear\], we can describe a geometric intuition behind the proof of Lemma \[degree after stretch\]. In fact, we can consider as a representation of a geometric fact that the vertical stretch by a factor $C$ scales the area of an arbitrary parallelogram by the same factor. Classification of quotient bundles ================================== The main statement and its consequences {#statement and consequences} --------------------------------------- $ $ Let us state our main theorem, which gives a complete classification of all quotient bundles of a given vector bundle on ${\mathcal{X}}$. \[classification of quotient bundles\] Let ${\mathcal{E}}$ be a vector bundle on ${\mathcal{X}}$. Then a vector bundle ${\mathcal{F}}$ on ${\mathcal{X}}$ is a quotient bundle of ${\mathcal{E}}$ if and only if the following conditions are satisfied: 1. \[rank inequalities for quotients\] ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{F}}^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$. 2. \[equal rank condition for quotient bundles\] ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{F}}^{\leq \mu}$ whenever equality holds in \[rank inequalities for quotients\]. Moreover, if we align the Harder-Narasimhan polygons ${\mathrm{HN}}({\mathcal{E}})$ and ${\mathrm{HN}}({\mathcal{F}})$ so that their right endpoints lie at the origin, the conditions \[rank inequalities for quotients\] and \[equal rank condition for quotient bundles\] are equivalent to the following conditions: 1. \[dual slopewise dominance for quotients\] For each $i = 1, \cdots, \operatorname{rank}({\mathcal{F}})$, the slope of ${\mathrm{HN}}({\mathcal{F}})$ on the interval $[-i, -i+1]$ is greater than or equal to the slope of ${\mathrm{HN}}({\mathcal{E}})$ on this interval. 2. \[dual slopewise dominance equality condition for quotient bundles\] If both ${\mathrm{HN}}({\mathcal{E}})$ and ${\mathrm{HN}}({\mathcal{F}})$ have vertices at some integer $-j$, then the slope of ${\mathrm{HN}}({\mathcal{F}})$ on $[-j, -j+1]$ is greater than or equal to the slope of ${\mathrm{HN}}({\mathcal{E}})$ on $[-j-1, j]$ unless ${\mathrm{HN}}({\mathcal{E}})$ and ${\mathrm{HN}}({\mathcal{F}})$ agree on $[-j, 0]$. (right) at (0, 0); (q0) at (-1,2); (q1) at (-2.5, 3.4); (q2) at (-6, 4.8); (q3) at (-9, 4); (p0) at (-2, 1.5); (p1) at (-4.5, 2); (p2) at (-6, 1.3); (p3) at (-7, 0.1); (right) – (q0) – (q1) – (q2) – (q3); (right) – (p0) – (p1) – (p2) – (p3); (q0) circle \[radius=0.05\]; (q1) circle \[radius=0.05\]; (q2) circle \[radius=0.05\]; (q3) circle \[radius=0.05\]; (right) circle \[radius=0.05\]; (p0) circle \[radius=0.05\]; (p1) circle \[radius=0.05\]; (p2) circle \[radius=0.05\]; (p3) circle \[radius=0.05\]; (-3, -0.4) – (-3, 5); (-3.5, -0.4) – (-3.5, 5); (-6, -0.4) – (-6, 5); at (-2.8,-0.8) [$-i+1$]{}; at (-3.7,-0.8) [$-i$]{}; at (-6,-0.8) [$-j$]{}; (q3) ++(-0.8, 0.05) node [${\mathrm{HN}}({\mathcal{E}})$]{}; (p3) ++(-0.8, 0.05) node [${\mathrm{HN}}({\mathcal{F}})$]{}; (right) ++(0.3, -0.05) node [$O$]{}; We will discuss our proof of Theorem \[classification of quotient bundles\] in the subsequent sections. In this section we explain some classification results as consequences of Theorem \[classification of quotient bundles\]. Our first corollary of Theorem \[classification of quotient bundles\] dualizes the statement of Theorem \[classification of quotient bundles\] to classify *almost* all subbundles of a given vector bundle on ${\mathcal{X}}$. \[almost classification of subbundles\] Let ${\mathcal{E}}$ be a vector bundle on ${\mathcal{X}}$. Then a vector bundle ${\mathcal{D}}$ on ${\mathcal{X}}$ is (isomorphic to) a subbundle of ${\mathcal{E}}$ if the following conditions are satisfied: 1. \[rank inequalities for subbundles\] ${\mathrm{rk}}({\mathcal{E}}^{\geq \mu}) \geq {\mathrm{rk}}({\mathcal{D}}^{\geq \mu})$ for every $\mu \in {\mathbb{Q}}$. 2. \[equal rank condition for subbundles\] ${\mathcal{E}}^{\geq \mu} \simeq {\mathcal{D}}^{\geq \mu}$ whenever equality holds in \[rank inequalities for subbundles\]. Moreover, if we align ${\mathrm{HN}}({\mathcal{D}})$ and ${\mathrm{HN}}({\mathcal{E}})$ so that their left endpoints lie at the origin, the conditions \[rank inequalities for subbundles\] and \[equal rank condition for subbundles\] are equivalent to the following conditions: 1. \[slopewise dominance for subbundles\] For each $i = 1, \cdots, \operatorname{rank}({\mathcal{D}})$, the slope of ${\mathrm{HN}}({\mathcal{D}})$ on the interval $[i, i+1]$ is less than or equal to the slope of ${\mathrm{HN}}({\mathcal{E}})$ on this interval. 2. \[slopewise dominance equality condition for subbundles\] If both ${\mathrm{HN}}({\mathcal{D}})$ and ${\mathrm{HN}}({\mathcal{E}})$ have vertices at some integer $j$, then the slope of ${\mathrm{HN}}({\mathcal{D}})$ on $[j-1, j]$ is less than or equal to the slope of ${\mathrm{HN}}({\mathcal{E}})$ on $[j, j+1]$ unless ${\mathrm{HN}}({\mathcal{D}})$ and ${\mathrm{HN}}({\mathcal{E}})$ agree on $[0, j]$. (left) at (0, 0); (q0) at (1,2); (q1) at (2.5, 3.4); (q2) at (6, 4.8); (q3) at (9, 4); (p0) at (2, 1.5); (p1) at (4.5, 2); (p2) at (6, 1.3); (p3) at (7, 0.1); (left) – (q0) – (q1) – (q2) – (q3); (left) – (p0) – (p1) – (p2) – (p3); (q0) circle \[radius=0.05\]; (q1) circle \[radius=0.05\]; (q2) circle \[radius=0.05\]; (q3) circle \[radius=0.05\]; (left) circle \[radius=0.05\]; (p0) circle \[radius=0.05\]; (p1) circle \[radius=0.05\]; (p2) circle \[radius=0.05\]; (p3) circle \[radius=0.05\]; (3, -0.4) – (3, 5); (3.5, -0.4) – (3.5, 5); (6, -0.4) – (6, 5); at (2.8,-0.8) [$i-1$]{}; at (3.5,-0.8) [$i$]{}; at (6,-0.8) [$j$]{}; (q3) ++(0.8, 0.05) node [${\mathrm{HN}}({\mathcal{E}})$]{}; (p3) ++(0.8, 0.05) node [${\mathrm{HN}}({\mathcal{D}})$]{}; (right) ++(-0.3, -0.05) node [$O$]{}; For the first part, we wish to show that there exists an injective bundle map ${\mathcal{D}}{\hookrightarrow}{\mathcal{E}}$ if ${\mathcal{D}}$ satisfies the conditions \[rank inequalities for subbundles\] and \[equal rank condition for subbundles\]. By means of dualizing, it suffices to show that there exists a surjective bundle map ${\mathcal{E}}^\vee {\twoheadrightarrow}{\mathcal{D}}^\vee$, or equivalently that ${\mathcal{D}}^\vee$ is a quotient bundle of ${\mathcal{E}}^\vee$. This follows from Theorem \[classification of quotient bundles\] since by Lemma \[rank and degree of dual bundle\] we can rewrite the conditions \[rank inequalities for subbundles\] and \[equal rank condition for subbundles\] as follows: 1. \[rank inequalities for dual quotients\] ${\mathrm{rk}}(({\mathcal{E}}^\vee)^{\leq -\mu}) \geq {\mathrm{rk}}(({\mathcal{D}}^\vee)^{\leq -\mu})$ for every $\mu \in {\mathbb{Q}}$. 2. \[equal rank condition for dual quotient bundles\] $({\mathcal{E}}^\vee)^{\leq -\mu} \simeq ({\mathcal{D}}^\vee)^{\leq -\mu}$ whenever equality holds in \[rank inequalities for dual quotients\]. It remains to verify equivalence between the conditions \[rank inequalities for subbundles, intro\], \[equal rank condition for subbundles, intro\] and the conditions \[slopewise dominance for subbundles\], \[slopewise dominance equality condition for subbundles\]. By reflecting the HN polygons ${\mathrm{HN}}({\mathcal{D}})$ and ${\mathrm{HN}}({\mathcal{E}})$ about the $y$-axis, we obtain the HN polygons ${\mathrm{HN}}({\mathcal{D}}^\vee)$ and ${\mathrm{HN}}({\mathcal{E}}^\vee)$ with their right endpoints at the origin. We thus find that the conditions \[slopewise dominance for subbundles\] and \[slopewise dominance equality condition for subbundles\] are equivalent to the following conditions: 1. \[dual slopewise dominance for dual quotients\] For each $i = 1, \cdots, \operatorname{rank}({\mathcal{D}}^\vee)$, the slope of ${\mathrm{HN}}({\mathcal{D}}^\vee)$ on the interval $[-i, -i+1]$ is greater than or equal to the slope of ${\mathrm{HN}}({\mathcal{E}}^\vee)$ on this interval. 2. \[dual slopewise dominance equality condition for dual quotient bundles\] If both ${\mathrm{HN}}({\mathcal{D}}^\vee)$ and ${\mathrm{HN}}({\mathcal{E}}^\vee)$ have vertices at some integer $-j$, then the slope of ${\mathrm{HN}}({\mathcal{D}}^\vee)$ on $[-j, -j+1]$ is greater than or equal to the slope of ${\mathrm{HN}}({\mathcal{E}}^\vee)$ on $[-j-1, j]$ unless ${\mathrm{HN}}({\mathcal{D}}^\vee)$ and ${\mathrm{HN}}({\mathcal{E}}^\vee)$ agree on $[-j, 0]$. Moreover, by Theorem \[classification of quotient bundles\] these conditions are equivalent to the conditions \[rank inequalities for dual quotients\] and \[equal rank condition for dual quotient bundles\], which are equivalent to the conditions \[rank inequalities for subbundles\] and \[equal rank condition for subbundles\] as already noted. We thus have equivalence between the conditions \[rank inequalities for subbundles, intro\], \[equal rank condition for subbundles, intro\] and the conditions \[slopewise dominance for subbundles\], \[slopewise dominance equality condition for subbundles\] as desired, thereby completing the proof. We remark that Corollary \[almost classification of subbundles\] does not give a complete classification of subbundles since the condition \[equal rank condition for subbundles\] is not necessary. The main underlying issue is that the cokernel of an injective bundle map is not necessarily a vector bundle. On the other hand, it is not hard to see that the condition \[rank inequalities for subbundles\] is necessary (see Proposition \[subbundles necessary condition\]). In fact, we conjecture that the condition \[rank inequalities for subbundles\] alone should give a complete classification of subbundles of ${\mathcal{E}}$. \[conjecture classification of subbundles\] Let ${\mathcal{E}}$ be a vector bundle on ${\mathcal{X}}$. Then a vector bundle ${\mathcal{D}}$ is (isomorphic to) a subbundle of ${\mathcal{E}}$ if and only if ${\mathrm{rk}}({\mathcal{E}}^{\geq \mu}) \geq {\mathrm{rk}}({\mathcal{D}}^{\geq \mu})$ for every $\mu \in {\mathbb{Q}}$. As another consequence of Theorem \[classification of quotient bundles\], we have a complete classification of finitely globally generated vector bundles on ${\mathcal{X}}$. A vector bundle ${\mathcal{E}}$ on ${\mathcal{X}}$ is generated by $n$ global sections if and only if the following conditions are satisfied: 1. \[nonpositivity of slopes for globally generated bundles\] ${\mathrm{HN}}({\mathcal{E}})$ has only nonnegative slopes, i.e., ${\mathcal{E}}^{<0} = 0$. 2. \[rank bound for globally generated bundles\] ${\mathrm{rk}}({\mathcal{E}}) \leq n$ with equality if and only if ${\mathcal{E}}\simeq {{\mathcal{O}}}^{\oplus n}$. A vector bundle ${\mathcal{E}}$ on ${\mathcal{X}}$ is generated by $n$ global sections if and only if there is a surjective bundle map $${{\mathcal{O}}}^{\oplus n} {\twoheadrightarrow}{\mathcal{E}},$$ which amounts to saying that ${\mathcal{E}}$ is a quotient bundle of ${{\mathcal{O}}}^{\oplus n}$. By Theorem \[classification of quotient bundles\], this is equivalent to the following two conditions: 1. \[rank inequalities for globally generated bundles\] ${\mathrm{rk}}(({{\mathcal{O}}}^{\oplus n})^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{E}}^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$. 2. \[equal rank condition for globally generated bundles\] $({{\mathcal{O}}}^{\oplus n})^{\leq \mu} \simeq {\mathcal{E}}^{\leq \mu}$ whenever equality holds in \[rank inequalities for globally generated bundles\]. We aim to prove that ${\mathcal{E}}$ satisfies these conditions if and only if it satisfies the conditions \[nonpositivity of slopes for globally generated bundles\] and \[rank bound for globally generated bundles\]. Note that $$({{\mathcal{O}}}^{\oplus n})^{\leq \mu} = \begin{cases} {{\mathcal{O}}}^{\oplus n} &\text{ if } \mu \geq 0\\ 0 &\text{ if } \mu <0 \end{cases}.$$ Hence the condition \[rank inequalities for globally generated bundles\] is satisfied if and only if ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \leq n$ for all $\mu \geq 0$ and ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) = 0$ for all $\mu<0$. The condition for $\mu \geq 0$ is equivalent to the inequality ${\mathrm{rk}}({\mathcal{E}}) \leq n$ whereas the condition for $\mu<0$ amounts to saying that ${\mathrm{HN}}({\mathcal{E}})$ has no negative slopes. Therefore the condition \[rank inequalities for globally generated bundles\] is equivalent to the condition \[nonpositivity of slopes for globally generated bundles\] together with the inequality ${\mathrm{rk}}({\mathcal{E}})\leq n$. Let us now prove that the conditions \[nonpositivity of slopes for globally generated bundles\] and \[rank bound for globally generated bundles\] together imply the conditions \[rank inequalities for globally generated bundles\] and \[equal rank condition for globally generated bundles\]. By our discussion in the preceding paragraph, it suffices to verify the condition \[equal rank condition for globally generated bundles\] assuming the conditions \[nonpositivity of slopes for globally generated bundles\] and \[rank bound for globally generated bundles\]. Note that the condition \[equal rank condition for globally generated bundles\] is always satisfied when $\mu <0$; in fact, for $\mu<0$ the equality in \[rank inequalities for globally generated bundles\] yields ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) = {\mathrm{rk}}(({{\mathcal{O}}}^{\oplus n})^{\leq \mu}) = 0$ and thereby implying ${\mathcal{E}}^{\leq \mu} = ({{\mathcal{O}}}^{\oplus n})^{\leq \mu} = 0$. Hence we only need to consider the case when $\mu \geq 0$. In this case, the inequality in \[rank inequalities for globally generated bundles\] can be written as ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \leq n$ as noted in the previous paragraph. Now suppose that we have an equality for some $\mu \geq 0$. Using the condition \[rank inequalities for globally generated bundles\] for $\mu = {\mu_\text{max}}({\mathcal{E}})$ we obtain $${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \leq {\mathrm{rk}}({\mathcal{E}}) = {\mathrm{rk}}({\mathcal{E}}^{\leq {\mu_\text{max}}({\mathcal{E}})}) \leq n.$$ Hence the equality ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) = n$ implies ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) = {\mathrm{rk}}({\mathcal{E}}) = n$. We thus find ${\mathcal{E}}^{\leq \mu} = {\mathcal{E}}\simeq {{\mathcal{O}}}^{\oplus n}$ by the condition \[rank bound for globally generated bundles\], thereby verifying the condition \[equal rank condition for globally generated bundles\] as desired. It remains to prove that the conditions \[rank inequalities for globally generated bundles\] and \[equal rank condition for globally generated bundles\] together imply the conditions \[nonpositivity of slopes for globally generated bundles\] and \[rank bound for globally generated bundles\]. By our discussion in the second paragraph, we only need to verify the equality condition in \[rank bound for globally generated bundles\] assuming the conditions \[rank inequalities for globally generated bundles\] and \[equal rank condition for globally generated bundles\]. Now suppose that we have an equality ${\mathrm{rk}}({\mathcal{E}}) = n$ in the condition \[rank bound for globally generated bundles\]. Then we have an equality in condition \[rank inequalities for globally generated bundles\] for $\mu = {\mu_\text{max}}({\mathcal{E}})$. Hence the desired isomorphism ${\mathcal{E}}\simeq {{\mathcal{O}}}^{\oplus n}$ follows from the condition \[equal rank condition for globally generated bundles\] with $\mu = {\mu_\text{max}}({\mathcal{E}})$. Slopewise dominance of vector bundles {#slopewise dominance} ------------------------------------- $ $ The rest of this paper will be devoted to proving Theorem \[classification of quotient bundles\]. In this section, we introduce and study the notion of *slopewise dominance* which will be crucial for our proof. \[def of slopewise dominance\] Let ${\mathcal{V}}$ and ${\mathcal{W}}$ be vector bundles on ${\mathcal{X}}$. Assume that their HN polygons ${\mathrm{HN}}({\mathcal{V}})$ and ${\mathrm{HN}}({\mathcal{W}})$ are aligned as usual so that their left endpoints lie at the origin. We say that ${\mathcal{V}}$ *slopewise dominates* ${\mathcal{W}}$ if for $i = 1, \cdots, {\mathrm{rk}}({\mathcal{W}})$, the slope of ${\mathrm{HN}}({\mathcal{W}})$ on the interval $[i-1, i]$ is less than or equal to the slope of ${\mathrm{HN}}({\mathcal{V}})$ on this interval. (left) at (0, 0); (q0) at (1,2); (q1) at (2.5, 3.4); (q2) at (6, 4.8); (q3) at (9, 4); (p0) at (2, 1.5); (p1) at (4.5, 2); (p2) at (7.5, 0.7); (left) – (q0) – (q1) – (q2) – (q3); (left) – (p0) – (p1) – (p2); (q0) circle \[radius=0.05\]; (q1) circle \[radius=0.05\]; (q2) circle \[radius=0.05\]; (q3) circle \[radius=0.05\]; (left) circle \[radius=0.05\]; (p0) circle \[radius=0.05\]; (p1) circle \[radius=0.05\]; (p2) circle \[radius=0.05\]; (3.5, -0.4) – (3.5, 4.8); (4, -0.4) – (4, 4.8); at (3.4,-0.8) [$i-1$]{}; at (4.1,-0.8) [$i$]{}; (q3) ++(0.8, 0.05) node [${\mathrm{HN}}({\mathcal{V}})$]{}; (p2) ++(0.8, 0.05) node [${\mathrm{HN}}({\mathcal{W}})$]{}; (left) ++(-0.3, -0.05) node [$O$]{}; Slopewise dominance of ${\mathcal{V}}$ on ${\mathcal{W}}$ implies that ${\mathrm{rk}}({\mathcal{V}}) \geq {\mathrm{rk}}({\mathcal{W}})$. The notion of slopewise dominance gives us a characterization of the condition \[rank inequalities for quotients\] in Theorem \[classification of quotient bundles\] (and the condition \[rank inequalities for subbundles\] in Corollary \[almost classification of subbundles\]). \[slopewise dominance and rank inequalities\] Let ${\mathcal{V}}$ and ${\mathcal{W}}$ be vector bundles on ${\mathcal{X}}$. 1. \[subbundle rank inequalities and slopewise dominance\] ${\mathcal{V}}$ slopewise dominates ${\mathcal{W}}$ if and only if ${\mathrm{rk}}({\mathcal{V}}^{\geq \mu}) \geq {\mathrm{rk}}({\mathcal{W}}^{\geq \mu})$ for every $\mu \in {\mathbb{Q}}$. 2. \[quotient rank inequantlies and dual slopewise dominance\] ${\mathcal{V}}^\vee$ slopewise dominates ${\mathcal{W}}^\vee$ if and only if ${\mathrm{rk}}({\mathcal{V}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{W}}^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$. We first note that \[quotient rank inequantlies and dual slopewise dominance\] follows from \[subbundle rank inequalities and slopewise dominance\] as a dual statement. In fact, by Lemma \[rank and degree of dual bundle\] we can rewrite the inequality ${\mathrm{rk}}({\mathcal{V}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{W}}^{\leq \mu})$ as ${\mathrm{rk}}(({\mathcal{V}}^\vee)^{\geq -\mu}) \geq {\mathrm{rk}}(({\mathcal{W}}^\vee)^{\geq -\mu})$. Hence we only need to prove \[subbundle rank inequalities and slopewise dominance\]. We now assume the inequality ${\mathrm{rk}}({\mathcal{V}}^{\geq \mu}) \geq {\mathrm{rk}}({\mathcal{W}}^{\geq \mu})$ for every $\mu \in {\mathbb{Q}}$ and assert that ${\mathcal{V}}$ slopewise dominates ${\mathcal{W}}$. For each $i = 1, \cdots, {\mathrm{rk}}({\mathcal{W}})$, we let $\mu_i$ be the slope of ${\mathrm{HN}}({\mathcal{W}})$ on the interval $[i-1, i]$. If some $\mu_i$ is greater than the slope of ${\mathrm{HN}}({\mathcal{V}})$ on $[i-1, i]$, convexity of HN polygons yields ${\mathrm{rk}}({\mathcal{V}}^{\geq \mu_i}) < i \leq {\mathrm{rk}}({\mathcal{W}}^{\geq \mu_i})$ which contradicts the inequality we assumed. We thus deduce that ${\mathcal{V}}$ slopewise dominates ${\mathcal{W}}$ as desired. Conversely, we claim the inequality ${\mathrm{rk}}({\mathcal{V}}^{\geq \mu}) \geq {\mathrm{rk}}({\mathcal{W}}^{\geq \mu})$ for every $\mu \in {\mathbb{Q}}$ assuming that ${\mathcal{V}}$ slopewise dominates ${\mathcal{W}}$. Suppose for contradiction that ${\mathrm{rk}}({\mathcal{V}}^{\geq \mu}) < {\mathrm{rk}}({\mathcal{W}}^{\geq \mu})$ for some $\mu$. Then for $i = {\mathrm{rk}}({\mathcal{W}}^{\geq \mu})$, the slope of ${\mathrm{HN}}({\mathcal{W}})$ on the interval $[i-1, i]$ is at least $\mu$ whereas the slope of ${\mathrm{HN}}({\mathcal{V}})$ on this interval is less than $\mu$. In particular, the slope of ${\mathrm{HN}}({\mathcal{W}})$ on $[i-1, i]$ is greater than the slope of ${\mathrm{HN}}({\mathcal{V}})$ on this interval, yielding a desired contradiction. By Lemma \[slopewise dominance and rank inequalities\], Conjecture \[conjecture classification of subbundles\] can be stated as follows: subbundles of a given vector bundle ${\mathcal{E}}$ on ${\mathcal{X}}$ are precisely vector bundles that are slopewise dominated by ${\mathcal{E}}$. The notion of slopewise dominance also yields an interesting inequality on degrees which will be useful to us. \[nonnegative degree for slopewise dominant pairs\] Let ${\mathcal{V}}$ and ${\mathcal{W}}$ be vector bundles on ${\mathcal{X}}$ such that ${\mathcal{V}}$ slopewise dominates ${\mathcal{W}}$. We have an inequality $$\deg({\mathcal{V}})^{{\geq 0}}\geq \deg({\mathcal{W}})^{{\geq 0}}.$$ We align ${\mathrm{HN}}({\mathcal{V}})$ and ${\mathrm{HN}}({\mathcal{W}})$ as in Definition \[def of slopewise dominance\] so that their left endpoints lie at the origin. (left) at (0, 0); (q0) at (1,2); (q1) at (2.5, 3.4); (q2) at (6, 4.8); (q3) at (9, 4); (p0) at (2, 1.5); (p1) at (4.5, 2); (p2) at (7.5, 0.7); (aux) at (4.5, 4.2); (left) – (q0) – (q1) – (q2) – (q3); (left) – (p0) – (p1) – (p2); (q0) circle \[radius=0.05\]; (q1) circle \[radius=0.05\]; (q2) circle \[radius=0.05\]; (q3) circle \[radius=0.05\]; (left) circle \[radius=0.05\]; (p0) circle \[radius=0.05\]; (p1) circle \[radius=0.05\]; (p2) circle \[radius=0.05\]; (4.5, -0.4) – (4.5, 4.2); (6, -0.4) – (6, 4.8); at (4.5,-0.8) [${\mathrm{rk}}({\mathcal{W}}^{{\geq 0}})$]{}; at (6,-0.8) [${\mathrm{rk}}({\mathcal{V}}^{{\geq 0}})$]{}; (q3) ++(0.8, 0.05) node [${\mathrm{HN}}({\mathcal{V}})$]{}; (p2) ++(0.8, 0.05) node [${\mathrm{HN}}({\mathcal{W}})$]{}; (left) ++(-0.3, -0.05) node [$O$]{}; (p1) ++(-0.05, 0.25) node [$({\mathrm{rk}}({\mathcal{W}}^{{\geq 0}}), \deg({\mathcal{W}}^{{\geq 0}})$]{}; (q2) ++(0.3, 0.3) node [$({\mathrm{rk}}({\mathcal{V}}^{{\geq 0}}), \deg({\mathcal{V}}^{{\geq 0}})$]{}; (aux) ++(-0.5, 0.25) node [$({\mathrm{rk}}({\mathcal{W}}^{{\geq 0}}), d)$]{}; We denote by $d$ the $y$-value of ${\mathrm{HN}}({\mathcal{V}})$ at ${\mathrm{rk}}({\mathcal{W}}^{{\geq 0}})$. Since ${\mathrm{HN}}({\mathcal{V}})$ lies above ${\mathrm{HN}}({\mathcal{W}})$ by slopewise dominance, we compare the $y$-values of ${\mathrm{HN}}({\mathcal{V}})$ and ${\mathrm{HN}}({\mathcal{W}})$ at ${\mathrm{rk}}({\mathcal{W}}^{{\geq 0}})$ and obtain $$\deg({\mathcal{W}}^{{\geq 0}}) \leq d.$$ On the other hand, we observe that the $y$-value of ${\mathrm{HN}}({\mathcal{V}})$ increases on the interval $[0, {\mathrm{rk}}({\mathcal{V}}^{{\geq 0}})]$. Since ${\mathrm{rk}}({\mathcal{W}}^{{\geq 0}}) \leq {\mathrm{rk}}({\mathcal{V}}^{{\geq 0}})$ by Lemma \[slopewise dominance and rank inequalities\], we compare the $y$-values of ${\mathrm{HN}}({\mathcal{V}})$ at ${\mathrm{rk}}({\mathcal{W}}^{{\geq 0}})$ and ${\mathrm{rk}}({\mathcal{V}}^{{\geq 0}})$ to find $$d \leq \deg({\mathcal{V}}^{{\geq 0}}).$$ We thus combine the two inequalities to obtain the desired inequality. By the same argument we can prove the inequality $\deg({\mathcal{V}})^{\geq \mu} \geq \deg({\mathcal{W}})^{\geq \mu}$ for all $\mu >0$. However, the inequality does not necessarily hold for $\mu <0$. In fact, when $\mu$ is sufficiently small the inequality is equivalent to $\deg({\mathcal{V}}) \geq \deg({\mathcal{W}})$, which doesn’t necessarily hold as shown by ${\mathcal{V}}= {{\mathcal{O}}}(1)^{\oplus 4} \oplus {{\mathcal{O}}}(-1)^{\oplus 4}$ and ${\mathcal{W}}= {{\mathcal{O}}}(1/3)$ in the following figure: (left) at (0, 0); (q0) at (4, 4); (q1) at (8, 0); (p0) at (3, 1); (left) – (q0) – (q1); (left) – (p0); (q0) circle \[radius=0.05\]; (q1) circle \[radius=0.05\]; (left) circle \[radius=0.05\]; (p0) circle \[radius=0.05\]; (q1) ++(0.9, 0.05) node [${\mathrm{HN}}({\mathcal{V}})$]{}; (p0) ++(0.9, 0.05) node [${\mathrm{HN}}({\mathcal{W}})$]{}; (left) ++(-0.3, -0.05) node [$O$]{}; A number of our reduction arguments will use the following decomposition lemma regarding slopewise dominance. \[existence of maximal common factor decomp\] Let ${\mathcal{V}}$ and ${\mathcal{W}}$ be vector bundles on ${\mathcal{X}}$ such that ${\mathcal{V}}$ slopewise dominates ${\mathcal{W}}$. Then we have decompositions $$\label{max common factor decomp} {\mathcal{V}}\simeq {\mathcal{U}}\oplus {\mathcal{V}}' \quad\quad \text{ and } \quad\quad {\mathcal{W}}\simeq {\mathcal{U}}\oplus {\mathcal{W}}'$$ satisfying the following properties: 1. \[slopewise dominance for complement part\] ${\mathcal{V}}'$ slopewise dominates ${\mathcal{W}}'$. 2. \[inequality for max slopes of complement parts\] If ${\mathcal{W}}' \neq 0$, we have ${\mu_\text{max}}({\mathcal{V}}')>{\mu_\text{max}}({\mathcal{W}}')$. 3. \[ineqaulities for min slope of common factor\] If ${\mathcal{U}}\neq 0$ and ${\mathcal{W}}' \neq 0$, we have ${\mu_\text{min}}({\mathcal{U}}) \geq {\mu_\text{max}}({\mathcal{V}}') > {\mu_\text{max}}({\mathcal{W}}')$. Assume that ${\mathrm{HN}}({\mathcal{V}})$ and ${\mathrm{HN}}({\mathcal{W}})$ are aligned as in Definition \[def of slopewise dominance\]. For $0 \leq x \leq {\mathrm{rk}}({\mathcal{W}})$, we define $d(x)$ to be the vertical distance between ${\mathrm{HN}}({\mathcal{V}})$ and ${\mathrm{HN}}({\mathcal{W}})$ at $x$. Note that $d(x)$ is nonnegative and increasing by slopewise dominance of ${\mathcal{V}}$ on ${\mathcal{W}}$. Let us take the maximum $r$ with $d(r) = 0$. The interval $[0, r]$ corresponds to the common part of ${\mathrm{HN}}({\mathcal{V}})$ and ${\mathrm{HN}}({\mathcal{W}})$. Moreover, unless $r = {\mathrm{rk}}({\mathcal{W}})$ the polygon ${\mathrm{HN}}({\mathcal{W}})$ should change its slope at $r$ so that the function $d(x)$ becomes positive after this point. We thus see that $r$ must be an integer. We take ${\mathcal{U}}$ to be the vector bundle on ${\mathcal{X}}$ whose HN polygon is given by the common part of ${\mathrm{HN}}({\mathcal{V}})$ and ${\mathrm{HN}}({\mathcal{W}})$, as illustrated by the red polygon in the figure below. We also take ${\mathcal{V}}'$ and ${\mathcal{W}}'$ to be vector bundles on ${\mathcal{X}}$ whose HN polygons are given by the complement subpolygons of ${\mathrm{HN}}({\mathcal{V}})$ and ${\mathrm{HN}}({\mathcal{W}})$, as illustrated by the blue and green polygons in the figure below. Note that these definitions are valid since $r$ is an integer. (left) at (0, 0); (q0) at (1,2.5); (q1) at (4, 4.5); (q2) at (6.5, 4.5); (q3) at (9, 2.5); (p0) at (q0); (p1) at (2.5, 3.5); (p2) at (5.5, 3); (p3) at (7.5, 0.5); (left) – (p0) – (p1); (p1) – (q1) – (q2) – (q3); (p1) – (p2) – (p3); (q0) circle \[radius=0.05\]; (q1) circle \[radius=0.05\]; (q2) circle \[radius=0.05\]; (q3) circle \[radius=0.05\]; (left) circle \[radius=0.05\]; (p0) circle \[radius=0.05\]; (p1) circle \[radius=0.05\]; (p2) circle \[radius=0.05\]; (p3) circle \[radius=0.05\]; (2.5, -0.4) – (2.5, 3.5); at (2.5,-0.8) [$r$]{}; (q3) ++(0.8, 0.05) node [${\mathrm{HN}}({\mathcal{V}})$]{}; (p3) ++(0.8, 0.05) node [${\mathrm{HN}}({\mathcal{W}})$]{}; (left) ++(-0.3, -0.05) node [$O$]{}; (p0) ++(-0.5, -0.6) node [${\mathcal{U}}$]{}; (q2) ++(1.5, -0.7) node [${\mathcal{V}}'$]{}; (p2) ++(1, -0.6) node [${\mathcal{W}}'$]{}; It remains to check the desired properties for ${\mathcal{U}}$, ${\mathcal{V}}'$ and ${\mathcal{W}}'$. By construction we have decompositions $${\mathcal{V}}\simeq {\mathcal{U}}\oplus {\mathcal{V}}' \quad\quad \text{ and } \quad\quad {\mathcal{W}}\simeq {\mathcal{U}}\oplus {\mathcal{W}}'.$$ Moreover, we obtain slopewise dominance of ${\mathcal{V}}'$ on ${\mathcal{W}}'$ from slopewise dominance of ${\mathcal{V}}$ on ${\mathcal{W}}$. If ${\mathcal{W}}' \neq 0$, we have $r < {\mathrm{rk}}({\mathcal{W}})$ and therefore deduce the strict inequality ${\mu_\text{max}}({\mathcal{V}}')>{\mu_\text{max}}({\mathcal{W}}')$ from the fact that $d(x)$ becomes positive after $r$. If ${\mathcal{U}}\neq 0$ and ${\mathcal{W}}' \neq 0$, we also have ${\mu_\text{min}}({\mathcal{U}}) \geq {\mu_\text{max}}({\mathcal{V}}')$ by convexity of ${\mathrm{HN}}({\mathcal{V}})$, thereby obtaining a combined inequality ${\mu_\text{min}}({\mathcal{U}}) \geq {\mu_\text{max}}({\mathcal{V}}') > {\mu_\text{max}}({\mathcal{W}}')$. We will also need the following duality of slopewise dominance for vector bundles of equal ranks. \[duality of slopewise dominance for equal rank case\] Let ${\mathcal{V}}$ and ${\mathcal{W}}$ be vector bundles on ${\mathcal{X}}$ with ${\mathrm{rk}}({\mathcal{V}}) = {\mathrm{rk}}({\mathcal{W}})$. Then ${\mathcal{V}}$ slopewise dominates ${\mathcal{W}}$ if and only if ${\mathcal{W}}^\vee$ slopewise dominates ${\mathcal{V}}^\vee$. We align the polygons ${\mathrm{HN}}({\mathcal{V}})$ and ${\mathrm{HN}}({\mathcal{W}})$ so that their left points lie at the origin. By reflecting ${\mathrm{HN}}({\mathcal{V}})$ and ${\mathrm{HN}}({\mathcal{W}})$ about the $y$-axis, we obtain the polygons ${\mathrm{HN}}({\mathcal{V}}^\vee)$ and ${\mathrm{HN}}({\mathcal{W}}^\vee)$ with their right points at the origin. (left) at (0, 0); (q0) at (1,2.5); (q1) at (2.5, 4); (q2) at (5, 4.5); (p0) at (2, 1.2); (p1) at (4, 1.2); (p2) at (5, 0.5); (left) – (q0) – (q1) – (q2); (left) – (p0) – (p1) – (p2); (q0) circle \[radius=0.05\]; (q1) circle \[radius=0.05\]; (q2) circle \[radius=0.05\]; (left) circle \[radius=0.05\]; (p0) circle \[radius=0.05\]; (p1) circle \[radius=0.05\]; (p2) circle \[radius=0.05\]; (3, -0.4) – (3, 4.8); (3.5, -0.4) – (3.5, 4.8); (5, -0.4) – (5, 4.8); at (2.9,-0.8) [$i-1$]{}; at (3.6,-0.8) [$i$]{}; at (5,-0.8) [$r$]{}; (q2) ++(0.8, 0.05) node [${\mathrm{HN}}({\mathcal{V}})$]{}; (p2) ++(0.8, 0.05) node [${\mathrm{HN}}({\mathcal{W}})$]{}; (left) ++(0, -0.3) node [$O$]{}; (-q0) at (-1,2.5); (-q1) at (-2.5, 4); (-q2) at (-5, 4.5); (-p0) at (-2, 1.2); (-p1) at (-4, 1.2); (-p2) at (-5, 0.5); (left) – (-q0) – (-q1) – (-q2); (left) – (-p0) – (-p1) – (-p2); (-q0) circle \[radius=0.05\]; (-q1) circle \[radius=0.05\]; (-q2) circle \[radius=0.05\]; (-p0) circle \[radius=0.05\]; (-p1) circle \[radius=0.05\]; (-p2) circle \[radius=0.05\]; (-3, -0.4) – (-3, 4.8); (-3.5, -0.4) – (-3.5, 4.8); (-5, -0.4) – (-5, 4.8); at (-2.8,-0.8) [$-i+1$]{}; at (-3.7,-0.8) [$-i$]{}; at (-5,-0.8) [$-r$]{}; (-q2) ++(-0.8, 0.05) node [${\mathrm{HN}}({\mathcal{V}}^\vee)$]{}; (-p2) ++(-0.8, 0.05) node [${\mathrm{HN}}({\mathcal{W}}^\vee)$]{}; Note that ${\mathcal{V}}, {\mathcal{W}}, {\mathcal{V}}^\vee$ and ${\mathcal{W}}^\vee$ all have equal rank by our assumption ${\mathrm{rk}}({\mathcal{V}}) = {\mathrm{rk}}({\mathcal{W}})$. We let $r$ denote this common rank of ${\mathcal{V}}, {\mathcal{W}}, {\mathcal{V}}^\vee$ and ${\mathcal{W}}^\vee$. With this setup, we can establish our assertion by proving equivalence of the following statements: 1. \[slopewise dominace of W dual on V dual\] ${\mathcal{W}}^\vee$ slopewise dominates ${\mathcal{V}}^\vee$. 2. \[slope comparison of dual polygons\] For each $i = 1, 2, \cdots, r$, the slope of ${\mathrm{HN}}({\mathcal{V}}^\vee)$ on the interval $[-i, -i+1]$ is less than or equal to the slope of ${\mathrm{HN}}({\mathcal{W}}^\vee)$ on this interval. 3. \[slope comparison of origianl polygons\] For each $i = 1, 2, \cdots, r$, the slope of ${\mathrm{HN}}({\mathcal{W}})$ on the interval $[i-1, i]$ is less than or equal to the slope of ${\mathrm{HN}}({\mathcal{V}})$ on this interval. 4. \[slopewise dominance of V on W\] ${\mathcal{V}}$ slopewise dominates ${\mathcal{W}}$. Equivalence between \[slopewise dominace of W dual on V dual\] and \[slope comparison of dual polygons\] is a consequence of the fact that the left points of ${\mathrm{HN}}({\mathcal{V}}^\vee)$ and ${\mathrm{HN}}({\mathcal{W}}^\vee)$ have the same $x$-values of $-r$ in our alignment; in fact, to compare the slopes as per Definition \[def of slopewise dominance\] we only have to align the left points at the same $x$-values. Equivalence between \[slope comparison of dual polygons\] and \[slope comparison of origianl polygons\] is immediate from the symmetry of our alignment. Equivalence between \[slope comparison of origianl polygons\] and \[slopewise dominance of V on W\] is evident by Definition \[def of slopewise dominance\]. Formulation of the key inequality {#reformulation of statement} --------------------------------- $ $ Our primary goal in this section is to reduce the statement of Theorem \[classification of quotient bundles\] to an inequality for which we can apply the results from §\[Geometric interpretation of degrees\]. We begin by establishing equivalence of the two characterizations of quotient bundles in the statement of Theorem \[classification of quotient bundles\]. \[classification of quotient bundles by slopewise dominance\] For arbitrary vector bundles ${\mathcal{E}}$ and ${\mathcal{F}}$ on ${\mathcal{X}}$, the conditions \[rank inequalities for quotients\] and \[equal rank condition for quotient bundles\] in Theorem \[classification of quotient bundles\] are respectively equivalent to the conditions \[dual slopewise dominance for quotients\] and \[dual slopewise dominance equality condition for quotient bundles\] in Theorem \[classification of quotient bundles\]. As in the statement of Theorem \[classification of quotient bundles\], we align the HN polygons ${\mathrm{HN}}({\mathcal{E}})$ and ${\mathrm{HN}}({\mathcal{F}})$ so that their right endpoints lie at the origin. By reflecting the HN polygons ${\mathrm{HN}}({\mathcal{E}})$ and ${\mathrm{HN}}({\mathcal{F}})$ about the $y$-axis, we obtain the HN polygons ${\mathrm{HN}}({\mathcal{E}}^\vee)$ and ${\mathrm{HN}}({\mathcal{F}}^\vee)$ with their left endpoints at the origin. Then we find that the condition \[dual slopewise dominance for quotients\] is equivalent to slopewise dominance of ${\mathcal{E}}^\vee$ on ${\mathcal{F}}^\vee$, which is equivalent to the condition \[rank inequalities for quotients\] by Lemma \[slopewise dominance and rank inequalities\]. We thus have equivalence between the condition \[rank inequalities for quotients\] and the condition \[dual slopewise dominance for quotients\]. Let us now assert that the conditions \[rank inequalities for quotients\] and \[equal rank condition for quotient bundles\] together imply the conditions \[dual slopewise dominance for quotients\] and \[dual slopewise dominance equality condition for quotient bundles\]. By our discussion in the preceding paragraph, we only need to verify the condition \[dual slopewise dominance equality condition for quotient bundles\] assuming the conditions \[rank inequalities for quotients\] and \[equal rank condition for quotient bundles\]. Suppose that both ${\mathrm{HN}}({\mathcal{E}})$ and ${\mathrm{HN}}({\mathcal{F}})$ have vertices at some integer $-j$ such that the slope of ${\mathrm{HN}}({\mathcal{F}})$ on $[-j, -j+1]$ is not greater than or equal to the slope of ${\mathrm{HN}}({\mathcal{E}})$ on $[-j-1, -j]$. Taking $\mu$ to be the slope of ${\mathrm{HN}}({\mathcal{F}})$ on $[-j, -j+1]$, we find $${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \leq j = {\mathrm{rk}}({\mathcal{F}}^{\leq \mu}).$$ Now the conditions \[rank inequalities for quotients\] and \[equal rank condition for quotient bundles\] respectively yields ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) = {\mathrm{rk}}({\mathcal{F}}^{\leq \mu}) = j$ and ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{F}}^{\leq \mu}$, thereby implying that ${\mathrm{HN}}({\mathcal{E}})$ and ${\mathrm{HN}}({\mathcal{F}})$ must agree on $[-j, 0]$. We thus verify the condition \[dual slopewise dominance equality condition for quotient bundles\] as desired. It remains to prove that the conditions \[dual slopewise dominance for quotients\] and \[dual slopewise dominance equality condition for quotient bundles\] together imply the conditions \[rank inequalities for quotients\] and \[equal rank condition for quotient bundles\]. By our discussion in the first paragraph, we only need to verify the condition \[equal rank condition for quotient bundles\] assuming the conditions \[dual slopewise dominance for quotients\] and \[dual slopewise dominance equality condition for quotient bundles\]. Suppose that ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) = {\mathrm{rk}}({\mathcal{F}}^{\leq \mu})$ for some $\mu \in {\mathbb{Q}}$. Taking $j = {\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) = {\mathrm{rk}}({\mathcal{F}}^{\leq \mu})$, we have the following observations: 1. The slopes of ${\mathrm{HN}}({\mathcal{E}})$ and ${\mathrm{HN}}({\mathcal{F}})$ on $[-j, -j+1]$ are less than or equal to $\mu$. 2. The slopes of ${\mathrm{HN}}({\mathcal{E}})$ and ${\mathrm{HN}}({\mathcal{F}})$ on $[-j-1, -j]$ are greater than $\mu$ unless $j = {\mathrm{rk}}({\mathcal{F}})$. Then we consequently find the following facts: 1. Both ${\mathrm{HN}}({\mathcal{E}})$ and ${\mathrm{HN}}({\mathcal{F}})$ have vertices at $-j$. 2. The slope of ${\mathrm{HN}}({\mathcal{F}})$ on $[-j, -j+1]$ cannot be greater than or equal to the slope of ${\mathrm{HN}}({\mathcal{E}})$ on $[-j-1, -j]$. Now the condition \[dual slopewise dominance equality condition for quotient bundles\] implies that ${\mathrm{HN}}({\mathcal{E}})$ and ${\mathrm{HN}}({\mathcal{F}})$ must agree on $[-j, 0]$. Hence we obtain an isomorphism ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{F}}^{\leq \mu}$, thereby verifying the condition \[equal rank condition for quotient bundles\] as desired. As our next step, we verify that the conditions \[rank inequalities for quotients\] and \[equal rank condition for quotient bundles\] in Theorem \[classification of quotient bundles\] are indeed necessary. \[quotient bundles necessary condition\] Given a vector bundle ${\mathcal{E}}$ on ${\mathcal{X}}$, every quotient bundle ${\mathcal{F}}$ of ${\mathcal{E}}$ should satisfy the conditions \[rank inequalities for quotients\] and \[equal rank condition for quotient bundles\] of Theorem \[classification of quotient bundles\]. Let $\mu$ be an arbitrary rational number, and consider the decomposition ${\mathcal{E}}\simeq {\mathcal{E}}^{\leq \mu} \oplus {\mathcal{E}}^{>\mu}$. Since any bundle map from ${\mathcal{E}}^{>\mu}$ to ${\mathcal{F}}^{\leq \mu}$ must be zero by Lemma \[zero hom for dominating slopes\], the composite surjective map ${\mathcal{E}}{\twoheadrightarrow}{\mathcal{F}}{\twoheadrightarrow}{\mathcal{F}}^{\leq \mu}$ should factor through ${\mathcal{E}}^{\leq \mu}$. We thus find ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{F}}^{\leq \mu})$, thereby verifying the condition \[rank inequalities for quotients\]. Let us now assume that ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) = {\mathrm{rk}}({\mathcal{F}}^{\leq \mu})$ for some $\mu \in {\mathbb{Q}}$. Then the kernel of the surjective map ${\mathcal{E}}^{\leq \mu} {\twoheadrightarrow}{\mathcal{F}}^{\leq \mu}$ must be zero since it is a subbundle of ${\mathcal{E}}^{\leq \mu}$ whose rank is equal to ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) - {\mathrm{rk}}({\mathcal{F}}^{\leq \mu}) = 0$. Hence we obtain an isomorphism ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{F}}^{\leq \mu}$, thereby verifying the condition \[equal rank condition for quotient bundles\]. We note that Proposition \[quotient bundles necessary condition\] has the following dual statement: \[subbundles necessary condition\] Given a vector bundle ${\mathcal{E}}$ on ${\mathcal{X}}$, every subbundle ${\mathcal{D}}$ of ${\mathcal{E}}$ should satisfy ${\mathrm{rk}}({\mathcal{E}}^{\geq \mu}) \leq {\mathrm{rk}}({\mathcal{D}}^{\geq \mu})$ for every $\mu \in {\mathbb{Q}}$, or equivalently ${\mathrm{rk}}(({\mathcal{E}}^\vee)^{\leq \mu}) \geq {\mathrm{rk}}(({\mathcal{D}}^\vee)^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$. Let $\mu$ be an arbitrary rational number, and consider the decomposition ${\mathcal{E}}= {\mathcal{E}}^{< \mu} \oplus {\mathcal{E}}^{\geq \mu}$. Since any bundle map from ${\mathcal{D}}^{\geq \mu}$ to ${\mathcal{E}}^{<\mu}$ must be zero by Lemma \[zero hom for dominating slopes\], the composite injective map ${\mathcal{D}}^{\geq \mu} {\hookrightarrow}{\mathcal{D}}{\hookrightarrow}{\mathcal{E}}$ should factor through ${\mathcal{E}}^{\geq \mu}$. We thus obtain the desired inequality ${\mathrm{rk}}({\mathcal{E}}^{\geq \mu}) \leq {\mathrm{rk}}({\mathcal{D}}^{\geq \mu})$. The equivalent inequality for ${\mathcal{D}}^\vee$ and ${\mathcal{E}}^\vee$ then follows from Lemma \[rank and degree of dual bundle\]. By Proposition \[classification of quotient bundles by slopewise dominance\] and Proposition \[quotient bundles necessary condition\], it remains to prove sufficiency of the conditions \[rank inequalities for quotients\] and \[equal rank condition for quotient bundles\] in Theorem \[classification of quotient bundles\]. For this, the notion of slopewise dominance yields the following important reduction: \[reduction on min slopes\] We may assume ${\mu_\text{min}}({\mathcal{F}})>{\mu_\text{min}}({\mathcal{E}})$ to prove sufficiency of the conditions \[rank inequalities for quotients\] and \[equal rank condition for quotient bundles\] in Theorem \[classification of quotient bundles\]. Let ${\mathcal{E}}$ and ${\mathcal{F}}$ be vector bundles on ${\mathcal{X}}$ satisfying the conditions \[rank inequalities for quotients\] and \[equal rank condition for quotient bundles\] in Theorem \[classification of quotient bundles\]. Note that ${\mathcal{E}}^\vee$ slopewise dominates ${\mathcal{F}}^\vee$ by Lemma \[slopewise dominance and rank inequalities\]. Then Lemma \[existence of maximal common factor decomp\] yields decompositions $$\label{max common factor decomps for duals of E and F} {\mathcal{E}}^\vee \simeq {\mathcal{U}}^\vee \oplus {\mathcal{E}}'^\vee \quad\quad \text{ and } \quad\quad {\mathcal{F}}^\vee \simeq {\mathcal{U}}^\vee \oplus {\mathcal{F}}'^\vee$$ satisfying the following properties: 1. \[slopewise dominance for dual complement factors\] ${\mathcal{E}}'^\vee$ slopewise dominates ${\mathcal{F}}'^\vee$. 2. \[inequality for dual complement factors\] If ${\mathcal{F}}'^\vee \neq 0$, we have ${\mu_\text{max}}({\mathcal{E}}'^\vee)>{\mu_\text{max}}({\mathcal{F}}'^\vee)$. 3. \[inequalities for dual common factor\] If ${\mathcal{U}}^\vee \neq 0$ and ${\mathcal{F}}'^\vee \neq 0$, we have ${\mu_\text{min}}({\mathcal{U}}^\vee) \geq {\mu_\text{max}}({\mathcal{E}}'^\vee) > {\mu_\text{max}}({\mathcal{F}}'^\vee)$. By dualizing, we obtain decompositions $$\label{max common factor decomps for E and F} {\mathcal{E}}\simeq {\mathcal{U}}\oplus {\mathcal{E}}' \quad\quad \text{ and } \quad\quad {\mathcal{F}}\simeq {\mathcal{U}}\oplus {\mathcal{F}}'.$$ (left) at (0, 0); (q0) at (-1,2.5); (q1) at (-4, 4.5); (q2) at (-6.5, 4.5); (q3) at (-9, 2.5); (p0) at (q0); (p1) at (-2.5, 3.5); (p2) at (-5.5, 3); (p3) at (-7.5, 0.5); (left) – (p0) – (p1); (p1) – (q1) – (q2) – (q3); (p1) – (p2) – (p3); (q0) circle \[radius=0.05\]; (q1) circle \[radius=0.05\]; (q2) circle \[radius=0.05\]; (q3) circle \[radius=0.05\]; (left) circle \[radius=0.05\]; (p0) circle \[radius=0.05\]; (p1) circle \[radius=0.05\]; (p2) circle \[radius=0.05\]; (p3) circle \[radius=0.05\]; (q3) ++(-0.8, 0.05) node [${\mathrm{HN}}({\mathcal{E}})$]{}; (p3) ++(-0.8, 0.05) node [${\mathrm{HN}}({\mathcal{F}})$]{}; (left) ++(0.3, -0.05) node [$O$]{}; (p0) ++(0.5, -0.6) node [${\mathcal{U}}$]{}; (q2) ++(-1.5, -0.7) node [${\mathcal{E}}'$]{}; (p2) ++(-1, -0.6) node [${\mathcal{F}}'$]{}; We assert that ${\mathrm{rk}}({\mathcal{E}}'^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{F}}'^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$ with equality only if ${\mathcal{E}}'^{\leq \mu} \simeq {\mathcal{F}}'^{\leq \mu}$. In fact, the inequality follows from slopewise dominance of ${\mathcal{E}}'^\vee$ on ${\mathcal{F}}'^\vee$ by Lemma \[slopewise dominance and rank inequalities\], so we only need to check the equality condition. When ${\mathcal{U}}=0$, we have ${\mathcal{E}}\simeq {\mathcal{E}}'$ and ${\mathcal{F}}\simeq {\mathcal{F}}'$ by and thus obtain the equality condition immediately from the condition \[equal rank condition for quotient bundles\] in Theorem \[classification of quotient bundles\] that we assume for ${\mathcal{E}}$ and ${\mathcal{F}}$. In addition, if ${\mathcal{F}}' = 0$ we have ${\mathrm{rk}}({\mathcal{F}}'^{\leq \mu}) = 0$ for every $\mu \in {\mathbb{Q}}$ and therefore find that the equality ${\mathrm{rk}}({\mathcal{E}}'^{\leq \mu}) = {\mathrm{rk}}({\mathcal{F}}'^{\leq \mu})$ holds only if ${\mathcal{E}}'^{\leq \mu} = {\mathcal{F}}'^{\leq \mu} = 0$. Hence it remains to consider the case when ${\mathcal{U}}\neq 0$ and ${\mathcal{F}}' \neq 0$. Now we can rewrite the property \[inequalities for dual common factor\] of the decompositions as $$\label{inequalities for max slope of common factor of E and F} {\mu_\text{max}}({\mathcal{U}}) \leq {\mu_\text{min}}({\mathcal{E}}') < {\mu_\text{min}}({\mathcal{F}}').$$ Moreover, for each $\mu \geq {\mu_\text{max}}({\mathcal{U}})$ the decompositions yield $$\label{slopewise max common factor decomps for E and F} {\mathcal{E}}^{\leq \mu} \simeq {\mathcal{U}}\oplus {\mathcal{E}}'^{\leq \mu} \quad\quad \text{ and } \quad\quad {\mathcal{F}}^{\leq \mu} \simeq {\mathcal{U}}\oplus {\mathcal{F}}'^{\leq \mu}.$$ Let us now assume that an equality ${\mathrm{rk}}({\mathcal{E}}'^{\leq \mu}) = {\mathrm{rk}}({\mathcal{F}}'^{\leq \mu})$ holds for some $\mu \in {\mathbb{Q}}$. We may also assume that $\mu \geq {\mu_\text{max}}({\mathcal{U}})$ since otherwise both ${\mathcal{E}}'^{\leq \mu}$ and ${\mathcal{F}}'^{\leq \mu}$ would be zero by . Then by we have $${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) = {\mathrm{rk}}({\mathcal{U}}) + {\mathrm{rk}}({\mathcal{E}}'^{\leq \mu}) = {\mathrm{rk}}({\mathcal{U}}) + {\mathrm{rk}}({\mathcal{F}}'^{\leq \mu}) = {\mathrm{rk}}({\mathcal{F}}^{\leq \mu}),$$ which yields ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{F}}^{\leq \mu}$ by the condition \[equal rank condition for quotient bundles\] in Theorem \[classification of quotient bundles\]. Hence we obtain the desired condition ${\mathcal{E}}'^{\leq \mu} \simeq {\mathcal{F}}'^{\leq \mu}$ from . Now observe from that a surjective bundle map ${\mathcal{E}}' {\twoheadrightarrow}{\mathcal{F}}'$ gives rise to a surjective bundle map ${\mathcal{E}}{\twoheadrightarrow}{\mathcal{F}}$ by direct summing with the identity map for ${\mathcal{U}}$. Now our discussion in the preceding paragraph implies that we can prove sufficiency of the conditions \[rank inequalities for quotients\] and \[equal rank condition for quotient bundles\] in Theorem \[classification of quotient bundles\] after replacing ${\mathcal{E}}$ and ${\mathcal{F}}$ with ${\mathcal{E}}'$ and ${\mathcal{F}}'$. We may further assume that ${\mathcal{F}}\neq 0$ after this replacement since a zero bundle is clearly a quotient bundle of any vector bundle. Then the replacement gives an additional condition ${\mu_\text{min}}({\mathcal{F}})>{\mu_\text{min}}({\mathcal{E}})$ by the property \[inequality for dual complement factors\] of the decompositions \[max common factor decomps for duals of E and F\], thereby yielding our desired reduction. Under the additional assumption ${\mu_\text{min}}({\mathcal{F}})>{\mu_\text{min}}({\mathcal{E}})$, the equality condition \[equal rank condition for quotient bundles\] in Theorem \[classification of quotient bundles\] is never satisfied when both ${\mathcal{E}}^{\leq \mu}$ and ${\mathcal{F}}^{\leq \mu}$ are nonzero. In fact, for nonzero ${\mathcal{E}}^{\leq \mu}$ and ${\mathcal{F}}^{\leq \mu}$ we have $${\mu_\text{min}}({\mathcal{E}}^{\leq \mu}) = {\mu_\text{min}}({\mathcal{E}}) > {\mu_\text{min}}({\mathcal{F}}) = {\mu_\text{min}}({\mathcal{F}}^{\leq \mu})$$ which implies that the condition ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{F}}^{\leq \mu}$ never holds. In this sense, we can consider our reduction in Lemma \[reduction on min slopes\] as taking care of the equality condition \[equal rank condition for quotient bundles\] in Theorem \[classification of quotient bundles\]. This point of view is also present in the proof where we obtained our reduction by discarding the “equality part" represented by the factor ${\mathcal{U}}$. We now state our key inequality for proving sufficiency of the conditions \[rank inequalities for quotients\] and \[equal rank condition for quotient bundles\] in Theorem \[classification of quotient bundles\]. \[key inequality\] Let ${\mathcal{E}}$, ${\mathcal{F}}$ and ${\mathcal{Q}}$ be vector bundles on ${\mathcal{X}}$ with the following properties: 1. \[slope condition on E and F\] ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{F}}^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$ with equality only when ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{F}}^{\leq \mu}$. 2. \[slope condition on E and Q\] ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{Q}}^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$ with equality only when ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{Q}}^{\leq \mu}$. 3. \[slope condition on F and Q\] ${\mathrm{rk}}({\mathcal{F}}^{\geq \mu}) \geq {\mathrm{rk}}({\mathcal{Q}}^{\geq \mu})$ for every $\mu \in {\mathbb{Q}}$. 4. \[min slope condition on E and F\] ${\mu_\text{min}}({\mathcal{E}}) < {\mu_\text{min}}({\mathcal{F}})$. Then we have an inequality $$\label{deg inequality for surj} \deg({\mathcal{E}}^\vee \otimes {\mathcal{Q}})^{{\geq 0}}+ \deg({\mathcal{Q}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}\leq \deg({\mathcal{E}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}+ \deg({\mathcal{Q}}^\vee \otimes {\mathcal{Q}})^{{\geq 0}}$$ with equality if and only if ${\mathcal{Q}}= {\mathcal{F}}$. \[example showing necessity of min slope condition\] We discuss an example which shows that our reduction in Lemma \[reduction on min slopes\] is crucial for the formulation of Proposition \[key inequality\]. Take ${\mathcal{E}}= {{\mathcal{O}}}^{\oplus 3}, {\mathcal{F}}= {{\mathcal{O}}}^{\oplus 2}$ and ${\mathcal{Q}}= {{\mathcal{O}}}$. Note that our choice does not satisfy the property \[min slope condition on E and F\]. However, we check the other properties \[slope condition on E and F\], \[slope condition on E and Q\] and \[slope condition on F and Q\] by Proposition \[quotient bundles necessary condition\] and Proposition \[subbundles necessary condition\] after observing that ${\mathcal{F}}$ and ${\mathcal{Q}}$ are quotient bundles of ${\mathcal{E}}$ while ${\mathcal{Q}}$ is a subbundle of ${\mathcal{F}}$. We now observe that all terms in are zero, thereby obtaining an equality even though ${\mathcal{Q}}\neq {\mathcal{F}}$. We thus see that the equality condition in Proposition \[key inequality\] can be broken without the assumption ${\mu_\text{min}}({\mathcal{E}}) < {\mu_\text{min}}({\mathcal{F}})$. We will prove Proposition \[key inequality\] in §\[proof of key inequality\]. Here we explain why establishing Proposition \[key inequality\] finishes the proof of Theorem \[classification of quotient bundles\]. Proposition \[key inequality\] implies sufficiency of the conditions \[rank inequalities for quotients\] and \[equal rank condition for quotient bundles\] in Theorem \[classification of quotient bundles\]. Let ${\mathcal{E}}$ and ${\mathcal{F}}$ be vector bundles on ${\mathcal{X}}$ satisfying the conditions \[rank inequalities for quotients\] and \[equal rank condition for quotient bundles\] in Theorem \[classification of quotient bundles\]. We further assume that ${\mu_\text{min}}({\mathcal{E}}) < {\mu_\text{min}}({\mathcal{F}})$ in light of Lemma \[reduction on min slopes\]. We wish to prove existence of a surjective bundle map ${\mathcal{E}}{\twoheadrightarrow}{\mathcal{F}}$ assuming Proposition \[key inequality\]. For this, it suffices to check that ${\mathcal{E}}$ and ${\mathcal{F}}$ satisfy the properties \[existence of nonzero bundle map from E to F\] and \[positive codim for Hom minus surj\] of Proposition \[dimension inequality for surj maps\]. The property \[existence of nonzero bundle map from E to F\] of Proposition \[dimension inequality for surj maps\] is immediate from our assumption ${\mu_\text{min}}({\mathcal{E}}) < {\mu_\text{min}}({\mathcal{F}})$ by Lemma \[zero hom for dominating slopes\]. Hence it remains to check the property \[positive codim for Hom minus surj\] of Proposition \[dimension inequality for surj maps\] for ${\mathcal{E}}$ and ${\mathcal{F}}$. Let ${\mathcal{Q}}$ be an arbitrary subbundle of ${\mathcal{F}}$ which also occurs as a quotient of ${\mathcal{E}}$. Then ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ satisfy the assumptions of Proposition \[key inequality\]; in fact, the properties \[slope condition on E and F\], \[slope condition on E and Q\] and \[slope condition on F and Q\] follow from Proposition \[quotient bundles necessary condition\] and Proposition \[subbundles necessary condition\] whereas the property \[min slope condition on E and F\] follows from our assumption. Since ${\mathcal{Q}}\neq {\mathcal{F}}$, Proposition \[key inequality\] thus yields a strict inequality $$\deg({\mathcal{E}}^\vee \otimes {\mathcal{Q}})^{{\geq 0}}+ \deg({\mathcal{Q}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}< \deg({\mathcal{E}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}+ \deg({\mathcal{Q}}^\vee \otimes {\mathcal{Q}})^{{\geq 0}}.$$ We thus verify the properties \[positive codim for Hom minus surj\] of Proposition \[dimension inequality for surj maps\] for ${\mathcal{E}}$ and ${\mathcal{F}}$, completing the proof. Proof of the key inequality {#proof of key inequality} --------------------------- $ $ We now aim to establish Proposition \[key inequality\]. For our convenience, let us introduce the following notation: \[definition of cEF(Q)\] For arbitrary vector bundles ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ on ${\mathcal{X}}$, we define $$c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}}) := \deg({\mathcal{E}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}+ \deg({\mathcal{Q}}^\vee \otimes {\mathcal{Q}})^{{\geq 0}}- \deg({\mathcal{E}}^\vee \otimes {\mathcal{Q}})^{{\geq 0}}-\deg({\mathcal{Q}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}.$$ Note that the inequality in Proposition \[key inequality\] can be stated as $c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}}) \geq 0$. In light of our discussion in §\[Geometric interpretation of degrees\], we may regard the quantity $c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}})$ as a measurement of the “difference" between the polygons ${\mathrm{HN}}({\mathcal{F}})$ and ${\mathrm{HN}}({\mathcal{Q}})$ when ${\mathcal{E}}$ is fixed. Our proof of Proposition \[key inequality\] will consist of a series of reduction steps as follows: 1. \[reduction to integer slopes\] We reduce the proof to the case when all slopes of ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ are integers. 2. \[reduction to equal ranks\] We further reduce the proof to the case ${\mathrm{rk}}({\mathcal{Q}}) = {\mathrm{rk}}({\mathcal{F}})$. 3. \[reduction to equal slopes\] After these reductions, we complete the proof by gradually “reducing" the slopes of ${\mathcal{F}}$ to the slopes of ${\mathcal{Q}}$. Throughout these reduction steps, we will establish the following key facts: 1. \[decreasing codimension over the entire reduction process\] The quantity $c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}})$ decreases to $0$ as we reduce ${\mathrm{rk}}({\mathcal{F}})$ to ${\mathrm{rk}}({\mathcal{Q}})$ and the slopes of ${\mathcal{F}}$ to the slopes of ${\mathcal{Q}}$. 2. \[equality condition for nonequal rank case\] When ${\mathrm{rk}}({\mathcal{Q}})<{\mathrm{rk}}({\mathcal{F}})$, the equality $c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}}) = 0$ never holds. 3. \[equality condition for equal rank case\] When ${\mathrm{rk}}({\mathcal{Q}}) = {\mathrm{rk}}({\mathcal{F}})$, the equality $c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}}) = 0$ holds only when ${\mathcal{Q}}= {\mathcal{F}}$. We will then obtain the desired inequality $c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}}) \geq 0$ from the first fact and the equality condition ${\mathcal{Q}}= {\mathcal{F}}$ follow from the second and the third fact. For curious readers, we provide some intuitions behind the key facts \[decreasing codimension over the entire reduction process\], \[equality condition for nonequal rank case\] and \[equality condition for equal rank case\] above and briefly describe how each property of ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ in Proposition \[key inequality\] will be used to establish these facts. The fact \[decreasing codimension over the entire reduction process\] relies on the inequalities ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{F}}^{\leq \mu})$, ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{Q}}^{\leq \mu})$ and ${\mathrm{rk}}({\mathcal{F}}^{\geq \mu}) \geq {\mathrm{rk}}({\mathcal{Q}}^{\geq \mu})$ for every $\mu \in {\mathbb{Q}}$. Note that these inequalities can be interpreted in terms of slopewise dominance by Lemma \[slopewise dominance relations for key inequality\]. Intuitively, these slopewise dominance relations enable us to “gradually reduce" ${\mathrm{HN}}({\mathcal{F}})$ to ${\mathrm{HN}}({\mathcal{Q}})$ in a way that the “difference" $c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}})$ of the two polygons always decreases. The fact \[equality condition for nonequal rank case\] is essentially a consequence of the assumption ${\mu_\text{min}}({\mathcal{E}}) < {\mu_\text{max}}({\mathcal{F}})$ that we added in light of Lemma \[reduction on min slopes\]. The key point is that, as we will see in the proof of Proposition \[key inequality reduction for equal rank\], this assumption prevents us from reaching to the condition ${\mathcal{Q}}= {\mathcal{F}}$ by cutting down ${\mathcal{F}}$. The fact \[equality condition for equal rank case\] comes from the equality conditions ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{F}}^{\leq \mu}$ and ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{Q}}^{\leq \mu}$ for the inequalities ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{F}}^{\leq \mu})$ and ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{Q}}^{\leq \mu})$. As we will see in Lemma \[decreasing c\_EF(Q) after max reduction\], these equality conditions ensure that $c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}})$ strictly decreases after the first reduction cycle in Step 3. Let us now make some preparations before proceeding to our reduction steps. We will frequently interpret the assumptions of Proposition \[key inequality\] in terms of slopewise dominance, as stated in the following lemma: \[slopewise dominance relations for key inequality\] Let ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ be as in the statement of Proposition \[key inequality\]. Then we have the following slopewise dominance relations: 1. ${\mathcal{E}}^\vee$ slopewise dominates ${\mathcal{F}}^\vee$. 2. ${\mathcal{E}}^\vee$ slopewise dominates ${\mathcal{Q}}^\vee$. 3. ${\mathcal{F}}$ slopewise dominates ${\mathcal{Q}}$. By Lemma \[slopewise dominance and rank inequalities\], each statement is equivalent to the corresponding inequality in the assumptions of Proposition \[key inequality\]. We also note that the assumptions of Proposition \[key inequality\] are invariant under certain transformations. \[assumptions of key inequality after vertical stretch\] Let ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ be as in the statement of Proposition \[key inequality\]. Choose a positive integer $C$, and let ${\tilde}{{\mathcal{E}}}, {\tilde}{{\mathcal{F}}}$ and ${\tilde}{{\mathcal{Q}}}$ be vector bundles on ${\mathcal{X}}$ whose HN polygons are obtained by vertically stretching ${\mathrm{HN}}({\mathcal{E}}), {\mathrm{HN}}({\mathcal{F}})$ and ${\mathrm{HN}}({\mathcal{Q}})$ by a factor $C$. Then we have the following properties of ${\tilde}{{\mathcal{E}}}, {\tilde}{{\mathcal{F}}}$ and ${\tilde}{{\mathcal{Q}}}$. 1. \[slope condition on E and F after vertical stretch\] ${\mathrm{rk}}({\tilde}{{\mathcal{E}}}^{\leq \mu}) \geq {\mathrm{rk}}({\tilde}{{\mathcal{F}}}^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$ with equality only when ${\tilde}{{\mathcal{E}}}^{\leq \mu} \simeq {\tilde}{{\mathcal{F}}}^{\leq \mu}$. 2. \[slope condition on E and Q after vertical stretch\] ${\mathrm{rk}}({\tilde}{{\mathcal{E}}}^{\leq \mu}) \geq {\mathrm{rk}}({\tilde}{{\mathcal{Q}}}^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$ with equality only when ${\tilde}{{\mathcal{E}}}^{\leq \mu} \simeq {\tilde}{{\mathcal{Q}}}^{\leq \mu}$. 3. \[slope condition on F and Q after vertical stretch\] ${\mathrm{rk}}({\tilde}{{\mathcal{F}}}^{\geq \mu}) \geq {\mathrm{rk}}({\tilde}{{\mathcal{Q}}}^{\geq \mu})$ for every $\mu \in {\mathbb{Q}}$. 4. \[min slope condition on E and F after vertical stretch\] ${\mu_\text{min}}({\tilde}{{\mathcal{E}}}) < {\mu_\text{min}}({\tilde}{{\mathcal{F}}})$. By construction, we have the following facts: 1. \[vertical stretch slopewise rank\] For ${\mathcal{V}}= {\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$, we have ${\mathrm{rk}}({\tilde}{{\mathcal{V}}}^{\leq \mu}) = {\mathrm{rk}}({\mathcal{V}}^{\leq \mu/C})$ and ${\mathrm{rk}}({\tilde}{{\mathcal{V}}}^{\geq \mu}) = {\mathrm{rk}}({\mathcal{V}}^{\geq \mu/C})$ for every $\mu \in {\mathbb{Q}}$. 2. \[vertical stretch equal rank condition\] For ${\mathcal{W}}= {\mathcal{F}}$ and ${\mathcal{Q}}$, we have ${\tilde}{{\mathcal{E}}}^{\leq \mu} \simeq {\tilde}{{\mathcal{W}}^{\leq \mu}}$ if ${\mathcal{E}}^{\leq \mu/C} \simeq {\mathcal{W}}^{\leq \mu/C}$. 3. \[vertical stretch min slope condition\] ${\mu_\text{min}}({\tilde}{{\mathcal{E}}}) = C \cdot {\mu_\text{min}}({\mathcal{E}})$ and ${\mu_\text{min}}({\tilde}{{\mathcal{F}}}) = C \cdot {\mu_\text{min}}({\mathcal{F}})$ Hence we deduce the properties \[slope condition on E and F after vertical stretch\] - \[min slope condition on E and F after vertical stretch\] from the corresponding properties of ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$. \[assumptions of key inequality after shear\] Let ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ be as in the statement of Proposition \[key inequality\]. For any integer $\lambda$, the vector bundles ${\mathcal{E}}(-\lambda), {\mathcal{F}}(-\lambda)$ and ${\mathcal{Q}}(-\lambda)$ satisfy the following properties: 1. \[slope condition on E and F after shear\] ${\mathrm{rk}}({\mathcal{E}}(-\lambda)^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{F}}(-\lambda)^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$ with equality only when ${\mathcal{E}}(-\lambda)^{\leq \mu} \simeq {\mathcal{F}}(-\lambda)^{\leq \mu}$. 2. \[slope condition on E and Q after shear\] ${\mathrm{rk}}({\mathcal{E}}(-\lambda)^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{Q}}(-\lambda)^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$ with equality only when ${\mathcal{E}}(-\lambda)^{\leq \mu} \simeq {\mathcal{Q}}(-\lambda)^{\leq \mu}$. 3. \[slope condition on F and Q after vertical stretch\] ${\mathrm{rk}}({\mathcal{F}}(-\lambda)^{\geq \mu}) \geq {\mathrm{rk}}({\mathcal{Q}}(-\lambda)^{\geq \mu})$ for every $\mu \in {\mathbb{Q}}$. 4. \[min slope condition on E and F after shear\] ${\mu_\text{min}}({\mathcal{E}}(-\lambda)) < {\mu_\text{min}}({\mathcal{F}}(-\lambda))$. Since the vector bundle ${{\mathcal{O}}}(-\lambda)$ has rank $1$ and degree $-\lambda$, tensoring a vector bundle with ${{\mathcal{O}}}(-\lambda)$ is the same as reducing all slopes by $\lambda$. Therefore we have the following observations: 1. \[shear slopewise rank\] For ${\mathcal{V}}= {\mathcal{E}}, {\mathcal{F}}$ or ${\mathcal{Q}}$, we have ${\mathrm{rk}}({\mathcal{V}}(-\lambda)^{\leq \mu}) = {\mathrm{rk}}({\mathcal{V}}^{\leq \mu+\lambda})$ and ${\mathrm{rk}}({\mathcal{V}}(-\lambda)^{\geq \mu}) = {\mathrm{rk}}({\mathcal{V}}^{\geq \mu+\lambda})$ for every $\mu \in {\mathbb{Q}}$. 2. \[shear equal rank condition\] For ${\mathcal{W}}= {\mathcal{F}}$ and ${\mathcal{Q}}$, we have ${\mathcal{E}}(-\lambda)^{\leq \mu} \simeq {\mathcal{W}}(-\lambda)^{\leq \mu}$ if ${\mathcal{E}}^{\leq \mu+\lambda} \simeq {\mathcal{W}}^{\leq \mu+\lambda}$. 3. \[shear min slope condition\] ${\mu_\text{min}}({\mathcal{E}}(-\lambda)) = {\mu_\text{min}}({\mathcal{E}})-\lambda$ and ${\mu_\text{min}}({\mathcal{F}}(-\lambda)) = {\mu_\text{min}}({\mathcal{F}})-\lambda$. We thus deduce the properties \[slope condition on E and F after shear\] - \[min slope condition on E and F after vertical stretch\] from the corresponding properties of ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$. We are now ready to carry out Step 1 and Step 2. \[key inequality reduction to integer slopes\] To prove Proposition \[key inequality\], we may assume that all slopes of ${\mathcal{E}}$, ${\mathcal{F}}$ and ${\mathcal{Q}}$ are integers. Let ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ be as in the statement of Proposition \[key inequality\]. Take $C$ to be the least common multiple of all denominators of the slopes of ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$, and let ${\tilde}{{\mathcal{E}}}, {\tilde}{{\mathcal{F}}}$ and ${\tilde}{{\mathcal{Q}}}$ be vector bundles on ${\mathcal{X}}$ whose HN polygons are obtained by vertically stretching ${\mathrm{HN}}({\mathcal{E}}), {\mathrm{HN}}({\mathcal{F}})$ and ${\mathrm{HN}}({\mathcal{Q}})$ by a factor $C$. Note that all slopes of ${\tilde}{{\mathcal{E}}}, {\tilde}{{\mathcal{F}}}$ and ${\tilde}{{\mathcal{Q}}}$ are integers by construction. We now use Lemma \[degree after stretch\] to obtain an identity $$c_{{\tilde}{{\mathcal{E}}}, {\tilde}{{\mathcal{F}}}}({\tilde}{{\mathcal{Q}}}) = C \cdot c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}})$$ which implies that the inequality for ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ follows from the corresponding inequality for ${\tilde}{{\mathcal{E}}}, {\tilde}{{\mathcal{F}}}$ and ${\tilde}{{\mathcal{Q}}}$. In addition, our construction translates the equality condition ${\mathcal{Q}}= {\mathcal{F}}$ for the former inequality to the equality condition ${\tilde}{{\mathcal{Q}}} = {\tilde}{{\mathcal{F}}}$ for the latter inequality. Now Lemma \[assumptions of key inequality after vertical stretch\] implies that we may prove Proposition \[key inequality\] after replacing ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ by ${\tilde}{{\mathcal{E}}}, {\tilde}{{\mathcal{F}}}$ and ${\tilde}{{\mathcal{Q}}}$, thereby yielding our desired reduction. \[key inequality reduction for equal rank\] It suffices to prove Proposition \[key inequality\] under the additional assumptions that ${\mathrm{rk}}({\mathcal{Q}}) = {\mathrm{rk}}({\mathcal{F}})$ and that all slopes of ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ are integers. Suppose that Proposition \[key inequality\] holds in the special case where the additional assumptions are satisfied. We assert that the general case of Proposition \[key inequality\] follows from this special case by induction on ${\mathrm{rk}}({\mathcal{F}}) - {\mathrm{rk}}({\mathcal{Q}})$. We assume that all slopes of ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ are integers in light of Proposition \[key inequality reduction to integer slopes\]. We first reduce our induction step to the case ${\mu_\text{min}}({\mathcal{F}}) = 0$. For this, we take $\lambda = {\mu_\text{min}}({\mathcal{F}})$ and consider the vector bundles ${\mathcal{E}}(-\lambda), {\mathcal{F}}(-\lambda)$ and ${\mathcal{Q}}(-\lambda)$. By construction we have ${\mu_\text{min}}({\mathcal{F}}(-\lambda)) = {\mu_\text{min}}({\mathcal{F}}) - \lambda = 0$. Moreover, our assumption implies that $\lambda$ is an integer, and consequently that all slopes of ${\mathcal{E}}(-\lambda), {\mathcal{F}}(-\lambda)$ and ${\mathcal{Q}}(-\lambda)$ are integers as well. We now apply Lemma \[degree after shear\] to get an identity $$c_{{\mathcal{E}}(-\lambda), {\mathcal{F}}(-\lambda)}({\mathcal{Q}}(-\lambda)) = c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}})$$ which implies that the inequality for ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ is equivalent to the corresponding inequality for ${\mathcal{E}}(-\lambda), {\mathcal{F}}(-\lambda)$ and ${\mathcal{Q}}(-\lambda)$. In addition, we translate the equality condition ${\mathcal{Q}}= {\mathcal{F}}$ for the former inequality to the equality condition ${\mathcal{Q}}(-\lambda) = {\mathcal{F}}(-\lambda)$ for the latter inequality. We also have ${\mathrm{rk}}({\mathcal{F}}) - {\mathrm{rk}}({\mathcal{Q}}) = {\mathrm{rk}}({\mathcal{F}}(-\lambda)) - {\mathrm{rk}}({\mathcal{Q}}(-\lambda))$ as tensoring with ${{\mathcal{O}}}(-\lambda)$ does not change ranks. Now Lemma \[assumptions of key inequality after shear\] implies that we may proceed to the induction step after replacing ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ by ${\mathcal{E}}(-\lambda), {\mathcal{F}}(-\lambda)$ and ${\mathcal{Q}}(-\lambda)$, thereby yielding our desired reduction. Let us now assume that ${\mu_\text{min}}({\mathcal{F}}) = 0$. For our induction step we assume ${\mathrm{rk}}({\mathcal{F}}) - {\mathrm{rk}}({\mathcal{Q}}) >0$, or equivalently ${\mathrm{rk}}({\mathcal{F}}) > {\mathrm{rk}}({\mathcal{Q}})$. Then we can write ${\mathcal{F}}= {\ddot}{{\mathcal{F}}} \oplus {{\mathcal{O}}}$ where ${\mu_\text{min}}({\ddot}{{\mathcal{F}}}) \geq 0$ and ${\mathrm{rk}}({\ddot}{{\mathcal{F}}}) \geq {\mathrm{rk}}({\mathcal{Q}})$. (left) at (0, 0); (q0) at (1,2); (q1) at (2, 3); (q2) at (3.5, 3.5); (q3) at (5, 3.5); (p0) at (1.5, 1); (p1) at (3, 1.3); (p2) at (4, 0.7); (left) – (q0) – (q1) – (q2) – (q3); (left) – (p0) – (p1) – (p2); (q0) circle \[radius=0.05\]; (q1) circle \[radius=0.05\]; (q2) circle \[radius=0.05\]; (q3) circle \[radius=0.05\]; (left) circle \[radius=0.05\]; (p0) circle \[radius=0.05\]; (p1) circle \[radius=0.05\]; (p2) circle \[radius=0.05\]; (4.5, -0.4) – (4.5, 3.6); at (4.4,-0.8) [${\mathrm{rk}}({\mathcal{F}})-1$]{}; (q3) ++(0.2, 0.3) node [${\mathrm{HN}}({\mathcal{F}})$]{}; (p2) ++(-0.2, -0.3) node [${\mathrm{HN}}({\mathcal{Q}})$]{}; (left) ++(-0.3, -0.05) node [$O$]{}; (0, ) – (1.5,); (0,0) circle \[radius=0.00\]; (left) at (0, 0); (q0) at (1,2); (q1) at (2, 3); (q2) at (3.5, 3.5); (q3) at (4.5, 3.5); (p0) at (1.5, 1); (p1) at (3, 1.3); (p2) at (4, 0.7); (left) – (q0) – (q1) – (q2) – (q3); (left) – (p0) – (p1) – (p2); (q0) circle \[radius=0.05\]; (q1) circle \[radius=0.05\]; (q2) circle \[radius=0.05\]; (q3) circle \[radius=0.05\]; (left) circle \[radius=0.05\]; (p0) circle \[radius=0.05\]; (p1) circle \[radius=0.05\]; (p2) circle \[radius=0.05\]; (4.5, -0.4) – (4.5, 3.6); (q3) – (5, 3.5); at (4.4,-0.8) [${\mathrm{rk}}({\mathcal{F}})-1$]{}; (q3) ++(0.2, 0.3) node [${\mathrm{HN}}({\ddot}{{\mathcal{F}}})$]{}; (p2) ++(-0.2, -0.3) node [${\mathrm{HN}}({\mathcal{Q}})$]{}; (left) ++(-0.3, -0.05) node [$O$]{}; We assert that the assumptions we have on ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ yield the corresponding conditions on ${\mathcal{E}}, {\ddot}{{\mathcal{F}}}$ and ${\mathcal{Q}}$. Since ${\mathcal{E}}$ and ${\mathcal{Q}}$ remain unchanged, we only need to check the following properties: 1. \[integer slopes for induction step\] the slopes of ${\ddot}{{\mathcal{F}}}$ are integers. 2. \[slope condition on E and F for induction step\] ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\ddot}{{\mathcal{F}}}^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$ with equality only when ${\mathcal{E}}^{\leq \mu} \simeq {\ddot}{{\mathcal{F}}}^{\leq \mu}$. 3. \[slope condition on F and Q for induction step\] ${\mathrm{rk}}({\ddot}{{\mathcal{F}}}^{\geq \mu}) \geq {\mathrm{rk}}({\mathcal{Q}}^{\geq \mu})$ for every $\mu \in {\mathbb{Q}}$. 4. \[min slopes condition for induction step\] ${\mu_\text{min}}({\mathcal{E}})<{\mu_\text{min}}({\ddot}{{\mathcal{F}}})$. The properties \[integer slopes for induction step\] and \[min slopes condition for induction step\] are immediate from our construction. The inequality ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\ddot}{{\mathcal{F}}}^{\leq \mu})$ in \[slope condition on E and F for induction step\] follows from the corresponding inequality ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{F}}^{\leq \mu})$ after observing that $${\mathrm{rk}}({\ddot}{{\mathcal{F}}}^{\leq \mu}) = \begin{cases} {\mathrm{rk}}({\mathcal{F}}^{\leq \mu}) - 1 &\text { if } \mu \geq 0\\ {\mathrm{rk}}({\mathcal{F}}^{\leq \mu}) & \text{ if } \mu < 0\end{cases}.$$ This observation further shows that equality in ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\ddot}{{\mathcal{F}}}^{\leq \mu})$ never holds for $\mu \geq 0$. Moreover, we have ${\ddot}{{\mathcal{F}}}^{\leq \mu} = 0$ for $\mu <0$ by the fact ${\mu_\text{min}}({\ddot}{{\mathcal{F}}}) \geq 0$, thereby deducing that equality in ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\ddot}{{\mathcal{F}}}^{\leq \mu})$ can hold only if ${\mathcal{E}}^{\leq \mu} = {\ddot}{{\mathcal{F}}}^{\leq \mu} = 0$. The remaining property \[slope condition on F and Q for induction step\] is equivalent to slopewise dominance of ${\ddot}{{\mathcal{F}}}$ on ${\mathcal{Q}}$ by Lemma \[slopewise dominance and rank inequalities\], so it follows from the following observations: 1. \[slopewise dominance of F on Q for induction step\] ${\mathcal{F}}$ slopewise dominates ${\mathcal{Q}}$ by Lemma \[slopewise dominance relations for key inequality\]. 2. \[relation between HN polygons for induction step\] ${\mathrm{HN}}({\ddot}{{\mathcal{F}}})$ is obtained from ${\mathrm{HN}}({\mathcal{F}})$ by removing the line segment over the interval $({\mathrm{rk}}({\mathcal{F}})-1, {\mathrm{rk}}({\mathcal{F}})]$. 3. Since ${\mathrm{rk}}({\mathcal{F}})>{\mathrm{rk}}({\mathcal{Q}})$, the removal process in \[relation between HN polygons for induction step\] does not affect slopewise dominance. Now since ${\mathrm{rk}}({\ddot}{{\mathcal{F}}}) - {\mathrm{rk}}({\mathcal{Q}}) < {\mathrm{rk}}({\mathcal{F}}) - {\mathrm{rk}}({\mathcal{Q}})$, our induction hypothesis yields $$\label{step 1 codim inequality for E, F', Q} c_{{\mathcal{E}}, {\ddot}{{\mathcal{F}}}}({\mathcal{Q}}) \geq 0$$ with equality if and only if ${\mathcal{Q}}\simeq {\ddot}{{\mathcal{F}}}$. For the desired inequality $c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}}) \geq 0$ we compute $$\begin{aligned} \deg({\mathcal{E}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}&= \deg({\mathcal{E}}^\vee \otimes ({\ddot}{{\mathcal{F}}} \oplus {{\mathcal{O}}}))^{{\geq 0}}\\ &= \deg({\mathcal{E}}^\vee \otimes {\ddot}{{\mathcal{F}}})^{{\geq 0}}+ \deg({\mathcal{E}}^\vee \otimes {{\mathcal{O}}})^{{\geq 0}}\\ &= \deg({\mathcal{E}}^\vee \otimes {\ddot}{{\mathcal{F}}})^{{\geq 0}}+ \deg({\mathcal{E}}^\vee)^{{\geq 0}},\\ \deg({\mathcal{Q}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}&= \deg({\mathcal{Q}}^\vee \otimes ({\ddot}{{\mathcal{F}}} \oplus {{\mathcal{O}}}))^{{\geq 0}}\\ &= \deg({\mathcal{Q}}^\vee \otimes {\ddot}{{\mathcal{F}}})^{{\geq 0}}+ \deg({\mathcal{Q}}^\vee \otimes {{\mathcal{O}}})^{{\geq 0}}\\ &= \deg({\mathcal{Q}}^\vee \otimes {\ddot}{{\mathcal{F}}})^{{\geq 0}}+ \deg({\mathcal{Q}}^\vee)^{{\geq 0}}.\end{aligned}$$ Then we have $$c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}}) = c_{{\mathcal{E}}, {\ddot}{{\mathcal{F}}}}({\mathcal{Q}}) - \deg({\mathcal{E}}^\vee)^{{\geq 0}}+ \deg({\mathcal{Q}}^\vee)^{{\geq 0}}.$$ Since ${\mathcal{E}}^\vee$ slopewise dominates ${\mathcal{Q}}^\vee$ as noted in Lemma \[slopewise dominance relations for key inequality\], we use Lemma \[nonnegative degree for slopewise dominant pairs\] to find $$\label{step 1 inequality for codim E, F, Q and codim E, F', Q} c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}}) \geq c_{{\mathcal{E}}, {\ddot}{{\mathcal{F}}}}({\mathcal{Q}})$$ with equality if and only if $\deg({\mathcal{E}}^\vee)^{{\geq 0}}= \deg({\mathcal{Q}}^\vee)^{{\geq 0}}$ or equivalently $\deg({\mathcal{E}})^{{\leq 0}}= \deg({\mathcal{Q}})^{{\leq 0}}$. Combining and we obtain the desired inequality $$\label{step 1 codim inequality for E, F, Q} c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}}) \geq 0.$$ It remains to check the equality condition for . From the equality conditions for and we get ${\mathcal{Q}}\simeq {\ddot}{{\mathcal{F}}}$ and $\deg({\mathcal{E}})^{{\leq 0}}= \deg({\mathcal{Q}})^{{\leq 0}}$. Since ${\mu_\text{min}}({\ddot}{{\mathcal{F}}}) \geq 0$ by construction, the condition ${\mathcal{Q}}\simeq {\ddot}{{\mathcal{F}}}$ implies that $\deg({\mathcal{Q}})^{{\leq 0}}= 0$. Hence we must have $\deg({\mathcal{E}})^{{\leq 0}}= 0$, which implies that ${\mathcal{E}}^{< 0} = 0$. We thus find ${\mu_\text{min}}({\mathcal{E}})>0 = {\mu_\text{min}}({\mathcal{F}})$, yielding a contradiction to our assumption. Therefore we conclude that the equality for never holds when ${\mathrm{rk}}({\mathcal{F}}) > {\mathrm{rk}}({\mathcal{Q}})$. We can get the same reduction by extending ${\mathcal{Q}}$ instead of cutting down ${\mathcal{F}}$ as we did in the proof. In some sense, it may be more natural to change ${\mathcal{Q}}$ than to change ${\mathcal{F}}$ since ${\mathcal{Q}}$ is introduced as an auxiliary vector bundle for the proof of Theorem \[classification of quotient bundles\]. However, an argument extending ${\mathcal{Q}}$ requires some additional work for a couple of reasons. First, establishing slopewise dominance of ${\mathcal{Q}}^\vee$ on ${\mathcal{E}}$ after extending ${\mathcal{Q}}$ needs some extra care whereas in our proof slopewise dominance of ${\mathcal{E}}$ on ${\mathcal{F}}$ after cutting down ${\mathcal{F}}$ was immediate. Second, extending ${\mathcal{Q}}$ requires to study three terms in $c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}})$, namely $\deg({\mathcal{Q}}^\vee \otimes {\mathcal{Q}})^{{\geq 0}}$, $\deg({\mathcal{E}}^\vee \otimes {\mathcal{Q}})^{{\geq 0}}$ and $\deg({\mathcal{Q}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}$, while in our proof we only had to study $\deg({\mathcal{E}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}$ and $\deg({\mathcal{Q}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}$. We now proceed to the final reduction step. Here we aim to reduce the slopes of ${\mathcal{F}}$ to the slopes of ${\mathcal{Q}}$ in a certain way that the quantity $c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}})$ can only decrease throughout the procedure. For a precise description of our procedure, we introduce the following construction: \[max slope reduction definition\] Let ${\mathcal{V}}$ and ${\mathcal{W}}$ be nonzero vector bundles on ${\mathcal{X}}$ with integer slopes such that ${\mathcal{V}}$ slopewise dominates ${\mathcal{W}}$. Let ${\overline}{{\mathcal{V}}}$ be the vector bundle on ${\mathcal{X}}$ obtained from ${\mathcal{V}}$ by reducing all slopes of ${\mathcal{V}}^{>{\mu_\text{max}}({\mathcal{W}})}$ to ${\mu_\text{max}}({\mathcal{W}})$. More precisely, we set $${\overline}{{\mathcal{V}}} := {{\mathcal{O}}}({\mu_\text{max}}({\mathcal{W}}))^{\oplus {\mathrm{rk}}({\mathcal{V}}^{>{\mu_\text{max}}({\mathcal{W}})})} \oplus {\mathcal{V}}^{\leq {\mu_\text{max}}({\mathcal{W}})}.$$ We say that ${\overline}{{\mathcal{V}}}$ is the *maximal slope reduction* of ${\mathcal{V}}$ to ${\mathcal{W}}$. (left) at (0, 0); (q0) at (1,2); (q1) at (2, 3); (q2) at (3.5, 3.5); (q3) at (5, 2.8); (p0) at (1.5, 1.2); (p1) at (3, 1); (p2) at (4, 0.2); (left) – (q0) – (q1); (q1) – (q2) – (q3); (left) – (p0) – (p1) – (p2); (q0) circle \[radius=0.05\]; (q1) circle \[radius=0.05\]; (q2) circle \[radius=0.05\]; (q3) circle \[radius=0.05\]; (left) circle \[radius=0.05\]; (p0) circle \[radius=0.05\]; (p1) circle \[radius=0.05\]; (p2) circle \[radius=0.05\]; (q3) ++(0.2, -0.4) node [${\mathrm{HN}}({\mathcal{V}})$]{}; (p2) ++(-0.2, -0.4) node [${\mathrm{HN}}({\mathcal{W}})$]{}; (left) ++(-0.3, -0.05) node [$O$]{}; (q0) ++(-0.5, 0.6) node [${\mathcal{V}}^{>{\mu_\text{max}}({\mathcal{W}})}$]{}; (q2) ++(0, 0.3) node [${\mathcal{V}}^{\leq {\mu_\text{max}}({\mathcal{W}})}$]{}; (0, ) – (1.5,); (0,0) circle \[radius=0.00\]; (left) at (0, 0); (q0) at (1,2); (q1) at (2, 3); (q2) at (3.5, 3.5); (q3) at (5, 2.8); (left) at (0, 0); (q1’) at (2, ); (q2’) at (3.5, +0.5); (q3’) at (5, -0.2); (p0) at (1.5, 1.2); (p1) at (3, 1); (p2) at (4, 0.2); (left) – (q0) – (q1); (q1) – (q2) – (q3); (left) –(q1’); (q1’) – (q2’) – (q3’); (p0) – (p1) – (p2); (q0) circle \[radius=0.05\]; (q1) circle \[radius=0.05\]; (q2) circle \[radius=0.05\]; (q3) circle \[radius=0.05\]; (left) circle \[radius=0.05\]; (q1’) circle \[radius=0.05\]; (q2’) circle \[radius=0.05\]; (q3’) circle \[radius=0.05\]; (p0) circle \[radius=0.05\]; (p1) circle \[radius=0.05\]; (p2) circle \[radius=0.05\]; (1, 1.5) – (1,1); (3.5, 2.8) – (3.5,2.3); (q3’) ++(0.2, -0.4) node [${\mathrm{HN}}({\overline}{{\mathcal{V}}})$]{}; (p2) ++(-0.2, -0.4) node [${\mathrm{HN}}({\mathcal{W}})$]{}; (left) ++(-0.3, -0.05) node [$O$]{}; The assumption that ${\mathcal{V}}$ and ${\mathcal{W}}$ have integer slopes is crucial in Definition \[max slope reduction definition\]. In fact, if ${\mu_\text{max}}({\mathcal{W}})$ is not an integer, reducing all slopes of ${\mathcal{V}}^{>{\mu_\text{max}}({\mathcal{W}})}$ to ${\mu_\text{max}}({\mathcal{W}})$ may not make sense as the resulting vector bundle should have an non-integer degree. For example, if we consider ${\mathcal{V}}= {{\mathcal{O}}}(1)^{\oplus 3}$ and ${\mathcal{W}}= {{\mathcal{O}}}\left(\frac{1}{2}\right)$, reducing all slopes of ${\mathcal{V}}$ to ${\mu_\text{max}}({\mathcal{W}}) = \frac{1}{2}$ should yield a semistable vector bundle of slope $\frac{1}{2}$ and rank $3$, which does not exist. On the other hand, slopewise dominance of ${\mathcal{V}}$ on ${\mathcal{W}}$ is not essential for the definition to make sense. However, there are a couple of reasons that we don’t consider the case when ${\mathcal{V}}$ does not slopewise dominates ${\mathcal{W}}$. First, our terminology doesn’t quite make sense in this case as ${\mathcal{V}}$ may have no slopes to reduce down to ${\mathcal{W}}$, for example when ${\mu_\text{max}}({\mathcal{V}})<{\mu_\text{min}}({\mathcal{W}})$. Second, we won’t need this case for our purpose since we will only apply the notion of maximal slope reduction to (some direct summands of) ${\mathcal{F}}$ and ${\mathcal{Q}}$ for which we have a slopewise dominance relation by Lemma \[slopewise dominance relations for key inequality\]. We note some basic properties of the maximal slope reduction. \[basic properties of max slope reduction\] Let ${\mathcal{V}}$ and ${\mathcal{W}}$ be nonzero vector bundles on ${\mathcal{X}}$ with integer slopes such that ${\mathcal{V}}$ slopewise dominates ${\mathcal{W}}$. Let ${\overline}{{\mathcal{V}}}$ denote the maximal slope reduction of ${\mathcal{V}}$ to ${\mathcal{W}}$. Then ${\overline}{{\mathcal{V}}}$ satisfies the following properties: 1. ${\mu_\text{max}}({\overline}{{\mathcal{V}}}) = {\mu_\text{max}}({\mathcal{W}})$. 2. ${\mathrm{rk}}({\overline}{{\mathcal{V}}}) = {\mathrm{rk}}({\mathcal{V}})$. 3. ${\mathcal{V}}= {\overline}{{\mathcal{V}}}$ if and only if ${\mu_\text{max}}({\mathcal{V}}) = {\mu_\text{max}}({\mathcal{W}})$. 4. ${\overline}{{\mathcal{V}}}$ slopewise dominates ${\mathcal{W}}$. 5. all slopes of ${\overline}{{\mathcal{V}}}$ are integers. All properties are immediate consequences of Definition \[max slope reduction definition\] We can now recursively define our procedure for reducing the slopes of ${\mathcal{F}}$ to the slopes of ${\mathcal{Q}}$ as follows: 1. \[slope reduction decomposition step\] Since ${\mathcal{F}}$ slopewise dominates ${\mathcal{Q}}$ as noted in Lemma \[slopewise dominance relations for key inequality\], we use Lemma \[existence of maximal common factor decomp\] to obtain decompositions $$\label{max common factor decomps for F and Q} {\mathcal{F}}\simeq {\mathcal{U}}\oplus {\mathcal{F}}' \quad\quad \text{ and } \quad\quad {\mathcal{Q}}\simeq {\mathcal{U}}\oplus {\mathcal{Q}}'$$ satisfying the following properties: 1. \[slopewise dominance for F’ and Q’\] ${\mathcal{F}}'$ slopewise dominates ${\mathcal{Q}}'$. 2. \[inequality for max slopes of F’ and Q’\] If ${\mathcal{Q}}' \neq 0$, we have ${\mu_\text{max}}({\mathcal{F}}')>{\mu_\text{max}}({\mathcal{Q}}')$. 3. \[ineqaulities for min slope of U\] If ${\mathcal{U}}\neq 0$ and ${\mathcal{Q}}' \neq 0$, we have ${\mu_\text{min}}({\mathcal{U}}) \geq {\mu_\text{max}}({\mathcal{F}}') > {\mu_\text{max}}({\mathcal{Q}}')$. 2. If ${\mathcal{Q}}' = 0$, we terminate the process. Otherwise, we go back to \[slope reduction decomposition step\] after replacing ${\mathcal{F}}$ by ${\mathcal{U}}\oplus {\overline}{{\mathcal{F}}}'$, where ${\overline}{{\mathcal{F}}}'$ denotes the maximal slope reduction of ${\mathcal{F}}'$ to ${\mathcal{Q}}'$. (left) at (0, 0); (q1) at (2.5, 3.5); (q2) at (4, 4); (q3) at (5, 3.7); (p0) at (0.5, 1.5); (p1) at (1.5, 2.5); (p2) at (3.5, 3); (p3) at (4.5, 2.5); (p4) at (5, 1.5); (left) – (p0) – (p1); (p1) – (q1) – (q2) – (q3); (p1) – (p2) – (p3) – (p4); (q1) circle \[radius=0.05\]; (q2) circle \[radius=0.05\]; (q3) circle \[radius=0.05\]; (left) circle \[radius=0.05\]; (p0) circle \[radius=0.05\]; (p1) circle \[radius=0.05\]; (p2) circle \[radius=0.05\]; (p3) circle \[radius=0.05\]; (p4) circle \[radius=0.05\]; (q3) ++(0.2, -0.4) node [${\mathrm{HN}}({\mathcal{F}})$]{}; (p4) ++(-0.2, -0.4) node [${\mathrm{HN}}({\mathcal{Q}})$]{}; (left) ++(-0.3, -0.05) node [$O$]{}; (p0) ++(-0.2, 0.3) node [${\mathcal{U}}$]{}; (q1) ++(0, 0.3) node [${\mathcal{F}}'$]{}; (p2) ++(0, -0.4) node [${\mathcal{Q}}'$]{}; (0, ) – (1.5,); (0,0) circle \[radius=0.00\]; (left) at (0, 0); (q1) at (2.5, 3.5); (q2) at (4, 4); (q3) at (5, 3.7); (q2’) at (4, ); (q3’) at (5, -0.3); (p0) at (0.5, 1.5); (p1) at (1.5, 2.5); (p2) at (3.5, 3); (p3) at (4.5, 2.5); (p4) at (5, 1.5); (left) – (p0) – (p1); (p1) – (q1) – (q2) – (q3); (p1) – (q2’) – (q3’); (p2) – (p3) – (p4); (q1) circle \[radius=0.05\]; (q2) circle \[radius=0.05\]; (q3) circle \[radius=0.05\]; (left) circle \[radius=0.05\]; (q2’) circle \[radius=0.05\]; (q3’) circle \[radius=0.05\]; (p0) circle \[radius=0.05\]; (p1) circle \[radius=0.05\]; (p2) circle \[radius=0.05\]; (p3) circle \[radius=0.05\]; (p4) circle \[radius=0.05\]; (3, 3.5) – (3,3); (q3’) ++(0.6, -0.4) node [${\mathrm{HN}}({\mathcal{F}})$]{}; (p4) ++(-0.2, -0.4) node [${\mathrm{HN}}({\mathcal{Q}})$]{}; (left) ++(-0.3, -0.05) node [$O$]{}; (p0) ++(-0.2, 0.3) node [${\mathcal{U}}$]{}; (q2’) ++(0, 0.3) node [${\overline}{{\mathcal{F}}}'$]{}; With this procedure defined, we will obtain the desired inequality $c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}}) \geq 0$ by establishing the following facts: 1. \[decreasing cEF(Q) during slope reduction\] The quantity $c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}})$ never increases throughout the process. 2. \[terminal condition for slope reduction\] The process eventually terminates with the condition ${\mathcal{Q}}= {\mathcal{F}}$ and $c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}}) = 0$. For the equality condition, we will further show that 1. \[strict increase after first slope reduction\] $c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}})$ strictly increases after the first cycle of the process, thereby deducing that the equality $c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}})= 0$ can be only achieved by starting with the terminal state ${\mathcal{Q}}= {\mathcal{F}}$. The main subtlety for our procedure arises from the fact that some of the assumptions we have on ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ may be lost during our process. In the following lemma, we give a list of all assumptions that are maintained throughout the process. \[assumptions of key inequality after max slope reduction\] Let ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ be nonzero vector bundles on ${\mathcal{X}}$ satisfying the following properties: 1. \[slope condition on E and F without equality condition\] ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{F}}^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$ 2. \[slope condition on E and Q\] ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{Q}}^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$ with equality only when ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{Q}}^{\leq \mu}$. 3. \[slope condition on F and Q\] ${\mathrm{rk}}({\mathcal{F}}^{\geq \mu}) \geq {\mathrm{rk}}({\mathcal{Q}}^{\geq \mu})$ for every $\mu \in {\mathbb{Q}}$. 4. \[integer slopes assumption for E, F and Q\] all slopes of ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ are integers. 5. \[equal rank assumption for F and Q\] ${\mathrm{rk}}({\mathcal{Q}}) = {\mathrm{rk}}({\mathcal{F}})$. Then all properties \[slope condition on E and F without equality condition\] - \[equal rank assumption for F and Q\] are invariant under replacing ${\mathcal{F}}$ by ${\overline}{{\mathcal{F}}}$, the maximal slope reduction of ${\mathcal{F}}$ to ${\mathcal{Q}}$. Let us first remark that the maximal slope reduction of ${\mathcal{F}}$ to ${\mathcal{Q}}$ makes sense. Indeed, the property \[slope condition on F and Q\] implies slopewise dominance of ${\mathcal{F}}$ on ${\mathcal{Q}}$ by Lemma \[slopewise dominance relations for key inequality\] while the property \[integer slopes assumption for E, F and Q\] says that ${\mathcal{F}}$ and ${\mathcal{Q}}$ have integer slopes. We now assert that the property \[slope condition on E and F without equality condition\] is a formal consequence of the other properties. Note that the property \[slope condition on F and Q\] is equivalent to slopewise dominance of ${\mathcal{F}}$ on ${\mathcal{Q}}$ by Lemma \[slopewise dominance and rank inequalities\]. Combining this with the property \[equal rank assumption for F and Q\], we obtain slopewise dominance of ${\mathcal{Q}}^\vee$ on ${\mathcal{F}}^\vee$ by Lemma \[duality of slopewise dominance for equal rank case\]. Hence Lemma \[slopewise dominance and rank inequalities\] now yields an inequality $${\mathrm{rk}}({\mathcal{Q}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{F}}^{\leq \mu}) \quad\quad \text{ for every } \mu \in {\mathbb{Q}}.$$ We then deduce the desired inequality ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{F}}^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$ by combining the above inequality with the inequality in the property \[slope condition on E and Q\]. Hence we only need to check the invariance of the other properties \[slope condition on E and Q\] - \[equal rank assumption for F and Q\]. The invariance of the property \[slope condition on E and Q\] is obvious since ${\mathcal{E}}$ and ${\mathcal{Q}}$ remain unchanged. The property \[slope condition on F and Q\] is equivalent to slopewise dominance of ${\mathcal{F}}$ on ${\mathcal{Q}}$ by Lemma \[slopewise dominance and rank inequalities\], so its invariance under replacing ${\mathcal{F}}$ by ${\overline}{{\mathcal{F}}}$ follows from Lemma \[basic properties of max slope reduction\]. The invariance of the properties \[integer slopes assumption for E, F and Q\] and \[equal rank assumption for F and Q\] also follow immediately from Lemma \[basic properties of max slope reduction\]. It ${\mathcal{Q}}$ is not semistable, the condition ${\mu_\text{min}}({\mathcal{E}})<{\mu_\text{min}}({\mathcal{F}})$ is also invariant under replacing ${\mathcal{F}}$ by ${\overline}{{\mathcal{F}}}$. We won’t need this fact, but we give a proof here for curious readers. We wish to show that ${\mu_\text{min}}({\mathcal{E}}) <{\mu_\text{min}}({\overline}{{\mathcal{F}}})$ if ${\mathcal{Q}}$ is not semistable. Since ${\overline}{{\mathcal{F}}}$ is obtained from ${\mathcal{F}}$ by reducing all slopes of ${\mathcal{F}}^{>{\mu_\text{max}}({\mathcal{Q}})}$ to ${\mu_\text{max}}({\mathcal{Q}})$, we have two possible values for ${\mu_\text{min}}({\overline}{{\mathcal{F}}})$ as follows: $$\label{possible values for min slope of max slope reduction} {\mu_\text{min}}({\overline}{{\mathcal{F}}}) = \begin{cases} {\mu_\text{min}}({\mathcal{F}}) & \text{ if } {\mu_\text{min}}({\mathcal{F}}) \leq {\mu_\text{max}}({\mathcal{Q}}), \\ {\mu_\text{max}}({\mathcal{Q}})& \text{ if } {\mu_\text{min}}({\mathcal{F}}) > {\mu_\text{max}}({\mathcal{Q}}).\end{cases}$$ Hence when ${\mu_\text{min}}({\mathcal{F}}) \leq {\mu_\text{max}}({\mathcal{Q}})$ the desired inequality ${\mu_\text{min}}({\mathcal{E}}) < {\mu_\text{min}}({\overline}{{\mathcal{F}}})$ is equivalent to the corresponding inequality ${\mu_\text{min}}({\mathcal{E}}) < {\mu_\text{min}}({\mathcal{F}})$ for ${\mathcal{E}}$ and ${\mathcal{F}}$. We now consider the case when ${\mu_\text{min}}({\mathcal{F}}) > {\mu_\text{max}}({\mathcal{Q}})$. Since ${\mathcal{E}}^\vee$ slopewise dominates ${\mathcal{Q}}^\vee$ by Lemma \[slopewise dominance relations for key inequality\], we have $$\label{min slope inequality for E and Q} {\mu_\text{max}}({\mathcal{E}}^\vee) \geq {\mu_\text{max}}({\mathcal{Q}}^\vee) \quad\quad \text{ or equivalently } \quad\quad {\mu_\text{min}}({\mathcal{E}}) \leq {\mu_\text{min}}({\mathcal{Q}}).$$ We also note that non-semistability of ${\mathcal{Q}}$ yields an inequality $$\label{min max slope inequality for non-semistable Q} {\mu_\text{min}}({\mathcal{Q}}) < {\mu_\text{max}}({\mathcal{Q}}).$$ We thus combine the above inequalities to find $${\mu_\text{min}}({\mathcal{E}}) \leq {\mu_\text{min}}({\mathcal{Q}}) < {\mu_\text{max}}({\mathcal{Q}}) = {\mu_\text{min}}({\overline}{{\mathcal{F}}}),$$ yielding the desired inequality. Lemma \[assumptions of key inequality after max slope reduction\] suggests that during our process we may lose the following assumptions: - ${\mu_\text{min}}({\mathcal{E}}) < {\mu_\text{max}}({\mathcal{F}})$ - the equality in ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{F}}^{\leq \mu})$ holds only when ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{F}}^{\leq \mu}$. Fortunately, losing either of these assumptions during our procedure will do no harm to our proof. In fact, the condition ${\mu_\text{min}}({\mathcal{E}})<{\mu_\text{min}}({\mathcal{F}})$ will be no longer necessary for the rest of our proof, while the equality condition ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{F}}^{\leq \mu}$ for the inequality ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{F}}^{\leq \mu})$ will be only necesary for establishing the fact that $c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}})$ strictly increases after the first cycle of our procedure. In other words, our proof will be valid as long as we begin our procedure with all assumptions in Proposition \[key inequality\]. Let us now prove the key inequality for Step 3. \[decreasing c\_EF(Q) after max reduction\] Let ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ be nonzero vector bundles on ${\mathcal{X}}$ with the following properties: 1. \[slope condition on E and F without equality condition, decreasing c\_EF(Q) after max redunction\] ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{F}}^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$ 2. \[slope condition on E and Q, decreasing c\_EF(Q) after max redunction\] ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{Q}}^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$ with equality only when ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{Q}}^{\leq \mu}$. 3. \[slope condition on F and Q, decreasing c\_EF(Q) after max redunction\] ${\mathrm{rk}}({\mathcal{F}}^{\geq \mu}) \geq {\mathrm{rk}}({\mathcal{Q}}^{\geq \mu})$ for every $\mu \in {\mathbb{Q}}$. 4. \[integer slopes assumption for E, F and Q, decreasing c\_EF(Q) after max redunction\] all slopes of ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ are integers. 5. \[equal rank assumption for F and Q, decreasing c\_EF(Q) after max redunction\] ${\mathrm{rk}}({\mathcal{Q}}) = {\mathrm{rk}}({\mathcal{F}})$. Let ${\overline}{{\mathcal{F}}}$ be the maximal slope reduction of ${\mathcal{F}}$ to ${\mathcal{Q}}$. Then we have an inequality $$\label{max slope reduction inequality for cEF(Q)} c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}}) \geq c_{{\mathcal{E}}, {\overline}{{\mathcal{F}}}}({\mathcal{Q}}).$$ Moreover, if the equality ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) = {\mathrm{rk}}({\mathcal{Q}}^{\leq \mu})$ for some $\mu \in {\mathbb{Q}}$ implies ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{F}}^{\leq \mu}$, then equality in holds only when ${\mu_\text{max}}({\mathcal{F}}) = {\mu_\text{max}}({\mathcal{Q}})$. Set $\lambda = {\mu_\text{max}}({\mathcal{Q}})$ and $r = {\mathrm{rk}}({\mathcal{F}}^{>\lambda})$. By definition, we may write $$\label{max slope decomps for max slope reduction} {\mathcal{F}}= {\mathcal{F}}^{> \lambda} \oplus {\mathcal{F}}^{\leq \lambda} \quad\quad \text{ and } \quad\quad {\overline}{{\mathcal{F}}} = {{\mathcal{O}}}(\lambda)^{\oplus r} \oplus {\mathcal{F}}^{\leq \lambda}.$$ Then we have $$\begin{aligned} \deg({\mathcal{E}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}&= \deg({\mathcal{E}}^\vee \otimes ({\mathcal{F}}^{> \lambda} \oplus {\mathcal{F}}^{\leq \lambda}))^{{\geq 0}}\\ &= \deg({\mathcal{E}}^\vee \otimes {\mathcal{F}}^{>\lambda})^{{\geq 0}}+ \deg({\mathcal{E}}^\vee \otimes {\mathcal{F}}^{\leq \lambda})^{{\geq 0}},\\ \deg({\mathcal{E}}^\vee \otimes {\overline}{{\mathcal{F}}})^{{\geq 0}}&= \deg({\mathcal{E}}^\vee \otimes ({{\mathcal{O}}}(\lambda)^{\oplus r} \oplus {\mathcal{F}}^{\leq \lambda}))^{{\geq 0}}\\ &= \deg({\mathcal{E}}^\vee \otimes {{\mathcal{O}}}(\lambda)^{\oplus r})^{{\geq 0}}+ \deg({\mathcal{E}}^\vee \otimes {\mathcal{F}}^{\leq \lambda})^{{\geq 0}},\\ \deg({\mathcal{Q}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}&= \deg({\mathcal{Q}}^\vee \otimes ({\mathcal{F}}^{> \lambda} \oplus {\mathcal{F}}^{\leq \lambda}))^{{\geq 0}}\\ &= \deg({\mathcal{Q}}^\vee \otimes {\mathcal{F}}^{> \lambda})^{{\geq 0}}+ \deg({\mathcal{Q}}^\vee \otimes {\mathcal{F}}^{\leq \lambda})^{{\geq 0}}, \\ \deg({\mathcal{Q}}^\vee \otimes {\overline}{{\mathcal{F}}})^{{\geq 0}}&= \deg({\mathcal{Q}}^\vee \otimes ({{\mathcal{O}}}(\lambda)^{\oplus r} \oplus {\mathcal{F}}^{\leq \lambda}))^{{\geq 0}}\\ &= \deg({\mathcal{Q}}^\vee \otimes {{\mathcal{O}}}(\lambda)^{\oplus r})^{{\geq 0}}+ \deg({\mathcal{Q}}^\vee \otimes {\mathcal{F}}^{\leq \lambda})^{{\geq 0}}. \end{aligned}$$ Thus we obtain $$\begin{aligned} \deg({\mathcal{E}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}- \deg({\mathcal{E}}^\vee \otimes {\overline}{{\mathcal{F}}})^{{\geq 0}}&= \deg({\mathcal{E}}^\vee \otimes {\mathcal{F}}^{>\lambda})^{{\geq 0}}- \deg({\mathcal{E}}^\vee \otimes {{\mathcal{O}}}(\lambda)^{\oplus r})^{{\geq 0}}, \label{max slope reduction difference of e,f terms}\\ \deg({\mathcal{Q}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}- \deg({\mathcal{Q}}^\vee \otimes {\overline}{{\mathcal{F}}})^{{\geq 0}}&= \deg({\mathcal{Q}}^\vee \otimes {\mathcal{F}}^{> \lambda})^{{\geq 0}}- \deg({\mathcal{Q}}^\vee \otimes {{\mathcal{O}}}(\lambda)^{\oplus r})^{{\geq 0}}. \label{max slope reduction difference of f,q terms}\end{aligned}$$ Let us write ${\overrightarrow{\mathrm{HN}}}({\mathcal{E}}) = (e_i), {\overrightarrow{\mathrm{HN}}}({\mathcal{F}}^{> \lambda}) = (f_j), {\overrightarrow{\mathrm{HN}}}({\mathcal{Q}}) = (q_k)$ and ${\overrightarrow{\mathrm{HN}}}({{\mathcal{O}}}(\lambda)^{\oplus r}) = ({\overline}{f})$, and set $f := \sum f_j$. Note that we can write ${\overline}{f} = \sum {\overline}{f}_j$ where ${\overline}{f}_j$ denotes the vector obtained by reducing the slope of $f_j$ to $\lambda$. By construction, we have the following observations: 1. \[equal x coordinates for max slope reduction HN vectors\] $f_x = {\mathrm{rk}}({\mathcal{F}}^{> \lambda}) = r = {\overline}{f}_x$ 2. \[comparison of y coordinates for max slope reduction HN vectors\] $f_y \geq {\overline}{f}_y$ with equality if and only if $f = {\overline}{f} = 0$. We now use Lemma \[degree in terms of HN vectors\] to write the right side of as $$\label{max slope reduction difference of e,f terms HN vectors} \deg({\mathcal{E}}^\vee \otimes {\mathcal{F}}^{>\lambda})^{{\geq 0}}- \deg({\mathcal{E}}^\vee \otimes {{\mathcal{O}}}(\lambda)^{\oplus r})^{{\geq 0}}= \sum_{e_i \preceq f_j} e_i \times f_j - \sum_{\mu(e_i) \leq \lambda} e_i \times {\overline}{f}.$$ Note that each $e_i$ with $\mu(e_i) \leq \lambda$ satisfies $e_i \preceq f_j$ for all $j$ since by construction we have $\mu(f_j) > \lambda$ for all $j$. We then find $$\label{max slope reduction e,f terms fundamental inequality} \sum_{\mu(e_i) \leq \lambda} e_i \times f_j \leq \sum_{e_i \preceq f_j} e_i \times f_j$$ as each term on the right hand side is nonnegative. Now yields $$\begin{aligned} \deg({\mathcal{E}}^\vee \otimes {\mathcal{F}}^{>\lambda})^{{\geq 0}}- \deg({\mathcal{E}}^\vee \otimes {{\mathcal{O}}}(\lambda)^{\oplus r})^{{\geq 0}}&\geq \sum_{\mu(e_i) \leq \lambda} e_i \times f_j - \sum_{\mu(e_i) \leq \lambda} e_i \times {\overline}{f}\\ &= \sum_{\mu(e_i) \leq \lambda} e_i \times \sum f_j - \sum_{\mu(e_i) \leq \lambda} e_i \times {\overline}{f} \\ &= \sum_{\mu(e_i) \leq \lambda} e_i \times (f - {\overline}{f}).\end{aligned}$$ Since $(f - {\overline}{f})_x = 0$ as noted in \[equal x coordinates for max slope reduction HN vectors\], we have $$\sum_{\mu(e_i) \leq \lambda} e_i \times (f - {\overline}{f}) = \left( \sum_{\mu(e_i) \leq \lambda} e_i\right)_x \cdot (f - {\overline}{f})_y = {\mathrm{rk}}({\mathcal{E}}^{\leq \lambda}) \cdot (f - {\overline}{f})_y.$$ We thus obtain an inequality $$\deg({\mathcal{E}}^\vee \otimes {\mathcal{F}}^{>\lambda})^{{\geq 0}}- \deg({\mathcal{E}}^\vee \otimes {{\mathcal{O}}}(\lambda)^{\oplus r})^{{\geq 0}}\geq {\mathrm{rk}}({\mathcal{E}}^{\leq \lambda}) \cdot (f - {\overline}{f})_y$$ which is equivalent by to an inequality $$\label{max slope reduction difference of e,f terms lower bound} \deg({\mathcal{E}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}- \deg({\mathcal{E}}^\vee \otimes {\overline}{{\mathcal{F}}})^{{\geq 0}}\geq {\mathrm{rk}}({\mathcal{E}}^{\leq \lambda}) \cdot (f - {\overline}{f})_y.$$ Let us now use Lemma \[degree in terms of HN vectors\] to write the right side of as $$\label{max slope reduction difference of f,q terms HN vectors} \deg({\mathcal{Q}}^\vee \otimes {\mathcal{F}}^{> \lambda})^{{\geq 0}}- \deg({\mathcal{Q}}^\vee \otimes {{\mathcal{O}}}(\lambda)^{\oplus r})^{{\geq 0}}= \sum_{q_k \preceq f_j} q_k \times f_j - \sum_{\mu(q_k) \leq \lambda} q_k \times {\overline}{f}.$$ Note that the conditions $q_k \preceq f_j$ and $\mu(q_k) \leq \lambda$ hold for all $j$ and $k$; indeed, by construction we have $\mu(q_k) \leq {\mu_\text{max}}({\mathcal{Q}}) = \lambda < \mu(f_j)$ for all $j$ and $k$. Hence we can simplify the above equation as $$\begin{aligned} \deg({\mathcal{Q}}^\vee \otimes {\mathcal{F}}^{> \lambda})^{{\geq 0}}- \deg({\mathcal{Q}}^\vee \otimes {{\mathcal{O}}}(\lambda)^{\oplus r})^{{\geq 0}}&= \sum q_k \times f_j - \sum q_k \times {\overline}{f} \\ &= \sum q_k \times \sum f_j - \sum q_k \times {\overline}{f}\\ &= \sum q_k \times (f - {\overline}{f}).\end{aligned}$$ Now, as in the previous paragraph, we use the fact $(f_j - f'_j)_x = 0$ from \[equal x coordinates for max slope reduction HN vectors\] to write $$\sum q_k \times (f - {\overline}{f}) = \left(\sum q_k\right)_x - (f - {\overline}{f})_y = {\mathrm{rk}}({\mathcal{Q}}) \cdot (f - {\overline}{f})_y$$ and consequently obtain an equation $$\deg({\mathcal{Q}}^\vee \otimes {\mathcal{F}}^{> \lambda})^{{\geq 0}}- \deg({\mathcal{Q}}^\vee \otimes {{\mathcal{O}}}(\lambda)^{\oplus r})^{{\geq 0}}= {\mathrm{rk}}({\mathcal{Q}}) \cdot (f - {\overline}{f})_y.$$ By , this equation is equivalent to $$\label{max slope reduction difference of f,q terms simple form} \deg({\mathcal{Q}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}- \deg({\mathcal{Q}}^\vee \otimes {\overline}{{\mathcal{F}}})^{{\geq 0}}= {\mathrm{rk}}({\mathcal{Q}}) \cdot (f - {\overline}{f})_y.$$ Note that we have $$c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}}) - c_{{\mathcal{E}}, {\overline}{{\mathcal{F}}}}({\mathcal{Q}}) = \big(\deg({\mathcal{E}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}- \deg({\mathcal{E}}^\vee \otimes {\overline}{{\mathcal{F}}})^{{\geq 0}}\big) - \big(\deg({\mathcal{Q}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}- \deg({\mathcal{Q}}^\vee \otimes {\overline}{{\mathcal{F}}})^{{\geq 0}}\big).$$ Hence and together yields an inequality $$\label{max slope reduction difference of cEF(Q) lower bound} c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}}) - c_{{\mathcal{E}}, {\overline}{{\mathcal{F}}}}({\mathcal{Q}}) \geq ({\mathrm{rk}}({\mathcal{E}}^{\leq \lambda}) - {\mathrm{rk}}({\mathcal{Q}})) \cdot (f - {\overline}{f})_y.$$ Now we observe ${\mathcal{Q}}= {\mathcal{Q}}^{\leq {\mu_\text{max}}({\mathcal{Q}})} = {\mathcal{Q}}^{\leq \lambda}$ and find $${\mathrm{rk}}({\mathcal{E}}^{\leq \lambda}) - {\mathrm{rk}}({\mathcal{Q}}) = {\mathrm{rk}}({\mathcal{E}}^{\leq \lambda}) - {\mathrm{rk}}({\mathcal{Q}}^{\leq \lambda}) \geq 0$$ where the inequality follows from the assumption \[slope condition on E and Q, decreasing c\_EF(Q) after max redunction\]. Since we also have $(f - {\overline}{f})_y \geq 0$ as noted in \[comparison of y coordinates for max slope reduction HN vectors\], we obtain $$\label{max slope redunction difference of cEF(Q) lower bound nonnegativity} ({\mathrm{rk}}({\mathcal{E}}^{\leq \lambda}) - {\mathrm{rk}}({\mathcal{Q}})) \cdot (f - {\overline}{f})_y \geq 0$$ We thus deduce the desired inequality from and . It remains to prove the last statement of Proposition \[decreasing c\_EF(Q) after max reduction\]. For the rest of the proof, we therefore assume that an equality ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) = {\mathrm{rk}}({\mathcal{F}}^{\leq \mu})$ for some $\mu \in {\mathbb{Q}}$ implies ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{F}}^{\leq \mu}$. Note that equality holds in if and only if equality holds in both and . The equality for gives us two cases to consider, namely 1. \[case of equal ranks upto max slope\] when ${\mathrm{rk}}({\mathcal{E}}^{\leq \lambda}) = {\mathrm{rk}}({\mathcal{Q}})$, 2. \[case of trivial max slope reduction\] when $(f-{\overline}{f})_y = 0$. We wish to show in both cases that the condition ${\mu_\text{max}}({\mathcal{F}}) = {\mu_\text{max}}({\mathcal{Q}})$ holds when equality in holds. We first investigate the case \[case of trivial max slope reduction\]. The defining condition $(f-{\overline}{f})_y = 0$ yields $f = 0$ by \[comparison of y coordinates for max slope reduction HN vectors\], thereby implying ${\mathrm{rk}}({\mathcal{F}}^{> \lambda}) = r = 0$ by \[equal x coordinates for max slope reduction HN vectors\]. The decompositions then yield ${\mathcal{F}}= {\overline}{{\mathcal{F}}}$, implying the condition ${\mu_\text{max}}({\mathcal{F}}) = {\mu_\text{max}}({\mathcal{Q}})$ by Lemma \[basic properties of max slope reduction\]. We thus see that the condition ${\mu_\text{max}}({\mathcal{F}}) = {\mu_\text{max}}({\mathcal{Q}})$ always holds in the case \[case of trivial max slope reduction\]. Let us now consider the case \[case of equal ranks upto max slope\]. We may assume that ${\mathcal{F}}^{> \lambda} \neq 0$, since otherwise we can argue as in the preceding paragraph to obtain the desired condition ${\mu_\text{max}}({\mathcal{F}}) = {\mu_\text{max}}({\mathcal{Q}})$. Suppose now that equality in holds. Then we must have equality in , which amounts to saying that every term on the right side of should appear on the left side of . In other words, every $e_i$ that satisfies $e_i \preceq f_j$ for some $j$ should also satisfy $\mu(e_i) \leq \lambda$. In particular, we obtain $\mu(e_i) \leq \lambda$ for all $e_i$ with $e_i \preceq f_1$. Since $\mu(f_1) = {\mu_\text{max}}({\mathcal{F}}^{>\lambda}) = {\mu_\text{max}}({\mathcal{F}})$, we deduce $$\label{E has no intermediate slopes for max slopep reduction} {\mathcal{E}}^{\leq {\mu_\text{max}}({\mathcal{F}})} = {\mathcal{E}}^{\leq \lambda}.$$ We then use the defining condition ${\mathrm{rk}}({\mathcal{E}}^{\leq \lambda}) = {\mathrm{rk}}({\mathcal{Q}})$ and the assumption \[equal rank assumption for F and Q, decreasing c\_EF(Q) after max redunction\] to find $${\mathrm{rk}}({\mathcal{E}}^{\leq {\mu_\text{max}}({\mathcal{F}})}) = {\mathrm{rk}}({\mathcal{E}}^{\leq \lambda}) = {\mathrm{rk}}({\mathcal{Q}}) = {\mathrm{rk}}({\mathcal{F}}) = {\mathrm{rk}}({\mathcal{F}}^{\leq {\mu_\text{max}}({\mathcal{F}})})$$ and consequently get an isomorphism $$\label{isomorphism between E and F up to max slope of F} {\mathcal{E}}^{\leq {\mu_\text{max}}({\mathcal{F}})} \simeq {\mathcal{F}}^{\leq {\mu_\text{max}}({\mathcal{F}})} = {\mathcal{F}}.$$ by our newest assumption. Moreover, we observe ${\mathcal{Q}}= {\mathcal{Q}}^{\leq {\mu_\text{max}}({\mathcal{Q}})} = {\mathcal{Q}}^{\leq \lambda}$ to rewrite the defining condition ${\mathrm{rk}}({\mathcal{E}}^{\leq \lambda}) = {\mathrm{rk}}({\mathcal{Q}})$ as ${\mathrm{rk}}({\mathcal{E}}^{\leq \lambda}) = {\mathrm{rk}}({\mathcal{Q}}^{\leq \lambda})$, thereby obtaining another isomorphism $$\label{isomorphism between E and Q up to max slope of Q} {\mathcal{E}}^{\leq \lambda} \simeq {\mathcal{Q}}^{\leq \lambda} = {\mathcal{Q}}$$ by the assumption \[slope condition on E and Q, decreasing c\_EF(Q) after max redunction\]. Now , and together imply that ${\mathcal{F}}\simeq {\mathcal{Q}}$, yielding the desired condition ${\mu_\text{max}}({\mathcal{F}}) = {\mu_\text{max}}({\mathcal{Q}})$. The following proposition translates the results of Lemma \[assumptions of key inequality after max slope reduction\] and Proposition \[decreasing c\_EF(Q) after max reduction\] in the setting of our slope reduction procedure. \[slope reduction single step\] Let ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ be nonzero vector bundles on ${\mathcal{X}}$ with the following properties: 1. \[slope condition on E and F without equality condition, slope reduction single step\] ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{F}}^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$ 2. \[slope condition on E and Q, slope reduction single step\] ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{Q}}^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$ with equality only when ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{Q}}^{\leq \mu}$. 3. \[slope condition on F and Q, slope reduction single step\] ${\mathrm{rk}}({\mathcal{F}}^{\geq \mu}) \geq {\mathrm{rk}}({\mathcal{Q}}^{\geq \mu})$ for every $\mu \in {\mathbb{Q}}$. 4. \[integer slopes assumption for E, F and Q, slope reduction single step\] all slopes of ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ are integers. 5. \[equal rank assumption for F and Q, slope reduction single step\] ${\mathrm{rk}}({\mathcal{Q}}) = {\mathrm{rk}}({\mathcal{F}})$. Consider the decompositions $$\label{max common factor decomps for F and Q in slope reduction step} {\mathcal{F}}\simeq {\mathcal{U}}\oplus {\mathcal{F}}' \quad\quad \text{ and } \quad\quad {\mathcal{Q}}\simeq {\mathcal{U}}\oplus {\mathcal{Q}}'$$ given by Lemma \[existence of maximal common factor decomp\]. Assume that ${\mathcal{Q}}' \neq 0$, and let ${\overline}{{\mathcal{F}}}'$ denote the maximal slope reduction of ${\mathcal{F}}'$ to ${\mathcal{Q}}'$. 1. \[invariance of assumptions in slope reduction steps\] The properties \[slope condition on E and F without equality condition, slope reduction single step\] - \[equal rank assumption for F and Q, slope reduction single step\] are invariant under replacing ${\mathcal{F}}$ by ${\tilde}{{\mathcal{F}}}:={\mathcal{U}}\oplus {\overline}{{\mathcal{F}}}'$. 2. \[statement of decreasing cEF(Q) in slope reduction steps\] We have an inequality $$\label{slope reduction step inequality for cEF(Q)} c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}}) \geq c_{{\mathcal{E}}, {\tilde}{{\mathcal{F}}}}({\mathcal{Q}}).$$ 3. \[equality condition for cEF(Q) in slope reduction steps\] If the equality ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) = {\mathrm{rk}}({\mathcal{F}}^{\leq \mu})$ for some $\mu \in {\mathbb{Q}}$ implies ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{F}}^{\leq \mu}$, then equality in never holds. (left) at (0, 0); (q1) at (2.5, 3.5); (q2) at (4, 4); (q3) at (5, 3.7); (p0) at (0.5, 1.5); (p1) at (1.5, 2.5); (p2) at (3.5, 3); (p3) at (4.5, 2.5); (p4) at (5, 1.5); (left) – (p0) – (p1); (p1) – (q1) – (q2) – (q3); (p1) – (p2) – (p3) – (p4); (q1) circle \[radius=0.05\]; (q2) circle \[radius=0.05\]; (q3) circle \[radius=0.05\]; (left) circle \[radius=0.05\]; (p0) circle \[radius=0.05\]; (p1) circle \[radius=0.05\]; (p2) circle \[radius=0.05\]; (p3) circle \[radius=0.05\]; (p4) circle \[radius=0.05\]; (q3) ++(0.2, -0.4) node [${\mathrm{HN}}({\mathcal{F}})$]{}; (p4) ++(-0.2, -0.4) node [${\mathrm{HN}}({\mathcal{Q}})$]{}; (left) ++(-0.3, -0.05) node [$O$]{}; (p0) ++(-0.2, 0.3) node [${\mathcal{U}}$]{}; (q1) ++(0, 0.3) node [${\mathcal{F}}'$]{}; (p2) ++(0, -0.4) node [${\mathcal{Q}}'$]{}; (0, ) – (1.5,); (0,0) circle \[radius=0.00\]; (left) at (0, 0); (q1) at (2.5, 3.5); (q2) at (4, 4); (q3) at (5, 3.7); (q2’) at (4, ); (q3’) at (5, -0.3); (p0) at (0.5, 1.5); (p1) at (1.5, 2.5); (p2) at (3.5, 3); (p3) at (4.5, 2.5); (p4) at (5, 1.5); (left) – (p0) – (p1); (p1) – (q1) – (q2) – (q3); (p1) – (q2’) – (q3’); (p2) – (p3) – (p4); (q1) circle \[radius=0.05\]; (q2) circle \[radius=0.05\]; (q3) circle \[radius=0.05\]; (left) circle \[radius=0.05\]; (q2’) circle \[radius=0.05\]; (q3’) circle \[radius=0.05\]; (p0) circle \[radius=0.05\]; (p1) circle \[radius=0.05\]; (p2) circle \[radius=0.05\]; (p3) circle \[radius=0.05\]; (p4) circle \[radius=0.05\]; (3, 3.5) – (3,3); (q3’) ++(0.6, -0.4) node [${\mathrm{HN}}({\tilde}{{\mathcal{F}}})$]{}; (p4) ++(-0.2, -0.4) node [${\mathrm{HN}}({\mathcal{Q}})$]{}; (left) ++(-0.3, -0.05) node [$O$]{}; (p0) ++(-0.2, 0.3) node [${\mathcal{U}}$]{}; (q2’) ++(0, 0.3) node [${\overline}{{\mathcal{F}}}'$]{}; Let us first verify that all constructions in the statement are valid. The validity of the decompositions relies on slopewise dominance of ${\mathcal{F}}$ on ${\mathcal{Q}}$, which follows from the property \[slope condition on F and Q, slope reduction single step\] by Lemma \[slopewise dominance and rank inequalities\]. For the validity of the maximal slopewise reduction of ${\mathcal{F}}'$ to ${\mathcal{Q}}'$, we verify slopewise dominance of ${\mathcal{F}}'$ on ${\mathcal{Q}}'$ by Lemma \[existence of maximal common factor decomp\] and integer slopes of ${\mathcal{F}}'$ and ${\mathcal{Q}}'$ by the property \[integer slopes assumption for E, F and Q, slope reduction single step\]. We assert that the properties \[slope condition on E and F without equality condition, slope reduction single step\] - \[equal rank assumption for F and Q, slope reduction single step\] yield the corresponding properties for ${\mathcal{E}}, {\mathcal{F}}'$ and ${\mathcal{Q}}'$ as follows: 1. \[slope condition on E and F’ without equality condition\] ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{F}}'^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$. 2. \[slope condition on E and Q’\] ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{Q}}'^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$ with equality only when ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{Q}}'^{\leq \mu}$. 3. \[slope condition on F’ and Q’\] ${\mathrm{rk}}({\mathcal{F}}'^{\geq \mu}) \geq {\mathrm{rk}}({\mathcal{Q}}'^{\geq \mu})$ for every $\mu \in {\mathbb{Q}}$. 4. \[integer slopes assumption for E, F’ and Q’\] all slopes of ${\mathcal{E}}, {\mathcal{F}}'$ and ${\mathcal{Q}}'$ are integers. 5. \[equal rank assumption for F’ and Q’\] ${\mathrm{rk}}({\mathcal{Q}}') = {\mathrm{rk}}({\mathcal{F}}')$. We only need to check the properties \[slope condition on E and Q’\] - \[equal rank assumption for F’ and Q’\] since the property \[slope condition on E and F’ without equality condition\] will then follow as a formal consequence of these properties as in the proof of Lemma \[assumptions of key inequality after max slope reduction\]. The property \[slope condition on E and Q’\] follows from the property \[slope condition on E and Q, slope reduction single step\] by writing ${\mathcal{Q}}' = {\mathcal{Q}}^{\leq \lambda}$ with $\lambda = {\mu_\text{max}}({\mathcal{Q}})'$. The property \[slope condition on F’ and Q’\] is equivalent to slopewise dominance of ${\mathcal{F}}'$ on ${\mathcal{Q}}'$ which follows from Lemma \[existence of maximal common factor decomp\]. The properties \[integer slopes assumption for E, F’ and Q’\] and \[equal rank assumption for F’ and Q’\] follow immediately from the corresponding properties \[integer slopes assumption for E, F and Q, slope reduction single step\] and \[equal rank assumption for F and Q, slope reduction single step\] by construction. With the properties \[slope condition on E and F’ without equality condition\] - \[equal rank assumption for F’ and Q’\] established, Lemma \[assumptions of key inequality after max slope reduction\] and Proposition \[decreasing c\_EF(Q) after max reduction\] now yield the following facts: 1. \[invariance of assumptions for F’ and Q’ in slope reduction steps\] The properties \[slope condition on E and F’ without equality condition\] - \[equal rank assumption for F’ and Q’\] are invariant under replacing ${\mathcal{F}}'$ by ${\overline}{{\mathcal{F}}}'$. 2. \[statement of decreasing cEF’(Q’) in slope reduction steps\] We have an inequality $$\label{slope reduction step inequality for cEF'(Q')} c_{{\mathcal{E}}, {\mathcal{F}}'}({\mathcal{Q}}') \geq c_{{\mathcal{E}}, {\overline}{{\mathcal{F}}}'}({\mathcal{Q}}').$$ 3. \[equality condition for cEF’(Q’) in slope reduction steps\] If the equality ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) = {\mathrm{rk}}({\mathcal{F}}'^{\leq \mu})$ for some $\mu \in {\mathbb{Q}}$ implies ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{F}}'^{\leq \mu}$, then equality in holds only when ${\mu_\text{max}}({\mathcal{F}}') = {\mu_\text{max}}({\mathcal{Q}}')$. We wish to deduce the statements \[invariance of assumptions in slope reduction steps\], \[statement of decreasing cEF(Q) in slope reduction steps\] and \[equality condition for cEF(Q) in slope reduction steps\] respectively from the above facts \[invariance of assumptions for F’ and Q’ in slope reduction steps\], \[statement of decreasing cEF’(Q’) in slope reduction steps\] and \[equality condition for cEF’(Q’) in slope reduction steps\]. Let us now prove the statement \[invariance of assumptions in slope reduction steps\]. Note that, as in the proof of Lemma \[assumptions of key inequality after max slope reduction\], we only need to show the invariance of the properties \[slope condition on E and Q, slope reduction single step\] - \[equal rank assumption for F and Q, slope reduction single step\]. The invariance of the property \[slope condition on E and Q, slope reduction single step\] is evident since ${\mathcal{E}}$ and ${\mathcal{Q}}$ remain unchanged. For the invariance of the remaining properties \[slope condition on F and Q, slope reduction single step\], \[integer slopes assumption for E, F and Q, slope reduction single step\] and \[equal rank assumption for F and Q, slope reduction single step\], we have to show that ${\mathcal{E}}, {\tilde}{{\mathcal{F}}}$ and ${\mathcal{Q}}$ satisfy the following properties: 1. \[slope condition on F and Q after slope reduction\] ${\mathrm{rk}}({\tilde}{{\mathcal{F}}}^{\geq \mu}) \geq {\mathrm{rk}}({\mathcal{Q}}^{\geq \mu})$ for every $\mu \in {\mathbb{Q}}$. 2. \[integer slopes assumption for E, F and Q, after slope reduction\] all slopes of ${\mathcal{E}}, {\tilde}{{\mathcal{F}}}$ and ${\mathcal{Q}}$ are integers. 3. \[equal rank assumption for F and Q, after slope reduction\] ${\mathrm{rk}}({\mathcal{Q}}) = {\mathrm{rk}}({\tilde}{{\mathcal{F}}})$. After writing ${\tilde}{{\mathcal{F}}} = {\mathcal{U}}\oplus {\overline}{{\mathcal{F}}}'$ by definition and also ${\mathcal{Q}}= {\mathcal{U}}\oplus {\mathcal{Q}}'$ as in , we deduce all of these properties from the invariance of the properties \[slope condition on F’ and Q’\], \[integer slopes assumption for E, F’ and Q’\] and \[equal rank assumption for F’ and Q’\] noted in \[invariance of assumptions for F’ and Q’ in slope reduction steps\] We move on to the statement \[statement of decreasing cEF(Q) in slope reduction steps\]. Since ${\mathcal{Q}}' \neq 0$ by our assumption, Lemma \[existence of maximal common factor decomp\] yields $$\label{order of slopes in max common factor decomps for slope reduction} {\mu_\text{min}}({\mathcal{U}}) \geq {\mu_\text{max}}({\mathcal{F}}') > {\mu_\text{max}}({\mathcal{Q}}') = {\mu_\text{max}}({\overline}{{\mathcal{F}}}') \quad\quad \text{ if } {\mathcal{U}}\neq 0.$$ Then by Corollary \[zero degree for completely dominating slopes\] we obtain $$\label{zero degree formula for max common factor decomps} \deg({\mathcal{U}}^\vee \otimes {\mathcal{F}}')^{{\geq 0}}= \deg({\mathcal{U}}^\vee \otimes {\overline}{{\mathcal{F}}}')^{{\geq 0}}= 0.$$ Note that does not require the condition ${\mathcal{U}}\neq 0$ from since it evidently holds when ${\mathcal{U}}= 0$. Now we use and the decompositions in to find $$\begin{aligned} \deg({\mathcal{E}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}&= \deg({\mathcal{E}}^\vee \otimes ({\mathcal{U}}\oplus {\mathcal{F}}'))^{{\geq 0}}\\ &= \deg({\mathcal{E}}^\vee \otimes {\mathcal{U}})^{{\geq 0}}+ \deg({\mathcal{E}}^\vee \oplus {\mathcal{F}}')^{{\geq 0}},\\ \deg({\mathcal{E}}^\vee \otimes {\tilde}{{\mathcal{F}}})^{{\geq 0}}&= \deg({\mathcal{E}}^\vee \otimes ({\mathcal{U}}\oplus {\overline}{{\mathcal{F}}}'))^{{\geq 0}}\\ &= \deg({\mathcal{E}}^\vee \otimes {\mathcal{U}})^{{\geq 0}}+ \deg({\mathcal{E}}^\vee \oplus {\overline}{{\mathcal{F}}}')^{{\geq 0}},\\ \deg({\mathcal{Q}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}&= \deg(({\mathcal{U}}\oplus {\mathcal{Q}}')^\vee \otimes ({\mathcal{U}}\oplus {\mathcal{F}}'))^{{\geq 0}}\\ &= \deg({\mathcal{U}}^\vee \otimes {\mathcal{U}})^{{\geq 0}}+ \deg({\mathcal{Q}}'^\vee \otimes {\mathcal{U}})^{{\geq 0}}+ \deg({\mathcal{U}}^\vee \otimes {\mathcal{F}}')^{{\geq 0}}+ \deg({\mathcal{Q}}'^\vee \otimes {\mathcal{F}}')^{{\geq 0}}\\ &= \deg({\mathcal{U}}^\vee \otimes {\mathcal{U}})^{{\geq 0}}+ \deg({\mathcal{Q}}'^\vee \otimes {\mathcal{U}})^{{\geq 0}}+ \deg({\mathcal{Q}}'^\vee \otimes {\mathcal{F}}')^{{\geq 0}}, \\ \deg({\mathcal{Q}}^\vee \otimes {\tilde}{{\mathcal{F}}})^{{\geq 0}}&= \deg(({\mathcal{U}}\oplus {\mathcal{Q}}')^\vee \otimes ({\mathcal{U}}\oplus {\overline}{{\mathcal{F}}}'))^{{\geq 0}}\\ &= \deg({\mathcal{U}}^\vee \otimes {\mathcal{U}})^{{\geq 0}}+ \deg({\mathcal{Q}}'^\vee \otimes {\mathcal{U}})^{{\geq 0}}+ \deg({\mathcal{U}}^\vee \otimes {\overline}{{\mathcal{F}}}')^{{\geq 0}}+ \deg({\mathcal{Q}}'^\vee \otimes {\overline}{{\mathcal{F}}}')^{{\geq 0}}\\ &= \deg({\mathcal{U}}^\vee \otimes {\mathcal{U}})^{{\geq 0}}+ \deg({\mathcal{Q}}'^\vee \otimes {\mathcal{U}})^{{\geq 0}}+ \deg({\mathcal{Q}}'^\vee \otimes {\overline}{{\mathcal{F}}}')^{{\geq 0}}. \end{aligned}$$ Therefore we have $$\begin{aligned} c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}}) - c_{{\mathcal{E}}, {\tilde}{{\mathcal{F}}}}({\mathcal{Q}}) &= \big(\deg({\mathcal{E}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}- \deg({\mathcal{E}}^\vee \otimes {\tilde}{{\mathcal{F}}})^{{\geq 0}}\big) - \big(\deg({\mathcal{Q}}^\vee \otimes {\mathcal{F}})^{{\geq 0}}- \deg({\mathcal{Q}}^\vee \otimes {\tilde}{{\mathcal{F}}})^{{\geq 0}}\big)\\ &= \big(\deg({\mathcal{E}}^\vee \otimes {\mathcal{F}}')^{{\geq 0}}- \deg({\mathcal{E}}^\vee \otimes {\overline}{{\mathcal{F}}}')^{{\geq 0}}\big) - \big(\deg({\mathcal{Q}}'^\vee \otimes {\mathcal{F}}')^{{\geq 0}}- \deg({\mathcal{Q}}'^\vee \otimes {\overline}{{\mathcal{F}}}')^{{\geq 0}}\big)\\ &= c_{{\mathcal{E}}, {\mathcal{F}}'}({\mathcal{Q}}') - c_{{\mathcal{E}}, {\overline}{{\mathcal{F}}}'}({\mathcal{Q}}').\end{aligned}$$ Hence the statement \[statement of decreasing cEF(Q) in slope reduction steps\] now follows directly from the fact \[statement of decreasing cEF’(Q’) in slope reduction steps\]. We now consider the final statement \[equality condition for cEF(Q) in slope reduction steps\]. In accordance with the statement, we suppose that the equality ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) = {\mathrm{rk}}({\mathcal{F}}^{\leq \mu})$ for some $\mu \in {\mathbb{Q}}$ implies ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{F}}^{\leq \mu}$. Since ${\mathcal{F}}'$ is a direct summand of ${\mathcal{F}}$, we have $$\label{slope condition on F and F' in slope reduction single step} {\mathrm{rk}}({\mathcal{F}}'^{\leq \mu}) \leq {\mathrm{rk}}({\mathcal{F}}^{\leq \mu}) \quad\quad \text{ for every } \mu \in {\mathbb{Q}}$$ with equality if and only if ${\mathcal{F}}'^{\leq \mu} = {\mathcal{F}}^{\leq \mu}$. We thus obtain a series of inequalities $${\mathrm{rk}}({\mathcal{F}}'^{\leq \mu}) \leq {\mathrm{rk}}({\mathcal{F}}^{\leq \mu}) \leq {\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \quad\quad \text{ for every } \mu \in {\mathbb{Q}}$$ combining and the inequality ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{F}}^{\leq \mu})$ in the property \[slope condition on E and F without equality condition, slope reduction single step\]. Moreover, the equality ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) = {\mathrm{rk}}({\mathcal{F}}'^{\leq \mu})$ for some $\mu$ implies ${\mathrm{rk}}({\mathcal{F}}'^{\leq \mu}) = {\mathrm{rk}}({\mathcal{F}}^{\leq \mu}) = {\mathrm{rk}}({\mathcal{E}}^{\leq \mu})$, which further implies ${\mathcal{F}}'^{\leq \mu} = {\mathcal{F}}^{\leq \mu} \simeq {\mathcal{E}}^{\leq \mu}$ by the equality condition of and our newly added assumption. Hence we deduce from the fact \[equality condition for cEF’(Q’) in slope reduction steps\] that equality in holds only if ${\mu_\text{max}}({\mathcal{F}}') = {\mu_\text{max}}({\mathcal{Q}}')$. However, our assumption ${\mathcal{Q}}' \neq 0$ implies ${\mu_\text{max}}({\mathcal{F}}') > {\mu_\text{max}}({\mathcal{Q}}')$ by Lemma \[existence of maximal common factor decomp\], thereby indicating that equality in never holds. Now the equation $$c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}}) - c_{{\mathcal{E}}, {\tilde}{{\mathcal{F}}}}({\mathcal{Q}}) = c_{{\mathcal{E}}, {\mathcal{F}}'}({\mathcal{Q}}') - c_{{\mathcal{E}}, {\overline}{{\mathcal{F}}}'}({\mathcal{Q}}')$$ that we established in the preciding paragraph implies that equality in never holds, completing the proof. We are finally ready to complete Step 3. Proposition \[key inequality\] holds under the additional assumptions that ${\mathrm{rk}}({\mathcal{Q}}) = {\mathrm{rk}}({\mathcal{F}})$ and that all slopes of ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ are integers. Let ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ be vector bundles on ${\mathcal{X}}$ satisfying the following properties: 1. \[slope condition on E and F, slope reduction initial state\] ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{F}}^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$ with equality only when ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{F}}^{\leq \mu}$. 2. \[slope condition on E and Q, slope reduction initial state\] ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{Q}}^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$ with equality only when ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{Q}}^{\leq \mu}$. 3. \[slope condition on F and Q, slope reduction initial state\] ${\mathrm{rk}}({\mathcal{F}}^{\geq \mu}) \geq {\mathrm{rk}}({\mathcal{Q}}^{\geq \mu})$ for every $\mu \in {\mathbb{Q}}$. 4. \[integer slopes assumption for E, F and Q, slope reduction initial state\] all slopes of ${\mathcal{E}}, {\mathcal{F}}$ and ${\mathcal{Q}}$ are integers. 5. \[equal rank assumption for F and Q, slope reduction initial state\] ${\mathrm{rk}}({\mathcal{Q}}) = {\mathrm{rk}}({\mathcal{F}})$. 6. \[min slope condition on E and F\] ${\mu_\text{min}}({\mathcal{E}}) < {\mu_\text{min}}({\mathcal{F}})$. We wish to prove the inequality which is equivalent to $c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}}) \geq 0$. Let us define a sequence $({\mathcal{F}}_n)$ of vector bundles on ${\mathcal{X}}$ as follows: 1. \[slope reduction initial state\] Set ${\mathcal{F}}_0 := {\mathcal{F}}$. 2. \[slope reduction inductive state\] For each $n \geq 0$, consider the decompositions $$\label{max common factor decomps for Fn and Q, slope reduction step} {\mathcal{F}}_n \simeq {\mathcal{U}}_n \oplus {\mathcal{F}}_n' \quad\quad \text{ and } \quad\quad {\mathcal{Q}}\simeq {\mathcal{U}}_n \oplus {\mathcal{Q}}_n'$$ given by Lemma \[existence of maximal common factor decomp\]. If ${\mathcal{Q}}_n' = 0$, we make ${\mathcal{F}}_n$ the final term of the sequence. Otherwise, we set $${\mathcal{F}}_{n+1} := {\mathcal{U}}_n \oplus {\overline}{{\mathcal{F}}}_n'$$ where ${\overline}{{\mathcal{F}}}_n'$ denotes the maximal slope reduction of ${\mathcal{F}}'_n$ to ${\mathcal{Q}}'_n$. An induction argument using Proposition \[slope reduction single step\] yields the following facts: 1. The sequence $({\mathcal{F}}_n)$ is well-defined with the following properties: 1. \[slope condition on E and Fn without equality condition, slope reduction process\] ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{F}}_n^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$ 2. \[slope condition on E and Q, slope reduction process\] ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) \geq {\mathrm{rk}}({\mathcal{Q}}^{\leq \mu})$ for every $\mu \in {\mathbb{Q}}$ with equality only when ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{Q}}^{\leq \mu}$. 3. \[slope condition on Fn and Q, slope reduction process\] ${\mathrm{rk}}({\mathcal{F}}_n^{\geq \mu}) \geq {\mathrm{rk}}({\mathcal{Q}}^{\geq \mu})$ for every $\mu \in {\mathbb{Q}}$. 4. \[integer slopes assumption for E, Fn and Q, slope reduction process\] all slopes of ${\mathcal{E}}, {\mathcal{F}}_n$ and ${\mathcal{Q}}$ are integers. 5. \[equal rank assumption for Fn and Q, slope reduction process\] ${\mathrm{rk}}({\mathcal{Q}}) = {\mathrm{rk}}({\mathcal{F}}_n)$. 2. \[statement of decreasing cEF(Q) in slope reduction process\] We have an inequality $$\label{slope reduction step inequality for cEF(Q)} c_{{\mathcal{E}}, {\mathcal{F}}_n}({\mathcal{Q}}) \geq c_{{\mathcal{E}}, {\mathcal{F}}_{n+1}}({\mathcal{Q}}).$$ 3. \[equality condition for cEF(Q) in slope reduction process\] If the equality ${\mathrm{rk}}({\mathcal{E}}^{\leq \mu}) = {\mathrm{rk}}({\mathcal{F}}_n^{\leq \mu})$ for some $\mu \in {\mathbb{Q}}$ implies ${\mathcal{E}}^{\leq \mu} \simeq {\mathcal{F}}_n^{\leq \mu}$, then equality in never holds. (left) at (0, 0); (q1) at (2.5, 3.5); (q2) at (4, 4); (q3) at (5, 3.7); (p0) at (0.5, 1.5); (p1) at (1.5, 2.5); (p2) at (3.5, 3); (p3) at (4.5, 2.5); (p4) at (5, 1.5); (left) – (p0) – (p1); (p1) – (q1) – (q2) – (q3); (p1) – (p2) – (p3) – (p4); (q1) circle \[radius=0.05\]; (q2) circle \[radius=0.05\]; (q3) circle \[radius=0.05\]; (left) circle \[radius=0.05\]; (p0) circle \[radius=0.05\]; (p1) circle \[radius=0.05\]; (p2) circle \[radius=0.05\]; (p3) circle \[radius=0.05\]; (p4) circle \[radius=0.05\]; (q3) ++(0.2, -0.4) node [${\mathrm{HN}}({\mathcal{F}}_n)$]{}; (p4) ++(-0.2, -0.4) node [${\mathrm{HN}}({\mathcal{Q}})$]{}; (left) ++(-0.3, -0.05) node [$O$]{}; (p0) ++(-0.2, 0.3) node [${\mathcal{U}}_n$]{}; (q1) ++(0, 0.3) node [${\mathcal{F}}_n'$]{}; (p2) ++(0, -0.4) node [${\mathcal{Q}}_n'$]{}; (0, ) – (1.5,); (0,0) circle \[radius=0.00\]; (left) at (0, 0); (q1) at (2.5, 3.5); (q2) at (4, 4); (q3) at (5, 3.7); (q2’) at (4, ); (q3’) at (5, -0.3); (p0) at (0.5, 1.5); (p1) at (1.5, 2.5); (p2) at (3.5, 3); (p3) at (4.5, 2.5); (p4) at (5, 1.5); (left) – (p0) – (p1); (p1) – (q1) – (q2) – (q3); (p1) – (q2’) – (q3’); (p2) – (p3) – (p4); (q1) circle \[radius=0.05\]; (q2) circle \[radius=0.05\]; (q3) circle \[radius=0.05\]; (left) circle \[radius=0.05\]; (q2’) circle \[radius=0.05\]; (q3’) circle \[radius=0.05\]; (p0) circle \[radius=0.05\]; (p1) circle \[radius=0.05\]; (p2) circle \[radius=0.05\]; (p3) circle \[radius=0.05\]; (p4) circle \[radius=0.05\]; (3, 3.5) – (3,3); (q3’) ++(1.1, 0) node [${\mathrm{HN}}({\mathcal{F}}_{n+1})$]{}; (p4) ++(-0.2, -0.4) node [${\mathrm{HN}}({\mathcal{Q}})$]{}; (left) ++(-0.3, -0.05) node [$O$]{}; (p0) ++(-0.2, 0.3) node [${\mathcal{U}}_n$]{}; (q2’) ++(0, 0.3) node [${\overline}{{\mathcal{F}}}_n'$]{}; (p3) ++(-0.3, -0.4) node [${\mathcal{Q}}'_{n+1}$]{}; We assert that the sequence $({\mathcal{F}}_n)$ is finite. It suffices to prove $$\label{increasing max common factor in slope reduction} {\mathrm{rk}}({\mathcal{U}}_n) < {\mathrm{rk}}({\mathcal{U}}_{n+1})$$ since we have ${\mathrm{rk}}({\mathcal{U}}_n) \leq {\mathrm{rk}}({\mathcal{Q}})$ by . To this end, we align the polygons ${\mathrm{HN}}({\mathcal{F}}_n)$ and ${\mathrm{HN}}({\mathcal{Q}})$ so that their left endpoints lie at the origin. The proof of Lemma \[existence of maximal common factor decomp\] shows that ${\mathcal{U}}_n$ represents the common part of ${\mathrm{HN}}({\mathcal{F}}_n)$ and ${\mathrm{HN}}({\mathcal{Q}})$. Moreover, since ${\overline}{{\mathcal{F}}}_n'$ is the maximal slope reduction of ${\mathcal{F}}'_n$ to ${\mathcal{Q}}'_n$, the polygons ${\mathrm{HN}}({\overline}{{\mathcal{F}}}_n')$ and ${\mathrm{HN}}({\mathcal{Q}}_n')$ with their left endpoints aligned have some nontrivial common part which we represent by a nonzero vector bundle ${\mathcal{T}}_n$. Let us now consider the decompositions $$\label{decomps after slope reduction} {\mathcal{F}}_{n+1} = {\mathcal{U}}_n \oplus {\overline}{{\mathcal{F}}}'_n \quad\quad \text{ and } \quad\quad {\mathcal{Q}}= {\mathcal{U}}_n \oplus {\mathcal{Q}}'_n$$ given by the definition of ${\mathcal{F}}_{n+1}$ and . The definition of ${\mathcal{F}}_{n+1}'$ assumes that ${\mathcal{Q}}_n' \neq 0$, so Lemma \[existence of maximal common factor decomp\] yields $${\mu_\text{min}}({\mathcal{U}}_n) > {\mu_\text{max}}({\mathcal{Q}}_n') = {\mu_\text{max}}({\overline}{{\mathcal{F}}}_n') \quad\quad \text{ if } {\mathcal{U}}_n \neq 0.$$ Hence the decompositions imply that the common part of ${\mathrm{HN}}({\mathcal{F}}_{n+1})$ and ${\mathrm{HN}}({\mathcal{Q}})$ (with their left endpoints at the origin) is represented by ${\mathcal{U}}_n \oplus {\mathcal{T}}_n$. By the proof of Lemma \[existence of maximal common factor decomp\] , this means $${\mathcal{U}}_{n+1} \simeq {\mathcal{U}}_n \oplus {\mathcal{T}}_n.$$ Now the inequality follows from the fact that ${\mathcal{T}}_n \neq 0$. Let $r$ be the index of the final term in the sequence $({\mathcal{F}}_n)$. We have ${\mathcal{Q}}_r' = 0$ by \[slope reduction inductive state\], so the decompositions become $${\mathcal{F}}_r \simeq {\mathcal{U}}_r \oplus {\mathcal{F}}_r' \quad\quad \text{ and } \quad\quad {\mathcal{Q}}\simeq {\mathcal{U}}_r.$$ Now the property \[equal rank assumption for Fn and Q, slope reduction process\] for $n = r$ implies ${\mathrm{rk}}({\mathcal{F}}_r') = 0$ or equivalently ${\mathcal{F}}_r' = 0$, thereby yielding $$\label{slope reduction final state} {\mathcal{F}}_r \simeq {\mathcal{Q}}.$$ Moreover, by Definition \[definition of cEF(Q)\] we find $$c_{{\mathcal{E}}, {\mathcal{F}}_r}({\mathcal{Q}}) = 0.$$ Now the fact \[statement of decreasing cEF(Q) in slope reduction process\] gives us a chain of inequalities $$\label{chain inequalities for slope reduction process} c_{{\mathcal{E}}, {\mathcal{F}}}({\mathcal{Q}}) = c_{{\mathcal{E}}, {\mathcal{F}}_0}({\mathcal{Q}}) \geq c_{{\mathcal{E}}, {\mathcal{F}}_1}({\mathcal{Q}}) \geq \cdots \geq c_{{\mathcal{E}}, {\mathcal{F}}_r}({\mathcal{Q}}) = 0,$$ thereby establishing the desired inequality . Our final task is to show that equality in holds if and only if ${\mathcal{Q}}= {\mathcal{F}}$. In fact, equality in evidently holds if ${\mathcal{Q}}= {\mathcal{F}}$, so it remains to show that equality in implies ${\mathcal{Q}}= {\mathcal{F}}$. Note that, by the property \[slope condition on E and F, slope reduction initial state\], whenever $r \geq 1$ the fact \[equality condition for cEF(Q) in slope reduction process\] yields a strict inequality $$c_{{\mathcal{E}}, {\mathcal{F}}_0}({\mathcal{Q}}) > c_{{\mathcal{E}}, {\mathcal{F}}_1}({\mathcal{Q}}).$$ Hence we deduce from that equality in holds only if $r = 0$ which implies $${\mathcal{F}}= {\mathcal{F}}_0 \simeq {\mathcal{Q}}$$ by . We thus complete the proof of Proposition \[key inequality\], and therefore the proof of Theorem \[classification of quotient bundles\].
--- abstract: 'We present a self-contained proof of the reflection principle for Brownian Motion.' address: - 'Department of Mathematics, University of South Carolina, Columbia, SC 29208 U.S.A.' - 'Department of Mathematics, University of South Carolina, Columbia, SC 29208 U.S.A.' author: - 'S. J. Dilworth' - Duncan Wright title: A Direct Proof of the Reflection Principle for Brownian Motion --- [^1] Introduction ============ The reflection principle proved below is one of the most important properties of Brownian Motion. So much so that any treatment of Brownian Motion would be incomplete without mentioning it and some of its many applications (see e.g.  [@S]). Most notable among these applications, using the hitting time $\tau_x=\inf\{t:B_t=x\}$, is that $$P(\tau_x\le t)= 2P(B_t\ge x),$$ which in turn yields that $X_t:=\max_{0\le s\le t}B_s$ and $|B_t|$ have the same distribution. This famous result is attributed to Louis Bachelier [@B p. 197], and also, in a later more rigorous treatment, to Paul Lévy [@L p. 293]. In fact it was Bachelier who first introduced the stochastic process, which later on became known as Brownian Motion, as a model for stock prices in his pioneering work in mathematical finance. Remarkably, [@B] precedes the rigorous construction of Brownian Motion by almost two decades. The reflection principle is invariably presented as a consequence of the Strong Markov Property. This approach has pedagogical value as it provides one of the first applications of the Strong Markov Property (see e.g. [@SP]). However, it has the drawback of being beyond the scope of less specialized texts and consequently the proof of the reflection principle is often omitted. We present here a short and direct proof requiring few prerequisites which is intended to make the reflection principle more accessible. Recall that a *Standard Brownian Motion* (SBM) on a probability space $(\Omega,\mathcal{F},P)$ is a gaussian process $(B_t)_{t \ge 0}$ (i.e., the finite-dimensional distributions are multivariate normal distributions), with $B_0=0$, continuous sample paths, $\mathbb{E} [B_t]=0$, and covariance function $\mathbb{E}[B_s B_t] = \min(s,t)$. The $\sigma$-algebra $\mathcal{F}_t$ is the smallest $\sigma$-algebra containing all $P$-null sets for which each $B_s$ ($0 \le s \le t$) is measurable. A stopping time with respect to the *standard Brownian filtration* $(\mathcal{F}_t)_{t\ge0}$ is a mapping $T \colon \Omega \rightarrow [0,\infty]$ satisfying $\{T \le t\} \in \mathcal{F}_t$ for each $t \ge 0$. $T$ is allowed to take the value $\infty$ with positive probability. A tool that is used in our proof of the reflection principle is the ‘uniqueness theorem’: the fact that the distribution of an $\mathbb{R}^n$-valued random vector $X$ is determined by its characteristic function $\phi_{X}(\lambda) := \mathbb{E}[\exp(i \mathbf{\lambda}\cdot X)]$ ($\lambda \in \mathbb{R}^n$) (see e.g. [@M p. 135]). The uniqueness theorem is used in a similar way to prove the Strong Markov Property in [@SP]. Our proof also uses standard properties of the conditional expectation operator with respect to a sub-$\sigma$-algebra $\mathcal{G}$, namely linearity and the fact that $\mathbb{E}[XY|\mathcal{G}] = X\mathbb{E}[Y|\mathcal{G}]$ for random variables $X,Y$ when $X$ is $\mathcal{G}$-measurable (see e.g.[@M p. 187]). The ‘independence of Brownian increments’ is used in the following intuitively obvious but slightly tricky to prove form: if $n \ge 1$ and $s < t_1<\dots <t_n$ and $f \colon \mathbb{R}^n \rightarrow \mathbb{R}$ is bounded and continuous, then, setting $V:=f(B_{t_1}-B_s,\dots, B_{t_n}-B_s)$, $$\label{eq: fact}\mathbb{E}[V|\mathcal{F}_s] = \mathbb{E}[V].$$ For completeness a short proof of this standard fact is given at the end. Reflection Principle ==================== (Reflection Principle) Let $(B_t)_{t\ge 0}$ be an SBM and let $T$ be a stopping time with respect to $(\mathcal F_t)_{t\ge 0}$. Define $$B_t^T:=\begin{cases} B_t, & 0\le t \le T \\ 2B_T-B_t, & t>T. \end{cases}$$ Then $(B_t^T)_{t\ge 0}$ is an SBM. Note that $(B^T_t)$ clearly has continuous sample paths. By the uniqueness theorem, to complete the proof it is enough to show, for each $n\ge 1$ and $0< t_1<\cdots <t_n<\infty$, and constants $\lambda_j\in\mathbb{R}$ ($1\le j\le n$), that $ \mathbb E[e^{iX^T}]= \mathbb E[e^{iX}], $ where $$X:=\sum_{j=1}^n \lambda_j B_{t_j}\quad \text{and}\quad X^T:=\sum_{j=1}^n \lambda_j B_{t_j}^T.$$ For notational convenience, set $t_0 :=0$ and $t_{n+1} :=\infty$. First, suppose $T$ takes only finitely many values $0\le a_1<\cdots <a_m <\infty $. For each $1\le r\le m$, choose $k_r$ such that $t_{k_r} \le a_r < t_{k_r+1}$ and let $$Y_r:=\sum_{j=1}^{k_r}\lambda_j B_{t_j} + \left( \sum_{j=k_r+1}^n \lambda_j\right) B_{a_r}\quad\text{and}\quad Z_r:=\sum_{j=k_r+1}^n \lambda_j (B_{t_j}-B_{a_r}).$$ Note that $Y_r$ is $\mathcal F_{a_r}$-measurable, $Z_r$ is independent of $\mathcal F_{a_r}$ by , and also that $$X= \sum_{r=1}^m (Y_r+Z_r)\mathbbm 1_{\{T=a_r\}}\quad\text{and}\quad X^T = \sum_{r=1}^m (Y_r-Z_r)\mathbbm 1_{\{T=a_r\}}.$$ Therefore $$\begin{aligned} \mathbb E[e^{iX^T}]&= \sum_{r=1}^m \mathbb E[e^{i(Y_r-Z_r)}\mathbbm 1_{\{T=a_r\}}]\\ %&= \sum_{r=1}^m \mathbb E[\mathbb E[e^{i(Y_r-Z_r)}\mathbbm 1_{\{T=a_r\}}|\mathcal F_{a_r}]]\\% &= \sum_{r=1}^m \mathbb E[e^{iY_r}\mathbbm 1_{\{T=a_r\}}\mathbb E[e^{-iZ_r} |\mathcal F_{a_r}]] \intertext{(since $e^{iY_r}\mathbbm 1_{\{T=a_r\}}$ is $\mathcal F_{a_r}$-measurable)} &= \sum_{r=1}^m \mathbb E[e^{iY_r}\mathbbm 1_{\{T=a_r\}}]\mathbb E[e^{-iZ_r}] \intertext{(by independence of $Z_r$ with respect to $\mathcal F_{a_r}$)} &= \sum_{r=1}^m \mathbb E[e^{iY_r}\mathbbm 1_{\{T=a_r\}}]\mathbb E[e^{iZ_r}] \intertext{(by symmetry of $Z_r$)} &=\sum_{r=1}^m \mathbb E[e^{i(Y_r+Z_r)}\mathbbm 1_{\{T=a_r\}}] %\intertext{(by reversing the steps)} = \mathbb E[e^{iX}] \end{aligned}$$ (by reversing the steps to get the first equality above). To extend the result to a general stopping time $T$, we simply approximate $T$ by stopping times $T_j$ which take only finitely many values. To make this precise, let $T_j(\omega) := 2^j$ if $T(\omega) > 2^j$ and $ T_j(\omega):= k2^{-j} $ if $(k-1)2^{-j}< T(\omega)\le k2^{-j}\le 2^j$. Then clearly $T_j\rightarrow T$ almost surely and, by continuity of the sample paths of $(B_t)$, $X^{T_j}\rightarrow X^T$ almost surely. Thus, by the bounded convergence theorem, $$\mathbb E[e^{iX^T}]=\lim_{j\rightarrow \infty}\mathbb E[e^{iX^{T_j}}]=\mathbb E[e^{iX}].$$ Finally, we prove . By definition of the conditional expectation operator, we have to show that, for all $A\in\mathcal F_s$,$$\label{eq:prove} \mathbb{E}[V \mathbbm{1}_A] = \mathbb{E}[V] P(A).$$The collection $\mathcal{G}$ of all $A \in \mathcal{F}_s$ for which holds is easily seen to be a *monotone class* (i.e., $\mathcal{G}$ is closed under countable increasing unions and decreasing intersections) containing the $P$-null sets. Moreover, given $m \ge 1$ and $0< s_1 <\dots< s_m \le s$, $\mathcal{G}$ contains the $\sigma$-algebra $\sigma(B_{s_1},\dots, B_{s_m})$, the smallest $\sigma$-algebra for which each $B_{s_j}$ ($1 \le j \le m$) is measurable: this follows from independence of Brownian increments. The union over all of these $\sigma$-algebras as $m$ and $(s_j)_{j=1}^m$ vary is an algebra whose augmentation by the $P$-null sets generates $\mathcal{F}_s$. The monotone class lemma (see e.g. [@M p. 4]) now gives $\mathcal{G} = \mathcal{F}_s$. [1]{} Louis Bachelier, [*Théorie mathématique du jeu*]{}, Annales Scientifiques de l’École Normale Supérieure [**18**]{} (1901), 143–209. Paul Lévy, [*Sur certains processus stochastiques homogènes*]{}, Comp. Math. [**7**]{} (1940), 283–339. Paul Malliavin, [*Integration and Probability*]{}, Springer-Verlag, New York, 1995. René L. Schilling and Lothar Partzsch, [*Brownian Motion An Introduction to Stochastic Processes*]{}, De Gruyter, 2012. Michael Steele, [*Stochastic Calculus and Financial Applications*]{}, Springer-Verlag, New York, 2001. [^1]: The first author was supported by The National Science Foundation under Grant Number DMS-1361461.
--- abstract: 'We conduct a model-independent effective theory analysis of hypercharged fields with various spin structures towards understanding the diboson excess found in LHC run I, as well as possible future anomalies involving $WZ$ and $WH$ modes. Within the assumption of no additional physics beyond the standard model up to the scale of the possible diboson resonance, we show that a hypercharged scalar and a spin 2 particle do not have tree-level $WZ$ and $WH$ decay channels up to dimension 5 operators, and cannot therefore account for the anomaly, whereas a hypercharged vector is a viable candidate provided we also introduce a $Z''$ in order to satisfy electroweak precision constraints. We calculate bounds on the $Z''$ mass consistent with the Atlas/CMS diboson signals as well as electroweak precision data, taking into account both LHC run I and II data.' author: - | Aqil Sajjad\ [ `sajjad@physics.harvard.edu`]{}\ [*Department of Physics, Harvard University, Cambridge, MA 02138, USA*]{} bibliography: - 'ref.bib' title: ' **Understanding diboson anomalies**' --- Introduction ============ The Atlas and CMS collaborations have recently reported several excesses in the diboson decay channels with a possible resonance around $2~{\rm TeV}$ in run 1 of the LHC [@Aad:2015owa; @Khachatryan:2014hpa; @CMS:2015gla]. The excesses include the $WZ$, $WW$ and $ZZ$ channels with local significances of $3.4\sigma$, $2.6\sigma$ and $2.9\sigma$, respectively with a resonance around $2~{\rm TeV}$ reported by Atlas and the $WH$ mode with a resonance around $1.8-1.9~{\rm TeV}$ with a deviation of $2.2\sigma$ according to CMS. The recently announced run 2 results on the other hand do not show any such excess but the data is not enough to rule out the effect at 95$\%$ confidence level [@ATLAS-CONF-2015-073; @CMS:2015nmz]. Specifically, the luminosities for the $8~{\rm TeV}$ run were $20.3~{\rm fb^{-1}}$ and $20~{\rm fb^{-1}}$ for Atlas and CMS, respectively, whereas those for the $13~{\rm TeV}$ data released in December were $3.2~{\rm fb^{-1}}$ and $2.6~{\rm fb^{-1}}$ for the two collaborations. Consequently, while the run II results put more stringent bounds on the possible $2~{\rm TeV}$ resonances, more data is needed to come to a definite conclusion about the excesses reported in run I. The tightest bound on the cross-section times branching ratio for the $WZ$ channel from LHC run I comes from the $WH$ data through the Goldstone equivalence theorem [@Hisano:2015gna], which gives a 95$\%$ confidence upper limit of about $7~{\rm fb}$ for a $2~{\rm TeV}$ resonance for a $8~{\rm TeV}$ $PP$ collision [@Khachatryan:2015bma]. This corresponds to about $54.7~{\rm fb}$ for a $13~{\rm TeV}$ experiment. In run II, the strictest constraint comes from the $WZ$ data, giving an upper bound of about $40~{\rm fb}$ for a $13~{\rm TeV}$ center of mass energy, which is smaller but still large enough to leave open the possibility that further data could lead to a discovery of a new particle. Beyond the $2~{\rm TeV}$ LHC run I diboson excess, the $WZ$ channel could also potentially arise in future experiments at other energies and will therefore also be an important part of future searches for new physics. In this backdrop, it is worth developing a framework for understanding such diboson excesses. The purpose of this paper is to offer a simple model-independent effective theory perspective for understanding charged resonances with diboson decays. The motivation for focusing on charged particles is partly that the largest reported statistical significance for the run I diboson excesses is for the $WZ$ channel, and partly that this involves a more constrained and therefore more interesting symmetry structure than does a simple neutral resonance (though of course what is more interesting can be a matter of perspective). Here we might point out that there is also the possibility of leakage between the $WZ$, $WW$ and $ZZ$ channels due to misidentification. One interesting work in this regard is [@Allanach:2015hba] which carries out a goodness of fit comparison for the various channels (see table V). The 3 fits they compare involve setting one of $WW$ or $WZ$ signal to be zero and fitting the data in terms of the remaining two modes (rows 1 and 2), or by setting the $WW$ and $ZZ$ to be nearly zero and explaining the data almost entirely in terms of $WZ$ (row 3). They find that all 3 fits have $\Delta \chi^2$ values less than 1, though setting $WZ$ to be zero gives a marginally better fit than the one in which $WW$ and $ZZ$ are both set to zero. With the 3 fits being compareable in quality, the diboson signal could be explained more or less equally well by either of the 3 combinations (i.e. $WZ$ with $ZZ$, $WW$ with $ZZ$ or almost entirely in terms of $WZ$) unless more data allows better discrimination. This means that there is considerable room for misidentification between the various channels, with a $W$ mistaken for a $Z$ and vice versa. With that being so, and with the reported statistical significance of the individual $WZ$ channel being the highest, the diboson excess could be explained entirely by a charged resonance decaying to $WZ$, which is the scenario we focus on through most of this work, though we also briefly address the possibility of an accompanying neutral resonance accounting for the reported $WW$ and $ZZ$ events. Our strategy will be to follow an effective theory approach. We will consider hypercharged fields that are singlets under the standard model $SU(2)_l$ group with different spin structures (scalar, spin 1 and spin 2) for the possible $2~{\rm TeV}$ particle and construct Lagrangian terms allowed by the symmetries. Since we are assuming $SU(2)$ singlets, the only way for these new fields to get an electric charge is for them to have hypercharge $\pm1$. For each spin case, we will start by assuming that there is no physics in addition to the standard model up to the $2~{\rm TeV}$ range except the possible resonance particle and relax this assumption only if we are forced to do so by some consistency requirements or existing experimental constraints. We will run into such an issue for the vector case where the electroweak precision bounds will force us to include a neutral $Z'$ in addition to $W'$. A $Z'$ could also potentially account for some of the $WW$ and $ZZ$ excess found in the LHC run I data, a possibility we will briefly discuss in the course of our analysis. It is worth mentioning that in [@Kim:2015vba], a somewhat similar effective theory framework has been used to investigate various spin structures for possible singlet resonances to account for the recently reported diboson anomaly. However, their study is strictly restricted to neutral candidates with the view that the reported $WZ$ excess could well be a $WW$ or $ZZ$ channel being mistaken as $WZ$ due to possible contamination [@Allanach:2015hba], whereas in this paper, we mainly focus on charged resonances. Moreover, in the analysis of a vector resonance [@Kim:2015vba] does not take into account electroweak precision bounds which require the introduction of a $W'$ in addition to $Z'$ in order to avoid large deviations of the $\rho$ parameter from unity. Another related work is [@Fichet:2015yia] which sets up the effective theory for spin 0 and 2 SM singlet resonances in the context of the diboson anomaly. Yet another alternative is to consider an $SU(2)_l$ triplet with vanishing hypercharge [@Thamm:2015csa]. As for the hypercharge case, we would like to acknowledge that [@Grojean:2011vu] is one of the earliest papers discussing the phenomenology of a $W'$ using an effective theory approach and even predicted the $WZ$ diboson decay channel back in 2011. We may also mention that some works have also considered explanations other than the diboson interpretation involving a $WW$, $ZZ$ or $WZ$ pair. These include the triboson scenario [@Aguilar-Saavedra:2015rna; @Aguilar-Saavedra:2015iew; @Bhattacherjee:2015svr] or the possibility that some BSM boson with a mass sufficiently close to $m_w$ and $m_z$ may have been misidentified as a $W$ or $Z$ [@Chen:2015xql; @Allanach:2015blv]. The organization of this paper will be as follows. In section 2, we consider a hypercharged scalar as a candidate for the possible $2~{\rm TeV}$ resonance. We show that such a scalar cannot account for the diboson anomaly since the symmetries of the standard model prohibit its decay to $WZ$ and $WH$ at tree-level at least up to dimension 5 operators. We also extend the discussion to the case of the 2 higgs doublet model and show that a hypercharged scalar along with the 2HDM cannot account for the $WZ$ excess either. We may also mention here that the 2HDM by itself cannot account for the diboson signal since the tree-level $WZ$ decay of the heavy charged higgs is well-known to be forbidden by the custodial symmetry [@Branco:2011iw; @Yagyu:2012qp] and there are only a few studies where possibilities involving extensions of the 2HDM have been considered [@Chen:2015xql; @Omura:2015nwa; @Sierra:2015zma]. In section 3, we discuss the possibility of a hypercharged vector $W'$ that quadratically mixes with $W$ as a possible explanation for the diboson signal. The underlying physics for such a vector particle may be an additional gauge field such as that in the $SU(2)_l \times SU(2)_r$ model [@Mohapatra:1974hk; @Mohapatra:1974gc; @Senjanovic:1975rk] which has also received considerable interest in the context of the diboson anomaly with [@Patra:2015bga; @Hisano:2015gna; @Cheung:2015nha; @Dobrescu:2015qna; @Gao:2015irw; @Brehmer:2015cia; @Dev:2015pga; @Das:2015ysz; @Aguilar-Saavedra:2015iew; @Shu:2015cxm; @Shu:2016exh] being some especially interesting works. [@Berlin:2016hqw] goes a step further by considering the left-right-symmetric model to simultaneously explain the $2~{\rm TeV}$ diboson excess as well as the $750~{\rm GeV}$ diphoton signal. We can of course also consider more complicated extensions of the SM gauge group such as those considered in [@Cao:2015lia; @Evans:2015cqq; @Aydemir:2015oob]. Alternatively, a hypercharged $W'$ may also arise from a composite theory [@Low:2015uha; @Carmona:2015xaa]. Working in our model independent effective theory approach, we show that a hypercharged $W'$ vector field can indeed account for the observed excess and calculate the relevant cross-section and decay rates. However, this scenario violates electroweak precision bounds on the $\rho$ parameter unless we also introduce a $Z'$ that quadratically mixes with $Z$. We calculate constraints on the $Z'$ mass and the $ZZ'$ mixing based on electroweak precision data. In section 4, we discuss the hypercharged spin 2 case and show that like the scalar, it too cannot have diboson decays to $WZ$ and $WH$, though the argument for this is slightly different. We thus conclude that within the assumption that there is no additional physics beyond the standard model up to the scale of the possible resonance ($2~{\rm TeV}$ in this case), only a vector resonance can possibly account for the recently reported $WZ$ and $WH$ anomalies, and therefore studies on this subject should focus their efforts accordingly. Hypercharged lorentz scalar =========================== We will consider this for the regular standard model as well as its extended version in which there are two higgs doublets and show that a hypercharged scalar cannot account for the diboson excess. A hypercharged scalar added to the regular standard model --------------------------------------------------------- We start by considering an $SU(2)_l$ singlet scalar $\phi$ with hyper charge 1 and try to construct interactions that give its decays into $WZ$ and $WH$. Throughout this paper, we will work in the notation where the higgs doublet $H$ transforms as $(2, -1/2)$ under the standard model $SU(2)_l \times U(1)$ group, and acquires a non-zero vacuum expectation value in its first component from electroweak symmetry breaking. With $H$ having hypercharge $-1/2$, we need $\phi$ coupling to two powers of $H$ to get a hypercharge singlet. Additionally, we throw in a pair of covariant derivatives in order to obtain couplings of $\phi$ $WZ$ and $WH$ (in any case, $\phi H\dot H$ is zero). We thus get the dimension 5 interaction \_[hh]{} = - HD\_D\^H +h.c \[d\_phi-interaction\] where $\Lambda$ is the scale associated with the underlying UV physics. This is the only (dimension 5) coupling of $\phi$ to two powers of $H$ since $\phi (D_\mu H) \dot (D^\mu H)$ is zero due to the anti-symmetry of the $SU(2)$ invariant dot product, and $(D_\mu \phi^*) H\dot D^\mu H$ is related to $\phi^* H\dot D_\mu D^\mu H$ through integration by parts. Naively, if we expand this in terms of the higgs components, we get $\phi W^\mu Z_\mu$ and $(\partial_\mu \phi) W^\mu (H+V)^2$ interactions, in which $V$ is the higgs vacuum expectation value. We may therefore be led to believe that we should get $WZ$ and $WH$ decays of $\phi$. However, if we use the equations of motion for the higgs doublet to eliminate $D_\mu D^\mu H$, we find that (\[d\_phi-interaction\]) is equal to ( Y\_u H |U\_r Q\_l +Y\_d |Q\_l D\_r H +Y\_l |L\_l e\_r H ) + h.c \[yukawa-factor-supressed-phiHQQ\] where $Q_l$ and $L_l$ are the left-handed quark and lepton $SU(2)$ doublets, $Y_u$, $Y_d$ and $Y_l$ are the Yukawa couplings for up and down type quarks and leptons, respectively, and there is an implicit quark generation index (and a CKM matrix for terms in which $u$ type quarks are coupled to $d$ type quarks when we switch to the mass eigen basis). The $\phi W^\mu Z^\mu$ and $(\partial_\mu \phi) W^\mu (H+V)^2$ terms are all gone and we do not get diboson decays of $\phi$ at least at tree-level. The absence of these decays can also be seen by working carefully with (\[d\_phi-interaction\]). The $(\partial_\mu \phi) W^\mu (H+V)^2$ term contains a mixing between $\phi$ and $W$. This results in an additional set of contributions to the diboson decay amplitude where $\phi$ first flips to a virtual $W$, which then decays to $WZ$ or $WH$ through the standard model $WWZ$ and $WWH$ couplings. And this additional set of contributions (through the virtual $W$) exactly cancel the contributions from the direct $\phi W^\mu Z_\mu$ and $(\partial_\mu \phi) W^\mu H$ interactions due to the custodial symmetry. We have thus found that a hypercharged scalar, at least by itself, cannot account for the observed anomaly as it does not have the required diboson decays at tree-level up to operators of dimension 5[^1]. We have not even addressed the other question of getting $pp \to \phi$ with a large enough cross-section. The issue on this front arises from the fact that we are unable to obtain Yukawa interactions between quark bilinears and $\phi$ except through non-renormalizable higgs couplings of the form $\phi H \dot \bar U P_r Q_l$ and $\phi H\dot Q_l P_r D_r$. The Yukawa interactions of $\phi$ to charged quark bilinears thus obtained are suppressed by $V/\Lambda$, which results in very small cross-sections for $pp \to \phi$ even if we are able to do some model building to get the couplings of the first generation quarks to be close to unity. If we try to write couplings of $\phi$ to a pair of right-handed quark fields, then Lorentz-invariance forces us to have currents, and we can only get couplings like $(D_\mu \phi) \bar U_r \gamma^\mu D_r$, which turns out to be further suppressed due to angular momentum conservation). However, at least in principle, it is possible that we might be able to produce $\phi$ from a $pp$ collision in a large enough number to be detectable in a next generation collider if not the LHC. But the absence of diboson decays of $\phi$ means that a stand-alone hypercharged scalar added to the standard model will have to be ruled out as a candidate for explaining any observed diboson signal even in next generation collider experiments. Extending to the 2 higgs doublet model -------------------------------------- We might be tempted to ask whether the above conclusion (i.e. the absence of $WZ$ and $WH$ decays) also holds for the 2 higgs version of the standard model since there, we can also write interactions in which $\phi$ (or its covariant derivative) couples to a product of the two higgs doublets (or their covariant derivatives) rather than the same doublet. We now show by working with the type II 2HDM that the answer is in the affirmative at least for the $WZ$ channel. For the type II 2HDM, our hypercharged scalar can have the cubic interactions with a pair of higgs fields \_[HH]{} H\_u\^H\_d +h.c \[phi-higgs-coupling-2hdm\] where $H_u$ and $H_d$ transform as $(2, 1/2)$ and $(2, -1/2)$ respectively under the $SU(2) \times U(1)$ gauge group and have the components H\_u = and H\_d = We can write the neutral components in terms of their vacuum expectation values and real and imaginary parts as H\^0\_u &=& (V\_u + X\_u + i Y\_u )\ H\^0\_d &=& (V\_d + X\_d + i Y\_d ) where $X_u$, $X_d$, $Y_u$ and $Y_D$ are all real scalar fields, and the vacuum expectation values $v_u$ and $v_d$ satisfy $\sqrt{v_u^2 +v_d^2} = v = 246~{\rm GeV}$. We also define the angle $\beta$ in terms of the equation $\tan\beta = V_u/V_d$. with the neutral components acquiring non-zero vacuum expectation values, (\[phi-higgs-coupling-2hdm\]) contains a quadratic mixing between $\phi$ and the charged higgs $H^\pm$ H\^- +h.c \[phi-charged-higgs-mixing\] where $H^\pm$ is the combination H\^= H\^\_u + H\^\_d We thus have a quadratic mixing through which $\phi$ inherits all the decays of the charged higgs. It is well-known from the literature on the 2HDM that the charged higgs boson does not have a tree-level decay to $WZ$ due to custodial symmetry (see [@Branco:2011iw; @Yagyu:2012qp] for a good overview). Moreover, there is also no $\phi G^\pm G^0$ term in (\[phi-higgs-coupling-2hdm\]), where $G^\pm$ and $G^0$ are the goldstone modes associated with the $W^\pm$ and $Z$ bosons, respectively, and are given by G\^= H\^\_u -H\^\_d and G\^0 = Y\_u -Y\_d Therefore, we conclude that $\phi$ does not have a $WZ$ decay at least at tree-level. As for $\phi \to WH$, the situation is slightly more subtle since the neutral scalar states in general have a different diagonalization from the charged and pseudoscalar states. (\[phi-higgs-coupling-2hdm\]) gives the coupling G\^- (x\_d -x\_u ) and unless the linear combination in parentheses is totally orthogonal to the light neutral higgs mode, we do get a $\phi \to WH$ contribution. That said, since the recently observed diboson excesses includes a larger $WZ$ signal, and since $\phi$ added to the 2HDM does not give any tree-level $WZ$ decay, we conclude that the 2HDM cannot account for the $WZ$ excess. However, this still leaves one more possibility involving the 2HDM which we now very briefly address. What if the quadratic mixing between $\phi$ and the charged higgs creates a heavy mass eigenstate with mass $2~{\rm TeV}$ and a light eigenstate whose mass is somewhere near $m_w$ and $m_z$. Could the observed excess be accounted for by the decay of the heavier eigenstate to $Z$ and the lighter mode misinterpreted as the $WZ$ channel? A somewhat similar scenario has been proposed in [@Chen:2015xql] for the pseudo scalar higgs where it was suggested that if we add a SM gauge singlet complex scalar to the 2HDM, then it is possible to generate mixings between the pseudo scalar component of the singlet with the massive neutral pseudo scalar higgs. If the lighter pseudo-scalar eigen state arising from this mixing has a mass sufficiently close to the $Z$ mass, then the decay of the charged higgs to a $W$ boson along with this lighter pseudo-scalar could potentially have been mistaken as $WZ$. However, while the scenario of the charged higgs of the 2HDM quadratically mixing with $\phi$ to give a light particle which may have been confused as $Z$ may seem appealing, it is not viable since this will also give an overly large contribution to the decay of the top quark to the lighter eigen state. The vector case =============== We now consider a vector field $W'$ with hypercharge $\pm 1$ [@Grojean:2011vu]. Such a field can only couple to right-handed fermion currents g\_r (W’\_|U\_r \^D\_r + W’\_|\_r \^e\_r) + h.c where we have also introduced right-handed neutrinos. For simplicity, we will assume that these interactions are flavour diagonal and all quark generations have the same coupling to $W'$. While our goal in this paper is to work in the effective theory framework, let us make some brief comments to motivate that such a theory is indeed possible. For a vector field to have a charge under an abelian gauge field, it either needs to be a non-abelian gauge field itself or a composite particle. The case of a $W'$ being a non-abelian gauge field can for instance arise from a $SU(2)_l \times SU(2)_r \times U(1)$ model [@Mohapatra:1974hk; @Mohapatra:1974gc; @Senjanovic:1975rk] where $W'$ is an $SU(2)_r$ gauge field which acts on right-handed fermion $SU(2)_r$ doublets. The higgs field is an $SU(2)_l\times SU(2)_r$ object with 2 of its components acquiring non-zero vacuum expectation values as discussed by [@Hisano:2015gna] in the context of the diboson anomaly. The higgs Yukawa terms which give masses to fermions are of the form $H_{i j} \bar f_{L, i} f_{R, j}$, where $L$/$R$ denote left/right handed and $i$ and $j$ are $SU(2)_l$ and $SU(2)_r$ indices. This requires the introduction of right-handed neutrinos in order to account for lepton masses. However, in a limit where one of the higgses is very heavy and can be integrated out, we get an effective theory in which the higgs is just an $SU(2)_l\times U(1)$ doublet and $W'$ is a hypercharged vector with no other symmetry indices. With $W'$ being an $SU(2)$ gauge field, there also has to be a $Z'$, though it is heavier than $W'$ because of $SU(2)_r \times U(1)$ symmetry breaking which also gives the $W'$ its mass. In the event of $W'$ being a composite field, we do not need to have an $SU(2)_l\times SU(2)_l$ higgs to account for fermion masses, and therefore we start with the regular standard model higgs doublet even in the full theory. One would also generally expect a $Z'$ in the composite case, though now the $W'$ and $Z'$ masses are not produced by the breaking of a gauge symmetry, and have different underlying dynamics. In short, the effective theory for a composite $W'$ and $Z'$ is somewhat similar to the $SU(2)_r\times SU(2)_l$ gauge theory, except that it does not necessitate having right-handed neutrinos at least from any symmetry requirements. It is of course another matter that the right-handed neutrino should be introduced regardless of that because of the non-zero mass for the neutrinos. Having argued that a hypercharged $W'$ is indeed plausible, let us now proceed to discuss its physics. As pointed out by [@Hisano:2015gna], a $W'$ needs to satisfy 2 sets of constraints: 1. The electroweak precision bounds which constrain the mixing between $W$ and $W'$. This mixing results in deviations of the $\rho$ parameter from unity, and are tightly bound [@Peskin:1991sw; @delAguila:2010mx; @Baak:2014ora]. 2. There are also the Drell-Yan bounds that the production cross-section times leptonic decay branching ratio for $W'$ ($\sigma(pp \to W') \times Br(W'\to LL)$) should be much smaller than $1 ~{\rm fb}$ [@Aad:2014cka; @ATLAS:2014wra; @Khachatryan:2014fba; @Khachatryan:2014tva]. To satisfy the first of these requirements, we will require that $Z'$ not be much heavier than $W'$. This way, the deviations of the $\rho$ parameter from 1 due to the $WW'$ are somewhat offset by effects due to the $ZZ'$ mixing. We will return to this shortly when we introduce $Z'$. As for the Drell-Yan constraints, these are satisfied if the right handed neutrinos are heavier than $W'$. Given that the lower bounds on right-handed neutrino masses are much larger anyway, the Drell-Yan bounds are already satisfied and we will not need to discuss them any further. Now, coming to the higgs interactions of $W'$, we now write the dimension 4 term i c\_ W\^[’+]{} HD\_H +h.c = W\^[’+]{} W\^- \_(H+V)\^2 + h.c \[WW’-interaction\] where we have expanded the higgs doublet in unitary gauge H = \[higgs-unitary-gauge\] with $V = 246 GeV$. This not only contains a quadratic mixing between $W'$ and $W$, but also has a $W' W H$ interaction. The $W'\to WH$ decay therefore has 2 contributions. One from the direct coupling and the other through the $WW'$ mixing which flips a $W'$ to a virtual $W$, which in turn decays to $WH$ through the standard model $WWZ$ or $WWH$ couplings. However, unlike the hypercharge scalar case, these two contributions do not cancel. As for the $WZ$ decay, there is no direct $W'WZ$ coupling and the only tree-level contribution therefore is through a virtual $W$ produced by the $WW'$ mixing. With $2~{\rm TeV}$ much larger than the $W$ and $Z$ masses, we can work in the limit where $m_w$, $m_z$ and $V$ are very small. This allows us to use the Goldstone equivalence theorem and we get the $W' \to WZ$ decay rate (W’WZ, WH) \[w-prime-wz-decay\] which for $m_{w'} = 2~{\rm TeV}$ gives $6.63 \, c_\pm^2 \, GeV$. The decay width for $W'$ to a pair of quarks in the massless quark limit is (W’u\_i|d\_j) = \[w-prime-qq-decay\] If $g_r$ is the same as the $W$ coupling to charged quark currents $e_w$, as is usually assumed for the $SU(2)_l \times SU(2)_r$ model to satisfy anomaly cancellation, then this gives $4.09 \, GeV$ for $m_{w'} = 2~{\rm TeV}$. The $WZ$, $WH$ and $u_i \bar d_j$ channels are the major decay modes of $W'$. Beyond these, the only other 2 body decay is the $W\gamma$ process, but it is highly suppressed because the photon does not have a longitudinal mode. Therefore, the leading order total decay width comes to about (W’) = + \[total-w-prime-decay-width\] Now, coming to the $pp \to W'$ process, we used CT14 PDFs [@Dulat:2015mca] for calculating the cross-section. For the $8 TeV$ $pp$ center of mass energy, we obtain the cross-sections \_[8 [TeV]{}]{}(pp W’\^) &= 1440.1 g\_r\^2  [fb]{}\ \_[13 [TeV]{}]{}(pp W’\^) &= 11.26 10\^3 g\_r\^2  [fb]{} which for $g_r = e_w$ give $74.06~{\rm fb}$ and $579~{\rm fb}$, respectively[^2]. From (\[w-prime-wz-decay\]), (\[total-w-prime-decay-width\]) and the assumption that $g_r$ is equal to the $W$ coupling to charged standard model fermions, we can obtain the branching ratios for $WZ/WH$ and the cross-sections for $W'$ production in a collision of 2 protons. Table \[interesting\_c\_values\] shows some interesting values of $c_\pm$ along with the corresponding branching ratio times cross-sections. $|c_\pm|$ $Br(W' \to WZ)$ $\Sigma_{8~{\rm TeV}}(pp \to WZ)$ in $fb$ $\Sigma_{13~{\rm TeV}}(pp \to WZ)$ in $fb$ ----------- ----------------- ------------------------------------------- -------------------------------------------- 1.00 0.260 19.2 150 0.464 0.0945 7.0 54.7 0.385 0.0691 5.12 40.0 0.193 0.0193 1.43 11.2 : Some interesting values of $c_\pm$ along with corresponding branching ratios and cross-section times branching ratios for the $PP\to W' \to WZ$ channel for a $2~{\rm TeV}$ resonance.[]{data-label="interesting_c_values"} Some comments about the table of $|c_\pm|$ values are in order. The $19.2~{\rm fb}$ cross-section times branching ratio value for $|c_\pm| =1$ for $8~{\rm TeV}$ falls within the range allowed by the run I $WZ$ data but is clearly ruled out by the run II results at 95$\%$ confidence level. In any case, as [@Hisano:2015gna] points out, CMS run I results also put a $7~{\rm fb}$ bound on the $WH$ cross-section times branching ratio [@Khachatryan:2015bma], which through the Goldstone equivalence theorem also imposes the same bound on the $WZ$ cross-section. The next $|c_\pm|$ value of $0.464$ in the table corresponds to this bound. Next is $|c_\pm| = 0.385$, giving the $40~{\rm fb}$ cross-section times branching ratio value for $13~{\rm TeV}$, which is the upper bound according to run II $WZ$ data [@ATLAS-CONF-2015-073; @CMS:2015nmz]. The run II data for the $WH$ channel, on the other hand, is less constraining and gives an upper bound of $60~{\rm fb}$ [@Atlas2WH], and therefore, we do not include it in our table of interesting data points. Now, as we will shortly see, through the quadratic mixing between $W'$ and $W$ in (\[WW’-interaction\]), all the above-mentioned values for $c_\pm$ result in a larger shift in $m_w$ than what is permitted by electroweak precision bounds, requiring the simultaneous introduction of a $Z'$ in the theory. The last line shows the threshold value of $|c_\pm| = 0.193$ for which the $\rho$ parameter lies at the boundary of the region allowed by precision data without the inclusion of a $Z'$. This corresponds to a cross-section times branching ratio of about $11.2~{\rm fb}$ for a $13~{\rm TeV}$ experiment. Since this is small but not totally negligible, this means that there is also a considerable region of parameter space where the $Z'$ is much heavier than the $W'$ and therefore does not appear in our effective theory at the $TeV$ or even $10~{\rm TeV}$ scale. We now address the issue of electroweak precision constraints in some detail and extract bounds on the mass of the $Z'$. The $WW'$ mixing term is W\^[’+]{} W\^- \_+h.c = m\_w\^2 W\^[’+]{} W\^- \_+h.c where we have taken $m_w^2$ as the tree-level value for the $W$ mass squared, which is equal to $\frac{e^2 V^2}{4 s_W^2}$. This allows writing the $WW'$ mass matrix as m\_w\^2 where $m_{w'} = 2~{\rm TeV}$. By diagonalizing this matrix, we get the leading order percentage shift in the $W$ mass squared = - \[W-mass-percentage-shift\] We can now relate this with deviations of the $\rho$ parameter from unity. The $\rho$ parameter is given by = Therefore, in terms of the Peskin-Takeuchi $T$ parameter [@Peskin:1991sw], we get T = -1 = - +… \[T-parameter-formula\] From electroweak precision measurements of the $T$ parameter [@Baak:2014ora], we have $T = 0.10 \pm 0.07$ for $U = 0$. This gives the bounds (since the $95$ percent confidence interval is roughly about $2\sigma$ around the mean), -0.04 &lt; T &lt; 0.24 \[T-bounds\] Now, from (\[T-parameter-formula\]) and (\[W-mass-percentage-shift\]), we get T = - if we assume $\Delta m_z^2 = 0$. And with $m_{w'}^2 = 2~{\rm TeV}$, this for any $|c_{\pm}| > 0.193$ is outside the $T$ bounds in (\[T-bounds\]). Since the more interesting values of $c_\pm$ for explaining the $2~{\rm TeV}$ diboson excess are above this threshold value as shown in table \[interesting\_c\_values\], this means that we must have a $Z'$ lurking nearby with a mixing with $Z$ such that the deviation in $m_z^2$ sufficiently offsets the effect of the shift in the $W$ mass. Specifically, we get the constraint -0.24 - &lt; &lt; 0.04 - \[algebraic-constraint-percentage-shift-Z\] Now, if $Z'$ has a quadratic mixing term with $Z$ of the form $m_{zz'}^2 Z'_\mu Z^\mu$ , the mass matrix for $Z$ and $Z'$ can be written as m\_z\^2 and diagonalizing this gives = - \[Z-mass-percentage-shift\] By combining (\[Z-mass-percentage-shift\]) with (\[algebraic-constraint-percentage-shift-Z\]), we obtain bounds on $m_{z'}$ and $m_{zz'}$ which are shown in figure \[Z-mass-constraint-plot\]. We focus on $m_{zz'}$ from $0~{\rm to}~V$ to keep the $ZZ'$ mixing small. The region between the two dashed red curves gives the $m_{z'}$ masses allowed by precision constraints for a given $m_{zz'}$ for $|c_\pm| = 0.464$, corresponding to a $WZ$ cross-section of $7~{\rm fb}$ for a center of mass $PP$ energy of $8~{\rm TeV}$ and about $54.7~{\rm fb}$ for $13~{\rm TeV}$. This was the upper bound on the $WZ$ mode from LHC run I. The blue curves on the other hand, give the $m_{z'}$ bounds corresponding to $|c_\pm| = 0.385$, which gives a $WZ$ cross-section times branching ratio of $40~{\rm fb}$ for the $13~{\rm TeV}$ case, which is the upper bound from run II data. The orange curve represents the lower bound on the $Z'$ mass for the threshold value of $c_\pm = 0.193$ below which we do not need to introduce a $Z'$ in the theory in order to satisfy precision constraints. This corresponds to a $WZ$ cross-section times branching ratio of $\sigma_{WZ}$ = $11.2~{\rm fb}$ for a $13~{\rm TeV}$ collision. For any cross-sections smaller than this value, the $z'$ mass must lie somewhere in the region above the orange curve, and this includes the uninteresting scenario that the recently reported excesses do not correspond to any new particle. The region below the red curves is disallowed even by run I. The region below the blue curves is ruled out at 95$\%$ confidence level by the run II data. The combined bound curves therefore lie somewhere in the narrow regions between the red and blue curves[^3]. ![Electroweak precision constraints on $m_{z'}$ as a function of $m_{zz'}$ from $0~{\rm to}~V$ for a $2~{\rm TeV}$ $W'$. The red, blue and orange curves correspond to $WZ$ cross-sections of $54.7~{\rm fb}$, $40~{\rm fb}$ and $11.2~{\rm fb}$, respectively, for a $13~{\rm TeV}$ collision. The red curves correspond to the upper bound for the $WZ$ cross-section from run I data, and the region below these curves is excluded as it pertains to larger cross-sections. The blue curves represent the upper bound on the cross-section set by run II $WZ$ data. The orange curve is the lower bound on $m_{z'}$ for a cross-section of $11.2~{\rm fb}$. For smaller cross-sections than this, a $z'$ is not needed to satisfy precision constraints. []{data-label="Z-mass-constraint-plot"}](2tev){width="80.00000%"} We can see that these precision constraints on the $Z'$ mass leave open a wide range of possibilities. For example, a $Z'$ in the $2-4~{\rm TeV}$ range which could potentially be detected at the LHC is very much consistent with the recently reported diboson anomaly. Such a $Z'$ that is slightly heavier than $W'$ could for instance arise from the left-right symmetric model. Interestingly, CMS did report an electron-positron excess at $2.9~{\rm TeV}$ [@CMS-DP-2015-039] in run I, though this was a very small event and taking it too seriously may be somewhat premature at this stage. There is also a large part of open parameter space where $Z'$ can be considerably heavier and therefore difficult to detect at the LHC, as well as the somewhat less likely region from the point of view of model building in which it may be lighter than $2~{\rm TeV}$. Then there is the possibility of a $2~{\rm TeV}$, which could also account for some part of the diboson excess with the somewhat bizarre miracle of $W'$ and $Z'$ masses being the same [^4]. In this case, all the three modes, namely $WZ$, $WW$, and $ZZ$ would be present in the actual physics. However, as we mentioned in the introduction, there is also considerable room for misidentification between the various channels due to the closeness of the $W$ and $Z$ masses, and the analysis of [@Allanach:2015hba] shows that fitting the data entirely in terms of $WZ$ also provides a reasonably good fit with $\Delta\chi^2$ of $0.8$. For this reason, we do not necessarily need a $2~{\rm TeV}$ to explain the diboson excess. However, taking one of the $WZ$ or $ZZ$ signals to be zero also provides fits of nearly similar quality, and therefore, it is also possible that the diboson signal could be coming entirely from a neutral $Z'$ [@Kim:2015vba] or through a mixture of mass degenerate $W'$ and $Z'$ particles decay into all the various diboson channels. That said, having a $W'$ and a $Z'$ with the same mass may require some model building as it is not entirely clear how such a scenario may arise. While the primary focus of this paper is the $2~{\rm TeV}$ excess found in LHC run I, our analysis is of course also applicable to any other value of the resonance. We therefore also show precision bounds on $m_{z'}$ for $m_{w'} = 1.6~{\rm TeV}$ and $2.4~{\rm TeV}$ In figures \[Z-mass-constraint-plot-1600-GeV\] and \[Z-mass-constraint-plot-2400-GeV\], respectively, just to illustrate how this works for 2 other $W'$ masses. While neither run of the LHC has found a noticeable excess at these values thus far, the tightest constraints come from run II $WH$ channel data, which gives upper bounds of $50~{\rm fb}$ and $20~{\rm fb}$, respectively, for the cross-section times branching ratios for these two masses for a $W'$ particle [@Atlas2WH]. In each of these plots, we show bounds on $m_{z'}$ with a blue pair of curves for $\sigma_{WZ}$ corresponding to the above-mentioned upper bounds set by the run 2 $WH$ data, and the orange line represents the threshold value of $|c_{\pm}|$ below which we do not need to introduce a $Z'$ in the theory in order to satisfy precision constraints. These threshold values of $|c_\pm|$ correspond to cross-section times branching ratios of $22.8~{\rm fb}$ and $5.62-20~{\rm fb}$ for $1.6~{\rm TeV}$ and $2.4~{\rm TeV}$, respectively, for a $13~{\rm TeV}$ experiment. We can see that even though no noticeable excess has been reported for these values of the $w'$ resonance, there is still a considerable region of parameter space that remains open. ![Electroweak precision constraints on $m_{z'}$ as a function of $m_{zz'}$ from $0~{\rm to}~V$ for a $1.6~{\rm TeV}$ $W'$. The blue curves represent the lower and upper bounds on $m_{z'}$ for a $\sigma_{WZ}$ of $50~{\rm fb}$, which is the upper bound set by run II $WH$ data, and the region below these curves is disallowed as it represents larger cross-sections. The orange curve is the lower bound on $m_{z'}$ for a cross-section of $22.8~{\rm fb}$. For any cross-sections smaller than this value, a $z'$ is not needed to satisfy precision constraints, and if a $z'$ does exist, then $m_{z'}$ must lie above the orange curve. []{data-label="Z-mass-constraint-plot-1600-GeV"}](1point6tev){width="80.00000%"} ![Electroweak precision constraints on $m_{z'}$ as a function of $m_{zz'}$ from $0~{\rm to}~V$ for a $2.4~{\rm TeV}$ $W'$. The blue curves represent the lower and upper bounds on $m_{z'}$ for a $\sigma_{WZ}$ of $20~{\rm fb}$, which is the upper bound set by run II $WH$ data, and the region below these curves is disallowed as it represents larger cross-sections. The orange curve is the lower bound on $m_{z'}$ for a cross-section of $5.62~{\rm fb}$. For any cross-sections smaller than this value, a $z'$ is not needed to satisfy precision constraints, and if a $z'$ does exist, then $m_{z'}$ must lie above the orange curve. []{data-label="Z-mass-constraint-plot-2400-GeV"}](2point4tev){width="80.00000%"} We conclude this section by listing down the dimension 4 interactions of $Z'$ allowed by symmetries. Continuing with our effective theory approach, we take $Z'$ to be a standard model gauge singlet to make it have no electromagnetic charge. We find that the couplings of $Z'$ to standard model fermions are somewhat less constrained than those of $W'$ as $Z'$ can couple to both left and right-handed fermions [@Kim:2015vba] |f\_i \^Z’\_(c\_l P\_l + c\_r P\_r) f\_i where $i$ is an index labling the various fermions in the standard model. As for the quadratic mixing of $Z'$ with $Z$, the symmetries allow 2 different mechanisms. One of these is kinetic mixing with the hypercharge gauge field as also noted by [@Kim:2015vba] - B\_ B\^ - Z’\_ Z’\^ - Z’\_ B\^ \[kinetic-mixing\] However, there is also the $Z'$ coupling to the higgs current which has not been considered in [@Kim:2015vba] i c\_0 Z’ H\^D\_H = - Z’\_Z\^(V+H)\^2 \[Z’-higgs-current-interaction\] and directly gives a mass mixing of the form $m_{zz'}^2 Z'_\mu Z^\mu$ when we replace the higgs fields with their vacuum expectation values. The former changes the kinetic energy and the latter directly modifies the mass matrix for the $Z$ and $Z'$. Since simultaneously diagonalizing the kinetic energy and mass terms is rather complicated, we can follow a two-step process. First, we can diagonalize (\[kinetic-mixing\]) and rescale $B_\mu$ by $\sqrt{1-\kappa^2}$ to obtain canonically normalized kinetic energy terms. We can then diagonalize the mass term in the next step. Since a detailed analysis of the parameter space is beyond the scope of this paper, we will not carry out this procedure here. We end our discussion of the vector case by noting that the above two quadratic mixing terms with $Z$ results in various diboson decay channels such as $WW$, $ZZ$, $HH$ and $ZH$ as we have mentioned earlier. This not only means that a $Z'$ could possibly also explain the $WW$ and $ZZ$ events in the $2~{\rm TeV}$ diboson excess, but also that searches for neutral diboson resonances should therefore be an integral part of any program for understanding the recently reported diboson anomalies. The spin 2 case =============== The Lagrangian for a massive spin 2 field is the same as the massive graviton (see [@Hinterbichler:2011tt] for an excellent review). The standard practise for gravity is to expand the metric around the Minkowski metric or some other static background as $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$. The dynamics of the graviton are then described by $h_{\mu\nu}$. In this paper, we will denote our hypercharged spin 2 field by $\Pi_{\mu\nu}$ in place of $h_{\mu\nu}$ to avoid confusion with the higgs. Now, if we follow our recipe of coupling our hypercharged fields with two powers of the higgs, we find that we are not able to write down any non-zero interactions. Since $\Pi^{\mu\nu}$ is symmetric, $\Pi^{\mu\nu} (D_\mu H) \dot D_\nu H$ is zero due to the anti-symmetry of the $SU(2)$ invariant dot product. The other possible terms to consider are $(D_\mu \Pi^{\mu\nu}) H\dot D_\nu H$ and $(D_\mu \Pi^{\mu\nu}) (D_\nu H)\dot H$, which are in fact related through integration by parts. Now, it is well-known in the literature on massive gravity (see the appendix for a quick derivation) that D\_\^ = 0 We are therefore forced to conclude that the diboson anomaly cannot be explained by a hypercharged spin 2 resonance. Conclusion ========== We have carried out a detailed effective theory analysis of hypercharged fields with various spin structures to investigate what type of particles could potentially account for the recently reported diboson excess. Working within the assumption that there is no additional physics beyond the standard model up to the scale of the possible diboson resonance, we have shown that a hypercharged scalar and a spin 2 particle do not have $WZ$ and $WH$ decay channels at tree-level (up to operators of at least dimension 5) and must therefore be ruled out as viable explanations for the anomaly. On the other hand, a hypercharged vector that quadratically mixes with $W$ not only has the required diboson decays but can also have a production cross-section in the right range to account for the $WZ$ and $WH$ excesses. However, electroweak precision bounds require that such a $W'$ be accompanied by a $Z'$ that quadratically mixes with $Z$. We have calculated constraints on the $Z'$ and its quadratic mixing with $Z$. These constraints allow the possibility of a $Z'$ that is slightly heavier than $W'$ as predicted by the $SU(2)_r\times SU(2)_l$ model, but also allow for a heavier $Z$ that may be difficult to detect at the LHC. There is also an open region of parameter space in which $Z'$ can be $2.0~{\rm TeV}$ or lighter, though it is not entirely clear if it is possible to come up with a model with such a spectrum. Like $W'$, $Z'$ too should have diboson decay modes due to its quadratic mixing with $Z$, except that these will involve the pairs $WW$, $ZZ$, $HH$ and $ZH$. The search for diboson signals can therefore serve as a very useful probe of new physics which will be of relevance even beyond the recently reported diboson excesses. Acknowledgments {#acknowledgments .unnumbered} =============== The author is especially grateful to Matthew Reece for his guidance and support throughout this project. Special thanks also to Prateek Agrawal, Prahar Mitra, Sabrina Pasterski, Abhishek Pathak, Matthew Schwartz and Taizan Watari for very helpful discussions. Derivation of $D_\mu \Pi^{\mu\nu} = 0$ for a spin 2 field ========================================================= Here we give a quick derivation of the equation $D_\mu \Pi^{\mu\nu} =0$ for a massive spin 2 field, which is well-known to experts on massive gravity but may not be familiar to readers outside that field. Readers interested in learning more on the subject may refer to [@Hinterbichler:2011tt] for a detailed review. The Lagrangian for a massive spin 2 field is the same as a massless graviton with the addition of the Fierz-Pauli mass term which is given by ( (\^ \_)\^2 -\^ \_ ) The equations of motion for $\Pi^{\mu\nu}$ are D\^2 \_ -D\_D\_\^\_-D\_D\_\^\_+\_ D\_D\_\^ +D\_D\_-\_ D\^2 -m\^2(\_ -\_ ) = 0 where $\Pi$ is the trace $\Pi^\mu _\mu$ and $D^2 = D_\mu D^\mu$. Acting on this with $D^\mu$, we get for non-zero $m^2$ m\^2 (D\_\^ -D\_) = 0 \[derivative-of-equation-of-motion\] Inserting this back into the equation of motion gives D\^2 \_ -D\_D\_-m\^2(\_ -\_ ) = 0 Taking the trace of this gives $\Pi = 0$. And plugging this result in (\[derivative-of-equation-of-motion\]) gives D\_\^ = 0 [^1]: We can consider higher dimensional operators like $\phi (H^\d D^\mu H) (H \dot D_\mu H)$, which may give the $\phi \to WZ$ decay at tree-level, but of course the decay rate will be highly suppressed. [^2]: These cross-sections include both $W'^+$ and $W'^-$ production since both contribute to the diboson signal. [^3]: That is, the lower $m_{z'}$ bound curve corresponding to the combined bound on the cross-times branching ratio will be somewhere between the lower red and blue curves, and the combined upper bound would be somewhere between the upper red and blue curves. [^4]: The existance of a neutral resonance with the same mass would not be such a miracle if we were considering an $SU(2)_l$ triplet but in this paper we are restricting our attention to $SU(2)_l$ singlets with hypercharge.
--- abstract: 'The correlation of the result lists provided by search engines is fundamental and it has deep and multidisciplinary ramifications. Here, we present automatic and unsupervised methods to assess whether or not search engines provide results that are comparable or correlated. We have two main contributions: First, we provide evidence that for more than 80% of the input queries —independently of their frequency— the two major search engines share only three or fewer URLs in their search results, leading to an increasing divergence. In this scenario (divergence), we show that even the most robust measures based on comparing lists is useless to apply; that is, the small contribution by too few common items will infer no confidence. Second, to overcome this problem, we propose the fist content-based measures —i.e., direct comparison of the contents from search results; these measures are based on the Jaccard ratio and distribution similarity measures (CDF measures). We show that they are orthogonal to each other (i.e., Jaccard and distribution) and extend the discriminative power w.r.t. list based measures. Our approach stems from the real need of comparing search-engine results, it is automatic from the query selection to the final evaluation and it apply to any geographical markets, thus designed to scale and to use as first filtering of query selection (necessary) for supervised methods.' author: - | Paolo D’Alberto\ \ \ Ali Dasdan\ \ \ bibliography: - 'biblio.bib' subtitle: '(Unsupervised Comparison of Search Engine Rankings)' title: 'On the Weakenesses of Correlation Measures used for Search Engines’ Results' --- Acknowledgments {#acknowledgments .unnumbered} =============== Under the hood of this machinery, we used several components and consulted very capable engineers: Suresh Lokia for the set of queries, Kexiang Hu for the scraping tool, Marcin Kadluczka for the high level fetching system for the retrieval of the documents in real time, and Amit Sasturkar and Swapnil Hajela for the word-view pipeline and document signature. We also thank Santanu Kolay for useful discussions on various aspects of this work and Ravi Kumar for discussions on the weighted form of Kendall’s tau. The community has spoken about and against this work. Here we share the anonymous considerations without our reply. Enjoy the drama.
--- abstract: 'We study both the wave-like behavior and particle-like behavior in a general Mach-Zehnder interferometer with its asymmetric beam splitter. A error-free measurement in the detector is used to extract the which-path information. The fringe visibility V and the which-path information $I_{path}$ are derived: their complementary relation $V+I_{path}\leq1$ are found, and the condition for the equality is also presented.' author: - Yanjun Liu - Jing Lu - Lan Zhou title: 'Complementarity via error-free measurement in a two-path interferometer' --- [^1] Introduction ============ The wave-like nature and particle-like nature are two mutual exclusion properties of quantum systems, and the appearance of these two properties are determined by the experimental instrument, which is known as Bohr’s complementarity principle [@Bohr]. The wave-particle duality is the well-known example used to exhibit the curious nature of the complementarity principle. In the recoiling-slit gedanken experiment introduced by Einstein and Bohr where a particle is sent through a movable slit placed before a double slit, Wootters and Zurek [@Woo] proposed their quantitative formulation of the wave-particle duality. In a two-path interferometer, such as Mach-Zehnder interferometer (MZI) [@Wolf], a complementarity was first found between a priori fringe visibility of the interference pattern and the predictability [@Yasin], which were determined by the initial state of the particle. Later on, the wave-like and particle-like nature were characterized by the visibility of the interferometer fringe and the path distinguishability [@Englert], respectively, and there is a trade-off between these quantities. Both predictability and distinguishing are quantities that measure the which-path knowledge. Since there are many ways to define the measure of which-path knowledge, the complementarity between the fringe visibility and the which-path knowledge has been studied greatly in theory and experiment [@Yasin; @Mandel; @Jaeger; @Horne; @Englert; @Rempe; @rrS; @Zawisky; @Kaszlikowski; @Bimonte; @Bramon; @Jakob; @BanO; @Masashi; @Han; @LiLi; @Fonseca; @Asad; @Angelo]. Form the information theory viewpoint, the achievement of knowledge is an information transmission process. It is natural to use information measure to characterize the particle nature of a quantum system in a two-path interferometer. Here, we employ the mutual information which is called which-path information (WPI). The information cannot be transmitted until a measurement is performed, for example, a detector is placed in one path of the MZI in Ref.[@Englert], and an error-minimum state distinguishing measurement [@Helstrom] is performed on the detector after the particle interacts with the detector to acquire the path distinguishability. Although such ambiguous measurement yields a conclusive outcome, there is nonvanishing probability of making a wrong guess since errors in the conclusive outcome are unavoidable. The other optimized measure strategies is the error-free discrimination [@Chefles] among nonorthogonal states, which allow a nonzero probability of inconclusive outcomes. In this paper, we study the trade-off between the fringe visibility and WPI in a two-path interferometer made of one symmetric beam splitter (BS) and one asymmetric BS. The error-free measurement is used to obtain the WPI. It is found that the magnitudes of the fringe visibility and the WPI are effected by the asymmetric BS and the input state of the particle. A bound between the fringe visibility and the WPI is also found. The paper is organized as follows. In Sec. , we introduce the setup that a quantum system display its wave-like behavior and particle-like behavior. In Sec. , the WPI is defined, and an unambiguous discrimination on the state of the detector is used to obtain the which-path information. In Sec. , we made our conclusion. \[Sec:2\]The setups and the state evolution =========================================== A general MZI, shown in Fig. \[fig1a.eps\], consists of two BSs and phase shifters (PSs). A beam of particles coming from either port a or b is first splitted by beam splitter BS1 into two beams and then these beams are recombined by BS2. So two paths a and b are available between BS1 and BS2. A particle taking path a(b) is denoted by the state $|a\rangle=\hat{a}^{\dagger}|0\rangle$ $(|b\rangle=\hat{b}^{\dagger}|0\rangle)$, where $\hat{a}^{\dagger}$ and $\hat{b}^{\dagger}$ are the corresponding creation operators in path a and b, and they satisfy $[\hat{a},\hat{b}]=0$, $[\hat{a},\hat{a}^{\dagger}]=1$, $[\hat{b},\hat{b}^{\dagger}]=1$. States $|a\rangle$ and $|b\rangle$ support a two-dimensional $H_{q}$. In this sense, a quantum bit is formed. The state of particle traveling in this interferometer is characterized by the change of a Bloch vector in a Bloch sphere. Before the particle incident into the general MZI, the state of the particle is described by the density matrix $$\rho _{in}^{Q}=\frac{1}{2}( 1+S_{x}\sigma _{x}+S_{y}\sigma _{y}+S_{z}\sigma _{z}),$$where $\sigma _{x},\sigma _{y},\sigma _{z}$ are the Pauli matrix and $\sigma_{z}=|b\rangle\langle b|-|a\rangle\langle a|$. Here, the initial Bloch vector $\overrightarrow{S}=(\overrightarrow{S_{x}},\overrightarrow{S_{y}},\overrightarrow{S_{z}})$. When $|\overrightarrow{S}|=1$, the particle is in a pure state, when $|\overrightarrow{S}|<1$ , the particle is in a mixed state. ![The schematic sketch of the general Mach-Zehnder interferometer with the second BS asymmetric, a detector is placed in path a.[]{data-label="fig1a.eps"}](fig1a.eps){height="5cm" width="8cm"} The effect of the BS on the state of the incoming particles is described by the operator $B(\beta)=\exp[i\beta(\hat{a}^{\dagger}\hat{b}+\hat{b}^{\dagger}\hat{a})]$, which preserves the total number of particles in this MZI. In the subspace spanned by basis $\{|a\rangle,|b\rangle\}$, the BS performs a rotation around the y axis by angle $\beta$, which is denoted by $$U_{B}(\beta)=\exp ( -i\frac{\beta }{2}\sigma _{y}) =( \begin{array}{cc} \sqrt{t} & -\sqrt{r} \\ \sqrt{r} & \sqrt{t} \end{array}).$$ Here, $r$ and $t$ respectively represent the reflection coefficient and transmission coefficient of the beam splitter. $$r=\sin^{2}{\frac{\beta}{2}} ,\text{ \ \ \ \ \ \ \ \ } t=\cos^{2}{\frac{\beta}{2}}.$$ The PS in path $d \in \{a,b \}$ is described by $P(\phi_{d})=\exp(i\phi_{d}\hat{d}^{\dagger}\hat{d})$. If the parameters $\phi_{a}$ and $\phi_{b}$ have the same magnitude but different sign, i.e., $\phi_{a}=-\phi_{b}=\phi$, a rotation around the $z$ axis by angle $\phi$ is realized by PS1 and PS2, $$U_{P}(\phi)=\exp ( -i\frac{\phi}{2}\sigma _{z}).$$ To acquire the WPI, a detector is usually placed in one of the paths (e.g. the a path). As long as the particle go through the general MIZ, a operator $$M=\frac{1+\sigma_{Z}}{2}I+\frac{1+\sigma_{Z}}{2}U,$$ performs on the detector, where U is unitary. The final state of the particle and the detector reads $$\begin{aligned} \rho _{f}&=&U_{B}(\beta)MU_{P}(\varphi)U_{B}(\frac{\pi}{2})\rho_{in}^{Q}\rho_{in}^{D} U_{B}^{\dagger}(\frac{\pi}{2})U_{P}^{\dagger}(\varphi)M^{\dagger}U_{B}^{\dagger}(\beta) \notag\\ &=&\frac{1}{4}( 1-S_{x}) ( 1+\sigma _{z}\cos \beta +\sigma _{x}\sin \beta ) \otimes \rho _{in}^{D} \notag\\ &&-\frac{1}{4}e^{-i\phi }( S_{z}-iS_{y}) ( \sigma _{z}\sin \beta -\sigma _{x}\cos \beta -i\sigma _{y}) \otimes \rho _{in}^{D}U^{\dagger} \notag\\ &&-\frac{1}{4}e^{i\phi}( S_{z}+iS_{y}) ( \sigma _{z}\sin \beta-\sigma _{x}\cos \beta +i\sigma _{y}) \otimes U\rho _{in}^{D}\notag\\ &&+\frac{1}{4}( 1+S_{x}) ( 1-\sigma _{z}\cos \beta -\sigma _{x}\sin \beta ) \otimes U\rho _{in}^{D}U^{\dagger},\end{aligned}$$ where $\rho_{in}^{D}$ is the initial state of the detector, and the BS1 is assumed to be symmetric. The probability that we finding the particle at the output port $a$ reads $$\begin{aligned} p(\phi)&=&tr_{QD}[ \frac{1}{2}( 1-\sigma _{z}) \rho _{f}] \notag\\ &=& \frac{1}{2}( 1+ S_{x}\cos\beta) \notag\\ &&+\frac{1}{2}\sqrt{S_{z}^{2}+S_{y}^{2}}\sin\beta |tr_{D}(U\rho _{in}^{D})| \cos ( \alpha +\gamma +\phi ) \label{2eq-01},\end{aligned}$$ where $\alpha$ and $\gamma$ are defined as $$\alpha= \arctan\frac{S_{y}}{S_{z}}, \text{ \ \ \ \ \ \ } \gamma= -i\ln\frac{tr_{D}(U\rho_{in}^{D})}{|tr_{D}(U\rho_{in}^{D})|}.$$ The fringe visibility which documents the wave-like property of the particle is defined via the probability in Eq. (\[2eq-01\]) as $$\begin{aligned} V&=&\frac{max P(\phi)-min P(\phi)}{max P(\phi)+min P(\phi)} ,\end{aligned}$$ where the maximum and minimum is achieved by adjusting $\phi$. And one can easily obtain that $$\begin{aligned} V&=&\frac{\sin \beta }{1+S_{x} \cos \beta }\sqrt{S_{z}^{2}+S_{y}^{2}}| tr_{D}( U\rho_{in} ^{D})| \label{2eq-02}.\end{aligned}$$ We note that the fringe visibility can also be defined by the probability at the output port $b$. However, the fringe visibility measured in either output port $a$ or $b$ is different expect the BS2 is symmetric (i.e. $\beta=\pi/2$). Eq. (\[2eq-02\]) shows that both the BS2 and the initial state have influence on the fringe visibility. It can be found that for a given $\beta$, more wave nature appears when the particle is in a pure state $(|\overrightarrow{S}|=1)$. To show the dependence of the wave nature of the BS2 and the initial state, we have plotted the fringe visibility V as a function of the $\beta$ and $S_{x}$ for $| tr_{D}(U\rho_{in} ^{D})|=1/3$ and $|\overrightarrow{S}|=1$ in Fig. \[fig2.eps\]. It can be observed for a given $S_{x}(\beta)$, the fringe visibility first increase and then decrease as $\beta(S_{x})$ increases, and the fringe visibility obtains the maximum $C \equiv |tr_{D}(U\rho_{in} ^{D})|$ when $\cos \beta=-S_{x}$. This is the reason the quantified wave-particle duality in [@Englert] is presented by choosing $S_{x}=0$ when $\beta=\pi/2$. The value of the fringe visibility is zero in the following situation: (1) The effect of the BS2 for the particle is full transmission or full reflection, corresponding to $\beta=0$ or $\pi$. (2) The particle travels only in a path or b path, corresponding to $S_{x}=1$ or $-1$. ![image](fig2.eps){height="8cm" width="14cm"} It is well-known that the wave-like property characterized by the fringe visibility is complementary to the particle-like property which gives rise to the WPI. A decrease of the fringe visibility predicts an increasing of the WPI. From Eq. (\[2eq-02\]), one can find that the BS2 besides the initial state affect the WPI, which indicates that the asymmetric BS2 introduce additional WPI [@LiLi]. Then, the set up of measuring the particle-like property is different form the one which is obtained simply by removing the BS2 in Fig. \[fig1a.eps\]. Actually, the particle-like property is measured by the setup with four input and output ports, which is shown in Fig. \[fig1b.eps\]. Since two paths c and d have been introduced, the initial density matrix for the total system reads $$\begin{gathered} \rho _{in}^{QD}=\rho _{in}^{Q}\otimes|00\rangle_{cd}\langle00|\otimes\rho _{in}^{D},\end{gathered}$$ where $|00\rangle_{cd}$ is vacuum states of the input ports c and d. Before the particle meets the BS2 and BS3, the state of the particle and the detector is the same to the state before the BS2 in Fig. \[fig1a.eps\]. The BS2 acts on the paths a and c, and BS3 acts on the paths b and d. The performance of BS2 and BS3 is denoted by $B_{2}=exp[(-\frac{\beta}{2})(\hat{a}^{\dagger}\hat{c}-\hat{c}^{\dagger}\hat{a})]$ and $B_{3}=exp[(-\frac{\beta}{2})(\hat{b}^{\dagger}\hat{d}-\hat{d}^{\dagger}\hat{b})]$ respectively. After the particle goes through the BS2 and BS3, the state for the particle appearing in either output a or d reads $$\begin{aligned} \rho^{QD} _{f}&=&\omega_{b} |d\rangle \langle d| \rho _{in}^{D} +\omega_{a} | a\rangle \langle a| U\rho _{in}^{D}U^{\dagger} \notag\\ &&+\frac{\sqrt{rt}}{1+S_{x}(t-r)}e^{i\phi}( S_{z}+iS_{y} ) | a\rangle \langle d| U\rho _{in}^{D} \notag\\ &&+\frac{\sqrt{rt}}{1+S_{x}(t-r)}e^{-i\phi}(S_{z}-iS_{y})| d\rangle \langle a| \rho _{in}^{D}U^{\dagger} \label{2eq-04}.\end{aligned}$$ ![Schematic representation of the setup with four input and output ports.[]{data-label="fig1b.eps"}](fig1b.eps){height="5cm" width="8cm"} Here, the *prior* probabilities $$\omega _{a} =\frac{t( 1+S_{x}) }{1+S_{x}(t-r) }, \text{ \ \ \ \ \ \ } \omega _{b} =\frac{r( 1-S_{x}) }{1+S_{x}(t-r)},$$ for finding the particle in output a and d respectively. Since the particle detected at output port a(d) is obtained by transmission (reflection) from path a(b), we change the letter d in Eq. (\[2eq-04\]) by b, which is rewritten as $$\begin{aligned} \rho^{QD} _{f}&=&\omega_{b} |b\rangle \langle b| \rho _{in}^{D} +\omega_{a} | a\rangle \langle a| U\rho _{in}^{D}U^{\dagger} \notag\\ &&+\frac{\sqrt{rt}}{1+S_{x}(t-r)}e^{i\phi}( S_{z}+iS_{y} ) | a\rangle \langle b| U\rho _{in}^{D} \notag\\ &&+\frac{\sqrt{rt}}{1+S_{x}(t-r)}e^{-i\phi}(S_{z}-iS_{y})| b\rangle \langle a| \rho _{in}^{D}U^{\dagger} \label{2eq-05}.\end{aligned}$$ \[Sec:3\] Information gain via error-free measurement ====================================================== After tracing over the degree of the particle in Eq. (\[2eq-05\]), we obtain the final state of the detector $$\begin{gathered} \rho^{D} _{f}= \omega_{b}\rho _{in}^{D} +\omega_{a}\rho _{out}^{D}.\end{gathered}$$ To obtain the WPI, we have to discriminate the states $\rho _{in}^{D}$ and $\rho _{out}^{D}\equiv U\rho _{in}^{D}U^{\dagger}$ with prior probabilities $\omega _{b}$ and $\omega _{a}$ in an optimal way. Here, we perform the error-free measurement on the detector. This kind of measurement gives two results: a conclusive one without any error and an inconclusive one. The conclusive result means which-path the particle takes is definitely known. Mathematically, to calculate the WPI, one has to introduce the positive operator-valued measure (POVM) $\{{\Pi _{k},k=a,b,0}\}$ with the resolution of the identity $\sum_{k}\Pi _{k}=I$, which leads to an inconclusive outcome 0 and two definitely results a and b. The unambiguous discrimination requires $$\rho _{in}^{D}\Pi_{a}=\rho_{out} ^{D}\Pi_{b}=0 \label{3eq-01}.$$ The joint probability that the particle travels on path $d \in \{a,b \}$ and the which-path result k indicated from the measurement of the detector reads $$\begin{aligned} Q(\mu,k)&=&Tr_{D}\langle \mu|\Pi_{k}\rho_{f}^{QD}|\mu\rangle \label{3eq-02}.\end{aligned}$$ Then, the amount of the WPI [@Thomas] obtained from the error-free measure is given by $$\begin{aligned} I_{path} &=&\underset{\mu =a,b}{\sum }\underset{k=a,b}{\sum }Q( \mu ,k) \log [ \frac{Q( \mu ,k) }{Q( \mu ) Q( k) }] \label{3eq-03}.\end{aligned}$$ For the sake of simplicity, we assume that the detector is initially in a pure state $\rho _{in}^{D}=| r\rangle\langle r|$. Since the unitary operator U is arbitrary, states $| r\rangle$ and $| s\rangle \equiv U| r\rangle$ can be assumed to be linearly independent, the POVM is constructed as $$\begin{aligned} \Pi _{a} &=&\alpha | r^{\perp}\rangle \langle r^{\perp}| , \notag\\ \Pi _{b} &=&\beta | s^{\perp}\rangle\langle s^{\perp}| , \notag \\ \Pi _{0}&=&( 1-\beta S^{2}) |r \rangle \langle r | +\beta SC| r\rangle \langle r^{\perp}| \notag\\ &&+\beta SC| r^{\perp}\rangle \langle r| +( 1-\beta C^{2}-\alpha) | \ r^{\perp}\rangle \langle r^{\perp}| ,\label{3eq-04}\end{aligned}$$ where states $| r\rangle(| s\rangle)$ are orthogonal to $$\begin{aligned} | r^{\perp}\rangle &=&\frac{1}{S}( |s\rangle -C| r\rangle ), \notag\\ | s^{\perp}\rangle &=&\frac{1}{S}( |r\rangle -C| s\rangle ),\end{aligned}$$ respectively. Capital letter $S=\sqrt{1-C^{2}}$, and $C =\langle r|s\rangle$, where the maximum value of the fringe visibility becomes the overlap between two linearly independent states. In Eq. (\[3eq-04\]), parameters $\alpha$ and $\beta$ are chosen to minimize the probability of failure $$\begin{aligned} Q &=& \omega _{b}Tr( \rho _{in}^{D}{\Pi }_{0}) +\omega _{a}Tr( \rho_{out} ^{D}{\Pi }_{0}) .\end{aligned}$$ By the Cauchy inequality and the resolution of the identity, we derive the lower bound on the probability of failure [@Feng] $$\begin{aligned} Q &\geq & 2\sqrt{ \omega _{a} \omega _{b}}F(\rho _{in}^{D},\rho_{out}^{D}) \label{3eq-05}, \end{aligned}$$ where the fidelity [@Nielsen] is defined as $$F =Tr|\sqrt{\rho _{in}^{D}\rho_{out}^{D}}| = C.$$ The lower bound of the failure probability is achieved if and only if $$\omega _{b}Tr( \rho _{in}^{D}\Pi _{0}) = \omega _{a}Tr( \rho_{out} ^{D}\Pi _{0}) =\sqrt{ \omega _{a} \omega _{b}}F( \rho _{in}^{D},\rho_{out} ^{D}) \label{3eq-06}.$$ ![image](fig3.eps){height="8cm" width="14cm"} ![image](fig4.eps){height="8cm" width="14cm"} ![image](fig5.eps){height="8cm" width="14cm"} The probability of the kth outcome, $tr(\Pi _{k}\rho _{k}^{D})$, is always real and nonnegative, which requires $0\leq\alpha,\beta\leq1$. The choice of the measurement that discriminates $| r\rangle$ and $| s\rangle$ unambiguously depends on the relation between the ratio $\sqrt{\omega _{a} / \omega _{b}}$ and the overlap C: \(1) When $\sqrt{\omega _{a} / \omega _{b}} \leq C$, the minimum probability of failure $Q=\omega _{a}+C^{2}\omega _{b}$ is achieved by selecting the following measurement operators $\Pi _{a} =0$, $\Pi _{b} =| s^{\perp}\rangle\langle s^{\perp}|$, $\Pi _{0} =| s\rangle\langle s|$. Here, state $| s\rangle$ is never detected, and the optimal POVM becomes a von Neumann projective measurement. Actually, in this case, $\omega _{b}>\omega _{a}$, the state of the detector is more likely to be in state $\rho _{in}^{D}$, so we make the failure direction along $\rho _{out}^{D}$ for obtaining the minimum probability of failure. The joint probability is obtained as $$\begin{aligned} Q( b,b)&=&(1-C ^{2})\frac{\sin ^{2}\frac{\beta }{2}( 1-S_{x}) }{1+S_{x}\cos \beta },\end{aligned}$$ $$\begin{aligned} Q( a,b)&=& 0,\end{aligned}$$ $$\begin{aligned} Q( a,0)&=&\frac{\cos^{2}\frac{\beta }{2}( 1+S_{x}) }{1+S_{x}\cos \beta },\end{aligned}$$ $$\begin{aligned} Q( b,0)&=& C ^{2}\frac{\sin ^{2}\frac{\beta }{2}( 1-S_{x}) }{1+S_{x}\cos \beta }.\end{aligned}$$ Then, the amount of WPI via the von Neumann projective measurement reads $$\begin{aligned} I_{path}&=& ( 1- C^{2})\frac{\sin ^{2}\frac{\beta }{2}( 1-S_{x}) }{1+S_{x}\cos \beta } \log \frac{1+S_{x}\cos \beta }{\sin ^{2}\frac{\beta }{2}( 1-S_{x}) } \label{3eq-08}.\end{aligned}$$ \(2) When $C\leq \sqrt{\omega _{a} / \omega _{b}}\leq 1/C$, the minimum probability of failure $Q=2C\sqrt{ \omega _{a} \omega _{b}}$ is achieved by choosing the following measurement operators $$\begin{aligned} \Pi_{a} &=&\frac{1}{S^{2}}( 1-C\tan \frac{\beta }{2}\sqrt{\frac{1-S_{x}}{1+S_{x}}}) | r^{\perp}\rangle \langle r^{\perp}| ,\end{aligned}$$ $$\begin{aligned} \Pi_{b} &=&\frac{1}{S^{2}}( 1-C\cot \frac{\beta }{2}\sqrt{\frac{1+S_{x}}{1-S_{x}}}) | s^{\perp}\rangle \langle s^{\perp}| ,\end{aligned}$$ $$\begin{aligned} \Pi_{0} &=& C\cot \frac{\beta }{2}\sqrt{% \frac{1+S_{x}}{1-S_{x}}}| r\rangle \langle r| \notag \\ &&+\frac{ C}{S}( 1-C\cot \frac{\beta }{2}\sqrt{\frac{% 1+S_{x}}{1-S_{x}}})| r\rangle \langle r^{\perp}| \notag \\ &&+\frac{C}{S}( 1-C\cot \frac{\beta }{2}\sqrt{\frac{1+S_{x}}{1-S_{x}}% }) | r^{\perp}\rangle \langle r| \notag \\ &&+[1-\frac{C^{2}}{S^{2}}( 1-C\cot \frac{\beta }{2}\sqrt{\frac{% 1+S_{x}}{1-S_{x}}}) \notag \\ &&-\frac{1}{S^{2}}( 1-C\tan \frac{\beta }{2}\sqrt{\frac{1-S_{x}}{1+S_{x}}}) ]| r^{\perp}\rangle \langle r^{\perp}| ,\end{aligned}$$ This measurement is more general than the von Neumann projective measurement. Via Eq. (\[3eq-02\]), the joint probability reads $$\begin{aligned} Q( a,a)&=&\frac{\cos ^{2}\frac{\beta }{2}( 1+S_{x}) }{1+S_{x}\cos \beta }% ( 1-C\tan \frac{\beta }{2}\sqrt{\frac{1-S_{x}% }{1+S_{x}}}),\end{aligned}$$ $$\begin{aligned} Q( b,b)&=&\frac{\sin ^{2}\frac{\beta }{2}( 1-S_{x}) }{1+S_{x}\cos \beta }% ( 1-C\cot \frac{\beta }{2}\sqrt{\frac{1+S_{x}% }{1-S_{x}}}),\end{aligned}$$ $$Q( a,b)=Q( b,a)=0,$$ $$\begin{aligned} Q( a,0)&=& C\frac{\cos ^{2}\frac{\beta }{2}( 1+S_{x}) }{1+S_{x}\cos \beta }% \tan \frac{\beta }{2}\sqrt{\frac{1-S_{x}}{1+S_{x}}},\end{aligned}$$ $$\begin{aligned} Q( b;0)&=& C\frac{\sin ^{2}\frac{\beta }{2}( 1-S_{x}) }{1+S_{x}\cos \beta }% \cot \frac{\beta }{2}\sqrt{\frac{1+S_{x}}{ 1-S_{x}}}.\end{aligned}$$ Then the amount of WPI obtained from the POVM measurement is calculated as $$\begin{aligned} I_{path}&=&\frac{\cos ^{2}\frac{\beta }{2}( 1+S_{x}) }{1+S_{x}\cos \beta }% ( 1-C\tan \frac{\beta }{2}\sqrt{\frac{1-S_{x}}{1+S_{x}}}) \notag\\ &&\times\log \frac{1+S_{x}\cos \beta }{\cos ^{2}\frac{\beta }{2}( 1+S_{x}) } \notag\\ &&+\frac{\sin ^{2}\frac{\beta }{2}( 1-S_{x}) }{1+S_{x}\cos \beta }% ( 1-C\cot \frac{\beta }{2}\sqrt{\frac{1+S_{x}}{1-S_{x}}}) \notag\\ &&\times\log \frac{1+S_{x}\cos \beta }{\sin ^{2}\frac{\beta }{2}( 1-S_{x}) } \label{3eq-09},\end{aligned}$$ according to its definition given by Eq. (\[3eq-03\]). \(3) When $\sqrt{\omega _{a} / \omega _{b}} \geq 1/C$, the minimum probability of failure $Q=\omega _{b}+C^{2}\omega _{a}$ is achieved by selecting the measurement operators $\Pi _{a} =| r^{\perp}\rangle\langle r^{\perp}|$, $\Pi _{b} =0$, $\Pi _{0} =| r\rangle \langle r|$. Here, state $| r\rangle$ is never detected, and the optimal POVM becomes a von Neumann projective measurement. In this case, $\omega _{a}>\omega _{b}$. The state of the detector is more likely to be in state $\rho _{out}^{D}$, so the failure direction is chosen along $\rho _{in}^{D}$ for obtaining the minimum probability of failure. The joint probability is obtained as $$\begin{aligned} Q( a,a)&=& (1-C ^{2})\frac{\cos ^{2}\frac{\beta }{2}( 1+S_{x}) }{1+S_{x}\cos \beta } ,\end{aligned}$$ $$Q( b,a)= 0,$$ $$\begin{aligned} Q( a,0)&=& C ^{2}\frac{\cos ^{2}\frac{\beta }{2}( 1+S_{x}) }{1+S_{x}\cos \beta },\end{aligned}$$ $$\begin{aligned} Q( b,0)&=&\frac{\sin ^{2}\frac{\beta }{2}( 1-S_{x}) }{1+S_{x}\cos \beta }.\end{aligned}$$ And, the amoumt of WPI is given by $$\begin{aligned} I_{path}&=& ( 1-C ^{2}) \frac{\cos ^{2}\frac{\beta }{2}( 1+S_{x}) }{1+S_{x}\cos \beta } \log \frac{1+S_{x}\cos \beta }{\cos ^{2}\frac{\beta }{2}( 1+S_{x}) } \label{3eq-10}.\end{aligned}$$ Eqs. (\[3eq-08\]), (\[3eq-09\]) and (\[3eq-10\]) show that the WPI is a piecewise function of the parameters $S_{x}$, $\beta$, and C. Although all the components of the Bloch vector determines the fringe visibility, only $S_{x}$ occurs in the expression of the WPI, indicating that $I_{path}$ is independent of the initial state of the quantum particle. In Fig.4, we plot the WPI as the function of $S_{x}$ and $\beta$ with the overlap $C=1/3$. The $I_{path}$ under the conditions $\sqrt{\omega _{a} / \omega _{b}} \in (0,C)$, $(C,C^{-1})$, and $(C^{-1},+\infty)$ is shown in the ranges with purple, brown, and cyan in Fig.4(a), respectively. The white range in Fig.4(a) indicates that $I_{path}$ is a discontinuous function of $S_{x}$ and $\beta$. It can be observed from Fig.4 that $I_{path}\leq1-C$ for any $S_{x}$ and $\beta$, and the position along the $S_{x}(\beta)$ axis that $I_{path}=1-C$ occurs varies with different given $\beta(S_{x})$. In Fig.5(6), we have plotted $I_{path}$ as the function of parameters $\beta(S_{x})$ and C for a given $S_{x}(\beta)$. One can also find that $I_{path}$ is less than or equal to $1-C$, $I_{path}$ decrease as C increase, the position that the peak occurs is fixed for different overlap C, i.e., the peak appears at $\beta=2\pi/3$ when $S_{x}=1/2$ in Fig.5(c), and $S_{x}=-1/2$ when $\beta=\pi/3$ in Fig.6(c). From Eq. (\[3eq-09\]), we find that the maximum of $I_{path}$ can be achieved once $\cos \beta=-S_{x}$. The wave-like and the particle-like property are quantitated by fringe visibility V in Eq. (\[2eq-02\]) and the WPI $I_{path}$ in Eq. (\[3eq-03\]). Since $V\leq C$ and $I_{path}\leq1-C$, we obtain the complementary relation $$\begin{aligned} V+I_{path}\leq1 \label{3eq-11},\end{aligned}$$ the equal sign holds in Eq. (\[3eq-11\]) when $\cos \beta=-S_{x}$. \[Sec:4\] conclusion ==================== We have investigated the complementarity of the fringe visibility and the WPI in a MZI with one asymmetric BS. Although the fringe visibility measured in either two output ports are different, there exists an upper limit, i.e. $V\leq C$. The upper bound $C=|tr_{D}(U\rho_{in} ^{D})|$ is determined by the initial state of the detector and the unitary operator performed on the detector. The maximum value of the fringe visibility C can be achieved when the quantum system is initially in pure state with $\cos \beta=-S_{x}$. To observe the particle-like behavior of this quantum system, a four-path interferometer must be introduced due to the asymmetrical BS2. The WPI is characterized by the WPI $I_{path}$, which is obtained via the unambiguous discrimination on the state of the detector. Although $I_{path}$ is dependent on the asymmetric BS and the initial state of the quantum system, the WPI is bounded by the following inequality, $I_{path}\leq1-C$. The maximum $I_{path}$ is achieved when $\cos \beta=-S_{x}$. It is also found that $V+I_{path}\leq1$. 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--- abstract: 'Due to a ferromagnetic in-chain coupling between Co$^{3+}$ ions at trigonal sites, chains Co$_2$O$_6$ are considered as large rigid spin moments. The antiferromagnetic Ising model on the triangular lattice is applied to describe an interchain ordering. An evolution of metastable states in a sweeping magnetic field is investigated by the single-flip technique. At the first approximation two steps in the magnetization curve and a plateau at $1/3$ of the saturation magnetization are found. Four steps in magnetization are determined in high-order approximations in agreement with experimental results.' author: - 'Yuri B. Kudasov' title: 'Step-like magnetization in a spin-chain system: Ca$_3$Co$_2$O$_6$' --- Among other spin-chain compounds, Ca$_3$Co$_2$O$_6$ has drawn recently considerable attention to their complex magnetic behavior . The most intriguing feature observed in Ca$_3$Co$_2$O$_6$ is a step-like shape of the magnetization curve [@drillon; @hardy2; @maignan]. The number of the steps in the curve depends strongly on a sweep rate of the external magnetic field and temperature [@drillon; @hardy2; @maignan]. Two steps become apparent in the temperature range from 12 K to 24 K [@maignan]. The first step takes place at the zero magnetic field. Then the magnetic moment remains constant at about $1/3$ of the full magnetization up to the magnetic field of 3.6 T where the second step occurs to the fully magnetized FM state. At least four equidistant steps are clearly visible below 10 K at a very low sweep rate. They are accompanied by a sizeable hysteresis. Similar phenomena were observed in other spin-chain compounds, e.g. Ca$_3$CoRhO$_6$ [@nijtaka1; @nijtaka2]. The structure of Ca$_3$Co$_2$O$_6$ consists of Co$_2$O$_6$ chains running along the $c$ axis. The Ca ions are situated between them. The chains are made up of alternating, face-sharing CoO$_6$ trigonal prisms and CoO$_6$ octahedra. The crystalline electric field splits the energy level of Co$^{3+}$ ions into the high-spin ($S=2$) and low-spin ($S=0$) states. The Co$^{3+}$ ions situated in the trigonal environment (CoI) are in the high-spin state and the octahedral Co sites (CoII) occurs in the low-spin state. In the last case the energy difference between the low-spin and high-spin states is very small and a tiny fraction of CoII sites is reported to be in the high-spin state. The crystalline electric field leads also to a very strong Ising-like anisotropy at the CoI sites. The chains form triangular lattice in the $ab$ plane that is perpendicular to the chains. An in-chain exchange interaction between magnetic CoI ions through the octahedra with non-magnetic CoII ions is ferromagnetic (FM). It causes the in-chain FM ordering of CoI ions at about 40 K. The interchain interaction is antiferromagnetic (AFM) and much weaker than the in-chain one. A partial AFM order of chains appears at 24 K. A weak feature concerned most probably with a transition in a new interchain order was also observed at around 12 K. This scenario of magnetic interactions in Ca$_3$Co$_2$O$_6$ is consistent with results of x-ray photoemission spectroscopy [@x-ray], neutron scattering [@neutron], magnetization and specific heat measurements [@hardy1; @drillon; @hardy2; @hardy3; @maignan; @martinez], nuclear magnetic resonance [@nmr], and theoretical calculations of indirect interactions between CoI sites [@fresard]. The model presented in Ref. [@drillon] deals with an in-chain structure of Ca$_3$Co$_2$O$_6$ and magnetization dynamics explained in terms of the quantum tunnelling. In the this Letter, we develop a new model for a description of the step-like magnetization in Ca$_3$Co$_2$O$_6$ at low temperatures, shifting the stress on the interchain magnetic order. The strong FM in-chain coupling makes it possible to consider a Ca$_2$O$_6$ chain as a large rigid spin formed by CoI ions. There are only two projections of the chain spin onto the $c$ axis due to the strong Ising-like anisotropy. Including the AFM coupling between the nearest-neighbor chain spins we arrive to the Ising Hamiltonian on the triangular lattice $$H=J\sum_{<ij>}{\sigma_i^z \sigma_j^z}-B\sum_{i}{\sigma_i^z} \label{Ising}$$ where $\sigma_i^z=\pm{1}$ is the $c$-axis projection of the $i$-th chain spin, $J>0$ is the parameter of the AFM interchain coupling, $B$ is the magnetic field, $<ij>$ denotes the summation over all the nearest-neighbor pairs on the triangular lattice. The strong dependence of the magnetization curve shape on the magnetic field sweep rate and temperature shows that the state of the system of the chain spins in the magnetic field is far from equilibrium. At the low sweep rate the system is rather in a metastable state than in the ground state. It is convenient to formulate necessary conditions of the metastability of the system in the following form $$\sigma_i^z h_i\leq 0 \label{meta}$$ where $$h_i = J\sum_{j(i)}{\sigma_j^z}-B$$ is the effective field for the $i$-th chain, $j(i)$ denotes summation over the nearest-neighbors of the $i$-th chain. We have used the unstrict inequality in Eq.(\[meta\]) keeping in mind a strong degeneracy of partially ordered AFM arrangements on the triangular lattice. A transition from one metastable state to another occurs through exited states. We obtain $\sigma_i^z h_i > 0$ at least for one chain in a exited state by definition. Let $\Delta E_i %%@ = 2 \sigma_i^z h_i> 0$ be the excitation energy per CoI site of the $i$-th chain. It follows from this that the probability of an exited chain at low temperature $T$ is extremely small $\propto exp(- N \Delta E_i / T)$ since the number of CoI sites in the chain ($N$) is considered to be large. That is why, we should investigate an evolution of metastable states in the slowly sweeping external magnetic field assuming $T=0$ quench. We perform the investigation of the evolution of the system using the single-flip technique that was applied earlier to nonequilibrium dynamics of the AFM Ising model on the the triangular lattice [@kim]. In the terms of the effective field, the spin-flip probability $A$ is taken in the following form $$\begin{aligned} A=\left \lbrace \begin{array}{cc} 0 & \text{ if }\sigma_i^z h_i < 0, \\ 1 & \text{ if }\sigma_i^z h_i \geq 0. \end{array}\right.\end{aligned}$$ In contrast to Ref.[@kim] where the spin-flip technique was applied to numerical Monte Carlo simulations, we investigate the evolution analytically. If a state under consideration is degenerate and different sequences of spin flips lead to different final states, one should take into account each possible sequence with equal probabilities. This assumption can be proven rigorously by Bogolubov’s quasiaverage technique. That is a small auxiliary random field should be added to the Hamiltonian (\[Ising\]) in order to lift the degeneracy. After that we take an average over a manifold of the auxiliary fields restoring equivalence of different chains and, then, let the amplitude of the auxiliary fields goes to zero. This approach is equivalent to the Monte Carlo technique in the limit of the large number of samples. We take the ground state of the triangular lattice as an initial state of the chain lattice at $B=0$. The ground state of triangular lattice at $B=0$ is strongly degenerate and it is impossible to represent it in an explicit form [@wannier]. On the other hand, we can produce a set of approximations for the ground state. Since the entropy density for the ground state was calculated exactly by Wannier [@wannier] as $$\begin{aligned} S = \frac{2}{\pi} \int^{\frac{\pi}{3}}_{0}{\ln (2 \cos \omega)} d\omega \approx 0.3383,\end{aligned}$$ we are able to compare the entropy density of the approximated state to the exact value in order to control the precision of our approximation. Previously, Maignan *et al*. [@maignan] performed a qualitative analysis of the magnetization curve starting with an initial state that consisted of alternating rows of spin-up and spin-down chains. The energy of such an arrangement equals the ground state energy but its statistical weight goes to zero in the limit of the infinite lattice. Therefore, arrangements of this type should be discarded [@wannier]. The first approximation to the ground state of the AFM Ising model on the triangular lattice is the honeycomb structure shown in Fig.\[f1\]. Two thirds of the total number of the chain spins are ordered in the AFM honeycomb structure whereas other chain spins placed in the centers of hexagons have arbitrary projections of chain spins onto $c$ axis. The entropy density of the first approximation equals $S_1=(1/3)ln 2\approx 0.231$. An arbirary small external magnetic field lifts degeneracy of chain spins in the centers of hexagons orienting them along the magnetic field ($\sigma_i^z=1$). We consider that the spin-up chains (the black circles) are directed parallel to the magnetic field. The grey circles become black in Fig.\[f1\]. This causes a step at $B=0$ with the height of $1/3$ where the full magnetization is taken as unity. Then, one third of the spin chains remain spin-down or antiparallel to the field. Since they are surrounded by 6 spin-up chains this configuration is stable up to the critical magnetic field $B_C=6J$ where a transition to the fully-polarized FM state takes place. The magnetization curve for the first approximation is shown in Fig.\[f4\]. It should be noticed that this curve is in an excellent agreement with the experimental data at the intermediate temperatures . ![\[f1\] The honeycomb magnetic structure. The black and white circles are spin-up and spin-down states, correspondingly, the gray circles can be either spin-up or spin-down states.](fig1_1.eps) ![\[f4\] The magnetic moment as a function of the dimensionless magnetic field $B/J$. The dash, dash-dot, and solid lines are the results of the first, second, and fourth approximations, correspondingly. The four nearest-neighbor configurations producing the critical spin-flip fields are shown. Note the break in the vertical axis.](fig4_3.eps) We can improve the approximation used above including tripod configurations [@wannier]. Since the chains placed in the centers of hexagons have an arbitrary projections of the spin onto $c$ axis, it occurs that three of them neighboring with the same chain of the honeycomb sublattice are in the same state. Then the configuration shown in Fig.\[f2\] can appear. The chain spin in the center of the tripod belongs to the honeycomb sublattice but, nevertheless, it can have an arbitrary $c$ projection of the chain spin. These states increase the entropy density up to $S=(5/12)ln2\approx 0.289$ [@wannier] and involve various types of configurations: isolated tripods and connected tripods. It is convenient to consider them separately. After straightforward calculations we obtain the probability of an isolated tripod $$P_2=(1/12)(1-1/2^3-1/2^4-1/2^5)^3 \approx 0.0397.$$ We include this type of configurations in the second approximation for the initial state. The entropy density of the second approximation is $S_2 \approx 0.259$. Applying the external magnetic field we again obtain a step at the zero magnetic field but the chain spins in the centers of hexagons sharing joint corners with the tripod center remain in the spin-down state because they are surrounded by 4 spin-up and 2 spin-down nearest-neighbors. The height of the step ($\Delta M(B/J)$) can be expressed through probability of the isolated-tripod configuration as $\Delta M(0)=(1/3)-2P_2$. The spin chains in the centers of the tripod hexagons flip at the new critical magnetic field $B_C=2J$. While the magnetic fields increases further, the curve occurs at the $1/3$-plateau and coincides with the curve obtained in the first approximation (see Fig. \[f4\]). The third approximation is to include configurations with isolated pairs of connected tripods. These configurations change the heights of the steps but give no new features in the magnetization curve. The probability of the pair of isolated tripod is $P_3 \approx %%@ 0.010$. The entropy density increases up to $S_3 \approx 0.273$. An isolated configuration of three tripods connected as a star is taken into account in the framework of the fourth approximation (see Fig.\[f3\]). The probability of this configuration is $P_4 \approx 0.0035$ and the entropy density in the fourth approximation is $S_4 \approx 0.280$. The key feature of this configuration is that there are chain spins that are surrounded by 5 spin-up and one spin-down nearest-neighbors. These chain spins remain stable up to the new critical magnetic field $B_C=4J$. To calculate the step of in the magnetization curve we have to calculate a number of various sequences of spin flips. The final value of the step is $\Delta M(4) \approx 0.008$ that is significantly smaller then $\Delta M(2) \approx 0.12$. There is a variety of more complex configurations then the tripod. However, it should be mentioned that further approximations should change the heights of the steps but they can not cause new features in the magnetization curve. As it follows from Eq.(\[meta\]) there exist only four critical magnetic fields related to the four configurations of the nearest-neighbors shown in Fig.\[f4\]. ![\[f2\] An arrangement containing the tripod configuration. The tripod is shown in the solid line. The three white circles close to the center of the tripod have 4 spin-up (the black circles) and 2 spin-down (the white circles) nearest-neighbors.](fig2_2.eps) ![\[f3\] Three tripods connected in the star arrangement. Three white circles close to the center of the star are surrounded by 5 spin-up and one spin-down nearest-neighbors.](fig3_1.eps) The magnetization curve obtained in the fourth approximation reproduces the key features of the experimental data at the very low sweep rate, namely, the four equidistant steps in the magnetization curve. In contrast with our results, the third step in the experimental curve is much larger than the second one. This quantitative discrepancy can be eliminated in higher order approximations that can be investigated both analytically, by the technique used in the present Letter, or numerically, applying Monte Carlo simulation [@kim]. Approximations of higher orders are of importance for quantitative calculations of the magnetization curve because the calculation convergence for the magnetic moment is slower than that for the configuration probability. Few questions on the magnetization of Ca$_3$Co$_2$O$_6$ are still unclear. There were observed weak smeared features in the experimental curve at high magnetic fields above the last step. They can stem from the small fraction of CoII sites that are the high-spin state, because they increase the chain spin and cause higher critical fields. The sizeable hysteresis also draws attention in the experimental curve. It depends drastically on the temperature and the magnetic field sweep rate. It should be mentioned that while the magnetic field sweeps down and crosses the highest critical field $B_C=6J$ the chain spins flip down at random and the system occurs in a new state that is different from that at the sweeping-up process. In conclusion, we have developed a new model for the step-like magnetization of Ca$_3$Co$_2$O$_6$ spin-chain compound. It can be applied also to other spin-chain compounds with the triangular lattice of chains, e.g. Ca$_3$CoRhO$_6$. Due to the in-chain FM coupling between Co$^{3+}$ ions at trigonal sites, Co$_2$O$_6$ chains are considered at low temperatures as a large rigid spins with the strong Ising-like anisotropy. The AFM Ising model on the triangular lattice is applied to the system of rigid FM-ordered chains. The crucial point of the model is the supposition that the system is out of equilibrium, because the dependence of the magnetic moment on the magnetic field in the ground state of the AFM Ising model on the triangular lattice is smooth with the exception of the step at the zero magnetic field [@mattis]. For the honeycomb AFM structure two steps were found in the theoretical magnetization curve in excellent agreement with experimental data at the intermediate temperatures. At higher approximations four equidistant steps were determined in accordance with experimental curves at the low temperatures and very low magnetic field sweep rate. The results obtained in the present Letter and the model of Ref.[@drillon] can be regarded as two limiting cases. The first deals with the nonequilibrium interchain ordering assuming the chain spins to be rigid. An in-chain fragmentation and the quantum tunnelling of the magnetic moment of the fragments are investigated in the second case totally neglecting the interchain ordering. I am grateful to C. Demangeat and M. Drillon for fruitful discussions and hospitality during my stay at Institut de Physique et Chimie des Matériaux de Strasbourg en Unité mixte de recherche. This study was partially supported by the INTAS grant (03-51-4778) “Hierarchy of scales in magnetic nanostructures”. [15]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , , , ****, (). , , , , , , , , ****, (). , , , , , , , ****, (). , , , , ****, (). , , , , , ****, (). , , , , , ****, (). , , , , , , , , , ****, (). , , , , , , , , , ****, (). , , , ****, (). , , , , , , ****, (). , , , , , ****, (). , , , , ****, (). , , , ****, (). , ****, (). , ** (, , ).
= -2cm = -1cm [**COVARIANT QUANTUM DYNAMICAL SEMIGROUPS:\ UNBOUNDED GENERATORS**]{} Steklov Mathematical Institute,\ Vavilova 42, 117966 Moscow, Russia\ E-mail: holevo@class.mi.ras.ru . [*A survey of probabilistic approaches to quantum dynamical semigroups with unbounded generators is given. An emphasis is made upon recent advances in the structural theory of covariant Markovian master equations. As an example, a complete characterizations of the Galilean covariant irreversible quantum Markovian evolutions is given in terms of the corresponding quantum master and Langevin equations. Important topics for future investigation are outlined.*]{} Introduction ============ Quantum dynamical semigroups are a noncommutative analog of (sub-) Markov semigroups in classical probability: while the latter are semigroups of maps in functional spaces, the former are semigroups of maps in operator algebras, having certain properties of positivity and normalization. In quantum statistical mechanics dynamical semigroups arise when one considers weak or singular coupling or low density limits for open quantum system interacting with surrounding, allowing to neglect the memory effects of the interaction [@spohn]. These semigroups satisfy differential equations that are noncommutative generalization of the Fokker-Planck or Chapman-Kolmogorov equations and represent the general solution of the Cauchy problem for such equations. Let ${\cal B}({\cal H})$ be the algebra of all bounded operators in a Hilbert space $\cal H$. We denote by $I$ the unit operator in $\cal H$, and by Id the identity map of ${\cal B}({\cal H})$. Since ${\cal B}({\cal H})$ is the dual Banach space of the space ${\cal T}({\cal H})$ of trace-class operators, it is supplied with the [*weak$^*$ topology*]{}. On norm-bounded sets this topology coincides with the weak operator topology (see e.g. [@brat] ). A bounded map $\Phi $ of ${\cal B}({\cal H} )$ into itself is [*completely positive*]{} (c. p.) if $$\sum \limits_{i,j}(\psi_{i} |\Phi [X_{i}^{*}X_{j}]\psi_{j}) \geq 0 \eqno(1)$$ for any finite sets $\{ \psi_j \} \in {\cal H} , \{ X_j \} \in {\cal B}({\cal H}).$ According to Stinespring’s theorem adapted to the case of ${\cal B}({\cal H})$ (see [@kraus]), a generic weak$^*$-continuous c. p. map has the representation $$\Phi [X] = L^* (X\otimes I_0 )L, \eqno(2)$$ where $L$ is a bounded operator from $\cal H$ to ${\cal H}\otimes{\cal H}_0$ and $I_0$ is the unit operator in an auxiliary Hilbert space ${\cal H}_0$. By using this representation, it is possible to show that such maps satisfying additional normalization condition $\Phi [I] = I$ represent irreversible evolutions of the open quantum system interacting via unitary operator with auxiliary system in a fixed initial state (see [@kraus], [@lewis]). If $I_0 = \int |\psi_x ><\psi_x |\mu (dx)$ is a resolution of identity in ${\cal H}_0$, then (2) implies $$\Phi [X] = \int L(x)^* XL(x)\mu (x),$$ where $L(x)=(I\otimes\psi_x )^* L$, and $I\otimes\psi_x$ is the operator from $\cal H$ to ${\cal H}\otimes{\cal H}_0$, mapping $\psi$ into $\psi\otimes\psi_x$. In particular, if the measure $\mu (dx)$ is discrete, we obtain the familiar representation of a normal c. p. map in ${\cal B}({\cal H} )$. By a [*dynamical semigroup*]{} in ${\cal B}({\cal H})$ (or [*quantum dynamical semigroup*]{}) we shall call a semigroup $\Phi_t ;~ t\geq 0,$ of weak$^*$ continuous completely positive maps in ${\cal B}({\cal H})$, satisfying $\Phi_0 = \mbox{Id}$, and $\Phi_t [I]\leq I$. Moreover, for any $X$ the function $t \rightarrow \Phi_t [X]$ is required to be weak$^*$-continuous. $\Phi_t$ is called [*unital*]{} if $\Phi_t [I] = I$. In the case of finite-dimensional $\cal H$ the weak$^*$ continuity is equivalent to the norm continuity; every quantum dynamical semigroup then has the form $\Phi_t = \mbox{exp}t{\cal L}$, where $\cal L$ is the [*generator*]{} of the semigroup. The generator is [*conditionally completely positive*]{} map, which means that inequality of the type (1) holds provided $\sum \limits_j X_j \psi_j = 0,$ and satisfies the normalization condition ${\cal L}[I]\leq 0$ (or ${\cal L}[I] = 0$ for unital semigroup). The semigroup is the unique solution of the [*backward*]{} and the [*forward Markovian master equations*]{} (M. m. e.) $$\frac{d}{dt}\Phi_t = {\cal L} \circ\Phi_t ;\qquad\frac{d}{dt}\Phi_t = \Phi_t\circ{\cal L},\eqno(3)$$ satisfying $ \Phi_0 = \mbox{Id}$. The conditional complete positivity of $\cal L$ is equivalent to the [*standard representation*]{} $${\cal L}[X] = \Phi [X] - K^* X - XK, \eqno(4)$$ where $ \Phi $ is a c. p. map of the form (2), and the normalization condition is equivalent to $L^* L\leq K^* + K$ (with equality for unital semigroups). Similar results hold for [*norm-continuous*]{} semigroups in infinite-dimensional $\cal H$ with all operators in question being [*bounded*]{}. The representation (4) for this case was established by Lindblad [@lindblad], and independently an equivalent representation was obtained by Gorini, Kossakowski and Sudarshan [@gorini] for dim${\cal H}< \infty$. A physical interpretation for the standard representation can be seen from the Dyson expansion of the solution of the forward   M. m. e. $$\Phi_t = {\hat \Phi}_t + \sum_{n=1}^{\infty}\int...\int_{0\leq t_1 \leq ...\leq t_n \leq t}{\hat \Phi}_{t_1} \circ\Phi\circ{\hat \Phi}_{t_2-t_1} ...\Phi\circ{\hat \Phi}_{t-t_{n}} dt_1 ...dt_{n} \eqno(5)$$ described as sequence of “spontaneous jumps” of the magnitude $\Phi$ occuring at times $t_1 \leq ... \leq t_n$ on the background of the “relaxing evolution” given by the semigroup ${\hat \Phi}_t [X] = \mbox{e}^{-K^* t}X\mbox{e}^{-K t}$. Let $g\rightarrow V_g$ be a unitary representation of a group $G$ in $\cal H$. The dynamical semigroup $ \Phi_t$ is called [*covariant*]{} if $$\Phi_t [V_g^* XV_g ] = V_g^* \Phi_t [X] V_g . \eqno(6)$$ The property of covariance reflects presence of certain symmetries in the interacting open quantum system and is important for applications. For example, covariance with respect to various subgroups of the orthogonal group is characteristic for Bloch type equations and is relevant for optical or magnetic resonance spectroscopy [@alicki] or theory of anisotropic relaxation of spin systems [@artem]. The combination of covariance and complete positivity imposes strong restrictions on the generator, in some cases defining practically uniquely the form of the corresponding M. m. e.. It can be shown [@hol1] that the generator of a covariant norm continuous semigroup admits the representation (4) in which $\Phi$ is covariant and $K$ commutes with $V_g$ (provided $G$ is amenable). This in general is not true for non-norm continuous case, see Section 4. Already in the mid-seventies when the structure of norm-continuous dynamical semigroups was well understood, it became clear that the non-norm continuous case, while being interesting from both physical and mathematical points of views, poses difficult problems. The generator of such a semigroup may be unbounded with domain not even necessarily being a \* - algebra. Therefore it was difficult to formulate a generalization of conditional complete positivity (or equivalent property) useful enough to obtain a kind of standard representation for $\cal L$ with all its important consequences. The very formulation of the standard representation needed clarification and it was not obvious whether there are “non-standard” unbounded generators. Among very few papers on the subject, Davies [@dav1] established a standard representation for semigroups on ${\cal B}({\cal H})$ having invariant pure state. Bratteli et al. [@brat] studied semigroups on rather general C$^*$-algebra covariant with respect to a compact Abelian group and satisfying rather strong restriction that $\cal L$ vanishes on the fixed point subalgebra, and showed a kind of Levy-Khinchin formula for $\cal L$. There were few papers on quasi-free dynamical semigroups on CCR algebra, generators of which are certainly unbounded and standard, e. g. [@emch], [@vanheuv]. Unbounded generators arise when the semigroup is covariant with respect to a non-compact symmetry group (such as translations or Galilei group, [@lanz]). While enormous attention was paid to the study of reversible evolutions generated by Schrödinger operators, much less is known about their irreversible Markovian counterparts. A substantial progress in this direction was achieved in the past few years by making use of profound analogies from the classical theory of Markov semigroups, as developed by Feller, Dynkin, Ito and McKeane, see e. g. [@feller], [@ito], or by direct use of classical probabilistic methods. This development concerns the following topics: - The minimal dynamical semigroup [@dav2], [@chebot], [@chf], [@hol2], [@sinha]. Existence and uniqueness of solutions of Markovian master equations [@hol3], [@hol9]; - The structure of (covariant) Markovian master equations [@hol2], [@hol6], [@hol3], [@hol7]; - Noncommutative excessive functions and arrival times [@bhat], [@hol4]. Non-standard generators [@hol4], [@hol5]; - Stochastic representations and hyperdissipativity [@hol9]. Relations to continuous measurement processes and nonlinear stochastic Schrödinger equations [@gat], [@hol9], [@kol]. Dilations to quantum Langevin equations [@hol7]. In what follows we shall concentrate on the second topic, restricting to brief comments concerning other topics; further details can be found in the references given above. The quantum Markovian master equations ====================================== The starting point of our approach, just as in the classical probability theory, is not a semigroup itself, but the differential equation it satisfies. This is also more natural for physical applications. A quantum M. m. e. must be an equation for matrix elements of the semigroup; thus we assume that there is a dense domain $\cal D \subset \cal H$, such that the following derivative $$\frac{d}{dt}\left.<\psi |\Phi_t [X]\phi > \right|_{t=0} = {\cal L}(\psi ; X; \phi ) \eqno(7)$$ exists for $\phi, \psi \in \cal D$, $X\in {\cal B}({\cal H}) $. The form ${\cal L}(\psi ; X; \phi )$ is called [*form-generator*]{}. It can be characterized by a number of nice properties including conditional complete positivity [@hol2], [@hol3]. These properties turn out to be equivalent to the [*standard representation*]{} $${\cal L} (\psi ;X ;\phi ) = <L \psi |(X\otimes I_0 )L \phi > - <K\psi |X\phi > - <\psi |XK\phi >, \eqno (8)$$ where $L, K$ are (unbounded) operators defined on $\cal D$ and satisfying the [*dissipativity condition*]{} $$||L\psi ||^2 \leq~ 2\mbox{ Re}<\psi |K\psi >,~~~\psi \in \cal D.$$ In particular, $K$ is accretive: $\mbox{ Re}<\psi |K\psi >\geq 0, \psi \in \cal D.$ The (backward) M. m. e. takes the form $$\frac{d}{dt}<\psi |\Phi_t [X]\phi > = {\cal L}(\psi ; \Phi_t [X]; \phi );~~ \phi ,\psi \in \cal D. \eqno (9)$$ The relation between the form-generator and the generator resembles relation between a formal differential operator and its closed extensions determined by certain boundary conditions. To see this let $\Psi_t = (\Phi_t )_*$ be the strongly continuous [*preadjoint semigroup*]{} in ${\cal T}({\cal H})$, such that ${\Psi_t}^* = \Phi_t$. Denoting its generator ${\cal L}_*$, one has $${\cal L} (\psi ;X ;\phi ) = \mbox{ Tr} {\cal L}_* [|\phi ><\psi |]X$$ for $\phi ,\psi\in \cal D$, $X\in{\cal B}({\cal H})$. The assumption that the derivative (7) exists for all $X$ is equivalent to dom${\cal L}_* \supset {\sf D}$, where $$\mbox{\sf D} =\mbox{ lin}\{|\phi ><\psi |: ~~\phi, \psi \in \cal D \} \eqno(10)$$ is a dense domain in ${\cal T}({\cal H})$. The M. m. e. (9) takes the form $$\frac{d}{dt}\mbox{ Tr}\rho \Phi_t [X] = \mbox{ Tr}{\cal L}_* [\rho ]\Phi_t [X], \qquad\rho\in\mbox{\sf D}, X\in {\cal B}({\cal H}). \eqno (11)$$ If is a core for ${\cal L}_*$, then this equation determines $\Phi_t$ uniquely, otherwise it may have non-unique solution. Under the condition that the closure of $K$ is maximal accretive, one can show that there exists a dynamical semigroup ${\Phi_t}^{\infty}$ giving the [*minimal solution*]{} of the equation (9) in the sense that for any other solution $\Phi_t$ the difference $\Phi_t - {\Phi_t}^{\infty}$ is completely positive. Of special interest is the case of a [*unital*]{} generator, satisfying $\mbox{ Tr}{\cal L}_* [\rho] \equiv 0, \rho\in$[D]{}, or $$||L \psi ||^2 = 2\mbox{ Re} <\psi |K\psi >,~~\psi \in \cal D.$$ In general ${\Phi_t}^{\infty}$ may not be unital; however if it is, then ${\Phi_t}^{\infty}$ is the unique solution of (9). The method of construction of the minimal dynamical semigroup developed in [@dav2] for resolvents, in [@chebot], [@chf] for associated integral equation, and in [@hol2], [@hol3] for the backward M. m. e., is the noncommutative extension of the Feller’s method [@feller]. It is based on a standard representation, i. e. on a decomposition of the relevant object into completely positive and relaxing parts. The starting point is the relaxing semigroup ${\hat \Phi}_t [X ] = \mbox{e}^{-{\bar K}^* t}X\mbox{e}^{-{\bar K}t}$ providing the unique solution of the equation (9) with $${\cal L} (\psi ;X ;\phi ) = - <K\psi |X\phi > - <\psi |XK\phi >,$$ which is then perturbed with the completely positive form $<L \psi |(X\otimes I_0 )L \phi >$ introducing spontaneous jumps on the background of the relaxing evolution ${\hat \Phi}_t$. It may be viewed upon as a generalization of the expansion (5) to the case of unbounded but completely positive perturbations. Just as in the classical case, “explosion” may occur if the infinite number of jumps happens during finite interval and the process reaches “boundary” in a finite time (this can never happen for a bounded generator). If $\Phi_t^{\infty}$ is not unital, then there is a positive probability of explosion, and additional “boundary conditions” are required to specify the solution, which amounts to certain maximal extension of ${\cal L}_*$ from [D]{}. Under the additional assumption that operator ${L}^*$ satisfies $$\sum_j ||L^* (\psi\otimes e_j )||^2 < \infty,~~~\psi \in {\cal D}^* ,$$ where $\{e_j\}$ is an orthonormal basis in ${\cal H}_0$, and ${\cal D}^* \subset\mbox{dom}K^*$ is a dense domain in ${\cal H}_0$, one can write also the [*forward*]{} Markovian master equation for the preadjoint semigroup $\Psi_t$: $$\frac{d}{dt} <\phi |\Psi_t [\rho ]\psi > = {\cal L}_* (\phi; \Psi_t [\rho ];\psi );~~ \phi ,\psi \in {\cal D}^*, \eqno (12)$$ where $${\cal L}_* (\phi ;\rho ;\psi ) = \mbox{Tr}\rho{\cal L}[|\psi><\phi|]$$ $$= \sum_j < L^* (\phi\otimes e_j )|\rho L^* (\psi \otimes e_j )> - <K^* \phi |\rho \psi > - <\phi |\rho K^* \psi >,$$ and $\cal L$ is the generator of $\Phi_t$ defined on [D]{}$^* = \{ |\psi><\phi| : \phi, \psi \in {\cal D}^*\}.$ Assuming $K^*$ to be maximal accretive one can prove that ${\Psi}_t^{\infty} = ({\Phi_t}^{\infty})_*$ is the minimal solution of the forward equation [@hol7], [@hol9]. However in general the forward and the backward equations are no longer equivalent. Thus the situation is similar to that for the Kolmogorov-Feller differential equations in the theory of Markov processes [@feller]. Going back to the problem of standard representation, we can make the following remarks. The fact that a form-generator has the standard representation (8) implies the possibility of decomposing the generator ${\cal L}_*$ into completely positive and relaxing parts only on the subspace [D]{} which need not be a core for ${\cal L}_*$. If explosion occurs, these two parts need not be separately extendable onto a core for ${\cal L}_*$. On the other hand, generators of different dynamical semigroups restricted to [D]{} can give rise to one and the same standard expression (8). One may formalize the notion of standard representation by saying that a dynamical semigroup is [*standard*]{} if it can be constructed as the minimal semigroup for some M. m. e., that is by a completely positive perturbation of a relaxing semigroup. In [@hol4] a possible noncommutative extension of “boundary conditions” for conservative form-generator was proposed as very singular completely positive perturbations vanishing on the dense domain [D]{}. By using such a perturbation the author gave a construction of non-standard dynamical semigroup on ${\cal B}({\cal H})$ [@hol4], [@hol5]. An example ========== Let $\xi_t, t\geq 0$ be stochastic process with stationary independent increments [@feller]. Roughly speaking, the (generalized) time derivative of $\xi_t$ is a continuous analog of a sequence of independent identically distributed random variables, that is a classical “noise” process. One of the beautiful results of probability theory is the Levy-Khinchin formula describing the possible form of the characteristic function of such process: $${\sf M}\mbox{exp}i\lambda\xi_t = \mbox{exp}t[i\beta\lambda - \frac{\alpha}{2}\lambda^2 +\int_{0<|y|}(\mbox{e}^{iy\lambda}-1-iy\lambda 1_h(y))\mu (dy)], \eqno(13)$$ where $\beta$ is real number, $\alpha\geq0$, $h$ is arbitrary but fixed positive number, $1_h(y)$ is the indicator of the set $|y|\leq h$, and $\mu (dy)$ is a positive measure on the set $\mbox{\bf R}\setminus\{0\}$, satisfying the condition $$\int_{0<|y|}[y^2 1_h(y)+(1-1_h(y))]\mu(dy)<\infty. \eqno(14)$$ In (13) the term $ i\beta\lambda -\frac{\alpha}{2}\lambda^2 $ corresponds to the Gaussian component of the process $\xi_t$, which is a continuous process. If in the integral term we take $\mu (dy ) =\mu~ \delta (y-y_0 )dy$ with $\mu >0$, then for $|y_0 |>h$ we obtain logarithm of the characteristic function of the Poisson process with the jumps of the magnitude $y_0$. Therefore for arbitrary measure $\mu (dy)$ the integral $\int_{h<|y|}(\mbox{e}^{iy\lambda}-1)\mu (dy)$ describes the mixture of independent Poisson processes with various magnitudes $y,~ |y|>h.$ It corresponds to the discontinuous (pure jump) component of the process $\xi_t$ (with magnitudes of jumps $|y|>h$). The value of $h$ is arbitrary but fixed, so the name “big jumps” is only conventional. The term related to “small jumps” (of magnitudes $|y|\leq h$) corresponds to the situation when infinitely many small jumps can accumulate during finite time, and one must include portions of linear drift between jumps in order that the total increment will remain finite. The process $\xi_t$ itself can be decomposed into three components – continuous Gaussian, Poisson “big jumps” and “small jumps”, according to Ito’s formula (see e. g. [@stoch]): $$d\xi_t = \beta dt+\sqrt{\alpha} dW_t+\int_{h<|y|}y\Pi(dy~dt)+\int_{0<|y|\leq h} {\tilde \Pi}(dy~dt), \eqno(15)$$ where $W_t$ is the standard Wiener process, $\Pi(dy~dt)$ is the Poisson random measure on ${\bf R}^2$ with the compensator $\mu (dy) dt$, so that $${\sf M}dW_t = 0,\quad {\sf M}\Pi(dy~dt) = \mu(dy)~dt, \eqno(16)$$ and ${\tilde \Pi}(dy~dt) ={\Pi}(dy~dt)- \mu(dy)~dt$ is the compensated random measure. Note that $\Pi([y_1, y_2], [t_1, t_2])$ is just the number of jumps of the process $\xi_t$ on the time interval $[t_1, t_2]$, which have magnitudes $y\in[y_1, y_2]$. Now consider the Hilbert space ${\cal H}=L^2({\bf R})$, and let $Q=x, P= i^{-1}\frac{d}{dx}$ be, respectively, the self-adjoint position and momentum operators for one-dimensional quantum system, so that $V_y=\mbox{exp}(iyQ), y\in{\bf R},$ and $U_x=\mbox{exp}(-ixP), x\in{\bf R},$ are the unitary groups in $\cal H$ satisfying the Weyl canonical commutation relation (CCR): $$U_x V_y = \mbox{exp}(-ixy) V_y U_x. \eqno(17)$$ Defining $$\Phi_t [X] = {\sf M}U_{\xi_t}^* X U_{\xi_t}, \quad t\geq 0, \eqno(18)$$ one easily sees that $\Phi_t$ is a unital dynamical semigroup in $\cal H$. Indeed, operators $\Phi_t$ are manifestly completely positive; the semigroup property follows from the fact that $\xi_t$ has stationary independent increments; the weak$^*$ continuity properties follow from the continuity properties of $U_x$ and of the expectation. The semigroup (18) represents the dynamics of quantum system in $\cal H$ interacting with the classical noise via unitary operators $\mbox{exp}(-i\xi_t P)$, averaged with respect to the distribution of the noise. To find the generator of this semigroup, one can use the Ito formula for $\mbox{exp}(i\xi_t P)$ (cf. [@hol8]): $$d\mbox{exp}(i\xi_t P)=\mbox{exp}(i\xi_t P) \{[i\beta P - \frac{\alpha}{2}P^2 +\int_{0<|y|\leq h}(\mbox{exp}(iyP)-1-iyP) \mu(dy)]dt$$ $$+ i\sqrt{\alpha}P dW_t + \int_{h<|y|}[\mbox{exp}(iyP)-1]\Pi(dy~dt) + \int_{0<|y|\leq h}[\mbox{exp}(iyP)-1]{\tilde \Pi}(dy~dt)\},$$ and the Ito product rule $$dW_t^2 = dt,\quad \Pi(dy~dt)^2 = \Pi(dy~dt), \eqno(19)$$ with all other products of stochastic differentials (including $dt$) equal to zero. Taking into account (16), one can obtain both backward and forward M. m. e. (9), (12) with ${\cal D}={\cal D}^* = C_0^2 ({\bf R})$, the subspace of twice continuously differential functions with compact support, where the form-generators correspond to the expression $${\cal L}[X] = i\beta[P,X]-\frac{\alpha}{2}[P,[P,X]] +\int_{0<|y|}(U_y^* X U_y - X -iy[P,X]1_h(y))\mu (dy), \eqno(20)$$ defined for $X\in$[D]{}. Here the first term is the Hamiltonian “drift”, the second term corresponds to the interaction with the Gaussian “white” noise and is typical for diffusion approximations, while the last term reflects the influence of the Poisson “shot” noises arising in low density limits. The generator (20) is bounded if and only if $\alpha=0, \beta=0$ and $\mu(dy)$ is a finite measure on ${\bf R}\setminus 0$. The Gaussian noise gives rise to the diffusive generator $${\cal L}[X] = -\frac{\alpha}{2}[P,[P,X]] = \frac{\alpha}{2}(2PXP - P^2 X - XP^2 ), \eqno(21)$$ with the obvious standard representation on [D]{}. A standard representation for the last term in (20) can be obtained by taking $$L = \int_{0<|y|}(I\otimes|y>)(U_y - I)\mu (dy),\quad K = \int_{0<|y|}(I - U_y - iyP1_h(y))\mu (dy),$$ where $\{ |y>\}$ is the canonical family of “ket” vectors in ${\cal H}_0 = L^2 ({\bf R},\mu)$. From the CCR it follows that the semigroup is covariant with respect to the representation $y\rightarrow V_y$ describing translations in the momentum space. As shown in [@hol2], for covariant M. m. e. the non-explosion in ${\cal B}({\cal H})$ is equivalent to the non-explosion in the fixed-point subalgebra ${\cal A}_V = \{X: V_g^* X V_g = X, g\in G\}$ of the representation $g\rightarrow V_g$. If this subalgebra is Abelian then the problem is reduced to the well-studied problem of non-explosion for a classical Markov process. In our example the fixed point algebra is the maximal Abelian subalgebra ${\cal A}_Q$ of operators of the form $X=f(Q)$; by the CCR $$\Phi_t [f(Q)] = {\sf M}f(Q+\xi_t),$$ and $${\cal L}f (x) = \beta\frac{df(x)}{dx}+\frac{\alpha}{2}\frac{d^2 f(x)}{dx^2} + \int_{0<|y|} [f(x+y)-f(x)-yf'(x)1_h(y)]\mu (dy)$$ is the generator of the semigroup corresponding to the process $\xi_t$ with stationary independent increments, for which explosion can never occur [@feller]. This is also strictly related to the additional property of covariance with respect to the space translations $x\rightarrow U_x$, shared by the semigroup (18). However the situation is different for more general momentum translation covariant M. m. e.. To see this, following [@hol3], consider the Hilbert space ${\cal H}= L^2 (l,\infty )$, the domain ${\cal D}=C_0^2 (l,\infty )$ consisting of continuously twice differentiable functions with compact support, vanishing at $l$, and the form-generator $${\cal L}(\phi, X, \psi )= <(P+L(Q))\phi | X(P+L(Q)\psi > -<K\phi | X\psi > - <\phi | XK\psi >, \eqno(22)$$ defined for $\phi,\psi\in{\cal D} $, where $K=\frac{P^2}{2}+PL(Q)+\frac{|L(Q)|^2}{2}$, and $L(Q)$ is a continuously differentiable complex function. This form-generator is covariant with respect to the representation $y\rightarrow V_y=\mbox{exp}(iyQ)$, and hence the corresponding minimal dynamical semigroup is also covariant [@hol2]. The restriction to the fixed point algebra ${\cal A}_Q$ corresponds to the classical diffusion on $(l,\infty )$ with the generator $${\cal L}f(x)=2\mbox{Im}L(x)\frac{df(x)}{dx}+\frac{1}{2}\frac{d^2 f(x)}{dx^2}.$$ Non-explosion means that both $l$ and $\infty$ are non-absorbing boundaries for this diffusion. The necessary and sufficient condition for this is Feller’s test [@ito], saying that the function $$\int_{x_0}^x \left[\mbox{exp}\int_x^y 4\mbox{Im}L(z)dz\right] dy,$$ where $x_0\in (l,\infty )$, must be non-integrable in the neighbourhoods of both $l$ and $\infty$. In particular, if $ L(x)\equiv 0$ (pure diffusion with no drift), then the probability of absorption at $l$ is positive, hence the minimal semigroup is non-unital and the solution of the M. m. e. is not unique. This minimal semigroup is the extension onto ${\cal B}({\cal H})$ of the Markov semigroup corresponding to the Brownian motion on $(l, \infty )$ killed at the boundary $l$. Other solutions of the backward M. m. e. are obtained by taking perturbations corresponding to various boundary conditions at $l$. An example of non-standard dynamical semigroup on ${\cal B}({\cal H})$ is constructed as a singular perturbation of this minimal semigroup corresponding to rebounding from $l$ to a fixed quantum state $\rho_0$ [@hol4], [@hol5]. Covariant evolutions and the group cohomology ============================================= Consider a backward M. m. e. given by a form-generator ${\cal L}(\phi, X, \psi)$. The standard representation (8) of the form-generator is not unique even if it is subjected to further condition of minimality [@hol3]. If $D$ is a unitary operator in ${\cal H}_0$, $a \in{\cal H}_0$, and $ b$ is a real number, then the operators $$L' = (I\otimes D)L + I\otimes a,\quad K' = K + (I\otimes a)^*(I\otimes D)L + [\frac{1}{2}\|a\|^2 - i b ]I,$$ where $(I\otimes a)$ is the operator from $\cal H$ to ${\cal H}\otimes{\cal H}_0$ acting as $(I\otimes a)\psi = \psi\otimes a$, give another standard representation for ${\cal L}(\phi, X, \psi)$ satisfying the minimality condition. The transformations $(D, a, b): (L, K)\rightarrow (L', K')$ form a kind of a “gauge group” (cf. also [@par]) under the multiplication law $$(D', a', b') (D, a, b) = (D'D, D' a+ a', b + b'- \mbox{Im}< a'|D' a >). \eqno(23)$$ We denote this group by $G({\cal L})$. It is endowed with the natural topology as a subset of the product ${\cal U} ({\cal H}_0 )\times{\cal H}_0\times {\bf R}$, where ${\cal U}({\cal H}_0)$ is the group of unitary operators in ${\cal H}_0$ with the weak operator topology. Let now the form-generator be [*covariant*]{} under a (projective) unitary representation $g\rightarrow V_g$ of a symmetry group $G$, namely, the domain $\cal D$ be invariant under $V_g$ and $${\cal L}(\phi, V_g^* XV_g, \psi) = {\cal L}(V_g\phi, X, V_g\psi),\quad \phi, \psi\in{\cal D}.$$ [**Theorem 1**]{}. [*There is a representation $g\rightarrow (D_g, a_g, b_g)$ of $G$ in $G({\cal L})$ such that $$(V_g^* LV_g, V_g^* KV_g) = (D_g, a_g, b_g)(L, K).\eqno(24)$$ If $V_g$ is a continuous representation of a topological group $G$ then the representation $(D_g, a_g, b_g)$ is continuous if and only if the scalar function $g\rightarrow{\cal L}(\phi, XV_g^*, V_g\psi)$ is continuous for all $\phi, \psi\in{\cal D}, X\in {\cal B}({\cal H})$.*]{} The proof of this theorem may be found in [@hol3], and here we discuss briefly the way it can be applied to find the form of a covariant generator for concrete symmetry groups $G$. From this theorem taking into account (24) it follows that $D_g$ is a unitary representation of $G$ and $ a_g$ is a first order cocycle for this representation in ${\cal H}_0$: $ a_{g'g} = D_{g'} a_g + a_{g'}$. Moreover, the real function $ b_g$ satisfies the coboundary equation $ b(g')+ b(g)- b(g'g) = \mbox{Im}< a_{g'}|D_{g'} a_g >$. Thus the structure of the covariant form-generator is determined by the low order cohomology of the group $G$, which was studied in detail for many interesting groups (see, e. g. [@cohom]). In particular, it is well known that the low order cohomology is trivial for compact groups, that is every cocycle is a coboundary for such groups, $ a_g = (D_g - I) a$ for some $ a\in{\cal H}_0$. It follows that in this case, similarly to the case of bounded generators, the covariant form-generator has the standard representation (8) where the c. p. component and the relaxing terms are separately covariant. That this is not the case for non-compact groups, can be easily seen from the example of the diffusive generator (21). Galilean covariant Markovian evolutions ======================================= Let $(\xi, \tau )\in {\bf R}^2$ be a point in the 2-dimensional non-relativistic space-time, and let $(x, v, t): (\xi, \tau ) \rightarrow (\xi',\tau' ) $ be the Galilei transformation $$\xi' = \xi +x+v\tau , \qquad\tau' =\tau +t, \eqno(25)$$ where $x\in {\bf R}$ is the space shift, $v\in {\bf R}$ the Galilean boost. For simplicity we consider zero-spin unit mass elementary system characterized by the Weyl operators $$W_{x,v} = \mbox{ exp}i(v Q - x P) = V_v U_x\mbox{exp}(\frac{i}{2}vx),$$ constituting irreducible representation of the CCR. A dynamical semigroup $\Phi_t$ is [*Galilean covariant*]{} [@lanz], [@hol7], if $$\Phi_t [W_{x,v}^* X W_{x,v}] = W_{x-vt,v}^* \Phi_t [X] W_{x-vt,v}.$$ Let ${\cal D} \subset\cal H$ be the dense domain $${\cal D} = \bigcap\limits_{x,v\in {\bf R} } {\rm dom} (v Q - x P)^2,$$ and let $\mbox{\sf D} \subset{\cal B} (\cal H )$ be the domain defined by the relation (10). We remark that $\cal D$ is invariant under $W_{x,v}$, and that [D]{} is norm-dense in ${\cal T}(\cal H )$ and ${\rm weakly}^*$ - dense in ${\cal B}(\cal H)$. We make the following assumption \(A) The domain [D]{} is contained both in ${\rm dom}{\cal L}_*$ and ${\rm dom}\cal L$. [**Theorem 2.**]{} [*A unital Galilean covariant dynamical semigroup satisfying the condition (A) has the generator $\cal L$ given by the following expression on [D]{} $${\cal L}[X] = i[\frac{P^2}{2},X]+ i[{\beta}_P P + {\beta}_Q Q, X]$$ $$- \frac{1}{2} \{ \alpha_{PP} [P,[P , X]] + \alpha_{PQ} [P , [Q ,X]] + \alpha_{QQ} [Q ,[Q , X]]\} \eqno (26)$$ $$+ \int\int_{x^2 + v^2 >0}\{ W_{x, v}^* X W_{x,v} - X - i[xP - vQ, X]1_h (x,v)\}\nu (dx~dv),$$ where $\beta_P ,\beta_Q \in {\bf R}$, the real matrix $$\left[ \begin{array}{cc}\alpha_{PP} & \alpha_{PQ} \\ \alpha_{PQ} & \alpha_{QQ}\end{array} \right]$$ is positive definite, $1_h (x,v)$ is the indicator of the set $x^2 + v^2 \leq h$ and $\nu(dx~dv)$ is a positive measure on ${\bf R}^2 \backslash \{ 0 \}$ satisfying the Levy condition $$\int\int_{x^2 + v^2 >0}\{(x^2+v^2)1_h (x,v)+[1-1_h(x,v)]\} \nu (dx~dv) < \infty.$$ Moreover, the domain [D]{} is a core for both ${\cal L}_*$ and $\cal L$ and the corresponding M. m. e. (9), (12) have $\Phi_t$ (resp. $\Psi_t$) as the unique solution.*]{} The last statement of the Theorem applies to particular Galilean covariant M. m. e. arising in various physical applications, such as quantum optics [@car], precision experiments [@brag], nonlinear quantum mechanics [@doebner] etc. The uniqueness of the solution of the M. m. e. is related to the fact that the fixed point algebra of the representation $(x,v)\rightarrow W_{x,v}$ is trivial, that is consists of multiples of the identity operator (cf. [@hol2]). A derivation of (26) can be based on Theorem 1. By subtracting from $\cal L$ the Hamiltonian term corresponding to the free motion we obtain a generator ${\cal L}_0$ satisfying the condition of Weyl covariance $${\cal L}_0 [W_{x,v}^* XW_{x,v}] = W_{x,v}^* {\cal L}_0 [X] W_{x,v} .$$ Let $L,K$ be the components of the standard representation of the corresponding form-generator. According to Theorem 1 there is a unitary representation $(x,v)\rightarrow D_{x,v}$ of the Abelian group ${\bf R}^2$ and the cocycle $ a_{x,v}$ in the Hilbert space ${\cal H}_0$ such that $L,K$ satisfy the covariance equations $$W_{x,v}^* LW_{x,v}=(I\otimes D_{x,v})L - I\otimes a_{-x,-v},$$ $$W_{x,v}^* KW_{x,v}= K - (I\otimes a_{-x,-v})^*L + [\frac{1}{2}\| a_{x,v}\|^2 - i b_{x,v}]I.$$ These equations can be solved by diagonalizing the representation $D_{x,v}$ and by using the structure of cocycles for representations of Abelian locally compact groups [@hol6]. The “Gaussian” part of the generator ${\cal L}_0$ arises from the identity subrepresentation of $D_{x,v}$ while the orthogonal complement gives the “jump” part. We conjecture that the assumption (A) can be deduced from the Galilean covariance itself, as we were able to deduce it from the Weyl covariance (see [@hol7], where an alternative proof of Theorem 2 is given). The generator (20) considered in Section 3 is a particular case of (26), provided we exclude the free Hamiltonian term. That generator arose from the semigroup (18) describing interaction of quantum system with the classical noise. It turns out to be possible to give a similar explicit description of the Galilean covariant quantum open systems, as systems interacting with specific classical noises. Let ${ \xi_t,\eta_t } $ be a classical stochastic process with stationary independent increments in ${\bf R}^2$, defined by the characteristic function of the Levy-Khinchin form $${\bf M}{\rm exp}i( \mu \xi_t - \lambda \eta_t) = {\rm exp}t\{ i(\mu\beta_P -\lambda\beta_Q)- \frac{1}{2}(\alpha_{PP} \mu ^2 + 2\alpha_{PQ}\mu \lambda + \alpha_{QQ}\lambda ^2 )$$ $$+ \int\int_{x^2 + v^2 >0} [{\rm e}^{i( \mu x - \lambda v)} - 1-i( \mu x - \lambda v)1_h(x,v)]\nu (dx~dv)\}, \eqno (27)$$ where $\beta_P ,\beta_Q ; \alpha_{PP} ,\alpha_{PQ} ,\alpha_{QQ}$ and $\nu (dx~dv)$ are taken from (26). Consider the stochastic differential equations $$dQ_t = \frac{P_t}{m}dt + d\xi_t,~~~dP_t = d\eta_t, \eqno (28)$$ with the initial conditions $Q_0 = Q, P_0 = P$. These will be the Heisenberg equations for our open quantum system. They correspond to the infinitesimal canonical transformation with the Hamiltonian $$dH_t = \frac{P^2}{2}dt+Pd\xi_t-Qd\eta_t.$$ Defining the chronologically ordered exponential $$U_t(\xi,\eta)={\cal T}\mbox{exp}(-i\int_0^t dH_s)$$ as the solution of the corresponding stochastic differential equation, we can prove (see [@hol7]) that $$\Phi_t [X] = {\sf M}U_t(\xi,\eta)^* XU_t(\xi,\eta).$$ This relation is a generalization of the representation (18) and the proof proceeds along similar lines by using the stochastic differential equation for $U_t(\xi, \eta)$ and the distribution of $\xi_t ,\eta_t$ defined by (27). Equations (28) are the Langevin equation giving the dilation of the dynamical semigroup $\Phi_t$ with the classical stationary independent increment processes as the driving noises. Discussion ========== The results described in the previous Section are due to the very restrictive nature of the full Galilean covariance. We obtain much broader and physically interesting class of quantum Markovian evolutions by omitting space translations and restricting only to Galilean boosts, that is to the fundamental symmetry of a non-relativistic particle in a potential field. The class of resulting evolutions is described in detail in [@hol7] for the case where the position space is the whole ${\bf R}^3$. Discussion at the end of Section 3 suggests that contrary to the case of full Galilean covariance, there is no automatic non-explosion, and boundary conditions should play an important role, especially for systems with restricted position domains. This case deserves much more detailed study. Other interesting problems are related to introducing spin degrees of freedom along with spatial ones and to gauge covariance. Another important distinction of the boost covariant evolutions is that they describe open systems interacting with quantum rather than classical noises. This means that the corresponding M. m. e. at least formally can be dilated to the Langevin equations (see [@hol7]) which are quantum stochastic differential equations driven by quantum Brownian motion or Poisson-type processes in the sense of [@par], but in general, with unbounded operator coefficients. For example, the Langevin equation dilating the diffusive M. m. e. defined by the form-generator (22) supplemented with the Hamiltonian term has the form $$df(Q_t)=i[\frac{P_t^2}{2},f(Q_t)]dt+f'(Q_t)i[(dA_t + L(Q_t)dt)^{\dagger} - \mbox{h. c.}] + \frac{1}{2}f''(Q_t)dt,$$ $$dP_t=U'(Q_t)dt+i[{\bar L}'(Q_t)(dA_t+\frac{1}{2}L(Q_t)dt) - \mbox{h.c.}],$$ where $U$ is the potential, $A^{\dagger}_t, A_t$ are creation-annihilation processes representing quantum Brownian motion and h. c. denotes hermitean conjugated terms. Remarkably, at least for the minimal solution of M. m. e. there always exists a representation via solutions of certain [*classical*]{} dissipative stochastic equation in the Hilbert space of the system. It provides a powerful probabilistic tool for study of the problem of non-explosion for quantum dynamical semigroups [@hol9], and of the nonlinear stochastic Schrödinger equation arising in the theory of continuous quantum measurement processes [@gat], [@hol9], [@kol]. 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--- abstract: 'This paper studies the existence and uniqueness of solution of Itô type stochastic differential equation $dx(t)=b(t, x(t), \om)dt+\si(t,x(t), \om) d B(t)$, where $B(t)$ is a fractional Brownian motion of Hurst parameter $H>1/2$ and $dB(t)$ is the Itô differential defined by using Wick product or divergence operator. The coefficients $b$ and $\si$ are random and anticipative. Using the relationship between the Itô and pathwise integrals we first write the equation as a stochastic differential equation involving pathwise integral plus a Malliavin derivative term. To handle this Malliavin derivative term the equation is then further reduced to a system of characteristic equations without Malliavin derivative, which is then solved by a careful analysis of Picard iteration, with a new technique to replace the Grönwall lemma which is no longer applicable. The solution of this system of characteristic equations is then applied to solve the original Itô stochastic differential equation up to a positive random time. In special linear and quasilinear cases the global solutions are proved to exist uniquely.' address: | Department of Mathematics,  University of Kansas\ 405 Snow Hall,   Lawrence,   KS 66045-2142\ Email:  yhu@ku.edu author: - Yaozhong Hu title: '**Itô stochastic differential equations driven by fractional Brownian motions of Hurst parameter $H>1/2$** ' --- Introduction ============ Let $T\in(0,\infty)$ be a given fixed number and let $\Om$ be the Banach space of continuous real-valued functions $f: [0, T]\rightarrow \RR$ with the supremum norm: $\|f\|=\sup_{0\le t\le T} |f(t)|$. For any $t\in [0, T]$ define the coordinate mapping $B(t): \Om\rightarrow \RR$ by $B(t)(\om)=\om(t)$. Let $\PP=\PP^H$ be the probability measure on the Borel $\si$-algebra $ \cF $ of $\Om$ such that $B=(B(t), 0\le t\le T)$ is a fractional Brownian motion of Hurst parameter $H\in (0, 1)$. Namely, on the probability space $(\Om, \cF, \PP)$, $B=(B(t), 0\le t\le T)$ is a centered (mean $0$) Gaussian process of covariance given by $$\EE\left(B(t)B(s)\right)=\frac12 \left(t^{2H}+s^{2H}-|t-s|^{2H}\right)\,.$$ Throughout the paper, we consider the case $H>1/2$. The natural filtration generated by $B(t)$ is denoted by $\cF_t$. For any $\be\in (0, H)$, it is known that almost surely, $B(t)$ is Hölder continuous of exponent $\be$. This means that there is a measurable subset of $\Om$ of probability one such that any element $\om$ in this set $B(\cdot, \om)$ is Hölder continuous of exponent $\be$. We shall work on this subset of $\Om$ and with an abuse of notation we shall denote this subset still by $\Om$. We also choose and fix such a $\be\in (1/2, H)$ throughout the paper. Fractional Brownian motions have been received a great attention in recent years. Stochastic integral, Itô formula, and many other basic results have been established. The stochastic differential equation of the form $$\label{e.1.1} dx(t)=b(t, x(t))dt+\si(t, x(t))\de B(t)\,, \ \ 0\le t\le T\,,\quad x(0)\quad \hbox{is given}\,,$$ has been studied by many authors and has found many applications in various fields, where $\de$ denotes the [*pathwise type integral*]{} defined by using Riemnan sum. Among many references we refer to [@BHOZ; @hustochastics; @misura] and in particular the references therein for more details. Lyons and his collaborators’ work on rough path analysis is a powerful tool in analyzing this type of equation (see [@FV; @lyonsqian] and the references therein). However, in the case when $B$ is the Brownian motion, the most studied equation is of Itô type. Namely, in Itô stochastic differential is used instead of the pathwise one. There are many reasons for the use of Itô stochastic differential in classical Brownian motion case. One reason is from the modeling point of view. If one uses to model the state of a certain system, then the term $b(t, x(t))$ represents [*all*]{} the “[*mean rate of change*]{}" of the system and the term $\si(t, x(t))\de B(t)$ is the “[*random perturbation*]{}", which has a zero mean contribution. When we use stochastic differential equations driven by fractional Brownian motion to model natural or social system, we also wish to separate the two parts: the part $b(t, x(t))$ represents [*all*]{} the mean rate of change and the part $\si(t, x(t))\de B(t)$ is merely the random perturbation, which should have a mean $0$. In another word, it is natural to require the [*mean*]{} of $\si(t, x(t))\de B(t)$ in to be zero. On the other hand, it is well-known from the work of [@duncan; @humams] that if $H\not=1/2$, then the pathwise type stochastic integral with respect to fractional Brownian motion may not be of zero mean. Namely, it is possible that $\EE \left[\int_0^T \si(t, x(t))\de B(t)\right] \not= 0$. Motivated by this phenomenon, an Itô type stochastic integral $\int_0^T \si(t, x(t))d B(t)$ is introduced with the use of Wick product in [@duncan; @huoksendal] (see [@BHOZ; @humams] and the references therein). This integral has the property that the expectation $\EE\left[ \int_0^T \si(t, x(t))d B(t)\right]$ is always equal to zero. This motivates to replace the pathwise integral in by the Itô one. In other words, we are led to consider the following [*Itô stochastic differential equation*]{} $$\label{e.ito-equation} dx(t)=b(t, x(t))dt+\si(t, x(t))d B(t)\,, \ \ 0\le t\le T\,,\quad x(0)\quad \hbox{is given}\,,$$ where $dB(t)$ denotes the [*Itô type stochastic differential*]{} (divergence type integral) defined in [@duncan] (see also [@BHOZ; @humams; @huoksendal] and references therein), $b$ and $\si$ are two real-valued functions from $ [0, T]\times \RR$ to $\RR$ satisfying some conditions that will be made precise later (we shall allow them to be random). To solve the above equation , a natural approach to try is the Picard iteration. To explain the difficulty let us define $x_n(t)$ by the following recursive formula (naive Picard iteration): $$\label{e.1.3} x_n(t) =x(0)+\int_0^t b(s, x_{n-1}(s) )dt+\int_0^t \si(s, x_{n-1}(s) )d B(s)\,, \quad 0\le t\le T\,,$$ where $ n=1, 2, \cdots $ and $x_0(t) :=x(0)$ for all $0\le t\le T$. Consider the above stochastic integral term on the right hand side. An Itô isometry formula states that $$\begin{aligned} &&\EE \left(\int_0^t \si(s, x_{n-1}(s) )d B(s)\right)^2\\ &=& \EE \left\{ \int_0^t\int_0^t \phi(u, v) \si(u, x_{n-1}(u) ) \si(v, x_{n-1}(v) ) dudv \right.\\ &&\quad \left. + \int_0^t\int_0^t \si_x (u, x_{n-1}(u) ) \si_x (v, x_{n-1}(v) ) {\mathbb D}^\phi_v x_{n-1}(u) {\mathbb D}^\phi_u x_{n-1}(v) dudv \right\} \,,\end{aligned}$$ where $\phi(u,v)=H(2H-1)|u-v|^{2H-2}$, $\si_x(t,x)$ denotes the partial derivative of $\si(t,x)$ with respect to $x$ and ${\mathbb D}^\phi_u$ is the Malliavin derivative (see forthcoming definition in next section). From this identity one sees that to bound the $L^2$ norm of $x_n $ one has to use the $L^2$ norm of $x_{n-1} $ plus the $L^2$ norm of the Malliavin derivative of $x_{n-1} $. In a similar way to bound the Malliavin derivative one has to use the second order Malliavin derivative, and so on. Thus, we see that the naive Picard iteration approximation cannot be applied to study the Itô stochastic differential equation . We shall use a different approach to study . To explain this approach we first use the relationship between pathwise and Itô stochastic integrals (established for example in [@duncan Theorem 3.12]. See also [@BHOZ] and [@humams]) to write the equation as $$\begin{aligned} x(t)&=&x(0)+\int_0^t b(s, x(s))ds+\int_0^t \si(s, x(s))\de B(s)\nonumber\\ &&\qquad -\int_0^t \si_x(s, x(s)) {\mathbb D}^\phi_s x(s) ds\,, \ \ 0\le t\le T\,.\label{e.1.4}\end{aligned}$$ Thus, the equation is reduced to an equation involved the pathwise integral plus a Malliavin derivative term. To understand the character of this equation, we consider heuristically the dependence on the random element $\om\in \Om$ of the random variable $x(t, \om)$ as a function of infinitely many variables (defined on $\Om$). We write it formally as $x(t, \om)=u(t, \tilde \ell_1, \cdots, \tilde \ell_n , \cdots)$, where $u(t, x_1, x_2, \cdots)$ is a function of infinitely many variable and $\ell_1, \cdots \ell_, \cdots$ are smooth deterministic functions such that $\langle \ell_i\,, \ell_j\rangle_{{\mathcal H}_\phi}=\begin{cases}1& \hbox{when $i=j$} \\ 0&\hbox{otherwise}\end{cases}$ (see the forthcoming definition for the Hilbert space ${\mathcal H}_\phi$. We usually assume that $\left\{\ell_1, \cdots \ell_, \cdots\right\}$ to be an orthonormal basis of ${\mathcal H}_\phi$) and $\tilde \ell_i=\int_0^T\ell_i(s) dB(s)$. Thus ${\mathbb D}_s^\phi x(s) =\sum_{i=1}^\infty \phi_i(s) \frac{\partial u}{\partial x_i} (s, x_1, x_2, \cdots) $ with $\phi_i(s)=\int_0^T \phi(s, r) \ell_i(r) dr$. With the above notations, the equation can be written as (we omit the explicit dependence of $u$ on $(x_1, x_2, \cdots)$) $$\begin{aligned} u(t)&=&u(0)+\int_0^t b(s, u(s))ds+\int_0^t \si(s, u(s))\de B(s)\nonumber\\ &&\qquad -\sum_{i=1}^\infty \int_0^t \phi_i(s) \si_x(s, u(s)) \frac{\partial u}{\partial x_i} ds\,, \ \ 0\le t\le T\,.\label{e.1.5}\end{aligned}$$ This is a first order hyperbolic partial differential equation driven by fractional Brownian motion for a function of infinitely many variables. We shall use the idea of characteristic curve approach from the theory of the first order (finitely many variables) hyperbolic equations (see for example [@li; @serre1; @serre2]). But since the classical theory is not directly applicable here we need to find the characteristic curve and prove the existence and uniqueness of the solution. Let us also point out that stochastic hyperbolic equations has also been studied by several authors (see e.g. [@Kunita]), which is different from ours. Now we explain our approach to solve . First, we construct the following coupled system of characteristic equations: $$\begin{cases}\Gamma(t)=\om+\int_0^t \si _x(s, z(s) )\int_0^\cdot \phi(s, u)du ds\,;\\ \\ z(t)= \eta(\om ) +\int_0^t b(s, z(s) )ds +\int_0^t \si(s, z(s) )\de B(s)\\ \qquad\qquad +\int_0^t\int_0^s \si(s, z(s) ) \si_x(u, z(u) ) \phi(s, u) duds\,, \end{cases} \label{e.1.6}$$ where $\si_x$ denotes the partial derivative with respect to $x$, $\Ga(t):\Om \rightarrow \Om$ and $z(t):\Om \rightarrow \RR$. This system of equations comes from the characteristic curve equation for the first order hyperbolic equation of a function of infinitely many variables (see [@li; @serre1; @serre2]). For this system of characteristic equations, we shall show the following statements. 1. We use the Picard iteration approach to show that the above system of equations has a unique solution. Since the fractional Brownian motion $B$ is not differentiable, the powerful Grönwall lemma cannot no longer be used. Additional effort is needed to solve the corresponding . We use a different contraction argument, presented in Section 4. We call this approach [*fractional Picard iteration*]{} and hope that this general contraction principle may also be useful in solving other equations involving Hölder continuous controls. Let us point out that has a global solution. 2. We show in Section 5 that $\Ga(t):\Om\rightarrow \Om$ defined by has an inverse $\La(t)$ when $t$ is sufficiently small (smaller than a positive random constant) and $x(t, \om)=z(t, \La(t, \om))$ satisfies (or ). To this end we need to use a new Itô formula which is quite interesting itself. This new Itô formula is presented in Section 3. 3. For general nonlinear equation we can only solve the equation up to a positive (random) time. This is because the inverse $\La(t)$ of $\Ga(t):\Om\rightarrow \Om$ exists only up to some random time (see one example given in Section 5). However, for linear or quasilinear equation we can solve the equation for all time $t\ge 0$. In particular, in one dimensional linear case, we can find the explicit solution. This is done in Section 6. For notational simplicity we only discuss one dimensional equation. The system of several equations can be handled in a similar way. It is only notationally more complex. On the other hand, our approach works for more general random anticipative coefficients with general anticipative random initial conditions. We present our work in this generality. This means we shall study a slightly more general equation (see in Section 5) instead of . There has been an intensive study on anticipative stochastic differential equations by using anticipative calculus (see [@Bu92; @Bu94; @nualart]). We hope our work can shed some lights to this topic as well. Some preliminary results are presented in Section 2 and some notations used in this paper are also fixed there. Preliminary =========== Fractional integrals and derivatives ------------------------------------ Denote $\left( -1\right) ^{\alpha }=e^{i\pi \alpha }$ and $ \Gamma \left( \alpha \right) =\int_{0}^{\infty }r^{\alpha -1}e^{-r}dr$. Let $a,b\in \mathbb{R}$ with $a<b$ and let $f\in L^{1}\left( a,b\right) $ and $\alpha >0.$ The left-sided and right-sided fractional Riemann-Liouville integrals of $f$ are defined by $$I_{a+}^{\alpha }f\left( t\right) =\frac{1}{\Gamma \left( \alpha \right) }\int_{a}^{t}\left( t-s\right) ^{\alpha -1}f\left( s\right) ds$$and $$I_{b-}^{\alpha }f\left( t\right) =\frac{\left( -1\right) ^{-\alpha }}{\Gamma \left( \alpha \right) }\int_{t}^{b}\left( s-t\right) ^{\alpha -1}f\left( s\right) ds,$$respectively if the above integrals exist, where $a\le t\le b$. The Weyl derivatives are defined as (if the integrals exist) $$D_{a+}^{\alpha }f\left( t\right) =\frac{1}{\Gamma \left( 1-\alpha \right) }\left( \frac{f\left( t\right) }{\left( t-a\right) ^{\alpha }}+\alpha \int_{a}^{t}\frac{f\left( t\right) -f\left( s\right) }{\left( t-s\right) ^{\alpha +1}}ds\right) \label{e.weyl-der-left}$$and $$D_{b-}^{\alpha }f\left( t\right) =\frac{\left( -1\right) ^{\alpha }}{\Gamma \left( 1-\alpha \right) }\left( \frac{f\left( t\right) }{\left( b-t\right) ^{\alpha }}+\alpha \int_{t}^{b}\frac{f\left( t\right) -f\left( s\right) }{\left( s-t\right) ^{\alpha +1}}ds\right)\,. \label{e.weyl-der-right}$$ For any $\lambda \in (0,1)$, denote by $C^{\lambda }(a,b)$ the space of $\lambda $-Hölder continuous functions on the interval $[a,b]$. We will make use of the notations$$\left\| x\right\| _{a,b,\beta }=\sup_{a\leq \theta <r\leq b}\frac{|x_{r}-x_{\theta }|}{|r-\theta |^{\beta }},$$and$$\Vert x||_{a,b}=\sup_{a\leq r\leq b}|x_{r}|,$$where $x:\mathbb{R}^{d}\rightarrow \mathbb{R}$ is a given continuous function. We refer to [@samko] for more details on fractional integrals and derivatives. Let $\pi: a=t_0<t_1<\cdots<t_{n-1}<t_n=b$ be a partition of $[a, b]$ and denote $|\pi|=\max_{0\le i\le n-1} (t_{i+1}-t_i)$. Assume that $f\in C^{\lambda }(a,b)$ and $g\in C^{\mu }(a,b)$ with $\lambda +\mu >1$. For these two functions, we define the Riemann sum $\displaystyle S_\pi(f|g)=\sum_{i=0}^{n-1} f(t_i)(g(t_{i+1})-g(t_i))$. From a classical result of Young [@young], we know that as $|\pi|\rightarrow 0$, the limit of $S_\pi(f|g)$ exists and is called the Riemann-Stieltjes integral $$\int_{a}^{b}fdg=\lim_{|\pi|\rightarrow 0} S_\pi(f|g)\,.$$ We also have the following proposition. \[p.integration-by-parts\] Suppose that $f\in C^{\lambda }(a,b)$ and $g\in C^{\mu }(a,b)$ with $\lambda +\mu >1$. Let $1-\mu<\al<{\lambda } $. Then the Riemann Stieltjes integral $\int_{a}^{b}fdg$ exists and it can be expressed as$$\int_{a}^{b}fdg=(-1)^{\alpha }\int_{a}^{b}D_{a+}^{\alpha }f\left( t\right) D_{b-}^{1-\alpha }g_{b-}\left( t\right) dt, \label{1.8}$$where $g_{b-}\left( t\right) =g\left( t\right) -g\left( b\right) $. We also note that in the convergent Riemann sum, we can also use $$\tilde S_\pi(f|g)=\sum_{i=0}^{n-1} f(\xi_i)(g(t_{i+1})-g(t_j))\,,$$ where $\xi_i$ is any point in $[t_i, t_{i+1}]$. Let $\Om$, ${\mathcal H}$ be two separable Banach spaces such that ${\mathcal H}$ is continuously embedded in $\Om$. Let $\BB$ be another separable Banach space. A mapping $F:\Om\rightarrow \BB$ is called ${\mathcal H}$-differentiable if there is a bounded linear mapping from ${\mathcal H}$ to $\BB$ (if such mapping exists, then it is unique and is denoted by ${\mathbb D}F(\om)$) such that $$\frac{F(\om+\vare h)-F(\om)}{\vare}={\mathbb D}F(\om)(h)+ o(\vare)\quad \hbox{as $\vare\rightarrow 0$}\,; \quad \forall \ \om\in \Om,\ h\in {\mathcal H}\,.$$ We can also considered ${\mathbb D}F(\om)$ as an element in the tensor product space $ \BB\otimes {\mathcal H}'$, where ${\mathcal H}' $ is the dual of ${\mathcal H}$. The directional derivative ${\mathbb D}_h F$ for any direction $h\in {\mathcal H}$ is defined as ${\mathbb D}_hF(\om)= {\mathbb D}F(\om)(h)=\langle {\mathbb D}F(\om), h\rangle_{{\mathcal H}',{\mathcal H}}$. To simplify notation we also write ${\mathbb D}_hF(\om)={\mathbb D}F(\om) h$. If ${\mathcal H}$ and $\BB$ are Banach spaces of real functions, and if there is a function $g(s, \om)$ such that ${\mathbb D}_hF(\om)= {\mathbb D}F(\om)(h) =\int_0^T f(s, \om)h(s)ds, \ \forall h\in {\mathcal H}$, then we denote ${\mathbb D}_sF(\om)=g(s, \om)$. It is easy to see that we have the following chain rule. If $F:\Om\rightarrow \BB_1$ is ${\mathcal H}_1$-differentiable and $G:\BB_1\rightarrow \BB_2$ is ${\mathcal H}_2$-differentiable such that ${\mathbb D}F(\om)$ is a bounded mapping from ${\mathcal H}_1$ to ${\mathcal H}_2$, then $G\circ F$ is also ${\mathcal H}_1$-differentiable and $${\mathbb D}G(F(\om))=({\mathbb D}G)(F(\om))\circ {\mathbb D}F(\om)\,.$$ For any function $f(t)=f(t, \om)$, where $(t, \om)\in [0, T]\times \Om $, which is Hölder continuous with respect to $t$ of exponent $\mu>1-H$, by Proposition \[p.integration-by-parts\], we can define the pathwise integral $ \int_0^t f(s)\de B(s)$ as the (pathwise) limit as $|\pi|=\max_{0\le i\le n-1} (t_{i+1}-t_i)\rightarrow 0$ of the following Riemann sum $$S_\pi(f) =\sum_{i=0}^{n-1} f(s_k) (B(s_{k+1})-B(s_k))\,, \label{e.riemann-sum-left}$$ where $\pi: 0=s_0<s_1<\cdots<t_{n_{n-1}}<t_n=t$ is a partition of $[0, t]$. This integral can also be given by $$\int_0^t f(s)\de B(s)=\int_0^t D^{1-\al} _{0+} f(s) D_{t-}^{ \al} B_{t-}(s) ds\,,$$ where $\al$ satisfies $ 1-\mu<\al<H$. If $f$ is Hölder continuous of exponent greater than $1-H$ and if $g$ is continuous, then $\eta(t)=\eta(0)+\int_0^t f(s) \de B(s)+\int_0^t g(s) ds$ is well-defined. For any continuous function $F$ on $[0, T]\times \RR$, which is continuously differentiable in $t$ and twice continuously in $x$ we have the following Itô formula: $$\begin{aligned} F(t, \eta(t)) &=& F(0, \eta(0)) +\int_0^t \left[ \frac{\partial }{\partial s} F(s, \eta(s)) +\frac{\partial }{\partial x}F (s, \eta(s)) g(s) \right] ds\nonumber\\ &&\qquad + \int_0^t \frac{\partial }{\partial x}F (s, \eta(s)) f(s) \de B(s) \,. \label{e.pathwise-ito}\end{aligned}$$ \[see for example [@hustochastics] and references therein.\]   We also notice that if $f:[0, T]\times \Om\rightarrow \RR$ is Hölder continuous of exponent $\mu>1-H$, then in the Riemann sum the left point $s_k$ can be replaced by any points $\xi_k$ in the subinterval. Namely, $$\int_0^T f(s) \de B_s=\lim_{|\pi|\rightarrow 0} \sum_{i=0}^{n-1} f(\xi_k) (B(s_{k+1})-B(s_k))\,, \label{e.def-pathwise-any-point}$$ where $\xi_k $ is any point in $[s_k, s_{k+1}]$. In the stochastic analysis of fractional Brownian motions of Hurst parameter $H>1/2$, usually, we take the above Banach space ${\mathcal H}$ to be the reproducing kernel Hilbert space ${\mathcal H}_\phi$: $${\mathcal H}_{\phi}=\left\{ f:[0, T]\rightarrow\RR\,, \|f\|_{{\mathcal H}_{\phi}}^2= \int_0^T\int_0^T f(u) f(v)\phi(u-v) dudv<\infty\right\}\,, \label{e.def-Hphi}$$ which is the completion of the space of smooth functions on $[0, T]$ with respect to the norm $\|\cdot\|_{{\mathcal H}_\phi}$, where $$\phi(u):=H(2H-1) |u|^{2H-2}\,.$$ The element in ${\mathcal H}_\phi$ may be generalized function (distribution) although we still write $f:[0, T]\rightarrow \RR$ in . We can define ${\mathbb D}_s F(\om)$ as usual and we denote $$\label{e.def-Dphi} {\mathbb D}^\phi_t F(\om) =\int_0^T \phi(t,s) {\mathbb D}_s F(\om) ds\,.$$ The expectation $\EE \int_a^bf(s) \de B(s)$ may generally not be zero. In [@duncan] (see also [@BHOZ; @humams]) we introduce an Itô stochastic integral by using the Wick product. We also established a relationship between pathwise and Itô integrals. Here, we can use this relationship to define Itô integral as $$\label{e.ito-pathwise-relation} \int_0^T f(t)dB(t)=\int_0^T f(t)\de B(t)-\int_0^T {\mathbb D}^\phi _t f(t)dt$$ if $f$ is Hölder continuous of exponent $\mu>1-H$ and $ {\mathbb D}^\phi _s f(s)$ exists and is integrable. It is easy to see that $\EE \left( \int_0^T f(t)dB(t)\right)=0$. The Itô formula and many other results for Itô stochastic integral have been established. Here, we explain that the Itô formula for Itô integral can also be obtained from and . \[p.2.2\] Let $$\eta(t)=\eta+\int_0^t f(s) dB_s+\int_0^t g(s) ds\,,$$ where $f$ is Hölder continuous with exponent greater than $1-H$ and $g$ is continuous. Assume that ${\mathbb D}^\phi_s f(s)$ exists and is a continuous function of $s$. Let $F:[0, T]\times \RR \rightarrow \RR$ be continuously differentiable in $t$ and twice continuously differentiable in $x$. Then $$\begin{aligned} F(t, \eta(t)) &=&F(0, \eta_0) +\int_0^t \frac{\partial F}{\partial x} (s, \eta(s))f(s)d B(s) \label{e.ito-ito-formula} \\ && +\int_0^t \left[ \frac{\partial F}{\partial s} (s, \eta(s)) + \frac{\partial F}{\partial x } (s, \eta(s)) g(s) + \frac{\partial^2 F}{\partial x^2} (s, \eta(s)) f(s) {\mathbb D}_s ^\phi \eta(s) \right] ds\,. \nonumber \end{aligned}$$ We briefly sketch the proof. First, by we see $$\eta(t)=\eta_0+\int_0^t f(s) \de B_s+\int_0^t \left[g(s)-{\mathbb D}^\phi_s f(s) \right] ds\,.$$ From it follows that $$\begin{aligned} F(t, \eta(t)) &=&F(0, \eta_0)+ \int_0^t \frac{\partial F}{\partial s} (s, \eta(s))ds +\int_0^t \frac{\partial F}{\partial x} (s, \eta(s))f(s)\de B(s) \nonumber\\ &&\qquad +\int_0^t \frac{\partial F}{\partial x} (s, \eta(s)) \left[ g(s)- {\mathbb D}^\phi_s f(s)\right] ds\\ &=&F(0, \eta_0)+ \int_0^t \frac{\partial F}{\partial s} (s, \eta(s))ds +\int_0^t \frac{\partial F}{\partial x} (s, \eta(s))f(s)d B(s) \nonumber\\ &&\qquad +\int_0^t \left\{ \frac{\partial F}{\partial x} (s, \eta(s)) \left[g(s)- {\mathbb D}^\phi_s f(s)\right] +{\mathbb D}^\phi_s \left(\frac{\partial F}{\partial x} (s, \eta(s))f(s) \right)\right\} ds\,. \end{aligned}$$ This is simplified to . Itô formulas ============ Denote $\cT=[0, T]$. If $X(t)=\eta+\int_0^t f(s)ds+\int_0^t g(s) \de B(s)$ and if $F$ is a function from $\cT\times \RR$ to $\RR$, then an Itô formula for $F(t, X(t))$ is given by , or if the integral is Itô type (see also [@young; @zahle; @hustochastics] and in particular the references therein). However, to show the existence and uniqueness of the solution to Itô stochastic differential equation we need an Itô formula of the following form: If $X$ is as above, $\Ga: \cT\times \Om\rightarrow \Om$ and $F: \cT\times \RR\times \Om$ to $\RR$, we want to find an Itô formula for $F(t, X(t), \Ga(t))$. Here and in what follows, we omit the explicit dependence on $\om$ when it is clear. \[t.3.1\] Let $h(t, u, \om)$, $(t,u,\om)\in \cT^2\times \Om$ be a continuous function of $t$ and $u$. Define a family of nonlinear transforms from $\Om$ to $\Om$ by $$\Ga(t, \om)=\om+ \int_0^\cdot h(t, u, \om) du\,,\quad t\in \cT\,.$$ Let $f:\cT\times \Om\rightarrow \RR$ be measurable such that for any $\om\in \Om$, $f:\cT \rightarrow \RR$ is Hölder continuous of order greater than $1-H$ so that $F=\int_0^T f(s) \de B(s)$ is well-defined. We have $$\begin{aligned} F\circ \Ga(t, \om) &=&\int_0^T f(s, \Ga(t, \om) )\de B(s) +\int_0^T f(s, \Ga(t, \om) ) h(t,s, \om) ds\,. \nonumber\\ \label{e.3.2} \end{aligned}$$ By a limiting argument we may assume that $f$ is of the form $$f(t, \om)=\sum_{k=0}^{n-1} a_k (\om) I_{[t_k, t_{k+1})}(t)\,,$$ where $0=t_0<t_1<\cdots<t_{n-1}<t_n=T$ is a partition of the interval $[0, T]$. Thus $$\begin{aligned} F(\om) &=& \sum_{k=0}^{n-1} a_k(\om)\left[B({t_{k+1}}, \om)-B({t_k}, \om)\right] \\ &=& \sum_{k=0}^{n-1} a_k (\om) \left[\om(t_{k+1})-\om(t_k) \right]\,.\end{aligned}$$ Thus $$\begin{aligned} F(\Ga(t,\om)) &=& \sum_{k=0}^{n-1} a_k(\Ga(t,\om) ) \Bigg\{ \om({t_{k+1}}) + \int_0^{t_{k+1}} h(t, s, \om)ds -\om({t_k}) -\int_0^{t_{k}} h(t, s, \om)ds \Bigg\} \\ &=& \sum_{k=0}^{n-1} \left\{a_k(\Ga(t,\om) ) ( \om({t_{k+1}})-\om({t_k})) + a_k(\Ga(t,\om) ) \int_{t_k} ^{t_{k+1}} h(t, s, \om)ds \right\} \\ &=&\int_0^T f(s, \Ga(t, \om) )\de B(s)+\int_0^T f(s, \Ga(t, \om) )h(t, s, \om)ds \,,\end{aligned}$$ which is . Let ${\mathcal H}\subseteq \Om$ be a Banach space continuously embedded in $\Om$. Now we state our new Itô formula. \[t.3.2\] Let measurable functions $\eta: \RR\times \Om\rightarrow \RR $, $f_0, f_1: \cT\times \RR\times \Om\rightarrow \RR $, $g_0, g_1:\cT\times \Om\rightarrow \RR$ satisfy $$\begin{cases} \hbox{ $f_0(s, x, \om) $ and $g_0(s, \om)$ are continuous in $s\in \cT$} \,;\nonumber\\ \hbox{$f_1(s,x,\om)$ and $g_1(s,\om)$ are H\"older continuous}\nonumber\\ \qquad \qquad\quad \hbox{ with respect to $s$ of order greater than $1-H$}\,;\nonumber\\ \hbox{$f_1(s,x,\om)$ is Lipschitz in $x$}\,. \end{cases}$$ Define $$\begin{aligned} F(t,x,\om) &=& \eta(x, \om)+\int_0^t f_0(s, x, \om) ds+\int_0^t f_1(s, x, \om) \de B(s)\end{aligned}$$ and $$\begin{aligned} G(t, \om) &=&\xi(\om)+\int_0^tg_0(s, \om) ds+\int_0^t g_1(s, \om) \de B(s)\,. \end{aligned}$$ Assume that $F$ and $\frac{\partial }{\partial x} F(t, x, \om)$ are Hölder continuous in $t$ of exponent greater than $1-H$ and Lipschitz in $x$, ${\mathcal H}$-differentiable in $\om$. Let $\xi:\Om\rightarrow \RR$ be measurable. Let $h$ and $\Ga$ be defined as in Lemma \[t.3.1\] and assume that $\Ga:[0, T]\times \Om\rightarrow \Om$ is continuously differentiable in $s$ with respect to the topology of ${\mathcal H}$ (namely, $\frac{d}{ds} \Ga(s)\in {\mathcal H}$). Then $$\begin{aligned} &&F(t, G(t), \Ga(t)) = \eta(\xi(\om), \om) +\int_0^t f_0(s, G(s), \Ga(s)) ds \nonumber \\ &&\quad \qquad+\int_0^t f_1(s, G(s), \Ga(s)) \de B(s) +\int_0^t f_1(s, G(s), \Ga(s)) h(s, s, \om) ds \nonumber \\ &&\quad \qquad +\int_0^t \frac{\partial }{\partial x}F(s, G(s), \Ga(s)) g_0(s) ds +\int_0^t \frac{\partial }{\partial x}F(s, G(s), \Ga(s)) g_1(s) \de B(s) \nonumber \\ &&\quad \qquad +\int_0^t ({\mathbb D}F)(s, G(s), \Ga(s)) \frac{d}{ds} \Ga(s) \, ds \,, \label{e.ito-formula} \end{aligned}$$ where and in what follows we denote $$({\mathbb D}F)(s, G(s), \Ga(s)) \frac{d}{ds} \Ga(s) :={\mathbb D}F(s, x, \om) \bigg|_{x=G(s)\!,\, \om =\Ga(s)} \frac{d}{ds} \Ga(s)\,.$$ Let $\pi:0=t_0<t_1<\cdots<t_n=t$ be a partition of $[0, t]$ and denote $|\pi|=\max_{0\le k\le n-1}(t_{k+1}-t_k)$. We have $$\begin{aligned} &&F(t, G(t), \Ga(t))-F(0, G(0), \Ga(0))\\ & &\qquad = \sum_{k=0}^{n-1} \left[ F(t_{k+1}, G(t_{k+1}), \Ga(t_{k+1}))- F(t_{k }, G(t_{k }), \Ga(t_{k }))\right]\\ & &\qquad = I_1+I_2+I_3\,,\end{aligned}$$ where $$\begin{aligned} I_1&=& \sum_{k=0}^{n-1} \left[ F(t_{k+1}, G(t_{k+1}), \Ga(t_{k+1}))-F(t_{k }, G(t_{k+1}), \Ga(t_{k+1}))\right] \,;\\ I_2&=& \sum_{k=0}^{n-1} \left[ F(t_{k}, G(t_{k+1}), \Ga(t_{k+1}))-F(t_{k }, G(t_{k }), \Ga(t_{k+1}))\right]\,; \\ I_3&=& \sum_{k=0}^{n-1} \left[ F(t_{k }, G(t_{k }), \Ga(t_{k+1}))-F(t_{k }, G(t_{k }), \Ga(t_{k }))\right]\,.\end{aligned}$$ Let us first look at $I_1$. Using Lemma \[t.3.1\], we have $$\begin{aligned} I_1 &=& \sum_{k=0}^{n-1} \int_{t_k}^{t_{k+1}} f_0(s, G(t_{k+1}), \Ga(t_{k+1})) ds\\ &&\qquad +\sum_{k=0}^{n-1} \int_{t_k}^{t_{k+1}} f_1(s, x, \om)\de B(s)\Big|_{ x=G(t_{k+1}), \om=\Ga(t_{k+1}) }\\ &=& \sum_{k=0}^{n-1} \int_{t_k}^{t_{k+1}} f_0(s, G(t_{k+1}), \Ga(t_{k+1})) ds +\sum_{k=0}^{n-1} \int_{t_k}^{t_{k+1}} f_1(s, G(t_{k+1}), \Ga(t_{k+1})) \de B(s) \\ &&\qquad + \sum_{k=0}^{n-1} \int_{t_k}^{t_{k+1}} f_1(s, G(t_{k+1}), \Ga(t_{k+1})) h(t_{k+1}, s) ds\,. \end{aligned}$$ From here it is easy to see that $$\begin{aligned} \lim_{|\pi|\rightarrow 0} I_1 &=& \int_0^t f_0(s, G(s), \Ga(s)) ds +\int_0^t f_1(s, G(s), \Ga(s)) \de B(s)\nonumber \\ &&\qquad +\int_0^t f_1(s, G(s), \Ga(s)) h(s, s, \om) ds \,. \label{proof-t.3.2-i1}\end{aligned}$$ Using the mean value theorem we have, denoting $G_{k, \theta}= G(t_k)+\theta \left[G(t_{k+1}) -G(t_k)\right]$, $$\begin{aligned} I_2 &=& \sum_{k=0}^{n-1} \int_0^1 \frac{\partial }{\partial x}F(t_{k}, G_{k, \theta} , \Ga(t_{k+1})) d\theta \left[ G(t_{k+1})-G(t_k)\right]\\ &=& \int_0^1 d\theta \left\{ \sum_{k=0}^{n-1} \int_{t_k}^{t_{k+1}} \frac{\partial }{\partial x}F(t_{k}, G_{k, \theta} , \Ga(t_{k+1})) \left[ g_0(s) ds +g_1(s)\de B(s)\right] \right\}\,. \end{aligned}$$ Since for any $\theta \in [0, 1]$, $G_{k, \theta}$ is any point between $G(t_k)$ and $G(t_{k+1})$, we see that for any $\theta\in [0, 1]$, $$\begin{aligned} &&\lim_{|\pi|\rightarrow 0} \sum_{k=0}^{n-1} \int_{t_k}^{t_{k+1}} \frac{\partial }{\partial x}F(t_{k}, G_{k, \theta} , \Ga(t_{k+1})) \left[ g_0(s) ds +g_1(s)\de B(s)\right] \nonumber\\ &&\qquad =\int_0^t \frac{\partial }{\partial x}F(s, G(s), \Ga(s)) g_0(s) ds +\int_0^t \frac{\partial }{\partial x}F(s, G(s), \Ga(s)) g_1(s) \de B(s)\,. \nonumber\end{aligned}$$ This implies $$\begin{aligned} \lim_{|\pi|\rightarrow 0} I_2 =\int_0^t \frac{\partial }{\partial x}F(s, G(s), \Ga(s)) g_0(s) ds +\int_0^t \frac{\partial }{\partial x}F(s, G(s), \Ga(s)) g_1(s) \de B(s)\,. \nonumber\\ \label{proof-t.3.2-i2} \end{aligned}$$ $I_3$ can be computed as follows. $$\begin{aligned} I_3 &=& \sum_{k=0}^{n-1} \int_{t_k}^{t_{k+1}} {\mathbb D}F(t_k, G(t_k), \Ga(s)) \frac{d}{ds} \Ga(s) \, ds \,. $$ From here it follow easily $$\begin{aligned} \lim_{|\pi|\rightarrow 0} I_3 &=& \int_0^t {\mathbb D}F(s, G(s), \Ga(s)) \frac{d }{d s}\Ga(s) ds \,. \label{proof-t.3.2-i3}\end{aligned}$$ Now we combine , and to prove the theorem. 1. $F(0, G(0), \Ga(0))=\eta(\xi(\om), \om)$. Thus, we can replace $ \eta(\xi(\om), \om)$ in by $F(0, G(0), \Ga(0)) $. 2. We may also write the above Itô formula in differential form as follows. $$\begin{aligned} &&dF(t, G(t), \Ga(t)) = f_0(t, G(t), \Ga(t))dt +f_1(t, G(t), \Ga(t))\de B(t)\nonumber\\ &&\qquad + \frac{\partial }{\partial x}F(t, G(t), \Ga(t)) \left[ g_0(t) dt + g_1(t) \de B(t)\right] \nonumber \\ &&\qquad + f_1(t, G(t), \Ga(t)) h(t, t, \om) dt + ({\mathbb D}F)(t, G(t), \Ga(t)) \frac{\partial }{\partial t} \Ga(t) dt\,. \nonumber\\\end{aligned}$$ If $F$ and $ G$ are given by Itô integrals, then what will be the Itô formula? Similar to the argument in the proof of Proposition \[p.2.2\] we can use the relationship to obtain an analogous Itô formula for Itô integrals. Let $$\begin{aligned} F(t, x, \om) &=& \eta(x, \om)+\int_0^t f_0(s,x,\om)ds+\int_0^t f_1(s, x, \om) dB(s)\\ G(t, \om) &=& \xi( \om)+\int_0^t g_0(s, \om)ds+\int_0^t g_1(s, \om) dB(s)\,.\end{aligned}$$ Let $h$ and $\Ga$ be defined as in Lemma \[t.3.1\]. We assume the conditions in Theorem \[t.3.2\] hold. Moreover, we also assume that ${\mathbb D}_sf_1^\phi(s)$ and ${\mathbb D}_sg_1^\phi(s) $ are continuously in $s$. From we have (we omit the explicit dependence on $\om$) $$\begin{aligned} F(t, x ) &=& \eta(x )+\int_0^t \left[ f_0(s,x) -{\mathbb D}_s^\phi f_1(s,x)\right] ds+\int_0^t f_1(s, x ) \de B(s)\\ G(t ) &=& \xi +\int_0^t \left[ g_0(s)-{\mathbb D}_s^\phi g_1(s)\right] ds+\int_0^t g_1(s )\de B(s)\,.\end{aligned}$$ By the Itô formula we have $$\begin{aligned} &&F(t, G(t), \Ga(t)) = \eta(\xi)+\int_0^t \left[ f_0-{\mathbb D}_s^\phi f_1\right] (s, G(s), \Ga(s)) ds \\ &&\qquad + \int_0^t f_1(s, G(s), \Ga(s))\de B(s)+\int_0^t f_1(s, G(s), \Ga(s)) h(s,s, \om ) ds\\ &&\qquad +\int_0^t \frac{\partial }{\partial x} F(s, G(s), \Ga(s))\left[ g_0 -({\mathbb D}_s^\phi g_1) \right] (s, \Ga(s)) ds\\ &&\qquad + \int_0^t \frac{\partial }{\partial x} F(s, G(s), \Ga(s))\ g_1 (s, \Ga(s)) \de B(s) \nonumber\\ &&\qquad +\int_0^t ({\mathbb D}F)(s, G(s), \Ga(s)) \frac{d}{ds}\Ga(s) ds\,. \end{aligned}$$ Using again the relationship between pathwise and Itô integral, we can rewrite the above identity as $$\begin{aligned} &&F(t, G(t), \Ga(t))\nonumber\\ &&= \eta(\xi)+\int_0^t \left\{ {\mathbb D}_s^\phi [f_1(s, G(s), \Ga(s))]+\left[ f_0 -{\mathbb D}_s^\phi f_1\right] (s, G(s), \Ga(s)) \right\} ds \nonumber\\ && + \int_0^t f_1(s, G(s), \Ga(s)) d B(s)+\int_0^t f_1(s, G(s), \Ga(s)) h(s,s, \om ) ds \nonumber\\ &&\ +\int_0^t \frac{\partial }{\partial x} F(s, G(s), \Ga(s))\left[ g_0 -({\mathbb D}_s^\phi g_1) \right] (s, \Ga(s)) ds \nonumber\\ && + \int_0^t {\mathbb D}_s^\phi \left[ \frac{\partial }{\partial x} F(s, G(s), \Ga(s))\ g_1 (s, \Ga(s))\right] ds\nonumber\\ && + \int_0^t \frac{\partial }{\partial x} F(s, G(s), \Ga(s))\ g_1 (s, \Ga(s)) d B(s) \nonumber\\ && +\int_0^t ({\mathbb D}F)(s, G(s), \Ga(s)) \frac{d}{ds}\Ga(s) ds\,. \end{aligned}$$ An iteration principle ====================== After transforming the original equation into the system of equations (or in next section for general random coefficient case), we can now use the Picard iteration method to solve the new differential system (pathwise). But the second equation in involves $\int_0^t \si(s, z(s), \Ga(s))\de B(s)$. Since $B(s)$ is not differentiable, one cannot no longer use the powerful Grönwall lemma. One way to get around this difficulty is to use the Besov spaces (see e.g. [@nualartrascanu]). Here, we propose to use the Hölder spaces which seems to be simpler. The idea is motivated by the works [@hunualartabel; @hunualarttams]. In this section we present a general contraction principle, which may be useful in solving other equations driven by Hölder continuous functions. We call this approach the fractional Picard iteration. In next section we shall use this general contraction principle to solve and subsequently to solve . Let $\BB$ be a separable Banach space with norm $\|\cdot\|$ (in case we need to specify we write $\|\cdot\|_\BB$). We denote by $\BB[0, T]$ the Banach space of all continuous functions from $[0, T]$ to $\BB$ with the sup norm $\|x\|_{0, T}=\sup_{0\le t\le T}\|x(t)\|_\BB$. For any $0\le a<b\le T$ and an element $x\in \BB[0, T]$, we define $$\|x\|_{a, b, \be} =\sup_{a\le s<t\le b} \frac{\|x(t)-x(s)\|}{|t-s|^\be}$$ if the above right hand side is finite. We also use the notation $\|x\|_{a, b}=\sup_{a\le t\le b}\|x(t)\| $ and in the case when $a=b$, we denote $\|x\|_{a, a}= \|x(a)\| $. Given $\De\in (0, T]$ and $\be\in (0, 1]$ we denote $$\||x\||_{\De, \be} =\sup_{0\le t\le T}|x(t)|+\sup_{0\le s<t\le T, t-s\le \De} \frac{\|x(t)-x(s)\|}{|t-s|^\be} \,.$$ Denote $$\BB^{\De, \be}[0, T]=\left\{ x\in \BB[0, T]\,; \ \||x\||_{\De, \be}<\infty\right\}\,.$$ As in Theorem 1.3.3 of [@kufner], it is easy to verify that $\||x\||_{\De, \be}$ is a norm and $\BB^{\De, \be}[0, T]$ is a Banach space with respect to this norm. When we need to emphasize the interval we may also add the interval into the notation, namely, we may write $\||x\||_{a, b, \De, \be}$. If $\De$ is clear, we omit the dependence on $\De$ and write $\BB^{ \be}[0, T]= \BB^{\De, \be}[0, T]$. We shall consider a mapping $F$ from $ \BB[0, T]$ into itself. Thus, for any element $x\in \BB[0, T]$, $F(x)$ is a function from $[0, T]$ to $\BB$. We can thus write such function as $F(t, x)$. We say $F$ is progressive if for any $a\in [0, T]$, $\left\{F(t, x)\,,\ 0\le t\le a\right\}$ depends only on $\{x(t), 0\le t\le a\}$. In other words, if $x(t)=y(t)$ for all $t\in [0, a]$, then $F(t,x)=F(t, y)$ for all $t\in [0, a]$. Here is the main theorem of this section. \[t.fix-point-theorem\] Let $\BB$ be a separable Banach space and let $F $ be a progressive mapping from $ \BB[0, T] \rightarrow \BB [0, T]\ $ such that $F(0, x)\in \BB$ is independent of $x$ (it is equivalent to say that $F(0, x)\in \BB $ is independent of $x(0)$). Suppose that there are constants $\kappa$, $\Delta>0$, $\gamma$, $\be \in (0, 1]$ and there is a positive function $h:\RR^4\rightarrow \RR$, increasing in all of its arguments, such that the following statements are true. 1. For any $0\le a<b\le T$ with $b-a\le \De$ and for any $x \in \BB ^{\De, \be }[0, T]$ we have $$\|F ( x )\|_{a, b, \be } \le \kappa \big(1+\|x \|_{0, a} + \|x \|_{a,b,\be } (b-a)^\ga \big)\,. \label{e.F-holder-bound} $$ 2. For any $0\le a<b\le T$ with $b-a\le \De$ and for any $x_1, x_2 \in \BB ^{\De, \be }[0, T]$ we have $$\begin{aligned} \|F ( x_1 )- F ( x_2)\|_{a, b, \be } & \le & \bar h_{a, b}(x_1, x_2) \bigg\{ \|x_1-x_2\|_{0, a} \nonumber\\ &&\quad + \|x_1 -x_2\|_{a,b,\be } (b-a)^\ga \bigg\} \,, \label{e.F-diff-holder-bound}\end{aligned}$$ where $$\begin{aligned} \bar h_{a, b}(x_1, x_2) = h\big(\|x_1 \|_{0, a}, \|x_2 \|_{0, a}, \|x_1 \|_{a, b, \be },\|x_2 \|_{a, b, \be }\big)\,. \label{e.def-barh} \end{aligned}$$ Then the mapping $F :\BB^{\be }[0, T]\ \rightarrow \BB^{\be }[0, T] $ has a unique fixed point $x\in \BB[0, T]$. This means that there is a unique $x\in \BB[0, T]$ such that $x(t)=F(t, x )$ for all $t\in [0, T]$. Moreover, there is a $\tau_0>0$ such that $$\begin{cases} \|x \|_{0, T } \le c_2 e^{c_1 \kappa^{1/\ga}T } (1+\|F(0)\|) \,, \\ \sup_{0\le a< b\le T, b-a\le \tau_0} \|x \|_{a, b, \be } \le c_2 e^{c_1 \kappa^{1/\ga}T } (1+\|F(0)\|) \,, \end{cases} \label{e.bound-holder-x-theorem}$$ where $c_1$ and $c_2$ are two constants depending only on $\De$. We divide the proof into several steps. [*Step 1*]{}.   First, we prove that there is a $\tau_1\in (0, \Delta]$ (the choice of $\tau_1$ will be made more precise later) such that $F$ has a unique fixed point on the interval $[0, \tau_1]$. To this end we use Picard iteration. We define $ x_0 (t) = F(0, x) $ for all $t\in [0, \tau_1]$ which is an element in $\BB$ by our assumption that $F(0, x)$ is independent of $x$. We also define for $n=0, 1, 2, \cdots$ $$x_{n+1}(t) =F (t, x_ { n } )\,, \quad t\in [0, \tau_1]\,.$$ It is easy to see by the assumption that $x_n(0)=F(0,x )$ for all $n\ge 0$. From the assumption , we have $$\begin{aligned} \|x_{n+1}\|_{0, \tau_1, \be} &\le& \kappa (1+\|x_n(0)\|+\|x_n\|_{0, \tau_1, \be} \tau_1^\ga)\\ &\le& \kappa (1+\|F(0)\|+\|x_n\|_{0, \tau_1, \be} \tau_1^\ga)\,.\end{aligned}$$ Let $$\tau_1\le \frac{1}{(2\kappa)^{1/\ga}}\wedge \De \,. \label{e.cond1-tau}$$Then we have $$\|x_{n+1}\|_{0, \tau_1, \be } \le \kappa (1+ \|F(0, x)\|)+\frac12 \|x_n\|_{0, \tau_1, \be} \,. \label{e.proof-x(n+1)by-xn}$$ By induction, we have $$\sup _{n\ge 0} \|x_n\|_{0, \tau_1, \be}\le 2\kappa (1+ \|F(0)\|)\,.$$ Now by the fact that $\|x_n\|_{0, \tau_1}\le \|x_n(0)\|+\|x_n \|_{0, \tau_1, \be} \tau_1^\ga$, we see that $$\sup _{n\ge 0} \|x_n\|_{0, \tau_1 }\le \|F(0)\| + 2\kappa (1+\|F(0)\|) \tau_1^ \ga \le 2 (1+\|F(0)\|) \,.$$ By the definition of $\bar h$ we see that $$\sup _n \bar h_{0, \tau_1} (x_{n-1}, x_n)\le M_1<\infty$$ for some positive constant $M_1\in (0, \infty)$. Notice that $ x_n(0)=x_{n-1}(0)$. Thus condition gives $$\|F(x_{n+1})-F(x_n)\|_{0, \tau_1, \be} \le M_1 \|x_{n+1}-x_n\|_{0, \tau_1, \be} \tau_1^\ga \,.$$ Choose $$\tau_1\le \frac{1}{(2M_1)^{1/\ga}}\wedge \frac{1}{(2\kappa)^{1/\ga}}\wedge \De\,. \label{e.cond2-tau}$$ Then we have $$\begin{aligned} \| x_{n+1} - x_n \|_{0, \tau_1, \be} &=& \|F(x_{n })-F(x_{n-1})\|_{0, \tau_1, \be} \\ &\le& \frac12 \left \| x_{n } - x_{n-1} \right \|_{0, \tau_1, \be} \,. \end{aligned}$$ Since $x_n(0)=F(0)$ for all $n$ this means that $\{x_n\}$ is a Cauchy sequence in $\BB^\be [0, \tau_1]$ and it converges to an element $x\in \BB^\be [0, \tau_1]$. Obviously, this limit $x$ is the unique solution to $x(t) =F(t, x)$ for $t\in [0, \tau_1]$. Clearly, the limit satisfies $$\sup _{n\ge 0} \|x \|_{0, \tau_1 } \le 2 (1+ \|F(0)\|) \, ,\quad \|x \|_{0, \tau_1, \be } \le 2 \kappa (1+ \|F(0)\|) \, .$$ [*Step 2*]{}.   Now we explain the inductive argument to construct a unique solution on the interval $ [0, T\wedge T_{k+1} ]$ from a solution on $ [0, T\wedge T_k ]$, where $T_k=\tau_1+\cdots+\tau_k$. For any positive integer $k\ge 1$ assume that there is a unique solution $x(t), t\in [0, T_k]$ satisfying $x(t)=F(t, x), t\in [0, T_k]$. We want to construct a unique solution $x(t), t\in [0, T_{k+1}]$ satisfying $x(t)=F(t, x), t\in [0, T_{k+1}]$ for some $\tau_{k+1}>0$ (see below for the definition of $\tau_{k+1}$). To simplify notation we assume $ T_{k+1}\le T$ (or we replace $T_{k+1}$ by $T_{k+1}\wedge T$). Define the following sequence (still use $x_n$) $$\begin{cases} & x_0(t)= \begin{cases} x(t)\,, &\qquad\qquad\quad \hbox{when $0\le t\le T_k$}\,, \\ x(T_k)\,, &\qquad\qquad\quad \hbox{when $T_k \le t\le T_{k+1}$} \,, \end{cases} \nonumber\\ & x_{n+1}(t)=F(t, x_n)\,, \qquad\qquad \hbox{for all $0\le t\le T_{k+1}$} \,, \label{e.zn-sup} \end{cases}$$ where $n=0, 1, \cdots$. Since $x(t), t\in [0, T_k]$ is the unique solution to $x(t)=F(t, x), t\in [0, T_k]$, we see that $x_n(t)=x(t)$ for all $t\in [0, T_k]$. With exactly the same argument as for , we have for any positive integer $k\ge 1$, $$\begin{aligned} \|x_{n+1}\|_{T_k, T_{k+1}, \be } &\le& \kappa (1+ \|x\|_{0, T_k} ) +\frac12 \|x_n\|_{ T_k, T_{k+1}, \be } $$ under the condition $$\tau_{k+1}\le \frac{1}{(2\kappa)^{1/\ga}}\wedge \De\,.$$ \[We can take $\tau_{k+1}= \frac{1}{(2\kappa)^{1/\ga}}\wedge \De$\]. This can be used (by induction on $n$) to prove $$\sup_{n\ge 0} \|x_n\|_{ T_k, T_{k+1}, \be } \le 2 \kappa (1+ \|x\|_{0, T_k })=:M_{k+1}^{(1)}\,. \label{e.proof.bound-holder-xn}$$ As a consequence, we have $$\begin{aligned} \sup_{n\ge 0} \|x_n\|_{0, T_{k+1} } &\le& \|x\|_{0, T_k} + \|x_n\|_{ T_k, T_{k+1}, \be } \tau_{k+1}^\ga \nonumber\\ &\le& (2 \kappa \tau_{k+1} ^\be+1) (1+\|x\|_{0, T_k }) \nonumber\\ &\le& 2 (1+\|x\|_{0, T_k })=:M_{k+1}^{(2)} \label{e.x-uniform-bound} \,. \end{aligned}$$ Now letting $M_{k+1}:=h(M_{k+1}^{(2)}, M_{k+1}^{(2)}, M_{k+1}^{(1)}, M_{k+1}^{(1)} )$, we have by $$\begin{aligned} \|x_{n+1}-x_n\|_{ T_k, T_{k+1}, \be} &=& \|F(x_{n })-F(x_{n-1})\|_{ T_k, T_{k+1}, \be} \nonumber\\ &\le& M_{k+1} \|x_{n }-x_{n-1}\|_{ T_k, T_{k+1}, \be} \tau_{k+1}^\ga \nonumber\\ &\le& \frac12 \|x_{n }-x_{n-1}\|_{ T_k, T_{k+1}, \be} \end{aligned}$$ if $$\tau_{k+1}\le\frac{1}{M_{k+1}^{1/\ga}}\wedge \frac{1}{(2\kappa)^{1/\ga }}\wedge \De\,. \label{e.cond2-tau=k+1}$$ Thus, under the above condition \[e.cond2-tau=k+1\], $\{x_n\}$ is a Cauchy sequence in $\BB [0, T_{k+1}]$. It has a unique limit $x$ which satisfies $x(t)=F(t, x)$ for all $t\in [0, T\wedge T_{k+1}]$. Indeed, the fact $x(t)$ satisfies $x(t)=F(t, x)$ for all $t\in [0, T\wedge T_{k}]$ follows from the inductive assumption. On $[T\wedge T_k, T\wedge T_{k+1}]$, $x_n$ is a Cauchy sequence in $B^\be [T\wedge T_k, T\wedge T_{k+1}]$ and $F$ is continuous on $B^\be [T\wedge T_k, T\wedge T_{k+1}]$ by the assumption . It is also easy to verify from and that the solution satisfies $$\begin{cases} \|x \|_{ T_k, T_{k+1}, \be } \le 2 \kappa (1+\|x \|_{0, T_k})\,; \\ \|x \|_{0, T_{k+1} } \le 2 (1+ \|x \|_{0, T_k})\,. \end{cases} \label{e.4.100}$$ [*Step 3*]{}. Denote $T_\infty=T\wedge(\tau_1+\tau_2+\cdots)$. By induction argument, we can construct a unique solution $x(t)$ on $t\in [0, T_\infty]$ for the equation $x(t)=F(t, x)$. We want to show $T_\infty=T$. To do this, the idea is to show that $\tau_k\ge \tilde \tau_0$ for some $\tilde \tau_0>0$ and for all $k\ge 1$. From the same argument as for and we have $$\begin{cases} & \|x \|_{ k\tau, (k+1)\tau, \be } \le 2 \kappa (1+ \|x \|_{0, k\tau})\,; \\ & \|x \|_{0, (k+1)\tau } \le 2 (1+ \|x \|_{0, k\tau }) \end{cases}\label{e.proof-x-bound}$$ as long as $\displaystyle \tau\le \frac{1}{(2\kappa)^{1/\ga}}$ and $(k+1)\tau\le T_\infty$. In fact, to obtain the above bounds, we only need to use the condition . We choose $\displaystyle \tau= \frac{1}{(2\kappa)^{1/\ga}}\wedge \Delta $ and divide the interval $[0, T_\infty]$ into $N$ sub-intervals, where $$N=\left[\frac{T_\infty}{\tau}\right]+1=\left[\frac { T_\infty } {\De}\right] \vee \left[ T_\infty (2\kappa)^{1/\ga} \right]+1\,.$$ Denote $A_k= \|x \|_{0, k\tau }$. The second inequality in can be written as $$A_{k+1}\le 2 +2 A_k\,, \quad k=0, 1, 2, \cdots$$ An elementary induction argument yields $$\begin{aligned} A_k &\le& 2+2^2+\cdots +2^k+2^k A_0\\ &\le& 2^{k+1} +2^{k } A_0\,. \end{aligned}$$ Thus, we see that $$\begin{aligned} \|x \|_{0, T_\infty } &\le& 2^{N+1}+2^N \|F(0)\| \nonumber\\ &\le& c_2 e^{c_1 \kappa^{1/\ga}T } (1+\|F(0)\|)\ \label{e.bound-uniform-x-proof}\end{aligned}$$ for some constants $c_1$ and $c_2$ dependent only on $\De$. This together with the first inequality in yields $$\|x \|_{ k\tau, (k+1)\tau, \be } \le c_2 e^{c_1 \kappa^{1/\ga}T } (1+\|F(0)\|) \label{e.bound-holder-x-proof}$$ for any $k$ such that $(k+1)\tau\le T_\infty$ and for $\displaystyle \tau= \frac{1}{(2\kappa)^{1/\ga}}\wedge \De$. [*Step 4*]{}.   Denote $$\begin{cases} \tilde M_1&=\tilde M_2= c_2 e^{c_1 \kappa^{1/\ga}T } (1+\|F(0)\|) \,;\nonumber\\ \tilde M &= h(\tilde M_2, \tilde M_2, \tilde M_1, \tilde M_1)\,. \nonumber \end{cases}$$ Then from and we see that $M_{k+1}^{(1)}\le \tilde M_1$ and $M_{k+1}^{(2)}\le \tilde M_2$ for all $k$. Since $h$ is increasing in all of its arguments, this means that we can choose $\tau_k$ such that $$\tau_k\ge \tilde \tau:\displaystyle = \frac{1}{\tilde M^{1/\ga}}\wedge \frac{1}{(2\kappa)^{1/\ga}}\wedge \De\,, \quad \forall \ k\ge 1\,.$$ Since $\tilde \tau$ is independent of $k$, we see that $T_\infty=T$. The first inequality is a straightforward consequence of . With possibly a different choice of $c_1$ and $c_2$, we can write as $$\|x \|_{ k\tau_0, (k+1)\tau_0, \be } \le c_2 e^{c_1 \kappa^{1/\ga}T } (1+\|F(0)\|)\,.$$ If $a, b\in [k\tau_0, (k+1)\tau_0]$, then we see easily that the second inequality in holds. If $a\in [(k-1)\tau _0, k\tau_0]$ and $b\in [k\tau_0, (k+1)\tau_0]$, then $$\begin{aligned} \frac{ \|x(b)-x(a)\|}{|b-a|^\be} &\le& \frac{ \|x(b)-x(k\tau_0)\|+\|x(k\tau)-x(a)\|}{|b-a|^\be}\\ &\le& \frac{ \|x(b)-x(k\tau_0)\|} {|b-k \tau_0|^\be}+\frac{\|x(k\tau)-x(a)\|}{|k\tau_0-a|^\be}\\ &\le& \|x\|_{(k-1)\tau _0, k\tau_0, \be} +\|x\|_{k\tau _0, (k+1)\tau_0, \be}\\ &\le& 2 c_2 e^{c_1 \kappa^{1/\ga}T }(1+\|F(0)\|)\,.\end{aligned}$$ Up to a different choice of constant $c_2$, we prove the second inequality in . If $\bar h$ has some particular form, one may obtain some stability results for the solutions. Namely, if $x_1$ and $x_2$ are two solutions with different initial conditions, or with different $F$, one may bound $\|x_2-x_1\|$ (see analogous results [@hunualartabel; @hunualarttams]). However, we shall not persuade this problem. General stochastic differential equations ========================================= Reduction of the equation ------------------------- Let $b\,, \si:[0, T]\times \RR\times \Om\rightarrow \RR$ be a measurable mapping. We shall specify the conditions on them later. Let $\eta$ be a given random variable. The main objective of this paper is to study the following Itô stochastic differential equation $$\begin{cases} dx(t)=b(t, x(t), \om)dt+\si(t, x(t), \om)dB(t)\,, \quad 0\le t\le T\,, \\ x(0)=\eta\,, \end{cases} \label{e.5.1}$$ where $dB(t)$ is the Itô differential. We can use the argument as in the introduction (see e.g. ) to reduce the above equation , now with random coefficients. Using the relationship between the Itô and pathwise stochastic integrals and the chain rule for derivative we have $$\begin{aligned} \int_0^t \si(s, x(s), \om)dB(s) &=&\int_0^t \si(s, x(s), \om)\de B(s) -\int_0^t {\mathbb D}^\phi_s \left[\si(s, x(s), \om) \right]ds \\ &=& \int_0^t \si(s, x(s), \om)\de B(s) -\int_0^t {\mathbb D}^\phi_s \left[\si\right] (s, x(s), \om)ds \\ &&\qquad -\int_0^t \si_x (s, x(s), \om){\mathbb D}^\phi_s x(s) ds \,,\end{aligned}$$ where $\si_x$ denotes the partial derivative of $\si$ with respect to $x$, and ${\mathbb D}^\phi_s \left[\si\right]$ denotes the partial derivative of $\si$ with respect to the random element $\om$. Thus, the equation may be written as $$\begin{aligned} x(t)&=&\eta+\int_0^t \tilde b(s, x(s), \om)ds+ \int_0^t \si(s, x(s), \om)\de B(s)\nonumber\\ &&\qquad -\int_0^t \tilde \si (s, x(s), \om){\mathbb D}^\phi_s x(s) ds\,, \label{e.5.5}\end{aligned}$$ where $$\begin{cases}\tilde b(s, x, \om):=b(s, x, \om)- {\mathbb D}^\phi_s \left[\si\right] (s, x, \om)\\ \tilde \si(s, x, \om)= \si_x(s, x, \om)\,. \end{cases}$$ As explained in the introduction, this equation can be considered as a first order nonlinear hyperbolic equation of infinitely many variables, driven by fractional Brownian motion, where $\om \in \Om$ is considered as an infinite dimensional variable. We shall use the elementary characteristic curve method. The characteristic equation will be an equation in $\Om$ which takes the form of the first equation of the following system of equations. This means that to solve the above equation we will first solve the following coupled system of equations (which we call it the system of characteristic equations corresponding to ). $$\begin{cases}\Gamma(t)=\om+\int_0^t \tilde \si (s, z(s), \Gamma(s))\int_0^\cdot \phi(s, u)du ds\,;\\ \\ z(t)= \eta(\om ) +\int_0^t \tilde b(s, z(s), \Gamma(s))ds +\int_0^t \si(s, z(s), \Gamma(s))\de B(s)\\ \qquad\qquad +\int_0^t\int_0^s \si(s, z(s), \Ga(s)) \tilde \si (u, z(u), \Ga(u)) \phi(s, u) duds\,. \end{cases} \label{e.5.6}$$ We shall show that the solution to equation can be used to express the solution of . However, first we need to show that has a (unique) solution. Solution to the reduced equation -------------------------------- In this section we prove that the system has a unique solution. When the Hurst parameter $H>1/2$ and in the absence of ${\mathbb D}^\phi_s x(s)$, the equation has been studied by many authors (see [@nualartrascanu; @zahle; @hustochastics] for a recent study and also for some more references). We only mention two works. In [@nualartrascanu], Besov spaces are used to accommodate the solutions. In [@hunualartabel], the solution is shown to be Hölder continuous and the stability with respect to Hölder norm is also studied in that paper (see [@hunualarttams] for a similar study when the Hurst parameter $H\in (1/3, 1/2]$). Here, we shall use the Hölder spaces together with the general contraction principle established in Section 4 to prove the existence and uniqueness of the solution. Our idea to solve the equation seems also new even in the classical case (namely, in the absence of ${\mathbb D}^\phi_s x(s)$ in ). The system of (two) equations will be solved for any fixed $\om\in \Om$. This means that we are going to find pathwise solution of by using Theorem \[t.fix-point-theorem\]. To this end, we rewrite the equation with a replacement of $\Ga$ of by $\Ga+\om$ (we use the same notation $\Ga(t)$ without ambiguity). $$\begin{cases}\Gamma(t)= \int_0^t \tilde \si (s, z(s), \Gamma(s)+\om)\int_0^\cdot \phi(s, u)du ds\,;\\ \\ z(t)= \eta(\om ) +\int_0^t \tilde b(s, z(s), \Gamma(s)+\om)ds +\int_0^t \si(s, z(s), \Gamma(s)+\om)\de B(s)\\ \qquad\qquad +\int_0^t\int_0^s \si(s, z(s), \Ga(s)+\om) \tilde \si(u, z(u), \Ga(u)+\om) \phi(s, u) duds\,. \end{cases} \label{e.to-solve}$$ Before we solve , we need to explain the space that the solution stay. To find such a space to accommodate the above $\Ga$ we introduce the following Banach space: $$\begin{aligned} {\mathcal H}&=&{\mathcal H}_p=\bigg\{ h: \cT\rightarrow \RR\,; \ \hbox{$h$ is absolutely continuous} \nonumber\\ &&\qquad\hbox{such that}\ \int_0^T |\dot h(s)|^p ds<\infty\bigg\}\,,\end{aligned}$$ where $p$ is a number such that $p\in (1, \frac{1}{2-2H})$ (we shall fix such a number throughout the remaing part of this paper), and where the norm is defined by $$\|h\|_{\mathcal H}=\|h\|_{{\mathcal H}_p} =\left( \int_0^T | \dot h(s)|^p ds\right)^{1/p}\,.$$ It is straightforward to see that any $h\in {\mathcal H}$ is an element of $\Om$ and $$\|h\|_\Om\le c_{p, T} \|h\|_{\mathcal H}\,.$$ The principle to choose the Banach space ${\mathcal H}$ is as follows. First, we need that $\frac{d}{dt}\Ga(t)\in {\mathcal H}$. Secondly, we want the norm of ${\mathcal H}$ is as strong as possible so that the coefficients $\sigma$, $\tilde \si$, and $\tilde b$ are differentiable on $\om$ with respect to this norm $\|\cdot\|_{\mathcal H}$. Namely, with respect to $\om$, the coefficients $\si$ and $b$ and the initial condition $\eta$ satisfy $$\left| \si( \om+h)-\si(\om)\right|\le C \|h\|_{\mathcal H}\,, \quad \forall \ h\in {\mathcal H}\,. \label{e.5.8}$$ (Similar inequality for $\tilde b$, $\tilde \si$, and the initial conditions). Of course, the larger the norm of ${\mathcal H}$, the broader the condition will be. In the analysis of nonlinear Wiener functionals, it is known that many interesting random variables do not satisfy with ${\mathcal H}=\Om$ (see the example of Lévy area in [@hugaussian]). But usually is satisfied when ${\mathcal H}$ is the Cameron-Martin norm, which is given by $\|h\|_{{\mathcal H}_\phi}^2:=\int_0^T\int_0^T \phi(u-v) h(u) h(v) dudv$, in our case of frcational Brownian motion. Namely, for many random variables in stochastic analysis, such as the solution of a stochastic differential equation, we have $$\left| f( \om+h)-f(\om)\right|\le C \|h\|_{{\mathcal H}_\phi} \,, \quad \forall \ h\in {\mathcal H}_\phi \,. \label{e.5.9}$$ An inequality of Littlewood-Paley type ([@mmv]) states $$\|h\|_{{\mathcal H}_\phi }\le C_H \|h\|_{{\mathcal H}_q}\,, \quad \forall \ h\in {\mathcal H}_q\,,\quad \hbox{with $q:=1/H$}\,.$$ When $H>2/3$, we have $q=\frac{1}{H}< \frac{1}{2-2H}$. In this case, we see that the implies when we choose $p$ close to $\frac{1}{2-2H}$. This means that the condition is satisfied for many random variables we encounter when $H>2/3$. Before we proceed to solve , we state some assumptions on the coefficients $b$ and $\si$. \[h.b-sigma\] Let $\cL$ be a positive constant. The measurable functions $b, \si:\cT\times \RR\times \Om\rightarrow \RR$ satisfy the following conditions. 1. $b$ is continuously differentiable in $x$ and satisfies $$\begin{cases} |b(t,x,\om)| \le \cL(1+|x|)\,;\\ \left|\frac{\partial }{\partial x} b(t,x,\om)\right| \le \cL \,;\\ \left\|{\mathbb D}b(t,x,\om)\right\|_{\mathcal H}\le \cL \,. \end{cases}$$ 2. $\si(t,x,\om)$ is twice continuously differentiable in $x$ with bounded first and second derivative and satisfies $$\begin{cases} |\si(t,x,\om)| \le \cL(1+|x|)\,; \\ |\frac{\partial }{\partial t} \si(t,x,\om)- \frac{\partial }{\partial t} \si(t, y, \tilde \om)| \le \cL (|x-y|+\|\om-\tilde \om\|_{\mathcal H})\,;\\ |\frac{\partial}{\partial x} \si(t,x,\om)|+|\frac{\partial^2}{\partial x^2} \si(t,x,\om)| \le \cL\,; \\ |{\mathbb D}_t^\phi \si(t,x,\om)| \le \cL(1+|x|)\,;\\ \|{\mathbb D}\si(t,x,\om) \|_{\mathcal H}+\|{\mathbb D}^2 \si(t,x,\om) \|_{{\mathcal H}^2} \le \cL\,;\\ \|{\mathbb D}\frac{\partial}{\partial x} \si(t,x,\om) \|_{\mathcal H}\le \cL\,. \end{cases}$$ From now on we denote $\BB={\mathcal H}\oplus \RR$ and we denote by $\BB[0, T]$ the space of all continious functions from $[0, T]$ ro $\BB$. Similarly, we will also use the notation ${\mathcal H}[0, T]$. $\RR[0, T]$ is then the space $C([0, T])$ of all continuous functions from $[0, T]$ to $\RR$. Define a mapping from $\BB[0, T]$ to $\BB[0, T]$ as follows. $$\begin{cases} F_1(t, \Ga, z):= \int_0^t \tilde \si (s, z(s), \Gamma(s)+\om)\int_0^\cdot \phi(s, u)du ds\,;\\ \\ F_2(t, \Ga, z):= \eta(\om ) +\int_0^t \tilde b(s, z(s), \Gamma(s)+\om)ds +\int_0^t \si(s, z(s), \Gamma(s)+\om)\de B(s)\\ \qquad\qquad\qquad +\int_0^t\int_0^s \si(s, z(s), \Ga(s)+\om) \tilde \si(u, z(u), \Ga(u)+\om) \phi(s, u) duds\,. \end{cases} \label{e.def-F}$$ We also write $F_i(\Ga, z)=F_i(t, \Ga, z)$, $i=1, 2$. It is easy to see that for any $(\Ga, z)\in \BB[0, T]$, $(F_1(\Ga, z), F_2(\Ga,z))$ is also in $\BB[0, T]$. \[l.uniform-bound-f1\] For any $\tau \in \cT$, if $(\Ga, z)\in \BB[0,\tau ]$, then $F_1(\Ga, z)\in {\mathcal H}[0, \tau ] $ and $$\begin{aligned} \|\frac{d}{dt}F_1(\Gamma, z)\|_{\mathcal H}&\le& \kappa \,, \label{e.5.17}\\ \|F_1(\Ga, z)\|_{ 0, \tau }& \le& \kappa \tau \,, \label{e.5.18}\end{aligned}$$ where and in what follows $\kappa=c_{p, H, T, \cL}$ is a constant depending only on $p, H$, $T$ and the bound $\cL$ of the coefficients $b$ and $\si$, which may vary at different occurrences. From the definition of $F_1$ we see $$\frac{d}{dt} F_1(\Ga, z)= \tilde \si (t, z(t), \Gamma(t)+\om)\int_0^\cdot \phi(t, u)du \,.$$ Recall that we fix $p<\frac{1}{2-2H}$. Since $\tilde \si$ is bounded and $\phi(t,u)=H(2H-1) |t-u|^{2H-2}$, we have $$\begin{aligned} \| \frac{d}{dt} F_1(\Ga, z)\|_{\mathcal H}^p \le \kappa \int_0^T \phi(t,u)^p du = \kappa \,. \end{aligned}$$ This proves . Similarly, since $F_1(0)=0$, we have $$\begin{aligned} \| F_1(\Ga, z)\|_{ 0, \tau}\le \| \frac{d}{dt} F_1(\Ga, z)\|_{ 0, \tau }\tau &=& \kappa\tau \,, \end{aligned}$$ proving . \[l.uniform-bound-f2\] For any $\tau\in \cT$, if $(\Ga, z)\in \BB[0, \tau]$, then $F_2(\Ga, z)\in C[0, \tau]$ and for any $0\le a<b\le \tau$, $$\begin{aligned} \|F_2\|_{a,b, \be} & \le& \kappa \left(1+ \|B\|_{a, b, \be}\right) \Big\{ 1+\|z\|_{0, a}\nonumber\\ &&\qquad + \|z\|_{a, b, \be}(b-a)^{ \be} +\| \Ga\|_{a, b, \be } (b-a) ^\be \Big\}\,. \label{e.sup-f2} \end{aligned}$$ First, we write $$F_2=\eta(\om)+F_{21}+F_{22}+F_{23}\,,$$ where $$\begin{cases} F_{21}(t)= \int_0^t \tilde b(s, z(s), \Gamma(s)+\om )ds\,;\nonumber\\ F_{22}(t)= \int_0^t \si(s, z(s), \Gamma(s)+\om)\de B(s)\,; \nonumber\\ F_{23}(t) = \int_0^t\int_0^s \si(s, z(s), \Ga(s)+\om) \tilde \si(u, z(u), \Ga(u)+\om) \phi(s, u) duds\,. \end{cases}$$ From the assumption on $b$ and ${\mathbb D}_s^\phi \si$, we see that for any $0\le a<b\le \tau$, we have $$\begin{aligned} |F_{21}(b) -F_{21}(a)| &=& \int_a^b |\tilde b(s, z(s), \Gamma(s)+\om)|ds\\ &\le& \kappa \int_a^b \left[1+|z(s)|\right] ds\\ &\le& \kappa \left[ 1+\|z\|_{ a, b }\right] (b-a) \,. \end{aligned}$$ This implies that $$\begin{aligned} \|F_{21}\|_{ a, b , \be} &\le& \kappa \left[1+ \|z\|_{ a, b } \right] \nonumber\\ &\le& \kappa \left[1+|z(a)|+ \|z\|_{ a, b, \be } (b-a)^\be \right] \,. \label{e.sup-f21} \end{aligned}$$ Now we consider $F_{23}$. We have $$\begin{aligned} \left|F_{23}(b)-F_{23}(a)\right| &=&\int_a^b\int_0^s |\si(s, z(s), \Ga(s)+\om) \tilde \si(u, z(u), \Ga(u)+\om) \phi(s, u)| duds \\ &\le& \kappa \int_a^b\int_0^s \left[ 1+ | z(s)| \right] \phi(s, u) duds \\ &\le& \kappa \left[ 1+ \|z\|_{ a, b } \right] \int_a^b\int_0^s \phi(s, u) duds \\ &\le& \kappa \left[ 1+ \|z\|_{ a, b } \right] (b^{2H}-a^{2H})\\ &\le& \kappa \left[ 1+ \|z\|_{ a, b } \right] (b-a)\,. \end{aligned}$$ This implies $$\|F_{23}\|_{ a, b , \be}\le \kappa \left[ 1+ \|z\|_{ 0, a } + \|z\|_{ a, b } (b-a)^\be \right] \,. \label{e.sup-f23}$$ $F_{22} $ is more complicated to handle because the fractional Brownian motion $B$ is not differentiable. Denote $\si_r=\si(r, z(r), \Ga(r)+\om)$. We have for an $\al\in (1-\be, \be)$, $$\begin{aligned} &&\left|\int_a^b \si(r, z(r), \Ga(r)+\om) \de B(r)\right| = \left| \int_a^b D_{b-}^{1-\al} B_{b-}(r) D_{a+} ^\al \si_rdr \right| \\ &&\qquad \le \kappa \|B\|_{a, b, \be} \Big| \int_a^b (b-r)^{\al+\be-1} \Big\{ \frac{\si_r}{(r-a)^\al }+\int_a^r \frac{\si_r-\si_\rho}{(r-\rho)^{\al+1}}d\rho\Big\}dr\Big|\\ &&\qquad \le \kappa \|B\|_{a, b, \be} \Big\{ (1+\|z\|_{a, b }) (b-a)^\be +(\|z\|_{a,b,\be} +\|\Ga\|_{a,b,\be}) (b-a)^{ 2\be} \Big\} \,. \end{aligned}$$ This implies $$\begin{aligned} \|F_{22}\|_{a, b, \be} &\le & \kappa \|B\|_{a, b, \be} \Big\{ 1+\|z\|_{0, a} + ( \|z\|_{a, b, \be}+\|\Ga\|_{a,b,\be}) (b-a)^{ \be}\Big\} \,. \label{e.sup-f22} \end{aligned}$$ Combining , and , we prove the lemma. To bound the Hölder norm of the difference, we first need the following simple general result. \[l.one-comp-holder\] Let $B_1$ and $B_2$ be two Banach spaces with norms $\|\cdot\|_1$ and $\|\cdot\|_2$ and let $f: B_1\rightarrow B_2$ be twice continuously (Frechet) differentiable with bounded first and second derivatives. 1. If $x_1, x_2, y_1, y_2\in B_1$, then $$\begin{aligned} &&\|f(y_2)-f(y_1)-f(x_2)+f(x_1)\|_2 \le \|f' \|_\infty \| y_2-y_1- x_2+ x_1\|_1 \nonumber\\ &&\qquad \qquad + \|f''\|_\infty \left[ \|y_1-x_1\|_1 +\|y_2-x_2\|_1 \right]\| x_2-x_1\|_1 \,. \label{e.four-point}\end{aligned}$$ 2. Let $ x_1, x_2: [a, b]\rightarrow B_1$ be Hölder continuous of order $\be$. Then for any $a\le s<t\le b$, we have $$\begin{aligned} &&|f(x_2(t))-f(x_1(t))-f(x_2(s)) +f(x_1(s))|_2 \le \|f' \|_\infty \| x_2- x_1\|_{s, r, \be } (r-s)^\be \nonumber\\ &&\qquad \qquad + \|f''\|_\infty \left[ \|x_1\|_{s,r, \be}+\|x_2\|_{s,r, \be}\right]\| x_2-x_1\|_{s,r} (r-s)^\be \,. \nonumber\\ \label{e.5.one-comp-holder}\end{aligned}$$ This inequality may be well-known. We include a short proof for the completeness. Using the mean value theorem we have $$\begin{aligned} && f(y_2)-f(y_1)-f(x_2)+f(x_1) \\ && =\int_0^1 f'( (1-\theta) y_1+\theta y_2 )d\th (y_2-y_1 ) \\ && -\int_0^1 f'((1-\theta) x_1+\theta x_2 )d\th ( x_2-x_1) \\ && =\int_0^1 f'((1-\theta) y_1+\theta y_2 )d\th (y_2-y_1- x_2+x_1) \\ && +\int_0^1\left[ f'((1-\theta) y_1+\theta y_2 )- f'((1-\theta) x_1+\theta x_2 ) \right] d\th (x_2-x_1)\\ &&=\int_0^1 f'(y_1+\theta (y_2-y_1))d\th (y_2-y_1- x_2+x_1) \\ && +\int_0^1\int_0^1 f''\left(\upsilon \left[ y_1+ \theta (y_2-y_1) \right] +(1-\upsilon) \left[ (1-\theta) x_1+\theta x_2 \right] \right) d\th d\upsilon \\ &&\qquad (x_2-x_1) \otimes \left[ (1-\theta) (y_1-x_1 ) +\theta ( y_2 - x_2 ) \right]\,. \end{aligned}$$ This proves easily. The inequality is straightforward consequences of . \[l.F-diff-holder-norm-bound\] Denote $$F^{(i)}_j( t)=F_j(t, \Ga_i, z_i)\,,\quad i, j=1,2\,.$$ Then, we have $$\begin{aligned} &&\left\|\frac{d}{dt} F^{(2)}_1( t)-\frac{d}{dt}F^{(1)}_1(t)\right\|_{\mathcal H}\le \kappa \left[\|z_2 -z_1 \|_{0, \tau} +\|\Ga_2 -\Ga_1 \|_{0, \tau}\right] \nonumber\\ \label{e.holder-f1}\end{aligned}$$ and $$\begin{aligned} &&\left\| F_{2}^{(2)} -F_{2}^{(1)} \right\|_{a, b, \be} \le \kappa (1+\|B\|_{a,b, \be}) \big(1+\|z_1\|_{0, a}+\|z_2\|_{0,a}+\|z_1\|_{a, b, \be}\nonumber\\ &&\qquad + \|z_2\|_{a, b, \be} + \|\Ga_1\|_{a,b, \be}+\|\Ga_2\|_{a,b, \be}\big) \bigg[ |z_2(a)-z_1(a)|+\|\Ga_2(a)-\Ga_1(a)\| \nonumber\\ &&\qquad +\left[ \|z_2-z_1\|_{a, b, \be} +\| \Ga_2- \Ga_1\|_{a, b, \be}\right] (b-a)^{\be} \bigg] \,. \label{e.holder-f2}\end{aligned}$$ Let $(\Ga_1, z_1)$ and $(\Ga_2, z_2)$ be two elements in $\BB[0, \tau]$. We recall $$\begin{cases} F_1^{(i)}(t)= \om+\int_0^t \tilde \si (s, z_i(s), \Gamma_i(s)+\om )\int_0^\cdot \phi(s, u)du ds\,; \nonumber\\ F_2^{(i)}(t)= \eta(\om ) +\int_0^t \tilde b(s, z_i(s), \Gamma_i(s)+\om )ds +\int_0^t \si(s, z_i(s), \Gamma_i(s)+\om )\de B(s)\nonumber\\ \qquad\qquad +\int_0^t\int_0^s \si(s, z_i(s), \Ga_i(s)+\om ) \tilde \si(u, z_i(u), \Ga_i(u)+\om ) \phi(s, u) duds\,.\nonumber \end{cases}$$ To simplify notation we also denote $$\begin{aligned} b^{(i)}(s)&=& b(s, z_i(s), \Gamma_i(s)+\om )\,, \quad \tilde b^{(i)}(s)= \tilde b(s, z_i(s), \Gamma_i(s)+\om )\,,\\ \si ^{(i)}(s)&=& \si (s, z_i(s), \Gamma_i(s)+\om )\,, \quad \tilde \si ^{(i)}(s)=\tilde \si (s, z_i(s), \Gamma_i(s)+\om )\,. \end{aligned}$$ We have for any $t\in [0, \tau]$, $$\begin{aligned} &&\left\|\frac{d}{dt} F^{(2)}_1( t)-\frac{d}{dt}F^{(1)}_1(t)\right\|_{\mathcal H}=\left\|\int_0^\cdot \left[ \tilde \si^{(2)}(t)-\tilde \si^{(1)}(t)\right] \phi(t, u) du\right\|_{\mathcal H}\\ &&\qquad \le\kappa \left\|\int_0^\cdot\left[ |z_2(t)-z_1(t)| +\|\Ga_2 (t)-\Ga_1(t)\|_{\mathcal H}\right] \phi(t, u) du \right\|_{\mathcal H}\\ &&\qquad \le\kappa \left[\|z_2 -z_1 \|_{0, \tau} +\|\Ga_2 (t)-\Ga_1(t)\|_{0, \tau}\right] \left[ \int_0^T \phi(t, u)^p du \right]^{1/p} \\ &&\qquad \le \kappa \left[\|z_2 -z_1 \|_{0, \tau} +\|\Ga_2 (t)-\Ga_1(t)\|_{0, \tau}\right] \,. \end{aligned}$$ This is . As in the proof of Lemma \[l.uniform-bound-f1\] we denote $$\begin{cases} F_{21}^{(i)}(t)= \int_0^t \tilde b(s, z_i(s), \Gamma_i(s)+\om )ds\nonumber\\ F_{22}^{(i)}(t)= \int_0^t \si(s, z_i(s), \Gamma_i(s)+\om )\de B(s)\nonumber\\ F_{23}^{(i)}(t)=\int_0^t\int_0^s \si(s, z_i(s), \Ga_i(s)+\om ) \tilde \si(u, z_i(u), \Ga_i(u)+\om ) \phi(s, u) duds\,.\nonumber \end{cases}$$ For any $a, b\in [0, \tau]$, we have $$\begin{aligned} &&|F_{21}^{(2)}(b)-F_{21}^{(1)}(b)-F_{21}^{(2)}(a)+F_{21}^{(1)}(a)|\nonumber\\ &&\quad =\int_a^b |\tilde b_2(r)-\tilde b_1(r)|dr\nonumber\\ &&\quad \le\kappa \int_a^b\left[ |z_2(r)-z_1(r) |+\|\Ga_2(r)-\Ga_1(r)\|_{\mathcal H}\right] dr\nonumber\\ &&\quad \le \kappa \left[ |z_2 -z_1 |_{a,b}+\|\Ga_2 -\Ga_1 \|_{a,b}\right] (b-a)\,. \end{aligned}$$ This yields $$\begin{aligned} \|F_{21}^{(2)} -F_{21}^{(1)}\|_{a, b, \be} &\le& \kappa \left[ |z_2 -z_1 |_{a,b}+\|\Ga_2 -\Ga_1 \|_{a,b}\right] (b-a)^{1-\be} \nonumber\\ &\le& \kappa \left[ |z_2(a)-z_1(a)|+\|\Ga_2(a)-\Ga_1(a)\|\right](b-a)^{1-\be}\nonumber\\ && +\kappa \left[ |z_2 -z_1 |_{a,b, \be }+\|\Ga_2 -\Ga_1 \|_{a,b, \be }\right] (b-a) \nonumber\\ &\le& \kappa \left[ |z_2(a)-z_1(a)|+\|\Ga_2(a)-\Ga_1(a)\|\right] \nonumber\\ && +\kappa \left[ |z_2 -z_1 |_{a,b, \be }+\|\Ga_2 -\Ga_1 \|_{a,b, \be }\right] (b-a) ^\be \,. \label{e.holder-f21}\end{aligned}$$ Now we find the bounds for $F_{22}^{(i)}$. We have $$\begin{aligned} &&|F_{22}^{(2)}(b)-F_{22}^{(1)}(b)-F_{22}^{(2)}(a)+F_{22}^{(1)}(a)| \nonumber\\ &&\quad = \left|\int_a^b \left( \si^{(2)}(r) -\si^{(1)}(r) \right) \de B(r)\right|\nonumber\\ &&\quad = \left|\int_a^b D_{b-}^{1-\al} B_{t-} (r) D_{a+}^\al \left( \si^{(2)}(r) -\si^{(1)}(r) \right) dr\right|\nonumber\\ &&\quad = \frac{1}{\Ga(1-\al)} \Bigg|\int_a^b D_{b-}^{1-\al} B_{t-}(r) \Bigg( \frac{\si^{(2)}(r) -\si^{(1)}(r)}{(r-a)^\al} \nonumber\\ &&\qquad\quad +\al \int_a^r \frac{\si^{(2)}(r) -\si^{(1)}(r)-\si^{(2)}(\rho) +\si^{(1)}(\rho)}{ (r-\rho)^{\al+1}} d\rho \Bigg) dr\Bigg|\nonumber\\ &&\qquad \le \kappa \|B\|_{a, b, \be}\left[I_1+I_2\right]\,,\label{e.proof-bound-by-i1-i2}\end{aligned}$$ where $$\begin{aligned} I_1 &=& \int_a^b (b-r)^{\al+\be-1} \frac{\left|\si^{(2)}(r) -\si^{(1)}(r)\right|}{(r-a)^\al} dr\,; \label{e.def-i1} \\ I_2 &=& \int_a^b (b-r)^{\al+\be-1} \int_a^r \frac{ \left|\si^{(2)}(r) -\si^{(1)}(r)-\si^{(2)}(\rho) +\si^{(1)}(\rho)\right| }{ (r-\rho)^{\al+1}} d\rho dr\,. \nonumber\\ \label{e.def-i2} \end{aligned}$$ It is easy to see that $$I_1\le \kappa \left[ \|z_2-z_1\|_{a, b}+\|\Ga_2-\Ga_1\|_{a, b}\right] (b-a)^\be\,. \label{e.diff-f2-i1}$$ To bound $I_2$, we need the following identity. $$\begin{aligned} \si^{(2)}(r)-\si^{(1)}(r)-\si^{(2)}(\rho)+\si^{(1)}(\rho) =J_1+J_2+J_3+J_4+J_5\,,\end{aligned}$$ where $$\begin{cases} J_1 = \si(r, z_2(r), \Ga_2(r)+\om )-\si(r, z_1(r), \Ga_2(r)+\om)\nonumber\\ \qquad \qquad -\si(r, z_2(\rho), \Ga_2(r)+\om)+\si(r, z_1(\rho), \Ga_2(r)+\om)\,; \nonumber\\ J_2 = \si(r, z_1(r), \Ga_2(r)+\om)-\si(r, z_1(r), \Ga_1(r)+\om)\nonumber\\ \qquad \qquad -\si(r, z_1(r), \Ga_2(\rho)+\om)+\si(r, z_1(r), \Ga_1(\rho)+\om)\,;\nonumber\\ J_3 = \si(r, z_1(r), \Ga_2(\rho )+\om)-\si(r, z_1(r), \Ga_1(\rho )+\om)\nonumber\\ \qquad \qquad -\si(r, z_1(\rho), \Ga_2(\rho)+\om)+\si(r, z_1(\rho), \Ga_1(\rho)+\om)\,;\nonumber\\ J_4 = \si(r, z_1(\rho), \Ga_2(\rho )+\om)-\si(r, z_1(\rho), \Ga_2(r )+\om)\nonumber\\ \qquad \qquad -\si(r, z_2(\rho), \Ga_2(\rho)+\om)+\si(r, z_2(\rho), \Ga_2(r)+\om)\,;\nonumber\\ J_5 = \si(r, z_2(\rho), \Ga_2(\rho )+\om)-\si(\rho, z_2(\rho), \Ga_2(\rho )+\om)\nonumber\\ \qquad \qquad + \si(\rho, z_1(\rho), \Ga_1(\rho )+\om)-\si(r, z_1(\rho), \Ga_1(\rho )+\om)\,. \nonumber \end{cases}$$ From Lemma \[l.one-comp-holder\], we see that $$|J_1|\le \kappa \left[ \|z_2-z_1\|_{ \rho, r, \be} +\left(\|z_1\|_{\rho,r, \be}+ \|z_2\|_{\rho,r, \be}\right)\|z_2-z_1\|_{\rho, r}\right] (r-\rho)^\be \label{e.proof-j1}$$ and $$\|J_2\|\le \kappa \left[ \| \Ga _2- \Ga_1\|_{ \rho, r, \be } +\left(\| \Ga_1\|_{\rho,r, \be}+ \| \Ga_2\|_{\rho,r, \be }\right)\|\Ga_2-\Ga_1\|_{\rho, r, \be }\right] (r-\rho)^\be \,. \label{e.proof-j2}$$ Use the mean value theorem to obtain $$\begin{aligned} J_3&=&\int_0^1\int_0^1 {\mathbb D}\tilde \si( (1-\upsilon)z_1(\rho)+\upsilon z_1(r), (1-\th)\Ga_1(\rho)+\th \Ga_2(\rho)) d\th d\upsilon\nonumber \\ &&\qquad (z_1(r)-z_1(\rho)) (\Ga_2(\rho)-\Ga_1(\rho))\,. \end{aligned}$$ This shows $$|J_3|\le \kappa \|z_1\|_{\rho, r, \be}\| \Ga_2- \Ga_1\|_{\rho, r} (r-\rho)^\be \,. \label{e.proof-j3}$$ In a similar way we can obtain $$|J_4|\le \kappa \|\dot \Ga_2\|_{ \rho, r }\| z_2- z_1\|_{\rho, r} (r-\rho) \,. \label{e.proof-j4}$$ From the assumption on $\si$ it is to verify that $$|J_5|\le \kappa (r-\rho) \left[ \|z_2-z_1\|_{ \rho, r}+\|\Ga_2-\Ga_1\|_{\rho, r}\right] \,. \label{e.proof-j5}$$ Combining -, we have $$\begin{aligned} && \left|\si^{(2)}(r)-\si^{(1)}(r)-\si^{(2)}(\rho)+\si^{(1)}(\rho)\right| \nonumber\\ &&\qquad \le \kappa \bigg[\|z_2-z_1\|_{a, b, \be} +\| \Ga_2- \Ga_1\|_{a, b,\be }\nonumber\\ &&\qquad +\left(1+\|z_1\|_{a, b, \be}+ \|z_2\|_{a, b, \be}+\| \Ga_1\|_{a,b, \be }+\| \Ga_2\|_{a,b, \be }\right) \|z_2-z_1\|_{a,b}\nonumber\\ &&\qquad +\left(1+\|z_1\|_{a, b, \be}+ \|z_2\|_{a, b, \be}+ \| \Ga_1\|_{a, b, \be }\right. \nonumber\\ &&\qquad \left. +\| \Ga_2\|_{a,b, \be }\right) \|\Ga_2-\Ga_1\|_{a,b}\bigg] (r-\rho)^\be \,. \end{aligned}$$ Substituting the above inequality into yields $$\begin{aligned} &&I_2 \le \kappa \bigg[\|z_2-z_1\|_{a, b, \be} +\| \Ga_2- \Ga_1\|_{a, b, \be}\nonumber\\ &&\qquad +\left(1+\|z_1\|_{a, b, \be}+ \|z_2\|_{a, b, \be}+ \| \Ga_1\|_{a, b, \be } +\| \Ga_2\|_{a,b, \be }\right) \|z_2-z_1\|_{a,b}\nonumber\\ &&\qquad +\left(1+\|z_1\|_{a, b, \be}+\|z_2\|_{a, b, \be}+ \| \Ga_1\|_{a, b, \be }+\| \Ga_2\|_{a,b, \be}\right) \|\Ga_2-\Ga_1\|_{a,b}\bigg] (b-a)^{2\be} \,. \nonumber \\\end{aligned}$$ Substituting the bounds for $I_1$ and $I_2$ into we have $$\begin{aligned} &&\|F_{22}^{(2)}-F_{22}^{(1)}\|_{a,b,\be} \le \kappa \|B\|_{a,b, \be} \bigg[\|z_2-z_1\|_{a, b, \be} +\| \Ga_2- \Ga_1\|_{a, b, \be}\nonumber\\ &&\qquad +\left(1+\|z_1\|_{a, b, \be}+ \|z_2\|_{a, b, \be}+\|\Ga_1\|_{a,b, \be} +\|\Ga_2\|_{a,b, \be}\right) \|z_2-z_1\|_{a,b}\nonumber\\ &&\qquad +\left(1+\|z_1\|_{a, b, \be}+\|z_2\|_{a, b, \be}+ \| \Ga_1\|_{a, b,\be }+\|\Ga_2\|_{a,b, \be}\right) \|\Ga_2-\Ga_1\|_{a,b}\bigg] (b-a)^{\be} \nonumber \\ &&\le \kappa \|B\|_{a,b, \be} \left(1+\|z_1\|_{a, b, \be}+ \|z_2\|_{a, b, \be}+ \|\Ga_1\|_{a,b, \be}+\|\Ga_2\|_{a,b, \be}\right) \nonumber\\ &&\qquad \bigg[ |z_2(a)-z_1(a)|+\|\Ga_2(a)-\Ga_1(a)\| \nonumber\\ &&\qquad +\left[ \|z_2-z_1\|_{a, b, \be} +\| \Ga_2- \Ga_1\|_{a, b, \be}\right] (b-a)^{\be} \bigg] \,. \label{e.holder-f22}\end{aligned}$$ Finally, we turn to bound $\|F_{23}^{2}-F_{23}^{1}\|_{a, b, \be} $. We have $$\begin{aligned} &&\left| F_{23}^{2}(b)-F_{23}^{1}(b)-F_{23}^{2}(a)+F_{23}^{1}(a)\right|\\ &&\quad \le \int_a^b \int_0^s \left|\si(s, z_2(s), \Ga_2(s)) - \si(s, z_1(s), \Ga_1(s)) \right| |\tilde \si(u, z_2(u), \Ga_2(u))|duds\\ &&\qquad+\int_a^b \int_0^s \left|\tilde \si(u, z_2(u), \Ga_2(u)) - \tilde \si(u, z_1(u), \Ga_1(u)) \right| |\si (s, z_1(s), \Ga_1(s))|duds\\ &&\quad \le \kappa (b-a) \left\{ \|z_2-z_1\|_{a, b} +\|\Ga_2-\Ga_1\|_{a, b} +\|z_1\|_{a, b}\left[\|z_2-z_1\|_{0, b}+\| \Ga_2-\Ga_1\|_{0, b}\right]\right\} \,. \end{aligned}$$ This means $$\begin{aligned} &&\left\| F_{23}^{2} -F_{23}^{1} \right\|_{a, b, \be} \le \kappa (b-a) ^{1-\be} \big\{ \|z_2-z_1\|_{a, b} +\|\Ga_2-\Ga_1\|_{a, b} \nonumber \\ &&\qquad +\|z_1\|_{a, b}\left[\|z_2-z_1\|_{0, b}+\| \Ga_2-\Ga_1\|_{0, b}\right]\big\}\nonumber\\ &&\qquad \le \kappa (1 +\|z_1\|_{a, b}) \left[\|z_2-z_1\|_{0, b}+\| \Ga_2-\Ga_1\|_{0, b}\right] (b-a) ^{1-\be} \nonumber\\ &&\qquad \le \kappa (b-a) ^{1-\be} (1 +\|z_1\|_{a, b}) \bigg[|z_2 -z_1 |_{0,a}+ \| \Ga_2 -\Ga_1 \| _{0, a} \nonumber\\ &&\qquad +\left[ \|z_2-z_1\|_{a, b, \be }+\| \Ga_2-\Ga_1\|_{a, b, \be }\right](b-a)^\be \bigg]\, . \label{e.holder-f23}\end{aligned}$$ Combining , , and , we prove . Now we are ready to prove one of our main theorems of this section. \[t.5.7\] Let $T\in (0, \infty)$ be any given number. Assume the hypothesis \[h.b-sigma\]. Then, the equation has a unique solution. Moreover, there is a $\tau_0>0$ such that the solution satisfies $$\begin{aligned} \sup_{0\le t\le T}|z(t)| &\le& c_2 \exp\left\{ c_1 \|B\|_{0, T, \be}^{1/\be}\ \right\} \label{e.final-uniform-bound-solution} \\ \sup_{0\le a< b\le T, b-a\le \tau_0}|z |_{a, b, \be} &\le & c_2 \exp\left\{ c_1 \|B\|_{0, T, \be}^{1/\be} \right\}\,\label{e.final-holder-bound-solution} \end{aligned}$$ for some constants $c_1$ and $c_2$ dependent only on $\be, p, T$ and the bound $\cL$ for the coefficients $b$ and $\si$. Let $\BB={\mathcal H}\oplus \RR$. Then $F=(F_1, F_2)$ defined by is a mapping from (some domain of) $\BB[0, T]$ to $\BB[0, T]$. The inequality implies that there is a constant $c$ depending only on $\be, p, T$ and the bound $\cL$ for the coefficients $b$ and $\si$ such that $$\|F_1\|_{a, b, \be}\le c$$ for any $a, b\in [0, T]$. This together with implies that $F=(F_1, F_2)$ satisfies the condition (i) of Theorem \[t.fix-point-theorem\] with $\kappa$ there being replaced by $c \left(1+ \|B\|_{a, b, \be}\right)$. Lemma \[l.F-diff-holder-norm-bound\] implies that $F=(F_1, F_2)$ satisfies the condition (ii) of Theorem \[t.fix-point-theorem\] with the function $\bar h$ being given by $$\begin{aligned} \bar h&=&\bar h((\Ga_1,z_1), (\Ga_2,z_2)) = \cL (1+\|B\|_{a,b, \be}) \big(1+\|z_1\|_{0, a}+\|z_2\|_{0,a}\\ &&\qquad +\|z_1\|_{a, b, \be} + \|z_2\|_{a, b, \be} + \|\Ga_1\|_{a,b, \be}+\|\Ga_2\|_{a,b, \be}\big) \,.\end{aligned}$$ Thus, we can apply Theorem \[t.fix-point-theorem\] to prove that there is a $x \in \BB[0, T]$ satisfies the equation $x(t)=F(t, x )$. This means that $x$ satisfies the equation , hence it satisfies . The bounds and are immediate consequence of . Solution to the original equation --------------------------------- We need the following lemmas in the proof of the existence and uniqueness theorem for equation . \[l.diff-of-La\] Let $\Ga :\cT\times \Om\rightarrow \Om$ be continuously differentiable in $t$ and ${\mathcal H}$-differentiable in $\om$. If $\Ga(t):\Om\rightarrow \Om$ has an inverse $\La(t)$ and if $\La(t)$ is differentiable in $t$ in the Hilbert space ${\mathcal H}$, then $$\frac{\partial \La}{\partial t}(t, \om) =-({\mathbb D}\La)(t, \om)\frac{\partial \Ga}{\partial t}(t, \La(t, \om))\,.$$ Since $\La(t)$ is the inverse of $\Ga(t)$ we have $$\Ga(t, \La(t, \om))=\om\,, \quad \forall \ \om \in \Om\,.$$ Differentiating both sides with respect to $\om $, we have $$({\mathbb D}\Ga)(t, \La(t, \om))({\mathbb D}\La)(t, \om)=I\,,$$ where $I$ is an identity operator from ${\mathcal H}$ to ${\mathcal H}$. Therefore, we obtain $$\left[({\mathbb D}\Ga)(t, \La(t, \om))\right]^{-1}=({\mathbb D}\La)(t, \om)\,. \label{e.proof-nabla-G}$$ On the other hand, differentiating $\Ga(t, \La(t, \om))=\om$ with respect to $t$, we have $$\frac{\partial \Ga}{\partial t}(t, \om) +({\mathbb D}\Ga)(t, \La(t)(\om))\frac{\partial \La}{\partial t}(t, \om)=0\,.$$ Thus $$\frac{\partial \La}{\partial t}(t, \om) =-\left[({\mathbb D}\Ga)(t, \La(t, \om))\right]^{-1}\frac{\partial \Ga}{\partial t}(t, \La(t, \om))\,. \label{e.proof-diff-La}$$ Combining and we have $$\frac{\partial \La}{\partial t}(t, \om) =-({\mathbb D}\La)(t, \om)\frac{\partial \Ga}{\partial t}(t, \La(t, \om))\,.$$ This proves the lemma. \[l.reduced-to-original\] Let $\tau\in (0, T]$ be a positive number. Assume that $\Ga(t):\Om\rightarrow \Om$ defined by has an inverse $\La(t)$ for all $t\in [0, \tau]$ and assume that $\La(t)$ is differentiable in $t\in [0, \tau]$ in the Hilbert space ${\mathcal H}$. Let $z$ be defined by . Then $x(t)=z(t, \La(t)), t\in [0, \tau]$ satisfies the equation . If $\Ga(t)$ defined by has inverse $\La(t)$, then $$\begin{aligned} \om&=& \Gamma(t, \La(t))=\La(t)+\int_0^t \tilde \si (s, z(s), \Gamma(s))\int_0^\cdot \phi(s, u)du ds\Big|_{\om=\La(t)}\,. \end{aligned}$$ Or $$\begin{cases} \La(t) =\om+\int_0^\cdot h(t,u,\om) du\qquad \quad \hbox{with} \\ h(t,u, \om) =- \int_0^t \tilde \si (s, z(s), \Gamma(s)) \phi(s, u) ds\Big|_{\om=\La(t)}\,. \end{cases} \label{e.h-for-La}$$ On the other hand, from we have $$\begin{aligned} \frac{d}{dt}\Ga(t,\om)\big|_{\om=\La(t)} &=& \tilde \si(t, z(t), \Ga(t))\int_0^\cdot \phi(t, u) du\big|_{\om=\La(t)}\nonumber\\ &=& \tilde \si(t, x(t,\om), \om)\int_0^\cdot \phi(t, u) du \,. \label{e.diff-Ga-at-La} \end{aligned}$$ We apply the Itô formula to $z(t, \La(t))$ with $$\begin{aligned} f_0&=& \tilde b(s, z(s), \Gamma(s)) + \si(s, z(s), \Ga(s)) \int_0^s \tilde \si(u, z(u), \Ga(u)) \phi(s, u) du \\ f_1&=& \si(s, z(s), \Gamma(s))\ \end{aligned}$$ and with $h$ defined by . We shall use $\si(s, x(s))$ to denote $\si(s, x(s, \om), \om)$ etc. Noticing $z(s)\big|_{\om=\La(s)}=x(s)$, we have $$\begin{aligned} x(t) &=& z(t, \La(t))\nonumber\\ &=& \eta(\om) + \int_0^t \left(\int_0^s \tilde \si(u, z(u), \Ga(u)) \phi(s, u) du \right) \Big|_{\om=\La(s)} \si(s, x(s) ) ds\nonumber\\ &&\quad +\int_0^t \tilde b(s, x(s) ) ds + \int_0^t \si (s, x(s))\de B(s) \nonumber\\ &&\quad +\int_0^t {\mathbb D}z(s, \La(s))\frac{d}{ds}\La(s) ds +\int_0^t \si(s, x(s)) h(s,s,\om) ds\nonumber\\ &=&\eta(\om)+I_1+I_2+I_3+I_4+I_5 \,. \label{e.i1-4-for-x}\end{aligned}$$ From and since $\phi(s,u)=\phi(u,s)$, we see that $$I_1+I_5=0\,.\label{e.i1-i4}$$ By Lemma \[l.diff-of-La\] and then by , we have $$\begin{aligned} I_4 &=&\int_0^t ({\mathbb D}z)(s, \La(s))\frac{\partial }{\partial s} \La(s) ds\\ &=& -\int_0^t ({\mathbb D}z)(s, \La(s)) ({\mathbb D}\La)(s)\left(\frac{\partial }{ \partial s}\Ga\right) (s, \La(s))ds\\ &=& -\int_0^t \left[ {\mathbb D}x (s)\right] \tilde \si(s , x(s)) \int_0^\cdot \phi(s,u) du ds \,. \end{aligned}$$ This yields $$I_4=-\int_0^t \tilde \si(s , x(s)) {\mathbb D}^\phi_s x(s) ds\,.\label{e.i3}$$ Substituting and into we have $$\begin{aligned} x(t) &=&\eta(\om) + \int_0^t \tilde b(s, x(s) ) ds + \int_0^t \si(s, x(s))\de B(s)\nonumber\\ &&\qquad -\int_0^t \tilde \si(s , x(s)) {\mathbb D}^\phi_s x(s) ds\,. \end{aligned}$$ This is the lemma. Now we show that there is a positive $\tau$ such that $\Ga(t)$ has inverse $\La(t)$. Before we continue, we need the following simple inequality. Assume that $ B:[0, T]\rightarrow \RR$ is a Hölder continuous function of exponent $\be \in (0,1)$. Let $\BB_1 $ and $\BB_2$ be two Banach spaces and let $f:[0, T]\rightarrow \BB_1 $ and $g:[0, T]\rightarrow \BB_2$ be two Hölder continuous functions with exponent $\al \in ( 1-\be, 1)$. Then $$\begin{aligned} &&\|\int_0^\cdot f(s)\otimes g(s) dB(s)\|_{a, b, \be} \le \kappa \|B\|_{a, b, \be} \bigg\{ \|f\|_{a, b}\|g\|_{a,b} \nonumber \\ &&\qquad\qquad\qquad + \left[\|f\|_{a, b}\|g\|_{a, b, \be}+\|g\|_{a, b}\|f\|_{a, b, \be}\right] (b-a)^{\be } \bigg\} \,. \label{e.product-holder-bound}\end{aligned}$$ We refer to [@hugaussian], and in particular, the references therein for the tensor product. Since $\al+\be>1$, we can choose a $\la$ such that $\la<\al$ and $1-\la<\be$. For any $a, b\in [0, T]$, we have $$\begin{aligned} \left\|\int_a^b f(s)\otimes g(s) dB(s) \right\| &=& \left\| \int_a^b D_{a+}^{\la} \left[f(s)\otimes g(s)\right] D_{b-}^{1-\la} B_{b-}(s) ds \right\| \\ &\le &\kappa \|B\|_{a, b, \be} \int_a^b (b-s)^{\be+\la-1} \left\| D_{a+}^{\la} \left[f(s)\otimes g(s)\right] \right\| ds\,. \end{aligned}$$ From the definition of the Weyl derivative , we see essily $$\begin{aligned} &&\left\| D_{a+}^{\la} \left[f(s)\otimes g(s)\right] \right\|\le \kappa \bigg\{ \|f\|_{a, b}\|g\|_{a,b} (s-a)^{-\la} \\ &&\qquad \qquad + \left[\|f\|_{a, b}\|g\|_{a, b, \be}+\|g\|_{a, b}\|f\|_{a, b, \be}\right] (s-a)^{\be-\la} \bigg\}\,. \end{aligned}$$ Therefore, we have $$\begin{aligned} &&\left\|\int_a^b f(s)\otimes g(s) dB(s) \right\| \le \kappa \|B\|_{a, b, \be} \int_a^b (b-s)^{\be+\la-1} \bigg\{ \|f\|_{a, b}\|g\|_{a,b} (s-a)^{-\la} \\ &&\qquad + \left[\|f\|_{a, b}\|g\|_{a, b, \be}+\|g\|_{a, b}\|f\|_{a, b, \be}\right] (s-a)^{\be-\la} \bigg\} ds\\ &&\qquad \le \kappa \|B\|_{a, b, \be} \bigg\{ \|f\|_{a, b}\|g\|_{a,b} \\ &&\qquad + \left[\|f\|_{a, b}\|g\|_{a, b, \be}+\|g\|_{a, b}\|f\|_{a, b, \be}\right] (b-a)^{\be } \bigg\} (b-a)^\be \end{aligned}$$ which implies the lemma. By a Picard iteration procedure and by the bounds that we are going to obtain, we can show that ${\mathbb D}\Ga(t )$ and ${\mathbb D}z(t )$ exist under the hypothesis \[h.b-sigma\]. From the equation , we have $$\begin{aligned} {\mathbb D}\Ga(t) &=&I+\int_0^t \si_{xx}(s, z(s), \Ga(s)) {\mathbb D}z(s) \otimes \int_0^\cdot \phi(s, u) du ds\\ &&\qquad +\int_0^t {\mathbb D}\si_{x }(s, z(s), \Ga(s)) {\mathbb D}\Ga(s) \otimes \int_0^\cdot \phi(s, u) du ds \end{aligned}$$ and $$\begin{aligned} {\mathbb D}z(t) &=&{\mathbb D}\eta+\int_0^t \tilde b_{x }(s, z(s), \Ga(s)) {\mathbb D}z(s) ds+ \int_0^t {\mathbb D}\tilde b (s, z(s), \Ga(s)) {\mathbb D}\Ga(s) ds\\ && +\int_0^t \si_{x }(s, z(s), \Ga(s)) {\mathbb D}z(s) \de B(s)\\ &&+ \int_0^t {\mathbb D}\si (s, z(s), \Ga(s)) {\mathbb D}\Ga(s) \de B(s) +\si(\cdot, z(\cdot), \Ga(\cdot)) I_{[0, t]}(\cdot)\nonumber\\ &&+ \int_0^t \si_{x }(s, z(s), \Ga(s)) {\mathbb D}z(s) \int_0^s \si_{x }(u, z(u), \Ga(u )) \phi(s,u) du ds \\ && + \int_0^t {\mathbb D}\si (s, z(s), \Ga(s)) {\mathbb D}\Ga(s) \int_0^s \si_{x }(u, z(u), \Ga(u )) \phi(s,u) du ds \\ && + \int_0^t \si (s, z(s), \Ga(s)) \int_0^s \si_{xx }(u, z(u), \Ga(u )) {\mathbb D}z(u) \phi(s,u) du ds\\ &&+ \int_0^t \si (s, z(s), \Ga(s)) \int_0^s {\mathbb D}\si_{x }(u, z(u), \Ga(u )) {\mathbb D}\Ga(u) \phi(s,u) du ds\,. \end{aligned}$$ Similar to the lemmas \[l.uniform-bound-f1\] and \[l.uniform-bound-f2\], we can obtain \[l.bounds-for-nablas\] Under the hypothesis \[h.b-sigma\], there is a $\kappa_B$ depending on $\be, p, T$ and $\|B\|_{0, T, \be}$ such that $$\begin{aligned} \|\frac{d}{dt} {\mathbb D}\Ga \|_{0, t} &\le& \kappa_B \left(\|{\mathbb D}z \|_{0, t} +\|{\mathbb D}\Ga \|_{0, t} \right)\,; \label{e.5.58}\\ \|{\mathbb D}z\|_{a, b, \be} &\le& \kappa_B \big(1+\|{\mathbb D}z\|_{0, a} +\|{\mathbb D}\Ga\|_{0, a}\nonumber\\ &&\quad +\|{\mathbb D}z\|_{a, b, \be} (b-a)^\be +\|{\mathbb D}\Ga\|_{a, b, \be} (b-a)^\be \big)\,.\label{e.5.59}\end{aligned}$$ Let us denote the integral terms in the above expression for ${\mathbb D}z(t)$ by $I_k$, $k=1, 2, \cdots, 8$. Let us explain how to bound $I_3=\int_0^t \si_{x }(s, z(s), \Ga(s)) {\mathbb D}z(s) \de B(s)$. Since $z(s)$, $\Ga(s)$ are Hölder continuous (of exponent $\be$ with respect to $s$), $f(s):=\si_{x }(s, z(s), \Ga(s))$ is then also Hölder continuous. Now the inequality can be invoked to obtain $$\begin{aligned} \|I_3\|_{a, b, \be} &\le& \kappa \|B\|_{a, b, \be} \bigg\{ \|f\|_{a, b}\|{\mathbb D}z\|_{a,b} \nonumber \\ &&\qquad + \left[\|f\|_{a, b}\| {\mathbb D}z\|_{a, b, \be}+\|{\mathbb D}z\|_{a, b}\|f\|_{a, b, \be}\right] (b-a)^{\be } \bigg\} \nonumber\\ &\le& \kappa_B \bigg\{ \|{\mathbb D}z\|_{a,b} + \left[ \|{\mathbb D}z\|_{a, b, \be}+ \|{\mathbb D}z\|_{a, b, \be}\right] (b-a)^{\be } \bigg\}\end{aligned}$$ since both $\|f\|_{a, b}\|$ and $\|f\|_{a, b, \be}$ are bounded by $\kappa_B$. The other terms can be treated in exactly the same way. \[l.nabla-gamma-z-bounds\] Under the hypothesis, there is a $\kappa_B$ depending on $\be, p, T$ and $B_{0, T, \be}$ such that $$\|{\mathbb D}\Ga \|_{0, T}+\|{\mathbb D}z\|_{0, T}\le \kappa_B\,. \label{e.uniform-bound-nablas}$$ To show the existence of ${\mathbb D}\Ga(t)$ and ${\mathbb D}z(t)$ and to show the above bound we still use the idea in the proof of Theorem \[t.fix-point-theorem\]. First, we can show that the $x_n$ defined there are ${\mathcal H}$-differentiable and similar bounds holds as those in Lemma \[l.bounds-for-nablas\] hold recursively for all $x_n$. This can be used to obtain the uniform bounds for all $n$. Since the proof is analogous that of Theorem \[t.fix-point-theorem\], we shall not provide it here again. \[l.5.13\] Let $G:\Om\rightarrow {\mathcal H}$ be continuously ${\mathcal H}$-differentiable mapping such that $$\|{\mathbb D}G\|_{\mathcal H}\le c<1 \,.$$ Define $\Ga:\Om\rightarrow \Om$ by $$\Ga(\om)=\om+G(\om)\,.$$ Then $\Ga$ has a (unique) inverse $\La$ such that $\Ga(\La(\om))=\La(\Ga(\om))=\om$. We define $$\La_0(\om)=\om\,,\quad \La_{n+1}(\om)=\om-G(\La_n(\om))\,, \quad n=0, 1, 2, \cdots\,.$$ Then $$\begin{aligned} \left\|\La_{n+1}(\om)-\La_n(\om)\right\|_{\mathcal H}&=&\left\|G(\La_n(\om))-G(\La_{n-1}(\om))\right\|_{\mathcal H}\\ &\le& c \left\|\La_{n }(\om)-\La_{n-1}(\om)\right\|_{\mathcal H}\le \cdots \\ &\le& c ^n \left\|\La_{1 }(\om)-\La_{0}(\om)\right\|_{\mathcal H}\,.\end{aligned}$$ This means that $$\La_n(\om)-\om=\sum_{k=1}^n \left(\La_k(\om)-\La_{k-1}(\om)\right)$$ is a Cauchy sequence in ${\mathcal H}$. Thus $\La_n(\om)$ converges to an element $\La(\om) $ in $\Om$. From the construction of $\La_n$ we see that $\La$ satisfies $$\La(\om)=\om -G(\La(\om))\,.$$ Thus, $$\begin{aligned} \Ga(\La(\om)) &=& \La(\om)+G(\La(\om))\\ &=& \om -G(\La(\om))-G(\La(\om))=\om\,.\end{aligned}$$ This exactly means that $\La$ is the inverse of $\Ga$. Now we can state the main theorem of this paper. \[t.5.14\] Let $b, \si:[0, T]\times \RR\times \Om\rightarrow \RR$ satisfy the hypothesis . Then, the equation has a unique solution $x(t)$ up to some positive random time $\tau>0$. This means that there is a unique positive random time $\tau>0$ such that $$\begin{aligned} x (t\wedge\tau ) = \eta+\int_0^{t\wedge\tau } b(s, x (s), \om)d s+ \int_0^{t\wedge\tau } \si (s, x (s), \om)d B(s) \,,\ \forall \ t\in [0, T]\,. \nonumber\\ \label{e.5.88}\end{aligned}$$ According to Lemma \[l.reduced-to-original\] the remaining main task is to show the existence of an inverse $\La(t)$ of $\Ga(t)$. Since the existence and uniqueness of the solution of the system is known by Theorem \[t.5.7\], we only need to consider the first equation of the system , which is $$\Gamma(t)=\om+\int_0^t \tilde \si (s, z(s), \Gamma(s))\int_0^\cdot \phi(s, u)du ds\,.$$ Due to the presence of $z(t)$ in the above equation it is hard to prove that $\Ga(t)$ has an inverse for all $t\in [0, T]$. In fact, since $z(t)=z(t, \om)$ has been proved to exist, we may write the above equation as $$\Gamma(t)=\om+\int_0^t \hat \si (s, \om, \Gamma(s)) ds\,,$$ where $\hat \si(s,\om, \Ga)= \tilde \si (s, z(s, \om), \Gamma )\int_0^\cdot \phi(s, u)du ds$ is mapping from $[0, T]\times \Om\times \Om\rightarrow {\mathcal H}$. \[As earlier we may replace $\Ga(t)-\om$ by $\Ga(t)$ so that $\hat \si$ is a mapping from $[0, T]\times \Om\times {\mathcal H}\rightarrow {\mathcal H}$.\] Or we can write the following differential equation in ${\mathcal H}$: $$\dot \Gamma(t)= \hat \si (t, \om, \Gamma(t)) \,.$$ The dependence on $\om$ in the coefficient $\hat \si$ may prevent the solution $\Ga(t)$ to have inverse for all time $t$. To explain we give one example in one dimension ($\dim(\Om)=\dim({\mathcal H})=1$). We let $\hat \si(t, \om, \Ga)= -\om -\Ga$. Then the solution to the equation with initial condition $\Ga(0)=\om$ is explicitly given by $\Ga(t,\om)=2\om e^{-t} -\om$. But $\Ga(t, \om)=0$ when $t= \ln 2$. So, $\Ga(t, \om)$ is not invertible when $t= \ln 2$. Due to the above example, we are only seeking the inverse of $\Ga(t)$ when $t$ is sufficiently small (but strictly positive). We shall prove that $\Ga(t):\Om\rightarrow \Om$ has an inverse when $t$ is sufficiently small. Given an arbitrarily fixed positive number $R$, we define the following random time: $$\tau_R=\tau_R(\om)=\inf\left\{ t>0\,, \quad |B(t, \om)|>R\,, \|B\|_{0,t, \be }>R\right\}\,.$$ Since we have chosen a version of the Brownian motion $B$ such that $B(t)$ is Hölder continuous, we see that $0<\tau_R<\infty$ for all $\om\in \Om$. Now we define $$B_R(t,\om)=\begin{cases} B(t,\om)&\qquad \quad \hbox{when \ $0\le t\le \tau_R$}\\ B(\tau_R, \om) &\qquad \quad \hbox{when \ $ t\ge \tau_R$}\,. \end{cases}$$ It is clear that $$\sup_{0\le t\le T} |B_R(t)|\le R \quad{\rm and}\quad \|B_R\|_{a, b,\be}\le 2R \quad \hbox{ for any $0\le a<b\le T$}\,.$$ Now we consider the equation with $B$ replaced by $B_R$ and the corresponding solutions are replaced by $\Ga_R$ and $z_R$: $$\begin{cases} \Gamma_R(t)=\om+\int_0^t \tilde \si (s, z_R(s), \Gamma_R(s))\int_0^\cdot \phi(s, u)du ds\,;\\ \\ z_R(t)= \eta(\om ) +\int_0^t \tilde b(s, z_R(s), \Gamma_R(s))ds +\int_0^t \si(s, z_R(s), \Gamma_R(s))\de B_R(s)\\ \qquad\qquad +\int_0^t\int_0^s \si(s, z_R(s), \Ga_R(s)) \tilde \si(u, z_R(u), \Ga_R(u)) \phi(s, u) duds\,. \end{cases} \label{e.5.64}$$ By the inequality we see that $$\|{\mathbb D}\Ga_R \|_{0, T}+\|{\mathbb D}z_R\|_{0, T}\le c_R \,,$$ where $ c_R$ is a constant independent of $B$ (then independent of $\om$). This combined with Lemma \[l.bounds-for-nablas\] yields $$\|\frac{d}{ds}{\mathbb D}\Ga(s)\|_{0, T}\le c_R\,. \label{e.5.67}$$ On the other hand, we can write $${\mathbb D}\Ga(t)=I+G(t,\om)\,,\quad\hbox{where}\quad G(t,\om)=\int_0^t \frac{d}{ds}{\mathbb D}\Ga(s) ds\,.$$ The inequality implies that $$\|{\mathbb D}G(t, \om)\|\le c_Rt\le 1/2\,,\quad \hbox{if}\ \ t\le 1/(2c_R)\,.$$ Thus, from Lemma \[l.5.13\] it follows that $\Ga(t):\Om\rightarrow \Om$ has a (unique) inverse $\La(t)$ when $t\le t_0:=1/(2c_R)$. Thus, the system of equation has a unique solution such that $\Ga_R(t)$ has a (unique) inverse $\La_R(t)$. By Lemma \[l.reduced-to-original\] we see that $x_R(t)=z_R(t, \La_R(t))$ is a solution to $$\begin{aligned} x_R(t)&=&\eta+\int_0^t \tilde b(s, x_R(s), \om)ds+ \int_0^t \si(s, x_R(s), \om)\de B_R(s)\nonumber\\ &&\qquad -\int_0^t \tilde \si (s, x(s), \om){\mathbb D}^\phi_s x_R(s) ds\,, \quad 0\le t\le t_0\,. \end{aligned}$$ But when $t\le \tau_R$, $B_R(t)=B(t)$. Then when $t\le t_0\wedge\tau_R$, we have $$\begin{aligned} x_R(t)&=&\eta+\int_0^t \tilde b(s, x_R(s), \om)ds+ \int_0^t \si(s, x_R(s), \om)\de B(s)\nonumber\\ &&\qquad -\int_0^t \tilde \si (s, x(s), \om){\mathbb D}^\phi_s x_R(s) ds\,, \quad 0\le t\le t_0\wedge\tau_R\,. \end{aligned}$$ This can also be written as $$\begin{aligned} x_R(t\wedge\tau_R)&=&\eta+\int_0^{t\wedge\tau_R} b(s, x_R(s), \om)d s+ \int_0^{t\wedge\tau_R} \si (s, x_R(s), \om)d B(s) \,. \end{aligned}$$ The theorem is then proved. Linear and quasilinear cases ============================ Quasilinear case ---------------- If the diffusion coefficient $\si$ satisfies $$\si(s, x, \om)=a_1(s, \om) x+a_0(s, \om) \,, \label{e.si-quasilinear}$$ then we say equation is [*quasilinear stochastic differential equation* ]{}driven by fractional Brownian motion. Associated with this equation, the corresponding system of equations can be written as $$\begin{cases} \Gamma(t)=\om+\int_0^t a_1 (s, \Gamma(s))\int_0^\cdot \phi(s, u)du ds\\ \\ z(t)=\eta+\int_0^t \tilde b(s, z(s), \Gamma(s))ds +\int_0^t \si(s, z(s), \Gamma(s))\de B(s)\\ \ \qquad \qquad +\int_0^t \si(s, z(s), \Ga(s)) \int_0^s a_1(u, \Ga(u)) \phi(s, u) du ds\,, \end{cases} \label{e.quasilinear.transformed}$$ where $$\tilde b(t,x,\om)=\tilde b(s, x, \om):=b(s, x, \om)- x {\mathbb D}^\phi_s \si_1(s, \om) -{\mathbb D}^\phi_s \si_0 (s, \om)\,.$$ This system is decoupled. We can first solve the above first equation. Assume that $a_1(t,\om)$ is uniformly Lipschitz in $\om$ with respect to ${\mathcal H}$ norm. Namely, there is a positive constant $\kappa$ such that $$|a_1(t, \om+h)-a_1(t, \om)|\le \kappa \|h\|_{\mathcal H}\,, \qquad \forall \ t\in [0, T]\,,\ \, \om\in \Om\,, \ h\in {\mathcal H}\,.$$ Then the first equation in has a unique solution $\Ga(t)$. For all $t\in [0, T]$, $\Ga(t):\Om\rightarrow \Om$ has an inverse $\La(t)$. Moreover, the inverse $\La(t)$ is given by $\La(t)=\La(t,t)$, where $\left\{\La(\cdot, t)\right\}$ satisfies $$\La(s,t)=\om +\int_0^s a_1 (t-v, \La(v, t))\int_0^\cdot \phi(v, u)du dv\,, \quad 0\le s\le t\,. \label{e.inverse-gamma}$$ From the general dynamic system theory we see that for any $t_0\in [0, T]$, there is a unique solution $\Ga(t)=\Ga(t, t_0, \om)$ such that the first equation of has a unique solution for all $t\in [0, T]$ (even when $t<t_0$) such that $\Ga(t_0, t_0, \om)=\om$ and $ \Ga(t, t_0, \om)$ satisfies the flow property: $$\Ga(t, s, \Ga(s, t_0, \om))=\Ga(t+s, t_0, \om)\,, \ \forall \ t_0, s, t\in [0, T]\ \hbox{such that $s+t\in [0, T]$}\,.$$ This can be used to show the proposition easily. Once we obtain $\Gamma(t, \om)$ we can substitute it into the second equation in to obtain the following equation $$z(t)=\eta+\int_0^t \bb (s, z(s), \om)ds +\int_0^t \si(s, z(s), \Gamma(s))\de B(s)\,, \label{e.quasilinear6.6}$$ where $$\begin{aligned} \bb (s, z , \om) &=& \tilde b(s, z, \Ga(s))+\si(s, z , \Ga(s)) \int_0^s a_1(u, \Ga(u)) \phi(s, u) du \\ &=& b(s, x, \Ga(s))- x {\mathbb D}^\phi_s \si_1(s, \Ga(s)) -{\mathbb D}^\phi_s \si_0 (s, \Ga(s))\\ &&\qquad +\si(s, z , \Ga(s)) \int_0^s a_1(u, \Ga(u)) \phi(s, u) du\,. \end{aligned}$$ Now we can use Lemma \[l.reduced-to-original\] to obtain the following theorem. Let the diffusion coefficient $\si$ be given by . Let $\Ga$ be the unique solution to the first equation of . Let $z$ be the unique solution to the second equation of . Then has a unique solution $x(t)$ which is given by $$x(t)=z(t, \La(t))\,.$$ Moreover, there are positive constants $c_1$ and $c_2$, $\De\in [0, T]$, depending only on $p, \be, T$ such that for all $ \ 0\le a<b\le T, \ b-a\le \De$ $$\begin{cases}\sup_{0\le t\le T} |x(t)| \le c_2 \exp\left\{ c_1 \|B\|_{0, T, \be}^{1/\be} \right\} \\ \\ \|x\|_{a, b, \be} \le c_2 \exp\left\{ c_1 \|B\|_{0, T, \be}^{1/\be} \right\} \,. \end{cases} \label{e.quasilinear-bound}$$ The first inequality in is a direct consequence of and the above second inequality is the consequence of together with an easy bound for $\frac{d}{dt} \La(t)$. Let us now try to solve equation , which can be written as $$z(t)=\eta+\int_0^t \bb(s, z(s), \om)ds +\int_0^t \left[a_1(s, \Ga(s)) z(s) +a_0(s, \Ga(s))\right] \de B(s)\,.$$ Namely, $$z(t)-\int_0^t \left[a_1(s, \Ga(s)) z(s) +a_0(s, \Ga(s))\right] \de B(s)=\eta+\int_0^t \bb(s, z(s), \om)ds \,. \label{e.6.8}$$ Or $$dz(t)- \left[a_1(t, \Ga(t)) z(t) +a_0(t, \Ga(t))\right] \de B(s)= \bb(t, z(t), \om)dt \,. \label{e.6.9}$$ Let $$\begin{cases} A_1(t)=\exp\left\{-\int_0^t a_1(s, \Ga(s)) \de B(s)\right\} \\ A_2(t)= \int_0^t A_1(s) \ a_0(s, \Ga(s)) \de B(s)\,. \end{cases}\label{e.a1-a2}$$ Denote $$y(t)=A_1(t)z(t)- A_2(t)\,.$$ Then the equation can be written as $$\begin{aligned} dy(t) &=& A_1(t) \left\{ dz(t)- \left[a_1(t, \Ga(t)) z(t) +a_0(t, \Ga(t))\right] \de B(t) \right\}\nonumber\\ &=& A_1(t) \bb(t, z(t), \om) dt\nonumber\\ &=& \cB(t, y(t), \om) dt\,,\end{aligned}$$ where $$\cB(t, y)=A_1(t)\bb(t, A_1^{-1}(t) (y+A_2(t)), \om) \label{e.def-cB}$$ with $A_1^{-1}(t)=\exp\left\{\int_0^t a_1(r, \Ga(r)) \de B(r)\right\}$. This equation is a (pathwise) ordinary differential equation and can be solved by classical method. To summarize here is how we can solve the quasilinear equation of the following form $$dx(t)=b(t, x(t), \om) dt+\left[a_1(t, \om) x(t)+a_0(t, \om)\right] d B(t)\,, \quad x(0)=\eta(\om)\,.$$ 1. First we solve the first equation in to obtain $\Ga(t)$. 2. Then we solve to obtain the inverse $\La(t) =\La(t,t)$ of $\Ga(t)$. 3. Define $A_1$ and $A_2$ by . 4. Define $\cB(t,y)=\cB(t, y, \om)$ by and solve the (ordinary differential) equation $\dot y(t)=\cB(t, y(t))\,, \ y(0)=\eta $. 5. Let $z(t,\om)= A_1^{-1}(t, \om) (y(t,\om)+A_2(t, \om))$. 6. The solution $x(t)$ is then given by $x(t,\om)=z(t, \La(t,\om))$. If $b$ and $\si$ are deterministic, then the system of equations becomes $$\begin{cases}\Gamma(t)=\om+\int_0^t \tilde \si (s, z(s), \Gamma(s))\int_0^\cdot \phi(s, u)du ds\,;\\ z(t)= \eta(\om ) +\int_0^t \tilde b(s, z(s) )ds +\int_0^t \si(s, z(s) )\de B(s)\\ \qquad\qquad +\int_0^t\int_0^s \si(s, z(s) ) \tilde \si (u, z(u) ) \phi(s, u) duds\,. \end{cases} \label{e.6.14a}$$ This system of equations is also decoupled. One may first solve the above second equation to obtain $z(t,\om)$ and then substitute it into the above first equation to obtain $\Ga(t)$. However, the main diffculty remains to study the invertibility of $\Ga(t):\Om\rightarrow \Om$, which is hard as explained in the proof of Theorem \[t.5.14\]. Linear case ----------- If $\si$ is linear as in the previous subsection and if $b$ is also linear, i.e. $$\begin{cases} \si(s, x, \om)=a_1(s, \om) x+a_0(s, \om) \\ b(s,x,\om)=\be_1(s, \om) x+\be_0(s, \om) \,, \end{cases}$$ then the equation is called linear equation. $\Gamma$ and $\La$ can be found in the same way as in the quasilinear case. As we shall see that $z$ also satisfies a linear equation, we explain how to obtain the explicit form for $z(t)$. First, notice that the second equation in becomes $$\begin{aligned} z(t) &=&\eta+\int_0^t \be_1 (s, \Ga(s)) z(s) ds+\int_0^t \be_0(s, \Ga(s))ds\nonumber\\ &&\qquad -\int_0^t \left[{\mathbb D}^\phi_s a_1\right](s, \Ga(s)) z(s)ds -\int_0^t \left[{\mathbb D}^\phi_s a_0\right](s, \Ga(s)) ds \nonumber\\ &&\qquad +\int_0^t a_1(s, \Ga(s)) z(s)\de B(s)+\int_0^t a_0(s, \Ga(s)) \de B(s)\nonumber\\ &&\qquad +\int_0^t a_1(s, \Ga(s))\int_0^s a_1(u, \Ga(u)) \phi(s, u) du z(s) ds \nonumber\\ &&\qquad +\int_0^t a_0(s, \Ga(s))\int_0^s a_1(u, \Ga(u)) \phi(s, u) du ds\,. \label{e.linear-z}\end{aligned}$$ Introduce $$\begin{aligned} \Phi(t,s) &:=&\exp\bigg(\int_s^t \left[ \be_1(u, \Ga(u))-({\mathbb D}^\phi_u a_1)(u, \Ga(u))\right]du +\int_s^t a_1(u, \Ga(u))\de B(u)\nonumber\\ &&\qquad +\int_s^t a_1(u, \Ga(u))\int_0^u a_1(v, \Ga(v)) \phi(u, v) dv du \bigg)\,.\end{aligned}$$ Then the above equation can be solved explicitly as $$\begin{aligned} z(t) &=&\Phi(t, 0) \eta +\int_0^t \Phi(t, s)\left[\be_0(s, \Ga(s))-({\mathbb D}^\phi_s a_0)(s, \Ga(s))\right]ds\nonumber \\ &&\qquad+\int_0^t \Phi(t,s) a_0(s, \Ga(s)) \de B(s)\nonumber \\ &&\qquad +\int_0^t \Phi(t,s) a_0(s, \Ga(s)) \int_0^s a_1(v, \Ga(v)) \phi(s, v) dvds\,. \label{e.linear-z} \end{aligned}$$ Thus the solution becomes $$\begin{aligned} x(t) &=& z(t, \La(t))\,. \end{aligned}$$ If $b(s, x,\om)=b(s)x$ and $\si(s,x,\om)=a(s) x$ and $\eta=x_0 $, where $b(s)$ and $a(s)$ are deterministic function of $s$, then the first equation of becomes $$\Ga(t)=\om+\int_0^t a (s)\int_0^\cdot \phi(s,u) duds \,.$$ Thus $$\La(t)=\om-\int_0^\cdot h(t, u)du\,,$$ where $$h(t, u)=\int_0^t a (s) \phi(s, u) ds\,.$$ Since ${\mathbb D}^\phi a =0$, we have $$\Phi(t,s)=\exp\left\{ \int_s^t b (u) ds+\int_s^t a (u) \de B(u) +\int_s^t a(u)\int_0^u a(v) \phi(u,v) dv du\right\}\,.$$ Now since ${\mathbb D}^\phi b=0$ and $\be_0=a_0=0$, we have $$\begin{aligned} z(t) &=&\Phi(t, 0) x_0\,. $$ Since $\Phi(t,s)=\Phi(t,s, \om)$ still depends on $\om$. Using lemma \[t.3.1\], we have $$\begin{aligned} \Psi(t,s) &:=&\Phi(t, s, \La(t))\nonumber\\ &=& \exp\bigg\{ \int_s^t b(u) ds+\int_s^t a (u) \de B(u)-\int_s^tdu \int_u^t a (u) a (s) \phi(s,u)ds\bigg\} \nonumber \\ \label{e.def-Psi} \end{aligned}$$ Thus the solution to $$dx(t)=b(t) x(t) dt+ a(t)x(t) dB(t)$$ is given by $$\begin{aligned} x(t) &=&\Psi(t, 0) x_0 \nonumber\\ &=& \exp\bigg\{ \int_0^t b(u) ds+\int_0^t a (u) \de B(u)\nonumber\\ &&\qquad -\frac12 \int_0^t \int_0^t a (u) a (s) \phi(s,u) duds \bigg\} x_0\,. \end{aligned}$$ This is well-known (see for example [@BHOZ; @duncan; @humams; @hustochastics; @huoksendal]). In the ame way as above example, we can solve the following linear stochastic differential equation $$dx(t)=[b(t) x(t) +\be(t) ]dt+ [a(t)x(t) +\al(t)] dB(t)\,,\quad x(0)=x_0\,,$$ where $ x_0\in \RR $, $b(t), \be(t), a(t) , \al(t)$ are deterministic functions, to obtain $$\begin{aligned} x(t) &=&\Psi(t, 0) x_0+\int_0^t \Psi(t, s) \be (s) ds +\int_0^t \Psi(t,s) \al(s ) \de B(s)\nonumber \\ &&\qquad +\int_0^t \Psi(t,s) \al (s ) \int_0^s a (v ) \phi(s, v) dvds\,, \label{e.linear-z-solution} \end{aligned}$$ where $\Psi$ is given by . [17]{} Biagini, F.; Hu, Y.; Øksendal, B.; and Zhang, T. Stochastic calculus for fractional Brownian motion and applications. Springer, 2008. Buckdahn, R. Skorohod stochastic differential equations of diffusion type. Probab. Theory Related Fields 93 (1992), no. 3, 297-323. Buckdahn, R. Anticipative Girsanov transformations and Skorohod stochastic differential equations. Mem. Amer. Math. Soc. 111 (1994), no. 533. Duncan, T. E.; Hu, Y. and Pasik-Duncan, B. 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Differential equations driven by Hölder continuous functions of order greater than $1/2$. Stochastic analysis and applications, 399-413, Abel Symp., 2, Springer, Berlin, 2007. Hu, Y. and Nualart, D. Rough path analysis via fractional calculus. Trans. Amer. Math. Soc. 361 (2009), no. 5, 2689-2718. Hu, Y. and Øksendal, B. Wick approximation of quasilinear stochastic differential equations. Stochastic analysis and related topics, V (Silivri, 1994), 203-231, Birkhäuser, 1996. Hu, Y.; Øksendal, B. Fractional white noise calculus and applications to finance. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), no. 1, 1-32. Hu, Y. and Yan, J. A. Wick calculus for nonlinear Gaussian functionals. Acta Math. Appl. Sin. Engl. Ser. 25 (2009), no. 3, 399-414. Kufner, A.; John, O. and Fuč’k, S. Function spaces. Noordhoff International Publishing, Leyden; Academia, Prague, 1977. Kunita, H. First order stochastic partial differential equations. 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--- abstract: 'In this paper we will present the results of Artin–Markov on braid groups by using the [Gröbner-Shirshov ]{}basis. As a consequence we can reobtain the normal form of Artin–Markov–Ivanovsky as an easy corollary.' address: - 'Sobolev Institute of Mathematics, Novosibirsk 630090, Russia' - 'Faculty of mathematics and mechanics, Novosibirsk State University, Novosibirsk, 630090, Russia' - 'Faculty of Science, The Chiness University of Hong Kong, Hong Kong, China (SAR)' author: - 'L. A. Bokut$^*$' - 'V.V. Chaynikov$^*$' - 'K.P. Shum$^{**} $' title: Markov and Artin normal form theorem for braid groups --- [^1] [^2] Introduction ============ The theory of braid groups were first studied by E. Artin dated back to 1925 (see  [@Ar25]). Artin established generators and defining relations of the braid group and a faithful representation theorem of the braid group as a subgroup of the automorphism group of a free group. Later on, Markov  [@Mar45] (1945) and Artin  [@Ar47] (1947) enriched the Artin generators by the Burau elements ( [@Bur32]) and find a presentation of the braid group in the Artin-Burau generators. We call it the Artin-Markov presentation of the braid group in the Artin-Burau generators. The main result of ( [@Mar45]) and ( [@Ar47]) is the normal form theorem for the braid group. We call it the Artin-Markov-Ivanovsky normal form theorem for Markov credited the result to A. Ivanovsky; it looks that there is no paper published by A. Ivanovskii himself. However, the proof of the normal form theorem of the braid group given by Markov and Artin are rather complicated because the automorphism of the free groups are involved. In this note we will try to escape the technical details of the proof given by Markov  [@Mar45] and Artin  [@Ar25]. We will show by direct calculations (of compositions) that the Artin-Markov presentation of the braid group is the minimal [Gröbner-Shirshov ]{}basis of the braid group in the Artin-Burau generators and under an appropriate ordering of group words, namely, the inverse tower ordering. As a consequence, by the Composition-Diamond Lemma the Artin-Markov-Ivanovsky normal form in the braid group follows as an immediate corollary of our result. Basic notations and results =========================== In this section we give some basic notations and cite some useful results in the literature. We first let $\mathbf{X}$ be a linearly ordered set and $k$ a field. Let $k\langle \mathbf{X}\rangle$ the free associative algebra over $\mathbf{X}$ and $k$. On the set $\mathbf{X}^*$ of words generated by $\mathbf{X}$ we impose a well order $\leq$ compatible with the multiplication of words. We call this kind of order a monomial order. Now, let $f\in k\langle \mathbf{X}\rangle$ be a polynomial with leading word $\bar f$. We say that $f$ is monic if $\bar f$ occurs in $f$ with coefficient $1$. We now formulate the following definitions. Let $f$ and $g$ be two monic polynomials. - Let $w$ be a word such that $w = \bar fb= a \bar g$, with $\deg( \bar f) + \deg(\bar g) > \deg (w)$. Then we call the polynomial $(f,g)_w$ the intersection composition of $f$ and $g$ with respect to $w$ if $(f,g)_w = fb - ag$. - Let $w = \bar f = a\bar gb$. Then we call the polynomial $(f,g)_w = f - agb$ the inclusion composition of $f$ and $g$ with respect to $w$. In the above case, the transformation $ f\mapsto (f,g)_w = f -agb$ is called the elimination of the leading word (ELW) of g in f. - Let $\mathbf{S} \subset k\langle \mathbf{X}\rangle$. We call a composition $(f, g)_w$ trivial relative to $\mathbf{S}$ (and $w$) if $$(f,g)_w = \sum \alpha_i a_it_i b_i,$$ where $t_i \in \mathbf{S}$, $a_i, b_i \in \mathbf{X} ^*$, and $\bar {a_it_ib_i} < w$. In a notation, $(f, g)_w\equiv 0\bmod (\mathbf{S}, w)$. In particular, if $(f,g)_w$ goes to zero by using ELW of polynomials from $\mathbf{S}$, then $(f,g)_w$ is trivial relative to $\mathbf{S}$. We assume that $f_1$ and $f_2$ are some polynomials satisfying the condition $ f_1 \equiv f_2 \bmod (\mathbf{S}, w) $ if $ f_1-f_2 \equiv 0 \bmod (\mathbf{S},w)$. Then we call $\mathbf{S}$ a *[Gröbner-Shirshov ]{}  set (basis)* in $k\langle \mathbf{X}\rangle$ if any composition of polynomials from $\mathbf{S}$ is trivial relative to $\mathbf{S}$. From now on, we denote the algebra with set of generators $\mathbf{X}$ and set of defining relations $\mathbf{S}$ by $\langle \mathbf{X}{\mathord{\hskip 1pt|\hskip 1pt}}\mathbf{S}\rangle$. The following lemma and its applications to [Gröbner-Shirshov ]{} bases was due to Newman  [@Ne42], Shirshov ( [@Sh62]), Buchberger  [@Bu65],  [@Bu70] and Bergman  [@Be78]. The lemma given below was formulated by Bokut (see  [@Bo72],  [@Bo76]). \[Composition-Diamond Lemma\] Let $R=\langle \mathbf{X}{\mathord{\hskip 1pt|\hskip 1pt}}\mathbf{S}\rangle$. The set of defining relations $\mathbf{S}$ is a [Gröbner-Shirshov ]{} set if and only if the set $$Irr(S)= \{u \in \mathbf{X}^* \mid u \neq a\bar fb, \mbox{ for any }f\in \mathbf{S}\}$$ of $\mathbf{S}$-irreducible words consists of a linear basis of $R$. Let $\mathbf{S}$ be a [Gröbner-Shirshov ]{}basis in $k\langle \mathbf{X}\rangle$. Then $\mathbf{S}$ is called a *minimal* [Gröbner-Shirshov ]{} basis if for any $s\in \mathbf{S}$, $s$ is a linear combination of $\mathbf{S}\setminus\{s\}$-irreducible words. Any ideal of $k\langle \mathbf{X} \rangle$ has a unique minimal [Gröbner-Shirshov ]{}basis (i.e., a set of generators of the ideal). If a subset $\mathbf{S}$ of $k\langle \mathbf{X}\rangle$ is not a [Gröbner-Shirshov ]{} basis then one can add to $\mathbf{S}$ all nontrivial compositions of polynomials of $\mathbf{S}$, and continue this process (infinitely) many times in order to have a [Gröbner-Shirshov ]{}  set (basis) $\mathbf{S}^{\rm comp}$ that generates the same ideal as $S$. This procedure is called the Buchberger - Shirshov algorithm [@Sh62], [@Bu65], [@Bu70]. A polynomial $f$ is called semigroup relation if $f$ is of the form $u - v$, where $u,v \in \mathbf{X}^*$. If $S$ is a set of semigroup relations, then any nontrivial composition of $S$ has the same form. Consequently, the set $\mathbf{S}^{\rm comp}$ also consists of semigroup relations. Let $A = sgp\langle \mathbf{X}{\mathord{\hskip 1pt|\hskip 1pt}}\mathbf{S}\rangle$ be a semigroup presentation. Then $\mathbf{S} \subset k\langle \mathbf{S}\rangle$ and one can find a [Gröbner-Shirshov ]{} basis $ \mathbf{S}^{\rm comp}$. This set does not depend on $k$ and it consists of semigroup relations. In this case, we call $\mathbf{S}^{\rm comp}$ a [Gröbner-Shirshov ]{}basis of $A$. It is the same as a [Gröbner-Shirshov ]{} basis of the semigroup algebra $kA = \langle \mathbf{X}{\mathord{\hskip 1pt|\hskip 1pt}}\mathbf{S}\rangle$. We now introduce the concept of inverse tower ordering of words. Let $X=Y\dot{\bigcup}Z$, words $Y^*$ and the letters $Z$ are well ordered. Suppose that the order on $Y^*$ is monomial. Any word in $X$ has the form $ u=u_0z_1\dots u_{k-1}z_ku_{k}, $ where $k\geq 0$, $z_i\in Z$, $u_i\in Y^*$. Define the inverse weight of the word $u \in X$ by: $$inwt(u)=(k,u_k,z_k,\dots ,z_1,u_0).$$ Now we order inverse weights lexicographically and define $$u>v \Longleftrightarrow inwt(u)>inwt(v).$$ Then we call the above order the *inverse tower order*. Clearly, the above order is a monomial order. In case $Y=T\dotbigcup U$ and $Y^*$ are endowed with the inverse tower order, we call order of words on $X$ the inverse tower order of word relative the presentation $$X=(T\dotbigcup U)\dotbigcup Z.$$ In general, we can define the inverse tower order of $X$-words relative to the presentation $$X=(\dots (X^{(n)}\dotbigcup X^{(n-1)})\dotbigcup \dots )\dotbigcup X^{(0)},$$ where $X^{(n)}$-words are endowed by a monomial order. Let $ \Sigma=\{\sigma_1, \dots, \sigma_{n-1}\}$ be a finite alphabet. Then, the following group presentation define the $n$-strand braid group: $$B_n=\langle\Sigma \mid \sigma_{i+1}\sigma_i\sigma_{i+1}=\sigma_i\sigma_{i+1}\sigma_i, \sigma_i\sigma_j=\sigma_j\sigma_i, i-j>1\rangle$$ Here any index falls into the interval $[1,n-1]$ . In the braid group $B_n$, we now introduce a new set of generators. We call them the Artin-Burau generators. In the braid group $B_n$, we let $$s_{i,i+1}=\sigma_i^2,\ s_{i,j+1}=\sigma_j\dots \sigma_{i+1}\sigma_i^2\sigma_{i+1}^{-1} \dots \sigma_j^{-1},$$ where $1\leq i<j\leq n-1$. Form the set $$S_j=\{s_{i,j}, s_{i,j}^{-1}, 2< i < j <n \}$$ and $$\Sigma^{-1}=\{\sigma_1^{-1},\dots , \sigma_{n-1}^{-1}\}.$$ Then the set $$S=S_n\cup S_{n-1}\cup \dots \cup S_2\cup \Sigma^{-1}$$ generates $B_n$ as a semigroup. We call elements of $S$ the Artin-Burau generators of $B_n$. Observe that generators $\sigma_i$ are omitted as well as the trivial group relations on them. With the above notation Markov  [@Mar45] used $s_{i,i+1}\sigma_i^{-1}$ to replace $\sigma_i$, and $\sigma_i^{-2}=s_{i,i+1}^{-1}$ to replace $\sigma_i^{-1}\sigma_i=1$. Then we order the set $S$ in the following way: $$S_n<S_{n-1}< \dots <S_2<\Sigma^{-1},$$ Clearly, in the above chain, any letter of $S_n$ is less than any letter of $S_{n-1}$ and so on. Also we define for each $j$ $$s_{1,j}^{-1}<s_{1,j}<s_{2,j}^{-1}< \dots <s_{j-1,j}, \hbox{\quad and\quad}\sigma_1^{-1}<\sigma_2^{-1}< \dots \sigma_{n-1}^{-1}.$$ With above notation, we now able to order the $S$-words by using the inverse tower order, according to the fixed presentation of $S$ as the union of $S_j$ and $\Sigma^{-1}$. We order the $S_n$-words by the $deg-inlex$ order, i.e., we first compare the words by length and than by inverse lexicographical order, starting from their last letters. The following abbreviations are taken from  [@Mar45]. $$\sigma_{i,j+1}=\sigma_i^{-1}\dots \sigma_{j}^{-1}, 1\leq i\leq j\leq n-1, \sigma_{ii}=1.$$ Also we denote $ \{a,b\}=b^{-1}ab $. Main results ============ We first cite some crucial results from  [@Mar45],  [@Ar47]. The first lemma is fairly easy. ( [@Mar45 Lemma  3],  [@Ar47 p.119])\[L1\] The following relations hold in the braid group $B_n$ for ($\delta=\pm1$): $$\begin{aligned} &\sigma_k^{-1}s_{i,j}^{\delta}=s_{i,j}^{\delta}\sigma_k^{-1},\quad k\neq i-1,i,j-1,j, \label{E1}\\ &\sigma_i^{-1}s_{i,i+1}^{\delta}=s_{i,i+1}^{\delta}\sigma_i^{-1},\label{E2}\\ &\sigma_{i-1}^{-1}s_{i,j}^{\delta}=s_{i-1,j}^{\delta}\sigma_{i-1}^{-1},\label{E3}\\ &\sigma_{i}^{-1}s_{i,j}^{\delta}=\{s_{i+1,j}^{\delta},s_{i,i+1}\}\sigma_{i}^{-1},\label{E4}\\ &\sigma_{j-1}^{-1}s_{i,j}^{\delta}=s_{i,j-1}^{\delta}\sigma_{j-1}^{-1},\label{E5}\\ &\sigma_{j}^{-1}s_{i,j}^{\delta}=\{s_{i,j+1}^{\delta},s_{j,j+1}\}\sigma_{j}^{-1}.\label{E6}\\ \noalign{\hbox{Also we have}} &\sigma_{i-1}s_{i,j}^{\delta}=\{s_{i-1,j}^{\delta},s_{i-1,i}^{-1}\}\sigma_{i-1},\nonumber\\ &\sigma_{i}s_{i,j}^{\delta}=s_{i+1,j}^{\delta}\sigma_{i},\nonumber\\ &\sigma_{j-1}s_{i,j}^{\delta}=\{s_{i,j-1}^{\delta},s_{j-1,j}^{-1}\}\sigma_{j-1},\nonumber\\ &\sigma_{j}s_{i,j}^{\delta}=s_{i,j+1}^{\delta}\sigma_{j}.\nonumber\end{aligned}$$ \[R1\] We note that last relations from the above lemma will not in [Gröbner-Shirshov ]{}basis of the braid group $B_n$ (see below). We use them for a sketch of a proof of next Lemma \[L2\]. ( [@Mar45 Lemmas 6 and 7],  [@Ar47 Theorem 18])\[L2\] The following relations hold in the braid group $B_n$ for all $i<j<k<l$, $\varepsilon=\pm1$: $$\begin{aligned} &s_{j,k}^{-1}s_{k,l}^{\varepsilon}=\{s_{k,l}^{\varepsilon}, s_{j,l}^{-1}\}s_{j,k}^{-1},\label{E7}\\ &s_{j,k}s_{k,l}^{\varepsilon}=\{s_{k,l}^{\varepsilon}, s_{j,l}s_{k,l}\}s_{j,k},\label{E8}\\ &s_{j,k}^{-1}s_{j,l}^{\varepsilon}=\{s_{j,l}^{\varepsilon}, s_{k,l}^{-1}s_{j,l}^{-1}\}s_{j,k}^{-1}, \label{E9}\\ &s_{j,k}s_{j,l}^{\varepsilon}=\{s_{j,l}^{\varepsilon}, s_{k,l}\}s_{j,k}, \label{E10}\\ &s_{i,k}^{-1}s_{j,l}^{\varepsilon}=\{s_{j,l}^{\varepsilon},s_{k,l}s_{i,l} s_{k,l}^{-1}s_{i,l}^{-1}\} s_{i,k}^{-1},\label{E11}\\ &s_{i,k}s_{j,l}^{\varepsilon}=\{s_{j,l}^{\varepsilon}, s_{i,l}^{-1}s_{k,l}^{-1}s_{i,l}s_{k,l}\} s_{i,k}.\label{E12}\end{aligned}$$ Also for $j<i<k<l$ or $i<k<j<l$, and $\varepsilon, \delta =\pm1$ $$\begin{aligned} &s_{i,k}^{\delta}s_{j,l}^{\varepsilon}=s_{j,l}^{\varepsilon}s_{i,k}^{\delta}.\label{E13}\end{aligned}$$ We provide a proof of (\[E7\]) for $\varepsilon=1$ as a typical example. We use arguments given by Markov in  [@Mar45]. First we can easily see that the relation holds for $j=k-1$, $l=k+1$. We assume that (\[E7\]) holds for $j<k<l$. Then we prove it for $j-1$ and $l+1$ using Lemma \[L1\]. We deduce the following equalities by direct computation: $$\begin{aligned} &s_{j,k}^{-1}s_{k,l+1}=s_{j,k}^{-1}\{s_{k,l},\sigma_l^{-1}\} =\{\{s_{k,l},s_{j,l}^{-1}\},\sigma_l^{-1} \}s_{j,k}^{-1}=\{s_{k,l+1},s_{j,l+1}^{-1}\}s_{j,k}^{-1},\\ &s_{j-1,k}^{-1}s_{k,l}=\{s_{j,k}^{-1},\sigma_{j-1}\}s_{k,l} =\sigma_{j-1}^{-1}s_{j,k}^{-1}\sigma_{j-1} s_{k,l}=\sigma_{j-1}^{-1}s_{j,k}^{-1}s_{k,l}\sigma_{j-1}=\\ &\{\{s_{k,l},s_{j,l}^{-1}\}s_{j,k}^{-1},\sigma_{j-1}\} =\{s_{k,l},s_{j-1,l}^{-1}\}s_{j-1,k}^{-1}.\end{aligned}$$ This shows that (\[E7\]) holds. Finally, we have the following relations in the braid group $B_n$( see  [@Mar45 Lemma 5]). A proof is fairly simple. \[L3\] The following relations hold in the braid group $B_n$: $$\begin{aligned} &\sigma_{j}^{-1}\sigma_k^{-1}= \sigma_k^{-1}\sigma_{j}^{-1},\quad j<k-1,\label{E14}\\ &\sigma_{j,j+1}\sigma_{k,j+1}=\sigma_{k,j+1}\sigma_{j-1,j},\quad k<j,\label{E15}\\ &\sigma_i^{-2}=s_{i,i+1}^{-1}, \label{E16}\\end{aligned}$$ We now call the relations (\[E1\])–(\[E16\]) together with the trivial relations $$s_{i,j}^{\pm 1}s_{i,j}^{\mp 1}=1$$ the Artin-Markov relations $\mathbf{S}$ for the braid group $B_n$ in terms of the Artin-Burau generators. Using the above relations $\mathbf{S}$ together with the definition $\sigma_i=s_{ii+1}\sigma_i^{-1}$, we can deduce Artin’s relations for $B_n$. Namely, in relation (\[E15\]) we let $k=j-1$. Then we have $$\sigma_j^{-1}\sigma_{j-1}^{-1}\sigma_j^{-1}=\sigma_{j-1}^{-1}\sigma_j^{-1}\sigma_{j-1}^{-1}.$$ Also $$\sigma_i^{-1}\sigma_i=\sigma_i^{-1}s_{i,i+1}\sigma_i^{-1}=s_{i,i+1}\sigma_i^{-2} =s_{i,i+1}s_{i,i+1}^{-1}=1,$$ and the same for $\sigma_i\sigma_i^{-1}=1$. \[L4\] The following relations are deduced by the ELW of $\mathbf{S}$ (to be more precis, by the ELW of (\[E7\])– (\[E16\])): $$\begin{aligned} &\sigma_{i,j}\sigma_{k,j}=\sigma_{k,j}\sigma_{i-1,j-1},\quad k<i,\\ &\sigma_{i,j}\sigma_{k,j}=s_{i,k+1}^{-1}\sigma_{k+1,j}\sigma_{i,j-1},\quad i\leq k.\end{aligned}$$ Main Theorem ============ Using the Artin-Markov relations given in the Section 3, we establish the following theorem. \[T1\] The Artin-Markov relations form a minimal [Gröbner-Shirshov ]{} basis of the braid group $B_n$ in term of the Artin-Burau generators relative to the inverse tower order of words. There are no inclusion compositions of defining relations. We only need to check all possible intersection compositions. Let us do some for examples. Let us check a composition of intersection of two relations $f,g$ of the form (\[E8\]) relative to the ambiguity $$w=(ij)(jk)(kl),\quad i<j<k<l,$$ where $(ij)=s_{i,j}$. We have $$f=(ij)(jk)-\{(jk),(ik)(jk)\}(ij),\ g=(jk)(kl)-\{(kl), (jl)(kl)\}(jk).$$ We need to prove that $$\begin{aligned} &\{(jk),(ik)(jk)\}(ij)(kl)\equiv (ij)\{(kl),(jl)(kl)\}(jk) \bmod (\mathbf{S},w). \label{E17}\end{aligned}$$ In fact, by computation we deduce that $$\begin{aligned} &(ij)\{(kl),(jl)(kl)\}(jk)\equiv \{(kl),\{(jl),(il)(jl)\}(kl)\}(ij)(jk)\equiv \end{aligned}$$ $$\begin{aligned} &\{(kl),\{(jl),(il)(jl)\}(kl)\}\{(jk),(ik)(jk)\}(ij).\label{E18}\end{aligned}$$ For the left hand side of (\[E17\]) we have $$\begin{aligned} &\{(jk),(ik)(jk)\}(ij)(kl)\equiv\{(jk),(ik)(jk)\}(kl)(ij)\equiv\\ &(jk)^{-1}(ik)^{-1}(jk)(ik)(jk)(kl)(ij)\equiv\\ &(jk)^{-1}(ik)^{-1}(jk)(ik)\{(kl),(jl)(kl)\}(jk)(ij)\equiv\\ &(jk)^{-1}(ik)^{-1}(jk)\{\{(kl),(il)(kl)\} ,\{(jl),(il)^{-1}(kl)^{-1}(il)(kl) \}\\ &\{(kl),(il)(kl)\}\}(ik)(jk)(ij)\equiv(jk)^{-1}(ik)^{-1}(jk)\{(kl),\\ &\{(jl),(il)^{-1}(kl)^{-1}\}(kl)(il)(kl)\}(ik)(jk)(ij)\equiv\\ &(jk)^{-1}(ik)^{-1}(jk)\{(kl),(il)(jl)(kl)\}(ik)(jk)(ij)\equiv\\ &(jk)^{-1}(ik)^{-1}\{\{(kl),(jl)(kl)\},(il)\{(jl),(kl)\}\{(kl),(jl)(kl)\}\}(jk)(ik)(jk)(ij)\equiv\\ &(jk)^{-1}\{\{\{(kl),(il)^{-1}\},X\},\{(il),(kl)^{-1}(il)^{-1}\}X\}(ik)^{-1}(jk)(ik)(jk)(ij)\equiv\end{aligned}$$ $$\begin{aligned} &(X=\{(jl),(kl)(il)(kl)^{-1}(il)^{-1}\}\{(kl),(il)^{-1}\}=\{(kl),(il)^{-1}\}\{(jl),(kl)\})\end{aligned}$$ $$\begin{aligned} &(jk)^{-1}\{\{\{(kl),(il)^{-1}\},\{(jl),(kl)\}\},(il)(jl)(kl)\}(ik)^{-1}(jk)(ik)(jk)(ij)\equiv\\ &\{\{\{\{(kl),(jl)^{-1}\},(il)^{-1}\},\{\{(jl),(kl)^{-1}(jl)^{-1}\},\{(kl),(jl)^{-1}\}\}\},\\ &(il)\{(jl),(kl)^{-1}(jl)^{-1}\}\{(kl),(jl)^{-1}\}\}(jk)^{-1}(ik)^{-1}(jk)(ik)(jk)(ij)=\end{aligned}$$ $$\begin{aligned} &\{\{\{(kl),(jl)^{-1}(il)^{-1}\},(jl)\},(il)(jl)(kl)\}\{(jk),(ik)(jk)\}(ij). \label{E19}\end{aligned}$$ Words (\[E18\]) and (\[E19\]) are the same for $$\begin{aligned} &\{(kl),\{(jl),(il)(jl)\}(kl)\}=\{\{\{(kl),(jl)^{-1}(il)^{-1}\},(jl)\},(il)(jl)(kl)\}.\end{aligned}$$ Thus, (\[E17\]) is verified and the composition is checked. To check the compositions of relations $f, g$ of the forms ($\ref{E12}), (\ref{E2})$ respectively, we let $w=\sigma_q^{-1}(jk)(kl)$ and let $$\begin{aligned} f&=\sigma_q^{-1}s_{jk}-s_{jk}\sigma_q^{-1},\quad q\neq j-1,j,k-1,k,\\ g&=(jk)(kl)-\{(kl),(jl)(kl)\}(jk).\end{aligned}$$ Then we have $$(f,g)_w=\sigma_q^{-1}\{(kl),(jl)(kl)\}(jk)-(jk)\sigma_q^{-1}(kl).$$ If $q\neq l-1,l$ then it is clear that the composition is trivial. So, we need to consider the following cases: a\) $q=l$. We have $$\begin{aligned} \sigma_l^{-1}\{(kl),(j&l)(kl)\}(jk)\equiv\{\{(k,l+1),(l,l+1)\},\{(j,l+1),(l,l+1)\}\\[6pt] \{(k,l+1),(&l,l+1)\}\}(jk)\sigma_l^{-1}\equiv \{(k,l+1),(j,l+1)(k,l+1)(l,l+1)\}(jk)\sigma_l^{-1},\\[6pt] (jk)\sigma_l^{-1}(kl)&\equiv (jk)\{(k,l+1),(l,l+1)\}\sigma_l^{-1}\\ &\equiv\{(k,l+1),(j,l+1)(k,l+1)(l,l+1)\}(jk)\sigma_l^{-1}.\end{aligned}$$ Hence, the case is verified. b\) $q=l-1$. We have $$\begin{aligned} &\sigma_{l-1}^{-1}\{(kl),(jl)(kl)\}(jk)\equiv\{\{(k,l-1),(j,l-1)(k,l-1)\}(jk)\sigma_{l-1}^{-1},\\ &(jk)\sigma_{l-1}^{-1}(kl)\equiv (jk)(kl)\sigma_{l-1}^{-1}\equiv \{(k,l-1),(j,l-1)(k,l-1)\}(jk)\sigma_{l-1}^{-1}.\end{aligned}$$ Hence, the case is also verified. Finally, we need to check the composition of relations (\[E16\]). We first let $$\begin{aligned} f&=\sigma_j^{-1}\sigma_k^{-1}\dots \sigma_j^{-1}-\sigma_k^{-1}\dots \sigma_j^{-1}\sigma_{j-1}^{-1},\quad k<j,\\ g&=\sigma_j^{-1}\sigma_l^{-1}\dots \sigma_j^{-1}-\sigma_l^{-1}\dots \sigma_j^{-1}\sigma_{j-1}^{-1},\quad l<j,\\ \text{ and let } w&=\sigma_j^{-1}\sigma_k^{-1}\dots \sigma_j^{-1}\sigma_l^{-1}\dots \sigma_j^{-1}.\end{aligned}$$ Then, we have $$(f,g)_w=-\sigma_k^{-1}\dots \sigma_j^{-1}\sigma_{j-1}^{-1}\sigma_l^{-1}\dots \sigma_j^{-1}+ \sigma_j^{-1}\sigma_k^{-1}\dots \sigma_{j-1}^{-1}\sigma_l^{-1}\dots \sigma_j^{-1}\sigma_{j-1}^{-1},$$ We consider the following cases: a\) $l=j-1$. In this case, by Corollary \[L4\], we have $$\begin{aligned} &(f,g)_w\equiv -\sigma_k^{-1}\dots \sigma_j^{-1}s_{j-1,j}^{-1} \sigma_j^{-1}+ \sigma_j^{-1}\sigma_k^{-1}\dots \sigma_{j-2}^{-1}s_{j-1j}^{-1}\sigma_j^{-1}\sigma_{j-1}^{-1};\\[6pt] &\sigma_k^{-1}\dots \sigma_j^{-1}s_{j-1,j}^{-1} \sigma_j^{-1}\equiv s_{j,j+1}^{-1}\sigma_k^{-1}\dots \sigma_{j-1}^{-1}s_{jj+1}^{-1}\equiv s_{j,j+1}^{-1}s_{kj+1}^{-1}\sigma_k^{-1}\dots \sigma_{j-1}^{-1};\\[6pt] &\sigma_j^{-1}\sigma_k^{-1}\dots \sigma_{j-2}^{-1}s_{j-1j}^{-1}\sigma_j^{-1}\sigma_{j-1}^{-1}\equiv \sigma_j^{-1}s_{kj}^{-1}\sigma_k^{-1}\dots \sigma_{j-2}^{-1}\sigma_j^{-1}\sigma_{j-1}^{-1}\\ &\qquad \equiv\{s_{kj+1}^{-1},s_{jj+1}\}s_{jj+1}^{-1}\sigma_k^{-1}\dots \sigma_{j-1}^{-1}\equiv s_{j,j+1}^{-1}s_{kj+1}^{-1}\sigma_k^{-1}\dots \sigma_{j-1}^{-1},\end{aligned}$$ and the case is done. b\) $l<j-1$. We use again Corollary \[L4\] to obtain $$\begin{aligned} &\sigma_k^{-1}\dots \sigma_j^{-1}\sigma_{j-1}^{-1}\sigma_l^{-1}\dots \sigma_j^{-1}\equiv \sigma_k^{-1}\dots \sigma_j^{-1}\sigma_l^{-1}\dots \sigma_j^{-1}\sigma_{j-2}^{-1}\equiv \sigma_{kj+1}\sigma_{lj+1}\sigma_{j-2}^{-1}\\ &\equiv\sigma_{lj+1}\sigma_{k-1j}\sigma_{j-2}^{-1}\hbox{\;\ (when $l<k$), or }\equiv s_{kl+1}^{-1}\sigma_{l+1j+1} \sigma_{kj}\sigma_{j-2}^{-1} \hbox{\;\ (when $k\leq l$)}.\end{aligned}$$ If $l<k$, then $$\begin{gathered} \sigma_j^{-1}\sigma_k^{-1}\dots \sigma_{j-1}^{-1}\sigma_l^{-1}\dots \sigma_j^{-1}\sigma_{j-1}^{-1}\equiv \sigma_j^{-1}\sigma_{kj}\sigma_{lj}\sigma_j^{-1} \sigma_{j-1}^{-1}\\ \equiv\sigma_j^{-1}\sigma_{lj}\sigma_{k-1j-1}\sigma_j^{-1}\sigma_{j-1}^{-1} \equiv \sigma_j^{-1}\sigma_{lj+1}\sigma_{k-1j} \ \equiv \sigma_{lj+1}\sigma_{k-1j}\sigma_{j-2}^{-1},\end{gathered}$$ and we are done. If $l\geq k$, then $k\leq l<j-1$ and so we have $$\begin{gathered} \sigma_j^{-1}\sigma_k^{-1}\dots \sigma_{j-1}^{-1}\sigma_l^{-1}\dots \sigma_j^{-1}\sigma_{j-1}^{-1}\equiv \sigma_j^{-1}\sigma_{kj}\sigma_{lj}\sigma_j^{-1} \sigma_{j-1}^{-1}\\ \equiv \sigma_j^{-1}s_{kl+1}^{-1}\sigma_{l+1j}\sigma_{kj-1}\sigma_j^{-1}\sigma_{j-1}^{-1} \equiv \sigma_j^{-1}s_{kl+1}^{-1}\sigma_{l+1j+1}\sigma_{kj},\end{gathered}$$ and $$\sigma_j^{-1}s_{kl+1}^{-1}\sigma_{l+1j+1}\sigma_{kj}\equiv s_{kl+1}^{-1}\sigma_{l+1j+1} \sigma_{kj}\sigma_{j-2}^{-1},$$ as desired. Applying [Composition-Diamond]{} Lemma we obtain: The set of $\mathbf{S} $-irreducible words of $B_n$ corresponding to the above [Gröbner-Shirshov ]{} basis $\mathbf{S} $ consists of the words $$\begin{aligned} \label{E20} &f_nf_{n-1}\dots f_2\sigma_{i_nn}\sigma_{i_{n-1}n-1}\dots \sigma_{i_22},\end{aligned}$$ where $f_j$ are free irreducible words in $\{s_{ij}\mid i< j\},\ 2\leq j\leq n$. \[C1\] Every word of $B_n$ has a unique presentation in the form $(\ref{E20})$. Let $\Sigma_n$ be the symmetric group, i.e., $$\Sigma_n=\langle s_1,\dots ,s_{n-1} \mid s_i^2=1, s_{i+1}s_is_{i+1}=s_is_{i+1}s_i, s_is_j=s_js_i,\; i-j>1\rangle,$$ and let $$S_{i,i}=1\hbox{ and }S_{i,j+1}=s_is_{i+1}\dots s_j,\quad i<j.$$ The following lemma was proved in  [@Mar45 Theorem 4, Corollary 6]. It also follows from the fact that $$\{s_i^2=1,\, s_is_j=s_js_i,\, i-j>1,\, S_{j,j+1}S_{k,j+1}=S_{k,j+1}S_{j-1,j},\, k<j\}$$ is a [Gröbner-Shirshov ]{} basis of $\Sigma_n$ under the deg-inlex order of words in $\{s_i\}$ (see [@BoS01]). \[L5\] Every element of $\Sigma_n$ has a unique presentation in a form $$S_{i_n,n}S_{i_{n-1},n-1}\dots S_{i_2,2},$$ where $i_j\leq j$ and $2\leq j\leq n$. Let $P_n$ be the group of pure braids. This is the kernel of the natural homomorphism of $B_n$ onto $\Sigma_n$. From Theorem \[T1\], Corollary \[C1\] and Lemma \[L5\] it follows $P_n$ is a group with generators $\{s_{ij}\}$ and defining relations $(\ref{E7})$–$(\ref{E13})$ $($which, together with the trivial relations, form a minimal [Gröbner-Shirshov ]{} basis of $P_n$ relative the inverse tower order of words in the generators$)$. [99]{} E. Artin, *Theory der Zöpfe.* Abh. Math. Seminar., Hamburg Univ. [**4**]{} (1925), 47–72. E. Artin, *Theory of braids.* Ann. of Math., [**48**]{} (1947), 101–126. G. M. Bergman, *The diamond lemma for ring theory.* Adv. in Math. [**29**]{} (1978), 178–218. L. A. Bokut, *Unsolvability of the word problem, and subalgebras of finitely presented Lie algebras.* Izv. Akad. Nauk SSSR Ser. Mat. [**36**]{} (1972), 1173–1219. L. A. Bokut, *Imbeddings into simple associative algebras.* Algebra i Logika [**15**]{} (1976), 117–142, 245. L.A. Bokut and L.-S. Shiao, *On [Gröbner-Shirshov ]{} basis of Coxeter groups*, Commun. in Algebra [**29**]{} (2001), no. 9, 4305–4319. B. Buchberger. An algorithm for finding a basis for the residue class ring of a zero-dimensional polynomial ideal. (German). Ph. D. thesis, University of Insbruck, Austria, 1965. B. Buchberger, *An algorithmical criteria for the solvability of algebraic systems of equations.* (German). Aequationes Math. [**4**]{} (1970), 374–383. W. Burau, *Über Zopfinvarianten .* Abh. Math. Sem. Hamburg Univ. [**9**]{} (1932), 2, 117–124. A.A. Markov, Foundations of the algebraic theory of braids. Proceedings of the Steklov Mathematical Institute. [**16**]{} Moscow-Leningrad, 1945. M. H. A. Newman, *On theories with a combinatorial definition of “equivalence.”* Ann. of Math. [**43**]{} (1942), 223–243. A. I. Shirshov, *Some algorithm problems for Lie algebras.* (Russian) Sibirsk. Mat. Z. [**3**]{} (1962), 292–296. (Translation appears in ACM SIGSAM Bull. [**33**]{} (1999).) [^1]: $^*$Supported in part by the Russia’s Fund for Fundamental Research and the Leading Scientific Schools Fund (Russia). [^2]: $^{**}$Partially supported by a RGC(HK) grant 2003/04
--- abstract: 'Static and dynamic magnetic properties of a ferrimagnetic \[Fe(35Å)/Gd(50Å)\]$_{12}$ superlattice were investigated in a wide $4-300$ K temperature range using magneto-optical Kerr effect (MOKE) and ferromagnetic resonance (FMR) techniques. The multilayer structure was sputtered on a transparent glass substrate which made it possible to perform MOKE measurements on both Fe and Gd terminated sides of the superlattice. These experiments allowed us to detect a transition between field-aligned and canted magnetic states on both sides of the film and to distinguish between the bulk and surface twisted phases of the superlattice. As a result, the experimental $H-T$ magnetic phase diagram of the system was obtained. FMR studies at frequencies $7-36$ GHz demonstrated a complex evolution of absorption spectra as temperature decreased from room down to 4 K. Two spectral branches were detected in the sample. Theoretical simulations show that the observed spectral branches correspond to different types of inhomogeneous resonance modes in the multilayer with non-uniform magnetization precession inside Gd layers.' address: - 'P.L. Kapitza Institute for Physical Problems RAS, 119334 Moscow, Russia' - 'Institute of Solid State Physics RAS, 142432 Chernogolovka, Moscow region, Russia' - 'M.N. Mikheev Institute of Metal Physics UB RAS, 620137 Ekaterinburg, Russia' - 'Ural Federal University, 620002 Ekaterinburg, Russia' author: - 'A.B. Drovosekov' - 'A.O. Savitsky' - 'D.I. Kholin' - 'N.M. Kreines' - 'V.V. Proglyado' - 'M.V. Ryabukhina' - 'E.A. Kravtsov' - 'V.V. Ustinov' title: | Twisted magnetization states and inhomogeneous resonance modes\ in a Fe/Gd ferrimagnetic multilayer --- Fe/Gd multilayer ,ferrimagnetics ,magnetic properties ,ferromagnetic resonance 68.65.Ac ,75.70.Cn ,75.50.Gg ,76.50.+g Introduction ============ Layered structures based on transition (TM) and rare-earth (RE) ferromagnetic (FM) metals, like Fe/Gd, are model ferrimagnetic systems demonstrating a rich magnetic phase diagram with complex types of magnetic ordering [@Cam2015; @Cam1993]. Due to an antiferromagnetic (AFM) coupling at Fe-Gd interfaces and essentially different Curie temperatures of Fe and Gd (for bulk materials, $T_\mathrm{C}^\mathrm{Fe}=1043$ K and $T_\mathrm{C}^\mathrm{Gd}=293$ K) a so-called “compensation point” $T_\mathrm{comp}$ can exist in the system. At $T=T_\mathrm{comp}$ magnetic moments of Fe and Gd layers are equal to each other and the total magnetization of the system vanishes. Below $T_\mathrm{comp}$, the magnetic moment in Gd subsystem exceeds that in Fe subsystem, while above $T_\mathrm{comp}$, opposite situation takes place. As a result, in weak fields applied in the film plane, a collinear magnetic phase is realized with Fe magnetization vector oriented parallel (at $T > T_\mathrm{comp}$) or antiparallel to the field direction (at $T < T_\mathrm{comp}$). As the magnetic field increases and exceeds some critical value, such field-aligned phases become unstable and a transition to canted magnetic state occurs. Moreover, due to a relatively weak exchange stiffness of Gd, the external magnetic field initiates essentially non-uniform distribution of magnetization inside Gd layers (twisted state). The above-discussed complex behaviour of the Fe/Gd system was described theoretically by Camley *et al.*, using the mean-field approach \[3–5\], and observed experimentally by different techniques in a number of works \[6–9\]. At the same time it was predicted theoretically that even more complicated situation takes place when a finite Fe/Gd superlattice is considered. In this case, two types of twisted magnetic states are possible in the system: surface twist and bulk twist [@LePage1990]. Starting from the field-aligned state in weak external field, an increase of the field leads first to distortion of the collinear state near the superlattice surface (at $H=H_\mathrm{s}$). At higher fields ($H>H_\mathrm{b}$) the bulk twisted state is realized. Fig.\[magn\_profiles\] represents schematically the corresponding magnetization distributions calculated for different field values at $T>T_\mathrm{comp}$ [@Drov2017]. It is important to note that the surface twist phase arises at the outermost layer of the superlattice when its magnetization is directed opposite to the applied field. Thus, the surface twist phase arises on Gd-terminated side of the superlattice at $T>T_\mathrm{comp}$ and on Fe-terminated side of the superlattice at $T<T_\mathrm{comp}$. ![image](Fig1){width="90.00000%"} Direct experimental observation of such surface twisted states comes to difficulties since it requires simultaneous probing bulk and surface magnetic states of the superlattice. A few works were devoted to this problem. Haskel *et al.* [@Hask2003] demonstrated surface twist effects in a Fe-terminated \[Fe/Gd\]$_{15}$/Fe multilayer, using grazing-incidence x-ray magnetic circular dichroism. Kravtsov *et al.* [@Kra2009] used simultaneous refinement of polarized neutron and resonant x-ray magnetic reflectometry data to directly obtain magnetization depth profiles in a \[Fe/Gd\]$_5$ multilayer. In both cases the complexity of the used methods makes it difficult to perform detailed studies of stability regions of bulk and surface twisted phases as a function of temperature and magnetic field. Magneto-optical Kerr effect (MOKE) is a relatively simple and sensitive method to obtain direct information about the surface magnetic state of the multilayer. The penetration depth of visible light into metal is about $\sim100$ Å which is comparable with typical thickness of individual layers in the superlattice. Thus, MOKE signal provides information about magnetization in several upper layers of the superlattice. Hahn *et al.* [@Hahn1995] used MOKE to study surface magnetic states in a \[Fe/Gd\]$_{15}$ structure. Since the samples were sputtered on non-transparent Si substrates, authors compared MOKE signals from superlattices terminated by Fe and Gd layers. The difference of the MOKE curves for two samples was explained by the surface magnetic twist arising in case of the surface layer magnetization oriented opposite to the field direction. In our previous work [@Drov2017] we studied static magnetization curves of a \[Fe/Gd\]$_{12}$ multilayer. Comparing the experimental data with mean-field calculations, we found indications of field-dependent phase transitions between field-aligned, surface- and bulk twisted states. However, the static magnetometry provides only the net magnetic moment of the entire multilayer and the surface effects are manifested too weakly. In this work we use MOKE to obtain more precise knowledge on the surface magnetic states in the superlattice. The investigated \[Fe/Gd\]$_{12}$ multilayer is grown on a transparent glass substrate which allows direct probing magnetic states on both sides of the structure. As a result, we determine the stability regions of bulk and surface twisted states in the superlattice, depending on temperature and magnetic field. The experimental phase diagram is compared with calculations based on the mean-field model [@Drov2017]. Studies of magnetization dynamics in RE/TM systems attract attention due to a recent idea to use such materials for realization of ultrafast magnetic switching, promising for potential applications in magnetic storage devices \[14–16\]. A number of works were devoted to investigations of ferromagnetic resonance (FMR) in TM/Gd multilayers \[17–26\]. Room temperature studies \[17–20\] demonstrated the importance of spin pumping into RE metal to explain a large FMR line width in TM/RE systems. Several groups reported about the effect of line broadening and shift of the absorption peak to lower fields at cooling the system below room temperature. Such behaviour was observed for Co/Gd [@Patrin2006; @Demirtas2010], Py/Gd [@Khod2017], Fe/Gd [@Drov2017] and Fe/Cr/Gd [@Drov2015; @Drov2018] multilayers. In most of the cited works only one “high-temperature” resonance peak was detected. This peak became much weaker or even disappeared as temperature decreased below $T_\mathrm{C}^\mathrm{Gd}$ which effect was explained by non-local damping mechanisms in the system [@Drov2017; @Khod2017]. In a short letter [@Svalov2001], Svalov *et al.* reported about experimental observation of a second absorption peak below $T_\mathrm{C}^\mathrm{Gd}$ in a Co/Gd multilayer. Similar behaviour was observed in our previous works for the Fe/Gd system [@Drov2017]. Theoretical simulations showed that the observed absorption peaks corresponded to different types of inhomogeneous resonance modes in the multilayer. In this work we perform more detailed investigation of temperature evolution of the resonance spectra in the Fe/Gd superlattice. In contrast to the work [@Drov2017], here we pay special attention to the transformation of the spectra in the vicinity of $T_\mathrm{C}^\mathrm{Gd}$. In particular, we note that the behaviour of the high-temperature resonance peak is strongly dependent on the pumping frequency. To explain this result and identify the observed resonance modes, the experimental data are compared with model calculations based on Landau-Lifshitz equations describing magnetization dynamics in the system. Sample and experimental details =============================== The \[Fe(35Å)/Gd(50Å)\]$_{12}$ superlattice was prepared on a glass substrate using high vacuum magnetron sputtering technique. Two chromium layers with thickness 50 Å and 30 Å served as buffer and cap layers respectively. X-ray diffraction studies performed in [@Drov2017] demonstrated well-defined layered structure of the sample with interfacial root mean square roughness of about 1–2 atomic monolayers. Magnetic properties of the multilayer were studied using MOKE and FMR techniques in the $4-300$ K temperature range in magnetic fields up to 10 kOe applied in the film plane. Longitudinal MOKE studies of the surface magnetization were performed on both sides of the film, using a 635 nm semiconductor laser. In our experimental geometry the MOKE signal was proportional to the component of magnetization parallel to the applied field. FMR measurements were carried out using a conventional field-sweep technique on a laboratory developed transmission type spectrometer at different frequencies in the range $7-36$ GHz. ![MOKE curves measured at 155 K from two sides of the film: 1) from the glass substrate side (Fe-terminated side of the superlattice) and 2) from the film surface (Gd-terminated side of the superlattice). Comparing the curves, different types of magnetic ordering can be identified.[]{data-label="Kerr"}](Fig2){width="\columnwidth"} ![MOKE curves obtained at different temperatures on Gd-terminated (a) and Fe-terminated (b) sides of the superlattice. Black arrows show transitions from field-aligned to canted state of the surface magnetization.[]{data-label="Kerr-T"}](Fig3){width="\columnwidth"} Results and discussion ====================== Magneto-optical Kerr effect --------------------------- Static magnetometry of the investigated sample performed in [@Drov2017] showed that Gd layers had reduced Curie temperature, $T_\mathrm{C}^\mathrm{Gd}\approx200$ K, comparing with the bulk value 293 K. The system demonstrated the compensation point at $T_\mathrm{comp}\approx90$ K. Testing MOKE experiments on Fe and Gd thin films showed that both Fe and Gd layers should contribute to the total Kerr effect for the combined Fe/Gd layered system. Under our experimental conditions, the MOKE signal from Gd is comparable with that from Fe (about two times smaller at low temperature) but has opposite sign. Thus, we expect different signs of MOKE for Gd- and Fe-aligned states in the investigated multilayer. Fig. \[Kerr\] shows the experimental MOKE hysteresis loops measured at $T=155$ K from two sides of the superlattice. For both curves, a flat part in the region of weak fields means that the magnetic moment of the outermost layer remains collinear to the external field. Positive sign of the MOKE signal at $H>0$ indicates the Fe-aligned state. At some higher field the MOKE signal decreases, indicating that the magnetization of the outermost layer begins to rotate. Note that on Gd-terminated side this rotation starts in weaker field ($H=H_\mathrm{s}$) than on Fe-terminated side ($H=H_\mathrm{b}$). Thus, we can conclude that in magnetic fields $H_\mathrm{s}<H<H_\mathrm{b}$ the surface twist state is realized on Gd-terminated side of the superlattice. In higher fields $H>H_\mathrm{b}$, a transformation to the bulk twisted phase occurs. Similar analysis of the MOKE curves was performed for different temperatures in the range 4–300 K (see Fig. \[Kerr-T\]) and the resulting phase diagram of the system was obtained (Fig. \[Phases\]). At $T>T_\mathrm{C}^\mathrm{Gd}$ we observe simple rectangular hysteresis loops without any signs of possible phase transitions. At lower temperatures the shape of the MOKE curves changes. The compensation point $T_\mathrm{comp}\approx90$ K can be clearly detected as temperature where an inversion of the hysteresis loop occurs (Fig. \[Kerr-T\]), i.e. different orientation of Fe magnetization is realized in weak fields above and below $T_\mathrm{comp}$. It is also clearly seen that at $T>T_\mathrm{comp}$ the rotation of magnetization starts in weaker fields on Gd-terminated side of the superlattice. On the contrary, at $T<T_\mathrm{comp}$ this rotation begins in weaker fields on Fe-terminated side of the multilayer. Unfortunately, in the region of low temperatures the increasing hysteresis smears the phase transitions and prevents accurate determination of the critical fields. As a result, the experimental error is increasing. Nevertheless, the observed behaviour is in agreement with the theoretical prediction that the surface twist phase arises on the side of the superlattice when the magnetization of the outermost layer is directed opposite to the applied field. In the phase diagram Fig. \[Phases\], the experimental stability regions for different phases are compared with the result of mean-field calculations (see [@Drov2017] for details). We note a good agreement between the experiment and the model. ![Resulting $H-T$ phase diagram of the investigated Fe/Gd superlattice. Points are obtained from MOKE data on two sides of the multilayer. Lines are calculations within the mean-field approach [@Drov2017]. The dashed line corresponds to a situation when Gd magnetization vanishes in the middle of Gd layer.[]{data-label="Phases"}](Fig4){width="\columnwidth"} ![image](Fig5){width="\textwidth"} ![image](Fig6){width="\textwidth"} Ferromagnetic resonance ----------------------- Fig. \[spectra\] demonstrates the temperature evolution of experimental resonance spectra at several different frequencies. At room temperature, one relatively narrow ($\Delta H\sim100$ Oe) absorption peak is observed. As temperature decreases, this “high-temperature” (HT) peak broadens and its position changes. We note that the direction of the line shift depends on frequency. At high frequencies ($f\gtrsim12$ GHz) the HT peak shifts towards lower fields (Fig. \[spectra\]b,c). The same behaviour was observed earlier for different types of TM/Gd multilayers \[21–23\]. This effect can be qualitatively described, considering a strongly coupled layered ferrimagnet (see Appendix in the end of this paper), so this behaviour can be considered as “normal”. In our case, however, another situation takes place at low frequencies ($f\lesssim12$ GHz). Here we observe the shift of the HT peak towards higher fields (Fig. \[spectra\]a). This result is opposite to the behaviour reported in the previous works \[21–23\] and clearly contradicts to the simple approximation of strongly coupled FM layers. At all frequencies under study, the HT peak disappears below $T\approx160$ K. At the same time a second “low-temperature” (LT) peak arises in the region of high fields. As temperature decreases, this peak shifts towards lower fields and becomes more pronounced. At high frequencies it can be clearly detected down to lowest temperature, however, at $f=7.65$ GHz it again disappears below $T\approx60$ K (Fig. \[spectra\]). Fig. \[fvsH\]a demonstrates the resulting experimental frequency-*vs*-field $f(H)$ dependencies at different temperatures. At room temperature the $f(H)$ curve for HT-mode can be qualitatively described by Kittel-like equation for FM film (see Appendix, Eq. ). However at lower temperatures the shape of the $f(H)$ curve changes strongly and the simple Kittel’s formula clearly becomes inapplicable. This means that the approximation of uniform magnetization precession within the structure is not valid. Taking into account a large exchange stiffness of Fe layers and a strong coupling at Fe-Gd interface, we can suppose that inhomogeneous precession occurs inside the Gd layers. To describe such inhomogeneous resonance modes theoretically we use the approach of the work [@Drov2017]. To model the non-uniform magnetization precession inside Gd layers they are divided into elementary “atomic” sublayers coupled with each other. The static magnetization in each sublayer is calculated using the mean-field model while the dynamics is described by Landau-Lifshitz equations (LLE) with relaxation terms. For relaxation terms, we consider Gilbert damping in Fe and Gd layers as well as diffusion-type damping in Gd (see [@Drov2017] for details and model parameters). As a result, we calculate the complex eigenfrequencies of the system $\omega=\omega^\prime+i\omega^{\prime\prime}$. The corresponding eigenvectors represent the depth profiles of magnetization precession in the superlattice. The damping of the calculated resonance modes can be characterized by quality factor (Q-factor) $Q=\omega^\prime/2\omega^{\prime\prime}$. The larger Q-factor is, the more intensive resonance peak is expected. Following our previous work [@Drov2017], we consider only the modes with in-phase precession of Fe layers and perform modelling for one period of the superlattice. Fig. \[fvsH\]b demonstrates the resulting calculated dependencies $f(H)$ at different temperatures (for illustrative purposes, only the modes with $Q>0.5$, are shown). The model predicts the existence of two spectral branches with different types of magnetic precession inside Gd layers (Fig. \[fvsH\]c). The HT-mode has a gap in the spectrum at low temperatures and corresponds to strongly non-uniform precession inside Gd layers. The LT-mode is quasi-uniform. Its frequency vanishes at $H=H_\mathrm{b}$, i.e. at phase transition from field-aligned to twisted magnetic state. In general, the behaviour of calculated curves $f(H)$ repeats qualitatively the experimental dependencies, except the temperature region $\approx200-225$ K (i.e. slightly above $T_\mathrm{C}^\mathrm{Gd}$) and in weak magnetic fields $H\lesssim1.5$ kOe. Above $T=225$ K the model predicts the crossing of two spectral branches. One branch with increasing dependence $f(H)$ corresponds to preferable precession of Fe layers. This branch has large Q-factor and is observed experimentally. The second branch with decreasing dependence $f(H)$ corresponds to preferable precession of inner part of Gd layers. This branch has small Q-factor and is not observed experimentally. Below $T=225$ K the model predicts the repulsion of these two crossing modes. As a consequence, a gap in the spectrum opens. Experimentally, however, such a gap arises only at $T\lesssim180$ K (Fig. \[fvsH\]a,b). Despite this discrepancy, Fig. \[fvsH\]b helps to understand different behaviour of the HT peak at frequencies below and above $f\approx12$ GHz, i.e different direction of the line shift at cooling the system below room temperature (Fig. \[spectra\]). The critical value 12 GHz corresponds to the frequency where the effect of modes repulsion arises. Fig. \[Hres-T\] shows the resulting experimental and calculated temperature dependencies of the resonance fields $H_\mathrm{res}(T)$ at different frequencies. It can be seen that the experimental and theoretical curves demonstrate not only qualitative but also a certain quantitative agreement. The noticeable discrepancy observed for HT-mode at 17.2 GHz below $T\approx230$ K is connected with the above-discussed inadequate description of the mode-repulsion region. ![Temperature dependencies of the resonance field at different frequencies. Points are experimental data, lines are calculations. Solid, dashed, and dotted lines correspond to different Q-factor of resonance modes.[]{data-label="Hres-T"}](Fig7){width="\columnwidth"} It is interesting to note that at low frequency ($f=7.65$ GHz) the model predicts the existence of minimum in the $H_\mathrm{res}(T)$ dependence for the LT-mode. This minimum is connected with the fact that the LT-mode frequency vanishes at $H=H_\mathrm{b}$ (i.e. $H_\mathrm{res}\rightarrow H_\mathrm{b}$ when $f\rightarrow0$). Since $H_\mathrm{b}$ turns to zero at $T_\mathrm{comp}$, we could expect the minimum of $H_\mathrm{res}(T)$ at this temperature. Experimentally, however, we did not manage to detect the absorption line below $T_\mathrm{comp}$. The reason for this can be the large damping of the corresponding resonance mode. Indeed, our calculations show that the Q-factor of the LT-mode increases below $T_\mathrm{comp}$ at $f=7.65$ GHz (see Fig. \[Hres-T\]). To summarize, we achieved a reasonable agreement between the experiment and model calculations. The model describes many features of the experimental spectra and helps to identify the types of the observed resonance modes. The main discrepancy between the experiment and model arises in the vicinity of $T_\mathrm{C}^\mathrm{Gd}$ where the calculated spectra are very sensitive to magnetic parameters of the system and can be strongly influenced by structural inhomogeneities of the real superlattice. Conclusion ========== In this work we demonstrated the realization of non-collinear magnetic states and inhomogeneous magnetization dynamics in a Fe/Gd artificial layered ferrimagnet. We have shown that both static and dynamic properties of the system are described taking into account essentially non-uniform magnetization distribution inside Gd layers. Using the magneto-optical Kerr effect, we defined the regions of stability for surface and bulk twisted states of the investigated multilayer. The resulting experimental $H-T$ phase diagram is in a good agreement with calculations based on the mean-field model. Ferromagnetic resonance spectra obtained in this work reveal a complex temperature evolution with two spectral branches that can not be explained in terms of uniform magnetic precession within the superlattice. The performed theoretical simulations of magnetization dynamics in the system show that the observed resonance modes correspond to different types of inhomogeneous precession inside Gd layers. In the end we would like to emphasize that the nanostructured ferrimagnets provide possibility to study such complex magnetic phenomena under easily achievable experimental conditions: in magnetic fields up to 1 T and at microwave frequencies. The traditional ferrimagnetic crystals would require magnetic fields and frequencies that are several orders of magnitude larger. In this respect, the artificial structures can be considered as suitable model objects for experimental investigations of non-collinear magnetic phases and inhomogeneous magnetization dynamics in ferrimagnets. Acknowledgments {#acknowledgments .unnumbered} =============== The work is partially supported by the Russian Foundation for Basic Research (grants No.16-02-00061, No.18-37-00182), by the Ministry of Education and Science of the Russian Federation (grant No.14-Z-50.31.0025), and by the Basic Research Program of the Presidium of Russian Academy of Sciences. Research in Ekaterinburg was performed in terms of the State assignment of Federal Agency of Scientific Organizations of the Russian Federation (theme “Spin” No. AAAA-A18-188 020290104-2). FMR frequency of a strongly coupled layered ferrimagnet ======================================================= Let us consider two FM layers with different magnetic moments $\mu_1 > \mu_2$. We suppose that these layers are strongly AFM coupled (the exchange energy is infinity). In this case, the magnetic field **H** applied in the film plane aligns $\boldsymbol{\mu}_1$ and $\boldsymbol{\mu}_2$ parallel and antiparallel to the field direction respectively. Considering Zeeman and demagnetizing energy of both layers, the total energy of the system can be written as $$E=-\mathbf{H}\left(\boldsymbol{\mu}_1+\boldsymbol{\mu}_2\right) + 2\pi\left[ \frac{\left(\boldsymbol{\mu}_1 \cdot \mathbf{z}\right)^2}{V_1} + \frac{\left(\boldsymbol{\mu}_2 \cdot \mathbf{z}\right)^2}{V_2} \right],$$ where **z** is a unit vector normal to the film plane, $V_1$ and $V_2$ are volumes of layers. Taking into account that $-\boldsymbol{\mu}_2\upuparrows\boldsymbol{\mu}_1$, the energy expression can be rewritten in the form $$E=-\mathbf{H}\boldsymbol{\mu} + 2\pi\frac{\mu_1^2V/V_1+\mu_2^2V/V_2}{(\mu_1-\mu_2)^2} \cdot \frac{\left(\boldsymbol{\mu} \cdot \mathbf{z}\right)^2}{V},$$ where $\boldsymbol{\mu}=\boldsymbol{\mu}_1+\boldsymbol{\mu}_2$ and $V=V_1+V_2$. Now it has the form of magnetic energy for a single FM film with modified demagnetizing factor. Thus, the FMR frequency of the system is defined by modified Kittel’s formula $$\omega=\gamma_\mathrm{eff} \sqrt{H\left(H+4\pi M_\mathrm{eff} \right)}, \label{A1}$$ where $$4\pi M_\mathrm{eff}=4\pi\frac{\mu_1^2/V_1+\mu_2^2/V_2}{\mu_1-\mu_2}, \label{A2}$$ and $\gamma_\mathrm{eff}$ is a net gyromagnetic ratio of two coupled layers [@Wan1953] $$\gamma_\mathrm{eff}=\frac{\mu_1-\mu_2}{\mu_1/\gamma_1-\mu_2/\gamma_2}, \label{A3}$$ where $\gamma_1$ and $\gamma_2$ are gyromagnetic ratios of individual layers. If $\gamma_1\approx\gamma_2$, Eqs. ,  predict increasing FMR frequency when $\mu_2$ is increasing. This behaviour is opposite to the case of amorphous or crystal ferrimagnetic film when the effective demagnetizing field is defined by simple expression $4\pi M_\mathrm{eff}=4\pi(M_1-M_2)$, where $M_{1,2}$ are magnetizations of FM sublattices [@Wan1953]. 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[**The Hybrid Idea of (Energy Minimization) Optimization Methods Applied to Study Prion Protein Structures Focusing on the $\beta$2-$\alpha$2 Loop**]{}\ [Jiapu Zhang$^\text{ab*}$]{}\ [**Abstract:**]{} [*In molecular mechanics, current generation potential energy functions provide a reasonably good compromise between accuracy and effectiveness. This paper firstly reviewed several most commonly used classical potential energy functions and their optimization methods used for energy minimization. To minimize a potential energy function, about 95% efforts are spent on the Lennard-Jones potential of van der Waals interactions; we also give a detailed review on some effective computational optimization methods in the Cambridge Cluster Database to solve the problem of Lennard-Jones clusters. From the reviews, we found the hybrid idea of optimization methods is effective, necessary and efficient for solving the potential energy minimization problem and the Lennard-Jones clusters problem. An application to prion protein structures is then done by the hybrid idea. We focus on the $\beta$2-$\alpha$2 loop of prion protein structures, and we found (i) the species that has the clearly and highly ordered $\beta$2-$\alpha$2 loop usually owns a 3$_{10}$-helix in this loop, (ii) a “$\pi$-circle" Y128–F175–Y218–Y163–F175–Y169–R164–Y128(–Y162) is around the $\beta$2-$\alpha$2 loop.\ *]{} [**Key words:**]{} [*Hybrid idea; computational optimization methods; energy minimization; potential energy; Lennard-Jones clusters; application to protein structures; prion proteins; the $\beta$2-$\alpha$2 loop.*]{}\ Introduction ============ In molecular mechanics, current potential energy functions provide a reasonably good accuracy to structural data obtained from X-ray crystallography and nuclear magnetic resonance (NMR), and dynamic data obtained from spectroscopy and inelastic neutron scattering and thermodynamic data. Currently, AMBER, CHARMM, GROMOS and OPLS/AMBER are among the most commonly used classical potential energy functions [@abraham_etal_GROMACS_2014; @amber14; @locatellis2008; @paquetv2015; @stote_etal1999]. The energy, $E$, is a function of the atomic positions, $R$, of all the atoms in the system these are usually expressed in term of Cartesian coordinates. The value of the energy is calculated as a sum of bonded (or internal) terms $E_{bonded}$, which describe the bonds, angles and bond rotations in a macromolecule, and a sum of non-bonded (or external) long-range terms $E_{non-bonded}$ [@abraham_etal_GROMACS_2014; @amber14; @stote_etal1999]: [$$\begin{aligned} \label{potential_energy_formula} E_{potential} & = & E_{bonded} + E_{non-bonded} \nonumber\\ & = & (E_{bond-stretch} + E_{angle-bend} + E_{rotate-along-bond} \nonumber\\ & & (+ E_{Urey-Bradley} + E_{improper} + U_{CMAP} ) ) \nonumber\\ & & + (E_{van-der-Waals} + E_{electrostatic} + E_{hydrogen-bonds}). $$ ]{} For example, for AMBER and CHARMM force fields [@amber14; @stote_etal1999] respectively, the potential energy functions are [@amber14]: $$\begin{aligned} \label{AMBER_potential_energy_formula} E_{AMBER} &=& \sum_{bonds} k_b(b-b_0)^2 + \sum_{angles} k_{\theta}(\theta - \theta_{eq})^2 +\sum_{dihedrals} (V_n/2) [1+\cos (n\phi -\gamma )] \nonumber \\ & &+ \sum_{i=1}^{N-1} \sum_{j=i+1}^N \left [ \frac{A_{ij}}{R_{ij}^{12}} -\frac{B_{ij}}{R_{ij}^6} + \frac{q_iq_j}{\varepsilon R_{ij} } \right ],\end{aligned}$$ $$\begin{aligned} \label{CHARMM_potential_energy_formula} E_{CHARMM} &=& \sum_{bonds} k_b(b-b_0)^2 + \sum_{angles} k_{\theta} (\theta - \theta_0)^2 +\sum_{dihedrals} k_{\phi} [1+\cos (n\phi -\delta )] \nonumber \\ & & +\sum_{Urey-Bradley} k_u (u-u_0)^2 +\sum_{impropers} k(\omega -\omega_0)^2 +\sum_{\phi, \psi} V_{CMAP}(\phi, \psi ) \nonumber \\ & &+ \sum_{nonbonded} \left ( \varepsilon \left [ \left ( \frac{R_{min_{ij}}}{R_{ij}} \right )^{12} - \left ( \frac{R_{min_{ij}}}{R_{ij}} \right )^{6} \right ] + \frac{q_iq_j}{\varepsilon R_{ij}} \right ),\end{aligned}$$ where $b, \theta, \phi, R_{ij}, u, \omega, \psi$ are basic variables ($b$ is the bond length of two atoms, $\theta$ is the angle of three atoms, $\phi$ is the dihedral angle of four atoms, $R_{ij}$ is the distance between atoms $i$ and $j$), and all other mathematical symbols are constant parameters specified in various force fields respectively. This paper will discuss how to effectively and efficiently use computational optimization methods to solve the minimization problem of the potential energy in Eq. (\[potential\_energy\_formula\]), i.e. $$\begin{aligned} \label{minimize_potential_energy} min \quad E_{potential}.\end{aligned}$$ Firstly, for Eq. (\[minimize\_potential\_energy\]), we consider why we should perform energy minimization (EM). There are a number of reasons: 1. \(i) To remove nonphysical (or bad) contacts / interactions. For example, when a structure that has been solved via X-ray crystallography, in the X-ray crystallization process, the protein has to be crystallized so that the position of its constituent atoms may be distorted from their natural position and contacts with neighbors in the crystal can cause changes from the in vitro structure; consequently, bond lengths and angles may be distorted and steric clashes between atoms may occur. Missing coordinates obtained from the internal coordinate facility may be far from optimal. Additionally, when two sets of coordinates are merged (e.g., when a protein is put inside a water box) it is possible that there are steric clashes / overlap presented in the resulting coordinate set (www.charmmtutorial.org). 2. \(ii) In molecular dynamics (MD) simulations, if a starting configuration is very far from equilibrium, the forces may be excessively large and the MD simulation may fail [@abraham_etal_GROMACS_2014]. 3. \(iii) To remove all kinetic energy from the system and to reduce the thermal noise in the structures and potential energies [@abraham_etal_GROMACS_2014]. 4. \(iv) Re-minimize is needed if the system is under different conditions. For example, in quantum mechanics / molecular mechanics (QM/MM) one part of the system is modeled in QM while the rest is modeled in MD, re-minimize the system with a new condition is needed (www.charmmtutorial.org). To perform EM is to make the system reaching to a equilibration state. EM of Eq. (\[minimize\_potential\_energy\]) can be challenging, as there are many local minima that optimization algorithms might get stuck in without finding the global minima - in most cases, this is what will actually happen. Thus, how much and how far we should minimize should be well considered. Over-minimization can lead to unphysical “freezing" of the structure and move too much from its original conformation; if not minimized enough and exactly, for example, the normal mode calculation cannot arrive at the bottom of its harmonic well. However, in MD, because the output of minimization is to be used for dynamics, it is not necessary for the optimization to be fully converged but a few hundreds or tens of local optimization search are good and kind enough. To make enough local optimization, usually, after we put the protein into a solvent (e.g. waters), first we restrain the protein by holding the solute fixed with strong force and only optimize the solvent, next holding the solute heavy atoms only, and then holding the CA atoms only, and lastly remove all restraints and optimize the whole system. Secondly, for Eq. (\[minimize\_potential\_energy\]), we consider what optimization algorithms we should use. In packages of [@abraham_etal_GROMACS_2014; @bhandarkar_etal_NAMD_2012; @amber14] etc, the following three local search optimization methods have been used. 1. \(i) SD (steepest descent) method is based on the observation that if the real-valued function $E(x)$ is defined and differentiable in a neighborhood of a point $x_0$ then $E(x)$ decreases fastest if one goes from $x_0$ in the direction of the negative gradient of $E(x)$ at $x_0$. SD method is the simplest algorithm, it simply moves the coordinates in the negative direction of the gradient (hence in the direction of the force - the force is the (negative) derivative of the potential), without consideration of build ups in previous steps - this is the fastest direction making the potential energy decrease. SD is robust and easy to implement. But SD is not the most efficient especially when closer to minimum and in the vicinity of the local minimum. This is to say, SD does not generally converge to a local minimum, but it can rapidly improve the conformation when system is far from a minimum - quickly remove bad contacts and clashes. 2. \(ii) Conjugate gradient (CG) method is a method adds an orthogonal vector to the current direction of optimization search and then moves them in another direction nearly perpendicular to this vector. CG method is fast-converging and uses gradient information from previous steps. CG brings you very close to the local minimum, but performs worse far away from the minimum. CG is slower than SD in the early stages of minimization, but becomes more efficient closer to the energy minimum. In GROMACS CG cannot be used with constraints and in this case SD is efficient enough. When the forces are truncated according to the tangent direction, making it impossible to define a Lagrangian, CG method cannot be used to find the EM path. 3. \(iii) L-BFGS method is a Quasi-Newton method that approximates the reverse of Hessian matrix $[\nabla^2 E(x)]^{-1}$ of $E(x)$ for the Newton method search direction $-[\nabla^2 E(x)]^{-1} \nabla E(x)$. L-BFGS method is mostly comparable to CG method, but in some cases converges 2$\sim$3 times faster with super-linear convergent rate (because it requires significantly fewer line search steps than Polak-Ribiere CG). L-BFGS of Nocedal approximates the inverse Hessian by a fixed number of corrections from previous steps. In practice L-BFGS converges faster than CG. 4. \(iv) The combination of CG and LBFGS, so-called lbfgs-TNCG-BFGS method is a preconditioned truncated Newton CG method, it requires fewer minimization steps than Polak-Ribiere CG method and L-BFGS method, but L-BFGS can sometimes be faster in the terms of total CPU times. If a global optimization is required, approaches such as simulated annealing (SA), parallel tempering method (super SA, also called replica exchange [@zhang2011c]), Metropolis algorithms and other Monte Carlo methods, Simplex method, Nudged Elastic Band method, different deterministic methods of discrete or continuous optimization etc may be utilized. The main idea of SA refinement is to heat up the system such that the molecule of interest has enough energy to explore a wide range of configurational space and get over local optimal energy barriers. Relatively large structural rearrangements are permitted at these high temperatures. As the temperature is cooled gradually, the structural changes proceed in smaller steps, continuing to descend toward the global energy minimum. For solving Eq. (\[minimize\_potential\_energy\]), without considering $E_{hydrogen-bonds}=\sum_{i=1}^{N-1} \sum_{j=i+1}^N \left [ \frac{C_{ij}}{R_{ij}^{12}} -\frac{D_{ij}}{R_{ij}^{10}} \right ]$, about 95% of the CPU time of calculations is spent at $$\begin{aligned} \label{minimize_LJ_potential_energy} min \quad E_{van-der-Waals}=\sum_{i=1}^{N-1} \sum_{j=i+1}^N \left [ \frac{A_{ij}}{R_{ij}^{12}} -\frac{B_{ij}}{R_{ij}^6} \right ],\end{aligned}$$ where $C_{ij}, D_{ij}$ are constants. In [@locatellis2008], this problem is also called Lennard-Jones (LJ) Atomic Cluster Optimization problem (where within the field of atomic clusters only nonbonded interactions are accounted for and particles are considered to be charge-free; e.g. real clusters of metals like gold, silver, and nickel). It is very necessary to up to date review some effective and efficient computational methods for solving Eq. (\[minimize\_LJ\_potential\_energy\]). There are numerous algorithms to solve Eq. (\[minimize\_LJ\_potential\_energy\]); here we just list the ones in The Cambridge Energy Landscape Database (http://doye.chem.ox.ac.uk/jon/structures/LJ.html) which can obtain the best global structures: 1. Hoare and Pal’s work [@hoarep1971a; @hoarep1971b; @hoarep1972] may be the early most successful results on LJ problem. The idea is using build-up techniques to construct the initial solutions which are expected to represent low energy states, and using those initial solutions as starting points for a local search method to relax to the optimal solution [@hoarep1971b]. The starting seed is the regular unit tetrahedron with atoms at the vertexes, the obvious global optimal solution for $N=4$. Beginning with this tetrahedron, Hoare and Pal (1971, 1972) added one atom at a time to construct a sequence of polytetrahedral structures and at last got good results up to $N=66$ [@hoarep1971a; @hoarep1971b; @hoarep1972]. For example, for $N=5$ its globally optimal trigonal bi-pyramid (bi-tetrahedron) structure is gotten by adding an atom at the tetrahedral capping position over a triangular face; following the bi-tetrahedron structure, the optimal structure of $N=6$ is tri-tetrahedron (another known optimal structure for $N=6$ is octahedron (using tetrahedral capping over triangular faces and half-octahedral capping over square faces), which is not a polytetrahedron); for $N=7$ its best structure constructed is the pentagonal bi-pyramid, a structure with a five-fold axis of symmetry. Many computer science data structure procedures such as greedy forward growth operator and reverse greedy operator can make the build-up technique work well. The application of methods of studying noncrystalline clusters to the study of “spherical" face centred cubic (fcc) microcrystallites was described in [@hoarep1972]. In [@hoarep1971a] the chief geometrical features of the clustering of small numbers of interacting particles were described. 2. The data structure of Northby [@northby1987] in finding the good starting solution is the lattice based structure. The lattice structures consist of an icosahedral core and particular combinations of surface lattice points. A class of icosahedral packings was by constructed in [@mackay1962] adding successively larger icosahedral shells in layers around a core central atom; this icosahedral lattice can be described as 20 slightly flattened tetrahedrally shaped fcc units with 12 vertices on a sphere centered at the core atom. Atoms within each triangular face are placed in staggered rows in a two dimensional hexagonal close-packed arrangement. Each atom in the interior of a face in a given shell is a tetrahedral capping position relative to three atoms in the underlying shell. Northby (1987) relaxed the structure of [@mackay1962] to get his IC and FC multilayer icosahedral lattice structures [@northby1987]. The IC lattice can be referred to the FORTRAN code in [@xue1994b]; it consists of all those sites which will comprise the outer shell of the next complete Mackay [@mackay1962] icosahedron. FC lattice is a slight modification of IC lattice in that its outer shell maintains icosahedral symmetry and consists of points at the icosahedral vertices and the stacking fault positions of the outer IC shell. Basing on the IC and FC lattices, Northy (1987) gave his algorithm first finding a set of lattice local minimizers and then relaxing those lattice minimizers by performing continuous minimization starting with those lattice minimizers [@northby1987]. The algorithm was summarized as Algorithm 1 and Algorithm 2 of [@xue1994b]. 3. The great majority of the best known solutions of Northy [@northby1987] are icosahedral in character. The hybridization of global search and local search methods, usually, is more effective to solve the large scale problem than the global search method or local search method working alone. Catching those two ideas, Romero et al. (1999) combined a genetic algorithm with a stochastic search procedure on icosahedrally derived lattices [@romerobg1999; @barrongrs1999]. The structures of the optimal solutions gotten in [@romerobg1999] are either icosahedral or decahedral in character. The best results of [@wolf1998] for N = 82, 84, 86, 88, 92, 93, 94, 95, 96, 99, 100 were gotten by using a genetic algorithm alone. Deaven et al. (1996) also using the genetic algorithm got the optimal value known for the magic number $N=$ 88 [@deaventmh1996]. 4. The successful works to improve Northby’s results in [@northby1987] were mainly done by Xue [@xue1994a; @xue1994b], Leary [@leary1997], and Doye et al. [@doyewb1995; @doyew1995]. 1. Xue (1994a) introduced a modified version of the Northby algorithm [@xue1994a]. He showed that in some cases the relaxation of the outer shell lattice local minimizer with a worse potential function value may lead to a local minimizer with a better value. In Northby’s algorithm [@northby1987] the lattice search part is a discrete optimization local search procedure, which makes a lattice move to its neighboring lattice with O($N^{\frac{5}{3}}$) time complexity. In [@xue1994a] Xue (1994a) introduced a simple storage data structure to reduce the time complexity to O($N^{\frac{2}{3}}$) per move; and then used a two-level simulated annealing algorithm within the supercomputer CM-5 to be able to solve fastly the LJ problem with sizes as large as 100,000 atoms. In [@xue1994b] by employing AVL trees [@horowitzs1990] data structure Xue (1994b) furthermore reduced the time complexity to O($\log N$ ) if NN (nearest neighbor) potential function is used. Xue (1994b) relaxed every lattice local minimizer found instead of relaxing only those lattice local minimizers with best known potential function value by a powerful Truncated Newton local search method [@xue1994b], and at last got the best results known for $N=$ 65, 66, 134, 200, 300. 2. Leary (1997) gave a successful Big Bang Algorithm [@leary1997] for getting the best values known of $N =$ 69, 78, 88, 107, 113, 115. In [@leary1997], the FCC lattice structure is discussed and its connections are made with the macrocluster problem. It is also concluded in [@leary1997] that almost all known exceptions to global optimality of the well-known Northby multilayer icosahedral conformations for microclusters are shown to be minor variants of that geometry. The Big Bang Algorithm contains 3 steps: Step 1 is an initial solution generating procedure which randomly generates each coordinate of the initial solution with the independently normal distribution; Step 2 is to generate the new neighborhood solution by discrete-typed fixed step steepest descent method, which is repeated until no further progress is made; Step 3 is to relax the best solution gotten in Step 2 by a continuous optimization method–conjugate gradient method. 3. Doye et al. (1995) investigated the structures of clusters by mapping the structure of the global minimum as a function of both cluster size and the range of the pair potential which is appropriate to the clusters of diatomic molecule, C$_{60}$ molecule, and the ones between them both [@doyew1995]. For the larger clusters the structure of the global minimum changes from icosahedral to decahedral to fcc as the range is decreased [@doyew1995]. In [@doyewb1995], Doye et al. (1995) predicted the growth sequences for small decahedral and fcc clusters by maximisation of the number of NN contacts. 5. Calvo et al. (2001) gave some results on quantum LJ Clusters in the use of Monte Carlo methods [@calvodw2001]. 6. Xiang et al. (2004a) presented an efficient method based on lattice construction and the genetic algorithm and got global minima for $N =$ 310$\sim$561 [@xiangjcs2004] In 2004, Xiang et al. (2004b) continued to present global minima for $N =$ 562$\sim$1000 [@xiangccs2004]. 7. Barron-Romero (2005) found the best solutions for $N =$ 542–3, 546–8 in the use of a modified peeling greedy search method [@barronromero2005]. 8. Takeuchi (2006) found best solutions for $N =$ 506, 521, 537–8 and 541 by a clever and efficient method “using two operators: one modifies a cluster configuration by moving atoms to the most stable positions on the surface of a cluster and the other gives a perturbation on a cluster configuration by moving atoms near the center of mass of a cluster" [@takeuchi2006]. 9. Lai et al. (2011a) found best solutions for $N =$ 533 and 536 using the dynamic lattice searching method with two-phase local search and interior operation [@laixh2011a; @laixh2011b; @yexh2011]. 10. Algorithms to get the structures at the magic numbers $N=$ 17, 23, 24, 72, 88 (the exceptions to [@romerobg1999]): 1. Freeman et al. (1985) presented the best value for $N=$ 17 when the thermodynamic properties of argon clusters were studied by a combination of classical and quantum Monte Carlo methods [@freemand1985]. The poly-icosahedral growth of Farges et al. (1985) starts from a 13-atom primitive icosahedron containing a central atom and 12 surface atoms [@farges_etal1985]. On each one of the five tetrahedral sites, surrounding a particular vertex, a new atom is added and finally a sixth atom is placed on top to create a pentagonal cap. In this way a 19-atom structure being made of double interpenetrating icosahedra, which is a 13-atom icosahedra sharing 9 atoms, is obtained; i.e., for three pentagonal bipyramids each one shares an apex with its nearest neighbour. In this way a 23-atom model consisting of three interpenetrating icosahedra is gotten for the best value known. 2. Wille (1987) used the SA method yielding low-lying energy states whose distribution depends on the cooling rate to find the best solution known for $N=$ 24 [@wille1987]. 3. Coleman et al. (1997) proposed a build-up process to construct the optimal solution structures. The HOC (half octahedral cap) structure of the optimal solution for $N=$ 72 is found by a prototype algorithm designed using the anisotropic effective energy simualted annealing method at each build-up stage ([@colemansw1997]). 4. Wales & Doye (1997) gave the lowest values known for $N =$ 192, 201 [@walesd1997]. Their method is so-called basin-hopping method, in which first the transformed function $\tilde{f} (x) = \min \{ f(x) \}$ was defined and performed starting from $x$ by the PR conjugate gradient method and then the energy landscape for the function $\tilde{f} (x)$ was explored using a canonical Monte Carlo simulation. 5. Leary (2000) has developed techniques for moving along sequences of local minima with decreasing energies to arrive at good candidates for global optima and got the best value known on $N=$ 185. Now we have the outline of some successful optimization methods used to solve Eq.s (\[minimize\_potential\_energy\])$\sim$(\[minimize\_LJ\_potential\_energy\]). We have found the hybrid idea of optimization methods was not emphasized very much (especially for solving Eq. (\[minimize\_LJ\_potential\_energy\])). Thus, in Section 2 of this paper we will emphasize the hybrid idea of optimization methods by introducing our own hybrid methods used to solve Eq.s (\[minimize\_potential\_energy\])$\sim$(\[minimize\_LJ\_potential\_energy\]). Section 3 will present our recent results of applying the hybrid idea of SD and CG and SD again to do EM of some NMR and X-ray prion protein structures in the PDB Bank (www.rcsb.org); interesting findings will be reported in this Section. Why we choose prion proteins in this study is due to prions effect humans and almost all animals for a major public health concern (e.g. milks and meats we daily drink and eat). At last, in Section 4, we give a concluding remark on the effective and efficient hybrid idea of optimization methods. The hybrid idea and some hybrid optimization methods ==================================================== In this Section, we use how to construct molecular structures of prion amyloid fibrils at AGAAAAGA segment as an example to illuminate the hybrid idea and some hybrid optimization methods we designed. Neurodegenerative amyloid diseases such as Alzheimer’s, Parkinson’s and Hungtington’s all featured amyloid fibrils. Prions also cause a number of neurodegenerative diseases too. All these amyloid fibrils in 3-dimensional quaternary structure have 8 classes of steric zippers, with strong van der Waals interactions between $\beta$-sheets and hydrogen bonds between $\beta$-strands. Currently, there is no structural information about prion AGAAAAGA amyloid fibrils because of unstable, noncrystalline and insoluble nature of this region, though numerous laboratory experimental results have confirmed this region owning an amyloid fibril forming property (initially described in 1992 by Stanley B. Prusiner’s group). We also did accurate computer calculations on this region and confirmed the amyloid fibril property in this palindrome region [@zhanghwwz2012; @zhangz2013]. In [@zhang2011a], we constructed three models, model 1 belongs to Class 7 (antiparallel, face=back, up-up) and models 2–3 belong to Class 1 (parallel, face-to-face, up-up) of steric zippers. The models were firstly optimized by SD and then followed by CG. SD has fast convergence but it is slow when close to minimums. CG is efficient but its gradient RMS and GMAX gradient do not have a nice convergence. When the models could not be optimized furthermore, we employed standard SA method (that simulates the annealing process of crystal materials with Monte Carlo property). After SA, we refined the models by SD and then CG again. SA is a global search optimization method [@zhang2014] that can make local optimal jump out of / escape from the local trap. We found the refinement results in a loss of potential energy nearly the same magnitude as that of SA; this implies to us SA is very necessary and very effective in our molecular modeling. Numerical results show to us the hybrid is very necessary, effective and efficient. When the gradient or its generalizations of the target/objective function $E(x)$ are very complex in form or they are not known, derivative-free methods benefit optimization problems. In [@zhangsw2011], we introduced derivative-free discrete gradient (DG) method [@bagirovkm2014] into the derivative-free global search SA optimization method or genetic algorithms (GAs, which simulate the process natural competitive selection, crossover, and mutation of species), and designed hybrid methods SADG, GADG. In implementation, the hybrids of DG + SADG / GADG + DG were used, and at last SD+CG + SA + SD+CG of Amber package [@amber14] were used to refine the models. We found the hybrids work very well, and more precise best solutions for$N =$ 39, 40, 42, 48, 55, 75, 76, and 97 were found and their figures show that their structures are more stable than the ones currently best solutions known. We also found the hybrid of evolutionary computations with simulate annealing SA-SAES($\mu +\lambda $), SA-SACEP perform better than evolutionary computations or SA work alone [@zhang2011b]. Canonical dual theory in some sense is the hybrid of the primal and the dual. In [@zhanggy2011], we solved the dual problem and then got the solutions for the primal problem. We found the refinement using AMBER package is not necessary. This implies to us the hybrid of primal and dual in canonical dual theory is good enough and effective. As said in Section 1, in some cases, CG cannot be used to find the EM path; this point will also be shown in next Section (see Tab. \[energy\_variations\_during\_energy\_minimizations\]). Thus, in [@zhanghwwz2012], we specially studied and implement the LBFGS method designed by us and then hybridize it with the LBFGS method of AMBER package. We found the hybrid is very necessary and effective. By our numerical experiences shown above, the hybrid idea is very necessary, effective and efficient for some hybrid optimization methods in known packages or designed by us. In next Section, we will apply the hybrid idea to do some practical works for some important prion protein NMR and X-ray structures deposited in the PDB Bank. An application to prion protein structures, focusing on the $\beta$2-$\alpha$2 loop =================================================================================== Before we use the structure taken from PDB Bank, usually we need to relax it, in order to remove bad contacts and also fix up hydrogen positions. Fairly short local optimization is sufficient to refine and relax the structure. We will use SD-CG-SD local optimization methods. In SD, its search direction is a n-dimensional search and its step-length search is a 1-dimensional search. In CG, the search is usually in a 2-dimensional subspace and conjugacy is a good property only associated with exact line search [@sunz2001]. Using the hybrid of SD and CG is also in order to remove all these (dimensional) unbalances. In our EM here, the free package Swiss-PdbViewer 4.1.0 (spdbv.vital-it.ch) that has been developed for 20 years is used, we set 3000 steps for SD, then 3000 steps for CG, and then 3000 steps for SD again, Bonds, Angles, Torsions, Improper, Non-bonded and Electrostatic are considered, 12.000 $\mathring{\text{A}}$ is chosen for the Cutoff, stop SD or CG when delta E between two steps is below 0.005 kJ/mol, and stop SD or CG when Force acting on any atom is below 1.000. We found, for the research of prion proteins, the S2-H2 loop (and its interactions with the C-terminal of H3) is a focus [@biljan_etal2012a; @biljan_etal2012b; @biljanigrzpl2011; @calzolai_etal2000; @christen_etal2009; @christen_etal2012; @dambergercphw2011; @gossertblfw2005; @ile_etal2010; @leeahkssy2010; @wenlxpyhl2010; @wenlyxpxl2010; @kong_etal2013; @perezdw2010; @perezw2008; @sweeting_etal2013; @zahngvsw2003; @zhangszss2000; @kurt_etal2014a; @kurt_etal2014b; @huangc2015]. All prion protein structures have high similarity in three $\alpha$-helices (H1, H2, H3) and two $\beta$-strands (S1, S2), but there is a great difference just at this S2-H2 loop: 1. \(i) structure with disordered S2-H2 loop: 1. mousePrP (1AG2.pdb at 25 $\mathring{\text{}}$C), 2. humanPrP (1QLX.pdb), 3. bovinePrP (1DWY.pdb), 4. SyrianHamsterPrP (1B10.pdb), 5. dogPrP (1XYK.pdb) (- resist to prion infection), 6. catPrP (1XYJ.pdb), 7. sheepPrP (1UW3.pdb), 8. mousePrP\[N174T\] (1Y15.pdb), 9. humanPrP\[Q212P\]-M129 (2KUN.pdb), 10. humanPrP-M129 (1QM1.pdb), 11. rabbitPrP\[S173N\] (2JOH.pdb), 12. rabbitPrP\[I214V\] (2JOM.pdb), 13. rabbitPrP\[S170N\] (4HLS.pdb), 14. rabbitPrP\[S174N\] (4HMM.pdb), 15. rabbitPrP\[S170N,S174N\] (4HMR.pdb), 2. \(ii) structure with highly and clearly ordered S2-H2 loop: 1. mousePrP (2L39.pdb at 37 $\mathring{\text{}}$C), 2. mousePrP\[V166A\] (2KFO.pdb), 3. mousePrP\[D167S\] (2KU5.pdb at 20 $\mathring{\text{}}$C), 4. mousePrP\[D167S,N173K\] (2KU6.pdb), 5. mousePrP\[Y169G\] (2L1D.pdb), 6. mousePrP\[Y169A\] (2L40.pdb), 7. mousePrP\[S170N\] (2K5O.pdb), 8. mousePrP\[S170N,N174T\] (1Y16.pdb), 9. mousePrP\[F175A\] (2L1E.pdb), 10. mousePrP\[Y225A,Y226A\] (2KFM.pdb), 11. mousePrP\[Y169A,Y225A,Y226A\] (2L1K.pdb at 20 $\mathring{\text{}}$C), 12. elkPrP (1XYW.pdb), 13. pigPrP (1XYQ.pdb), 14. BankVolePrP (2K56.pdb), 15. TammarWallabyPrP (2KFL.pdb), 16. rabbitPrP (2FJ3.pdb, 3O79.pdb), 17. horsePrP (2KU4.pdb), where elk and Bank Vole can be infected by prions though they have a highly and clearly ordered S2-H2 loop, and the codes in the brackets are the PDB codes in the PDB Bank. For all these NMR and X-ray structures we did SD-CG-SD relaxation and the variations of the EMs are listed in Tab. \[energy\_variations\_during\_energy\_minimizations\]. From Tab. \[energy\_variations\_during\_energy\_minimizations\], we can see the energy decreases from SD to CG and from CG to SD. For mousePrP\[Y169A\], CG is not working well, but it adjusts the SD methods so that it make SD-CG-SD work very well in the second round. \[energy\_variations\_during\_energy\_minimizations\] \[hydorgen\_bonds\_in\_loop\_optimized\] \[salt\_bridges\_in\_loop\_optimized\] ![[]{data-label="charge_distributions"}](Fig02_ClearLoop_noClearLoop.eps){width="5.2in"} ![[]{data-label="surface_electrostatic_distributions"}](ordered_disordered.eps){width="5.2in"} After the SD-CG-SD relaxation of all the structures, now these optimized structures can be used to obtain some helpful structural information \[e.g. (i) hydrogen bonds (see Tab. \[hydorgen\_bonds\_in\_loop\_optimized\]), (ii) electrostatic charge distributions on the protein structure surface (see Fig. \[charge\_distributions\] and Tab. \[cliques\_of\_positively\_charged\_residues\]), (iii) salt bridges (see Tab. \[salt\_bridges\_in\_loop\_optimized\]), and (iv) $\pi$-$\pi$-stacking and $\pi$-cations (see Tab. \[pi\_stacks\]); here why we consider the information of (i)$\sim$(iv) is due to “the performance of protein biological function is driven by a number of non-covalent interactions such as hydrogen bonding, ionic interactions, Van der Waals forces, and hydrophobic packing" (en.wikipedia.org/wiki/Protein\_structure)\] at the S2-H2 loop, in order to furthermore understand the S2-H2 loop: why some species has a clearly and highly ordered S2-H2 loop and why some species just has a disordered S2-H2 loop. (i) Using the VMD package (www.ks.uiuc.edu/Research/vmd/), with Tab. \[hydorgen\_bonds\_in\_loop\_optimized\], we may observed that the species owning the disordered S2-H2 loop usually does not have a 3$_{10}$-helix in the S2-H2 loop (except for sheepPrP, rabbitPrP\[S170N\]-X-ray, rabbitPrP\[S174N\]-X-ray, and rabbitPrP\[S170N,S174N\]-X-ray), but the species that has the clearly and highly ordered S2-H2 loop usually owns a 3$_{10}$-helix, constructed by the following hydrogen bond(s) respectively: 1. V166–Y169 - mousePrP at 37 $\mathring{\text{}}$C, mousePrP\[D167S,N173K\], mousePrP\[F175A\], rabbitPrP-X-ray, 2. R164–Q168 - mousePrP\[V166A\], 3. P165–Q168 - mousePrP\[Y225A,Y226A\], elkPrP, BankVolePrP, 4. P165–Q168, I166–Y169 - TammarWallabyPrP. \(ii) Seeing Fig. \[charge\_distributions\] and Tab. \[cliques\_of\_positively\_charged\_residues\], we may know that at the S2-H2 loop it is mainly covered by the electrical cloud of negatively (in red color) charged residues \[except for mousePrP\[D167S,N173K\], rabbitPrP-NMR and horsePrP etc (Fig. \[surface\_electrostatic\_distributions\])\], with positively (in blue) charged residues R164 (for all species) and H177 (for mousePrP, humanPrP-M129, mousePrP\[D167S,N173K\] only) at the N-terminal end and C-terminal end of S2-H2 loop respectively (we found there is a salt bridge R164–D178 linking this loop of rabbitPrP-NMR, rabbitPrP-X-ray, horsePrP, dogPrP, elkPrP and buffaloPrP for long time MD simulations [@zhang2010; @zhang2011d; @zhang2011e; @zhangl2011; @zhangz2014; @zhangwz2015; @zhang2015a; @zhang2011f]). From Tab. \[cliques\_of\_positively\_charged\_residues\], we might see that the negatively charged S2-H2 loop might have long distance nuclear overhauser effect (NOE) interactions with the positively charged residues such as K204, R208, K/R220, R227, R228, R229, and R230 at the C-terminal end of H3. (iii) The salt bridges in Tab. \[salt\_bridges\_in\_loop\_optimized\] might be not very strong and will be quickly broken in a long time MD simulations [@zhang2011e]. (iv) Lastly, we present some bioinformatics of $\pi$-$\pi$-stacking and $\pi$-cations (one kind of van der Waals interactions) at the S2-H2 loop. Seeing Tab. \[pi\_stacks\], we may know at S2-H2 loop and its contacts with the C-terminal end of H3 there are the following $\pi$-$\pi$-stacks Y169–F175, F175–Y218, Y163–Y218, and the following $\pi$-cations R164–Y169, R164–Y128, which clearly contribute to the clearly and highly ordered S2-H2 loop structures [@zhang2015b]. For buffaloPrP, we found another two $\pi$-stackings: Y163–F175–Y128 [@zhang2015b; @zhang2015a]. Thus, for PrPs, we found an interesting “$\pi$-circle" Y128–F175–Y218–Y163–F175–Y169–R164–Y128(–Y162) around the S2-H2 loop. \[cliques\_of\_positively\_charged\_residues\] \[pi\_stacks\] A concluding remark =================== In optimization, especially for solving large scale or complex or both optimization problems, the hybrid of optimization (local search or global search) methods is very necessary, and very effective and efficient for solving optimization problems. In molecular mechanics, to optimize its potential energy, even just one part of it e.g. the Lennard-Jones potential, is still a challenge to optimization methods; the hybrid idea is very helpful and useful. An application to prion protein structures is then done by the hybrid idea. Focusing on the $\beta$2-$\alpha$2 loop of prion protein structures, we found (i) the species that has the clearly and highly ordered $\beta$2-$\alpha$2 loop usually owns a 3$_{10}$-helix in this loop, (ii) a “$\pi$-circle" Y128–F175–Y218–Y163–F175–Y169–R164–Y128(–Y162) is around the $\beta$2-$\alpha$2 loop. In conclusion, this paper proposes a hybrid idea of optimization methods to efficiently solve the potential energy minimization problem and the LJ clusters problem. We first reviewed several most commonly used classical potential energy functions and their optimization methods used for energy minimization, as well as some effective computational optimization methods used to solve the problem of Lennard-Jones clusters. In addition, we applied this hybrid idea to construct molecular structures of prion amyloid fibrils at AGAAAAGA segment, by which we provided the additional insight for the $\beta$2-$\alpha$2 loop of prion protein structures. 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--- abstract: 'The existence of stationary solutions of the Einstein-Vlasov-Maxwell system which are axially symmetric but not spherically symmetric is proven by means of the implicit function theorem on Banach spaces. The proof relies on the methods of [@akr14] where a similar result is obtained for uncharged particles. Among the solutions constructed in this article there are rotating and non-rotating ones. Static solutions exhibit an electric but no magnetic field. In the case of rotating solutions, in addition to the electric field, a purely poloidal magnetic field is induced by the particle current. The existence of toroidal components of the magnetic field turns out to be not possible in this setting.' author: - Maximilian Thaller title: Rotating Clouds of Charged Vlasov Matter in General Relativity --- Introduction ============ The Einstein-Vlasov-Maxwell system (EVM-system) describes an ensemble of charged particles whose motion is governed by gravity and an electro-magnetic field but which do not interact via collisions. In the framework of General Relativity gravity is described by the curvature of the manifold, the space-time, on which the particles live. Both the space-time curvature and the electro-magnetic field are generated collectively by the particles themselves. In contrast to the Einstein-Vlasov system, which only takes into account gravity, particles described by the EVM-system are not freely falling, i.e. their trajectories are not geodesics. In this article the existence of stationary, rotating solutions of the EVM-system is proven by means of the implicit function theorem. The proof is a generalisation of [@akr14], where the existence of rotating, stationary solutions of the Einstein-Vlasov system with uncharged particles is proved, to the case where the particles are charged and hence induce an electro-magnetic field. In the context of kinetic theory this method has already been used in [@r00] to show the existence of stationary, rotating solutions of the Vlasov-Poisson system. The idea of this method is to introduce a parameter $\lambda$ to the system which can “turn on” rotation and to perturb the system around a spherically symmetric, static solution without rotation. To this end one considers a functional $\mathfrak F : \mathcal X \times [-\delta,\delta]\to \mathcal X$, where $\mathcal X$ is a suitable function space which will contain the solution and $[-\delta, \delta]$ is the interval in which the parameter $\lambda$ will lie. The operator is constructed such that if $\mathfrak F(\zeta,\lambda) = 0$ then $\zeta$ is a collection of functions which constitute a solution of the Vlasov-Poisson system with the parameter $\lambda$. The solution $\zeta_0$, corresponding to $\lambda=0$, is known and we have $\mathfrak F(\zeta_0,0)=0$. The main part of the work consists in showing that the implicit function theorem can be applied. Then it follows that to each $\lambda\in (-\delta,\delta)$ there exists $\zeta_\lambda \in \mathcal X$ such that $\mathfrak F(\zeta_\lambda, \lambda)=0$. This collection $\zeta_\lambda$ of functions consequently solves the Vlasov-Poisson system and this solution is axially symmetric but not spherically symmetric. It is in the nature of this method that the obtained rotating solutions have small overall angular momentum. In [@akr11] a similar method with a different set up has been used to show the existence of axially but not spherically symmetric, static solutions of the Einstein-Vlasov system. In this context it was used that the Vlasov-Poisson system is the non-relativistic limit of the Einstein-Vlasov system, in the sense that a solution of the Einstein-Vlasov system converges to a solution of the Vlasov-Poisson system if the speed of light $c$ goes to infinity. So besides $\lambda$, the speed of light $c$ has been introduced to the system as a second parameter. Perturbing off a spherically symmetric, static solution of the Vlasov-Poisson system in those two parameters $\lambda$ and $c$ yields an axially but not spherically symmetric, static solution of the Einstein-Vlasov system. The deviation from spherical symmetry is small but by a scaling argument the solution can be made fully relativistic, i.e. $c=1$. In [@akr14] further technical insights made it possible to include rotation into the picture. Lichtenstein developed a method based on the implicit function theorem to construct rotating fluid bodies [@l18; @l33] in Newtonian gravity. This approach has later been reformulated in a modern mathematical language [@h94] and improved [@h95]. In [@abs08; @abs09] the authors use an implicit function argument to construct axially symmetric static and rotating elastic bodies in Einstein gravity. In a series of papers of which the last one is [@cdkkmr18] the authors construct stationary solutions of the Einstein equations with negative cosmological constant without any symmetries. Many different matter models can be included, such as a scalar field, Maxwell, or Yang-Mills. Space-times with rotating, charged matter configurations have been studied in the literature by analytical and numerical means, see e.g. [@bbgn95; @cpl00; @fr12]. An important motivation for these studies is the modelling of rotating stars or neutron stars with a magnetic field. In these articles the matter is modelled as a perfect fluid and different shapes of the magnetic field can be observed depending on the assumptions on the fluid, like an equation of state or conductivity properties. For example rotating solutions with no poloidal magnetic field can be constructed, cf. [@fr12]. These works can serve as a source of intuition for the study of rotating clouds of Vlasov matter. There is however an important difference. When studying a perfect fluid, the Einstein-Euler system (which describes a space-time containing matter of the type of a perfect fluid) has to be supplemented by an equation of state which captures the physical properties of the fluid under consideration. Depending on the choice of the equation of state, different matter configurations and different electro-magnetic fields can be constructed. For Vlasov matter however there is much less variety in the physical properties of the solutions that can be obtained. The basic assumptions on the particles’ behaviour and how the energy and the angular momentum is distributed among the particles (this is sometimes referred to as a [*microscopic equation of state*]{}) already determines the macroscopic character of the solutions. It turns out that rotating solutions of the EVM-system must have a poloidal magnetic field but no toroidal magnetic field. We briefly mention that in the non-relativistic setting a variety of different axially symmetric solutions can be constructed explicitly, cf. for example [@galactic_dynamics]. A well studied class of these solutions are disk solutions which serve as models for disk shaped galaxies and which are used to study some physical properties of these galaxies. The so called Morgan & Morgan disk solutions, introduced in [@mm69], are important in this context. In [@rap12] the authors construct comparable axially symmetric solutions in Newtonian gravity with general relativistic corrections. Surprisingly these general relativistic corrections account for changes of the solutions far from the galaxy core – a region where it was expected that Newtonian gravity describes the physics well and general relativistic effects do not play a significant role. This observation adds to the motivation of studying axially symmetric configurations of collisionless particles in the fully general relativistic picture. The present article generalises [@akr14] to the case of charged particles, i.e.  solutions of the EVM-system are constructed by perturbing off a non-trivial, spherically symmetric, static solution of the Vlasov-Poisson system. It is assumed that the particles are charged with a particle charge $q$, i.e. an electro-magnetic field is included into the framework. A priori this can be done in two different ways. Either one considers $q$ as a third (a priori small) parameter which “turns on” charge. In this case one still perturbs off a spherically symmetric, static, uncharged solution of the Vlasov-Poisson system. The other way is to use the fact that in the non-relativistic limit the Maxwell equations reduce to the Poisson equation as well and one perturbs around a charged solution of the Vlasov-Poisson system. It turns out that the first approach is easier from a technical point of view since the operator $\mathfrak F$ that the implicit function theorem will be applied to is changed only insignificantly by the included Maxwell equations. However, the result would be restricted to small particle charge parameters $q$. In the second approach arbitrary values $0 \leq q < m_p$ of the particle charge parameter can be treated, where $m_p$ denotes the mass of the particles. In this case the operator $\mathfrak F$ has additional terms. In this article the second approach is presented. In an axially symmetric, static setting the EVM-system reduces to a system of coupled, non-linear Poisson equations in different dimensions and a first order PDE. The solution of this system consists in a collection of functions which we denote $\zeta$. For the construction of a well defined solution operator $\mathfrak F$ one has to assure for that the source terms of these Poisson equations are sufficiently regular. However, after the variable substitution $A_\varphi = \varrho^2 a$ one obtains for the $\varphi$-component of the electro-magnetic four potential $A$ the equation $$\label{intro_poisson_a} \Delta_5 a = \frac{2}{1+h} \frac{a \partial_\varrho h}{\varrho} + \frac{2}{4\pi^2 c^2} \frac{a \partial_\varrho \nu}{\varrho} + \dots.$$ On the right hand side only some a priori problematic terms are written out explicitly. The functions $\nu$ and $h$ are part of the collection $\zeta$ of solution functions of the EVM-system. These terms are a priori problematic because they are singular at the axis $\varrho = 0$. Looking a bit closer one notices that the right member of equation (\[intro\_poisson\_a\]) is not singular if $h$ and $\nu$ are axially symmetric functions of a certain regularity. However, by dividing by $\varrho$ one “looses derivatives”. For this reason the function space $\mathcal X$ has to be chosen such that the individual functions of the collection $\zeta$ have a hierarchy in regularity. For equation (\[intro\_poisson\_a\]) for example one needs that $h$ and $\nu$ are of higher regularity than $a$. This article is a generalisation of [@akr14] and the proof follows the same scheme. Including charge into the framework does not only increase the number of equations in the system but it also increases significantly the number of terms in each equation. Some of these terms require some care in the analysis but clearly not all of them. Still all required properties of the system have to be checked term by term. In order to make the presentation more concise this article resorts more to shorthands and schematic or symbolic notation than [@akr11; @akr14]. In the next section the EVM-system will be introduced. Then, in Section \[sect\_result\], the result of this article will be stated and an outline of the proof will be given. The rest of the article is devoted to the introduction of the technical setup, the definition of the relevant objects, i.e. function spaces and solution operators, and the proofs of important properties of these operators. The Einstein-Vlasov-Maxwell system ================================== A solution of the Einstein-Vlasov-Maxwell system (EVM-system) for particles with mass $m_p\geq 0$ and charge $0 \leq q < 1$ is a Lorentzian metric $g\in T^*\mathscr M \otimes T^*\mathscr M$ defined on a four dimensional manifold $\mathscr M$, a particle distribution function $f \in C^1(T\mathscr M; \mathbb R_+)$, defined on the tangent bundle of $\mathscr M$, and an electro-magnetic field tensor $F\in\Lambda^2(T \mathscr M)$ such that the EVM-system, $$\begin{aligned} G_{\mu\nu} &= \frac{8\pi}{c^4} \left(T_{\mu\nu} + \tau_{\mu\nu}\right), \label{eq_einstein} \\ T_{\mu\nu} &= g_{\mu\alpha} g_{\nu\beta} \frac{c}{m_p} \int_{\mathscr P_x} f(x,p) p^\alpha p^\beta \, \mathrm{dvol}_{\mathscr P_x}, \label{eq_em_tensor} \\ \tau_{\mu\nu} &= \frac{1}{4\pi} \left(-\frac 1 4 g_{\mu\nu} F_{\alpha\beta} F^{\alpha\beta} + F_{\nu\alpha} F_{\mu}^{\;\;\alpha} \right), \label{el_em_tensor} \\ \mathfrak T (f) &=0, \label{eq_vlasov} \\ \mathrm dF &= 0, \label{maxwell_eq_1} \\ \nabla_\alpha F^{\alpha\beta} &= - 4\pi qJ^\beta,\quad J^\beta =\frac 1 c \int_{\mathscr P_x} f(x,p)p^\beta \mathrm{dvol}_{\mathscr P_x}, \label{maxwell_eq_2} \end{aligned}$$ is satisfied. Here $G_{\mu\nu}$ is the Einstein tensor and we choose units such that $G=1$ ($G$ is the gravitational constant) but we leave $c$ as parameter in the system. We give a brief explanation of the involved quantities, consult however e.g. [@sz14] for a more detailed introduction to the EVM-system. The particle distribution function $f=f(x,p)$ describes the particle number density at a certain point in $x \in \mathscr M$ with a certain four-momentum $p \in T_x\mathscr M$. The particle number can be obtained via integration. The quantity $m_p$, defined by the relation $$\label{m_s_r} g_{\mu\nu}(x) p^\mu p^\nu = -c^2 m_p^2, \qquad x \in \mathscr M, p \in T_x\mathscr M$$ is interpreted as the particles’ rest mass. It can be shown that it stays constant along the characteristic curves of the Vlasov equation (\[eq\_vlasov\]). Consequently the particle distribution function $f$ describing an ensemble of particles where all particles have the same rest mass $m_p$ can be assumed to be supported on the mass shell $\mathscr P_{m_p}$, a seven dimensional submanifold of $T\mathscr M$ which is defined to be $$\mathscr P_{m_p} = \{(x,p)\in T\mathscr M \,:\, g_{\mu\nu}(x)p^\mu p^\nu = -c^2 m_p^2, \; p\,\mathrm{is\,future\,pointing}\}.$$ In the remainder of this article we assume $m_p=1$ for all particles, and we denote the corresponding mass shell simply by $\mathscr P$. The volume form $\mathrm{dvol}_{\mathscr P_x}$ on the mass shell fibre $\mathscr P_x$ over $x\in \mathscr M$ is given by $$\mathrm{dvol}_{\mathscr P_x} = \frac{\sqrt{|\det(g_{\mu\nu}(x))|}}{-p_0}\, \mathrm d p^1 \wedge \mathrm dp^2 \wedge \mathrm dp^3,$$ and the transport operator $\mathfrak T$ is given by $$\label{def_transport_op} \mathfrak T = p^\mu \partial_\mu + \left(q F^\gamma{}_{\mu} \, p^\mu - \Gamma^\gamma_{\alpha\beta} p^\alpha p^\beta\right) \partial_{p^\gamma}.$$ It is tangent to any mass shell $\mathscr P$ [@sz14]. Assume that we have a solution $(g,f,F)$ of the EVM system and that on $\mathscr M$ we have coordinates $t$, $x^1$, $x^2$, $x^3$, where $t$ is the time coordinate. Assume further that $\partial_t$ is a Killing field. Then the solution is asymptotically flat if the boundary conditions $$\label{bc_as_flat} \lim_{|x|\to \infty} g = \eta, \quad \lim_{|x|\to \infty} f = 0, \quad \lim_{|x|\to \infty} F = 0$$ are satisfied, where $\eta$ denotes the Minkowski metric. The result {#sect_result} ========== In this article we prove the following result. \[main\_theorem\] There exist asymptotically flat, stationary solutions $(g, f, F) \in (T^*\mathscr M \otimes T^*\mathscr M) \times C_c^1 (\mathscr P; \mathbb R_+) \times \Lambda^2(\mathscr M)$ of the EVM-system (\[eq\_einstein\])–(\[maxwell\_eq\_2\]) with particle charge parameters $q\in [0,1)$, which are axially symmetric but not spherically symmetric. Such a solution has no toroidal magnetic field and it has a poloidal magnetic field if and only if the solution is not static, i.e. rotating. The proof which is given at this place is rather an outline of the poof, the technical details are given in the subsequent sections. The proof follows the same structure as in [@akr14] where the existence of stationary, rotating, axially symmetric solutions is proved for uncharged particles. Each step is however a bit more involved and some arguments have to be formulated differently due to the additional Maxwell equations. We comment on the modifications in the respective sections. [*Step 1: Elimination of the Vlasov equation.*]{} For the particle distribution function we use the ansatz $f(x,p) = \phi(E(x,p))\psi(\lambda, L(x,p))$, see (\[ansatz\_f\]) below. So the particle distribution depends only on the particle energy $E(x,p)$ and the $z$-component of the angular momentum $L(x,p)$, see the definitions (\[formula\_e\]) and (\[formula\_l\]) below. Since the quantities $E$ and $L$ are conserved along its characteristics the Vlasov equation is automatically satisfied for such an ansatz, cf. Section \[sect\_char\] below. Furthermore, we introduce a parameter $\lambda$ which “turns on” the dependency of $f$ on $L$. This means that if $\lambda = 0$ then $\psi \equiv 1$, i.e. for each value of the $z$-component of the angular momentum there are equally many particles. [*Step 2: Reduction of the remaining system.*]{} First we express the EVM-system (\[eq\_einstein\])–(\[maxwell\_eq\_2\]) in cylindrical coordinates. The assumptions that the solution is asymptotically flat, axially symmetric, and time independent yield simplifications of the system of equations. We call this simplified system the [*reduced EVM-system*]{}, cf. Definition \[def\_red\_evm\] below, and it is stated in Section \[sect\_red\_sys\], equations (\[final\_eq\_nu\])–(\[bc\_center\]), below, where any value of $c\in (0,\infty)$ is admitted. The solution of the reduced EVM-system is determined by the collection $\zeta = (\nu, h, \xi, \omega, A_t, a) \in \mathcal X$ of six functions, defined in a suitably chosen function space $\mathcal X$ (defined in Section \[sect\_function\_space\] below). Proposition \[prop\_equivalent\] below states that a solution of the reduced EVM-system with any parameter $c$ can be converted into an axially symmetric, stationary solution of the EVM-system with the parameter $c=1$. [*Step 3: Introduction of the solution operator $\mathfrak F$.*]{} A solution of the reduced EVM-system with parameters $\gamma:= c^{-2}, \lambda \in [0,1) \times (-1,1)$ is then obtained as perturbation of a spherically symmetric solution of the Vlasov-Poisson system. This spherically symmetric solution of the Vlasov-Poisson system we denote by $\zeta_0\in\mathcal X$. To this end in Section \[sect\_def\_f\] an operator $\mathfrak F: \mathcal X \times [0,1) \times (-1,1) \to \mathcal X$ with the following properties is defined. Firstly, a collection of functions $\zeta\in\mathcal X$ is a solution of the reduced EVM-system with parameters $\gamma$, $\lambda$ if and only if $\mathfrak F[\zeta; \gamma, \lambda] = 0$. (The “if”-direction is essential.) Secondly, $\mathfrak F[\zeta_0; 0, 0] = 0$. In Section \[sect\_well\_defined\] we show that this operator is well defined. The mentioned properties are shown in Proposition \[prop\_consistent\] and Lemma \[lem\_n\_zero\] below. [*Step 4: Application of the implicit function theorem.*]{} The aim is to apply the implicit function theorem on Banach spaces, cf. for example [@d85 Theorem 15.1]. This theorem implies the existence of $\delta>0$ such that there exists a mapping $\mathfrak Z : [0,\delta) \times (-\delta, \delta) \to \mathcal X$ such that for all $(\gamma,\lambda)\in [0,\delta)\times (-\delta,\delta)$ we have $$\mathfrak F(\mathfrak Z(\gamma,\lambda); \gamma, \lambda) = 0,$$ i.e. $\mathfrak Z(\gamma,\lambda)$ is a solution of the reduced EVM-system with parameters $\gamma, \lambda$. This solution $\mathfrak Z(\gamma,\lambda)$ then gives rise to a solution of the EVM-system with the asserted properties, by Proposition \[prop\_equivalent\]. The implicit function theorem can be applied in this way if the operator $\mathfrak F$ is continuous at $(\zeta_0; 0, 0)$, if its Fréchet derivative $\mathfrak L := D\mathfrak F[\zeta_0;0,0]: \mathcal X \to \mathcal X$ at the point $(\zeta_0; 0, 0)\in \mathcal X \times [0,\delta) \times (-\delta,\delta)$ exists and is continuous, and if this Fréchet derivative $\mathfrak L$ is a bijection. These properties are established in Section \[sect\_frechet\_dir\]. Proposition \[prop\_appli\] below contains the details how it is made sure that the boundary conditions for an asymptotically flat solutions are satisfied. [*Step 5: Characterisation of the electro-magnetic field.*]{} The assertion that the solution comprises a poloidal magnetic field if and only if the solution is rotating follows from the structure of the reduced EVM-system, see Remark \[rem\_mag\_rot\]. For the assertion that there is no toroidal magnetic field, see Lemma \[lem\_no\_tor\]. Axial symmetry {#sect_axial} ============== Let $x^i$, $i=1,\dots,n$ be coordinates on $\mathbb R^n$. A function $f:\mathbb R^n \to \mathbb R$ is axially symmetric around the $x^n$-axis if and only if there exists a function $\hat f: [0,\infty) \times \mathbb R \to \mathbb R$ such that $$f\left(x^1,\dots,x^n\right) = \hat f\left(\varrho(x^1, \dots, x^{n-1}), x^n\right),$$ where $$\label{def_rho} \varrho(x^1, \dots, x^{n-1}) := \sqrt{\left(x^1\right)^2 + \dots + \left(x^{n-1}\right)^2}.$$ By abuse of notation, we will use the same symbol for the original function on $\mathbb R^2$, $\hat f$ in this example, and the induced axially symmetric functions $f$ on $\mathbb R^n$ for different dimensions $n$. \[rem\_even\] At some places in the analysis presented in this article it will be useful to view an axially symmetric function $f:\mathbb R^n \to \mathbb R$ as a function in $\varrho$ and $z$ defined on $\mathbb R^2$, by extending it as even function to negative values of $\varrho$. The obtained function on $\mathbb R^2$ then has the same regularity as the axially symmetric function on $\mathbb R^n$. We now introduce a coordinate gauge and the functions in terms of which we will formulate the reduced EVM-system. Consider the four dimensional manifold $\mathscr M$ which is assumed to be homeomorphic to $\mathbb R^4$ and which is equipped with the cylindrical coordinates $t$, $\varrho$, $z$, $\varphi$. A stationary Lorentzian metric is characterised by the four time independent, axially symmetric functions $\nu,\mu,\omega:\mathscr M \to \mathbb R$ and $H: \mathscr M \to \mathbb R_+$. It can be written in the form $$\label{ansatz_metric} g = -c^2 e^{\frac{2\nu(\varrho, z)}{c^2}}\mathrm dt^2 + e^{2\mu(\varrho, z)}\mathrm d\varrho^2 + e^{2\mu(\varrho, z)}\mathrm dz^2 + \varrho^2H(\varrho, z)^2 e^{-\frac{2\nu(\varrho, z)}{c^2}} \left(\mathrm d\varphi - \omega(\varrho, z) \mathrm dt\right)^2,$$ cf. [@b72] for details. The electro-magnetic field tensor $F$ is given as the exterior derivative of the electro-magnetic four potential $A\in\Lambda^1(\mathscr M)$, i.e. $F = dA$. With respect to the coordinate co-basis of $t$, $\varrho$, $z$, $\varphi$ the electro-magnetic potential $A$ takes the form $$A = A_t \mathrm dt + A_\varrho \mathrm d\varrho + A_\varphi \mathrm d\varphi + A_z \mathrm dz.$$ We assume that all components are time independent and axially symmetric. In terms of the electro-magnetic field tensor $F$ the electric field $E\in \Lambda^1(\mathscr M)$ and the magnetic field $\mathscr B \in \Lambda^1(\mathscr M)$ are defined as follows. The electric field $E$ is defined by the splitting $F = E \wedge \mathrm dt + B$, where the two form $B$ includes no term with $\mathrm dt$. The magnetic field is defined by the splitting $\star F = \mathscr E - \mathscr B \wedge \mathrm dt$, where $\star : \Lambda^2(\mathscr M) \to \Lambda^2(\mathscr M)$ is the Hodge star operator and $\mathscr E$ is a two-form with no $\mathrm dt$-term. Cf. [@f12] for details. Define $\beta := \partial_z A_{\varrho} - \partial_\varrho A_{z}$. Then a calculation yields that the toroidal magnetic field component $\mathscr B_\varphi$ takes the form $$\label{tor_mag} \mathscr B_\varphi = 2c e^{-2\mu} \varrho H \beta,$$ and the poloidal magnetic field components, $\mathscr B_\varrho$ and $\mathscr B_z$, contain only the $t$- and the $\varphi$-component of $A$. In fact a calculation yields $$\begin{aligned} \mathscr B_\varrho &=- \frac{2 e^{- 2\nu/c^2}}{c \varrho H} \left( c^2 e^{4\nu/c^2} A_{\varphi,z} - \varrho^2 H^2 \omega (A_{t,z} + \omega A_{\varphi,z}) \right), \\ \mathscr B_z &= \frac{2 e^{- 2\nu/c^2}}{c \varrho H} \left( c^2 e^{4\nu/c^2} A_{\varphi,\varrho} - \varrho^2 H^2 \omega (A_{t,\varrho} + \omega A_{\varphi,\varrho}) \right).\end{aligned}$$ Next we introduce the parameter $\gamma = \frac{1}{c^2}$ and the orthonormal frame $e_a = e_a{}^\alpha \partial_\alpha$, $\alpha = t,\varrho, z, \varphi$, where the non-trivial matrix elements are $$\label{frame_matrix} e_0{}^t = e^{-\gamma \nu}, \quad e_0{}^\varphi = e^{-\gamma \nu} \omega, \quad e_1{}^\varrho = e^{-\mu}, \quad e_2{}^z = e^{-\mu}, \quad e_3{}^\varphi = \frac{e^{\gamma \nu}}{\varrho H}.$$ The corresponding co-frame reads $\alpha^a = e^a{}_\alpha \mathrm dx^\alpha$, where $(e^a{}_\alpha) = (e_a{}^\alpha)^{-1}$ (the inverse matrix), and via the relation $p^\mu \partial_\mu = v^\mu e_\mu$ this frame introduces the new momentum variables $v^0, v^1, v^2, v^3$, given by $$\label{def_frame} v^0 = e^{\gamma\nu} p^t, \quad v^1 = e^\mu p^\varrho, \quad v^2 = e^\mu p^z, \quad v^3 = \varrho H e^{-\gamma\nu} \left(p^\varphi - \omega p^t\right).$$ In the remainder of this article we work with the coordinates $$ \label{the_coords} t \in \mathbb R, \quad \varrho \in [0, \infty), \quad \varphi \in [0,2\pi), \quad z \in \mathbb R, \quad (v^0, v^1, v^2, v^3) \in \mathbb R^4$$ on the tangent bundle $T\mathscr M$. In these frame coordinates the mass shell relation (\[m\_s\_r\]) becomes $$-c^2 = -c^2 \left(v^0\right)^2 + \left(v^1\right)^2 + \left(v^2\right)^2 + \left(v^3\right)^2$$ and on $\mathscr P$ we consequently have $$\label{mass_shell_frame} v^0 = \sqrt{1+\gamma |v|^2}, \quad \mathrm{where}\, |v| = \sqrt{\left(v^1\right)^2 + \left(v^2\right)^2 + \left(v^3\right)^2}.$$ The method of characteristics {#sect_char} ============================= The Vlasov equation (\[eq\_vlasov\]) can be dealt with by the method of characteristics which is now described. \[lem\_conserved\_quan\] The quantities $E$ and $L$, defined on the tangent bundle $T \mathscr M$, by $$\begin{aligned} L &:= \varrho H e^{ -\gamma\nu} v^3 - q A_\varphi, \label{formula_l} \\ E &:= \frac{e^{\gamma\nu} v^0 - 1}{\gamma} + \omega \varrho H e^{ -\gamma\nu} v^3 + q A_t, \label{formula_e}\end{aligned}$$ are conserved along the characteristic curves of the Vlasov equation, i.e. $$\mathfrak TE = 0, \quad \mathfrak TL = 0.$$ The assertion of this lemma can be shown via a direct calculation and it is moved to the appendix. Unlike the uncharged case, in the charged case the characteristic curves of the Vlasov equations are not the lifts of the geodesics to $T\mathscr M$. Consequently the conserved quantities cannot be obtained by $g(X, p)$, where $X$ is a Killing vector field and $p$ is the canonical momentum. However, this structure can still be recognised in the present case. If we define $$\begin{aligned} \tilde E &:= -g(\partial_t, p), \label{tilde_e} \\ \tilde L &:= g(\partial_\varphi, p), \label{tilde_l}\end{aligned}$$ it turns out that the quantities $E$ and $L$ can be obtained from $\tilde E$ and $\tilde L$ by taking into account a suitable correction due to the electro-magnetic field. We have $$E = \tilde E - \frac{1}{\gamma} + qA_t, \qquad L = \tilde L - qA_\varphi.$$ \[cor\_char\] Every function $f:\mathscr P \to \mathbb R_+$ which can be expressed as $$\label{product_strucutre} f(t,\varrho,\varphi,z,v^0,v^1,v^2,v^3) = \phi(E) \tilde \psi(L)$$ with some functions $\phi, \tilde \psi \in C^1(\mathbb R ; \mathbb R_+)$, solves the Vlasov equation (\[eq\_vlasov\]) and is axially symmetric and time independent. Since $\mathfrak T E = \mathfrak T L = 0$ we have by the chain rule $\mathfrak Tf = 0$. The remaining asserted properties of $f$ are inherited from the metric functions $\nu$, $\mu$, $H$, and $\omega$. A more general statement than Corollary \[cor\_char\] is true, for ansatz functions that do not have the product structure (\[product\_strucutre\]). The corollary is however stated this way because in this article only ansatz functions of the form (\[product\_strucutre\]) are considered. From now on we work with the ansatz $$\label{ansatz_f} f(x,v) = \phi\left(E \right)\psi(\lambda, L),$$ where $E$ and $L$ are the conserved quantities, given in (\[formula\_e\]) and (\[formula\_l\]), respectively, and $\lambda \in [0,1]$ is the parameter which “turns on” anisotropy in momentum of the particle distribution. The functions $\phi$ and $\psi$ are assumed to fulfil the assumptions listed below. For an integrable function $U$ and $\phi \in C^1(\mathbb R;\,\mathbb R_+)$, where $\mathrm{supp}(\phi)\subset (-\infty, E_0]$ for some $0\leq E_0 < \infty$, we define $$\begin{aligned} \rho_U(r) := \int_{\mathbb R_v^3} \phi\left(\frac{|v|^2}{2} + U(r)\right) \,\mathrm dv^1 \mathrm dv^2 \mathrm dv^3, \label{def_rho_u} \\ \alpha_U(r) := \int_{\mathbb R_v^3} \phi'\left(\frac{|v|^2}{2} + U(r)\right) \,\mathrm dv^1 \mathrm dv^2 \mathrm dv^3. \label{def_alpha_u}\end{aligned}$$ We assume that the functions $\phi$ and $\psi$ in (\[ansatz\_f\]) have the following properties. 1. $\phi\in C^2(\mathbb R)$ and there exists $E_0 > 0$ such that $\phi(E)=0$ for $E\geq E_0$ and $\phi(E) > 0$ for $E < E_0$. \[cond\_phi\] 2. The ansatz $f(x,v)=\phi\left(\frac 12 |v|^2 + U_N(x)\right), x, v \in\mathbb R^3$, leads to a compactly supported, spherically symmetric steady state $(f_N, U_N)$ of the Vlasov-Poisson system for particles with mass $1-q^2$, i.e., there exists a solution $U_N \in C^2(\mathbb R^3)$, of the equation $\Delta U_N = 4\pi (1-q^2) \rho_{N}(x)$, $U_N(0)=0$, where we used the shorthand $\rho_N := \rho_{U_N}$. This solution is spherically symmetric, $U_N(x) = U_N(|x|)$, and the support of $\rho_N \in C_c^2(\mathbb R^3)$ is the closed ball $\overline B_{R_N}(0)$ where $U_N(R_N)=E_0$ and $U_N(r) < E_0$ for $0\leq r < R_N < \infty$, and $U_N(r) > E_0$ for $r> R_N$. 3. We have $6+ 4\pi (1-q^2) r^2 \alpha_N(r) > 0$ for all $r\in[0,\infty)$. 4. $\psi\in C_c^\infty (\mathbb R^2)$ is compactly supported, $\psi \geq 0$, $\partial_L \psi(\lambda, 0) = 0$ for $\lambda \in \mathbb R$, and $\psi(0, L) = 1$ on an open neighbourhood of the set $\{L = L_N(x,v) \, | \, (x,v) \in \mathrm{supp}(f_N)\}$, where $L_N := \varrho v^3$ is the $z$-component of the Newtonian angular momentum. There exist ansatz functions $\phi \in C_c^2(\mathbb R)$ and $\psi\in C_c^\infty((-1/2, 1/2) \times \mathbb R)$ satisfying the upper conditions. Consider the polytropes $\phi(E) = [E_0-E]_+^k$ for $k\in[2,7/2)$. Condition (1) is clearly satisfied. Condition (2) is also satisfied, cf. [@bfh86; @rr00]. By the same proof as for [@akr11 Lemma 7.1] it can be shown that the third condition is satisfied for polytropes with exponent $k$ sufficiently close to $7/2$. To this end one uses the equation $\Delta U_N = 4\pi (1-q^2) \rho_N$ instead of $\Delta U_N = 4\pi \rho_N$. Then merely the constant $4\pi$ has to be replaced by $4\pi(1-q^2)$ in the proof of [@akr11 Lemma 7.1]. It is essential that $1-q^2 > 0$, the precise value is however irrelevant for the argument. The reduced system of equations {#sect_red_sys} =============================== Before the reduced system of equations is presented some notation and shorthands shall be introduced. Partial derivatives $\partial_\varrho \nu$, $\partial_z A_\varphi$, etc. will be denoted as $\nu_{,\varrho}$, $A_{\varphi,z}$, etc. We define the functions $\xi, h, a$ by the following changes of variables: $$\begin{aligned} \xi &= \mu + \gamma \nu, \\ H &= 1 + h, \\ A_\varphi &= \varrho^2 a.\end{aligned}$$ Further, we call $(\nu, h, \xi, \omega, A_t, a)$ the [*solution functions*]{} and in the remainder of this article we will use the shorthand $$\zeta := (\nu, h, \xi, \omega, A_t, a).$$ We do not include the components $A_\varrho$ and $A_z$ of the four-potential $A$ into the solution functions $\zeta$ since it will turn out that in the current setting they must vanish everywhere, cf. Lemma \[lem\_no\_tor\] below. In [@akr11; @akr14], where the existence of axially symmetric solutions of the Einstein-Vlasov system with uncharged particles is proven, a reduced system of equations is considered as well. The reduced EVM-system presented below coincides with the reduced system in [@akr14] if the charge parameter $q$ is set to zero. When the Maxwell equations are added to the framework not only the number of equations increases but also the number of terms in the Einstein equations increases by a multiple. For this reason, below, we are going to introduce [*source functions*]{} to collect these terms. This allows to present the reduced system in a compact way and also facilitates the presentation of the subsequent analysis. Moreover, we will introduce [*matter functions*]{} which basically consist in combinations of components $T_{\mu\nu}$ of the Vlasov part of the energy momentum tensor, as in [@akr14]. In the subsequent analysis it will be necessary to show different properties of the matter functions and the source functions, like regularity with respect to the coordinates $\varrho$ and $z$, decay properties, symmetries, or Fréchet differentiability with respect to the solution functions $\zeta$. This means that at some occasions the source functions and the matter functions have to be seen as functions of $\varrho$ and $z$ which are parameterised by the solution functions. At other occasions they have to be seen as functions which take both the coordinate $\varrho$ and the solution functions $\zeta$ (and their derivatives) as arguments. Moreover, for the analysis of the matter functions several different integral representations will be necessary. In order to give a clear presentation we deem it favourable to resort to symbolic notation in a larger extent than in [@akr14]. Now we define the matter functions. These matter functions depend on the solution functions $\nu$, $h$, $\xi$, $\omega$, $A_t$, $a$. At different places in this article we want to see them either as functions taking the evaluated solution functions as argument ($M_i^{(\gamma,\lambda)}$ below) or as families of functions which are parameterised by the solution functions ($\mathfrak M_i[\zeta; \gamma, \lambda]$ below) and which only depend on $(\varrho,z)$. We define $$\begin{aligned} M_1^{(\gamma,\lambda)}(\varrho, \zeta) &:= 4\pi e^{2(\xi-\gamma\nu)} \int_{\mathbb R_v^3} \phi(E)\psi(\lambda, L) \frac{1 + 2 \gamma |v|^2}{\sqrt{1+ \gamma |v|^2}} \, \mathrm d^3v, \label{def_hat_m_1} \\ M_2^{(\gamma,\lambda)}(\varrho, \zeta) &:= 8\pi \gamma^2 (1+h) e^{2(\xi-\gamma\nu)} \int_{\mathbb R_v^3} \phi(E)\psi(\lambda, L) \frac{(v^1)^2+(v^2)^2}{\sqrt{1+\gamma |v|^2}} \, \mathrm d^3v, \label{def_hat_m_2}\\ %M_3^{(\gamma,\lambda)}(\varrho, \zeta) &:= 0, \label{def_hat_m_3} \\ M_4^{(\gamma,\lambda)}(\varrho, \zeta) &:= -\frac{16\pi \gamma}{\varrho(1+h)} e^{2\xi - 4\gamma\nu}\int_{\mathbb R_v^3} \phi(E)\psi(\lambda, L) v^3\,\mathrm d^3v, \label{def_hat_m_4} \\ M_5^{(\gamma,\lambda)}(\varrho, \zeta) &:= 4\pi q e^{2\xi - 3\gamma\nu} \int_{\mathbb R_v^3} \phi(E)\psi(\lambda, L) \left(e^{2\gamma\nu} + \frac{\gamma \varrho (1+h) \omega v^3}{\sqrt{1+\gamma |v|^2}}\right) \, \mathrm d^3v, \label{def_hat_m_5} \\ M_6^{(\gamma,\lambda)}(\varrho, \zeta) &:= -\frac{4\pi q \gamma (1+h)}{\varrho} e^{2\xi-3\gamma\nu} \int_{\mathbb R_v^3} \phi(E)\psi(\lambda, L) \frac{v^3}{\sqrt{1+\gamma |v|^2}}\, \mathrm d^3v, \label{def_hat_m_6}\end{aligned}$$ where $\mathrm d^3v = \mathrm d v^1 \mathrm dv^2 \mathrm dv^3$ and $E$ and $L$ are seen as functions of $\varrho$, $\zeta$ and $v^1,v^2, v^3$, according to the formulas (\[formula\_e\]) and (\[formula\_l\]) whereas $\xi$, $\nu$, $h$, $\varrho$ are seen as variables. Moreover let $$\label{def_all_ms} \mathfrak M_i[\zeta; \gamma, \lambda](\varrho, z) := M_i^{(\gamma,\lambda)}(\varrho, \zeta(\varrho, z)), \quad i = 1, 2, 4, 5, 6.$$ We remark that if $\psi$ is even in $L$, then $$\label{m4_m6_0} M_4^{(\gamma,\lambda)}(\varrho,\zeta) = M_6^{(\gamma,\lambda)}(\varrho,\zeta) = 0, \quad \mathrm{if}\;\omega=0.$$ This follows immediately since the integrand in $M_4^{(\gamma,\lambda)}$ and $M_6^{(\gamma,\lambda)}$ is antisymmetric in $v^3$. In the same spirit as the matter functions we define for $\gamma\in [0,1]$ the source functions $g_i^{(\gamma)} : \mathbb R^{19} \to \mathbb R$, $i=1, \dots, 6$. The source functions take the solution functions $\zeta_i$, $i=1,\dots,6$ and their derivatives $\zeta_{i,\varrho}$ and $\zeta_{i,z}$, $i=1,\dots,6$ as separate arguments, i.e. they are considered as independent variables. We denote $$\begin{aligned} \zeta_{,\varrho} = (\zeta_{1,\varrho}, \dots, \zeta_{6,\varrho}) = (\partial_{\varrho} \zeta_1, \dots, \partial_\varrho \zeta_6), \quad \zeta_{,z} = (\zeta_{1,z}, \dots, \zeta_{6,z}) = (\partial_{z} \zeta_1, \dots, \partial_z \zeta_6).\end{aligned}$$ Then the source functions are defined to be $$\begin{aligned} g_1^{(\gamma)}(\varrho, \zeta, \zeta_{,\varrho}, \zeta_{,z}) &:= -\frac{ h_{,\varrho} \nu_{,\varrho} + h_{,z} \nu_{,z}}{1+ h} + \frac{\varrho^2}{2}(1+ h)^2 e^{-4\gamma \nu} \left( \omega_{,\varrho}^2 + \omega_{,z}^2\right) \label{for_g_1} \\ &\quad -\gamma^2 e^{-2\gamma\nu} \left( ( A_{t,\varrho} + 2\omega\varrho a + \omega \varrho^2 a_{,\varrho})^2 + ( A_{t,z} + \omega \varrho^2 a_{,z})^2 \right) \nonumber \\ &\quad - \gamma \frac{e^{2\gamma \nu}}{(1+ h)^2} \left((2 a+\varrho a_{,\varrho})^2 + \varrho^2 a_{,z}^2 \right), \nonumber\end{aligned}$$ $$\begin{aligned} &g_3^{(\gamma)}(\varrho, \zeta, \zeta_{,\varrho}, \zeta_{,z}) := \left(\left(1 + \partial_{\varrho}(\varrho h)\right)^2 + \varrho^2 h_{,z}^2\right)^{-1} \label{for_g_3} \\ &\times \bigg( (1+\partial_{\varrho} (\varrho h)) \nonumber \\ &\qquad \times \left[ \frac{\varrho}{2}( h_{\varrho\varrho} - h_{zz}) + h_{,\varrho} -\gamma^2 (1+ h) \varrho ( \nu_{,z}^2 - \nu_{,\varrho}^2) - \gamma\varrho^3 (1+ h)^3 e^{-4\gamma \nu} ( \omega_{,\varrho}^2 - \omega_{,z}^2) \right] \nonumber \\ &\qquad + \varrho h_{,z} \left[\partial_{,\varrho} (\varrho h_{,z}) + 2 \gamma^2 (1+ h) \varrho \nu_{,\varrho} \nu_{,z} + \frac 12 \gamma e^{-4 \gamma \nu} \varrho^3 (1+ h)^3 \omega_{,\varrho} \omega_{,z}\right] \nonumber \\ &\qquad - 2\gamma^3 e^{-2 \gamma \nu} (1+ h) \varrho^2 h_{,z} \left(\left( A_{t,\varrho} + 2\varrho \omega a + \varrho^2 \omega a_{,\varrho} \right) \left( A_{t,z} + \varrho^2 \omega a_{,z}\right) \right) \nonumber \\ &\qquad + \gamma^3 e^{-2 \nu \gamma} (1+ h) \varrho (1+\partial_{,\varrho}(\varrho h)) \left(\left( A_{t,z}+ \varrho^2 \omega a_{,z} \right)^2 - \left( A_{t,\varrho} + 2\varrho \omega a + \varrho^2 \omega a_{,\varrho}\right)^2 \right) \nonumber \\ &\qquad + \gamma^2 \varrho^3 e^{2 \gamma \nu} \left(2\frac{ h_{,z}}{1 + h} (2 a a_{,z} + \varrho a_{,\varrho} a_{,z}) + \left(1 + \varrho \frac{h_{,\varrho}}{1+h} \right) \left(a_{,\varrho}^2 - a_{,z}^2\right)\right) \bigg), \nonumber\end{aligned}$$ $$\begin{aligned} g_4^{(\gamma)}(\varrho, \zeta, \zeta_{,\varrho}, \zeta_{,z}) &:= - \left(3\frac{ h_{,\varrho} \omega_{,\varrho} + h_{,z} \omega_{,z}}{1+ h} - 4\gamma (\nu_{,\varrho} \omega_{,\varrho} + \nu_{,z} \omega_{,z}) \right) \label{for_g_4} \\ &\quad \;\, + 4 \gamma^2 \frac{e^{2\gamma\nu}}{(1+ h)^2} \Big( \frac{2}{\varrho} A_{t,\varrho} a + A_{t,\varrho} a_{,\varrho} + A_{t,z} a_{,z} + 4 \omega a^2 \nonumber \\ &\hspace{5cm} + 2\omega \varrho a a_{,\varrho} + \omega \varrho^2 a_{,\varrho}^2 + \omega \varrho^2 a_{,z}^2\Big), \nonumber \end{aligned}$$ $$\begin{aligned} &g_5^{(\gamma)}(\varrho, \zeta, \zeta_{,\varrho}, \zeta_{,z}) \label{for_g_5} \\ &\;\; := 2\gamma \left(\nu_{,\varrho} A_{t,\varrho} + \nu_{,z} A_{t,z}\right) +4\gamma\omega \left(2\varrho \nu_{,\varrho} a + \varrho^2 \nu_{,\varrho} A_{,\varrho} + \varrho^2 \nu_{,z} a_{,z}\right) \nonumber \\ &\qquad - \frac{h_{,\varrho} A_{t,\varrho} + h_{,z} A_{t,z}}{1+ h} -2\omega \frac{2\varrho h_{,\varrho} a + \varrho^2 h_{,\varrho} a_{,\varrho} + \varrho^2 h_{,z} a_{,z}}{1+ h} \nonumber \\ &\qquad - \left(2\varrho a \omega_{,\varrho} + \varrho^2 a_{,\varrho} \omega_{,\varrho} + \varrho^2 a_{,z} \omega_{,z}\right) -2 (2\omega a + \varrho \omega a_{,\varrho}) \nonumber \\ &\qquad -\gamma \omega \varrho^2 (1+ h)^2 e^{-4\gamma\nu} \left( \omega_{,\varrho}( A_{t,\varrho} + 2 \varrho \omega a + \varrho^2 \omega a_{,\varrho}) + \omega_{,z}( A_{t,z} + \varrho^2 \omega a_{,z})\right), \nonumber\end{aligned}$$ $$\begin{aligned} g_6^{(\gamma)}(\varrho, \zeta, \zeta_{,\varrho}, \zeta_{,z}) &:= \gamma (1+ h)^2 e^{-4\gamma \nu} \left( \omega_{,\varrho} ( A_{t,\varrho} + 2\varrho \omega a + \varrho^2 \omega a_{,\varrho}) + \omega_{,z} ( A_{t,z} + \varrho^2 \omega a_{,z})\right) \label{for_g_6}\\ &\qquad + \frac{\frac 2 \varrho h_{,\varrho} a + h_{,\varrho} a_{,\varrho} + h_{,z} a_{,z}}{1+ h} + \frac{\gamma}{4\pi^2} \left(\frac{2}{\varrho} \nu_{,\varrho} a + \nu_{,\varrho} a_{,\varrho} + \nu_{,z} a_{,z}\right). \nonumber\end{aligned}$$ Furthermore we define $$\label{def_mathfrak_g} \mathfrak g_i[\zeta; \gamma](\varrho,z) := g_i^{(\gamma)}(\varrho, \zeta(\varrho,z), \zeta_{,\varrho}(\varrho,z), \zeta_{,z}(\varrho,z)), \quad i = 1,3,\dots, 6$$ as families of source functions which depend only on $\varrho$ and $z$ but which are parameterised by the solution functions $\zeta$. Moreover we define the operators $$\Delta_n := \partial_{\varrho\varrho} + \frac{n-2}{\varrho} \partial_\varrho + \partial_{zz}, \quad n=3,4,5.$$ As the notation indicates, these operators correspond to the Laplace operator for axially symmetric functions in three, four, and five dimensions. We consider the following boundary value problem, consisting in the Einstein equations, $$\begin{aligned} \Delta_3 \nu(\varrho,z) &= \mathfrak g_1[\zeta; \gamma](\varrho,z) + \mathfrak M_1[\zeta; \gamma, \lambda](\varrho,z) \label{final_eq_nu} \\ \Delta_4 h(\varrho,z) &= \mathfrak M_2[\zeta; \gamma, \lambda](\varrho,z), \label{eq_b} \\ \xi_{,\varrho}(\varrho,z) &= \mathfrak g_3[\zeta; \gamma](\varrho,z), \label{eq_xi} \\ \Delta_5 \omega(\varrho,z) &= \mathfrak g_4[\zeta; \gamma](\varrho,z) + \mathfrak M_4[\zeta; \gamma, \lambda](\varrho,z), \label{eq_omega}\end{aligned}$$ poloidal Maxwell equations, $$\begin{aligned} \Delta_3 A_t(\varrho,z) &= \mathfrak g_5[\zeta; \gamma](\varrho,z) + \mathfrak M_5[\zeta; \gamma, \lambda](\varrho,z), \label{eq_a0}\\ \Delta_5 a(\varrho, z) &= \mathfrak g_6[\zeta; \gamma](\varrho,z) + \mathfrak M_6[\zeta; \gamma, \lambda](\varrho,z), \label{final_eq_a}\end{aligned}$$ toroidal Maxwell equations, $$\begin{aligned} \left(\frac{h_{,z}}{1+h} + 2(\gamma \nu_{,z}-\xi_{,z})\right)\left(A_{z,\varrho} - A_{\varrho,z}\right) + \partial_z \left(A_{z,\varrho} - A_{\varrho,z}\right) &= 0, \label{tor_maxwell_1} \\ \left(\frac 1 \varrho + \frac{h_{,\varrho}}{1+h} + 2(\gamma \nu_{,\varrho} - \xi_{,\varrho})\right)\left(A_{\varrho,z} - A_{z,\varrho}\right) + \partial_\varrho \left(A_{\varrho,z} - A_{z,\varrho}\right) &= 0, \label{tor_maxwell_2}\end{aligned}$$ and the boundary conditions, $$\label{bc_infinity} \lim_{|(\varrho,z)| \to \infty} (|\nu| + |\xi| + |\omega| + |h| + |A_t| + |A_\varrho| + |A_z| + |a|)(\varrho,z) = 0$$ at spatial infinity and $$\begin{aligned} \xi(0,z) &= \ln(1+h(0,z)), \qquad z\in\mathbb R \label{bc_center} \end{aligned}$$ at the centre of symmetry. The connection between equations (\[final\_eq\_nu\])–(\[bc\_center\]) and the EVM-system is addressed in Proposition \[prop\_equivalent\] below. \[rem\_mag\_rot\] If the ansatz function $f = \phi(E)\psi(\lambda, L)$ for the matter distribution satisfies in addition to the conditions listed on page that $\psi$ is even in $L$, then the equations (\[final\_eq\_nu\])–(\[bc\_center\]) possess solutions such that $\omega \equiv a \equiv 0$, i.e. static solutions without rotation. Note that the corresponding matter functions vanish, cf. (\[m4\_m6\_0\]). So the equations exhibit the physical connection between rotation and the magnetic field. Intuitively one would think of this connection in the following way. If there is no overall rotation, i.e. $\omega \equiv 0$, then there is consequently no electric current and no magnetic field is induced. If there is rotation, however, the moving charges induce a poloidal magnetic field. Inspecting equations (\[eq\_omega\]) and (\[final\_eq\_a\]), we see that $\omega\equiv a \equiv 0$ is a solution, whereas it is not possible that only one of these functions is zero everywhere because they appear mutually as source terms in the equation of each other. \[lem\_no\_tor\] For each continuous solution of (\[final\_eq\_nu\])–(\[bc\_center\]) the combination $\beta = A_{\varrho,z} - A_{z,\varrho}$ vanishes everywhere, i.e. there is no toroidal magnetic field. (The toroidal component of the magnetic field is given in (\[tor\_mag\])). If we consider the quantity $\beta = A_{\varrho,z} - A_{z,\varrho}$, then equations (\[tor\_maxwell\_1\])–(\[tor\_maxwell\_2\]) read $\nabla \beta = -\beta \nabla\left( \ln(\varrho(1+h)) + 2(\gamma \nu + \xi)\right)$. This admits the solution $$\beta = C e^{-(\ln(\varrho(1+h))+2(\gamma\nu-\xi))}.$$ Since $-(\ln(\varrho(1+h))+2(\gamma\nu-\xi)) \to \infty$, as $\varrho \to 0$ we deduce that $C=0$ since otherwise the toroidal component of the magnetic field would diverge as $\varrho \to 0$, hence the assumption of a regular $\{ \varrho = 0 \}$-axis would be violated. Taking account for the fact that the magnetic field is purely poloidal, we exclude the corresponding equations (\[tor\_maxwell\_1\])–(\[tor\_maxwell\_2\]) from our notion of the reduced EVM system, i.e. we make the following definition. \[def\_red\_evm\] The reduced EVM-system with parameters $\gamma$, $\lambda$ is defined as equations (\[final\_eq\_nu\])–(\[final\_eq\_a\]), equipped with the boundary conditions (\[bc\_infinity\])–(\[bc\_center\]). The axially symmetric solutions of the EVM-system which are constructed in this article are obtained as perturbations around spherically symmetric solutions of the Vlasov-Poisson system. For this reason we discuss the non-relativistic limit of the EVM-system, i.e. the limit where $\gamma \to 0$. Define for the spherically symmetric steady state of the Vlasov Poisson system for particles of mass $1-q^2$ the potential at infinity $U_\infty$ by $$U_\infty := \lim_{|x|\to \infty} U_N(x).$$ Then by condition (2) on $\phi$ we clearly have $U_\infty > E_0$ and there exists $R \in (R_N, \infty)$ such that $$\label{def_cap_r} U_N(r) > \frac{E_0 + U_\infty}{2}, \quad \mathrm{for\, all} \; r > R.$$ (Recall that $R_N$ is such that $U_N(R_N)=E_0$.) It turns out, that in the limit $\gamma\to 0$, only the equations (\[final\_eq\_nu\]) and (\[eq\_a0\]) of the reduced EVM-system remain non-trivial and they reduce to the Poisson equations $$\begin{aligned} \Delta \nu_N &= 4\pi\rho_{\nu_N + q A_N}, \label{newt_1}\\ \Delta A_N &= -4\pi q \rho_{\nu_N + q A_N}, \label{newt_2}\end{aligned}$$ where we use the notation $\rho_{\nu_N + q A_N}$, introduced in (\[def\_rho\_u\]), on the right hand side. See the proof of Lemma \[lem\_n\_zero\] for details. The system (\[newt\_1\])–(\[newt\_2\]) equipped with the boundary conditions $$\nu_N(0)=0, \quad A_N(0)=0,$$ and the equation $$\label{poisson_eq_un} \Delta U_N = 4\pi (1-q^2) \rho_N, \quad U_N(0)=0$$ are equivalent in the sense that a solution of (\[newt\_1\])–(\[newt\_2\]) gives rise to a solution of (\[poisson\_eq\_un\]) via $U_N = \nu_N + q A_N$ and a solution of (\[poisson\_eq\_un\]) gives rise to a solution of (\[newt\_1\])–(\[newt\_2\]) via $\nu_N = (1-q^2)^{-1} U_N$, $A_N = -q(1-q^2)^{-1} U_N$. In Lemma \[lem\_sol\_decay\] below we will furthermore see that the limits $\nu_\infty = \lim_{|x|\to \infty} \nu$ and $A_\infty = \lim_{|x|\to \infty} A_t$ exist for any $\gamma\in(0,\infty)$ and that in the limit $\gamma\to 0$ there holds $A_\infty = -q\nu_\infty$, which is consistent. We are going to linearise around a solution of the system in the limit $(\gamma, \lambda)\to (0,0)$. We denote this solution by $\zeta_0$, i.e. $$\zeta_0 = (\nu_N,0,0,0,A_N,0).$$ \[lem\_comp\_supp\] If $\gamma > 0$ is sufficiently small, then the matter quantities of a solution $\zeta$ of the reduced EVM-system are supported within a ball of radius $R$ around the origin. The particle energy $E$ converges to the Newtonian particle energy $E_N$, given by $$\label{newtonian_energy} E_N := \frac{|v|^2}{2} + \nu_N + \omega L_N + qA_N, \quad L_N=\varrho v^3$$ in the non-relativistic limit where $\gamma \to 0$. Using the expansions $e^x = 1+ x+ \dots$ and $\sqrt{1+x} = 1+\frac 12 x + \dots$ we obtain $$\begin{aligned} E &= \frac{e^{\gamma \nu} \sqrt{1+\gamma |v|^2}-1}{\gamma} + \omega\tilde L + qA_t \label{newtonian_l_e} \\ &= \frac{|v|^2}{2} + \nu + \left(\frac{\nu^2}{2} - \frac{|v|^4}{4} + \frac{\nu|v|^2}{2}\right)\gamma + \dots + \omega \tilde L + qA_t \end{aligned}$$ and since $\nu \to \nu_N$, $A_t \to A_N$, $\omega \to 0$, we see $E \to E_N$ as $\gamma \to 0$. which is the Newtonian particle energy with potential $U_N = \nu_N + qA_N$. Now, since $\|\nu + qA_t - U_N\|_\infty \to 0$, as $\gamma \to 0$, there is $\gamma_0 > 0$ such that for all $0 \leq \gamma \leq \gamma_0$ we have $E > \nu + qA_t > E_0$ for all $|x| > R$. The function space of the solution {#sect_function_space} ================================== In this paragraph the function spaces are defined in which a solution $\zeta = (\nu, h, \xi, \omega, A_t, a)$ of the reduced EVM-system will be constructed. In [@akr11; @akr14] the considered function spaces contain axially symmetric functions on $\mathbb R^3$. Taking account for the fact that the reduced EVM-system is formulated as Poisson equations in different dimensions we define the function spaces for functions in the according dimensions. Furthermore, for the analysis of the source terms of these Poisson equations a hierarchy in regularity among the individual solution functions is needed, cf. Lemma \[lem\_reg\] below. For this reason the assumed regularity is a bit stronger than in [@akr14]. Let $\alpha\in(0, 1/2)$ be a fixed parameter and $Z_R = \{(x^1, x^2, x^3) \in \mathbb R^3 \, : \, \varrho(x^1,x^2) \leq R\}$. We define the following spaces of axially symmetric functions, $$\begin{aligned} \mathcal X_1 &:= \{ \nu \in C^{3,\alpha}(\mathbb R^3)\, |\, \nu = \nu(\varrho, z) = \nu(\varrho, -z),\; \mathrm{and} \; \| \nu\|_{\mathcal X_1} < \infty\}, \\ \mathcal X_2 &:= \{ h \in C^{3,\alpha}(\mathbb R^4)\, |\, h = h(\varrho, z) = h(\varrho, -z),\; \mathrm{and} \; \| h \|_{\mathcal X_2} < \infty\}, \\ \mathcal X_3 &:= \{ \xi \in C^{1,\alpha}(Z_R)\, |\, \xi = \xi(\varrho,z) = \xi(\varrho, -z),\; \mathrm{and} \; \| \xi \|_{\mathcal X_3} < \infty\}, \\ \mathcal X_4 &:= \{ \omega \in C^{2,\alpha}(\mathbb R^5)\, |\, \omega = \omega(\varrho, z) = \omega(\varrho, -z), \; \mathrm{and} \; \| \omega\|_{\mathcal X_4} < \infty\},\end{aligned}$$ and $$\mathcal X := \mathcal X_1 \times \mathcal X_2 \times \mathcal X_3 \times \mathcal X_4 \times \mathcal X_1 \times \mathcal X_4.$$ Let $\beta \in (0,1)$ be another fixed parameter. Then the corresponding norms are defined to be $$\begin{aligned} \| \nu \|_{\mathcal X_1} &:= \| \nu \|_{C^{3,\alpha}(\mathbb R^3)} + \left\|(1 + |x|)^{1+\beta} \nabla \nu\right\|_\infty, \\ \| h \|_{\mathcal X_2} &:= \| h \|_{C^{3,\alpha}(\mathbb R^4)} + \left\|(1+|x|)^3 \nabla h \right\|_\infty, \\ \| \xi \|_{\mathcal X_3} &:= \| \xi \|_{C^{1,\alpha}(Z_R)}, \\ \| \omega \|_{\mathcal X_4} &:= \| \omega \|_{C^{2,\alpha}(\mathbb R^5)} + \|(1+|x|)^3 \omega\|_\infty + \|(1+|x|)^4 \nabla \omega\|_\infty, \label{def_norm_x4}\end{aligned}$$ and $$\|\zeta\|_{\mathcal X} := \| \nu \|_{\mathcal X_1} + \| h \|_{\mathcal X_2} + \| \xi \|_{\mathcal X_3} + \| \omega \|_{\mathcal X_4} + \| A_t \|_{\mathcal X_1} + \| a \|_{\mathcal X_4}.$$ Finally we define $$\mathcal U := \{(\zeta,p) \in \mathcal X \times [0,\delta) \times (-\delta,\delta))\, |\, \| \zeta -\zeta_0) \|_{\mathcal X} < \delta_0\},$$ where $\delta_0 > 0$ is sufficiently small such that for all $(\zeta;\gamma,\lambda) \in \mathcal U$, we have $1+h(\varrho,z) > 1/2$ for all $(\varrho,z) \in [0,\infty) \times \mathbb R$. Solutions of the reduced system solve the full EVM-system ========================================================= In this article we construct solutions to the reduced EVM-system (\[final\_eq\_nu\])–(\[bc\_center\]). These solutions to the reduced EVM-system correspond to spherically symmetric, time independent solutions of the EVM-system (\[eq\_einstein\])–(\[maxwell\_eq\_2\]). The relations between these systems is the subject of the following proposition. As already mentioned, this article generalises [@akr14] to the case of charged particles and the reduced system treated here coincides with the reduced system considered in [@akr14] if the charge parameter $q$ is set to zero. \[prop\_equivalent\] A solution $\zeta \in \mathcal X$ of the reduced EVM-system (\[final\_eq\_nu\])–(\[bc\_center\]) with parameters $\lambda$, $\gamma$ gives rise to a time independent, axially symmetric solution $(g,f,A)$ of the EVM-system (\[eq\_einstein\])–(\[maxwell\_eq\_2\]) where $g$ is of the form (\[ansatz\_metric\]) and $f$ is of the form (\[ansatz\_f\]). Before we prove Proposition \[prop\_equivalent\] we establish the following scaling law. (Scaling law) \[lem\_scaling\]\ Let $(\nu, h, \xi, \omega, f, A_t, a)$ be a solution of the reduced EVM-system (\[eq\_einstein\])–(\[maxwell\_eq\_2\]) with parameters $(\lambda, c) \in (-1,1) \times (0,\infty)$. Then the functions $\tilde \nu, \tilde h, \tilde \xi, \tilde \omega, \tilde f, \tilde A_t, \tilde a$, given by $$\begin{gathered} \left(\tilde \nu(\varrho, z), \tilde h(\varrho, z), \tilde \xi(\varrho, z), \tilde \omega(\varrho, z), \tilde A_t(\varrho, z), \tilde a(\varrho, z)\right) \\ = \left(\frac{1}{c^2} \nu(c\varrho, cz), h(c\varrho, cz), \xi(c\varrho, cz), \omega(c\varrho, cz), \frac{1}{c^2} A_t(c\varrho, cz), a(c\varrho, cz)\right)\end{gathered}$$ and $$\label{scaling_f} \tilde f(\varrho, z, p^\varrho, p^z, p^\varphi) = c^3 f(c\varrho, cz, cp^\varrho, cp^z, p^\varphi)$$ satisfy the reduced EVM-system with parameters $(\lambda, 1)$. We check the laws for $A_t$ and $a$. For the other functions, cf. [@akr14]. For the Laplace operator we have the transformation law $$\Delta \tilde A_t (\varrho, z) = (\Delta A_t)(c\varrho, cz).$$ Then we use the Maxwell equations (\[eq\_a0\]) and (\[final\_eq\_a\]) for $A_t$ and $a$, respectively. Note that for example $$\left(\nabla A_t\right)(c\varrho, cz) = \frac{1}{c} \nabla (A_t(c\varrho, cz)) = c \nabla \tilde A_t(\varrho, z).$$ For the matter function corresponding to $A_t$ we obtain the expression $$\begin{aligned} &\mathfrak M_5[\zeta; \gamma, \lambda](c\varrho, cz) \\ &= -4\pi q e^{(2\tilde \xi-3 \tilde \nu)(\varrho,z)} \int_{\mathbb R_v^3} f\left(c\varrho, cz, p^\varrho(c\varrho, cz, v^1), p^z (c\varrho, cz, v^2), p^\varphi(c\varrho, cz, v^3)\right) \nonumber \\ &\hspace{5cm} \times \left(e^{2\tilde\nu(\varrho,z)} + \frac{\varrho(1+\tilde h(\varrho,z)\omega(\varrho,z) v^3}{c\sqrt{1+\gamma |v|^2}}\right) \, \mathrm dv^1 \mathrm dv^2 \mathrm dv^3 \nonumber\end{aligned}$$ and for $a$ we have the matter function $$\begin{gathered} \mathfrak M_6[\zeta; \gamma, \lambda](c\varrho, cz) = \frac{4\pi q}{c} \varrho (1 + \tilde h(\varrho,z)) e^{(2\tilde \xi- 3 \tilde \nu)(\varrho,z)} \\\times \int_{\mathbb R_v^3} f\left(c\varrho, cz, p^\varrho(c\varrho, cz, v^1), p^z (c\varrho, cz, v^2), p^\varphi(c\varrho, cz, v^3)\right) \frac{v^3}{\sqrt{1+\gamma |v|^2}}\, \mathrm dv^1 \mathrm dv^2 \mathrm dv^3.\end{gathered}$$ Now, applying the change of variables $v^i \to w^i = v^i/c$, $i=1,2,3$, and using the scaling law (\[scaling\_f\]) one recovers the original matter functions with $\tilde f$ instead of $f$. First we describe how the reduced EVM-system can be derived from the EVM-system. We start with the equations (\[final\_eq\_nu\])–(\[eq\_omega\]) which –without electro-magnetic field terms of course– have been considered in [@akr14]. Write down all Einstein equations in the coordinates $t,\varrho, \varphi, z$ and take into account the symmetries by substituting the ansatz (\[ansatz\_metric\]) for $g$. Suitable combinations of the Einstein equations yield the equations (\[final\_eq\_nu\])–(\[eq\_omega\]) for $\nu$, $h$, $\xi$, and $\omega$. For equation (\[final\_eq\_nu\]) take the combination $$\label{combination_nu} \frac 12 \left(e^{2\xi - 4\gamma\nu}(G_{tt} + 2\omega G_{t\varphi}) + \frac{1}{\gamma} (G_{\varrho\varrho} + G_{zz}) + e^{2\xi}\left(\frac{1}{\gamma \varrho^2 (1+h)^2} + \omega^2 e^{-4\gamma\nu}\right) G_{\varphi\varphi} \right).$$ For equation (\[eq\_b\]) take $(1+h)(G_{\varrho\varrho} + G_{zz})$, for equation (\[eq\_omega\]) take $\frac{2 e^{2\xi}}{\varrho^2 (1+h)^2} (G_{t\varphi} + \omega G_{\varphi\varphi})$, and for equation (\[eq\_xi\]) take $$\label{combination_xi} (1+h+\varrho h_{,\varrho}) \frac{(1+h)\varrho}{2} (G_{\varrho\varrho} - G_{zz}) + \varrho^2 h_z (1+h) G_{\varrho z}.$$ It is important to take the right combination of Einstein equations for the method to work and we follow [@akr11]. The components $G_{\mu\nu}$ of the Einstein tensor and the components $\tau_{\mu\nu}$ of the electro-magnetic part of the energy momentum tensor yield the left members and the source functions of equations (\[final\_eq\_nu\])–(\[eq\_omega\]). The matter functions $M_i^{(\gamma,\lambda)}$, $i=1,2,4$ are obtained as explained now. First, using the ansatz (\[ansatz\_f\]) for the particle distribution function $f$ and the orthonormal frame (\[def\_frame\]) one can write the components of the kinetic part $T_{\mu\nu}$ of the energy momentum tensor, defined in (\[eq\_em\_tensor\]), as the integral expression. $$T_{\mu\nu} = \int_{\mathbb R_v^3} \phi(E) \psi(\lambda, L) \frac{p_\mu p_\nu}{\sqrt{1+\gamma |v|^2}}\, \mathrm dv^1 \mathrm dv^2 \mathrm dv^3. \label{em_tensor_v}$$ For this formula the mass shell relation (\[mass\_shell\_frame\]) needs to be used. Furthermore, the variables $p_\mu$, $\mu=0,\dots, 3$ can in terms of the frame components $v^1$, $v^2$, $v^3$, be expressed as $$\label{pd_via_v} \begin{aligned} p_0 &= -\frac{e^{\gamma\nu}}{\gamma}\sqrt{1+\gamma |v|^2} - e^{-\gamma\nu}\varrho (1+h) \omega v^3, \\ p_\varrho &= e^\mu v^1, \quad p_z = e^\mu v^2, \quad p_\varphi = e^{-\gamma\nu} \varrho (1+h) v^3. \end{aligned}$$ Now taking the corresponding combinations of $T_{\mu\nu}$ and substituting the expressions (\[pd\_via\_v\]) for the $p$-variables one obtains after simplification the matter functions. These matter functions coincide with the corresponding matter terms in [@akr14], the only difference consists in the quantities $E$ and $L$. The matter quantity $M_3^{(\gamma,\lambda)}$ vanishes due to the symmetry $T_{\varrho\varrho} = T_{zz}$. The equations (\[eq\_a0\]) and (\[final\_eq\_a\]) for $A_t$ and $a$, respectively, are new with respect to [@akr14] and they are obtained by suitable combinations of the Maxwell equation $\nabla_\alpha F^{\alpha \beta} = -4\pi q J^\beta$ for $\beta = t$ and $\beta = \varphi$. These combinations are $$\begin{aligned} &\frac{1}{\gamma} e^{2\xi} \nabla_\alpha F^{\alpha t} - \omega \varrho^2 (1+h)^2 e^{2\xi-4\gamma\nu}\left(\omega \nabla_\alpha F^{\alpha t} - \nabla_\alpha F^{\alpha\varphi}\right), \label{combination_a0} \\ &(1+h)^2 e^{2\xi-4\gamma\nu}\left(\omega \nabla_\alpha F^{\alpha t} - \nabla_\alpha F^{\alpha\varphi}\right), \label{combination_a}\end{aligned}$$ respectively. The matter functions $M_5^{(\gamma,\lambda)}$ and $M_6^{(\gamma,\lambda)}$ are obtained by taking the respective combinations of the components of the matter current $J^\beta$, defined in (\[maxwell\_eq\_2\]). Using the orthonormal frame (\[def\_frame\]) it can be written as $$J^\beta = \gamma \int_{\mathbb R_v^3} \phi(E) \psi(\lambda, L) \frac{p^\beta}{\sqrt{1+\gamma |v|^2}}\, \mathrm dv^1 \mathrm dv^2 \mathrm dv^3. \label{matter_current_v}$$ The variables $p^\mu$, $\mu= 0,\dots,3$, are given in terms of the frame coordinates as $$\label{frame_reverse} p^0 = e^{-\gamma\nu} v^0, \quad p^1 = e^{-\mu} v^1, \quad p^2 = e^{-\mu} v^2, \quad p^3 = e^{-\gamma\nu} \omega v^0 + \frac{e^{\gamma\nu}}{(1+h)\varrho} v^3.$$ So far it has been proved that a solution of the EVM-system implies a solution of the reduced EVM-system since the latter one is obtained by linear combinations of certain components of the former one. It remains to verify that the converse is also true, i.e. that a solution to the reduced EVM-system with parameter $c \in [1,\infty)$ implies an axially symmetric, time independent solution of the EVM-system with $c=1$. First we note that by the scaling laws (Lemma \[lem\_scaling\]) a solution to the reduced EVM-system with $c=1$ can always be obtained. The Maxwell equations are already fulfilled since the number of equations has not been reduced. For the Einstein equations however the number of equations has been reduced, so situation is less clear. We define the quantity $$E_{\mu\nu} = G_{\mu\nu} - \frac{8\pi}{c^4} \left(T_{\mu\nu} + \tau_{\mu\nu}\right), \quad \mu, \nu = t, \varrho, z, \varphi.$$ The non-trivial components are $E_{tt}$, $E_{\varrho\varrho}$, $E_{zz}$, $E_{\varphi \varphi}$, $E_{t\varphi}$, and $E_{\varrho z}$. The other components are trivially zero since the Einstein tensor vanishes under the symmetry assumptions incorporated into the metric ansatz (\[ansatz\_metric\]). It remains to show that the components $E_{tt}$, $E_{\varrho\varrho}$, $E_{zz}$, $E_{\varphi \varphi}$, $E_{t\varphi}$, and $E_{\varrho z}$ vanish, too. This can be done by using the same argument as given in [@akr14 Section 6] since the Einstein part of the reduced EVM-system that we are working with consists in the same linear combinations of Einstein equations which has been considered in [@akr14]. A subtlety, which has to be dealt with, consists in the fact that $\xi$ is only $C^{1,\alpha}$, whereas Einstein’s equations are of second order. Since in the present setup $\xi$ has the same regularity as in the setup of [@akr14] the arguments of [@akr14] apply however. Finally, the boundary conditions (\[bc\_infinity\]) clearly imply the boundary conditions (\[bc\_as\_flat\]). Definition of the solution operator $\mathfrak F$ {#sect_def_f} ================================================= The equations (\[final\_eq\_nu\]), (\[eq\_b\]), (\[eq\_omega\])–(\[final\_eq\_a\]) of the reduced EVM-system are semi-linear Poisson equations. For this reason the solution operators corresponding to these equations are basically given in terms of the Greens function of the Laplace operator. If $q$ is set to zero, the solution operator introduced here coincides with the solution operator defined in [@akr14]. First, we recall some facts about the Poisson equation. Define for $n\geq 3$ the $n$-dimensional Greens function $G^n_y(x)$ of the Laplace operator $\Delta_n$ by $$\label{def_g_y} G^n_y(x) = \frac{1}{(n-2)|\mathbb S^{n-1}|} \frac{1}{|x-y|^{n-2}},$$ where $|\mathbb S^{n-1}|$ is the volume of the $(n-1)$-dimensional unit sphere. For later convenience we also define $$\hat G^n_y(x) = \frac{1}{(n-2)|\mathbb S^{n-1}|} \left( \frac{1}{|x-y|^{n-2}} - \frac{1}{|y|^{n-2}}\right)$$ and the functionals $$\label{def_greens_gn} G_n[f](x) := \int_{\mathbb R^n} G^n_y(x) f(y)\, \mathrm dy \quad \mathrm{and} \quad \hat G_n[f](x) := \int_{\mathbb R^n} \hat G^n_y(x) f(y)\, \mathrm dy.$$ Then, in the sense of distributions, the solution of the Poisson equation $-\Delta_n u = f$ for $f\in L_{\mathrm{loc}}^1(\mathbb R^n)$ on $\mathbb R^n$, $n\geq 1$ is given by $u(x) = G_n[f](x)$, cf. [@ll00 Theorem 6.21]. Now we give the definition of $\mathfrak F$. To this end we first define the operators $\mathfrak G_i :\mathcal U \to \mathcal X_i$, $i=1,\dots,6$ (by $\mathcal X_5$ and $\mathcal X_6$ we understand $\mathcal X_1$ and $\mathcal X_4$, respectively). We define $$\begin{aligned} \mathfrak G_i[\zeta; \gamma, \lambda] &:= G_3[\mathfrak g_i[\zeta; \gamma]] + \hat G_3[\mathfrak M_i[\zeta; \gamma, \lambda]], &i &= 1,5, \label{def_g15}\\ \mathfrak G_2[\zeta; \gamma, \lambda] &:= G_4[\mathfrak M_2[\zeta; \gamma, \lambda]], && \label{def_g2}\\ \mathfrak G_3[\zeta; \gamma, \lambda] &:= \ln(1+h(0,z)) + \int_0^\varrho \mathfrak g_3[\zeta; \gamma](s,z) \, \mathrm ds, &&\label{def_g3} \\ \mathfrak G_i[\zeta; \gamma, \lambda] &:= G_5[\mathfrak g_i[\zeta; \gamma] + \mathfrak M_i[\zeta; \gamma, \lambda]], &i &= 4,6. \label{def_g46} %\mathfrak G_5[\zeta; \gamma, \lambda](\varrho,z) &:= G_3[\mathfrak g_5]\left(x^{(3)}_{(\varrho, z)}\right) - \hat G_3[\mathfrak M_5]\left(x^{(3)}_{(\varrho, z)}\right), \label{def_g5} \\ %\mathfrak G_6[\zeta; \gamma, \lambda](\varrho,z) &:= G_5[\mathfrak g_6 - \mathfrak M_6]\left(x^{(5)}_{(\varrho, z)}\right), \label{def_g6}\end{aligned}$$ Then we write compactly $$\mathfrak G[\zeta; \gamma, \lambda] := (\mathfrak G_1[\zeta; \gamma, \lambda], \dots, \mathfrak G_6[\zeta; \gamma, \lambda]).$$ Furthermore we define $$\mathfrak F:\mathcal U \to \mathcal X,\quad (\zeta; \gamma, \lambda) \mapsto \mathfrak F[\zeta; \gamma, \lambda] := \zeta - \mathfrak G[\zeta; \gamma, \lambda].$$ \[lem\_rel\_greens\] Let $(\zeta; \gamma, \lambda) \in \mathcal U$. Then $\mathfrak G_i[\zeta; \gamma, \lambda]$ is axially symmetric and even in the $x^n$-coordinate (also referred to as $z$-coordinate) for all $i=1,\dots,6$. Clearly $\mathfrak g_i[\zeta; \gamma]$ and $\mathfrak M_i[\zeta; \gamma, \lambda]$ are axially symmetric and even in $z$ if $\zeta$ is. Consider the following prototype term. Let $f:\mathbb R^n \to \mathbb R^n$ be an axially symmetric function that is even in $x^n=z$. One can check straight forwardly that $G_n[f]$ is axially symmetric and even in $z$ by performing and appropriate change of variables in the integral, i.e. we have for $A\in SO(n-1)$ $$G_n[f](A \cdot (x^1, \dots, x^{n-1})^\intercal, -x^n) = G_n[f](x).$$ The operators $\mathfrak G_1$ and $\mathfrak G_5$ have been defined such that the Fréchet derivative of $\mathfrak F$ with respect to $\nu$, $A_t$, at $(\zeta_0;0,0)$ is zero at $(\varrho,z)=0$. Observe the $\hat G$ in equation (\[def\_g15\]). This property is important in the proof that the Fréchet derivative at $(\zeta_0; 0, 0)$ is a bijection, cf. Lemma \[lem\_bijection\] below. \[prop\_consistent\] Let $\zeta\in\mathcal X$ and $(\gamma,\lambda) \in [0,\delta) \times (-\delta,\delta)$. Then $\mathfrak F[\zeta; \gamma, \lambda] = 0$ if and only if $\zeta$ restricted to $\{\varrho \geq 0\}$ is a solution of the reduced EVM-system (\[final\_eq\_nu\])–(\[final\_eq\_a\]) with parameters $\gamma$, $\lambda$. The statement is clear for $\mathfrak G_i$ and $\zeta_i$, $i=1,2,4,5,6$ since by Lemma \[lem\_rel\_greens\] these operators are the solution operators to the semi-linear Poisson equations (\[final\_eq\_nu\]), (\[eq\_b\]), (\[eq\_omega\])–(\[final\_eq\_a\]). For the operator $\mathfrak G_3$, we observe that differentiation of $\mathfrak G_3[\zeta;\gamma,\lambda](\varrho,z)$ with respect to $\varrho$ directly yields the right hand side of the $\xi$-equation (\[eq\_xi\]). \[lem\_n\_zero\] Recall $\zeta_0 = (\nu_N,0,0,0,A_N,0)$. We have $\mathfrak F[\zeta_0;0,0] = 0$. We adopt the notation $\rho_N := \rho_{U_N}$, $\alpha_N := \alpha_{U_N}$. The Einstein equations (\[eq\_b\])–(\[eq\_omega\]) for $h$, $\xi$, and $\omega$ are trivially satisfied for $\zeta=\zeta_0$. So it remains to consider equation (\[final\_eq\_nu\]) for $\nu$. The source function $\mathfrak g_1[\zeta_0; 0, 0]$ is zero. For the matter function $\mathfrak M_1$ a calculation yields $\mathfrak M_1[\zeta_0; 0, 0](\varrho,z) = 4\pi \rho_N(r)$, where $r=\sqrt{\varrho^2 + z^2}$. This is the energy density induced by the ansatz (\[ansatz\_f\]) in the Newtonian case. We see that the Maxwell equation (\[final\_eq\_a\]) for $a$ is satisfied with $\gamma=0$ and $a\equiv \omega\equiv h \equiv 0$. Concerning the Maxwell equation (\[eq\_a0\]) for $A_t$, we see that it reduces to $$\Delta_3 A_t = -4\pi q \rho_N(r).$$ So $U_N = \nu_N + qA_N$ solves the Poisson equation $$\Delta U_N(r) = 4\pi (1-q^2) \rho_N(r).$$ Note also that we are using the assumption $\psi(0,L)=1$. So we actually obtain $$U_N(r) = \mathfrak G_1[\zeta_0;0,0](\varrho,z) + q \mathfrak G_5[\zeta_0;0,0](\varrho,z)$$ and the assertion follows. $\mathfrak F$ is well defined {#sect_well_defined} ============================= We have to verify that for all $(\zeta; \gamma, \lambda) \in \mathcal U$ the functions $\mathfrak G_i[\zeta; \gamma, \lambda]$ satisfy the regularity conditions and the decay behaviour stated in the definition of $\mathcal X$, for $i=1,\dots,6$. Before we prove the regularity properties of $\mathfrak G[\zeta; \gamma, \lambda]$ we collect a few facts on axially symmetric functions, proven in [@akr11] and [@akr14]. (Lemma 7.1 in [@akr14]) \[lem\_axially\]\ Let $u:\mathbb R^n \to \mathbb R$ be axially symmetric and $u(x) = \tilde u(\varrho,z)$ where $\tilde u:[0,\infty)\times \mathbb R \to \mathbb R$. Let $k\in \{1,2,3\}$ and $\alpha \in (0,1)$. Then 1. $u\in C^k(\mathbb R^n) \Leftrightarrow \tilde u \in C^k([0,\infty) \times \mathbb R)$ and all derivatives of $\tilde u$ of order up to $k$ which are of odd order in $\varrho$ vanish for $\varrho = 0$, 2. $u\in C^{0,\alpha}(\mathbb R^n) \Leftrightarrow \tilde u \in C^{0,\alpha}([0,\infty) \times \mathbb R)$. (Lemma 3.2 in [@akr11]) \[lem\_axially\_2\]\ Let $\varphi=\varphi(\varrho,z)\in C^4(\mathbb R^2)$ be odd in $\varrho$ and define $$\zeta(\varrho,z) := \left\{\begin{array}{ll} \varphi(\varrho,z)/\varrho, & \varrho \neq 0, \\ \partial_\varrho \varphi(0,z), & \varrho = 0.\end{array}\right.$$ Then $\zeta\in C^3(\mathbb R^2)$ and all derivatives of $\zeta$ up to order $3$ which are of odd oder in $\varrho$ vanish for $\varrho = 0$. By abuse of notation, $\zeta\in C^3(\mathbb R^3)$. Next we establish regularity of the matter functions. Let $(\zeta; \gamma, \lambda) \in \mathcal U$. Then the functions $\mathfrak M_i[\zeta; \gamma, \lambda]$, $i=1,2,4,5,6$, if extended to negative values of $\varrho$ and thus seen as functions on $\mathbb R^2$, are even in $\varrho$. That the matter functions are even in $\varrho$ has already been observed in [@akr14] and the new matter functions $M_5^{(\gamma,\lambda)}$ and $M_6^{(\gamma,\lambda)}$ can be treated with the same ideas. We perform in the integrals of the formulas (\[def\_hat\_m\_1\])–(\[def\_hat\_m\_6\]) for the matter functions $M_i^{(\gamma, \lambda)}$, $i = 1,2,4,5,6$, a change of variables, given by $$\eta = \frac{e^{\gamma\nu}\sqrt{1+\gamma |v|^2} -1}{\gamma}, \quad s = (1+h) e^{-\gamma\nu} v^3.$$ Let $$m(\eta, h, \nu) := (1+h)e^{-\gamma \nu} \sqrt{\frac{e^{-2\gamma \nu}(\gamma \eta + 1)^2-1}{\gamma}}.$$ Then the domain of integration can be parameterised by $\eta \in ((e^{\gamma\nu}-1)/\gamma,\infty)$, $s\in(-m,m)$. Further, for a function $g = g(s, \eta, h, \nu, \varrho\omega)$, which will be chosen among the choices $$1 + 4\gamma\eta+2\gamma^2 \eta^2, \quad m^2-s^2, \quad s(1+\gamma\eta), \quad s, \quad 1+\gamma\eta + \gamma\omega\varrho s,$$ we define $M_{(\gamma, \lambda)}$ to be the operator which assigns to $g$ the function $$\begin{aligned} &M_{(\gamma, \lambda)}[g] : \mathbb R^2 \times \left(- \frac 12, \infty\right) \times \mathbb R^3 \to \mathbb R, \\ &(\varrho, \nu, h, \omega, A_t, a) \mapsto M_{(\gamma, \lambda)}[g](\varrho, \nu, h, \omega, A_t, a) \label{def_capital_m} \\ &\hspace{1cm} = \int_{\frac{e^{\gamma\nu} - 1}{\gamma}}^\infty \int_{-m(\eta, h, \nu)}^{m(\eta, h, \nu)} \phi(\eta + \varrho\omega s + q A_t) \psi(\lambda, \varrho s - q \varrho^2 a)\, g(s, \eta, h, \nu, \varrho\omega)\, \mathrm ds \mathrm d\eta. \end{aligned}$$ The range $(-1/2,\infty)$ of $h$ is motivated by the definition of the set $\mathcal U$ of functions that we consider. Then the matter functions $M_i^{(\gamma,\lambda)}$ can be written in the form $$\begin{aligned} M_1^{(\gamma,\lambda)}(\varrho, \zeta) &= \frac{8\pi^2}{1+h} e^{2\xi - 4\gamma\nu} M_{(\gamma, \lambda)}\left[ 2(1+\gamma\eta)^2 - e^{2\gamma\nu} \right](\varrho, \nu, h, A_t, a), \label{m1_mg} \\ M_2^{(\gamma,\lambda)}(\varrho, \zeta) &= \frac{16 \pi^2 \gamma^2}{(1+h)^2} e^{2\xi} M_{(\gamma, \lambda)}\left[m^2 - s^2\right](\varrho, \nu, h, A_t, a), \label{m2_mg}\\ M_4^{(\gamma,\lambda)}(\varrho, \zeta) &= -\frac{32 \pi^2 \gamma}{\varrho (1+h)^3}e^{2\xi} M_{(\gamma, \lambda)}[s(1 + \gamma\eta)](\varrho, \nu, h, A_t, a), \label{m4_mg} \\ M_5^{(\gamma,\lambda)}(\varrho, \zeta) &= \frac{8\pi^2 q}{1+h} e^{2(\xi-\gamma\nu)} M_{(\gamma, \lambda)}\left[1 + \gamma\eta + \gamma\omega \varrho s\right](\varrho, \nu, h, A_t, a), \label{m5_mg} \\ M_6^{(\gamma,\lambda)}(\varrho, \zeta) &= -\frac{8\pi^2 q \gamma}{\varrho(1+h)} e^{2(\xi-\gamma\nu)} M_{(\gamma, \lambda)}[s](\varrho, \nu, h, A_t, a). \label{m6_mg}\end{aligned}$$ Given these representations (\[m1\_mg\])–(\[m6\_mg\]) of the matter functions we observe the following fact. If $g(s, \eta, h, \nu, \varrho\omega)$ is even or odd in $s$ then $M_{(\gamma,\lambda)}[g](\varrho, \nu, h, A_t, a)$ is even or odd in $\varrho$, respectively. To see this we substitute $-\varrho$ for $\varrho$ in the formula (\[def\_capital\_m\]) for $M_{(\gamma,\lambda)}[g](\varrho, \nu, h, \omega, A_t, a)$ and make then the change of variables $s \to \hat s = -s$. If $g$ is even in $s$ we obtain the same expression as for “$+\varrho$”, whereas if $g$ is odd in $s$ we obtain its negative. Then we observe that $\mathfrak M_i[\zeta; \gamma, \lambda]$ is even in $\varrho$ for all $i\in\{1,2,4,5,6\}$. Consider for example $\mathfrak M_5[\zeta; \gamma, \lambda]$, given by $$\begin{aligned} &\mathfrak M_5[\zeta; \gamma, \lambda](\varrho, z) \\ &\qquad = \frac{8\pi^2 q}{1+h(\varrho, z)} e^{2(\xi-\gamma\nu)(\varrho, z)} M_{(\gamma, \lambda)}\left[1 + \gamma\eta\right](\varrho, \nu(\varrho, z), h(\varrho, z), A_t(\varrho, z), a(\varrho, z)) \\ &\qquad \quad + \frac{8\pi^2 q \gamma \omega(\varrho, z) \varrho }{1+h(\varrho, z)} e^{2(\xi-\gamma\nu)(\varrho, z)}M_{(\gamma, \lambda)}\left[s\right](\varrho, \nu(\varrho, z), h(\varrho, z), A_t(\varrho, z), a(\varrho, z)).\end{aligned}$$ Here we view $\zeta \in \mathcal X$ as even functions in $\varrho$, cf. Remark \[rem\_even\]. By the observation on $M_{(\gamma, \lambda)}$ which is mentioned above the first term is a product of functions that are even in $\varrho$. For the second term we observe that the fraction is odd in $\varrho$ since it contains $\varrho$ as explicit factor. The second factor is also odd in $\varrho$ by the upper observation. So in total the second term is even in $\varrho$. (Regularity of the matter functions) \[lem\_reg\_m\]\ Let $\phi \in C_c^\kappa(\mathbb R), \psi\in C^\infty_c(\mathbb R^2)$, and $\gamma \in [0,1]$, $\lambda\in [-1/2,1/2]$, where $\kappa \geq 1$. Further, let $g\in C^\sigma(\mathbb R^5)$, for $\sigma \geq 1$. Then all partial derivatives up to order $\min\{\kappa+1,\sigma\}$ of the function $M_{(\gamma, \lambda)}[g]$, defined in (\[def\_capital\_m\]), exist and are continuous. Furthermore, if $$g(s, \eta, h, \nu,\varrho\omega) |_{\eta = l(s, \nu, h)} = 0,$$ where $ l(s, \nu, b)$ is defined as $$\label{def_little_l} l(s, \nu, h) := \frac 1 \gamma \left(e^{\gamma \nu} \sqrt{1+\gamma\frac{s^2 e^{2 \gamma \nu}}{(1+h)^2}}-1\right),$$ then all partial derivatives up to order $\min\{\kappa+2,\sigma\}$ of $M_{(\gamma, \lambda)}[g]$ exist and are continuous. We write down the integral representation (\[def\_capital\_m\]) of $M_{(\gamma, \lambda)}[g]$ with respect to the new integration variable $\hat \eta := \eta + \varrho \omega + q A_t$. We obtain $$\begin{gathered} \label{exp_cm_1} M_{(\gamma, \lambda)}[g](\varrho, \nu, h, \omega, A_t, a) \\= \int_{-\infty}^\infty \int_{l(s, \nu, h)+\varrho\omega + q A_t}^\infty \phi(\hat\eta)\psi(\lambda, \varrho s- q \varrho^2 a) g(s, \hat\eta - \varrho\omega- q A_t, h, \nu, \varrho\omega)\, \mathrm d\hat \eta \mathrm ds.\end{gathered}$$ We write this in a schematic form in order to make the analysis clearer. Let $x=(x_1, \dots, x_6)$. In the following this vector represents $(\varrho, \nu, h, \omega, A_t, a)$. We write $$\label{exp_cm_2} M_{(\gamma, \lambda)}[g](x) = \int_{-\infty}^\infty \int_{\ell(s, x)}^\infty \phi(\hat\eta) \, \hat \psi(s, x) \, \hat g(s, \hat \eta, x)\, \mathrm d\hat\eta \mathrm ds,$$ Where $\ell$, $\hat \psi$, and $\hat g$ are defined in the obvious way such that the expressions (\[exp\_cm\_1\]) and (\[exp\_cm\_2\]) agree, i.e. $$\begin{aligned} \ell(s,x) &= l(s, x_2, x_3) + x_1 x_4 + q x_5, \\ \hat \psi(s, x) &= \psi(\lambda, x_1 s - q x_1^2 x_ 6), \\ \hat g(s, \hat \eta, x) &= g(s, \hat \eta - x_1 x_4 - q x_5, x_3, x_2, x_1x_4).\end{aligned}$$ Note that $\ell \in C^\infty(\mathbb R^3)$, since $l \in C^\infty(\mathbb R^3)$ already. To see the latter remind that $h > -1/2$ is assumed on the domain of $M_{(\gamma, \lambda)}$. We have for $i=1,\dots,6$ $$\begin{aligned} \partial_{x_i} M_{(\gamma, \lambda)}[g](x) &= \int_{-\infty}^\infty \int_{\ell(s,x)}^\infty \phi(\hat\eta) \, \partial_{x_i}\left(\hat \psi(s, x) \hat g(s, \hat \eta, x)\right)\, \mathrm d\hat\eta \mathrm ds \\ &\quad + \int_{-\infty}^\infty \phi(\ell(s,x)) \, \hat \psi(s, x) \, \hat g(s, \ell(s,x), x)\, \partial_{x_i} \ell(s,x)\, \mathrm ds. \nonumber\end{aligned}$$ Now we see that each additional derivative $\partial_{x_j}$, $j=1,\dots,6$ leads to a derivative acting on $\phi$, unless $\hat g(s, \ell(s,x), x) = 0$. In this case, only if there are three or more derivatives, there act one or more derivatives on $\phi$. Since $\hat\psi, \ell\in C^\infty$, and $\phi$ and $\psi$ are compactly supported, the regularity of $\phi$ and $g$ determines the regularity of $M_{(\gamma, \lambda)}[g]$ in the asserted way. Now we check the regularity properties of $\mathfrak G[\zeta; \gamma, \lambda]$. \[lem\_reg\] Let $(\zeta; \gamma, \lambda)\in \mathcal U$. Then we have $\mathfrak G_1[\zeta; \gamma, \lambda], \mathfrak G_2[\zeta; \gamma, \lambda], \mathfrak G_5[\zeta; \gamma, \lambda] \in C^{3,\alpha}(\mathbb R^2)$, $\mathfrak G_3[\zeta; \gamma, \lambda] \in C^{1,\alpha}(Z_R)$, and $\mathfrak G_4[\zeta; \gamma, \lambda], \mathfrak G_6[\zeta; \gamma, \lambda] \in C^{2,\alpha}(\mathbb R^2)$. By [@ll00 Theorem 10.3] the regularity of the axially symmetric solution functions $\mathfrak G_i[\zeta; \gamma, \lambda]$, $i=1,2,4,5,6$ follows from the regularity of the right members of the semi-linear Poisson equations (\[final\_eq\_nu\]), (\[eq\_b\]), (\[eq\_omega\])–(\[final\_eq\_a\]). These right members consist in the source functions $\mathfrak g_i[\zeta; \gamma]$ and the matter functions $\mathfrak M_i[\zeta; \gamma, \lambda]$. This regularity is now established. We have already observed that all matter functions $\mathfrak M_j[\zeta; \gamma, \lambda]$, $j=1,2,4,5,6$ and all source functions $\mathfrak g_i[\zeta; \gamma, \lambda]$, $i = 1,4,5,6$, if extended to negative values of $\varrho$ and thereby seen as functions on $\mathbb R^2$, are even in $\varrho$ and $z$. So by Lemma \[lem\_axially\] it suffices to establish the necessary regularity in $\varrho$ and $z$. We start by analysing the matter functions $\mathfrak M_j[\zeta; \gamma, \lambda]$, $j \in \{1,2,4,5,6\}$. By inspection of the formulas (\[m1\_mg\]), (\[m2\_mg\]), and (\[m5\_mg\]) and using Lemma \[lem\_reg\_m\] (which yields that all the $M_{(\gamma, \lambda)}[g]$ are at least $C^3$ in $\varrho$ and $z$), we see that the regularity of $\mathfrak M_1[\zeta; \gamma, \lambda]$, $\mathfrak M_2[\zeta; \gamma, \lambda]$, and $\mathfrak M_5[\zeta; \gamma, \lambda]$ is at least that of $\xi$, i.e. $C^{1,\alpha}(\mathbb R^2)$. In the formulas (\[m4\_mg\]) and (\[m6\_mg\]) for $\mathfrak M_6[\zeta; \gamma, \lambda]$ and $\mathfrak M_4[\zeta; \gamma, \lambda]$, respectively, we have the factors $$\begin{aligned} &\frac{1}{\varrho} M_{(\gamma, \lambda)}[s(1 + \gamma \eta)](\varrho, \nu(\varrho, z), h(\varrho, z), A_t(\varrho, z), a(\varrho, z)), \label{already} \\ &\frac{1}{\varrho} e^{2(\xi-\gamma\nu)} M_{(\gamma, \lambda)}[s](\varrho, \nu, h, A_t, a). \label{new_sim}\end{aligned}$$ Since, as already observed, $M_{(\gamma, \lambda)}[g](\varrho, \nu(\varrho, z), h(\varrho, z), A_t(\varrho, z), a(\varrho, z))$ is odd in $\varrho$ if $g$ is odd in $s$ Lemma \[lem\_axially\_2\] can be applied and this yields a regularity of $C^3$ in $\varrho$ and $z$, so in particular $C^{2,\alpha}(\mathbb R^2)$. The term (\[already\]) emerged already in the uncharged case treated in [@akr14], the term (\[new\_sim\]) is new but similar. In the charged case, there appear some more problematic terms with factors $\varrho^{-1}$ in the source functions $\mathfrak g_4[\zeta; \gamma, \lambda]$ and $\mathfrak g_6[\zeta; \gamma, \lambda]$. Except for these problematic terms the source functions $\mathfrak g_i[\zeta; \gamma, \lambda]$ consist in products, sums, and compositions of functions which are at least $C^{1,\alpha}$ (namely the solution functions $\zeta$ and their derivatives which are chosen in $\mathcal X$). Consequently $\mathfrak g_1[\zeta; \gamma, \lambda] , \mathfrak g_2[\zeta; \gamma, \lambda], \mathfrak g_5[\zeta; \gamma, \lambda]$ are already in $C^{1,\alpha}(\mathbb R^2)$. It remains to consider the terms with $\varrho^{-1}$. These terms are $$\label{crit_fun} \frac{A_{t,\varrho} a}{\varrho}, \quad \frac{\nu_{,\varrho} a}{\varrho}, \quad \frac{h_{,\varrho} a}{\varrho}.$$ We view $A_t$, $a$, $\nu$, and $h$ now as functions in $\varrho$, $z$ on $\mathbb R^2$ that are even in $\varrho$, cf. Remark \[rem\_even\]. The functions $A_{t,\varrho} a$, $\nu_{,\varrho} a$, and $h_{,\varrho} a$ are odd in $\varrho$ and in $C^{2,\alpha}(\mathbb R^2)$, so in particular in $C^{2}(\mathbb R^2)$. So, by Lemma \[lem\_axially\_2\], the functions (\[crit\_fun\]) are in $C^1(\mathbb R^2)$ and consequently also in $C^{0,\alpha}(\mathbb R^2)$. This is sufficient to prove the asserted regularity. Finally we consider the operator $\mathfrak G_3[\zeta; \gamma, \lambda]$. The asserted regularity is easy to see since the source function $\mathfrak g_3[\zeta; \gamma, \lambda]$ is obviously sufficiently regular, i.e. $C^{0,\alpha}$. Next we check the decay properties of $\mathfrak G[\zeta; \gamma, \lambda]$. First we recall a technical lemma. (Lemma 5.1 in [@akr14]) \[lem\_decay\]\ Let $f\in C^{0,\alpha}(\mathbb R^n)$, $n\geq 3$, fulfil $|f| \leq C (1+|x|)^{-(n+\epsilon)}$ for some constant $C>0$ and $\epsilon > 0$. Then $G_n[f] \in C^{2,\alpha}(\mathbb R^n)$, where $G_n[f]$ is defined in (\[def\_greens\_gn\]), and there exists a constant $\tilde C>0$ such that for all multi indices $\sigma$, $|\sigma|\leq 2$, and for all $x\in\mathbb R^n$ we have $$|\partial^\sigma G_n[f](x) | \leq \frac{\tilde C}{(1+|x|)^{n+|\sigma|-2}}.$$ \[lem\_sol\_decay\] Let $(\zeta; \gamma, \lambda)\in\mathcal U$. Then, there exists a constant $C>0$ such that for all $(\varrho,z)\in\mathbb R^2$, the following bounds hold: $$\begin{aligned} (\partial_\varrho + \partial_z) \mathfrak G_i[\zeta; \gamma, \lambda](\varrho,z) &\leq C \left(1+\sqrt{\varrho^2 + z^2}\right)^{-2}, \quad i=1,5, \\ (\partial_\varrho + \partial_z) \mathfrak G_2[\zeta; \gamma, \lambda](\varrho,z) &\leq C \left(1+\sqrt{\varrho^2 + z^2}\right)^{-3}, \\ (\partial_\varrho + \partial_z) \mathfrak G_j[\zeta; \gamma, \lambda](\varrho,z) &\leq C \left(1+\sqrt{\varrho^2 + z^2}\right)^{-4}, \quad j = 4,6, \\ \mathfrak G_j[\zeta; \gamma, \lambda](\varrho,z) &\leq C \left(1+\sqrt{\varrho^2 + z^2}\right)^{-3}, \quad j = 4,6.\end{aligned}$$ Furthermore the limits $$\label{def_lim_inf} \nu_{\infty}^{\gamma, \lambda} := \lim_{|(\varrho,z)|\to\infty} \mathfrak G_1[\zeta; \gamma, \lambda](\varrho,z), \quad A_{\infty}^{\gamma, \lambda} := \lim_{|(\varrho,z)|\to\infty} \mathfrak G_5[\zeta; \gamma, \lambda](\varrho,z)$$ exist. By Lemma \[lem\_decay\] it suffices to check that the source functions $\mathfrak g_i[\zeta; \gamma, \lambda]$, $i=1,4,5,6$ and the matter functions $\mathfrak M_j[\zeta; \gamma, \lambda]$, $j=1,2,4,5,6$ have the right decay behaviour. In fact the matter functions do not have to be taken into account here, because they are of compact support, cf. Lemma \[lem\_comp\_supp\]. The source functions have to be investigated term by term. Since these terms consist in products of derivatives of the functions $\zeta_j$, $j=1,\dots,6$, it is easy to see that the necessary decay is available. We illustrate this with the example of $\mathfrak g_1[\zeta; \gamma, \lambda]$. We have $$\begin{aligned} \mathfrak g_1[\zeta; \gamma, \lambda] &= -\frac{ h_{,\varrho} \nu_{,\varrho} + h_{,z} \nu_{,z}}{1+ h} + \frac{\varrho^2}{2}(1+ h)^2 e^{-4\gamma \nu} \left( \omega_{,\varrho}^2 + \omega_{,z}^2\right) \\ &\quad -\gamma^2 e^{-2\gamma\nu} \left( ( A_{t,\varrho} + 2\omega\varrho a + \omega \varrho^2 a_{,\varrho})^2 + ( A_{t,z} + \omega \varrho^2 a_{,z})^2 \right) \nonumber \\ &\quad - \gamma \frac{e^{2\gamma \nu}}{(1+ h)^2} \left((2 a+\varrho a_{,\varrho})^2 + \varrho^2 a_{,z}^2 \right). \nonumber\end{aligned}$$ We consider the first term $(h_{,\varrho} \nu_{,\varrho})/(1+h)$. Since $h\in \mathcal X_2$, $h > -1/2$ and $\nu \in \mathcal X_1$ we have $$\frac{|h_{,\varrho}(\varrho, z) \nu_{,\varrho}(\varrho,z)|}{1+h} \leq 2\frac{ \left\| (1+|x|)^{3} \nabla h \right\|_\infty \left\| (1+|x|)^{1+\beta} \nabla \nu \right\|_\infty}{ \left(1+|x|\right)^{4+\beta}}\leq \frac{C}{\left(1+|x|\right)^{4+\beta}}.$$ The remaining terms are treated in a similar fashion. Finally, by inspecting the formula (\[def\_g15\]) for the solution operators $\mathfrak G_1$ and $\mathfrak G_5$ corresponding to $\nu$ and $A_t$, respectively, we see that $$\begin{aligned} &\mathfrak G_1[\zeta; \gamma, \lambda](\varrho,z) + \frac{1}{|\mathbb S^2|} \int_{\mathbb R^3} \frac{\mathfrak M_1[\zeta; \gamma, \lambda](\varrho_y, z_y)}{|y|} \mathrm dy, \\ &\mathfrak G_5[\zeta; \gamma, \lambda](\varrho,z) + \frac{1}{|\mathbb S^2|} \int_{\mathbb R^3} \frac{\mathfrak M_5[\zeta; \gamma, \lambda](\varrho_y, z_y)}{|y|} \mathrm dy\end{aligned}$$ decay towards spatial infinity, also by Lemma \[lem\_decay\]. Note that in Lemma \[lem\_decay\] we have seen that for the functions $\nu$ and $A_t$ the decay is improved, form $(1+|x|)^{-(1+\beta)}$ to $(1+|x|)^{-2}$, i.e. assuming the weaker decay of $\nu, A_t \in \mathcal X_1$ we obtain the stronger decay of $\mathfrak G_1[\zeta; \gamma, \lambda]$, $\mathfrak G_5[\zeta; \gamma, \lambda]$. This is important in the proof that the Fréchet derivative of these components at $(\zeta_0; 0,0)$ is a compact operator, which in turn plays a role in the proof that this derivative is a bijection, cf. Lemma \[lem\_bijection\] below and [@akr11 Lemma 6.2]. All required properties of $\mathfrak G[\zeta; \gamma, \lambda]$ are now verified, thus the operator $\mathfrak F$ is well defined. The Fréchet derivative of $\mathfrak F$ {#sect_frechet_dir} ======================================= We denote the functions $\nu$, $h$, $\xi$, $\omega$, $A_t$, $a$ constituting the collection $\zeta$ by $\zeta_1, \dots, \zeta_6$, if convenient. The Fréchet derivative of $\mathfrak G_i$ with respect to $\zeta_j$ at $(\zeta; \gamma, \lambda)$ is a linear operator from $\mathcal X_j$ to $\mathcal X_i$, $i,j=1,\dots,6$. Here and in the remainder of the article by $\mathcal X_5$ and $\mathcal X_6$ we mean $\mathcal X_1$ and $\mathcal X_4$, respectively, since these are the function spaces corresponding to $\zeta_5$ and $\zeta_6$, respectively. We denote the Fréchet derivative by $$D_{\zeta_j} \mathfrak G_i[\zeta; \gamma, \lambda] : \mathcal X_j \to \mathcal X_i, \quad \delta \zeta_j \mapsto \left(D_{\zeta_j} \mathfrak G_i[\zeta; \gamma, \lambda]\right)\delta \zeta_j.$$ The operators $\mathfrak G_i:\mathcal U \to \mathcal X_i$, $i=1,\dots,6$ are continuous and continuously Fréchet differentiable with respect to $\nu, \xi, h, \omega, A_t, a$. The operators $\mathfrak G_i$, $i=1,2,4,5,6$ are of similar structure and we will start by analysing these operators. Schematically one can write these operators as sums of expressions of the form $$\mathfrak G_\Phi[\zeta; \gamma,\lambda](\varrho,z) = \int_{\mathbb R^n} G^n_y(|\varrho|,0,\dots, 0, z) \, \Phi^{(\gamma, \lambda)}(\varrho(y), \zeta(y), \zeta_{,\varrho}(y), \zeta_{,z}(y)) \, \mathrm dy$$ where the function $\Phi^{(\gamma, \lambda)}:\mathbb R^{19}\to\mathbb R$ is a placeholder for either $g_i^{(\gamma)}$ or $M_i^{(\gamma, \lambda)}$. In order to write this in a compact and handy way we define the functional $\tilde G_n$ (which is slightly different from $G_n$, cf. the definition (\[def\_greens\_gn\]) of $G_n$) by $$\tilde G_n\!\left[\Phi^{(\gamma,\lambda)},\zeta \right]\!(\varrho, z) := \int_{\mathbb R^n} G^n_y(|\varrho|,0,\dots, 0, z) \, \Phi^{(\gamma, \lambda)}(\varrho(y), \zeta(y), \zeta_{,\varrho}(y), \zeta_{,z}(y)) \, \mathrm dy.$$ We will check now that the Fréchet derivative of $\mathfrak G_\Phi$ with respect to $\zeta_j$ is given by $$\begin{gathered} \label{f_der_dphi} \left(D_{\zeta_j} \mathfrak G_\Phi[\zeta; \gamma, \lambda] \delta \zeta_j\right)(\varrho, z) \\= \tilde G_n\!\left[\left(\partial_{\zeta_j}\Phi^{(\gamma, \lambda)} \delta \zeta_j\right) + \left(\partial_{\zeta_{j,\varrho}}\Phi^{(\gamma, \lambda)} \, \partial_\varrho\left(\delta \zeta_{j}\right) \right) + \left(\partial_{\zeta_{j,z}} \Phi^{(\gamma, \lambda)} \, \partial_z\left(\delta \zeta_{j}\right)\right), \zeta\right].\end{gathered}$$ So we have to check that $$\begin{aligned} &\Big\| \tilde G_n[\Phi^{(\gamma, \lambda)},\zeta+\delta \zeta_j] - \tilde G_n[\Phi^{(\gamma, \lambda)},\zeta] \\ &\quad - \tilde G_n\!\left[\left(\partial_{\zeta_j}\Phi^{(\gamma, \lambda)} \delta \zeta_j\right) + \left(\partial_{\zeta_{j,\varrho}}\Phi^{(\gamma, \lambda)} \, \partial_\varrho\left(\delta \zeta_{j}\right) \right) + \left(\partial_{\zeta_{j,z}} \Phi^{(\gamma, \lambda)} \, \partial_z\left(\delta \zeta_{j}\right)\right), \zeta\right]\Big\|_{\mathcal X_\Phi} \nonumber \\ &= o\left(\|\delta \zeta_j\|_{\mathcal X_\Phi}\right). \nonumber \end{aligned}$$ Here $\mathcal X_\Phi$ is the function space corresponding to $\Phi^{(\gamma,\lambda)}$. I.e. if $\Phi^{(\gamma,\lambda)}$ is for example $M_1^{(\gamma,\lambda)}$ then $\mathcal X_\Phi$ is $\mathcal X_1$. Define $m$ as the number how often functions in $\mathcal X_\Phi$ are continuously differentiable, i.e. the largest number such that $\mathcal X_\Phi \subset C^{m,\alpha}$. By the standard elliptic estimate [@ll00 Theorem 10.3] and the inclusion $C^{m+1} \subset C^{m,\alpha}$ it suffices to check $$\begin{aligned} &\sum_{|\sigma|\leq {m-1}} \bigg\| \partial^\sigma \bigg( \Phi^{(\gamma, \lambda)}(\cdot, \zeta + \delta \zeta_j, \nabla (\zeta+\delta \zeta_j)) - \Phi^{(\gamma, \lambda)}(\cdot, \zeta, \nabla \zeta) \label{cond_diffbar} \\ & \hspace{2.5cm} - \partial_{\zeta_j}\Phi^{(\gamma, \lambda)}(\cdot,\zeta,\nabla\zeta) \delta \zeta_j - \partial_{\zeta_{j,\varrho}}\Phi^{(\gamma, \lambda)}(\cdot,\zeta,\nabla\zeta) \, \partial_\varrho\left(\delta \zeta_{j}\right) \nonumber \\ & \hspace{6.1cm} - \partial_{\zeta_{j,z}} \Phi^{(\gamma, \lambda)}(\cdot,\zeta,\nabla\zeta) \, \partial_z \left(\delta \zeta_{j}\right) \bigg) \bigg\|_\infty \leq o( \| \delta \zeta_j \|_{\mathcal X_i}). \nonumber\end{aligned}$$ It turns out that (\[cond\_diffbar\]) holds if the functions $\Phi^{(\gamma, \lambda)}$ are sufficiently regular, i.e. in $C^{m}$ to be precise. Now, $\Phi^{(\gamma, \lambda)}$ is either a source function $g_i^{(\gamma)}$ or a matter function $M_i^{(\gamma,\lambda)}$. The source functions are smooth in all of the variables $\zeta$, $\zeta_{,\varrho}$, and $\zeta_{,z}$, since they involve only the exponential function and addition, multiplication and division by $1+h$. Note here that $1+h > \frac 12$ if $(\zeta; \gamma, \lambda)\in\mathcal U$. For the matter functions $M_i^{(\gamma, \lambda)}$, $i=1,2,4,5,6$, defined in equations (\[def\_hat\_m\_1\])–(\[def\_hat\_m\_6\]), we note that they do not depend on derivatives of $\zeta$ and that the regularity is determined by the functions $M_{(\gamma,\lambda)}[g]$ which are all $C^3$ by Lemma \[lem\_reg\_m\] and this is sufficient. The operator $\mathfrak G_3$ is easier to treat since the expression (\[def\_g3\]) can be expanded explicitly in powers of $\delta h$, $\delta \nu$, $\delta \omega$, $\delta A_t$, and $\delta a$. Note again that $1+h$ is bounded away from zero for all $(\zeta; \gamma, \lambda)\in \mathcal U$. In the next step we calculate the Fréchet derivatives of $\mathfrak G_i$, $i=1,\dots,6$ and evaluate them at $(\zeta_0;0,0)$. The parts of $\mathfrak G_i$, $i = 1,\dots,6$ involving the source functions $\mathfrak g_i$ can be expanded directly, i.e. we calculate the Fréchet derivative at $(\zeta_0; 0, 0)$ by replacing $\mathfrak g_i[\zeta;\gamma](\varrho, z)$ in the integral expressions (\[def\_g15\])–(\[def\_g46\]) with the $\epsilon$-derivatives of $\mathfrak g_i[\zeta + \epsilon \delta\zeta_j;\gamma](\varrho, z)$ evaluated at $\epsilon = 0$ and then at $(\zeta_0;0,0)$. The non-zero derivatives are $$\begin{aligned} \left[\partial_\epsilon \mathfrak g_1[\zeta + \epsilon \delta h;\gamma](\varrho, z) \Big|_{\epsilon = 0}\right]_{(\zeta; \gamma, \lambda)=(\zeta_0;0,0)} &= - (\nabla U_N \cdot \nabla \delta h)(\varrho, z), \\ \left[\partial_\epsilon \mathfrak g_3[\zeta + \epsilon \delta h;\gamma](\varrho, z) \Big|_{\epsilon = 0}\right]_{(\zeta; \gamma, \lambda)=(\zeta_0;0,0)} &= \frac{\varrho}{2} (\partial_{\varrho\varrho} \delta h - \partial_{zz} \delta h)( \varrho, z) + \partial_\varrho \delta h(\varrho, z), \\ \left[\partial_\epsilon \mathfrak g_5[\zeta + \epsilon \delta h;\gamma](\varrho, z) \Big|_{\epsilon = 0}\right]_{(\zeta; \gamma, \lambda)=(\zeta_0;0,0)} &= - (\nabla A_N \cdot \nabla \delta h)(\varrho, z).\end{aligned}$$ The notation here should be interpreted as $\zeta + \epsilon \delta h = (\nu, h + \epsilon \delta h, \xi, \omega, A_t, a)$. For the parts involving the matter functions we use formula (\[f\_der\_dphi\]), where $\Phi^{(\gamma, \lambda)}$ is replaced by the matter functions $M_i^{(\gamma, \lambda)}$, $i=1,\dots,6$, given in (\[def\_hat\_m\_1\])–(\[def\_hat\_m\_6\]). The matter functions $M_i^{(\gamma, \lambda)}$, $i=1,\dots,6$ depend only on $\zeta$ and not on its derivatives. First we consider the matter functions $$\begin{aligned} M_1^{(\gamma,\lambda)}(\varrho, \zeta) &:= 4\pi e^{2(\xi-\gamma\nu)} \int_{\mathbb R_v^3} \phi(E)\psi(\lambda, L) \frac{1 + 2\gamma |v|^2}{\sqrt{1+\gamma |v|^2}} \, \mathrm d^3v, \\ M_2^{(\gamma,\lambda)}(\varrho, \zeta) &:= 8\pi \gamma^2 (1+h) e^{2(\xi-\gamma\nu)} \int_{\mathbb R_v^3} \phi(E)\psi(\lambda, L) \frac{(v^1)^2+(v^2)^2}{\sqrt{1+\gamma |v|^2}} \, \mathrm d^3 v,\\ M_4^{(\gamma,\lambda)}(\varrho, \zeta) &:= -\frac{16\pi \gamma}{\varrho(1+h)} e^{2\xi}\int_{\mathbb R_v^3} \phi(E)\psi(\lambda, L) v^3\,\mathrm d^3 v, \\\end{aligned}$$ of the Einstein equations, given in (\[def\_hat\_m\_1\])–(\[def\_hat\_m\_4\]), where $\mathrm d^3v = \mathrm dv^1\mathrm dv^2 \mathrm dv^3$. If one calculates the derivative of $M_i^{(\gamma, \lambda)}(\varrho, \zeta)$, $i=1,2,4$, with respect to any of the arguments $\nu, h, \xi, \omega, A_t, a$ one obtains back an expression with the same structure, possibly with the function $\partial_{\zeta_j}(\phi(E)\psi(\lambda,L))$ instead of $\phi(E)\psi(\lambda,L)$ in the integral. In the limit $\gamma\to 0$ only the terms where the $\gamma$-factors cancel will remain. Thus $\partial_{\zeta_j}M_i^{(\gamma, \lambda)}(\varrho, \zeta) |_{\gamma=0} = 0$ for $i=2,4$, and $$\begin{aligned} \partial_{\xi} M_1^{(\gamma, \lambda)}(\varrho, \zeta) \Big|_{\gamma=0} &= 2 \mathfrak M_1[\zeta; \gamma, \lambda] \Big|_{\gamma=0}, \\ \partial_{\zeta_j} M_1^{(\gamma, \lambda)}(\varrho, \zeta) \Big|_{\gamma=0} &= 4\pi e^{2(\xi - \gamma\nu)} \int_{\mathbb R_v^3} \partial_{\zeta_j} \left( \phi(E) \psi(\lambda,L) \right) \sqrt{1 + \gamma |v|^2}\, \mathrm d^3 v\Big|_{\gamma=0}, \label{second_d_m}\end{aligned}$$ where $j=1,2,4,5,6$. Consider now the term $\partial_{\zeta_j}(\phi(E)\psi(\lambda,L))$ in (\[second\_d\_m\]). First we observe that the assumption $\psi(0,L) = 1$ implies $\partial_L \psi(0,L) = 0$. This yields already $$\begin{aligned} \lim_{(\gamma,\lambda) \to (0,0)} \left[\partial_{\zeta_j}(\phi(E) \psi(\lambda, L))\right]_{\zeta = \zeta_0} &= \lim_{(\gamma,\lambda) \to (0,0)} \left[\psi(\lambda, L) \partial_{\zeta_j}\phi(E) \right]_{\zeta = \zeta_0} \\ &= \lim_{(\gamma,\lambda) \to (0,0)} \left[\partial_{\zeta_j}\phi(E) \right]_{\zeta = \zeta_0}. \label{ext_psi_ass}\end{aligned}$$ If we now set $\zeta=\zeta_0$ and consider the limit $(\gamma, \lambda) \to (0, 0)$ only the derivatives with respect to $\nu$ and $A_t$ are non-vanishing. The derivative with respect to $a$ vanishes due to (\[ext\_psi\_ass\]) and the fact that $E$ is independent of $a$. The derivatives with respect to $h$, $\xi$, $\omega$ vanish by symmetry. This can be seen as follows. We have $$E|_{\zeta = \zeta_0} = \frac{e^{\gamma \nu_N} \sqrt{1+\gamma |v|^2}- 1}{\gamma} + q A_N,$$ and therefore $$\label{lim_p_p} \lim_{(\gamma, \lambda) \to (0, 0)} \phi \left(E|_{\zeta = \zeta_0}\right) \psi \left( \lambda, L|_{\zeta = \zeta_0} \right) = \phi\left(\frac{|v|^2}{2} + \nu_N + q A_N\right)$$ where the Newtonian limit (\[newtonian\_l\_e\]) of the energy and the assumption $\psi(0, L)=1$ on $\psi$ has been used. Observe that the limit (\[lim\_p\_p\]) is even in $v^1$, $v^2$, $v^3$. Consider next the derivatives $$\partial_h E = \varrho \omega e^{-\gamma\nu} v^3, \quad \partial_\xi E = 0, \quad \partial_\omega E = \varrho (1+h) e^{-\gamma\nu} v^3.$$ These derivatives are either zero or odd in $v^3$. Integration over an odd-in-$v^3$ function yields zero. For the derivatives with respect to $\nu$ and $A_t$ the same principles apply, however not all terms vanish. Consider for example $\partial_\nu M_1^{(\gamma, \lambda)}(\varrho, \zeta)$. One obtains $$\begin{aligned} \partial_\nu M_1^{(\gamma,\lambda)}(\varrho, \zeta) &= -8\pi \gamma e^{2(\xi-\gamma\nu)} \int_{\mathbb R_v^3} \phi(E)\psi(\lambda, L) \frac{1 + 2\gamma |v|^2}{\sqrt{1+\gamma |v|^2}}\, \mathrm d^3v \\ & \quad + 4\pi e^{2(\xi-\gamma\nu)} \int_{\mathbb R_v^3} \phi(E)' \psi(\lambda, L) \left(e^{\gamma\nu} + \frac{\gamma \omega \varrho (1+h) v^3}{\sqrt{1+\gamma |v|^2}} \right) \left(1 + 2\gamma |v|^2\right)\, \mathrm d^3v \\ & \quad - 4\pi \gamma \varrho (1+h) e^{2\xi - 3\gamma\nu} \int_{\mathbb R_v^3} \partial_\nu \phi(E) \partial_L\psi(\lambda, L) v^3 \frac{1 + 2\gamma |v|^2}{\sqrt{1+\gamma |v|^2}}\, \mathrm d^3v.\end{aligned}$$ So the derivatives of the matter function $M_1^{(\gamma, \lambda)}$ of the Einstein equations which are non-vanishing at $(\zeta_0; 0, 0)$ are $$\begin{aligned} \partial_{\nu} M_1^{(\gamma, \lambda)} \Big|_{(\zeta_0; 0,0)} &= 4\pi \alpha_N, \\ \partial_{\xi} M_1^{(\gamma, \lambda)} \Big|_{(\zeta_0; 0,0)} &= 8\pi \rho_N, \\ \partial_{A_t} M_1^{(\gamma, \lambda)} \Big|_{(\zeta_0; 0,0)} &= 4\pi q \alpha_N,\end{aligned}$$ where $\rho_N$ and $\alpha_N$ are defined in (\[def\_rho\_u\]) and (\[def\_alpha\_u\]), respectively. Next we consider the matter functions $$\begin{aligned} M_5^{(\gamma,\lambda)}(\varrho, \zeta) &:= 4\pi q e^{2\xi-3\gamma\nu} \int_{\mathbb R_v^3} \phi(E)\psi(\lambda, L) \left(e^{2\gamma\nu} + \frac{\gamma \varrho (1+h) \omega v^3}{\sqrt{1+\gamma |v|^2}}\right) \, \mathrm d^3 v, \\ M_6^{(\gamma,\lambda)}(\varrho, \zeta) &:= -4\pi q \gamma \varrho (1+h) e^{2\xi-3\gamma\nu} \int_{\mathbb R_v^3} \phi(E)\psi(\lambda, L) \frac{v^3}{\sqrt{1+\gamma |v|^2}}\, \mathrm d^3 v,\end{aligned}$$ of the Maxwell equations in the representations given in (\[def\_hat\_m\_5\])–(\[def\_hat\_m\_6\]). The first observation is that if $\gamma=0$ then all terms but the first one of $M_5^{(\gamma, \lambda)}$ vanish. So we only need to discuss the derivatives of $$4\pi q e^{2\xi-\gamma\nu} \int_{\mathbb R_v^3} \phi(E) \psi(\lambda,L) \, \mathrm d^3 v.$$ By the same reasoning as above we obtain $$\begin{aligned} \partial_{\nu} M_5^{(\gamma, \lambda)} \big|_{(\zeta_0; 0,0)} &= 4 \pi q \alpha_N, \\ \partial_{\xi} M_5^{(\gamma, \lambda)} \big|_{(\zeta_0; 0,0)} &= 8 \pi q \rho_N, \\ \partial_{A_t} M_5^{(\gamma, \lambda)} \big|_{(\zeta_0; 0,0)} &= 4 \pi q^2 \alpha_N.\end{aligned}$$ We denote the Fréchet derivative of $\mathfrak F$ with respect to $\zeta$, at $(\zeta_0;0,0)$, by $\mathfrak L$, i.e. $$\begin{aligned} &\mathfrak L := D\mathfrak F[\zeta_0; 0,0]: \mathcal X \to \mathcal X, \label{def_l} \\ &\delta \zeta \mapsto \mathfrak L(\delta \zeta) = (\delta \nu - \mathfrak L_1(\delta \nu, \delta h, \delta \xi, \delta A_t), \delta h, \delta \xi - \mathfrak L_3(\delta h), \delta \omega, \delta A_t - \mathfrak L_5(\delta\nu, \delta \xi, \delta A_t), \delta a), \nonumber\end{aligned}$$ where $$\begin{aligned} \mathfrak L_1(\delta \nu, \delta h, \delta \xi, \delta A_t) &= -\mathfrak L_1^{(1)}(\delta \nu + q \delta A_t) - \mathfrak L_1^{(2)} (\delta \xi) + \mathfrak L_1^{(3)}(\delta h), \\ \mathfrak L_3(\delta h) &= \delta h(0,z) + \int_0^\varrho \left(\frac{s}{2} (\partial_{\varrho\varrho} \delta h - \partial_{zz} \delta h)(s,z) + \partial_\varrho \delta h(s, z) \right) \mathrm ds,\\ \mathfrak L_5(\delta \nu, \delta h, \delta \xi, \delta A_t) &= q \mathfrak L_1^{(1)}(\delta \nu + q \delta A_t) + q \mathfrak L_1^{(2)} (\delta \xi) + q \mathfrak L_5^{(3)}(\delta h),\end{aligned}$$ where $$\begin{aligned} \mathfrak L_1^{(1)}(\delta u) &= \int_{\mathbb R^3} \left(\frac{1}{|x-y|} - \frac{1}{|y|} \right) \alpha_N(|y|) \delta u(\varrho_y, z_y) \, \mathrm dy, \\ \mathfrak L_1^{(2)}(\delta \xi) &= 2 \int_{\mathbb R^3} \left(\frac{1}{|x-y|} - \frac{1}{|y|} \right) \rho_N(|y|) \delta \xi(\varrho_y, z_y) \, \mathrm dy, \\ \mathfrak L_1^{(3)}(\delta h) &= \frac{1}{4\pi} \int_{\mathbb R^3} \frac{1}{|x-y|} \nabla \nu_N(|y|) \cdot \nabla (\delta h)(\varrho_y, z_y) \, \mathrm dy, \\ \mathfrak L_5^{(3)}(\delta h) &= \frac{1}{4\pi} \int_{\mathbb R^3} \frac{1}{|x-y|} \nabla A_N(|y|) \cdot \nabla (\delta h)(\varrho_y, z_y) \, \mathrm dy.\end{aligned}$$ The shorthands $\rho_N = \rho_{U_N}$ and $\alpha_N = \alpha_{U_N}$ are defined in (\[def\_rho\_u\]) and (\[def\_alpha\_u\]), respectively, where $U_N = \nu_N + q A_N$, and the functions $\nu_N$ and $A_N$ are defined as the solutions of the system (\[newt\_1\])–(\[newt\_2\]). \[lem\_bijection\] $\mathfrak L$ is a bijection. First we prove that $\mathfrak L$ is injective. Since $\mathfrak L$ is linear it suffices to show that $\mathrm{ker}(\mathfrak L) = 0$. Let $\delta \zeta \in \mathcal X$ such that $\mathfrak L(\delta \zeta) = 0$. From the definition of $\mathfrak L$ in (\[def\_l\]) we immediately read off $\delta h = \delta \omega = \delta a = 0$. Consequently $\mathfrak L_3(\delta h)=0$ and therefore also $\delta \xi = 0.$ Since $\delta h = \delta\xi = 0$ and thus $\mathfrak L_1^{(3)}(\delta h) = \mathfrak L_5^{(3)}(\delta h) = \mathfrak L_1^{(2)}(\delta \xi) =0$ we can furthermore read off $\delta A_t = -q \delta \nu$. We finish the proof of injectivity by showing that $\delta \nu + q \delta A_t = 0$. To simplify notation we denote in the following $\delta u = \delta\nu + q\delta A_t \in \mathcal X_1$. Those two identities will then imply $(1-q^2)\delta u=0$ and therefore $\delta \nu=0$ and $\delta A_t = 0$. Adding the first and $q$ times the fifth component of $\mathfrak L(\delta\zeta) = 0$ yields $$\delta u = - \left(1-q^2\right) \int_{\mathbb R^3} \left(\frac{1}{|x-y|} - \frac{1}{|y|} \right) \alpha_N(|y|) \, \delta u(\varrho_y, z_y) \, \mathrm dy.$$ This is a solution of $$\begin{aligned} \Delta (\delta u) &= \left( 1 - q^2 \right) \alpha_N \, \delta u, \label{zero_p_eq_1} \\ (\delta u)(0) &= 0. \label{zero_p_eq_2}\end{aligned}$$ In [@akr11 Section 6] it has been shown that this is the only solution of (\[zero\_p\_eq\_1\])–(\[zero\_p\_eq\_2\]), provided that $6+4\pi r^2 (1-q^2) \alpha_N(r) > 0$ which is assumed. Next we show that $\mathfrak L$ is surjective. Let $b=(b_1,\dots, b_6)\in \mathcal X$ be given. The aim is now to construct $\delta \zeta = (\delta \nu, \delta h, \delta \xi, \delta\omega, \delta A_t, \delta a)\in\mathcal X$ such that $$\label{ldzg} \mathcal L(\delta \zeta) = b.$$ By inspecting the formula (\[def\_l\]) of $\mathfrak L$ we immediately see that we have to choose $\delta h = b_2$, $\delta \omega = b_4$, $\delta a = b_6$. In the third component of (\[ldzg\]) we obtain $$\delta \xi = b_3 + \mathfrak L_3(\delta h),$$ which is in $\mathcal X_3$ since $\mathfrak L_3(\delta h) \in \mathcal X_3$. It remains to construct $\delta \nu$ and $\delta A_t$. Note first that $\mathfrak L_1^{(2)}(\delta\xi), \mathfrak L_1^{(3)}(\delta h), \mathfrak L_5^{(3)}(\delta h) \in \mathcal X_1$ (recall $\mathcal X_5 = \mathcal X_1$). We add the first component of (\[ldzg\]) and $q$ times the fifth component of (\[ldzg\]). We obtain $$\delta u - \left(1-q^2\right) \mathfrak L_1^{(1)}(\delta u) = (b_1 + q b_5) - \left(1-q^2\right) \mathfrak L_1^{(2)}(\delta \xi) + \left(\mathfrak L_1^{(3)} + q^2 \mathfrak L_5^{(3)}\right) (\delta h).$$ This equation has a solution $\delta u \in \mathcal X_1$ since the operator $\mathfrak L_1^{(1)}$ is compact. This has been established in [@akr11 Lemma 6.2]. Then, considering the first component of (\[ldzg\]) again, we can construct $\delta \nu$ via $$\delta \nu = b_1 - \mathfrak L_1^{(1)}(\delta u) - \mathfrak L_1^{(2)}(\delta \xi) +\mathfrak L_1^{(3)}(\delta h).$$ Finally, we obtain $\delta A_t$ via $\delta A_t = \frac{1}{q} (\delta u-\delta\nu)$. Application of the implicit function theorem ============================================ In the preceding sections we have established that the solution operator $\mathfrak F$ fulfils the assumptions of the implicit function theorem for Banach spaces. Now we can prove the following proposition. \[prop\_appli\] There exist solutions $\zeta = (\nu, h, \xi, \omega, A_t, a)$ to the reduced EVM-system (\[final\_eq\_nu\])–(\[final\_eq\_a\]) with parameters $\gamma \in [0,\delta)$, $\lambda \in (-\delta, \delta)$ if $\delta$ is chosen sufficiently small that satisfy the boundary conditions (\[bc\_infinity\]) and (\[bc\_center\]). The solution $\zeta = (\nu, h, \xi, \omega, A_t, a)$ exists by virtue of the implicit function theorem. The functions $\omega$, $\xi$, $h$, and $a$ fulfil the boundary condition $$\lim_{|(\varrho, z)\to \infty} (|\omega| + |\xi| + |h| + |a|) = 0$$ by construction. For $\omega$ and $a$ see the definition (\[def\_norm\_x4\]) of the norm of the space $\mathcal X_4$. Analogously, with Lemma \[lem\_decay\], it follows that $h$ fulfils the boundary condition. By inspecting the structure (\[def\_g3\]) of the solution operator $\mathfrak G_3$ one easily sees that the boundary condition $$ \label{bc_center_2} \xi(0,z) = \ln(1+h(0,z))$$ is satisfied, too. For the boundary condition of $\xi$ at infinity one infers first from (\[bc\_center\_2\]) that $\lim_{|z|\to\infty}\xi(0,z) = 0$, and then the decay as $\varrho\to\infty$ can be deduced from the decay of the integrand of the solution operator $\mathfrak G_3$, cf. formula (\[def\_g3\]) and [@akr11 Prop. 2.3]. The solution functions $\nu$, $A_t$ obtained from the implicit function theorem do however a priori not satisfy the boundary condition $$\lim_{|(\varrho, z)| \to \infty} (|\nu| + A_t) = 0$$ and we define $$\nu_\infty^{\gamma, \lambda} := \lim_{|(\varrho, z)| \to \infty} |\nu|, \quad A_\infty^{\gamma, \lambda} := \lim_{|(\varrho, z)| \to \infty} |A_t|.$$ A rescaling is necessary. The functions $$\nu - \nu_\infty^{\gamma, \; \lambda}, \; \mu + \gamma \nu_\infty^{\gamma, \lambda}, \; h, \; e^{-\gamma \nu_\infty^{\gamma, \lambda}}, \; e^{-\gamma\nu_\infty^{\gamma, \lambda}} (A_t-A_\infty^{\gamma,\lambda}), \; e^{\gamma \nu_\infty^{\gamma, \lambda}} a$$ then fulfil the reduced EVM-system with the boundary conditions which correspond to an asymptotically flat solution. Appendix {#appendix .unnumbered} ======== Recall the definition of the transport operator, $$\mathfrak T = p^\mu \partial_\mu + \left(q F^\gamma{}_{\mu} \, p^\mu - \Gamma^\gamma_{\alpha\beta} p^\alpha p^\beta\right) \partial_{p^\gamma}.$$ Now it shall be expressed with respect to the frame coordinates (\[def\_frame\]). First we derive the form of $\mathfrak T$ with respect to a general orthonormal frame $e_a = e_a{}^\alpha \partial_{x^\alpha}$ (and corresponding co-frame $\alpha^b = e^b{}_\beta \mathrm dx^\beta$). Using the definitions $$\Gamma_{\alpha\beta}^\gamma = \mathrm dx^\gamma \left(\nabla_\beta \partial_\alpha \right), \quad \Gamma_{ab}^c = \alpha^c \left(\nabla_{e_b} e_a \right)$$ for $\Gamma_{\alpha\beta}^\gamma$ and $\Gamma_{ab}^c$ one derives the transformation law $$\label{trafo_christoffels} \Gamma_{ab}^c = e^c{}_\alpha e_b{}^\beta \partial_\beta e_a{}^\alpha + e^c{}_\gamma e_b{}^\beta e_a{}^\alpha \Gamma_{\alpha\beta} ^\gamma.$$ Furthermore the change of variables $$x^\mu \mapsto y^\mu = x^\mu, \quad p^\nu \mapsto v^a = e^a{}_\nu p^\nu$$ entails the replacements $$\label{trafo_derivatives} \partial_{x^\mu} = \partial_{y^\mu} + e_b{}^\alpha v^b \partial_\mu e^a{}_\alpha \partial_{v^a}, \quad \partial_{p^\nu} = e^a{}_\nu \partial_{v^a}.$$ This yields $$\mathfrak T = v^a e_a{}^\alpha \partial_\alpha + \left(q F^c{}_a v^a- \Gamma_{ab}^c v^a v^b \right) \partial_{v^c}.$$ In order to obtain the explicit expression for the transport operator $\mathfrak T$ with respect to the frame coordinates (\[the\_coords\]) we apply the transformation laws (\[trafo\_christoffels\]) and (\[trafo\_derivatives\]) to the frame (\[frame\_matrix\]), where the Christoffel symbols $$\Gamma_{\alpha\beta}^\gamma = \frac 12 g^{\gamma\delta} \left( \partial_\alpha g_{\beta\delta} + \partial_\beta g_{\delta\alpha} - \partial_\delta g_{\alpha\beta} \right)$$ are calculated from the ansatz (\[ansatz\_metric\]) for the metric. The transport operator is then explicitly given by $$\begin{aligned} \mathfrak T &= v^0 e^{-\gamma\nu} \partial_t + e^{-\mu} (v^1 \partial_\varrho + v^2 \partial_z) + \left(v^0 e^{-\gamma\nu} \omega + v^3 \frac{e^{\gamma\nu}}{\varrho H}\right) \partial_\varphi \\ &\quad -q e^{-\mu - \frac{\nu}{c^2}} \left(\left( A_{t,\varrho} + \omega A_{\varphi,\varrho} \right) \Omega_{01}^V + \left(A_{t,z} + \omega A_{\varphi,z}\right) \Omega_{02}^V \right) \\ &\quad + \frac{q}{\varrho H} e^{-\mu + \frac{\nu}{c^2}} \left(A_{\varphi,\varrho}\Omega_{13}^V + A_ {\varphi,z} \Omega_{23}^V \right) + q e^{-2\mu} \left(A_{\varrho,z} - A_{z,\varrho}\right) \Omega_{21}^V \\ &\quad + e^{-\mu} \frac{v^3}{c^2} \left(\nu_{,\varrho} \Omega_{13}^V + \nu_{,z} \Omega_{23}^V \right) - e^{-\mu} v^0 \left(\nu_{,\varrho} \Omega_{01}^V + \nu_{,z} \Omega_{02}^V \right) + e^{-\mu} \left(v^2 \mu_{,\varrho} - v^1 \mu_{,z}\right) \Omega_{21}^V \\ &\quad+ e^{-\mu} \frac{v^3}{H} \left(H_{,\varrho} \Omega_{31}^V + H_{,z} \Omega_{32}^V \right) + \frac{v^3}{\varrho} e^{-\mu} \Omega_{31}^V - e^{-\mu - 2\frac{\nu}{c^2}} \varrho H v^3 \left(\omega_{,\varrho}\Omega_{01}^V + \omega_{,z} \Omega_{02}^V \right)\end{aligned}$$ where we use the shorthands $$\begin{aligned} \Omega_{ij}^V := v^i \partial_{v^j} - v^j \partial_{v^i}, \quad \Omega_{0i}^V := \frac{v^i}{c^2} \partial_{v^0} + v^0 \partial_{v^i}.\end{aligned}$$ Now, the transport operator can be applied to the quantities $$\begin{aligned} L &= \varrho H e^{ -\gamma\nu} v^3 - q A_\varphi, \\ E &= \frac{e^{\gamma\nu} v^0 - 1}{\gamma} + \omega \varrho H e^{ -\gamma\nu} v^3 + q A_t,\end{aligned}$$ where we note that $E$ only depends on the variables $\varrho$, $z$, $v^0$, and $v^3$, and $L$ only depends on the variables $\varrho$, $z$, and $v^3$. 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--- abstract: '[The method of Lagrangian Descriptors (LDs) has been applied in many different contexts, specially in geophysical flows. In this paper we analyze their performance in incompressible flows. We construct broad families of systems where this diagnostic fails in the detection of barriers to transport. Another aim of this manuscript is to illustrate the same deficiencies in the recent diagnostic proposed by Craven and Hernández.]{}' author: - '[Alfonso Ruiz-Herrera]{}[^1]' title: '**Performance of Lagrangian Descriptors and Their Variants in Incompressible Flows**' --- [**The method of Lagrangian Descriptors (LDs) is a procedure to detect barriers to transport. Originally proposed by Mancho, Wiggins and their co-workers, this method has a basic heuristic law: invariant manifolds (hyperbolic structures) are detected via singular features of the $M$-function. This function is defined in terms of the arc length of the orbits of the system. In this paper we present a large family of incompressible flows in which LDs do not detect these structures. Actually, we observe that in some cases all singular features are evanescent at the invariant manifolds. Although this method has been intensively applied in geophysical flows, the conclusion is that it is not founded on solid mathematical grounds. Generally speaking, the singularities associated with the $M$-function in a system are unrelated to the detection of invariant manifolds. Recently, Craven and Hernández have extended the method of LDs to the context of thermalized chemical reactions. In the second part of this manuscript we generalize the previous conclusions to this setting.**]{} Introduction ============ The method of Lagrangian Descriptors (LDs) is a diagnostic to detect barriers to transport in fluid dynamics or invariant manifolds in dynamical systems [@mendoza-mancho; @PRL]-[@lopesino]. This method has been mainly applied by Mancho, Wiggins, and their co-workers in different contexts involving transport phenomena [@mendoza-mancho; @PRL]-[@wiggins]. For instance, Mendoza and Mancho studied the skeleton of the flow for the Kuroshio current [@mendoza; @npg2012] or de la Cámara [*et al.*]{} provided barriers to transport in the Antarctic polar vortex [@camara; @1]. The conclusions of those papers are nevertheless doubtful because there is no mathematical foundation of LDs beyond heuristic arguments in simple systems. In this paper, we provide some families of incompressible flows where LDs do not detect the expected structures. LDs were originally designed to compute separatrices in flows with general dependence on time, a computation typically performed via Finite Time Lyapunov Exponents or the methods of Haller and his co-workers (variational and geodesic theories of Lagrangian Coherent Structures) [@haller]. The heuristic law of LDs is that invariant manifolds are associated with some singular features of the contour-lines of the $M$-function (this function is defined in terms of the arc length of the trajectories of the system). Generally speaking, such law seems questionable. For instance, a prerequisite for reliable predictions is the independence of the observer or objectivity [@chaos; @intro]. Unfortunately, as Haller nicely emphasized [@haller; @M; @haller], LDs are not objective. In a previous work [@ruizherrera], we presented several pathologies and examples where LDs do not detect barriers to transport. Although those examples included Hamiltonian systems, they were rather involved and non-autonomous, (with an aperiodic time dependence). One of the aims of this paper is to present simple families of incompressible flows for which LDs do not detect the invariant manifolds. Note that the performance of LDs in higher dimension is rather limited. The method is already questionable in $$\label{S25I}\left\{\begin{array}{lll} x_{1}'=0.5 x_{1}\\ x_{2}'= 1.5 x_{2}\\ x'_{3}=-2 x_{3}.\\ \end{array}\right.$$ In this system, the contour-lines of the $M$-function (for large values of $\tau$) are hyperplanes parallel to $\{x_{3}=0\}$. Thus there is no established link between the dynamical skeleton of (\[S25I\]) and the structure of the $M$-function over finite time. Recently, Lopesino [*et al.*]{} [@lopesino] have obtained some theoretical results in order to justify the performance of LDs in $$\label{S2512}\left\{\begin{array}{lll} x_{n+1}= \lambda x_{n}\\ y_{n+1}= \frac{1}{\lambda}y_{n}\;\;\;\;{\rm with}\;\;\lambda>1. \end{array}\right.$$ The analysis of (\[S2512\]) can be obtained with elementary tools but the application of LDs seems to suggest that the results in [@lopesino] could be extended to larger classes of systems. In the next section we conclude that this is not the case because the method does not work for a small perturbation of system (\[S2512\]). Another variant of the method of LDs has been presented by Cranven and Hernández to compute the invariant manifolds in thermalized chemical reactions [@craven; @and; @hernandez]-[@craven; @and; @hernandez; @1]. This new approach, reminiscent to the original paper by Madrid and Mancho [@mancho; @chaos], consists in minimizing the arc length of trajectories forwards in time to detect the stable manifold. The objective of this technique was to describe manifolds that separate reactive and non-reactive trajectories in the Langevin equation [@craven; @and; @hernandez]-[@craven; @and; @hernandez; @1]. Again, the presentation is in a heuristic style and the theoretical aspects are not developed. We will describe some simple examples in the context of the Langevin equation where invariant manifolds cannot be found via their method. An outline of the paper is as follows: Section 2 reviews the literature related to the method of LDs for the reader’s convenience. Section 3 deals with the examples of incompressible flows. The analysis is restricted to this class of systems but our results can be applied to a broader range of dynamical systems, in particular, in most linear systems. Section 4 deals with the Langevin equation in connection with the method employed by Craven and Hernández. Finally, we discuss the implications of our analysis in practical situations. Lagrangian Descriptors: Definitions and Mathematical Foundation =============================================================== LDs aim to describe the global dynamical picture of the geometrical structures for dynamical systems. In particular, they were used to detect stable and unstable manifolds and elliptic regions. This method in continuous systems typically involves the map defined as $$M(\tau; x_{0},t_{0})=\int_{t_{0}-\tau}^{t_{0}+\tau}\|v(x(t;t_{0},x_{0}),t)\|dt$$ where $x(t;t_{0},x_{0})$ is the unique solution satisfying $x(t_{0})=x_{0}$ for a smooth velocity field $$\label{eq} x'(t)=v(x(t),t).$$ The $M$-function was introduced by Madrid and Mancho in [@mancho; @chaos] and represents the arc length described by the trajectory starting at $x(t_{0})$ when it evolves forwards and backwards in time for a period $\tau$. This map is smooth except at equilibria. In discrete planar systems, we define for each $N\in\NN$, $$\label{Mp} MD_{p}(x_{0},y_{0})=\sum_{i=-N}^{N-1}|x_{i+1}-x_{i}|^{p}+|y_{i+1}-y_{i}|^{p}\;\;\;\;\;{\rm with}\;0<p<1.$$ with $\{x_{n}, y_{n}\}_{n=-N}^{n=N}$ an orbit of length $2N+1$ generated by a two-dimensional map. It is clear that $MD_{p}$ is non-smooth if, for some iteration, $x_{n+1}=x_n$ or $y_{n+1}=y_n$. Therefore, there are unbounded behaviours of the derivatives of $MD_{p}$ in those points independently of the dynamical behaviour of the system. As mentioned in page 3531 in [@mancho; @CNS], the heuristic law of LDs is: “singular contours of LDs correspond to invariant manifolds". We recall that in a steady system with a saddle equilibrium, the stable (resp. unstable) manifold is defined as the set of points attracting to that equilibrium forwards (resp. backwards) in time [@palis]. The concept of [*singular feature*]{} is never presented by Mancho, Wiggins, and their co-workers in a precise way which undoubtedly leads to ambiguity. Moreover, there is a lack of the definition of the structures it purports to detect. For the reader’s convenience, we consider the example discussed in page 3534 in [@mancho; @CNS], specifically $$\label{Sl1}\left\{\begin{array}{lll} x'=\lambda x\\ y'=-\lambda y\;\;\;\;{\rm with}\;\;\lambda>0. \end{array}\right.$$ The structure of $M$ for $\tau=20$ and $t_{0}=0$ displays “singular features" in the axes, the invariant manifolds of $(0,0)$ for system (\[Sl1\]). See FIG 1 (in this manuscript) or Figure 1 (c) or 2 (a) in [@mancho; @CNS], (see also Section 2.1.2 in [@mancho; @CNS]). ![ Contour-lines of $M$ in $(-1,1)\times(-1,1)$ associated with system (\[Sl1\]) for $\lambda=1$ and $\tau=20$. The reader can check that this figure coincides with Fig. 1 (c) and Fig. 2 (a) in [@mancho; @CNS] (for $\tau=10$).](fig1) Recently, Lopesino [*et al.*]{} [@lopesino] have provided mathematical theorems to support the performance of the discrete version of LDs. Specifically, they stated the following result. Let $$\label{S252}\left\{\begin{array}{lll} x_{n+1}=\lambda x_{n}\\ y_{n+1}=\frac{1}{\lambda} y_{n} \end{array}\right.$$ with $\lambda>1$. \[t10\] (Theorem 1 in [@lopesino]) Consider a vertical line perpendicular to the unstable manifold of the origin. In particular consider an arbitrary point $x=\bar{x}$ and a line parallel to the $y$ axis passing through this point. Then the derivative of $MD_{p}$ with $p<1$, along this line becomes unbounded on the unstable manifold of the origin. The proof is just a straightforward check that $MD_{p}(x_{0},y_{0})=(|x_{0}|^{p}+|y_{0}|^{p})f(\lambda, N, x_{0},y_{0})$ with $f$ a smooth map. We observe that Theorem \[t10\] is a consequence of the diagnostic itself because, as mentioned, $MD_{p}$ is non smooth if, for some iteration, $$\label{contonta} x_{n+1}=x_n\;\;\; or\;\;\; y_{n+1}=y_n.$$ Therefore, Theorem \[t10\] gives no specific mechanism for detecting invariant manifolds: it only works for systems satisfying (\[contonta\]). We discuss this fact with a concrete example. Consider $$\label{S}\left\{\begin{array}{lll} x_{n+1}= 2 x_{n}\\ y_{n+1}=\frac{1}{2} y_{n}+g(x_{n}) \end{array}\right.$$ with $g:\mathbb{R}\longrightarrow [0,\infty)$ a smooth function satisfying that $g(x)=0$ for all $x\not\in [0,1]$ and $g(x)>0$ if $x\in(0,1)$. The underlying map in (\[S\]) is area preserving and $(0,0)$ is a global saddle point. Fix $\bar{x}>2$, it is straightforward to prove that given a line parallel to the $y$ axis passing through this point, the derivative of $MD_{p}$ with $p<1$, along this line becomes unbounded at $y=0$. Note that $(\bar{x},0)$ is not a point at the invariant manifolds of (\[S\]). Thus, even under a slight modification of (\[S252\]), Theorem \[t10\] will give a false positive for an unstable or stable manifold. System (\[S252\]) also proves that Theorem \[t10\] does not provide a mechanism to approximate invariant manifolds when $N\longrightarrow\infty$, (this property was mentioned in Section 2.1.2 in [@lopesino]). Two remarks are in order: - In practical applications, Mancho, Wiggins, and their co-workers routinely use the contour-structure of the $M$-function (or $MD_{p}$ in discrete systems) to detect invariant manifolds, see [@mendoza-mancho; @PRL]-[@wiggins]. To support the applicability of LDs, their arguments consist in proving that certain derivatives of $M$ or $MD_{p}$ are unbounded as $\tau\longrightarrow \infty$, (see the theorems in [@lopesino] or Section 2.1.2 in [@mancho; @CNS]). Those arguments have no significance on the dynamical behaviour of the systems or the contour structure of the $M$-function because $M$ and $MD_{p}$ are typically unbounded as $\tau\longrightarrow\infty$. - In a recent work [@ruizherrera], we gave precise families of examples in which LDs fail in the detection of invariant manifolds. In our systems, the invariant manifolds were placed in the axes and the contour-lines of $M$ converge to horizontal lines as $\tau\longrightarrow \infty$, (the points inside these horizontal lines are indistinguishable for the function $M$). In this scenario, one can just deduce that the stable and unstable manifold are horizontal lines independently of the definition of singular point under consideration. Lagrangian descriptors and Incompressible Flows ================================================ In this section we illustrate that the method of LDs fails in the detection of invariant manifolds in 2D and 3D incompressible flows. Consider $$\label{Sl}\left\{\begin{array}{lll} x'=f(x)\\ y'=- y f'(x) \end{array}\right.$$ where $f:\mathbb{R}\longrightarrow \mathbb{R}$ is of class $\mathcal{C}^{2}$ with $f(0)=0$ and satisfying the following conditions: C1 : $f(x)>0$ if $x\in(0,\infty)$ and $f(x)<0$ if $x\in (-\infty,0)$. C2 : $f$ is bounded. C3 : $f'(0)\geq f'(x)>0$ for all $x\in\mathbb{R}$. We observe that (\[Sl\]) has a global saddle at the origin where the $x$-axis (resp $y$-axis) is the un-stable (resp. stable) manifold. However, the structure of $M$ is smooth in a neighbourhood of the $y$-axis as $\tau\rightarrow \infty$. Therefore, there are no singular features in a neighbourhood of the $y$-axis, (see FIG 2). ![Contour-lines of $M$ in $(-0.1,0.1)\times(-0.5,0.5)$ associated with system (\[Sl\]) for $f(x)=\tanh x$ and $\tau=20$. One observes a smooth pattern in a neighbourhood of the $y$-axis .](NEW) Next, we give an analytic proof to justify this fact. \[tmanin\] Assume that $f:\mathbb{R}\longrightarrow \mathbb{R}$ satisfies [**C1-C3**]{} and C4 : $f'(x)=k$ for all $x\in[-a,a]$ with $k,a>0$. Then, the contour lines of $M$ converge to horizontal lines as $\tau\longrightarrow \infty$ in a neighbourhood of the $y$-axis. **C4** can be relaxed in a great deal, (e.g. we can consider $f''$ bounded in $\mathbb{R}$, $f''(x)>0$ in $(-\infty,0)$ and $f''(x)<0$ in $(0,\infty)$). However, the mathematical proof of Theorem \[tmanin\] is much more involved. A prototypical function satisfying **C1**-**C4** is $$\label{f}f(x)= \left\{ \begin{array}{lcc} \arctan(x+1)-1 & if & x \leq -1 \\ \\ x & if & -1 < x < 1 \\ \\ \arctan(x-1)+1 & if & x \geq 1. \end{array} \right.$$ To prove Theorem \[tmanin\], we need the following lemma. \[L2\] Assume the conditions of Theorem \[tmanin\]. Given $(x_{0},y_{0})\in[-a,a]\times (0,\infty)$ then $$\lim_{\tau\longrightarrow \infty}\frac{M(\tau;x_{0},y_{0})}{M(\tau;0,y_{0})}=1.$$ **Proof.** First of all, we note that the solution of (\[Sl\]) with initial condition $(x_{0},y_{0})$ is given by $$(x(t;x_{0},y_{0}),y(t;x_{0},y_{0}))=(x(t;x_{0}),y_{0}e^{-\int _{0}^{t}f'(x(s;x_{0}))ds})$$ where $x(t;x_{0})$ is the solution of $x'=f(x)$ with initial condition $x_{0}$. By simple computations and using **C2** and **C3**, $$M(\tau;0,y_{0})=y_{0} (e^{f'(0)\tau}-e^{-f'(0)\tau}),$$ $$M(\tau;x_{0},y_{0})=\int _{-\tau}^{\tau}\sqrt{x'(t;x_{0})^{2}+y_{0}^2 e^{-2\int_{0}^{t}f'(x(s;x_{0}))ds}f'(x(t;x_{0}))^{2}}dt$$ $$\leq \int _{0}^{\tau}\sqrt{f(x(t;x_{0}))^{2}+y_{0}^2 f'(0)^2}dt+ \int _{-\tau}^{0}\sqrt{f(x(t;x_{0}))^{2}+y_{0}^2 e^{-2f'(0)t}f'(0)^{2}}dt$$ $$\leq \int _{0}^{\tau}\sqrt{m^{2}+y_{0}^2 f'(0)^2}dt+ \int _{-\tau}^{0}\sqrt{m^{2}+y_{0}^2 e^{-2f'(0)t}f'(0)^{2}}dt$$ for all $(x_{0},y_{0})$ with $y_{0}>0$ where $m$ is a bound of $f$. Therefore, $$\limsup_{\tau\longrightarrow \infty}\frac{M(\tau,x_{0},y_{0})}{M(\tau,0,y_{0})}\leq 1$$ because of $$\lim _{\tau\longrightarrow \infty}\frac{\tau \sqrt{m^{2}+y_{0}^2 f'(0)^2}}{y_{0} (e^{f'(0)\tau}-e^{-f'(0)\tau})}=0,$$$$\lim _{\tau\longrightarrow \infty}\frac{\int _{-\tau}^{0}\sqrt{m^{2}+y_{0}^2 e^{-2f'(0)t}f'(0)^{2}}dt}{y_{0}(e^{f'(0)\tau}-e^{-f'(0)\tau})}=1.$$ The second limit is a simple application of the L’Hôpital rule. By **C1** and **C4**, $$-a\leq x(t;x_{0})\leq a$$ for all $t<0$ and $-a\leq x_{0}\leq a$. Thus, $f'(x(t;x_{0}))=f'(0)=k$ for all $t<0$ and $-a\leq x_{0}\leq a$. Now, we conclude that $$\liminf_{\tau\longrightarrow \infty}\frac{M(\tau;x_{0},y_{0})}{M(\tau;0,y_{0})}\geq 1$$ since $$M(\tau;x_{0},y_{0})=\int _{-\tau}^{\tau}\sqrt{x'(t;x_{0})^{2}+y_{0}^2 e^{-2\int_{0}^{t}f'(x(s;x_{0}))ds}f'(x(t;x_{0}))^{2}}dt$$ $$\geq \int _{-\tau}^{0}\sqrt{x'(t;x_{0})^{2}+y_{0}^2 e^{-2\int_{0}^{t}f'(x(s;x_{0}))ds}f'(x(t;x_{0}))^{2}}dt\geq y_{0}( e^{ f'(0) \tau}-1).$$ **Proof of Theorem \[tmanin\].** For each $x_{0}\in[-a,a]$, we define $y_{0}^{\tau}$ such that $(x_{0},y_{0}^{\tau})\in \gamma ^{\tau}_{(0,y_{0})}$, where $\gamma_{(0,y_{0})}^{\tau}$ is the contour-line of $M$ passing through $(0,y_{0})$ for $\tau$. First we observe that for all $\tau>0$ large enough, $y_{0}^{\tau}$ exists since $$M(\tau;x_{0},0)\leq 2 m \tau < M(\tau;0,y_{0})=y_{0}(e^{f'(0)\tau}-e^{-f'(0)\tau})$$ where $m$ is a bound of $f$ and $$M(\tau;x_{0},2 y_{0})\geq 2 y_{0}(e^{f'(0)\tau}-1)\geq M(\tau;0,y_{0}).$$ By Lemma \[L2\], $$M(\tau,x_{0},y_{0})=M(\tau,0,y_{0})+f(\tau,x_{0},y_{0})$$ with $$\lim_{\tau\longrightarrow \infty} \frac{f(\tau,x_{0},y_{0})}{e^{f'(0)\tau}-e^{-f'(0)\tau}}=0$$ for all $(x_{0},y_{0})\in K$. Therefore, by using that $M(\tau,0,y_{0})=M(\tau,x_{0},y_{0}^{\tau})$, we have that $$y_{0}(e^{f'(0)\tau}-e^{-f'(0)\tau})= y_{0}^{\tau}(e^{f'(0)\tau}-e^{-f'(0)\tau})+f(\tau,x_{0},y_{0}^{\tau}).$$ From this expression, $$y_{0}=y_{0}^{\tau} + \frac{f(\tau,x_{0},y_{0}^{\tau})}{e^{f'(0)\tau}-e^{-f'(0)\tau}}$$ and clearly, $\lim_{\tau\longrightarrow \infty}y_{0}^{\tau}=y_{0}$. Collecting all the information, we have proved that $$\label{condition1} \gamma_{(0,y_{0})}^{\tau}\cap ([-a,a]\times (0,\infty))\rightarrow [-a,a]\times \{y_{0}\}.$$ In other words, the contour lines in a neighbourhood of the stable manifold tend to horizontal segments. Next we show the inability of the method of LDs to detect invariant manifolds in linear systems. \[t2\] Consider $$\label{S25}\left\{\begin{array}{lll} x_{1}'=\lambda_{1} x_{1}\\ x_{2}'=\lambda_{2} x_{2}\\ \vdots\\ x'_{N-1}=\lambda_{N-1}x_{N-1}\\ x'_{N}=-\lambda_{N} x_{N} \end{array}\right.$$ with $\lambda_{i}>0$ and $\lambda_{N}>\max\{\lambda_{i}:i=1,...,N-1\}$. Then the contour-surfaces of $M$ in a neighbourhood of the $x_{N}$-axis converge to hyper-planes parallel to $x_{N}=0$ as $\tau\longrightarrow \infty$. **Proof.** Denote by $\{e_{i}:i=1,...,N\}$ the usual basis of $\mathbb{R}^{N}$. It is clear that $$M(\tau;x_{i}e_{i})=x_{i}(e^{\lambda_{i}\tau}-e^{-\lambda_{i}\tau})$$ and $$M(\tau;x_{N}e_{N})\leq M(\tau;(x_{1},...,x_{N}))\leq M(\tau;x_{1}e_{1})+\ldots + M(\tau;x_{N}e_{N})$$ for all $(x_{1},\ldots,x_{N})\in\mathbb{R}^{N}$. Therefore, for $(x_{0},...,x_{N})\in \mathbb{R}^{N-1}\times (0,\infty)$, we have that $$\lim_{\tau\longrightarrow \infty}\frac{M(\tau;(x_{1},...,x_{N}))}{M(\tau;x_{N}e_{N})}=1$$ for all $(x_{1},...,x_{N})\in K $. By repeating the proof of the previous theorem, we get that $$\label{condition1} \gamma_{x_{N}e_{N}}^{\tau}\cap ([-a,a]^{N-1}\times (0,\infty))\rightarrow [-a,a]^{N-1}\times \{y_{N}\}$$ with $a>0$. We have denoted by $\gamma_{x_{N}e_{N}}^{\tau}$ the contour-surface of $M$ for $\tau$ passing through $x_{N}e_{N}$. That is, the contour surfaces of $M$ in a neighbourhood of the stable manifolds tend to a hyperplane parallel to $x_{N}=0$. We observe that Theorem \[t2\] can be applied in incompressible flows, for instance $\lambda_{1}=\lambda_{2}=1$ and $\lambda_{3}=2$. Minima of $M$ do not detect invariant manifolds =============================================== Craven and Hernández in [@prl] have employed a modified version of the method of Lagrangian Descriptors to study the dynamical skeleton of Langevin equation $$\label{1} m \ddot{q}=-\gamma \dot{q} -\frac{\partial V}{\partial q}(q,t)+ \sqrt{2\sigma}\xi_{a}(t)\;\;\; \;\;\;\;m, \sigma, \gamma > 0$$ where $\xi_{a}(t)$ is a noise input and $V(q,t)$ represents the underlying potential. The authors claim that the stable manifold (at $t_{0}$) in (\[1\]) is determined [*holding the coordinate $q=C$ constant*]{} (transversal direction) [*and minimizing with respect to $\dot{q}_{0}$*]{} $$\label{Lf} L_{f}((C,\dot{q}_{0}),t_{0})_{\tau}=\int _{t_0}^{t_{0}+\tau}\|{\bf{\dot{q}}_{c}}(C,\dot{q}_{0},t_{0},t)\|dt$$ where ${\bf{q_{c}}}(C,\dot{q}_{0},t_{0},t)=(q_{c}((C,\dot{q}_{0}),t_{0},t),\dot{q}_{c}((C,\dot{q}_{0}),t_{0},t))$ is the solution of (\[1\]) satisfying ${\bf q_{c}}(C,\dot{q}_{0},t_{0},t_{0})=(C,\dot{q}_{0})$. See Figure 2 (g) in [@prl] for a pictorial explanation of this heuristical law. In that figure, one observes that the authors associate the global minima of $L_{f}$ with the stable manifold, see also [@craven; @and; @hernandez; @craven; @and; @hernandez; @1]. The objective of this section is to show that this recipe does not provide a suitable mechanism to detect invariant manifolds in Langevin equations. First we consider $$\label{2} \left\{\begin{array}{lll} x'=-x+y\\ y'=h(y) \end{array}\right.$$ with $$h(y)= \left\{\begin{array}{lll} y-y^2&if\;\;y\geq0\\ y+y^2&if\;\;y<0. \end{array}\right.$$ In (\[2\]), the $x$-axis separates the basis of attraction of equilibria $(1,1)$ and $(-1,-1)$. Holding $x=x_{0}$ constant with $x_{0}>1$, the global minimum of $L_{f}((x_{0},y),0)_{\tau}$ is attained at (aprox.) $1$, not $0$ (the location of the stable manifold), (see FIG. 3 left). In fact $\frac{d}{dy}L_{f}((x_{0},y),0)_{\tau}\Big|_{y=0}=\frac{-e^{\tau}+e^{-\tau}}{2}.$ Notice that $(x_{0},1)$ is not a point at the invariant manifolds. This is the typical dynamical situation studied in [@prl], a manifold which separates reactive and non-reactive trajectories. We have taken system (\[2\]) because the solutions and all quantities are computable with simple mathematical skills. A particular counter-example of type (\[1\]) with similar pathologies is $$\label{r1} \ddot{q}=-\dot{q}+0.1 q(1-q^{2}).$$ This equation has two attractors, namely $(1,0), (-1,0)$, and the origin is a saddle point. The graph of $L_{f}((1.1,\dot{q}),0)_{\tau}$ is illustrated in FIG.3 right. The global minimum $L_{f}((1.1,\dot{q}),0)_{\tau}$ is attained in the interval $(-0.1,0.1)$ for all $\tau$. However, $(1.1,6.17)$ (aprox.) is the point of the stable manifold. ![Representation of $L_{f}((1.1,y),0)_{2}$ in $y\in[-2,2]$ for system (\[2\]) (left) and $L_{f}((1.1,q'),0)_{20}$ in $[-10,10]$ for equation (\[r1\]). The global minimum is not attained at the stable manifold.](F2 "fig:") ![Representation of $L_{f}((1.1,y),0)_{2}$ in $y\in[-2,2]$ for system (\[2\]) (left) and $L_{f}((1.1,q'),0)_{20}$ in $[-10,10]$ for equation (\[r1\]). The global minimum is not attained at the stable manifold.](DIBU "fig:") Discussion ========== The objective of this paper has been to illustrate that singularities of the contour-lines of $M$ do not generally detect invariant manifolds in incompressible flows. We have constructed systems in which the stable and unstable manifolds are placed in the axes but the contour lines of $M$ converge to horizontal lines as $\tau\longrightarrow\infty$ (the points inside the segments are indistinguishable for the function $M$). Hence, if the contour-lines of $M$ for large values were useful in the detection of invariant manifolds, one would deduce that the stable and unstable manifolds are horizontal lines (independently of the definition of singularity under consideration). In real applications, LDs involve finite (large) values of $\tau$. As mentioned above, from a mathematical point of view, the contour-structure of the $M$ function is smooth for all $\tau$. Therefore, the possible “singularities" would appear at $\tau=\infty$. Notice that if the contour-lines of $M$ converge to horizontal lines as $\tau\longrightarrow\infty$, horizontal lines will appear (in our computer) for $\tau$ large enough. Except for the simple system $$\label{2} \left\{\begin{array}{lll} x'=-\lambda x\\ y'=\lambda y, \end{array}\right.$$ there are no mathematical basis to justify the applicability of LDs. Therefore, the conclusions inferred in [@mendoza-mancho; @PRL]-[@wiggins] need some type of updating. Of course, the $M$-function always creates certain patterns when plotted over initial conditions. However, as emphasized in this manuscript, this fact does not imply that the output has dynamical significance. In [@balibrea; @response], Mancho, Wiggins, and their co-workers claim that criticisms along the lines of [@ruizherrera] or the present paper are not relevant because the conclusions of LDs are not based on the contour-structure of the $M$-function. This claim is inaccurate. It seems that they misrepresent what they have actually done. To check that Mancho, Wiggins and their co-workers routinely use the contour-structure of $M$ in their work, the reader can see figures 1-4 in [@mendoza-mancho; @PRL]; figures 1-3, 5,7,9 in [@camara; @1] (the contour-lines of the derivative of $M$ are missing); see figures of section 4 (applications) in [@lopesino]; see figures 1-5 and section 4.1 in [@mancho; @avion] (this is their last application) and so on. In [@balibrea; @response], they suggested that one has to look for points at which certain directional derivatives of $M(x_0, t_0, \tau )$ do not exist, inspired by Lopesino [*et al*]{} [@lopesino]. As mentioned in Section II, the results in [@lopesino] give no a recipe to detect invariant manifolds for general systems. Moreover, those results were given to extract information on the contour-lines of $MD_{p}$ (see the applications and introduction in [@lopesino]). The function $M(x_0, t_0, \tau )$ is smooth (except at equilibria) and is usually unbounded as $\tau\longrightarrow\infty$. The absence of discontinuity of $M$, as in the systems discussed in [@balibrea; @response], appears in most systems everywhere. For instance, in system $$\label{3} \left\{\begin{array}{lll} x'=- y\\ y'= x, \end{array}\right.$$ the $M$-function is given by $M(\tau,x,y)=2\tau \sqrt{x^2+y^2}$ and the partial derivatives are unbounded if $x\not=0$ or $y\not=0$ when $\tau\longrightarrow\infty$. Funding Statement {#funding-statement .unnumbered} ================= This research was supported by Spanish grant MTM2014-56953-P. 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--- abstract: 'We introduce a family of noncommutative $4$-spheres, such that the instanton projector has its first Chern class trivial: $ch_1(e) = B \chi + b \xi$. We construct for them a 4-dimensional cycle and calculate explicitly the Chern-Connes paring for the instanton projector.' author: - Andrzej Sitarz title: Dynamical Noncommutative Spheres --- Introduction ============ The construction of noncommutative spheres based on homological principles was proposed by Connes [@Connes2000], the basic assumption is that the algebra is generated by the elements of an projector (or unitary matrix over the algebra in the odd case) and its Chern classes in Hochschild homology vanish in all dimensions smaller than the dimension of the manifold. Connes proved that in dimension $2$ only commutative solutions appear. First noncommutative examples of solutions in dimension three and four were constructed in [@CoLa] then a systematic analysis of this type of solutions as well as construction of all three-dimensional solutions were given in [@CoDuVio]. All constructed examples of noncommutative three (and four) are of good homological dimension (related to Hochschild or cyclic homology). Moreover, they seem to be (and in some cases certainly are) nice examples of noncommutative spin geometries, as defined by Connes [@Connes2]. In this paper we introduce a variation of the noncommutative deformation of a four-sphere. With a subtle generalization of the deformation parameter we shall obtain a family of objects indexed by smooth functions on an interval, a special case of a constant function corresponding to the isospectral deformation [@CoLa]. The deformation in question goes beyond the so far considered models fo noncommutative spheres like $SU_q(2)$ and its suspension (see [@DLM]), deformations based on suspensions (and their twists) of Podles spheres ([@Si0; @BG]) or the above mentioned isospectral deformations. It rather extends the original ideas of Matsumoto [@Matsumoto] who first considered (in $C^*$-algebraic setup) the three-spheres[^1], studied later in [@CoLa]; in fact he described an entire family of their generalizations (we shall mention them later). In the paper we present the construction of the deformation, define the instanton projector, differential calculus over the deformed spheres, we construct a four-dimensional cycle, calculate Chern classes of the instanton projector and the corresponding Chern numbers. The name [*dynamical*]{}, which we use for the deformation has been motivated by possible physical applications: although we work here with a deformation of a compact manifold, it is easy to generalize the procedure to construct such deformations of $\R^4$ or $M \times \R$. With the natural interpretation of the coordinate as time we obtain [*time-dependent noncommutativity*]{}, an idea, which could be motivated, for instance, in string theory from considerations of branes in a non-static $B$-field. Preliminaries ============= We shall begin by recalling the main steps of the construction of isospectral deformations, as done in [@CoLa]. Let $M$ be a compact manifold, $\CA = C^\infty(M)$ and let the two torus $T^2$ act on $\CA$. Since any smooth function (with respect to the action of the torus) could be presented as a doubly infinite norm convergent series of homogeneous elements, where $f$ is homogeneous of of bidegree $n_1,n_2$ iff $$(u_1, u_2) \acts f = (u_1)^{n_1} (u_2)^{n_2} f,$$ for $u_1,u_2 \in T^2$, one might introduce a deformed algebra as using a left (or right) twist maps: $$\sum_{n_1, n_2} T_{n_1,n_2} = T \mapsto l(T) = \sum_{n_1, n_2} T_{n_1,n_2} \lambda^{n_2 \delta_1}$$ where $\lambda$ is a complex number of module $1$ and $\delta_1, \delta_2$ are the generators representations of the (projective) unitary representation of the action of the torus. Then we have the lemma: There exists an associative product on the vector space of smooth functions $\CA$, $(x,y) \mapsto x*y$ such that $$l(x) l(y) = l(x*y).$$ For the homogeneous elements $(x,y)$ of order $n_1, n_2$ and $m_1,m_2$, respectively, it is: $$x*y = \lambda^{n_1 m_2} xy.$$ From the algebraic point of view the constructed deformation is a cocycle deformation of the algebra through the twist from the Cartan subalgebra of its symmetry group. This description was developed in [@Sit] and used to demonstrate that twisted isometry of the algebra is the Hopf-algebra isometry of deformed spectral triple, a dual approach to symmetries was suggested in [@Varilly], whereas a systematic approach to $\theta$-deformations is presented in [@CoDuVio]. Dynamical twists ---------------- We shall introduce here a generalization of the above deformation, which we shall study in details in a particular case of the sphere. Our assumptions are as in the situation discussed earlier: we work with smooth functions on a compact oriented manifold, such that its symmetry groups contains a torus and we assume that the (smooth) action of the isometry group is (projectively) lifted to the Hilbert space. Let $f$ be a smooth function, and let $T(f)_{n_1,n_2}$ be its component of the Fourier series with respect to the action of $T^2$: $$T(f) = \sum_{n_1,n_2} T(f)_{n_1,n_2},$$ where the series is norm converging (norms of homogeneous elements, which are the elements of this series, are of rapid decay). Let $H$ be a self-adjoint element of the algebra $C^\infty(M)$ (real smooth function), which is of bidegree $(0,0)$, so it is invariant with respect to the action of the two-torus. Let us define a map, which shall assign to every element $f \in C^\infty(M)$ an element of the deformed algebra: $$T_H(f) = \sum_{n_1,n_2} T(f)_{n_1,n_2} e^{2\pi i n_2 H \delta_1}.$$ Let us observe that the series is again infinite norm convergent (we modify each element by multiplication with an operator of norm $1$) and since $H$ commutes with the action of torus the definition well posed (that is the bidegree of an element is stable under multiplication by any function of $H$). So we have a lemma: \[dytwi\] If $f,g$ are homogeneous operators of degrees $(n_1,n_2)$ and $(m_1,m_2)$, respectively, then: $$T_H(f) T_H(g) = T_H(f \ast g),$$ where $$f \ast g = e^{2\pi i H n_2 m_1} (f g).$$ Similarly, one may define an opposite deformation: $$T_H^o(f) = \sum_{n_1,n_2} e^{2\pi i n_1 H \delta_2} T(f)_{n_1,n_2} .$$ such that $[T_H^0(x), T_H(y)]=0$ if $[x,y]=0$ The proof of the lemma follows directly the proof of lemma 4 of [@CoLa]. We shall present now two basic examples of this type of deformation. Let $T^3$ be a three-torus, consider the natural action of two-torus $T^2 \subset T^3$ on $C^\infty(T^3)$. If we denote the unitary generators of $C^\infty(T^3)$ by $U,V,W$, then $W$ remains the invariant element under the action of $T^2$. If we make the choice of $H=\theta$ as a constant we obtain a product of a noncommutative torus $T^2_\theta$ with $S^1$. However, the simplest nontrivial choice of $e^{2\pi i H} = W$ gives us the algebra relations: $$UV = W VU, \;\;\;\; [U,W] = [V,W] = 0. \label{Heis}$$ Clearly, the first relation can be generalized to: $$UV = f(W) VU.$$ where $f(W)$ is a suitable smooth function $f: S^1 \to S^1$, however, in the particular case $f(W)=W$ the algebra is the group algebra of the discrete 3-dimensional Heisenberg group [@AnPa]. We shall study the properties of this algebra, in particular the explicit construction of the $K$-cycle and Chern-Connes pairing in a separate paper [@Sit2] Let us consider a $\ast$-algebra generated by elements $a,a^\ast,b,b^\ast$ and $t=t^\ast$ subject to the following set of relations: $$\begin{array}{lll} [a, t] =0, & & [a^\ast, t] = 0, \\ {}[b, t] =0, & & [b^\ast, t] = 0, \\ {}[a ,a^\ast] =0, & \phantom{xxxxx} & [b, b^\ast] =0, \\ ab = \lambda(t) ba, & & a b^\ast = \bar{\lambda}(t) b^\ast a, \\ a^\ast b = \bar{\lambda}(t) a^\ast b, & & a^\ast b^\ast = \lambda(t) b^\ast a^\ast, \end{array} \label{algebra}$$ where $\lambda(t)$ is a unitary element, $\lambda(t) \bar{\lambda}(t) =1$, expressed as a function of the central element $t$, so we may assume: $$\lambda(t) = e^{-i\phi(t)},$$ where $\phi$ is a smooth real function of $-1 \geq \leq 1$. Furthermore, we have the restriction: $$a a^\ast + b b^\ast + t^2 = 1, \label{sphere}$$ which is the relation defining the (noncommutative) 4-sphere. One could easily verify that the above set of relations is consistent, for any choice of the function $\phi$, the particular example of $\phi=\theta=\const$ being the isospectral deformation of the sphere. Passing from algebraic (polynomial) algebra to the algebra of smooth functions one can easily observe that the algebra describes the dynamical deformation of the four-sphere as presented in Lemma \[dytwi\], with $H = \phi(t)$ (the parameter $t$ corresponds to the choice of presentation of $S^4$ as a suspension of $S^3$). We shall denote this algebra by $\CS^4_\lambda$, let us observe that the center of the algebra in question contains $t$, $a a^\ast$ and $b b^\ast$ but could be much bigger depending on the function $\lambda$. Instanton bundles over $\CS^4_\lambda$ ====================================== One of the most appealing feature of the construction of [@CoLa] was the existence of the instanton bundle over the deformed algebra. This was shown by the construction of the projector $e$ with vanishing lower Chern classes and $ch_2(e)$ giving rise to a Hochschild cocycle over the algebra. The projector in our case is unmodified: $$e = \frac{1}{2} \left( \begin{array}{llll} 1+t & 0 & a & b \\ 0 & 1+t & - \lambda(t) b^\ast & a^\ast \\ a^\ast & - \bar{\lambda}(t) b & 1-t & 0 \\ b^\ast & a & 0 & 1-t \end{array} \right), \label{proj}$$ the only significant distinction for the $\lambda=\const$ case is that no longer all the entries of the projector are the generators of the algebra. Of course, since $\lambda$ is not a constant parameter one may easily verify that the Chern homology elements constructed out of $e$ shall not be the same as in $\lambda=\const$ case. In particular, we have: $$ch_1(e) = t \ts x_i \ts y_i - x_i \ts t \ts y_i + x_i \ts y_i \ts t,$$ where: $$x_i \ts y_i = b \ts b^* - b^* \ts b + \lambda b^* \ts \overline{\lambda} b - \overline{\lambda} b \ts \lambda b^*.$$ It is easy to verify that $b\, ch_1(e)$ vanishes, however $B\, ch_1(e)$ does not: $$B \, ch_1(e) = 1 \ts ch_1(e).$$ We shall postpone further discussion of the Chern classes until the last section of the paper, when it shall be clear that although $ch_1(e)$ does not vanish, its class is trivial. In fact, using the the natural construction of differential structures on the deformed sphere and the natural trace on the algebra we shall give explicit formula for the volume form, which arises naturally from the Chern class $e\,de\,de\,de\,de$ and calculate the Chern number of the above projector $e$. The differential calculus on $S^4_\lambda$ ========================================== Unlike in the $\lambda=\const$ case we have no clear indication for the construction of differential calculi. We shall look for a guiding principle of the smallest calculi, which, when restricted to commutative subalgebras, remains classical and for $\lambda=1$ gives the correct limit of the differential structures on a four-sphere. Before we begin let us observe that the commutation relations between algebra generators $a,a^*,b,b^*$ could be rewritten as $$x^i x^j = A_{ij} \, x^j x^i, \;\;\; 1 \leq i,j \leq 4, \label{comm}$$ where there is no summation in the formula, $x^i$ denote the generating monomials $a,a^*,b,b^*$ and the matrix $A_{ij}$ is $t$-dependent, in our case: $$A = \left( \begin{array}{llll} 1 & 1 & \lambda(t) & \bar{\lambda}(t) \\ 1 & 1 & \bar{\lambda}(t) &\lambda(t) \\ \bar{\lambda}(t) &\lambda(t) & 1 & 1 \\ \lambda(t) & \bar{\lambda}(t) &1 & 1 \end{array} \right).$$ We make an Ansatz that the bimodule of one forms is generated by $dx^i$ and a central one-form $dt$, with quadratic the bimodule commutation rules: $$x^i \, dx^j = A_{ij} dx^j \, x^i + \oh B_{ij} dt (x^i \, x^j). \label {diff1}$$ We assume as well, that $t\, dx^i = dx^i\, t$. It is easy to see that such relations are consistent with the algebra commutation rules. Further, if we differentiate (\[comm\]) and use (\[diff1\]) we obtain the following relation between $B$ and $A$: $$\oh ( B_{ij} - B_{ji} ) = \frac{1}{A_{ij}} \dot{A}_{ij},$$ We shall restrict ourselves only to the antisymmetric solution for $B$, which are explicitly given by the above formula. Expressing the relations (\[diff1\]) in terms of the generators we have: $$\begin{array}{lll} a\, da = da\, a, &\hbox to 1cm{\hfil}& b\, db = db\, b, \\ a\, da^\ast = da^\ast a, & &b\, db^\ast = db^\ast\, b, \end{array}$$ and $$\begin{array}{l} a\, db = \lambda(t)\, db\, a + \oh \dot{\lambda}(t) \bar{\lambda}(t)\, dt\, ab, \\ a\, db^* = \bar{\lambda}(t)\, db^*\, a - \oh \dot{\lambda}(t) \bar{\lambda}(t)\, dt\, ab^* \\ b\, da = \bar{\lambda}(t)\, da\, b - \oh \dot{\lambda}(t) \bar{\lambda}(t)\, dt\, ba, \\ b \,da^* = \lambda(t) \, da^* \, b + \oh \dot{\lambda}(t) \bar{\lambda}(t) \, dt \, ba^* \\ a^\ast\, db = \bar{\lambda}(t)\, db\, a^\ast - \oh \dot{\lambda}(t)\bar{\lambda}(t)\, dt\, a^\ast b, \\ b^\ast\, da = \lambda(t)\, da\, b^\ast + \oh \dot{\lambda}(t)\bar{\lambda}(t)\, dt\, ab^\ast, \\ a^\ast\, db^\ast = \lambda(t)\, db^\ast\, a^\ast + \oh \dot{\lambda}(t)\bar{\lambda}(t)\, dt\, a^\ast b^\ast, \\ b^\ast\, da^\ast = \bar{\lambda(t)}\, da^\ast\, b^\ast - \oh \dot{\lambda}(t)\bar{\lambda}(t)\, dt\, a^\ast b^\ast. \end{array} \label{diff-e}$$ We shall not forget that by differentiating the constraint (\[sphere\]) we have (after using (\[diff1\])): $$a\, da^* + a^*\, da + b\, db^* + b^*\, db + 2 t dt = 0, \label{diff2}$$ Note that the left-hand side of (\[diff2\]) side is a central element of the bimodule of one forms and therefore the restriction (\[diff2\]) is compatible with the (\[diff-e\]). Now, we are prepared to construct the full differential algebra. Let $\Omega_u(\CS^4_\lambda)$ be a universal differential algebra, and let $\CJ_1 \subset \Omega^1_u(\CS^4_\lambda)$ be the kernel of the projection map $\pi: \Omega^1_u(\CS^4_\lambda) \mapsto \Omega^1(\CS^4_\lambda)$. Then the differential algebra $\Omega(\CS^4_\lambda)$ is a $\Z$-graded algebra obtained as a quotient of $\Omega_u(\CS^4_\lambda)$ by the differential ideal generated by $\CJ^1 +d \CJ^1$. Clearly, the subbimodule $\CJ^1$ is in our case defined by relations (\[diff1\]) and (\[diff2\]). Thus by differentiating them we obtain the first set of rules: $$\begin{aligned} && dx^i \, dt = - dt \, dx^i, \label{diff3} \\ && dt\, dt = 0. \\ && dx^i\, dx^j = - A_{ij} dx^j \, dx^i + \oh B_{ij} A_{ij} dt\, dx^j \, x^i - \oh B_{ij} dt\, dx^i \, x^j \end{aligned}$$ We immediately see that in the differential algebra $\Omega(\CS^4_\lambda)$ all generators $dx^i$ and $dt$ are nilpotent, and $da,da^*$, $db,db^*$ are pairwise skew-symmetric: $$da\, da^* = - da^* \, da, \;\;\;\;\; db\, db^* = - db^* \, db. \label{diff4}$$ For the remaining relations we have: $$\begin{array}{l} da\, db + \lambda(t)\, db\, da = \oh \dot{\lambda}(t)\, dt\, db\, a -\oh \dot{\lambda}(t) \bar{\lambda}(t) \, dt\,da\, b, \\ db\, da^\ast + \lambda(t)\, da^\ast\, db = \oh \dot{\lambda}(t)\, dt\, da^\ast\, b - \oh \dot{\lambda}(t)\bar{\lambda}(t)\, dt\, db\,a^\ast. \end{array}$$ Before we prove more results on the differential algebra we introduced, let us observe interesting relations: $$\begin{array}{l} b \, da\, da^* = da\, da^*\, b - \oh \dot{\lambda}(t) \bar{\lambda}(t)\, dt\, ( a\, da^*\, b + a^* \, da\, b) = \\ \phantom{xxxx} = da\, da^*\, b + \oh \dot{\lambda}(t) \bar{\lambda}(t)\, dt \, ( db^*\, b^2 + db\, bb^*). \end{array}$$ where in the last step we used (\[diff2\]). By differentiating it we obtain: $$db \, da\, da^* = da\, da^*\, db - \oh \dot{\lambda}(t) \bar{\lambda}(t) \, dt \, db\, db^* b.$$ Similar result can also be proven for $db^*$: $$db^* \, da\, da^* = da\, da^*\, db^* - \oh \dot{\lambda}(t) \bar{\lambda}(t) \, dt \, db\, db^* b.$$ and for products of $da\, db\, db^*$. In particular, we can see that: $$\begin{array}{l} da\, da^*\, db \, db^* = db\, da\, da^*\, db^* = db^*\, da\, da^*\, db, \\ da\, da^*\, db \, db^* = da\, db\, db^*\, da^* = da^*\, db\, db^*\, da, \\ da\, da^*\, db \, db^* = db \, db^*\, da\, da \end{array} \label{3forms}$$ Next we shall prove that the differential algebra has a finite dimension: The differential algebra $\Omega(\CS^4_\lambda)$ has dimension $4$, for all $n >4$ we have $\Omega^n(\CS^4_\lambda)=0$. Clearly, it is sufficient to show that $dt\, da\, da^*\, db\, db^*$ vanishes. Let us consider the relation (\[diff2\]) and multiply it from the left by a two-form $\oh t da \, da^*$ and from the right by $db \, db^*$. Using the associativity of the product together with relations (\[diff3\]) and the fact that all one generating one-forms are nilpotent we obtain: $$t^2 dt\, da\, da^* \, db \, db^* = 0. \label{d1}$$ Similarly, if we multiply (\[diff2\]) from the left by $dt\, da\, a^*$ and by $db \, db^*$ from the right we obtain: $$a a^* \, dt\, da\, da^*\, db\, db^* = 0. \label{d2}$$ Finally, multiplying it by $b \, dt\, da\, da^*$ from the left and by $db^*$ from the right we get: $$b b^* \, dt\, da\, da^*\, db\, db^* = 0. \label{d3}$$ By adding the three identities (\[d1\])-(\[d3\]) and using the constraint (\[sphere\]) we obtain the desired result. So far we have shown that the maximal degree of forms is $4$, it appears however that the structure is exactly as in the “classical” case and we are able to demonstrate that there exist one generating four-form: The bimodule of differential forms of degree $4$ is a free bimodule module over the algebra. The generating form $\omega$ can be chosen as: $$\omega = \frac{1}{4} \left( t \, da\, da^* \, db \, db^* - 2 a\, dt\, da^*\, db \, db^* + 2 dt \, da\, da^* \, db \, b^* \right), \label{volf}$$ where the factor $\frac{1}{4}$ was chosen so that it would correspond to the volume form on $S^4$ in the classical limit. Consider $t\, \omega$. Using the commutation rules of $dt$ with other one-forms (\[diff3\]) as well as the fact that $t$ is central we might rewrite it conveniently as: $$t \, \omega = \frac{1}{4}( t^2 \, da\, da^* \, db \, db^* + a \, da^* \, (2t\, dt) \,db \, db^* + da\, da^* \, (2t dt) \, db \, b^*) = \ldots$$ Next, using (\[diff2\]) and keeping in mind that $dt$ and $dx^i$ are nilpotent we get: $$\ldots = \frac{1}{4}(t^2 + a a^* + bb^*) da\, da^* \, db \, db^* = \frac{1}{4} da\, da^* \, db \, db^*.$$ where we have used first the fact that $a, a^*$ commute with $da,da^*$ (and similar property of $b,b^*$ and their differentials) as well as the defining relation (\[sphere\]). Similarly one may verify the identities: $$\begin{aligned} && a \, \omega = \frac{1}{2} \, dt\, da \, db\, db^*, \\ && a^* \, \omega = - \frac{1}{2} \, dt\, da^* \, db\, db^*, \\ && \omega\, b= \frac{1}{2} \, dt\, da \, da^* \, db, \\ && \omega \, b^* = - \frac{1}{2} \, dt\, da \, da^* \, db^*.\end{aligned}$$ The form $\omega$ is central, i.e. it commutes with all elements of the algebra. As this result is not evident though it follows from an easy algebraic calculation we shall demonstrate it only for $[b, \omega]$. First, observe that only the first component in the sum (\[volf\]) might give a nontrivial contribution as the remaining two contain $dt$ and then the nontrivial permutation rules of generators through differentials are homogeneous and will cancel out. $$\begin{array}{l} [b, \omega] = \frac{1}{4}( b t\, da\, da^* \, db\, db^* - t\, da\, da^* \, db\, db^* b) = \\ \phantom{xxxx} = \frac{1}{4} t \dot{\lambda}(t) \bar{\lambda}(t) \, \left( \bar{\lambda}(t)\, da\, dt \, (ba^*) - dt\, da^* \, (ab) \right) \,db \, db^* = \ldots \end{array}$$ now, if we permute $t$ and use (\[diff2\]) to substitute a nontrivial one-form for $t\, dt$, still using the fact that the one forms are nilpotent: $$\begin{array}{l} \phantom{xxxx} = \ldots \frac{1}{16} \dot{\lambda}(t) \bar{\lambda}(t) \, \left( - \bar{\lambda}(t)\, da\, (a\, da^*) \, (ba^*) + a^* \, da\, da^* \, (ab) \right) \,db \, db^* = \\ \phantom{xxxx} = \frac{1}{16} \dot{\lambda}(t) \bar{\lambda}(t) \, \left( - da\, (a da^*) \, (a^* b) + a^* \, da \, da^* \, (ab) \right) \,db \, db^* = 0. \end{array}$$ Before we proceed with the construction of the integral of $4$-forms, let us observe the properties of a trace on the algebra itself. \[grtrace\] Let $\int$ be the standard (normalized) integral on $S^4$ and $\eta$ be a linear map on $\CS^4_\lambda$, which maps an element of $\CS^4_\lambda$ to an element of $C(S^4)$, with the identification of every element with $a,a^*$ to the left of $b,b^*$ with the corresponding function on $S^4$. Then $x \mapsto \int \eta (x)$ is a trace on $\CS^4_\lambda$. Clearly we have a linear map, it remains only to show the cyclicity. First, note that the integral on $S^4$ is nontrivial on functions depending only on $aa^*$ and $bb^*$. Therefore, we might restrict ourselves to such case. Let us take two monomials $p,q$ in $a,a^*,b,b^*$ such that their product is a monomial of $aa^*$ and $bb^*$. Then we shall prove that $\eta(pq) = \eta(qp) $. Let $p= a^{\alpha_p} (a^*)^{\beta_p} b^{\gamma_p} (b^*)^{\delta_p}$ and $q= a^{\alpha_q} (a^*)^{\beta_q} b^{\gamma_q} (b^*)^{\delta_q}$. First, we calculate $pq$ using (\[algebra\]): $$p\,q = \lambda(t)^{\gamma_p \beta_q + \delta_p \alpha_q} \bar{\lambda}(t)^{\gamma_p \alpha_q + \delta_p \beta_q} a^{\alpha_p +\alpha_q} (a^*)^{\beta_p+\beta_q} b^{\gamma_p+ \gamma_q} (b^*)^{\delta_p+\delta_q},$$ since $\bar{\lambda} = \lambda^{-1}$ we might rewrite the formula as: $$\eta(p\, q) = \lambda(t)^{(\gamma_p-\delta_p) (\beta_q - \alpha_q) } \eta(p) \eta(q).$$ On the other hand, for $qp$ we have: $$q\, p = \lambda(t)^{\gamma_q \beta_p + \delta_q \alpha_p} \bar{\lambda}(t)^{\gamma_q \alpha_p + \delta_q \beta_p} a^{\alpha_q +\alpha_p} (a^*)^{\beta_q+\beta_p} b^{\gamma_q+ \gamma_p} (b^*)^{\delta_q+\delta_p},$$ which gives: $$\eta(q\, p) = \lambda(t)^{(\gamma_q-\delta_q) (\beta_p - \alpha_p) } \eta(p) \eta(q).$$ Now, it is easy to see that both coefficients are equal, since by our assumption that the product depends only on $aa^*$ and $bb^*$: $$\alpha_p + \alpha_q = \beta_p + \beta_q, \;\;\;\; \gamma_p + \gamma_q = \delta_p + \delta_q,$$ and thus: $$(\gamma_p-\delta_p) (\beta_q - \alpha_q) = (\gamma_q-\delta_q) (\beta_p - \alpha_p).$$ We now define the integral on $4$-forms. There exist a linear functional on $\Omega^4(\CS^4_\lambda)$ such that $\int (d\rho) =0$ for every $\rho \in \Omega^3(\CS^4_\lambda)$ and $\int \omega = \frac{8}{3} \pi^2$. We begin by defining the integral. Since we know that every four-form $\theta$ could be written as $\theta = x \omega$ we shall set $$\int \theta = \int \eta(x).$$ Note that since $\omega$ is central, $x\, \omega = \omega\, x$, we have in effect a linear map $\eta: \Omega^{4}(\CS^4_\lambda) \mapsto \Omega^4(S^4)$. We shall demonstrate that there exists also the extension of map $\eta: \Omega^3(\CS^4_\lambda) \mapsto \Omega^3(S^4)$ such that the following diagram is commutative: $$\begin{CD} \Omega^3(\CS^4_\lambda) @>d>> \Omega^{4}(\CS^4_\lambda) \\ @V\eta VV @VV\eta V \\ \Omega^3(S^4) @>d>> \Omega^{4}(S^4) \end{CD}$$ To define the map $\eta$ on three forms we shall use their following presentation as a linear space: \[order\] Every $3$-form (over polynomials) could be presented (though not in unique way) as a finite sum of elements of the type: $$\rho = t^\alpha p(a,a^*) \, \chi_i \, q(b,b^*)$$ where $\alpha=0,1$ and $\chi_i$ are forms of the type: $$\begin{aligned} da \, da^* \, db, & da \, da^* \, dt, & da \, da^* \, db^*, \\ da \, db \, db^*, & da^* \, db \, db^*, & dt \, db \, db^*, \\ dt \, da \, db, & & dt \, da \, db^*, \\ dt \, da^* \, db, & & dt \, da^* \, db^* .\end{aligned}$$ Of course, these forms are not independent (when we consider them in the bimodule of three-forms). However, it is important that we can map them to $\Omega^3(S^4)$ by setting first $\eta(\chi_i)$, for instance: $$\eta ( da\, da^*\, db) = d\eta(a)\, d\eta(a^*)\, d\eta(b),$$ and then: $$\eta(\rho) = \eta(t)^\alpha \, p(\eta(a),\eta(a^*)) \, \eta(\chi) \, q(\eta(b), \eta(b^*)).$$ To see that the map is well-defined (as a linear map) let us observe that by using the so ordered product of functions and differentials we see no nontrivial commutation rules. Thus, the characterization of $\Omega^3(\CS^4_\lambda)$ and $\Omega^3(S^4)$ as a linear space are exactly the same. Now, using the presentation (\[order\]) we can easily see that $\eta(d \rho) = d \eta(\rho)$ for every three-form $\rho$. Indeed, the external derivative vanishes on all three-forms $\chi$ and on functions depending only of $a,a^*,t$ and respectively, on $b,b^*,t$ we have standard differentiation: $$d \eta( p(a,a^*,t) ) = \eta( dp(a,a^*,t) ),$$ and $$d \eta( q(b,b^*,t) ) = \eta( dq(b,b^*,t) ).$$ Since again, we multiply by $a,a^*$ and its differentials from the left and $b,b^*$ and its differentials from the left – we encounter no commutators between $a,a^*$ and their differentials and $b,b^*$ and their differentials. Hence, noncommutativity plays no role in the map $\eta$ and the action of the external derivative. Using the constructed differential structures and the trace we have: $\Omega^*(\CS^4_\lambda)$ is a differential graded algebra with a closed graded trace $\int : \Omega^4(\CS^4_\lambda) \to \C$. So far we have showed the existence of a closed trace on $\Omega^4(\CS^3)$. Because of its particular form (\[grtrace\]) it is evident that $\int x \rho = \int \rho x$ for every four-form $\rho$ and $x \in \CS^4_\lambda$. Now, let us take a three-form $\beta$ and a one-form $x\, dy$: $$\begin{array}{l} \int (x\, dy\, \beta + \beta \, x \, dy) = \int ( dy \, \beta \, x + \beta \,d(xy) - \beta \, dx \, y) \\ \phantom{xxx} = \int ( d( y \, \beta \, x) - y \, d(\beta \, x) + \beta\, d(xy) + d(\beta \, x) y - (d\beta) \, xy = \\ \phantom{xxx} = \int \left( d( y \, \beta \, x) - [ y, d(\beta\, x)] +d ( \beta\, xy) \right) = 0. \end{array}$$ Similarly, we proceed for two-forms. As an immediate corollary we have: Let $\psi$ be a multilinear functional defined as: $$\psi(a_0, a_1, a_2, a_3, a_4) = \int a_0 \, da_1 \, da_2 \, da_3 \, da_4.$$ then $\psi$ is a cyclic cocycle. Having a cyclic cocycle enables us to calculate the Chern-Connes pairing with the instanton projector, which we introduced earlier (\[proj\]). The Chern character ------------------- Let us consider the construction of an element of $\Omega^4(\CS^4_\lambda)$ out of the projector $e$: $$\hbox{ch}(e) = - \frac{1}{8 \pi^2} \hbox{Tr} (e\, de\, de\, de\, de),$$ where the trace is over matrix indices of $e$. We shall use the block form of $e$ and the rules of differential calculi to facilitate the calculations. Let us denote: $$q = \left( \begin{array}{ll} a & b \\ -\lambda(t) b^* & a^* \end{array} \right),$$ then we can write $e$ and $de$ as block matrices: $$e = \oh \left( \begin{array}{ll} t+1 & q \\ q^* & 1-t \end{array} \right),$$ $$de = \oh \left( \begin{array}{ll} dt & dq \\ dq^* & -dt \end{array}\right).$$ where $1 \pm t$ and $\pm dt$ denote diagonal matrices. Using this fact and that $(dt)^2=0$ and $dt$ anticommutes with the rest of the one-forms, we obtain: $$de\, de\, de\, de = \frac{1}{16} \left( \begin{array}{ll} (dq\, dq^*)^2 & 4\, dt\, dq\, dq^*\, dq \\ -4\, dt \, dq^* \, dq\, dq^* & (dq\, dq^*)^2 \end{array} \right).$$ Therefore for the trace of $e\, ed\, de\, de\, de$ we shall have: $$\begin{array}{l} \ldots =\frac{1}{32} \hbox{Tr} \left( (1+t) (dq\, dq^*)^2 + (1-t) (dq^* \, dq)^2 + \right. \\ \phantom{xxxxxxx} - \left. 4 q\, dt\, dq^* \, dq\, dq^* + 4q^* \, dt\, dq\, dq^* \, dq \right), \end{array} \label{chern1}$$ where the trace is now over two-dimensional matrices. As a next step let us calculate $dq\, dq^*$ and $dq^* \, dq$: [$$\begin{array}{l} dq \, dq^* = \\ = \left( \begin{array}{ll} da\, da^* + db\, db^* & 2 db\, da - \oh \dot{\lambda} \bar{\lambda}\, dt ( db\, a + \bar{\lambda} da\, b) \\ 2 da^*\, db^* - \oh \dot{\lambda} \, dt ( db^* \, a^* + \bar{\lambda} da^*\, b^*) & da^* \, da + db^*\, db + \dot{\lambda} \bar{\lambda}\, dt\, (db b^* + db^* \, b) \end{array} \right). \end{array}$$ ]{} Now, we shall calculate the diagonal part of $(dq \, dq^*)^2$, the element from the top-left corner, $\{(dq \, dq^*)^2\}_{11}$. is: $$\begin{array}{l} \{(dq \, dq^*)^2\}_{11} = (da\, da^* + db\, db^*)^2 + 4 db\, da\, da^*\, db^* + \\ \phantom{x} - \dot{\lambda} \bar{\lambda}\, dt ( -a\, da^* \, db \, db^* + da\, da^* db^* \, b + da\, da^* db\, b^* - a^*\, da\, db\, db^*) = \ldots \end{array}$$ In the last expression, using (\[diff2\]) we can substitute $-a\,da^* - a\, da$ by $ b\, db^* + b^* \, db + 2t\, dt$, then, however, we shall encounter at least one element of the type $(dt)^2$, $(db)^2$ or $(db^*)^2$ and therefore it shall vanish. Moreover, using the previously derived rules (\[3forms\]) we see that in the end we obtain: $$\ldots = 6 da\, da^* db\, db^*.$$ Quite similarly, for the other diagonal element of $(dq\, dq^*)^2$ we shall have: $$\begin{array}{l} \{(dq \, dq^*)^2\}_{22} = 4 da^*\, db^* \, db \, da + (da^*\, da + db^* \, db)^2 = \\ \phantom{xxxx} = 6 da\, da^* db\, db^*. \end{array}$$ The calculation for the sum of the diagonal elements of $(dq^* \, dq)^2$ yields (we skip the intermediate technical steps, which are same as in the previous example): $$\hbox{Tr} (dq^* \, dq)^2 = - 12 da\, da^* \, db \, db^*.$$ Coming back to our expression (\[chern1\]) it is easy to demonstrate that $-4 \hbox{Tr}( q\, dt\, dq^* \, dq\, dq^*)$ and $ 4 \hbox{Tr} (q^*\, dt\, dq \, dq^*\, dq)$ give the same contributions, which together add up to: $$24\, dt \left( -a\, da^*\, db\, db^* - da\, da^* \, db^*\, b + da\, da^* \, db \, b^* + a^* \, da\, db \, db^* \right).$$ Summing it all together and using again (\[diff2\]) we obtain: $$\begin{array}{l} \hbox{ch}(e) = - \frac{1}{8\pi^2} \int \hbox{Tr}(e\, de\, de\, de\,de) = \\ \phantom{xxxx} - \frac{1}{8\pi^2} \frac{1}{32} 24 \int \left( t \, da \, da^*\, db\, db^* - 2 a\, dt\, da^* \, db\, db^* + 2 dt\, da\, da^*\, db b^* \right) = \\ \phantom{xxxx} = - \frac{3}{32\pi^2} 4 \int \omega = - \frac{3}{8\pi^2} \frac{8}{3} \pi^2 = -1. \end{array}$$ where we have used the normalization of the integral of $1$ over $S^4$ giving the volume of four-sphere. As an immediate corollary we have: The element $e\, de\,de\,de\,de$ gives a nontrivial cohomology class of the complex $\Omega(\CS^4_\lambda)$. Now, we shall come back to the first Chern form: $$\hbox{ch}_1(e) = - \frac{1}{2\pi i} \hbox{Tr} ( e\, de \, de ),$$ which, evidently, does not vanish: $$- \frac{1}{2\pi i} \hbox{Tr} ( e\, de \, de ) = - \frac{1}{2\pi i} 2 \dot{\lambda}(t) \bar{\lambda}(t)\, dt ( b \, db^* + b^*\, db), \label{ch1f}$$ however, it is in the trivial cohomology class. If $\lambda= e^{- i \phi(t)}$ for a real function $\phi$ then: $$\hbox{ch}_1(e) = \frac{1}{\pi} d \left( \phi(t) ( b \, db^* + b^*\, db) \right).$$ What does it mean? Let us remind that the $ch_1(e)$ in the reduced $(b,B)$ double complex was clearly a cycle. Furthermore, one might easily observe that it was depending only on the commutative subalgebra generated by $t$, $b$ and $b^*$, which we shall denote by $\C[b,b^*,t]$ (we might equally well describe the algebra as the subalgebra of smooth functions on $S^4$ invariant under the action of $\delta_2$ - and it is the algebra of smooth functions on a three-dimensional closed ball). Since it is a regular commutative algebra we might use the results relating Hochschild and homology of with the de Rham complex. There exists an element $\chi \in C_1(\C[b,t])$ and $\xi \in C_3(\C[b,t])$ such that: $$ch_1(e) = B \chi + b \xi. \label{coh}$$ First, let us observe that since $b \, ch_1(e) = 0$ we might map $ch_1(e)$ to $\Omega^2(\C[t,b])$, the image being exactly the two-form (\[ch1f\]). This form is exact, as we have demonstrated explicitly. If we take the one form in $\Omega^1(\C[b,t])$, $\chi_0$, $d \chi_0 = ch_1(e)$, by using the commutative diagram relating Hochschild homology with differential forms (see Proposition 2.3.4, p.69, [@Loday]) we obtain the desired cycle $\chi = \pi^{-1}(\chi_0)$. Then the Hochschild class of $B \chi$ is the same as this of $ch_1(e)$, so the difference is in the image of $b$, and then by chosing any suitable cycle $\xi$ we get (\[coh\]). Therefore, although $ch_1(e)$ does not vanish identically, we still are almost in the same situation. By correcting slightly $ch_2(e)$ we are again able to obtain a Hochschild cycle of dimension $4$, which corresponds to the volume form: $$v = ch_2(e) + B \xi.$$ Indeed: $$\begin{array}{l} b v = b \, ch_2(e) + b B \, \xi = B ch_1(e) - B b \xi \\ \phantom{xxxx} = B(ch_1(e) - b \xi) = B (B \chi) = 0. \end{array}$$ Conclusions =========== The construction presented in this paper extends the notion of noncommutative spheres to objects defined through instanton bundles, whose first Chern class does not vanish but is homologically trivial. Our aim was to demonstrate that such solutions exists, are easily obtained by a slight generalization of the [*twisted*]{} noncommutative spheres. We demonstrated as well the existence of 4-dimensional differential calculus (a 4-dimensional cycle) and calculated explicitly the Chern-Connes pairing. Of course, it is possible to consider further generalizations going in this direction, for instance one might consider (in the same spirit) the Matsumoto [@Matsumoto] 3-spheres defined through generators as: $$\begin{array}{lll} {}[a ,a^\ast] =0, & \phantom{xxxxx} & [b, b^\ast] =0, \\ ab = \lambda ba, & & a b^\ast = \bar{\lambda} b^\ast a, \\ a^\ast b = \bar{\lambda} a^\ast b, & & a^\ast b^\ast = \lambda b^\ast a^\ast, \end{array}$$ and $$a a^\ast + b b^\ast = 1,$$ where $\lambda(t)$ is a unitary element from the center of the algebra, $\lambda(t) \bar{\lambda}(t) =1$, for instance: $$\lambda = \lambda (bb^*).$$ Similarly as for the four-sphere one may view this algebra as generated by the matrix elements of is generator of $K_1$ class: $$U = \left( \begin{array}{ll} a & b \\ - \lambda b^* & a^* \end{array} \right),$$ Now, it is easy to verify that the Chern character of the generator $U$ for this algebra is: $$ch_{\frac{1}{2}}(U) = b \ts b^* - b^* \ts b + \lambda b^* \ts \bar{\lambda} b - \bar{\lambda} b \ts \lambda b^*.$$ Again, although this Chern character does not vanish, since it is over a commutative subalgebra we see that the same argument as in the case of 4-sphere applies and it is sufficient to study the image of $ch_{\frac{1}{2}}(U)$ in the de Rham complex: $$\pi(ch_{\frac{1}{2}}(U)) = - \frac{1}{2 \pi i} bb^*( \lambda\, d \bar{\lambda} - \bar{\lambda} \, d \lambda).$$ If $\lambda = e^{2 \pi i f(bb^*)}$ for some smooth real function $f$ we get: $$\pi(ch_{\frac{1}{2}}(U)) = - 2 bb^* f'(bb^*) d (bb^*).$$ To proceed further we need to identify the commutative algebra we are working with and it is easy to see that these are functions on a disk. For this reason the above one-form, which is closed is also exact - so again, within the de Rham complex the lower Chern character is of trivial cohomology class. Although we have concentrated in this paper only on the case of four-dimensional spheres (motivated by the instanton algebra construction of [@CoLa]) there are numerous examples of other deformation of this type (one of which we already mentioned). Clearly, the procedure might be as well generalized to higher-dimensional spheres. Their applications to physical theories (allowing, for instance, for a change of commutativity with time) shall be discussed elsewhere [@Sit2]. [**Acknowledgements**]{}\ The author would like to thank Michel Dubois-Violette for discussion and remarks, Piotr Hajac for thorough discussions on Matsumoto spheres, H-J.Schneider and J.Wess for kind invitation to their seminars and the entire Munich group (Lehrstuhl J.Wess) for hospitality. [C]{} J.Anderson, W.Paschke, [*The rotation algebra*]{}, Houston J.Math. 15, 1, 1–26, (1989) T.Brzeziński, C.Gonera, [*Noncomutative 4-spheres based on all Podleś 2-spheres and beyond.*]{} Lett.Math.Phys. 54, no. 4, 315–321, (2000) A.Connes, [*Noncommutative geometry Year 2000.*]{} GAFA 2000 (Tel Aviv, 1999). Geom. Funct. Anal. 2000, Special Volume, Part II, 481–559, arXiv:math.QA/0011193, A.Connes [*A short survey of noncommutative geometry.*]{} J. Math. Phys. 41, no. 6, 3832–3866, (2000) A.Connes, M. Dubois-Violette, [*Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples.*]{} arXiv:math.QA/0107070 A.Connes, G.Landi, [*Noncommutative manifolds, the instanton algebra and isospectral deformations.*]{} Comm. Math. Phys. 221, no. 1, 141–159, (2001) L.Dabrowski, G.Landi and T.Masuda, [*Instantons on the quantum 4-spheres $S_q^4$,*]{} Comm.Math.Phys. 221, 161, (2001) J-L.Loday, [*Cyclic Homology*]{}, Springer Verlag, Berlin-Heidelberg 1992, K.Matsumoto, [*Noncommutative three-dimensional spheres.*]{} Japan.J.Math. (N.S.) 17, no.2, 333–356, (1991)\ K.Matsumoto, [*Noncommutative three-dimensional spheres. II. Noncommutative Hopf fibering.*]{} Yokohama Math.J. 38 , no.2, 103–111 (1991)\ K.Matsumoto, [*Noncommutative $3$-spheres.*]{} Current topics in operator algebras (Nara, 1990), 234–245, World Sci. Publishing, River Edge, NJ, 1991\ K.Matsumoto, J.Tomiyama, [*Noncommutative lens spaces.*]{} J.Math.Soc.Japan 44, no.1, 13–41, (1992) A.Sitarz, [*More Noncommutative 4-Spheres*]{}, Lett.Math.Phys. 55, 127–131, (2001) A.Sitarz [*Twists and spectral triples for isospectral deformations.*]{} Lett.Math.Phys. , 58, 69–79, (2001) A.Sitarz, [*in preparation*]{} J.Varilly, [*Quantum symmetry groups of noncommutative spheres.*]{} Comm. Math. Phys. 221 no. 3, 511–523, (2001) [^1]: The original definition of Matsumoto three-spheres uses different generators, however, in $C^*$-algebraic formulations invertible transformations between generators of [@Matsumoto] and [@CoLa] could be easily constructed explicitly: these are however only continuous but not smooth.
--- abstract: | Let $n$ be a nonzero integer and $a_1<a_2<\cdots<a_m$ positive integers such that $a_ia_j+n$ is a perfect square for all $1\leq i<j\leq m$. It is known that $m\leq 5$ for $n=1$. In this paper we prove that $m\leq 31$ for $|n|\leq 400$ and $m<15.476\, \log{|n|}$ for $|n|>400$. author: - | [Andrej Dujella]{}\ University of Zagreb, Croatia --- Introduction ============ Let $n$ be a nonzero integer. A set of $m$ positive integers $\{a_1,a_2,\ldots,a_m\}$ is called *a $D(n)$-$m$-tuple* (or *a Diophantine $m$-tuple with the property $D(n)$*) if $a_ia_j+n$ is a perfect square for all $1\leq i<j\leq m$. Diophantus himself found the $D(256)$-quadruple $\{1,\,33,\,68,\,105\}$, while the first $D(1)$-quadruple, the set $\{1,\,3,\,8,\,120\}$, was found by Fermat (see [@Dic; @Dio]). In 1969, Baker and Davenport [@B-D] proved that this Fermat’s set cannot be extended to a $D(1)$-quintuple, and in 1998, Dujella and Pethő [@D-P] proved that even the Diophantine pair $\{1,3\}$ cannot be extended to a D(1)-quintuple. A famous conjecture is that there does not exist a $D(1)$-quintuple. We proved recently that there does not exist a $D(1)$-sextuple and that there are only finitely many, effectively computable, $D(1)$-quintuples (see [@D-jnt; @D-fin]). The question is what can be said about the size of sets with the property $D(n)$ for $n\neq 1$. Let us mention that Gibbs [@Gibbs1] found several examples of Diophantine sextuples, e.g. $\{99,\,315,\,9920,\,32768,\,44460,\,19534284\}$ is a $D(2985984)$-sextuple. Define $$M_n=\sup \{ |S| \,:\, \mbox{$S$ has the property $D(n)$} \}.$$ Considering congruences modulo $4$, it is easy to prove that $M_{n}=3$ if $n\equiv 2 \!\!\pmod{4}$ (see [@Bro; @G-S; @M-R]). On the other hand, if $n\not\equiv 2\!\!\pmod{4}$ and $n\not\in\{-4,\,-3,\,-1,\,3,\,5,\,8,\,12,\,20\}$, then $M_n\geq 4$ (see [@D-acta1]). In [@D-size], we proved that $M_n \leq 32$ for $|n|\leq 400$ and $$M_n < 267.81 \,\log{|n|} \,(\log\log{|n|})^2 \quad \mbox{for $|n| > 400$}.$$ The purpose of the present paper is to improve this bound for $M_n$, specially in the case $|n| > 400$. We will remove the factor $(\log\log{|n|})^2$, and also the constants will be considerably smaller. The above mentioned bounds for $M_n$ were obtained in [@D-size] by considering separately three types (large, small and very small) of elements in a $D(n)$-$m$-tuple. More precisely, let $$\begin{aligned} A_n \!\!&=&\!\! \sup \{ |S\cap [|n|^3, +\infty \rangle |\,:\, \mbox{$S$ has the property $D(n)$} \}, \\ B_n \!\!&=&\!\! \sup \{ |S\cap \langle n^2, |n|^3 \rangle |\,:\, \mbox{$S$ has the property $D(n)$} \}, \\ C_n \!\!&=&\!\! \sup \{ |S\cap [1,n^2] |\,:\, \mbox{$S$ has the property $D(n)$} \}.\end{aligned}$$ In [@D-size], it was proved that $A_n\leq 21$ and $B_n < 0.65 \,\log{|n|} +2.24$ for all nonzero integers $n$, while $C_n < 265.55 \,\log{|n|}\,(\log\log{|n|})^2 +9.01 \,\log\log{|n|}$ for $|n| > 400$ and $C_n \leq 5$ for $|n|\leq 400$. The combination of these estimates gave the bound for $M_n$. In the estimate for $A_n$, a theorem of Bennett [@Ben] on simultaneous approximations of algebraic numbers was used in combination with a gap principle, while a variant of the gap principle gave the estimate for $B_n$. The bound for $C_n$ (number of “very small” elements) was obtained using the Gallagher’s large sieve method [@Gal] and an estimate for sums of characters. In the present paper, we will significantly improve the bound for $C_n$ using a result of Vinogradov on double sums of Legendre’s symbols. Let us mention that Vinogradov’s result, in a slightly weaker form, was used recently, in similar context, by Gyarmati [@Gya] and Sárközy & Stewart [@Sar-Ste]. We will prove the following estimates for $C_n$. \[pr:1\] If $|n| > 400$, then $C_n < 11.006 \,\log{|n|}$. If $|n|\geq {10}^{100}$, then $C_n < 8.37 \,\log{|n|}$. More detailed analysis of the gap principle used in [@D-size] will lead us to the slightly improved bounds for $B_n$. \[pr:2\] For all nonzero integers $n$ it holds $B_n < 0.6114\, \log{|n|} + 2.158$. If $|n| > 400$, then $B_n < 0.6071\, \log{|n|} + 2.152$. By combining Propositions \[pr:1\] and \[pr:2\] with the above mentioned estimate for $A_n$, we obtain immediately the following estimates for $M_n$. \[tm:1\] If $|n| \leq 400$, then $M_n \leq 31$. If $|n| > 400$, then $M_n < 15.476, \log{|n|}$. If $|n|\geq {10}^{100}$, then $M_n < 9.078\, \log{|n|}$. Three lemmas ============ \[l:V\] Let $p$ be an odd prime and $\gcd(n,p)=1$. If $A,B \subseteq \{0,1, \ldots, p-1\}$ and $$T = \sum_{x\in A} \sum_{y\in B} \Big( \frac{xy+n}{p} \Big),$$ then $|T| < \sqrt{p|A|\cdot |B|}$. See [@Vin Problem V.8.c)]. ------------------------------------------------------------------------ \[l:G\] If all but $g(p)$ residue classes [mod]{} $p$ are removed for each prime $p$ in a finite set ${\cal S}$, then the number of integers which remain in any interval of length $N$ is at most $$\label{gal} \Big( \sum_{p\in{\cal S}} \log{p} -\log{N}\Big) \Big/ \Big( \sum_{p\in{\cal S}} \frac{\log{p}}{g(p)} - \log{N}\Big)$$ provided the denominator is positive. See [@Gal]. ------------------------------------------------------------------------ \[l:3\] If $\{a,b,c\}$ is a Diophantine triple with the property $D(n)$ and $ab+n=r^2$, $ac+n=s^2$, $bc+n=t^2$, then there exist integers $e$, $x$, $y$, $z$ such that $$ae+n^2=x^2, \quad be+n^2=y^2, \quad ce+n^2=z^2$$ and $$c=a+b+\frac{e}{n} + \frac{2}{n^2}(abe+rxy).$$ See [@D-size Lemma 3]. ------------------------------------------------------------------------ Proof of Proposition 1 ====================== Let $N\geq n^2$ be a positive integer. Since $|n|>400$, we have $N>1.6\cdot 10^5$. Let $D =\{a_1,a_2,\ldots,a_m\} \subseteq \{1,2, \ldots, N\}$ be a Diophantine $m$-tuple with the property $D(n)$. We would like to find an upper bound for $m$ in term of $N$. We will use the Gallagher’s sieve (Lemma \[l:G\]). Let $${\cal S}=\{ p \,:\, \mbox{$p$ is prime, $\gcd(n,p)=1$ and $p\leq Q$} \},$$ where $Q$ is sufficiently large. For a prime $p\in {\cal S}$, let $C$ denotes the set of integers $b$ such that $b \in \{0,1,2,\dots, p-1\}$ and there is at least one $a\in D$ such that $b\equiv a \!\!\pmod{p}$. Then $\Big( \frac{xy+n}{p} \Big) \in \{0,1\}$ for all distinct $x,y\in C$. Here $\Big( \frac{.}{p} \Big)$ denotes the Legendre symbol. If $0\in C$, then $\Big( \frac{n}{p} \Big) =1$. For given $x\in C\setminus \{0\}$, we have $\Big( \frac{xy_0+n}{p} \Big)$ for at most one $y_0\in C$. If $y\neq x,y_0$, then $\Big( \frac{xy+n}{p} \Big)=1$. Therefore, $$\begin{aligned} T &=& \sum_{x,y\in C} \Big( \frac{xy+n}{p} \Big) = \sum_{x\in C} \Big( \sum_{y\in C} \Big( \frac{xy+n}{p} \Big) \Big) \\ &\geq & \sum_{x\in C} (|C|-3) \geq |C|(|C|-3).\end{aligned}$$ On the other hand, Lemma \[l:V\] implies $$T < |C| \cdot \sqrt{p}.$$ Thus, $|C| < \sqrt{p} +3$ and we may apply Lemma \[l:G\] with $$g(p) = \min \{ \lfloor \sqrt{p} \rfloor +3, p\}.$$ Let us denote the numerator and denominator from (\[gal\]) by $E$ and $F$, respectively. By [@R-S Theorem 9], we have $$E=\sum_{p\in {\cal S}} \log{p} - \log N<\theta(Q) < 1.01624\,Q.$$ The function $f(x)=\frac{\log x}{\min \{\sqrt{x}+3,x\}}$ is strictly decreasing for $x>25$. Also, if $Q\geq 118$, then $f(p)\geq f(Q)$ for all $p\leq Q$. For $p\in \mathcal{S}$ it holds $\gcd(n,p)=1$. This condition comes from the assumptions of Lemma \[l:V\]. However, we will show later that $n$ can be divisible only by small proportion of the primes $\leq Q$. Assume that $n$ is divisible by at most 5% of primes $\leq Q$. Then, for $Q\geq 118$, we have $$\begin{aligned} \label{z2} F &\geq& \sum_{p\in{\cal S}} f(p) - \log N \geq \frac{\log Q}{\sqrt{Q} +3} \cdot |{\cal S}| - \log{N} \nonumber \\ &\geq& \frac{\log Q}{\sqrt{Q} +3} \cdot \frac{19}{20} \pi(Q) - \log{N} > \frac{0.95\, Q}{\sqrt{Q}+3} - \log N.\end{aligned}$$ Since $F$ has to be positive in the applications of Lemma \[l:G\], we will choose $Q$ of the form $$\begin{aligned} Q= c_1 \cdot \log^2{N}.\end{aligned}$$ We have to check whether our assumption on the proportion of primes which divide $n$ is correct. Suppose that $n$ is divisible by at least 5% of the primes $\leq Q$. Then $|n|\geq p_1p_2\cdots p_{\lceil \pi(Q)/20 \rceil}$, where $p_i$ denotes the $i$-th prime. By [@R-S 3.5 and 3.12], we have $p_{\lceil \pi(Q)/20 \rceil} > R$, where $$R= \frac{1}{20}\frac{Q}{\log{Q}} \log\Big(\frac{1}{20}\frac{Q}{\log{Q}} \Big).$$ Assume that $c_1\geq 6$. Then $Q>860$ and $R>11.77$. From [@R-S 3.16], it follows that $$\label{z1} \log{|n|}>\sum_{p\leq R} \log{p} >R\Big(1-\frac{1.136}{\log{R}} \Big).$$ Furthermore, $\frac{1}{20}\frac{Q}{\log{Q}} >Q^{0.273}$ and $R>0.0136\,Q$. Hence, (\[z1\]) implies $\log{R}>7.793$ and therefore $$\log N \geq 2\log |n| > 0.01466\, Q \geq 0.08796\, \log^2 N,$$ contradiction the assumption that $N>1.6\cdot 10^5$. Therefore, we have that $n$ is divisible by at most 5% of the primes $\leq Q$, and hence we have justifies the estimate (\[z2\]). Under the assumption that $c_1\geq 6$, the inequality (\[z2\]) implies $$F> 0.861\, \sqrt{Q} - \log N = (0.861\,\sqrt{c_1} -1) \log N$$ and $$\frac{E}{F} < \frac{1.017\, c_1}{0.861\,\sqrt{c_1}-1} \cdot \log N.$$ For $c_1=6$ we obtain $$\label{400} \frac{E}{F} < 5.503\, \log N.$$ Assume now that $N\geq {10}^{200}$ and $c_1\geq 4$. Then $Q> 848303$ and we can prove in the same manner as above that $n$ is divisible by at most 1% of the primes $\leq Q$. This fact implies $$\frac{E}{F} < \frac{1.017 c_1}{0.986\sqrt{c_1}-1} \cdot \log N.$$ For $c_1=4.11$ we obtain $$\label{100} \frac{E}{F} < 4.185 \log N.$$ Setting $N=n^2$ in (\[400\]) and (\[100\]), we obtain the statements of Proposition \[pr:1\]. ------------------------------------------------------------------------ Proof of Proposition 2 ====================== We may assume that $|n| > 1$. Let $\{a,b,c,d\}$ be a $D(n)$-quadruple such that $n^2 < a<b<c<d$. We apply Lemma \[l:3\] on the triple $\{b,c,d\}$. Since $b>n^2$ and $be+n^2 \geq 0$, we have that $e\geq 0$. If $e=0$, then $d=b+c+2\sqrt{bc+n} < 2c+2\sqrt{c(c-1)+n} < 4c$, contradicting the fact that $d>4.89\, c$ (see [@D-size Lemma 5]). Hence $e\geq 1$ and $$\label{d} d > b+c+\frac{2bc}{n^2}+ \frac{2t\sqrt{bc}}{n^2}.$$ Lemma \[l:3\] also implies $$\label{c} c \geq a+b+2r.$$ From $r^2\geq ab-\sqrt[4]{ab}$ and $ab\geq 30$ it follows that $r > 0.96\, a$, and (\[c\]) implies $c>3.92\, a$. Similarly, $bc\geq 42$ implies $t > 0.969\, \sqrt{bc}$ and, by (\[d\]), $d> b+c+ 3.938\, \frac{bc}{n^2} > 4.938\, c+b$. Assume now that $\{a_1,a_2,\ldots,a_m\}$ is a $D(n)$-$m$-tuple and $n^2 < a_1 < a_2 < \cdots <a_m <|n|^3$. We have $$\begin{aligned} a_3 > 3.92\, a_1, \quad a_{i} > 4.938\, a_{i-1} + a_{i-2}, \quad \mbox{for $i=4,5,\ldots,m$}.\end{aligned}$$ Therefore, $a_m > {\alpha}_m a_1$, where the sequence $({\alpha}_k)$ is defined by $$\begin{aligned} \label{alpha} {\alpha}_k = 4.938 {\alpha}_{k-1}+{\alpha}_{k-2}, \quad {\alpha}_2 = 1, \,\, {\alpha}_3 = 3.92.\end{aligned}$$ Solving the recurrence (\[alpha\]), we obtain ${\alpha}_k \approx \beta {\gamma}^{k-3}$, with $\beta \approx 3.964355$, $\gamma \approx 5.132825$. More precisely, $$|{\alpha}_k - \beta {\gamma}^{k-3}| < \frac{1}{\beta {\gamma}^{k-3}} \,.$$ From $|n|^3 - 1 \geq a_m > {\alpha}_m a_1 \geq {\alpha}_m (n^2+1)$, it follows ${\alpha}_m \leq |n|-\frac{1}{|n|}$ and $\beta {\gamma}^{m-3} < |n|$. Hence, $$\label{m} m < \frac{1}{\log{\gamma}} \log{|n|} + 3 - \frac{\log{\beta}}{\log{\gamma}}.$$ For the above values of $\beta$ and $\gamma$ we obtain $$m < 0.6114\, \log{|n|} + 2.158.$$ Assume now that $|n| > 400$. Then $bc>ab>{400}^4$, which implies $c>3.999999\, a$ and $d>4.999999\, c + b$. Therefore, in this case the relation (\[m\]) holds with $\beta \approx 4.042648$, $\gamma \approx 5.192581$, and we obtain $$m < 0.6071\, \log{|n|} + 2.152.$$ ------------------------------------------------------------------------ The constants in Theorem \[tm:1\] can be improved, for large $|n|$, by using formula (2.26) from [@R-S] in the estimate for the sum $\sum_{p \in {\cal S}} f(p)$. In that way, it can be proved that for every $\varepsilon > 0$, $F > (2-\varepsilon) \sqrt{Q} - \log N$ holds for sufficiently large $Q$. Also, in the proof of Proposition \[pr:2\], for sufficiently large $|n|$ we have $c>(4-\varepsilon)a$ and $d>(5-\varepsilon)c+b$, which leads to $\displaystyle{ B_n < \bigg(\frac{1}{\log(\frac{5+\sqrt{29}}{2})}+\varepsilon \bigg) \,\log |n|}$. These results imply that for every $\varepsilon > 0$ there exists $n(\varepsilon)$ such that for $|n|>n(\varepsilon)$ it holds $$M_n < \bigg(2+\frac{1}{\log(\frac{5+\sqrt{29}}{2})}+\varepsilon \bigg) \,\log |n|.$$ The author is grateful to the referees for valuable comments, and in particular for pointing out a gap in the proof of Proposition \[pr:1\] in the first version of the manuscript. [99]{} [Department of Mathematics\ University of Zagreb\ Bijenička cesta 30, 10000 Zagreb\ Croatia\ [*E-mail address*]{}: [duje@math.hr]{}]{}
--- abstract: 'It is well known, that the Alamouti scheme is the only space-time code from orthogonal design achieving the capacity of a multiple-input multiple-output (MIMO) wireless communication system with $n_T=2$ transmit antennas and $n_R=1$ receive antenna. In this work, we propose the $n$-times stacked Alamouti scheme for $n_T=2n$ transmit antennas and show that this scheme achieves the capacity in the case of $n_R=1$ receive antenna. This result may regarded as an extension of the Alamouti case. For the more general case of more than one receive antenna, we show that if the number of transmit antennas is higher than the number of receive antennas we achieve a high portion of the capacity with this scheme. Further, we show that the MIMO capacity is at most twice the rate achieved with the proposed scheme for all SNR. We derive lower and upper bounds for the rate achieved with this scheme and compare it with upper and lower bounds for the capacity. In addition to the capacity analysis based on the assumption of a coherent channel, we analyze the error rate performance of the stacked OSTBC with the optimal ML detector and with the suboptimal lattice-reduction (LR) aided zero-forcing detector. We compare the error rate performance of the stacked OSTBC with spatial multiplexing (SM) and full-diversity achieving schemes. Finally, we illustrate the theoretical results by numerical simulations.' author: - 'Aydin Sezgin,  and Oliver Henkel, [^1][^2] [^3]' title: 'Stacked OSTBC: Error Performance and Rate Analysis ' --- Introduction ============ Recent information theoretic results have demonstrated that the ability of a system to support a high link quality and higher data rates in the presence of Rayleigh fading improves significantly with the use of multiple transmit and receive antennas [@Telatar99; @FoschiniGans98]. Since then there has been considerable work on a variety of schemes [@TirkHotWichBuch] which exploit multiple antennas at both the transmitter and receiver in order to either obtain transmit and receive diversity and therefore increase the reliability of the system, e.g., orthogonal space-time block codes (OSTBC) and space-time trellis codes [@TarokhSeCa98; @Alamouti; @TarokhJafarkCalder99] or achieve the theoretical bounds [@Foschini96] derived in [@Telatar99; @FoschiniGans98]. Interested readers are referred to [@TirkHotWichBuch], where a detailed analysis of different schemes is given. The performance of OSTBC with respect to mutual information has been analyzed (among others) in [@HassibiHoch2002; @Paulraj; @Sandhu; @Bauch] and it was shown that the capacity is achieved only in the case of $n_T=2$ transmit, the well known Alamouti scheme [@Alamouti], and $n_R=1$ receive antennas due to the rate loss inherent in OSTBC with higher number of transmit antennas. Recently, it was shown in [@SezgJorICASSP05] that due to this rate loss, OSTBC with odd number of antennas are always outperformed by OSTBC with even number of antennas, restricting even more the deployment of OSTBC. On the one hand, we have the OSTBC with low complexity and low rates. On the other hand, we have the space-time trellis codes, which achieve higher spectral efficiency in addition to high performance with respect to frame error rates. However, the decoding complexity of space-time trellis codes is increasing exponentially with the number of transmit antennas and the transmission rate. In order to achieve higher spectral efficiency combined with low complexity maximum likelihood detectors, [@Jafarkhani01; @PapadiasFoschi01; @TirkkonenHottinenNON; @A.Sezgin2004; @SezginInfoTheory] designed quasi-orthogonal space-time block codes (QSTBC) with transmission rate one for more than two transmit antennas. Other approaches aimed at reducing the decoding complexity of space-time trellis codes. For instance, a layered space-time architecture was proposed in [@TarokhNagSesCa99], where the transmit antennas were partitioned into two-antenna groups and on each group space-time trellis codes were used as component codes. In order to further decrease the complexity of this layered space-time architecture, [@A.F.Naguib1998; @N.Prasad2001; @N.Prasad2003] used the Alamouti scheme as component code for each group in combination with a suboptimal successive group interference suppression detection strategy. The outage probability of this scheme was analyzed in [@SezginJorsCapHighRate] for $n_T>n_R$ and an upper bound was derived. An asymptotic analysis of the rate achievable with this scheme is performed in [@LiYeCapStackedAlamou]. For $n=2$, this transmission scheme is also referred to as double-space-time transmit diversity (DSTTD) and was proposed as one possible candidate for high speed downlink packet access (HSDPA) in 3GPP and beyond [@TexasInstrum]. It is obvious that reducing the computational complexity of the detector without sacrificing much performance is an important issue. There is a huge amount of suboptimal detectors with low complexity in the literature, linear detectors like zero-forcing (ZF) or minimum mean square error (MMSE) and nonlinear detectors like e.g. VBLAST [@WolnianskyFoschiniGoldenValen98]. Unfortunately, these detectors significantly sacrifice performance in terms of bit-error-rate (BER). Recently, lattice reduction (LR) aided detection in combination with suboptimal detectors was proposed by Yao and Wornell in order to improve the performance of multi antenna systems [@YaoWornell]. The lattice reduction algorithm proposed in [@YaoWornell] is optimal, but works only for MIMO systems with two transmit and two receive antennas. The impact of receive antenna correlation on the performance of LR-aided detection was analyzed in [@MengPanYouKimLLLforVLBAST]. In [@WindpassingerLLL], the work of [@YaoWornell] was extended to systems with more transmit and receive antennas, using the sub-optimal LLL [@LLL] lattice reduction algorithm. In [@WuebbenLLL], the LR-aided schemes in [@WindpassingerLLL] were adopted to the MMSE criterion. Note that the error rate curves of all these LR detectors are parallel to those for maximum likelihood (ML) detection with only some penalty in power efficiency. In this work, we show that the stacked Alamouti scheme is capable to achieve the capacity in combination with the optimal maximum likelihood detector for the case of $n_T=2n$ transmit antennas and $n_R=1$ receive antennas. This was also shown for the basic Alamouti scheme with $n_T=2$ and $n_R=1$ [@HassibiHoch2002]. Our result may therefore be regarded as an extension of the Alamouti scheme to $n_T>2$. Furthermore, we show that in the case of more than one receive antenna and if $n_T>n_R$ the stacked Alamouti scheme is capable to achieve a significant portion of the capacity and approaches the capacity if $n_T\gg n_R$. For any $n_T$, $n_R$, we show that the MIMO capacity is at most twice the rate achieved with the proposed scheme for all SNR. However, achieving high portions of the capacity does not guarantee good performance in terms of error probability. Thus, we compare the error-rate performance of the proposed scheme with spatial multiplexing (SM), a rate oriented space-time transmission schemes which achieve a high portion of the capacity of MIMO systems, and with the aforementioned diversity-oriented QSTBC by deploying LR-aided linear ZF and ML detectors at the receiver, respectively. The remainder of this paper is organized as follows. In Section \[seq:system\_model\], we introduce the system model and establish the notation. The structure of the stacked Alamouti scheme and the equivalent channel model are shown in section \[sec:Diversity\_analy\]. The analysis of the mutual information is presented in section \[sec:MutInfo\]. LR-aided linear ZF detection is shortly described in section  \[sec:Schemes\] including the analysis of the probability density function of the condition number of the equivalent channel generated by the different transmission schemes (SM,QSTBC, and stacked OSTBC). Section \[seq:simulations\] provides simulation results, followed by some concluding remarks in Section \[sec:Conclusion\]. System model {#seq:system_model} ============ We consider a system with $n_T$ transmit and $n_R$ receive antennas. Our system model is defined by $$\label{eq:System} \mathbf{Y} = \mathbf{G}_{n_T}\mathbf{H}^T+ \mathbf{N}\;,$$ where $\mathbf{G}_{n_T}$ is the ($T \times n_T $) transmit matrix, $\mathbf{Y}=[\mathbf{y}_1,\dots,\mathbf{y}_{n_R}]$ is the ($T \times n_R$) receive matrix, $\mathbf{H}=[\mathbf{h}_1,\dots,\mathbf{h}_{n_T}]$ is a ($n_R \times n_T $) matrix characterizing the coherent channel, and $\mathbf{N}=[\mathbf{n}_1,\dots,\mathbf{n}_{n_R}]$ is the complex ($T \times n_R$) white Gaussian noise (AWGN) matrix, where an entry $\{n_{ti}\}$ of $\mathbf{N}$ ($1\leq i \leq n_R$) denotes the complex noise at the $i$th receiver for a given time $t\,(1\leq t \leq T)$. The real and imaginary parts of $n_{ti}$ are independent and $\mathcal{N}$(0,$n_T/(2\mathrm{SNR})$) distributed. An entry of the channel matrix is denoted by {$h_{ij}$}. This represents the complex gain of the channel between the $j$th transmit ($1\leq j \leq n_T$) and the $i$th receive ($1\leq i \leq n_R$) antenna, where the real and imaginary parts of the channel gains are independent and normal distributed random variables with $\mathcal{N}$(0,1/2) per dimension. The channel matrix is assumed to be constant for a block of $T$ symbols and changes independently from block to block. The average power of the symbols transmitted from each antenna is normalized to be $\nicefrac{1}{n_T}$, so that the average power of the received signal at each receive antenna is one and the signal-to-noise ratio (SNR) is $\rho$. It is further assumed that the transmitter has no channel state information (CSI) and the receiver has perfect CSI. Code construction {#sec:Diversity_analy} ================= A space time block code is defined by its transmit matrix $\mathbf{G}_{n_T}$ with entries $\{x_j\}_{j=1}^{p}$, which are elements of the vector $\mathbf{x}=[x_1,\dots,x_p]^T$ with $x_1,\dots,x_{p} \in \mathcal{C}$, where $\mathcal{C} \subseteq \mathbb{C}$ denotes a complex modulation signal set with unit average power, e.g. $M$-PSK.. The rate $R$ of a space-time code is defined as $R=p/T$. In this paper, we focus on the rate $n_T/2$ stacked Alamouti scheme. Starting with the well known (basic) Alamouti scheme [@Alamouti] for $n_T=2$ transmit antennas $$\nonumber \mathbf{G}_{2}(x_1,x_2) = \left[ \begin{array}{*{2}{r}} x_1 & x_2 \\ x_2^* & -x_1^* \\ \end{array} \right]\;,$$ the transmit matrix of the stacked Alamouti scheme with $n_T=2n$ is constructed in the following way $$\begin{aligned} \nonumber &\mathbf{G}_{n_T}\left(\{x_j\}_{j=1}^{n_T}\right) \\ & =\left[\mathbf{G}_{2}(x_1,x_2), \mathbf{G}_{2}(x_3,x_4),\dots,\mathbf{G}_{2}(x_{n_T-1},x_{n_T})\right].\nonumber\end{aligned}$$ For the case of $n=2$, i.e. $n_T=4$ transmit antennas we have $$\nonumber \mathbf{G}_{4}(\{x_j\}_{j=1}^{4}) = \left[ \begin{array}{*{4}{r}} x_1 & x_2 & x_3 & x_4 \\ x_2^* & -x_1^* & x_4^* & -x_3^* \\ \end{array} \right]\;,$$ which is also referred to as DSTTD[@TexasInstrum]. After some manipulations (particularly complex-conjugating) the system model in (\[eq:System\]) can be rewritten as $$\label{eq:system_H_vorne} \mathbf{y}'=\mathbf{H}'\mathbf{x}+ \mathbf{n}'\;,$$ where $\mathbf{y}'$, $\mathbf{n}'\in \mathbb{C}^{2n_R}$ and $\mathbf{H}' \in \mathbb{C}^{2n_R\times n_T}$. The equivalent channel equals $$\begin{aligned} \mathbf{H}'=[(\mathbf{H}_{1}')^T,\dots,(\mathbf{H}_{i}')^T,\dots,(\mathbf{H}_{n_R}')^T]^T, \nonumber\end{aligned}$$ where $\mathbf{H}_i'$ is given as $$\label{eq:neuer_Kanal} \mathbf{H}_i'= \left[ \mathbf{H}_{i,1}',\mathbf{H}_{i,3}',\dots,\mathbf{H}_{i,n_T-1}' \right],$$ where $$\nonumber \mathbf{H}_{i,j}' = \left[ \begin{array}{*{2}{c}} h_{ij} & h_{i(j+1)} \\ -h_{i(j+1)}^* & h_{ij}^* \\ \end{array} \right]\;.$$ Mutual Information {#sec:MutInfo} ================== The instantaneous capacity $I$ of a MIMO system with $n_T$ transmit and $n_R$ receive antennas is given as [@Telatar99; @FoschiniGans98] $$\label{eq:MutInfoNrGeneral} I = \log_2\det\left(\mathbf{I}_{n_T} +\frac{\rho}{n_T}\mathbf{H}^H\mathbf{H}\right)\;.$$ In the following two subsections, we derive lower and upper bounds for both the ergodic capacity and the average rate achievable with the proposed stacked scheme in order to yield lower and upper bounds on the ratio of the ergodic capacity to the average rate of the stacked OSTBC. In the third subsection, we characterize the absolute loss of the average rate of the stacked OSTBC to the ergodic capacity. Upper bounds on the ergodic capacity and the average rate of stacked OSTBC {#sec:MutInfoUB} -------------------------------------------------------------------------- By applying the trace-determinant inequality $\det(\mathbf{A})^{1/n}\leq \frac{1}{n}\mathrm{tr}(\mathbf{A})$, we arrive at a simple upper bound on the instantaneous capacity given as $$\label{eq:UppBoundCapa} I \leq I_{ub}= L \log_2 \bigg(1+\frac{\rho}{n_TL}\underbrace{\sum_{j=1}^{n_T}\sum_{i=1}^{n_R}|h_{ji}|^2}_{\lambda}\bigg)\;,$$ where $L$ is equal to $L=\min(n_T,n_R)$. Averaging the upper bound in  over all channel realizations results in [@Gradshteyn] ($C=\mathbb{E}[I]$ denotes ergodic capacity) $$\begin{aligned} \label{eq:UppBoundAnaly} C \leq C_{ub} =\mathbb{E}\left[I_{ub}\right] = & \frac{L}{\ln(2)}\sum\limits_{k=0}^{n_Tn_R-1} \left(\frac{n_TL}{\rho}\right)^{n_Tn_R-k-1}\\ & e^{\frac{n_TL}{\rho}} \Gamma\left(1-(n_Tn_R-k),\frac{n_TL}{\rho}\right) \nonumber.\end{aligned}$$ Note that for high SNR, the slope of the upper bound is equal to $L$. In addition to this upper bound, we compare the rate achieved with the stacked scheme with the following upper bound $$\label{eq:BoundGrant} C \leq C_{\mathrm{Jen}} =\log_2\left(\sum_{i=0}^L \binom{L}{i}\frac{K!}{(K-i)!}\left(\frac{\rho}{n_T}\right)^i\right),$$ derived in [@GrantUppBound] by using Jensen’s inequality, where $K=\max(n_T,n_R)$. In the following, we analyze the performance of the stacked scheme with respect to mutual information and derive upper bounds for the average rate of the stacked scheme. We first analyze the case of $n_R=1$ receive antennas and then generalize the analysis to the case of arbitrary number of receive antennas. ### Case $n_R=1$ In case of $n_R=1$, the achievable rate of the stacked Alamouti scheme is $$\nonumber I_{sA} = \frac{1}{2}\log_2\det\left(\mathbf{I}_{n_T} +\frac{\rho}{n_T}(\mathbf{H}_1')^H\mathbf{H}_1'\right)\;.$$ Using the determinant equality $\det(\mathbf{I}+\mathbf{AB})=\det(\mathbf{I}+\mathbf{BA})$, after some manipulations we arrive at $$\label{eq:MutInfoNReq1} I_{sA} = \log_2\left(1 +\frac{\rho}{n_T} \sum\limits_{j=1}^{n_T}|h_{j1}|^2\right)\;,$$ which equals the capacity of a MIMO system with $n_T$ transmit and $n_R=1$ receive antennas [@Telatar99], i.e. as long as $n_R=1$, the capacity is achieved for arbitrary $n=n_T/2$. Note that in [@TirkHotWichBuch p.199] a Taylor series expansion is performed for the capacity and the mutual information achievable with certain schemes such as the stacked OSTBC. After comparing the first two expansion coefficients (the linear term and the second order coefficients) it is shown that the stacked OSTBC reaches second-order capacity for $n_R=1$, i.e. the second-order coefficient of the mutual information of the stacked OSTBC is equal to the second-order coefficient of the capacity. Although essential features of the mutual information can be already seen from the first and second-order coefficients (especially at low SNR), our result above may regarded as more general, since the exact capacity and mutual information expressions are analyzed. Further note that the result above may be regarded as an extension of the results in [@HassibiHoch2002]. There it was shown, that the basic Alamouti scheme with $n_T=2$ and $n_R=1$ achieves the capacity. ### Case of $n_T=4$ and $n_R=2$ (DSTTD) In the case of $n_T=4$ transmit and $n_R=2$ receive antennas, the equivalent channel is given by $$\nonumber \mathbf{H}'= \left[% \begin{array}{cccc} h_{11} & h_{12} & h_{13} & h_{14} \\ -h^*_{12} & h^*_{11} &-h^*_{14} & h^*_{13} \\ h_{21} & h_{22} & h_{23} & h_{24} \\ -h^*_{22} & h^*_{21} &-h^*_{24} & h^*_{23} \\ \end{array}% \right].$$ The achievable rate in this case is given as $$\nonumber I_{sA} = \frac{1}{2}\log_2\det\left(\mathbf{I}_{n_T} +\frac{\rho}{n_T}\left[% \begin{array}{cccc} \lambda_1 & 0 & \alpha_1 & \alpha_2 \\ 0 & \lambda_1 & -\alpha_2^* & \alpha_1^* \\ \alpha_1^* & -\alpha_2 & \lambda_2 & 0 \\ \alpha_2^* & \alpha_1 & 0 & \lambda_2 \\ \end{array}% \right]\right),$$ where $\lambda_i=\sum_{j=1}^{n_T}|h_{ij}|^2$, $\alpha_1=h_{11}h_{21}^*+h_{12}h_{22}^*+h_{13}h_{23}^*+h_{14}h_{24}^*$, and $\alpha_2=-h_{11}h_{22}+h_{12}h_{21}-h_{13}h_{24}^*+h_{14}h_{23}$. Using Fischer’s inequality $$\nonumber \det\left(\left[% \begin{array}{cc} \mathbf{A} & \mathbf{B^H} \\ %Oliver B-->B^H \mathbf{B} & \mathbf{D} \\ %Oliver C-->B \end{array}% \right]\right)\leq \det(\mathbf{A})\det(\mathbf{D})$$ yields $$\nonumber I_{sA}\leq \log_2\left(\left(1+\frac{\rho}{n_T}\lambda_1\right)\left(1+\frac{\rho}{n_T}\lambda_2\right)\right).$$ By using the arithmetic-geometric inequality, we arrive at $$\nonumber I_{sA}\leq 2\log_2\left(1+\frac{\rho}{2n_T}||\mathbf{H}||^2\right).$$ This upper bound equals to twice the rate of a full code rate OSTBC for $n_T=4$ transmit and $n_R=2$ receive antennas with a power penalty of $3$ dB. In this particular case a more precise statement can be made due to the following strict form of Fischer’s inequality [@hen.wun-itg05] \[lem.strictFischer\] Let $\mathbf{P}=\left[ \begin{array}{cc} \mathbf{A} & \mathbf{B^H} \\ \mathbf{B} & \mathbf{D} \\ \end{array}% \right]$ ($\mathbf{A},\mathbf{D}$ square, nonempty) be positive definite. Then $$\text{$\mathbf{B}$ has full rank} \quad\Rightarrow\quad \det\mathbf{P} < (\det \mathbf{A}) (\det \mathbf{D})$$ Let $\mathbf{R}\succ0$ denote positive definiteness, and $\mathbf{R}\succ \mathbf{S}$ defined by $(\mathbf{R}-\mathbf{S})\succ 0$. Then [@HornJohnson 7.7.6] $\mathbf{P}\succ0 \Leftrightarrow (\mathbf{A}\succ0, \mathbf{D}\succ \mathbf{BA}^{-1}\mathbf{B}^H)$. Thus for arbitrary $\mathbf{B}$ holds $\mathbf{D}-(\mathbf{D}-\mathbf{BA}^{-1}\mathbf{B}^H)=\mathbf{BA}^{-1}\mathbf{B}^H\succeq0$ and becomes strict if $\mathbf{B}$ has full rank. Since $\big(0\prec \mathbf{S}\prec \mathbf{R} \Rightarrow \det \mathbf{S}< \det \mathbf{R}\big)$ we obtain $\det \mathbf{P}=(\det\mathbf{A})(\det[\mathbf{D}-\mathbf{BA}^{-1}\mathbf{B}^H]) < (\det \mathbf{A})(\det \mathbf{D})$, if $\mathbf{B}$ has full rank. From $\det\mathbf{B}=|\alpha_1|^2+|\alpha_2|^2$ it follows, that apart from the set of events $\{\alpha_1=\alpha_2=0\}$ of measure zero, $\mathbf{B}$ has full rank, thus the upper bound for $I_{sA}$ is strict with probability one. ### Case of arbitrary $n_R$ The available portion of the mutual information achievable with $n_R\geq 1$ for the stacked Alamouti scheme is $$\label{eq:exactMutInfoStackAlam} I_{sA} = \frac{1}{2}\log_2\det\left(\mathbf{I}_{n_T} +\frac{\rho}{n_T}(\mathbf{H}')^H\mathbf{H}'\right)\;.$$ Following the derivation above for arbitrary $n_R$ results in $$\label{eq:UbbBoundStAl} I_{sA}\leq I_{sA}^{ub}=\frac{L_1}{2}\log_2 \left(1+\frac{2\rho}{n_TL_1}||\mathbf{H}||^2\right),$$ where $L_1=\min(n_T,2n_R)$. By averaging (\[eq:UbbBoundStAl\]) over all channel realizations, an upper bound on the average rate $R_{sA}^{ub}\geq \mathbb{E}[I_{sA}]$ of the stacked Alamouti scheme similar to  may be obtained $$\begin{aligned} \label{eq:UppBoundExactNRBel} R_{sA} \leq R_{sA}^{ub} =\mathbb{E}\left[I_{ub}\right] = & \frac{L_1}{2\ln(2)}\sum\limits_{k=0}^{n_Tn_R-1} \left(\frac{n_TL_1}{2\rho}\right)^{n_Tn_R-k-1}\\ & e^{\frac{n_TL_1}{2\rho}} \Gamma\left(1-(n_Tn_R-k),\frac{n_TL_1}{2\rho}\right)\nonumber,\end{aligned}$$ which can be approximated using $\log_2(1+x)\approx \log_2(x)$ for $x\gg 1$ by $$\nonumber R_{sA}^{ub}\approx \frac{L_1}{2}\log_2\left(\frac{2\rho}{n_TL_1}\right)+\frac{L_1}{2\ln(2)}\left(\sum\limits_{p=1}^{n_Tn_R-1}\frac{1}{p}-\gamma\right).$$ Note that the approximation gets better for higher SNR and may be inaccurate for low SNR. Further note that, for high SNR, the slope of the upper bound  and its approximation is equal to $\nicefrac{L_1}{2}$. Lower bounds on the ergodic capacity and the average rate of stacked OSTBC {#sec:MutInfoLB} -------------------------------------------------------------------------- Similarly to the last subsection, here we derive lower bounds for the ergodic capacity and the average rate of the stacked OSTBC. Due to the peculiar property of stacked OSTBC, lower bounds are obtained in the procedure for the following cases: (i) $n_T\leq n_R$, (ii) $n_R < n_T < 2n_R$, (iii) $2n_R\leq n_T \leq 4n_R$, and (iv) $4n_R < n_T$. First of all, from [@OymanNBPaulraj] we obtain the following lower bound on the ergodic capacity $$\label{eq:BoundOyman} C \geq C_{lb}= \sum_{j=1}^{L}\log_2\left(1+\frac{\rho}{n_T}\exp\left(\sum_{p=1}^{K-j}\frac{1}{p}-\gamma\right)\right),$$ where $\gamma\approx 0.57721566$ is Euler’s constant. In order to derive an upper bound on the ratio of the ergodic capacity to the average rate achieved with the stacked scheme, we need a lower bound for the average rate of the stacked scheme. To this end, we rewrite  as follows $$\begin{aligned} \label{eq:MutInfoStAlReW} I_{sA} = \frac{1}{2}\log_2\det\left(\mathbf{I}_{n_T} +\frac{\rho}{n_T}(\mathbf{H})^H\mathbf{H}+\frac{\rho}{n_T}(\mathbf{H}'_e)^H\mathbf{H}'_e\right),\end{aligned}$$ where $\mathbf{H}$ is the actual MIMO channel, which is obtained by taking the odd rows of the equivalent channel $\mathbf{H}'$ and $\mathbf{H}_e$ is obtained by taking the even rows of $\mathbf{H}'$. The relation between the actual channel $\mathbf{H}$ and $\mathbf{H}_e$ is described in the following proposition. \[prop:RelHandHe\] Let $\mathbf{H}_e$ be the even and $\mathbf{H}$ the odd rows of $\mathbf{H}'$ given in , respectively. Then the following holds 1. $\nonumber \mathbf{H}_e=\mathbf{H}^*\mathbf{J} $, where[^4] $$\nonumber \mathbf{J}=\mathbf{I}_{\frac{n_T}{2}}\otimes\left[% \begin{array}{cc} 0 & 1 \\ -1 & 0 \\ \end{array}% \right].$$ 2. $ \mathbb{E}\left[\mathbf{H}\mathbf{H}_e^H\right]=\mathbb{E}\left[\mathbf{H}\mathbf{J}^T\mathbf{H}^T\right]=\mathbf{0} $. The proof is straightforward and uninformative and thus it is omitted. Eq.  can be rewritten as $$\begin{aligned} \nonumber I_{sA} & =\frac{1}{2}\log_2\Bigg(\det\left(\mathbf{I}_{n_T} +\frac{\rho}{n_T}(\mathbf{H})^H\mathbf{H}\right) \times\\ &\det\left(\mathbf{I}_{n_T}+\frac{\rho}{n_T}\mathbf{H}'_e\left(\mathbf{I}_{n_T} +\frac{\rho}{n_T}(\mathbf{H})^H\mathbf{H}\right)^{-1}(\mathbf{H}'_e)^H\right) \Bigg) \nonumber\\ &= \frac{1}{2}\log_2\det\left(\mathbf{I}_{n_T} +\frac{\rho}{n_T}(\mathbf{H})^H\mathbf{H}\right) +\frac{1}{2}\;\times \nonumber \\ & \log_2\det\Big(\mathbf{I}_{n_T} + \frac{\rho}{n_T}\mathbf{H}'_e\left(\mathbf{I}_{n_T} +\frac{\rho}{n_T}(\mathbf{H})^H\mathbf{H}\right)^{-1}(\mathbf{H}'_e)^H\Big)\;.\nonumber\end{aligned}$$ Since $\mathbf{H}'_e\left(\mathbf{I}_{n_T} +\frac{\rho}{n_T}(\mathbf{H})^H\mathbf{H}\right)^{-1}(\mathbf{H}'_e)^H$ is a positive semidefinite matrix, it follows immediately that the rate achieved with the stacked Alamouti is lower bounded by $$\begin{aligned} \nonumber I_{sA} \geq \frac{1}{2}\log_2\det\left(\mathbf{I}_{n_T} +\frac{\rho}{n_T}(\mathbf{H})^H\mathbf{H}\right),\end{aligned}$$ which is half the capacity of a MIMO system with $n_T$ transmit and $n_R$ receive antennas. Another lower bound is obtained for the case $n_T\leq n_R $ by applying Minkowski’s determinant inequality [@HornJohnson p.482] ($\det(\mathbf{A}+\mathbf{B}) \geq (\det(\mathbf{A})^{\frac{1}{n}}+\det(\mathbf{B})^{\frac{1}{n}})^{n}$, $\mathbf{A}\succ0, \mathbf{B}\succeq0$) to  $$\begin{aligned} R_{sA}& =\mathbb{E}\left[\frac{1}{2}\log_2\det\left(\mathbf{I}+\frac{\rho}{n_T}(\mathbf{H}')^H\mathbf{H}'\right)\right] \nonumber \\ & \geq \frac{n_T}{2}\mathbb{E}\left[\log_2\left(1+\rho\det\left(\frac{1}{n_T}(\mathbf{H}')^H\mathbf{H}'\right)^{\frac{1}{n_T}}\right)\right]\nonumber \\ & =\frac{n_T}{2}\mathbb{E}\left[\log_2\left(1+\rho\det\left(\frac{1}{n_T}(\mathbf{H}^H \mathbf{H}+ \mathbf{H}_e^H\mathbf{H}_e)\right)^{\frac{1}{n_T}}\right) \right].\nonumber \end{aligned}$$ Applying again Minkowski’s determinant inequality results in $$\begin{aligned} R_{sA} \geq \frac{n_T}{2}\mathbb{E}\Bigg[\log_2\Big(1+\rho\det\left(\frac{1}{n_T}\mathbf{H}^H\mathbf{H}\right) ^{1/n_T} \nonumber \\ +\rho\det\left(\frac{1}{n_T}\mathbf{H}_e^H\mathbf{H}_e\right)^{\frac{1}{n_T}}\Big)\Bigg].\nonumber\end{aligned}$$ Since $\mathbf{H}_e$ is obtained simply by conjugating and exchanging some elements of the actual matrix $\mathbf{H}$, it can be shown that the eigenvalues of $(\mathbf{H}_e)^H(\mathbf{H}_e)$ are the same as the eigenvalues of $\mathbf{H}^H(\mathbf{H})$. Therefore, the lower bound is equal to $$\begin{aligned} R_{sA}\geq \frac{n_T}{2}\mathbb{E}\left[\log_2\left(1+\rho\exp\ln\left(2\det\left(\frac{1}{n_T}\mathbf{H}^H\mathbf{H}\right)^{\frac{1}{n_T}} \right)\right)\right].\nonumber\end{aligned}$$ Since $\log_2(1+ce^x)$ is a convex function in $x$ for $c>0$ and by applying Jensen’s inequality it holds that $\mathrm{E}\left[\log_2(1+ce^x)\right]\geq \log_2(1+c\exp(\mathrm{E}\left[x\right]))$, we have $$\begin{aligned} R_{sA}& \geq \frac{n_T}{2}\log_2\left(1+\rho\exp\mathbb{E}\left[\ln\left(2\det\left(\frac{1}{n_T}\mathbf{H}^H\mathbf{H}\right)^{\frac{1}{n_T}} \right)\right]\right) \nonumber\\ & = \frac{n_T}{2}\log_2\left(1+\rho 2\exp\frac{1}{n_T}\mathbb{E}\left[\ln\left(\det\left(\frac{1}{n_T}\mathbf{H}^H\mathbf{H}\right) \right)\right]\right).\nonumber \end{aligned}$$ From [@OymanNBPaulraj; @Goodman], we know that $$\nonumber \mathbb{E}\left[\ln\left(\det\left(\frac{1}{n_T}\mathbf{H}^H\mathbf{H}\right) \right)\right]=\sum_{j=1}^{n_T}\mathbb{E}\left[\ln X_j\right]-n_T\ln n_T,$$ where the $X_j$ are independent, $\chi^2$ distributed independent variables with $ 2(n_R-j+1) $ degrees of freedom. Using this yields $$\begin{aligned} \nonumber R_{sA} & \geq \frac{n_T}{2}\log_2\left(1+\frac{\rho}{\frac{n_T}{2}} \exp\left(\frac{1}{n_T}\sum_{j=1}^{n_T}\mathbb{E}\left[\ln X_j\right]\right)\right). \end{aligned}$$ With $$\nonumber \mathbb{E}\left[\ln X_j\right]=\psi(n_R-j+1),$$ where $\psi(\cdot)$ is the digamma function, which may be rewritten for integer arguments as follows $$\nonumber \psi(x)=-\gamma+\sum_{p=1}^{x-1}\frac{1}{p}.$$ Using this results in the following lower bound for the average rate of the stacked scheme. $$\begin{aligned} \nonumber R_{sA}\geq & \frac{n_T}{2}\log_2\left(1+\frac{\rho}{\frac{n_T}{2}}\exp \left(\frac{1}{n_T}\sum_{j=1}^{n_T}\sum_{p=1}^{n_R-j}\frac{1}{p}-\gamma\right)\right)\\ & \quad[\text{case }n_T \leq n_R]. \nonumber\end{aligned}$$ Similar steps can be pursued for $n_T\geq 4 n_R$ resulting in the following lower bound $$\begin{aligned} \nonumber R_{sA}\geq & n_R\log_2\left(1+\frac{2\rho}{n_T}\exp\left(\frac{1}{2n_R}\sum_{j=1}^{2n_R} \sum_{p=1}^{\nicefrac{n_T}{2}-j}\frac{1}{p}-\gamma\right)\right)\\ & \quad[\text{case }n_T > 4n_R] \nonumber\end{aligned}$$ For the case of $n_T\geq 2n_R$ we rewrite as $$\begin{aligned} \label{eq:BlockStrukSOSTBCRate} I_{sA} =\frac{1}{2}\log_2\det\left(\mathbf{I}_{2n_R} +\frac{\rho}{n_T}\left[% \begin{array}{cc} \mathbf{H}\mathbf{H}^H & \mathbf{H}\mathbf{H}_e^H \\ \mathbf{H}_e\mathbf{H}^H & \mathbf{H}_e\mathbf{H}_e^H \\ \end{array}% \right]\right).\end{aligned}$$ Since $\mathbb{E}\left[\mathbf{H}\mathbf{H}_e^H\right]=\mathbf{0}$ from proposition \[prop:RelHandHe\], we may proceed as in [@FoschiniGans98] to arrive at a lower bound given as $$\begin{aligned} \nonumber I_{sA}\geq \frac{1}{2}\sum_{k=1}^{L_1}\log_2\left(1+\frac{\rho}{n_T}X_{k}\right),\end{aligned}$$ where $X_{k}$ are again independent, $\chi^2$ distributed independent variables with $ 2(K_1-k+1) $ degrees of freedom with $K_1=\max(2n_R,n_T)$. By following the same line of arguments as in [@OymanNBPaulraj], we arrive at $$\begin{aligned} \nonumber R_{sA} \geq R_{sA}^{lb}= & \frac{1}{2}\sum_{j=1}^{L_1}\log_2\left(1+\frac{\rho}{n_T} \exp\left(\sum_{p=1}^{K_1-j}\frac{1}{p}-\gamma\right)\right)\\ & \quad[\text{case }n_T \ge 2n_R] \nonumber\end{aligned}$$ In [@LiYeCapStackedAlamou], a similar (however, looser) lower bound was derived for this case in order to analyze the asymptotic performance (with respect to $\rho$) of stacked OSTBC. For the case of $n_R<n_T<2n_R$ we have $$\begin{aligned} R_{sA}& =\mathbb{E}\left[\frac{1}{2}\log_2\det\left(\mathbf{I}+\frac{\rho}{n_T}(\mathbf{H}')^H\mathbf{H}'\right)\right]\nonumber \\ & =\mathbb{E}\left[\frac{1}{2}\log_2\det\left(\mathbf{I}+\frac{\rho}{n_T}\left(\mathbf{H} \mathbf{H}^H+ \mathbf{H}_e\mathbf{H}_e^H\right)\right)\right]\nonumber \\ & = \mathbb{E}\left[\frac{1}{2}\log_2\det\left(\frac{1}{2}\mathbf{I}+\frac{\rho}{n_T}\mathbf{H} \mathbf{H}^H+ \frac{1}{2}\mathbf{I}+\frac{\rho}{n_T}\mathbf{H}_e\mathbf{H}_e^H\right)\right]. \nonumber\end{aligned}$$ Applying now Minkowski’s determinant inequality results in $$\begin{aligned} R_{sA}\geq \frac{1}{2}\mathbb{E}\left[\log_2\det\left(\mathbf{I}+\frac{2\rho}{n_T}\mathbf{H}\mathbf{H}^H\right)\right]\end{aligned}$$ and finally $$\begin{aligned} \nonumber R_{sA} \geq R_{sA}^{lb}= & \frac{1}{2}\sum_{j=1}^{L}\log_2\left(1+\frac{2\rho}{n_T} \exp\left(\sum_{p=1}^{K-j}\frac{1}{p}-\gamma\right)\right) \\ & \quad[\text{case } n_R< n_T < 2n_R]. \nonumber\end{aligned}$$ The lower bound results derived in this subsection are summarized in Table \[tab:LowBounds\] on the top of the next page. ----------------------------------------------------------------------------------------------------------------- Case Lower bound on $R_{sA}$ -------------------- -------------------------------------------------------------------------------------------- $n_T \leq n_R$ $\frac{n_T}{2}\log_2\left(1+\frac{\rho}{\frac{n_T}{2}}\exp \left(\frac{1}{n_T}\sum_{j=1}^{n_T}\sum_{p=1}^{n_R-j}\frac{1}{p}-\gamma\right)\right)$ $n_R < n_T < 2n_R$ $\frac{1}{2}\sum_{j=1}^{L}\log_2\left(1+\frac{2\rho}{n_T} \exp\left(\sum_{p=1}^{K-j}\frac{1}{p}-\gamma\right)\right)$ $2n_R \leq n_T $ $\frac{1}{2}\sum_{j=1}^{L_1}\log_2\left(1+\frac{\rho}{n_T} \exp\left(\sum_{p=1}^{K_1-j}\frac{1}{p}-\gamma\right)\right)$ $4n_R < n_T$ $n_R\log_2\left(1+\frac{2\rho}{n_T}\exp\left(\frac{1}{2n_R}\sum_{j=1}^{2n_R} \sum_{p=1}^{\nicefrac{n_T}{2}-j}\frac{1}{p}-\gamma\right)\right)$ ----------------------------------------------------------------------------------------------------------------- Note that for high SNR, most of the bounds have a slope equal to $\nicefrac{L_1}{2}$, which equals the slope of the upper bound . Only for the case $n_R < n_T < 2n_R$, the slope of the lower bound is equal to $\nicefrac{L}{2}$. In Fig. \[fig:RsAUpLowBounds\] on the top of the next page, the average rate, the upper bound  and the lower bounds from Table \[tab:LowBounds\] for $n_T=$ and $n_R=1,\dots,4$ are depicted. From the Fig., we observe that the upper bound in  and lower bounds track the average rate quite well. Only in the aforementioned case $n_R < n_T < 2n_R$, the slope of the lower bound differs from the exact performance and the upper bound. Note that for $n_R=1$, the upper bound coincides with the exact performance. ------------------------------------------------------------------------ Characterization of the absolute rate loss ------------------------------------------ In this subsection, we characterize the absolute rate loss of the stacked OSTBC to the ergodic capacity using Fischer’s inequality. First of all, we discuss the case of $n_T\geq 2n_R$. Note that the rate loss with the basic Alamouti scheme ($n_T=2$) was also analyzed in [@Sandhu; @HassibiHoch2002] using different approaches. Starting from , applying Fischer’s inequality and averaging over all channel realizations we arrive at $$\begin{aligned} R_{sA} \leq \mathbb{E}\Bigg[ & \frac{1}{2}\log_2\Bigg(\det\left(\mathbf{I}_{n_R} +\frac{\rho}{n_T}\mathbf{H}\mathbf{H}^H\right) \nonumber \\ & \times \left(\mathbf{I}_{n_R} +\frac{\rho}{n_T}\mathbf{H}_e\mathbf{H}_e^H\right)\Bigg)\Bigg] = C(\rho, n_T,n_R) \label{eq:AbsRatLossNTNR} \\ & \qquad[\text{case } n_T \geq 2n_R] \nonumber,\end{aligned}$$ i.e. as long as $n_T\geq 2n_R$, the average rate of the stacked OSTBC is only upper bounded by the ergodic capacity. Proceeding similarly for the case $n_T < 2n_R$ results in $$\begin{aligned} R_{sA} & =\frac{1}{2}\mathbb{E}\left[\log_2\det\left(\mathbf{I}_{n_T} +\frac{\rho}{n_T}\left[% \begin{array}{cc} \mathbf{\tilde{H}}^H\mathbf{\tilde{H}} & \mathbf{\tilde{H}}^H\mathbf{\tilde{H}}_e \\ \mathbf{\tilde{H}}_e^H\mathbf{\tilde{H}} & \mathbf{\tilde{H}}_e^H\mathbf{\tilde{H}}_e \\ \end{array}% \right]\right)\right] \nonumber\\ & \leq \mathbb{E}\left[\log_2\left(\det\left(\mathbf{I}_{\frac{n_T}{2}} +\frac{\rho}{n_T}\mathbf{\tilde{H}}^H\mathbf{\tilde{H}}\right)\right)\right]\nonumber \\ &=C\left(\frac{\rho}{2},\frac{n_T}{2},2n_R\right)<C\left(\rho,n_T,n_R\right) \;\;\;[\text{case } n_T < 2n_R],\label{eq:AbsRatLossNT22NR}\end{aligned}$$ where $\mathbf{\tilde{H}}$ is obtained by taking the odd columns of the equivalent channel $\mathbf{H}'$ and $\mathbf{\tilde{H}}_e$ is obtained by taking the even columns of $\mathbf{H}'$. From , we observe that for $n_T < 2n_R$ the average rate of the stacked OSTBC is upper bounded by the ergodic capacity of a system with $\frac{n_T}{2}$ transmit and $2n_R$ receive antennas with a power penalty of $3$ dB. We can characterize the gap in  and  due to the application of Fischer’s inequality. For $n_T < 2n_R$, we have then $$\begin{aligned} \Delta & = C\left(\frac{\rho}{2},\frac{n_T}{2},2n_R\right)-R_{sA} \nonumber\\ & =\frac{1}{2} \mathbb{E}\left[\log_2\left(\frac{\det\left(\mathbf{I}_{n_T} +\frac{\rho}{n_T}\mathbf{W}_D\right)}{\det\left(\mathbf{I}_{n_T} +\frac{\rho}{n_T}\left[ \begin{array}{cc} \mathbf{\tilde{H}}^H\mathbf{\tilde{H}} & \mathbf{\tilde{H}}^H\mathbf{\tilde{H}}_e \\ \mathbf{\tilde{H}}_e^H\mathbf{\tilde{H}} & \mathbf{\tilde{H}}_e^H\mathbf{\tilde{H}}_e \\ \end{array}% \right]\right)}\right)\right] \nonumber,\end{aligned}$$ where $$\begin{aligned} \nonumber \mathbf{W}_D=\left[% \begin{array}{cc} \mathbf{\tilde{H}}^H\mathbf{\tilde{H}} & \mathbf{0} \\ \mathbf{0} & \mathbf{\tilde{H}}_e^H\mathbf{\tilde{H}}_e \\ \end{array}% \right].\end{aligned}$$ Since the events of $\mathbf{\tilde{H}}_e^H\mathbf{\tilde{H}}$ having not full rank are of measure zero the strict form of Fischer’s inequality stated in Lemma \[lem.strictFischer\] shows, that the gap in  and  is non zero in general, i.e. $\Delta>0$, thus it is not possible to reach the upper capacity bounds. With $$\begin{aligned} \nonumber \mathbf{W}_{\mathrm{Off}}= \left[ \begin{array}{cc} \mathbf{\tilde{H}}^H\mathbf{\tilde{H}} & \mathbf{\tilde{H}}^H\mathbf{\tilde{H}}_e \\ \mathbf{\tilde{H}}_e^H\mathbf{\tilde{H}} & \mathbf{\tilde{H}}_e^H\mathbf{\tilde{H}}_e \\ \end{array} \right] - \mathbf{W}_D,\end{aligned}$$ we can rewrite $$\begin{aligned} & \det\left(\mathbf{I}_{n_T} + \frac{\rho}{n_T} \left[ \begin{array}{cc} \mathbf{\tilde{H}}^H\mathbf{\tilde{H}} & \mathbf{\tilde{H}}^H\mathbf{\tilde{H}}_e \\ \mathbf{\tilde{H}}_e^H\mathbf{\tilde{H}} & \mathbf{\tilde{H}}_e^H\mathbf{\tilde{H}}_e \\ \end{array} \right]\right) \nonumber \\ & = \det\left(\mathbf{I}_{n_T} +\frac{\rho}{n_T}(\mathbf{W}_{\mathrm{Off}}+\mathbf{W}_{D})\right) \nonumber \\ & =\det\left(\mathbf{I}_{n_T} +\frac{\rho}{n_T}\mathbf{W}_{D}\right)\times \nonumber\\ & \det\left(\mathbf{I}_{n_T} +\left[\mathbf{I}_{n_T} +\frac{\rho}{n_T}\mathbf{W}_{D}\right]^{-1}\frac{\rho}{n_T}\mathbf{W}_{\mathrm{Off}}\right)\nonumber\end{aligned}$$ to arrive at $$\begin{aligned} \nonumber \Delta=-\frac{1}{2} \mathbb{E}\Bigg[\log_2\det\Bigg(\mathbf{I}_{n_T} +\underbrace{\left[\mathbf{I}_{n_T} +\frac{\rho}{n_T}\mathbf{W}_{D}\right]^{-1}\frac{\rho}{n_T}\mathbf{W}_{\mathrm{Off}}}_{\mathbf{A}}\Bigg)\Bigg].\end{aligned}$$ Using $$\begin{aligned} \nonumber \det(\mathbf{I}_{n_T}+\mathbf{A})=\exp\left(\sum_{k=1}^{L_1}\ln(1+\mu_{k})\right)\end{aligned}$$ yields $$\begin{aligned} \nonumber \Delta \leq \frac{1}{2\ln(2)}\mathbb{E}\left[\sum\limits_{k=1}^{L_1}\mu_{k}^2\right]\end{aligned}$$ where the inequality follows from Taylor series expansion $x-\frac{1}{2}x^2\leq \ln(1+x)$ around $x=0$ and the fact that $\mathrm{tr}(\mathbf{A})=0$, since $\mathbf{A}$ has zero block matrices on its diagonal. Its off-diagonal blocks have the form $\mathbf{B}=\left[\mathbf{I}_{n_T} +\frac{\rho}{n_T}\mathbf{\tilde{H}}^H\mathbf{\tilde{H}}\right]^{-1} \frac{\rho}{n_T}\mathbf{\tilde{H}}^H\mathbf{\tilde{H}_e}$ and $\mathbf{B}_e=\left[\mathbf{I}_{n_T} +\frac{\rho}{n_T}\mathbf{\tilde{H}_e}^H\mathbf{\tilde{H}_e}\right]^{-1} \frac{\rho}{n_T}\mathbf{\tilde{H_e}}^H\mathbf{\tilde{H}}$, respectively. Note that the matrices in brackets have the same eigenvalues. This implies that each eigenvalue of $\mathbf{A}$ appears twice, i.e. $\mu_{k}=\mu_{k+\nicefrac{L_1}{2}}$, $1 \leq k \leq \nicefrac{L_1}{2}$. Additionally applying the inequality [@HornJohnson (5.6.Ex.26)] $\mathrm{tr}(\mathbf{A}^2)\le ||\mathbf{A}||^2$ we obtain $\sum\limits_{k=1}^{L_1}\mu_{k}^2 =2\sum\limits_{k=1}^{L_1/2}\mu_{k}^2 \le 2 ||\mathbf{B}||_F^2$. Further we have $$||\mathbf{B}||_F^2=\mathrm{tr}\left\{ \left(\frac{\rho}{n_T}\right)^2 \mathbf{\tilde{H_e}}\mathbf{\tilde{H_e}}^H \mathbf{\tilde{H}} \left[\mathbf{I}_{n_T} +\frac{\rho}{n_T}\mathbf{\tilde{H}}^H\mathbf{\tilde{H}}\right]^{-2} \mathbf{\tilde{H}}^H \right\}$$ which can be interpreted as the trace of a product of two positive semi definite matrices $\mathbf{P}$, $\mathbf{Q}$. Using the fact, that $\mathbf{\tilde{H_e}}\mathbf{\tilde{H_e}}^H$ has the same ordered eigenvalues as $\mathbf{\tilde{H}}\mathbf{\tilde{H}}^H$ and the inequality $\mathrm{tr}(\mathbf{PQ})\le\sum_j \mu_k(\mathbf{P})\mu_k(\mathbf{Q})$ [@MarshallOlkin] yields $\sum\limits_{k=1}^{L_1}\mu_{k}^2\le L_1$ and we arrive at the final bound $$\begin{aligned} \nonumber \Delta \leq \frac{1}{\ln(2)}\mathbb{E}\left[ \sum\limits_{k=1}^{\nicefrac{L_1}{2}}\mu_{k}^2\right] \le \frac{L_1}{2 \ln(2) }.\end{aligned}$$ In addition to that, we have the loss between $C\left(\frac{\rho}{2},\frac{n_T}{2},2n_R\right)$ and $C\left(\rho,n_T,n_R\right)$. Approximating  and  for high SNR as $$\begin{aligned} & C_{\mathrm{Jen}} \left(\rho,n_T,n_R\right)= \log_2\left(1+ \sum_{i=1}^L \binom{L}{i}\frac{K!}{(K-i)!}\left(\frac{\rho}{n_T}\right)^i\right) \nonumber\\ \approx & \log_2\left(1+ \frac{K!}{(K-L)!} \left(\frac{\rho}{n_T}\right)^L\right) \nonumber\\ = & \log_2\left( \frac{(K-L)!}{K!}+ \left(\frac{\rho}{n_T}\right)^L\right)+\log_2\left(\frac{K!}{(K-L)!} \right) \nonumber\\ \approx & \log_2\left( 1+ \left(\frac{\rho}{n_T}\right)^L\right) \approx \log_2\left( \left( 1+ \frac{\rho}{n_T}\right)^L\right) \nonumber \\ = & L\log_2\left(1+\frac{\rho}{n_T}\right)\label{eq:ApproxCapNTNR}\end{aligned}$$ and $$\begin{aligned} & C\left(\frac{\rho}{2},\frac{n_T}{2},2n_R\right) \nonumber \\ \overset{(a)}{\geq} & \frac{n_T}{2} \log_2\left(1+\frac{\rho}{n_T}\exp\left(\frac{2}{n_T} \sum_{j=1}^{\frac{n_T}{2}}\sum_{p=1}^{2n_R-j}\frac{1}{p}-\gamma\right)\right) \nonumber\\ = & \frac{n_T}{2} \log_2\left(\exp\left(-\frac{2}{n_T} \sum_{j=1}^{\frac{n_T}{2}}\sum_{p=1}^{2n_R-j}\frac{1}{p}+\gamma\right)+\frac{\rho}{n_T}\right) \nonumber \\ & + \frac{n_T}{2\ln(2)} \left(\frac{2}{n_T} \sum_{j=1}^{\frac{n_T}{2}}\sum_{p=1}^{2n_R-j}\frac{1}{p}-\gamma\right) \nonumber \\ \approx & \left(\frac{n_T}{2}\right)\log_2\left(1+\frac{\rho}{n_T}\right),\label{eq:ApproxCapNT22NR}\end{aligned}$$ where $(a)$ follows from applying Jensen’s inequality to . With  and , the loss between $C\left(\frac{\rho}{2},\frac{n_T}{2},2n_R\right)$ and $C\left(\rho,n_T,n_R\right)$ is quite accurately described by $$\begin{aligned} \nonumber & C\left(\rho,n_T,n_R\right)-C\left(\frac{\rho}{2},\frac{n_T}{2},2n_R\right) \\ & \approx \left(L-\frac{n_T}{2}\right)\log_2\left(1+\frac{\rho}{n_T}\right), \quad[\text{case }n_T < 2n_R] \nonumber.\end{aligned}$$ Finally, the absolute loss for $n_T < 2n_R$ between the ergodic capacity of a MIMO system and the stacked scheme is given by $$\begin{aligned} \nonumber & \left(L-\frac{n_T}{2}\right)\log_2\left(1+\frac{\rho}{n_T}\right) \leq C\left(\rho,n_T,n_R\right)-R_{sA} \\ & \leq \frac{n_T}{2 \ln(2) } + \left(L-\frac{n_T}{2}\right)\log_2\left(1+\frac{\rho}{n_T}\right) \nonumber.\end{aligned}$$ The same procedure can be pursued for $n_T\geq 2n_R$ resulting in the following general characterization for any $n_T,n_R$ $$\begin{aligned} \max\left(0,L-\frac{n_T}{2}\right)\log_2\left(1+\frac{\rho}{n_T}\right) \leq C\left(\rho,n_T,n_R\right) &-R_{sA} \nonumber \\ \leq \frac{L_1}{2 \ln(2) } + \max\left(0,L-\frac{n_T}{2}\right)\log_2\left(1+\frac{\rho}{n_T}\right),\nonumber\end{aligned}$$ which is equal to $$\begin{aligned} & \left(L-\frac{L_1}{2}\right)\log_2\left(1+\frac{\rho}{n_T}\right) \leq C\left(\rho,n_T,n_R\right) &-R_{sA} \nonumber \\ & \leq \frac{L_1}{2 \ln(2) } + \left(L-\frac{L_1}{2}\right)\log_2\left(1+\frac{\rho}{n_T}\right).\label{eq:AbsLossAllg}\end{aligned}$$ From , we observe that as long as $n_T \geq 2n_R$, the absolute loss is only a constant, which depends only on the number of receive antennas. In case $n_T < 2n_R$ the absolute loss increases linearly with $\left(L-\frac{n_T}{2}\right)$. Suboptimal detection and condition number {#sec:Schemes} ========================================= In the previous sections, we have shown that the stacked OSTBC achieves significant portions of the ergodic capacity. This does not, however, guarantee good performance in terms of error probability, which will be investigated in this section. Note that in the analysis in the previous sections it was implicitly assumed, that an optimal maximum-likelihood detector is used at the receiver, which performs an exhaustive search over all possible transmit symbols at each detection step. Especially for higher number of transmit antennas, this becomes computationally prohibitive. If additionally high rates are requested, then higher order modulation sizes are necessary which increases the computational complexity even more. Thus, suboptimal detection schemes have to be employed reducing the detection complexity and thereby achieving reasonable error rate performance results. Therefore, in this section the impact of the suboptimal LR-aided linear ZF-detector on the performance of the stacked OSTBC is analyzed and compared to SM and QSTBC by resorting the equivalent channel representation. In order to apply the LR algorithm, the system model has to rewritten, which is done in the following subsections for the different transmission schemes. Afterwards, the LR-aided linear ZF-detection is described briefly. Spatial Multiplexing (SM) ------------------------- For SM, the transmit matrix $\mathbf{G}_{n_T}$ is reduced to $\mathbf{x}$, since $T=1$. In order to apply the suboptimal LR for SM, the system model in (\[eq:System\]) has to be rewritten as a real model [@WindpassingerLLL] of the form $$\nonumber \mathbf{y}_E = \left[% \begin{array}{cc} \Re\{\mathbf{x}\} \\ \Im\{\mathbf{x}\} \\ \end{array}% \right]^T\mathbf{H}^{SM}_E+ \mathbf{n}_E\;,$$ where $$\nonumber \mathbf{y}_E=\left[% \begin{array}{c} \Re\{\mathbf{y}\} \\ \Im\{\mathbf{y}\} \\ \end{array}% \right]^T\;,\mathbf{n}_E=\left[% \begin{array}{c} \Re\{\mathbf{n}\} \\ \Im\{\mathbf{n}\} \\ \end{array}% \right]^T\;,$$ and $$\nonumber \mathbf{H}^{SM}_E=\left[% \begin{array}{rr} \Re\{\mathbf{H}\} & \Im\{\mathbf{H}\} \\ -\Im\{\mathbf{H}\} & \Re\{\mathbf{H}\} \\ \end{array}% \right]\;.$$ In the following, we refer to $\mathbf{H}^{SM}_E$ as the equivalent channel for the SM scheme. QSTBC ----- Without loss of generality, in this subsection we shortly describe the QSTBC for $n_T=4$ transmit antennas [@SharmaPapadias02]. To generalization to higher number of transmit antennas is straightforward [@A.Sezgin2004]. The transmit matrix for $n_T=4$ transmit antennas is then given [@SharmaPapadias02; @A.Sezgin2004]. $$\begin{aligned} \mathbf{G}_{4}({\mathbf{x}}) = \left[ \begin{array}{*{4}{r}} x_1 & x_2 & x_3 & x_4 \\ x_2^* & -x_1^* & x_4^* & -x_3^* \\ x_3 & -x_4 & -x_1 & x_2 \\ x_4^* & x_3^* & -x_2^* & -x_1^* \\ \end{array}\nonumber \right]\;.\end{aligned}$$ After rewriting (\[eq:System\]), we arrive at (similar to the proposed scheme, (cf. ) $$\label{eq:EQ_SysModQSTBC} \mathbf{y}^Q=\mathbf{H}^Q\mathbf{x}+ \mathbf{n}^Q\;,$$ where $\mathbf{H}^Q=[(\mathbf{H}_1^Q)^T,\dots,(\mathbf{H}_i^Q)^T,\dots,(\mathbf{H}_{n_R}^Q)^T]^T$ and $(\mathbf{H}_i^Q)$ is given as $$\mathbf{H}_i^Q=\left[% \begin{array}{rrrr} h_{1i} & h_{2i} & h_{3i} & h_{4i} \\ -h_{2i}^* & h_{1i}^* & -h_{4i}^* & h_{3i}^* \\ -h_{3i} & h_{4i} & h_{1i} & -h_{2i} \\ -h_{4i}^* & -h_{3i}^* & h_{2i}^* & h_{1i}^* \\ \end{array}% \right] \nonumber \;.$$ For general $n_T$, we have to rewrite the system model in  as a real model similar to SM. For $n_T=4$, however, it is not necessary to resort to the real system model. Here, the system model can be decomposed such that the iterative optimal algorithm in [@YaoWornell] for a system with $n_T=2$ transmit antennas can be applied. For this we first perform channel-matched filtering as the first stage and noise pre-whitening as the second stage of preprocessing at the receiver resulting in two independent subsystems [@SharmaPapadias03], one of which $$\nonumber \tilde{\mathbf{y}}_o = \underbrace{\left[% \begin{array}{rr} \beta & \jmath\beta \\ \epsilon & -\jmath\epsilon \\ \end{array}% \right]}_{\mathbf{H}_E^{Q}} \left[\begin{array}{c} x_1 \\ x_3 \\ \end{array}\right] + \tilde{\mathbf{n}}_o \;,$$ is only a function of the elements of $\mathbf{x}$ with odd index, and the other one is only a function of the elements of $\mathbf{x}$ with even index, $$\nonumber \tilde{\mathbf{y}}_e = \underbrace{\left[% \begin{array}{rr} \beta & \jmath\beta \\ \epsilon & -\jmath\epsilon \\ \end{array}% \right]}_{\mathbf{H}_E^{Q}} \left[\begin{array}{c} x_4 \\ x_2 \\ \end{array}\right] + \tilde{\mathbf{n}}_e \;,$$ where $\mathbf{H}_E^{Q}$ is the $2\times 2$ equivalent channel for QSTBC, $\beta = \sqrt{\frac{\lambda+\alpha}{2}}$, $\epsilon = \sqrt{\frac{\lambda-\alpha}{2}}$, $\lambda = \sum_{i=1}^{n_R}\sum_{j=1}^{n_T} |h_{i,j}|^2$, and $\alpha = \sum_{i=1}^{n_R}2\mathrm{Im}(h_{i,1}^*h_{i,3}+h_{i,4}^*h_{i,2})$. Both subsystems can now be detected separately, which reduces the complexity of the receiver significantly. In order to get the best performance with respect to error rates and a decoupled system with scalar input and scalar output as in the case of OSTBC, the columns of $\mathbf{H}_E^{Q}$ have to be orthogonal. However, the probability that this occurs for $\mathbf{H}_E^{Q}$ is zero. For orthogonality, it follows from the scalar product of the columns of $\mathbf{H}_E^{Q}$ that $\alpha$ has to be zero. But since the channel entries $\{hji\}$ are mutually independent and identically distributed (i.i.d.) random complex Gaussian processes, the probability $P_r(\alpha=0)$ is equal to the probability $P_r(\sum_{i=1}^{n_R}2\mathrm{Im}(h_{i,1}^*h_{i,3}+h_{i,4}^*h_{i,2})=0)$, which in turn is zero. From this it follows that orthogonality and therefore a decoupled system can not be achieved. A disadvantage of this QSTBC is that in order to achieve the same transmission rate as SM, we have to compensate the rate loss by using a considerably higher constellation. But recall that higher constellations complicates amplification, synchronization, and detection. E.g., a transmission rate of 4 bits/sec/Hz for a system with $n_T=4$ transmit antennas is achieved by SM with BPSK, whereas 16QAM is required for the code rate one QSTBC. In [@PapadiasFoschi01; @SezginHenkelAsi05] it was shown that QSTBC approach the capacity in case of $n_R=1$, which is achieved in case of the stacked OSTBC as shown in section \[sec:MutInfoUB\]. For $n_R>1$, the performance of QSTBC in terms of mutual information degrades severely in contrast to the stacked OSTBC, which achieve at least half of the capacity as derived in section \[sec:MutInfoLB\]. Proposed scheme --------------- Given , the equivalent real signal model for the proposed stacked OSTBC is given as $$\nonumber \mathbf{y}'' = \mathbf{H}_E^{OS}\left[% \begin{array}{c} \Re\{\mathbf{x}\} \\ \Im\{\mathbf{x}\} \\ \end{array}% \right]+ \mathbf{n}''\;,$$ where $$\nonumber \mathbf{H}_E^{OS}=\left[% \begin{array}{rr} \Re\{\mathbf{H}'\} & -\Im\{\mathbf{H}'\} \\ \Im\{\mathbf{H}'\} & \Re\{\mathbf{H}'\} \\ \end{array}% \right]\;.$$ LR-aided linear ZF Detection ---------------------------- By applying the algorithm, the $m\times n$ equivalent channel $\mathbf{H}_E$ for each transmission scheme can be decomposed as $$\label{eq:H_E_decomp} \mathbf{H}_E=\mathbf{Q}\mathbf{R}\;,$$ where $\mathbf{R}$ is a $n\times n$ matrix with integer entries and $\mathbf{Q}$ is a $m\times n$ matrix, which is better conditioned than $\mathbf{H}_E$, i.e. the columns of $\mathbf{Q}$ are less correlated and shorter. A good indication for the correlation of a matrix is the so called condition number, which is defined as the ratio of the largest singular value of the matrix to the smallest. Using (\[eq:H\_E\_decomp\]), the equivalent signal model is then given as $$\nonumber \mathbf{y} = \mathbf{H}_E\mathbf{x}_r+ \mathbf{n}= \mathbf{QR}\mathbf{x}_r+ \mathbf{n}=\mathbf{Q}\mathbf{z}+ \mathbf{n}\;.$$ Now, by multiplying $\mathbf{Q}^{-1}$ from left to $\mathbf{y}$ we arrive at $$\nonumber \mathbf{\tilde{y}} =\mathbf{z}+ \mathbf{Q}^{-1}\mathbf{n}\;,$$ where the noise enhancement and coloring is relatively small, since $\mathbf{Q}^{-1}$ is also good conditioned. In order to get a estimation for the transmitted symbols, the following operation has to be applied $$\label{eq:DetecScalShiftQuant} \mathbf{\hat{x}} = C\left(\mathbf{R}^{-1}\mathcal{Q}_{\mathbb{Z}^n} \left[\frac{1}{C}\mathbf{\tilde{y}}-\mathbf{R}\frac{1}{2}\mathbf{1}_n\right]+ \frac{1}{2}\mathbf{1}_n\right)\;,$$ where $\mathbf{1}_n$ is a $n\times 1$ vector of ones, $C$ is a constant given as $C=\sqrt{\frac{6}{M-1}}$ and $\mathcal{Q}_{\mathbb{Z}^n}[\cdot]$ describes the component-wise quantization with respect to the infinite integer space $\mathbb{Z}$. However, this quantization can only be applied, if the transmit modulation signal set $\mathcal{C}$ is transformed to $\mathbb{Z}$, which is achieved with the scaling and shifting of $\mathbf{\tilde{y}}$ within the quantization operation in . Note that after this quantization, re-scaling and re-shifting, some points may lie outside the constellation. A suboptimal solution is to assign these points to the nearest point within the constellation. For BPSK, the effect of this assignment has a significant effect on the error rate performance, however, this gain diminishes with higher order modulations. Condition number {#sec:cond_number} ---------------- For illustration, ![Pdfs of channel cond. numbers with SM or the stacked OSTBC with and w/o LR for a $4\times 4$ system.[]{data-label="fig:Prob_Dens_Cond_NTHalf"}](fig2.eps) the probability density functions (pdfs) of the natural logarithm of the condition number of the channels for the stacked QSTBC and SM are depicted in Fig. \[fig:Prob\_Dens\_Cond\_NTHalf\]. From the Fig., we observe that the SM-channel is bad-conditioned and that LR has a great impact on the channel. For the stacked OSTBC, we observe that the impact of LR is not as significant as for SM. The pdf of the natural logarithm of the condition number for the QSTBC is depicted in Fig. \[fig:Prob\_Dens\_Cond\]. For comparison, the pdf for the stacked OSTBC is also plotted. In case of QSTBC, for some channels we have no gain with LR, since many samples of the equivalent channel generated with QSTBC have inherently low condition numbers such that the LR has no effect. Different from the QSTBC, for the stacked OSTBC there is a gain achieved by applying the LR for almost all samples of the equivalent channel model. Note that for orthogonal channels (e.g., with OSTBC), the pdf is a dirac impulse at position $0$. ![Pdfs of channel cond. numbers with the stacked OSTBC or QSTBC with and w/o LR.[]{data-label="fig:Prob_Dens_Cond"}](fig3.eps) Simulations {#seq:simulations} =========== In Fig. \[fig:capacity\_nr1\], the average rate of the stacked Alamouti scheme and the ergodic capacity of a MIMO system with $n_R=2$ and $n_T=2,4$ and $n_T=8$ is depicted. In case of $n_T=2$, we have the standard Alamouti scheme. From the Fig., we observe that the difference between the average rate of the stacked Alamouti scheme and the capacity diminishes significantly by increasing the number of transmit antennas. ![Ergodic capacity and average rates of the stacked OSTBC with $n_R=2$ receive and $n_T=2$,$n_T=4$ and $n_T=8$ transmit antennas.[]{data-label="fig:capacity_nr1"}](fig4.eps) In Fig. \[fig:capacity\_nr2\], the average rate of the stacked Alamouti scheme and the ergodic capacity with $n_T=4$ and $n_R=2,4$ and $n_T=8$ is depicted. ![Ergodic capacity and average rates of the stacked OSTBC with $n_T=4$ transmit and $n_R=2$,$n_R=4$ and $n_R=8$ receive antennas.[]{data-label="fig:capacity_nr2"}](fig5.eps) In contrast to the case of increasing number of transmit antennas, here we observe that the difference between the average rate of the stacked Alamouti scheme and the ergodic capacity increases by increasing the number of receive antennas. In Fig. \[fig:Ratio\], the ratio $C/R_{sA}$ is depicted for $n_T=8$ transmit and $n_R=2$ (bottom) to $n_R=9$ (top) receive antennas. For high SNR, we observe that as long as $n_T \geq 2n_R$ the ratio decreases as the SNR increases. In case $n_T< 2n_R$ the ratio increases steadily. As derived in section \[sec:MutInfoLB\], the ratio is upper bounded by $C/R_{sA}<2$ for any $n_R$, $n_T$. ![Ratio $C/R_{sA}$ for $n_T=8$ transmit and $n_R=2$ (bottom) to $n_R=9$ (top) receive antennas.[]{data-label="fig:Ratio"}](fig6.eps) In Fig. \[fig:RatioAnaly\], the ratio $C/R_{sA}$ is depicted for $n_T=8$ transmit and $n_R=4$, $n_R=6$ and $n_R=9$ receive antennas. In addition to that, we used our lower and upper bounds derived in the previous section in order to derive lower and upper bounds for the ratio $C/R_{sA}$, i.e. $$\begin{aligned} \frac{C_{lb}}{R_{sA}^{ub}}\leq \frac{C}{R_{sA}}\leq \frac{C_{\mathrm{Jen}}}{R_{sA}^{lb}}\end{aligned}$$ Based on the derivations in section \[sec:MutInfoLB\], we know that the ratio is upper bounded by $2$. Further, since the trivial lower bound is equal to $1$, we only depicted $1\leq C/R_{sA}\leq 2$. For $n_R=9$, we observe that both the lower and upper bound are getting tighter for higher SNR. At low SNR, the upper bound performs better than the lower bound. For $n_R=4$, $n_R=6$ and low SNR, we observe that the upper bound is quite loose in comparison to $n_R=9$. The lower bound for $n_R=4$ is not depicted here, since it is lower than the trivial lower bound of $1$. ![Ratio $C/R_{sA}$ for $n_T=8$ transmit and $n_R=4$, $n_R=6$ to $n_R=9$ receive antennas.[]{data-label="fig:RatioAnaly"}](fig7.eps) In Fig. \[fig:AbsLoss\], ![Absolute loss $\Delta$ for $n_T=6$ transmit and different numbers of receive antennas.[]{data-label="fig:AbsLoss"}](fig8.eps) the absolute loss $\Delta$ is depicted for $n_T=6$ transmit antennas and $n_R=2-4$ and $n_R=7$ receive antennas. From the figure, we observe that as long as $n_T \geq 2n_R$, the slope of the absolute loss tends to a constant for high SNR. This behavior is tracked quite well by the bound in , which is also depicted in the figure. In Fig. \[fig:BER\_R1\], ![BER for QSTBC and the stacked OSTBC with ML and LR-ZF, 4 bit/sec/Hz.[]{data-label="fig:BER_R1"}](fig9.eps) the BER of the stacked OSTBC with QAM and the QSTBC with 16-QAM is depicted for a transmission rate of 4 bits/sec/Hz. Note that in order to make a fair comparison of the three transmission schemes (i.e. QSTBC, SM, and stacked OSTBC), we analyzed a system with $n_T=n_R=4$ antennas, since for SM with suboptimal detectors it is necessary that $n_R\geq n_T$. From the figure, we observe, that the performance of the stacked OSTBC with LR-ZF detection is comparable with the optimal ML detection. In fact, the diversity gain of both detectors is equal and there is only a power penalty of about $1.7$dB of LR-ZF to ML. The gap between ML and LR-ZF detection is even smaller for QSTBC. Here, the power penalty is about $0.6$dB. Interestingly, the performance of the stacked OSTBC for both ML and LR-ZF detection is better than that of QSTBC in the SNR region shown in the figure. However, for very high SNR and low BER, the diversity gain of $n_Tn_R$ (contrary to diversity of $2n_R$ for the stacked OSTBC) for the QSTBC will show its effect and in can be expected that the performance of QSTBC gets better than that of the stacked OSTBC. For smaller $n_R$, this intersection point is expected be at lower SNR values. The bit error-rate performance of SM for BPSK and a transmission rate of 4 bits/sec/Hz is shown in Fig. \[fig:BER\_RNThalf\]. For comparison purposes, we also plotted the BER of the stacked QSTBC with QAM. Here, we observe that the BER performance with ML-detection of the stacked OSTBC is better than that of SM for all SNR values. In case of LR-ZF detection, SM performs only better than QSTBC for low SNR of about $2$dB. However, the gap in power efficiency between ML and LR-ZF is higher for the stacked QSTBC in comparison to SM with BSPK. Note that (as aforementioned) the small gap for SM is only due to the BPSK modulation. For higher modulation sizes, this gap is even higher. ![BER for SM and stacked OSTBC with ML and LR-ZF, 4 bit/sec/Hz.[]{data-label="fig:BER_RNThalf"}](fig10.eps) By increasing the transmission rate to $8$bit/sec/Hz, i.e. QAM for SM and 16QAM for the stacked OSTBC, we observe in Fig. \[fig:BER\_RNThalf\_16QAM\] that the gap between ML and LR-ZF is dramatically increased in case of SM to about $6$dB. On the other hand, the gap between ML and LR-ZF for the stacked OSTBC and 16QAM is reduced in comparison to the gap achieved with QAM (cf. Fig. \[fig:BER\_RNThalf\]) to about $1.3$dB. Although the performance of SM with ML detection is better than that of the stacked OSTBC for low and moderate SNR values, for high SNR values it is the other way around. ![BER for SM and stacked OSTBC with ML and LR-ZF, 8bit/sec/Hz.[]{data-label="fig:BER_RNThalf_16QAM"}](fig11.eps) The performance of the stacked OSTBC with LR-ZF detection is better for the whole SNR range in comparison to SM, which is of higher interest for practical applications, since the computational complexity of the ML detector is exponential in the transmission rate. Another disadvantage of SM is that we need at least as many receive as transmit antennas, i.e. $n_T\leq n_R$, whereas only $\frac{n_T}{2}$ receive antennas are necessary for the stacked OSTBC. Multiple receive antennas are only optional for the QSTBC . Conclusion {#sec:Conclusion} ========== In this paper, we analyzed the performance of stacked OSTBC in terms of the average rate. We showed, that the stacked scheme achieves the capacity of a MIMO system in the case of $n_R=1$ receive antennas. Further, we showed that the MIMO capacity is at most twice the rate achieved with the proposed scheme at any SNR. We derived lower and upper bounds for the rate achieved with this scheme and compared it with upper and lower bounds for the capacity. In addition to the capacity analysis, we also analyzed the error rate performance of the proposed scheme. To this end, we combined the stacked OSTBC with a zero-forcing (ZF) detector applying lattice-reduction (LR) aided detection, since this suboptimal detector achieves the same diversity as the optimal ML detector with only some penalty in power efficiency. We analyzed the effect of LR on the equivalent channel generated by the stacked OSTBC, for spatial multiplexing (SM) and QSTBC. We observed the highest gain for SM and a higher gain for the stacked OSTBC in comparison to the QSTBC. Finally, we illustrated the theoretical results by numerical simulations. From simulation results we observed that the stacked scheme approaches the ergodic capacity of a MIMO system by increasing the number of transmit antennas for a fixed number of receive antennas. Furthermore, we observed that as long as the number of transmit antennas is twice the number of receive antennas the ratio of the capacity to the rate of the proposed scheme improves by increasing the SNR. Regarding the simulation of the error rate performance, we observed that in the considered SNR region the stacked OSTBC performs better in terms of BER for ML as well as for LR-aided ZF-detection than SM and QSTBC in the setup given. Further, we observed that the gap between maximum-likelihood and LR-ZF detection is dramatically reduced in comparison to SM schemes, especially for higher transmission rates. 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[^2]: A.Sezgin has been with the Fraunhofer-Institute for Telecommunications, Heinrich-Hertz-Institut, Einsteinufer 37, D-10587 Berlin, Germany. He is now with the Information Systems Laboratory, Stanford University, CA 94305-9510, USA (e-mail:sezgin@stanford.edu). [^3]: O.Henkel is with the Fraunhofer-Institute for Telecomm., HHI, Einsteinufer 37, 10587 Berlin, Germany, (e-mail:henkel@hhi.de). [^4]: Notation: $\mathbf{A}^T$, $\mathbf{A}^H$, $\mathbf{A}^*$ means transpose, hermitian transpose, and complex conjugation, respectively
--- author: - | L. J. Bignell$^a$[^1] , D. Beznosko$^{a,c}$, M. V. Diwan$^a$, S. Hans$^b$, D. E. Jaffe$^a$, S. Kettell$^a$, R. Rosero$^b$, H. W. Themann$^{a,d}$, B. Viren$^a$, E. Worcester$^a$, M. Yeh$^b$, and C. Zhang$^a$\ Physics Department, Brookhaven National Laboratory,\ Upton NY, USA\ Chemistry Department, Brookhaven National Laboratory,\ Upton NY, USA\ Now at Department of Physics, Nazarbayev University,\ Astana , Kazakhstan\ Now at Center for Axion and Precision Physics Research, Institute for Basic Science,\ Daejeon, Republic of Korea\ E-mail: title: 'Characterization and Modeling of a Water-based Liquid Scintillator' --- =1 Introduction {#sec:Intro} ============ Water-based liquid scintillator (WbLS) is a recently developed scintillating material that has been identified as a candidate detector medium for the next generation of intensity frontier particle physics experiments [@Yeh2011]. WbLS is a stable scintillating emulsion with a large fraction of aqueous phase. With suitable chemical purification and material optimisation, WbLS may be produced with an optical attenuation of tens of meters in the photon energy range relevant to detection by bialkali photomultiplier tubes (PMTs) (figure \[fig:UVVIS\]). Such a long optical attenuation length would make Cherenkov imaging feasible in a large WbLS detector, which is not possible in large detectors filled with ordinary liquid scintillator. As a scintillating material, WbLS is sensitive to particle interactions below the Cherenkov threshold. The aqueous phase of WbLS may also allow the loading of metallic ions that are not easily incorporated into the non-polar organic solvents that are typical of liquid scintillators. This may permit novel schemes to incorporate metal ions at high concentrations in the WbLS material. We are currently evaluating the possibility to incorporate candidate isotopes for neutrinoless double beta decay, neutron absorbing isotopes, and high Z materials for gamma attenuation. The optical and chemical properties of WbLS suggest that it may be an effective detection medium to be used in a particle detector capable of a broad range of measurements. Neutrinoless double beta decay, high energy neutrino beam measurements, proton decay, and low energy neutrino physics have been proposed as plausible measurements using a large scale (30 – 100 kiloton) WbLS detector [@Alonzo2014etal]. In order to rigorously evaluate the suitability of WbLS for such an application, a comprehensive understanding of the performance of the material must be developed. Basic material properties such as the light yield, the optical attenuation length, and the emission spectrum must be measured. A model of the response of the WbLS to ionising radiation and optical and ultraviolet photons must also be developed and validated. In this study, we have performed several experimental tests aimed at elucidating these properties, and we have also developed a simulation model that successfully predicts the performance of the WbLS material in our measurement geometries. It is our intention to further develop and validate this simulation model using larger volume geometries to permit accurate study of the performance of a proposed large experiment that uses WbLS as its detection medium. ![The optical attenuation spectra for the 1% WbLS used in this study (solid black) and for a more recent formulation of 1% WbLS (dashed black). The 1% WbLS fluorescence spectrum (green) and photomultiplier tube (Hamamatsu R7723) quantum efficiency (red) are also indicated. The fluorescence spectrum was measured using an exciting wavelength of 290 nm.[]{data-label="fig:UVVIS"}](UV-VIS_Fluor_NoLegend_v3.pdf){width="0.8\columnwidth"} Experiment {#sec:Expt} ========== WbLS Samples ------------ WbLS samples composed of 0.4% (WbLS-1) and 1% (WbLS-2) liquid scintillator by mass were prepared, as well as pure liquid scintillator (LS) samples. The pure LS was identical to that used by the Daya Bay experiment [@An2012etal]. Table \[tab:ScintComp\] details the compositions of these samples. Sample Name Organic Phase Primary Fluor Secondary Fluor ------------- --------------- --------------- -------------------- WbLS-1 PC, 0.4% PPO, 0.4 g/L bis-MSB, 3 mg/L WbLS-2 PC, 0.99% PPO, 1.36 g/L bis-MSB, 7.48 mg/L LS LAB, 100% PPO, 2 g/L bis-MSB, 15 mg/L \[tab:ScintComp\] Optical Characterization Measurements ------------------------------------- Ultraviolet-visible absorption spectroscopy of water, WbLS-1, and WbLS-2 were taken using a Shimandzu UV-1800 spectrophotometer. The optical attenuation spectrum of WbLS-2 is shown in figure \[fig:UVVIS\], along with that of a more recent WbLS formulation that also consists of 1% scintillator, the fluorescence emission spectrum, and the photomultiplier quantum efficiency spectrum for the PMTs used in this study. The optical attenuation of the more recent formulation of a LAB-based WbLS is markedly less than WbLS-2, after the purification of all starting materials and the addition of a surface-scattering reduction agent. Fluorescence spectroscopy was carried out using a PTI fluoresence spectrometer. The excitation-emission spectrum of WbLS-2 is shown in figure \[fig:ExEmMap\], which exhibits an excitation wavelength dependence of the emission spectrum shape. ![Excitation-emission spectrum for WbLS-2.[]{data-label="fig:ExEmMap"}](WbLSExEmMapContour_v2.pdf){width="0.8\columnwidth"} Proton Beam Measurements {#sec:BeamInfo} ------------------------ Two beamline instruments were developed for this study, Detectors A and B (figure \[fig:NSRLExpts\]). ![A schematic drawing of the instruments used in the proton beam measurements, Detector A and Detector B. See the text for details.[]{data-label="fig:NSRLExpts"}](13ASchematic_v4){width="0.8\columnwidth"} ![A schematic drawing of the instruments used in the proton beam measurements, Detector A and Detector B. See the text for details.[]{data-label="fig:NSRLExpts"}](12CSchematic_v3){width="0.8\columnwidth"} Detector A was fabricated using black ABS polymer to supress light reflections from the walls. The 130 mm x 115 mm x 63.5 mm detector volume was filled with water during initial irradiations, then WbLS-2. Two PMTs (Hamamatsu R7723, 51 mm diameter) observed the liquid volume; one placed downstream from the beam so as to allow detection of Cherenkov light, and one placed upstream from the beam so as to suppress detection of Cherenkov light. To allow for any variability in the photomultiplier response, all measurements were repeated with the PMT locations relative to the beam direction swapped. The PMT response was found to be similar for each tube (figure \[fig:PMTresponse\]). We report the averaged values of the PMT response hereafter. Two 2x2x0.5 cm plastic scintillator hodoscopes (H1 and H2) were used for triggering and to define the beam location. Detector B consists of two right cylindrical 150 mm x 150 mm tubs. Tub 1 was fabricated using 6.35 mm thick white teflon and Tub 2 was fabricated using 6.35 mm thick aluminium coated with black perfluoroalkoxy alkane paint on its inner surface. The tubs were exposed to the proton beam for consecutive fillings of water, WbLS-1, WbLS-2, and LS. Two 2x2x0.5 cm plastic scintillator hodoscopes were used to define the beam. A 400 nm LED was placed just above each scintillating volume in Detector A to permit an in-situ single photoelectron PMT calibration. The low energy water irradiations were used for single photoelectron PMT calibrations in Detector B. All samples were investigated using proton beams with incident energies of 475 MeV and 2 GeV at the NASA Space Radation Laboratory at Brookhaven National Laboratory. A 210 MeV incident proton beam was additionally used to investigate all samples in Detector B. The beam intensity at all energies was made low enough that the probability of having more than a single proton incident per accelerator bunch was small. An analysis of our results where the two-proton events are separable from the 1 proton events indicates that the fraction of two-proton events varied as a function of beam energy and was less than 10% in all irradiations. We have estimated the influence of the two-proton events by resampling the simulated single proton distributions presented below. Double proton events have no effect on the results with a large (&gt;10) mean number of detected photoelectrons. The effect on the remaining data is estimated to be &lt;10% and is included in the systematic uncertainty discussed in section \[sec:Results\]. ![The measured photoelectron distributions for identical measurements using the two different photomultipliers (represented as the red and blue traces) in Detector A. The solid traces represent measurements taken downstream of the 2 GeV proton beam on a water target, and the dashed traces represent measurements taken upstream of the 475 MeV proton beam on a WbLS-2 target. The photomultiplier responses are similar for small and large numbers of photoelectrons.[]{data-label="fig:PMTresponse"}](PMTAblue_PMTBred_IdenticalMeasurements_withLegend_FatLines.pdf){width="0.8\columnwidth"} Instrument Sample Incident Proton Energy ------------ -------- ----------------------------- Detector A Water 2 GeV and 475 MeV Detector A WbLS-2 2 GeV and 475 MeV Detector B Water 210 MeV, 475 MeV, and 2 GeV Detector B WbLS-1 210 MeV, 475 MeV, and 2 GeV Detector B WbLS-2 210 MeV, 475 MeV, and 2 GeV Detector B LS 210 MeV, 475 MeV, and 2 GeV : A summary of the proton beam exposures. \[tab:BeamMeas\] The data readout in all measurements was achieved using a CAEN V1729A 14 bit waveform digitizer sampling at 1 gigasamples per second. The raw waveforms were stored for offline analysis. Data acquisition was triggered by coincident hodoscope events during the accelerator beam gate. Analysis ======== Signal Processing Algorithm --------------------------- Signal processing of the acquired waveforms was achieved using a custom algorithm which was developed for background subtraction, timing, and pulse area determination. The algorithm is based upon that used in the DarkSide experiment [@Alexander2013etal] and its operation is illustrated in Figure \[fig:Algo\]. A moving average of the waveform data is taken as the baseline. A data point is considered to be part of the baseline if the difference between that data point value and the value of the averaged baseline as it was `trig_pre_samples` data points ago is less than the algorithm’s threshold parameter, `max_amplitude`. If a trigger event occurs (`trig_start`), the algorithm waits until the waveform falls below the threshold level, then waits for an additional pile-up pulse for `untrigger_length` samples. If a pile-up event occurs, `trig_stop` is unset and the piled-up event is considered to be part of the same trigger. Otherwise, the algorithm linearly interpolates between the baseline values at the start and end of the pulse and continues to average the baseline and search for a new trigger. The pulse charge was measured as the summed pulse area between `trig_pre_samples` and `trig_post_samples`, less the baseline. We used `trig_pre_samples` = 3 ns and `trig_post_samples` = 10 ns. ![image](BLAlgoExplain_v2.pdf){width="80.00000%"} The use of a moving average of the baseline attenuates high frequency components of the signal pedestal noise; allowing a `max_amplitude` value of approximately 0.3 photoelectrons. Simulation Model ---------------- We have used Geant4 v10.0 [@Agostinelli2003etal] to simulate detectors A and B, the energy deposit in the hodoscopes and scintillator, and the optical photon measurements by the PMTs. For modeling the optical photon processes, the optical properties of the detector materials were required. Where possible we used measured values. Our measurements of the excitation and emission spectra and optical attenuation coefficient spectrum of the scintillators used in this study were incorporated into the model. The wavelength-shifting absorption coefficient was taken to be equal to the difference between the WbLS optical attenuation coefficient and the optical attenuation coefficient of water. The scintillation emission spectrum was assumed to be identical to that produced when excited by 290 nm photons. The refractive index of water and optical absorbance of water were taken from Segelstein [@Segelstein1981]. The refractive index and re-emission probability of liquid scintillator were taken from the Daya Bay simulation model [@Wang2009]. The refractive index of WbLS samples was calculated as a linear combination of water and pure liquid scintillator, according to the fraction of scintillating solvent present in the WbLS. The spectral dependence of the WbLS re-emission probability was assumed to be identical to that of pure liquid scintillator, although the amplitude of the re-emission probability was left as a free parameter whilst being constrained to a maximum value of 1. The 2 GeV measurements in Detector A were employed for this optimisation. We assumed that the re-emission probabilities of WbLS-1 and WbLS-2 are identical. The optical absorbance and refractive index data for the UV-transparent acrylic windows were taken from Band *et al.* [@Band2012]. The refractive index of the borosilicate glass PMT windows was calculated using the empirical Sellmeier equation parameters provided by Schott [@Schott2012]. The quantum efficiency spectrum was estimated using data provided by Hamamatsu. Several optical parameters were either unknown or poorly known, so that their values were assumed or left as free parameters. The black vessels were assumed to be perfectly unreflective, and the white vessel was assumed to be a wavelength-independent diffuse reflector with the reflectance allowed to vary. The water measurements were used to calibrate other model parameters to reproduce the measured results. The free parameters were the PMT photocathode radius for Detector A and the beam height relative to the centre of the detector vessel for Detector B. The beam height in the Detector B was estimated to be located at the centre of the tub with an uncertainty of $\pm$ 1 cm. The optimal beam height was found to be 0.69 $\pm$ 0.05 cm above the centre of the vessel. The optimal photocathode radius of 22 $\pm$ 1 mm is in fair agreement with the manufacturer’s specification of $>$23 mm. ### Wavelength Shifting Model For ordinary liquid scintillator with a scintillation yield of $10^{4}$ photons per MeV, the number of scintillation photons produced per centimeter for a minimum ionising particle experiencing a stopping power of 2 MeV/cm is $2 \times 10^{4}$. This vastly outnumbers the yield of Cherenkov photons, which is $\approx 600$ photons per centimeter in water integrated over the 200-500 nm spectral range. As more than half of these photons fall in the 200 to 300 nm spectral range, the Cherenkov light accounts $<$1.5% of the detectable light from a minimum ionising particle in ordinary liquid scintillator. WbLS has a much lower scintillation light yield than ordinary liquid scintillator, so that the number of Cherenkov photons may in fact be greater than the number of scintillation photons. This Cherenkov light may be directly detected or absorbed by the WbLS. Wavelength-shifting (WLS) of the absorbed light may occur with a probability described by the WbLS’s re-emission probability. Understanding the Cherenkov and scintillation light generation and WLS is therefore an important aspect of evaluating the suitability of WbLS for large detector geometries. This sensitivity to WLS of Cherenkov photons has motivated us to extend the Geant4 optical WLS model. Our modifications to the WLS model have allowed both the re-emission probability and the emission spectrum to depend upon the wavelength of the absorbed photon. Figure \[fig:WLSmodels\] compares the model predictions of the optical photon spectrum incident on the upstream PMT in Detector A due to a beam of 2 GeV protons, when it is filled with WbLS-2. Although a large proportion of wavelength-shifted light is predicted by both models, the original Geant4 model predicts 21% less WLS light. There is also some spectral distortion evident in the modified WLS model spectrum at around 450 nm due to the influence of the changes in the fluorescence emission spectrum that occur mainly for exciting wavelengths between about 350 nm and 400 nm (figure \[fig:ExEmMap\]). Our modified WLS model predicts a mean number of detected photons that is 17% greater than the Geant4 WLS model. ![The predicted optical photon spectrum in Detector A due to the scintillation (dashed traces) and wavelength-shifting (solid traces) processes, as measured by the upstream PMT for incident 2 GeV protons. The original Geant4 wavelength-shifting model results are represented using red traces, and the results from the modified wavelength-shifting model developed in this work are represented using blue traces.[]{data-label="fig:WLSmodels"}](OPspecAtPMT_2GeV_WbLS_upstream_MyWLSblue_G4WLSred_withLegend.pdf){width="0.8\columnwidth"} ### Scintillation Parameter Optimisation The scintillation light yields and ionisation quenching factors of all samples were taken as free model parameters. The light yields were estimated by performing a $\chi^2$ optimisation of the simulated photoelecton distribution to the measurement, using data from Detector B when irradiated by 475 MeV protons. The ionisation quenching factor optimisation was performed in a similar manner, using Detector B’s 210 MeV proton irradiation. ![image](NumPE_MeasVsSim_ALL_150813_v2_dashed.pdf){width="\textwidth"} ![Measured (red trace) and simulated (blue trace) photoelectron distribution for a 2 GeV proton beam incident upon water in Detector B, Tub 2. The range over which the minimisation to the measured data was performed is shown.[]{data-label="fig:DetBWater"}](NumPE_MeasVsSim_Water_2GeV_ModGeom3_dashed.pdf){width="0.8\columnwidth"} Results {#sec:Results} ======= Detector B ---------- The simulated and measured photoelectron (PE) distributions are given in figures \[fig:DetBResults\] and \[fig:DetBWater\]. Cuts were applied to the hodoscope pulse amplitude and timing, as well as the relative PMT-hodoscope timing to select single proton events from the incident beam. The cuts effectively removed the pedestal noise and did not bias the photoelectron distribution. The measurement of 475 MeV and 210 MeV protons on a water target are not shown as few events were observed during this measurement, and those events that were registered were of very low amplitude. We attribute the events observed in these measurements to the detection of Cherenkov emission from secondary electrons. The mean number of observed photoelectrons for all measurements and simulations are given in table \[tab:MeansDetB\]. In addition to the statistical uncertainty arising from the $\chi^2$ analysis of the simulated data, we estimate that there is an additional systematic uncertainty component of $\approx$ 10% with the main contributions due to uncertainties in the simulation model’s input data (such as the refractive index, PMT quantum efficiency, and re-emission probability), the simplified treatment of optical photon scattering, and the effect of two-proton events (Section \[sec:BeamInfo\]). This systematic component dominates the uncertainty of all simulated results. [|ccx[3cm]{}cc|]{} Sample & Proton Beam Energy & Most Probable Energy Deposit & N$_{PE}$, Measured & N$_{PE}$, Simulated\ Water & 210 MeV & 106.4 $\pm$ 0.3 MeV & 1.1 $\pm$ 0.0 & 1.5 $\pm$ 0.2\ Water & 475 MeV & 42.0 $\pm$ 0.2 MeV & 1.7 $\pm$ 0.0 & 1.3 $\pm$ 0.1\ Water & 2000 MeV & 26.6 $\pm$ 0.3 MeV & 2.8 $\pm$ 0.1 & 2.5 $\pm$ 0.3\ WbLS-1 & 210 MeV & 113.9 $\pm$ 0.3 MeV & 4.5 $\pm$ 0.1 & 4.4 $\pm$ 0.4\ WbLS-1 & 475 MeV & 42.2 $\pm$ 0.1 MeV & 3.6 $\pm$ 0.1 & 3.4 $\pm$ 0.3\ WbLS-1 & 2000 MeV & 27.4 $\pm$ 0.2 MeV & 17.0 $\pm$ 0.2 & 18.6 $\pm$ 1.9\ WbLS-2 & 210 MeV & 113.6 $\pm$ 0.3 MeV & 27.6 $\pm$ 0.3 & 27.3 $\pm$ 2.7\ WbLS-2 & 475 MeV & 42.1 $\pm$ 0.2 MeV & 16.7 $\pm$ 0.1 & 15.9 $\pm$ 1.6\ WbLS-2 & 2000 MeV & 27.5 $\pm$ 0.3 MeV & 30.8 $\pm$ 0.3 & 29.2 $\pm$ 2.9\ LS & 210 MeV & 96.6 $\pm$ 1.4 MeV & 2588 $\pm$ 21 & 2622 $\pm$ 262\ LS & 475 MeV & 32.5 $\pm$ 0.6 MeV & 1111 $\pm$ 10 & 1105 $\pm$ 110\ LS & 2000 MeV & 20.9 $\pm$ 0.4 MeV & 872 $\pm$ 15 & 933 $\pm$ 93\ \[tab:MeansDetB\] For Tub 2, the simulated results are generally in agreement with the measurements. The optimal values of light yield and ionisation quenching parameter are given in table \[tab:OptimalVals\]. We have used Birks’ semi-empirical ionisation quenching model [@Birks1964], and the quenching parameter refers to the material parameter of that model. The light yield of the LS sample is consistent with the typically measured value of the light yield of a liquid scintillator of $\approx 10^{4}$ photons/MeV. The quenching parameter for the LS sample falls in the lower range of the typically measured values of 0.07-0.2 mm/MeV [@Peron1996; @Torrisi2000; @Broda2002; @vonKrosigk2013]. The light yield values for the WbLS samples do not linearly scale with the fraction of scintillator, likely due to differences in fluor concentration and solvent type. Both WbLS samples are more susceptible to ionisation quenching than the LS sample. Indeed, the quenching parameter for both WbLS-1 and WbLS-2 is larger than any previously reported value that could be found by the authors for a liquid scintillator. The cause of these extraordinary values will be investigated in future studies. The 475 MeV and 210 MeV measurements in WbLS-1 resulted in small numbers of photoelectrons with broadened distributions relative to a Poisson distribution. It is also apparent that some low amplitude events with $\approx$ 1 PE contributed to the distribution, as evidence of these events are seen in some of the other measurements. The distribution broadening can be modeled by convolving the simulated distributions with a Gaussian as outlined in [@Balagura2006]. However the broadened distribution obtained using this technique were unable to reproduce the shape of the measured photoelectron distributions for all proton energies. Due to the poor agreement of distribution shapes, $\chi^2$ optimisation was quite sensitive to the selection or exclusion of the 1 PE bin in the analysis. The sensitivity of the light yield and quenching parameter to this effect is included in the statistical uncertainties for the WbLS-1 results reported in table \[tab:OptimalVals\]. The Tub 1 results are not presented, as it was not possible to obtain agreement with the simulation model that was consistent with the measured distributions across all beam energies for all materials, for any value of the tub reflectance. While the cause of this inconsistency is unknown, we have identified two possible causes that arise from simplifications in our simulation model. Firstly, there may be some wavelength-dependence to the tub reflectance. If this is the case, the effective reflectance for Cherenkov light is different from the effective reflectance of the scintillation and WLS emission, which may explain the discrepancies seen with different particle energies and samples, that have different proportions of Cherenkov and scintillation/WLS light. Another source of error that may contribute to the discrepancy is the model’s treatment of optical attenuation. In the model, the optical attenuation length – the sum of optical absorption and scattering – is treated as equivalent to the optical absorption length. Treating scattering as equivalent to absorption is not generally a problem in an unreflective detector with low PMT coverage – a photon that is scattered is likely to be absorbed at the wall. Therefore we do not expect this mechanism significantly influence the results in Tub 2. However, in a highly reflective detector such as Tub 1 the scattered photons are more probable to be detected, so that the conflation of scattering with absorption is a source of systematic error. This issue will need to be addressed prior to the application of the model to large detector geometries with greater PMT coverage. Material Light Yield (photons/MeV) Quenching Parameter (mm/MeV) ---------- ------------------------------------------- ------------------------------------------- WbLS-1 19.9 $\pm$ 1.1 (stat.) $\pm$ 2.0 (sys.) 0.70 $\pm$ 0.12 (stat.) $\pm$ 0.07 (sys.) WbLS-2 108.9 $\pm$ 0.8 (stat.) $\pm$ 10.9 (sys.) 0.44 $\pm$ 0.01 (stat.) $\pm$ 0.04 (sys.) LS 9156 $\pm$ 42 (stat.) $\pm$ 916 (sys.) 0.07 $\pm$ 0.01 (stat.) $\pm$ 0.01 (sys.) \[tab:OptimalVals\] Detector A ---------- The simulated and measured photoelectron (PE) distributions for Detector A are given in figure \[fig:DetAResults\]. Quality cuts to the data were applied in the same way as Detector B. The measurement of 475 MeV protons in water are not shown. The mean number of observed photoelectrons for all measurements and simulations are given in table \[tab:MeansDetA\]. Sample Incident Proton Energy Photomultiplier N$_{PE}$, Measured N$_{PE}$, Simulated -------- ------------------------ ----------------- -------------------- --------------------- Water 475 MeV Downstream 1.3 $\pm$ 0.0 1.6 $\pm$ 0.2 Water 475 MeV Upstream 1.2 $\pm$ 0.1 1.2 $\pm$ 0.1 Water 2000 MeV Downstream 33.0 $\pm$ 0.2 32.4 $\pm$ 3.2 Water 2000 MeV Upstream 1.2 $\pm$ 0.0 1.4 $\pm$ 0.1 WbLS-2 475 MeV Downstream 4.7 $\pm$ 0.0 4.6 $\pm$ 0.5 WbLS-2 475 MeV Upstream 4.6 $\pm$ 0.0 4.5 $\pm$ 0.5 WbLS-2 2000 MeV Downstream 21.5 $\pm$ 0.3 20.4 $\pm$ 2.0 WbLS-2 2000 MeV Upstream 7.7 $\pm$ 0.2 7.3 $\pm$ 0.7 \[tab:MeansDetA\] The simulated results used the optimised values of WbLS-2’s light yield and quenching from the Detector B analysis. Whilst fairly good agreement is obtained using these values, the shape of the simulated photoelectron distribution is in general broader than that of the measured distribution. The cause for this behaviour is not fully understood. We have also performed a $\chi^2$ optimisation of the simulated distribution to the measured one by adjusting the simulated light yield using the 475 MeV proton irradiation data. These results suggest an optimal light yield of 109.0 $\pm$ 1.0 and 110.2 $\pm$ 1.0 photons per MeV using the downstream and upstream PMTs, respectively. The uncertainties relate only the statistical uncertainty arising from the $\chi^2$ analysis of the simulated data, and we estimate an additional 10% systematic uncertainty. This good agreement with the Detector B analysis suggests that the model can consistently describe different measurement geometries. ![image](NumPE_MeasVsSim_ALL_150812_v2_dashed.pdf){width="\textwidth"} For the 2 GeV proton measurements, many more photons were observed in the downstream PMT than the upstream PMT due to the directional Cherenkov radiation. However, the number of photoelectrons measured in the downstream PMT at 2 GeV for WbLS was substantially less than the number of photoelectrons measured in water at the same energy, which suggests that some proportion of the detectable Cherenkov light is absorbed or scattered by the WbLS. Due to the black detector walls, the upstream detector is largely insensitive to the direct Cherenkov light (as shown in figure \[fig:DetAResults\](b)), and therefore this signal can be used to estimate the number of photoelectrons that are measured due to the isotropic light emission processes – scintillation and WLS. The absorption probability of the detectable Cherenkov light in the WbLS is given to first order by: $$P_{abs} = 1-\frac{N_{down}(WbLS, 2~GeV)-N_{up}(WbLS, 2~GeV)}{N_{down}(Water, 2~GeV)}$$ where $N_{down}$ and $N_{up}$ are the mean numbers of photoelectrons in the downstream and upstream photomultipliers for a given measurement, respectively. Our measurements indicate that $P_{abs} \approx 58\%$; that is, 58% of Cherenkov photons that were measured in the water sample were instead absorbed in the WbLS sample measurement. The large proportion of Cherenkov photons absorbed or scattered by the ${\mathcal O}(10 {\rm cm})$ path length in WbLS suggests that the optical attenuation length of the WbLS used in this measurement is too short to permit Cherenkov imaging in a very large detector. The optical attenuation length of more recent WbLS formulations has been improved by about 2 orders of magnitude (figure \[fig:UVVIS\]). To first order, the fraction of absorbed Cherenkov photons simply scales with the optical attenuation length, which suggests that the $\approx 58\%$ loss of Cherenkov photons would occur over a path length of ${\mathcal O}(10 {\rm m})$ using the more recent WbLS formulation; which is a length scale suitable for a large detector. Experimental measurements in a larger detector geometry are needed to more rigorously assess this model prediction. Our measurements can also give an estimate of the relative proportions of measured WLS and scintillation photons. The mean number of photons detected in the upstream PMT of Detector A for 2 GeV protons incident on WbLS-2 may be written as the sum of the detected scintillation and WLS light components: $$N_{up}(WbLS, 2~GeV) = N_{up}^{Scint}(WbLS, 2~GeV) + N_{up}^{WLS(Ckov)}(WbLS, 2~GeV) \label{eqn:Up2GeV}$$ where $N_{up}^{Scint}(WbLS, 2~GeV)$ and $N_{up}^{WLS(Ckov)}(WbLS, 2~GeV)$ are the mean number of detected scintillation photons and WLS Cherenkov photons, respectively. Any wavelength shifting of scintillation light is disregarded in this approximation. The mean number of photons detected in the upstream PMT of Detector A for 475 MeV protons incident on WbLS-2 may be written as a similar expression, though without any WLS Cherenkov light component, as 475 MeV protons are at the Cherenkov production threshold: $$N_{up}(WbLS, 475~MeV) = N_{up}^{Scint}(WbLS, 475~MeV)% + N_{up}^{WLS(Scint)}(WbLS, 475 MeV)$$ The 2 GeV and 475 MeV protons experience stopping powers in water of 2.0 and 2.8 MeV/cm, respectively. It is possible estimate the difference in scintillation light yield at the different energies using the Birks quenching correction: $$Q(E) = \frac{\frac{dE}{dx}}{1 + k_{B}\frac{dE}{dx}} \label{eqn:Birks}$$ where $E$ is the proton energy and $k_{B}$ is the quenching parameter. Note that equation \[eqn:Birks\] assumes a constant stopping power for the particles interacting in the tub. The corrected light yields can then be used to relate the number of scintillation photons measured using the 475 MeV proton beam with the 2 GeV proton beam: $$N_{up}(WbLS, 2~GeV) = \frac{Q(2 GeV)}{Q(475 MeV)}N_{up}(WbLS, 475~MeV) + N_{up}^{WLS(Ckov)}(WbLS, 2~GeV) \label{eqn:WLSCkov}$$ Equations \[eqn:Up2GeV\] and \[eqn:WLSCkov\] permit the calculation of the relative contributions of the scintillation and Cherenkov processes to the measured signal in the upstream PMT. Our results indicate that in the upstream PMT for incident 2 GeV protons; for every detected photon that was produced by scintillation $1.27 \pm 0.05$ photons are detected that are produced by WLS Cherenkov light. This value is consistent with the simulation prediction of 1.28. The relatively high proportion of re-emitted Cherenkov light suggests that the detection sensitivity of a WbLS-based detector will be improved significantly beyond what would be expected by the scintillation light yield alone for particles that exceed the Cherenkov threshold in the medium. However, the WLS Cherenkov light is less localised to the position of the energy deposit in the detector and modifies the linearity of the energy response of the scintillator. These complicating factors will need detailed study in a larger measurement geometry in order to evaluate their effect upon track reconstruction and calorimetry. Summary and Outlook {#sec:Conclusion} =================== The work outlined in this study has made a first measurement of the light production and propagation characteristics of water-based liquid scintillator. Two different WbLS concentrations and pure liquid scintillator were studied. The simulation model developed in this work appears to be fairly robust as it was able to reproduce the measured photoelectron distributions in kilogram-scale detectors for three different proton energies that each probed different light emission properties of the WbLS. Higher quality WbLS re-emission probability measurements and better treatment of scattering will be used to improve the simulation model. Measurements and model validation in larger test geometries are also required before our model can be used with confidence to predict the performance of a kiloton-scale detector. We have undertaken to develop a 1000 liter scale WbLS detector for this purpose. Acknowledgements {#acknowledgements .unnumbered} ================ We would like to thank Mike Sivertz, Adam Rusek, and Chiara La Tessa at the NASA Space Radiation Laboratory, as well as Ken Sexton for their assistance with this study. This research was supported by LDRD 12-033 of Brookhaven National Laboratory and by the U.S. Department of Energy, contract number KA2501032. [10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{} M. Yeh, S. Hans, W. Beriguete, R. Rosero, L. Hu, R. L. Hahn, M. V. Diwan, D. E. Jaffe, S. H. Kettell, and L. Littenberg, *[A]{} new water-based liquid scintillator and potential applications*, *Nucl. Instr. Meth. A*, vol. 660, no. 1, pp. 51–56, 2011. J. R. Alonso *et al.*, *Advanced scintillator detector concept ([ASDC]{}): A concept paper on the physics potential of water-based liquid scintillator*, [`arXiv`: 1409.5864](http://arxiv.org/abs/1409.5864). F. An *et al.*, *A side-by-side comparison of daya bay antineutrino detectors*, *Nucl. Instr. Meth. A*, vol. 685, pp. 78 – 97, 2012 [\[`arXiv`: 1202.6181\]](http://arxiv.org/abs/1202.6181) T. Alexander *et al.*, *[DarkSide]{} search for dark matter*, *JINST*, vol. 8, no. 11, p. C11021, Nov. 2013. 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P06004, 2012. *Optical Glass Data Sheets*, Schott North America, 2012, <http://www.schott.com/advanced_optics/us/abbe_datasheets/schott_datasheet_all_us.pdf>. J. B. Birks, *[T]{}he theory and practice of liquid scintillation counting*. Oxford, UK: Pergamon Press, 1964. M. Péron and P. Cassette, *[A]{} [C]{}ompton coincidence study of liquid scintillator response to low-energy electrons*, *Nucl. Instr. Meth. A*, vol. 369, no. 2-3, pp. 344–347, 1996. L. Torrisi, *Plastic scintillator investigations for relative dosimetry in proton-therapy*, *Nucl. Instr. Meth. B*, vol. 170, pp. 523–530, 2000. R. Broda, K. Maletka, T. Terlikowska, and P. Cassette, *[S]{}tudy of the influence of the [LS]{}-cocktail composition for the standardisation of radionuclides using the [TDCR]{} model*, *Appl. Rad. Isotop.*, vol. 56, pp. 285–289, 2002. B. von Krosigk, L. Neumann, R. Nolte, S. Röttger, and Z. K., *Measurement of the proton light response of various [LAB]{} based scintillators and its implication for supernova neutrino detection via neutrino-proton scattering*, [`arXiv`: 1301.6403](http://arxiv.org/abs/1301.6403) V. Balagura, M. Danilov, B. Dolgoshein, S. Klemin, R. Mizuk, P. Pakhlov, E. Popova, V. Rusinov, E. Tarkovsky, and I. Tikhomirov, *Study of scintillator strip with wavelength shifting fiber and silicon photomultiplier*, *Nucl. Instr. Meth. A*, vol. 564, no. 1, pp. 590 – 596, 2006, [\[`arXiv`: 0504194\]](http://arxiv.org/abs/physics/0504194). [^1]: Corresponding author.
When the strength of the electron-electron interaction $U$ is increased compared to the bare bandwidth $2D$, a metal-insulator transition (MIT) occurs [@Mott]. This phenomenon, known as the Mott transition, can take place in the absence of magnetic long-range order, and is still an outstanding problem in condensed-matter physics. From a theoretical point of view, a difficulty is the absence of an obvious order parameter to systematize the critical behavior of the observable quantities when the metal insulator transition is not accompanied by the onset of magnetic long range order. These issues are experimentally relevant to systems such as V$_2$O$_3$ and Ni(Se,S)$_2$ and are the subject of intensive experimental study. [@Imada:1998] In recent years, great progress has been made by using the dynamical mean-field theory (DMFT) [@Georges:1996]. This framework describes both paramagnetic metallic and paramagnetic insulating phases. The $U$-$T$ phase diagram ($T$ is the temperature) of the frustrated Hubbard model in the limit of large lattice coordination is qualitatively similar to that of the V$_2$O$_3$ and Ni(Se,S)$_2$ systems: A first-order phase-transition line ends in a second-order critical point, henceforth referred to as the Mott critical point, which is the main focus of this letter. We will use this framework to address the fundamental questions raised in the previous paragraph. There are two earlier qualitative ideas as to what should be the order parameter to describe the physics around the finite temperature Mott point. One idea is to connect the order parameter to the notions of “metallicity” or coherence. It can be traced back to the early paper of Brinkman and Rice [@Brinkman:1970] and is captured in a slave boson formalism where the metallic state has a non zero expectation value of a Bose field which describes the coherent propagation of one particle excitations. [@Kotliar:1986] In a very different picture, Castellani [*et al.*]{} viewed the metal as a liquid rich in doubly occupied sites, and the insulator as a liquid with few doubly occupied sites. The metal to insulator transition is viewed as a condensation of doubly occupied sites, and the order parameter is related to the Blume-Emery-Griffith model [@Castellani:1979]. The Landau approach presented here provides a synthesis of these ideas. It bridges naturally between a picture based on one particle excitations and a picture based on local collective excitations (or double occupancies). In agreement with Castellani [*et al.*]{} we find that the Mott transition has indeed an Ising-like character. On the other hand, we obtain a complementary description in terms of the one particle spectral function reminiscent of the slave boson picture. A simple and clear description of the critical behavior near the critical point emerges. It allows us to systematically derive the critical behavior of any observable quantity and to relate its non analytic dependence on $T$ and $U$ to that of the order parameter. Our results should be also of help in resolving some controversies on the solution of the Hubbard model in infinite dimensions [@Schlipf:1999; @Rozenberg:1999] by providing a theoretical framework in which to analyze numerical results on the finite temperature Mott transition. It can also be used to analyze results of photoemission and optical conductivity experiments. For simplicity, we focus on the single-band Hubbard model at half-filling, $$\hat{H}=-\frac{t}{\sqrt{z}}\sum_{\langle ij\rangle\sigma} c_{i\sigma}^{+}c_{j\sigma} +U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow}. \label{hubbard}$$ The first term describes the hopping between nearest neighbors on a lattice with coordination number $z$. The corresponding half bandwidth is our unit of energy, $D=2t=1$. The second term is an on-site interaction suppressing double occupancies by imposing an energy cost $U$ on each one. In the limit of infinite dimensions, $z\rightarrow\infty$, this model can be mapped onto a single-impurity Anderson model (SIAM) supplemented by a self-consistency condition. We adopt a semicircular density of states, which is realized on the Bethe lattice. The dynamical mean-field equations can be obtained by differentiating the Landau functional $$F_{\mbox{\scriptsize LG}}[\Delta]=-T\sum_n\frac{\Delta(i\omega_n)^2}{t^2} +F_{\mbox{\scriptsize imp}}[\Delta], \label{f_landau}$$ with respect to the hybridization function $\Delta(i\omega_n)$ of the SIAM, which has the meaning of a Weiss field. $i\omega_n$ are fermionic Matsubara frequencies, while $F_{\mbox{\scriptsize imp}}[\Delta]$ is the free energy of the SIAM, given by the action $S_{\mbox{\scriptsize imp}}=S_{\mbox{\scriptsize loc}}[\Delta=0] +\sum_{\sigma,n}f_{\sigma}^+(i\omega_n) \Delta(i\omega_n)f_{\sigma}(i\omega_n)$. Here, $S_{\mbox{\scriptsize loc}}[\Delta=0]$ is the action of the local $f$ level with the hybridization set to zero. The first term in Eq. (\[f\_landau\]) is the cost of forming the Weiss field $\Delta(i\omega_n)$ around a given site, while the second one is the free energy of an electron at this site in the presence of the Weiss field. Using the Green’s function of the SIAM, $G(i\omega_n)=(1/2T)\delta F_{\mbox{\scriptsize imp}}/\delta \Delta(i\omega_n)$, the mean-field equation reads $$\frac{t^2}{2T}\frac{\delta F_{\mbox{\scriptsize LG}}[\Delta]} {\delta\Delta(i\omega_n)}=t^2G(i\omega_n)[\Delta,\alpha] -\Delta(i\omega_n)=0. \label{mean-field}$$ Here, $\alpha=(U,T)$ comprises the control parameters. This Landau approach was used to describe the energetics of the Mott transition at zero temperature [@gabi]. We will show that near the finite temperature Mott point, the Weiss field has a singular dependence which can be parametrized by a single number which assumes the role of an effective order parameter for this transition. As in Landau theory, we [*assume*]{} that a finite temperature transition exists, and [*derive*]{} a complete description of the critical behavior near the transition as follows: First, we expand the mean-field equation (\[mean-field\]) around the critical point, $\alpha_c=(U_c,T_c)$, up to third order in the deviation of the hybridization function from its value at the critical point, $\delta\Delta=\Delta(\alpha_c+\delta\alpha)-\Delta(\alpha_c)$, and to first order in $\delta\alpha=(U-U_c,T-T_c)$. This expansion is well-behaved because the impurity model [*at finite temperatures*]{} depends smoothly on $\alpha$ and $\delta \Delta(i\omega_n)$. In order to carry out this expansion it is convenient to define a fluctuation matrix $$M_{nm}=\frac{t^2}{2T}\left.\frac{\delta^2F_{\mbox{\scriptsize LG}}[\Delta]}{\delta\Delta(i\omega_n)\delta\Delta(i\omega_m)} \right|_{\mbox{\scriptsize critical point}} \label{fluct_mat}$$ $M_{nm}$ has the form $-{\delta_{nm}}+K_{nm}$, where $K_{nm}$ is the Fourier transform of a kernel $ K(\tau, \tau')$ which is proportional to the connected correlation function of an operator $O(\tau)={{\int_0}^{\beta} du {f^+}(u+\tau) {f}(u)}$, $<O(\tau) O(\tau')> -<O(\tau)>< O(\tau')>$ where the average $<>$ is calculated with the action of an Anderson impurity model. It is well known that the correlation functions of the Anderson impurity model are [*bounded*]{}, and therefore the Kernel $K$ is square integrable $ {\int_0}^{\beta} {\int_0}^{\beta} d\tau d\tau' |K(\tau, \tau')|^2 < \infty $. Therefore it $K_{nm}$ is a Fredholm operator which and has a [*discrete*]{} spectrum of eigenvalues which we labeled by the index $l$. [@footnote] At half-filling, particle-hole symmetry guarantees that the order parameter $\Delta(i\omega)$ is odd and wholly imaginary. Accordingly, the fluctuation matrix is real and symmetric and has real eigenvalues $m_l$ belonging to eigenvectors $\phi_l(i\omega_n)$ which can be chosen to be purely imaginary and to form an orthonormal basis. The critical point, in this description of the problem, is signaled by the appearance of a single zero eigenvalue, $m_0=0$, which indicates the occurrence of a simple bifurcation. Next, we represent $\delta\Delta$ in the eigenbasis of the matrix (\[fluct\_mat\]), $\delta\Delta(i\omega_n)=\sum_l\eta_l\phi_l(i\omega_n)$, where all $\eta_l$ are real. By projecting the mean-field equation (\[mean-field\]) onto the eigenbasis $\phi_l$, we obtain an equation of the form $$\begin{aligned} &m_l\eta_l+F^{(0)}_l[\{\eta_{j\ne0}\}] +F^{(1)}_l[\{\eta_{j\ne0}\}]\eta_0& \nonumber\\ &+F^{(2)}_l[\{\eta_{j\ne0}\}]\eta_0^2 +F^{(3)}_l\eta_0^3=0&, \label{expansion}\end{aligned}$$ which holds for all $l$. $F^{(0)}_l$ is of order $\delta\alpha$. $F^{(1)}_l$ and $F^{(2)}_l$ have Taylor expansions in the $\eta_{j\ne0}$, where $F^{(1)}_l$ starts with the linear order. We solve Eq. (\[expansion\]) iteratively for all $\eta_{l\ne0}$ to obtain $\eta_{l\ne0}=a_l+c_l\eta_0^2+d_l\eta_0^3$. Here, $a_l$ is of first order in $\delta\alpha$, (which assures us that the leading singular dependence of the spectral function is proportional to $\phi_0$) further corrections have the form $b_l\eta_0$ with $b_l$ also of order $\delta\alpha$. By inserting this expression into the $l=0$ case of Eq. (\[expansion\]), we derive an effective equation for the zero-mode amplitude $\eta_0$. We can think of $\eta_0$ as the soft mode near the transition and the $\eta_{l\neq 0}$ as massive modes. The elimination of the massive modes renormalizes the coefficients of the effective action for the soft mode. In the resulting cubic equation for $\eta_0$, we eliminate the quadratic term by shifting $\eta_0$ by an appropriately chosen linear function in $\delta\alpha$, $\eta=\eta_0+\mbox{const}_1\times(T-T_c)+\mbox{const}_2\times(U-U_c)$. Close to the critical point, $\eta$ and $\eta_0$ are dominated by non analytic terms and are therefore essentially equal. We thus obtain an equation of state without quadratic term in $\eta$: $$p\eta+c \eta^3=h. \label{eqofstate}$$ Here, all quantities are real. As in Landau theory, a microscopic calculation of the Landau coefficients (p,c,h) is difficult. However we can extract exact information about the critical behavior from the knowledge that they are smooth functions of the control parameters, i.e. $c$ is finite at the critical point, whereas $p$ and $h$ are linear functions of $\delta\alpha$, $h=h_1(U-U_c)+h_2(T-T_c)$ and $p=p_1(U-U_c)+p_2(T-T_c)$. As a consequence, $\eta$ has a singular dependence on $U$ and $T$ near the critical point. At $U=U_c$, and for $T$ near $T_c$, $$\eta(U_c,T)\simeq \mbox{sign}(h_2/c) \mbox{sign}(T-T_c)|T-T_c|^{1/3}.$$ The mean-field equation (\[eqofstate\]) describes the Mott transition close to the critical point in terms of the order parameter $\eta$. In this form, the analogy with the liquid gas transition is evident. The Mott transition takes place on the line in the $U$-$T$ plane where $h$ vanishes and the system has full Ising symmetry. The critical point, $(U_c, T_c)$, divides this line into two half-lines. On the half-line where $T<T_c$, there are two solutions, $\eta=\pm\sqrt{|p/c|}$. We will see later that $\eta$ parametrizes the strength of the quasiparticle resonance of the single-particle spectrum (see Fig.  \[fig2\]). A positive or negative “field” $h$ increases or decreases this component of the spectral function, respectively. The field $h$ decreases when $U$ or $T$ is increased, because either increase eliminates the metallic coherence and thus reduces the value of $\eta$. We have used the sign convention whereby is positive. We now turn to various consequences of our construction. From Eq. (\[eqofstate\]), we can obtain the [*shape*]{} of the coexistence region near the critical point, where two solutions of the mean field equations coexist. It is centered symmetrically about the $h=0$ line, and its width along $T=\mbox{const}$ lines, $\Delta U$, scales with $(T_c-T)^{3/2}$. The constant of proportionality is given by $(4/\sqrt{c}\,|h_1|)[(p_2-p_1h_2/h_1)/3]^{3/2}$. An important quantity which is measured in numerical simulations is the double occupancy. It is connected to our order parameter $\eta$ as follows: $\langle d\rangle=(T/U)\sum_n\{[(i\omega_n+\mu) G(i\omega_n)-1]e^{i\omega_n0^+}-t^2G(i\omega_n)^2\}=\langle d\rangle_c+c_1^{(d)}\eta+c_2^{(d)}\eta^2$. In this expansion about the critical point, we have only retained the leading and next to leading nonanalytic terms responsible for the critical behavior. The susceptibility $\chi=\partial\langle d\rangle/\partial U$ diverges at the critical point. For example: $$\chi(U,T_c)\simeq (c_1^{(d)}/3)\mbox{sign}(h_1/c)|h_1/c|^{1/3}|U-U_c|^{-2/3}.$$ The double occupancy is related to the magnetization by the identity $\langle(n_{\uparrow}-n_{\downarrow})^2\rangle=1-2\langle d\rangle$. The magnetic response will therefore also exhibit nonanalytic dependences on the control parameters. There has been several numerical studies of the finite temperature Mott transition in this model. The Landau approach predicts the functional dependence of various quantities near the transition, and therefore the expressions derived in this paper, are useful for interpreting the numerical work. To illustrate how our approach sheds new light on previously obtained numerical data we compare in Fig. \[fig1\] the results for the double occupancy $\langle d \rangle$ obtained within the IPT and QMC calculations with $\Delta\tau=0.5/D$, after carrying out the shifts and the rescaling described in the figure caption. Within the statistical errors of the QMC calculation, the agreement is excellent. This surprising result is consistent with the Landau theory: different approximations for the solution of the impurity model reduce to the same Landau theory near the critical point, but with different values of the Landau coefficients. Therefore, with a suitable rescaling, the results near the critical point should agree with each other, and with a fit based on the Landau theory which is shown in the red line in figure 1. Small changes in the values of $\Delta\tau$ result in shifts of $U_c$, $T_c$, and $\langle d \rangle$ at criticality, but does not change the form of the critical behavior. We also note that the critical slowing down which has been observed in the iterative solutions of the mean field equations are a direct consequence of the presence of the soft mode $\eta$ described in the Landau approach. From our construction it is clear that $\eta$ provides the leading non analytic behavior of the Weiss field. In order to get a better feeling for its physical significance we have to understand how it can be probed experimentally. Since the order parameter is closely related to the amplitude of the quasiparticle peak, photoemission is an ideal tool to probe the temperature and pressure dependence of the order parameter near the critical point. This experimental technique, in the angle integrated mode, would also measure the convolution of the Fermi function with the analytically continued eigenfunction of the zero mode, $\mbox{Im}\phi_0(i\omega_n = \omega - i \delta)$. To visualize the shape of the spectral function near the critical point we must resort to calculations based on analytic methods such as IPT. =3.5in The inset of Fig.\[fig2\] shows the spectral function very near the critical point, computed within the IPT. =3.5in =3.5in It illustrates how the compromise between metallic and insulating features is realized. A finite $\eta$, depending on its sign, adds or subtracts spectral weight to the coherent low energy feature immersed in a constant backround in between the Hubbard bands. The zero mode is seen to affect mainly the low-energy part of the spectrum, which determines whether the system is metallic or insulating. The strong temperature dependence has been noticed in previous theoretical and experimental studies. [@Matsuura:1996] Its origin and connection to an order-parameter description of the Mott transition, however, had not been recognized until now. In the main panel of Fig. \[fig2\] we display the height of the quasiparticle peak $A_0=i\Delta(i0^+)/\pi t^2$, for $U\simeq U_c$, as a function of temperature in the vicinity of $T_c$. The rapid variation seen in the figure is consistent with the form $A_0=A_{0c}+c_1^{(A)}\eta+c_2^{(A)}\eta^2$ with coefficients $ c^{(A)}_i$ independent of U and temperature. Optical techniques are probably the best tool available to test the predictions of our theory. For instance, one may consider the integral of the optical conductivity up to some cuttoff, $N_{\mbox{\scriptsize eff}}(T)$. Since the optical conductivity in infinite dimensions is directly expressed in terms of the single-particle Green’s function, $N_{\mbox{\scriptsize eff}}(T)$ must also exhibit the singular temperature dependence near the transition. We would therefore expect the temperature variation of this quantity to be most visible for a relatively small cuttoff, displaying a rapid variation with $T$ similarly as for $A_0$. Since the singular dependence arises from the order parameter $\eta$, it should be possible to fit the Drude weight by $N_{\mbox{\scriptsize eff}}(T)=N_{\mbox{\scriptsize eff}}(T_c)+c_1^{(N)}\eta(T)+c_2^{(N)}\eta^2(T)$. $N_{\mbox{\scriptsize eff}}(T)$ has recently been measured in NiS$_{2-x}$Se$_x$ [@Takagi:1999], the observed strong temperature dependence of the effective number of carriers is consistent with our predictions. In summary, we derived an order parameter description of the Mott transition near its critical point in the $U$-$T$ plane. We showed that the critical behavior in proximity to this point is governed by an Ising-like Landau functional and is present in a large number of observable quantities. We predict that any physical quantity which is sensitive to the single-particle spectrum exhibits singular dependences on the control parameters close to the finite-temperature Mott point. The leading non analytic behavior of other physical quantities can be obtained along similar lines, i.e. by recognizing their coupling to the order parameter. This involves a few coefficients, (i.e. the $c^{(A)}$’s) which depend on the observable (and on the approximation method) and, as in Landau theory, should be taken as parameters. The dependence on temperature and on pressure is completely determined from the temperature or pressure dependence of the order parameter that follows from Eq. (\[eqofstate\]). ACKNOWLEDGMENT This work was supported by NSF 95-29138. E.L. was partially supported by the Deutsche Forschungsgemeinschaft. M.J.R. acknowledges support of Fundación Antorchas, CONICET (PID $N^o4547/96$), and ANPCYT (PMT-PICT1855). We thank R. Chitra for discussions and D. Vollhardt for useful comments on the mansucript. [10]{} N. F. Mott, [*Metal-Insulator Transitions*]{} (Taylor & Francis, 1974). M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. [**70**]{}, 1039 (1998). for a review see A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. [ **68**]{}, 13 (1996). W. F. Brinkman and T. M. Rice, Phys. Rev. B [**2**]{}, 4302 (1970). G. Kotliar and A. E. Ruckenstein, Phys. Rev. Lett. [**57**]{}, 1362 R. Fresard and P. Wolfle , Int. J. of Mod. Phys. B6, 685 (3087) (1986). C. Castellani, C. D. Castro, D. Feinberg, and J. Ranninger, Phys. Rev. Lett. [**43**]{}, 1957 (1979). J. Schlipf, M. Jarrell, P.G.J. van Dongen, N. Blümer, S. Kehrein, Th. Pruschke, D. Vollhardt, Phys. Rev. Lett. [**82**]{}, 4890 (1999). M. J. Rozenberg, R. Chitra, and G. Kotliar. Phys. Rev. Lett. [**83**]{}, 3498 (1999). W. Krauth cond-mat/9908221 G. Kotliar Eur. Jour. Phys. B, [**11**]{}, 27-39 (1999). A. Y. Matsuura [*et al.*]{}, Phys. Rev. B [**53**]{}, 7584 (1996). Th. Pruschke D. Cox and M. Jarrell, Phys. Rev. B 47, 3553 (1993). A. Georges and W. Krauth Phys. Rev. B 48, 7167 (1993). M. Rozenberg et. al. Phys. Rev. Lett. 75, 105 (1995). S. Miyasaka and H. Takagi (unpublished). This is only true at finite temperatures, which ensures that the limit of integration $\beta $ is finite. At zero temperature the spectrum of fluctuations is continuos and the transition at a point denoted $U_{c2}$ has a very different character and was described in Ref. [@gabi]. To accurately locate the critical point with IPT, we found it best to solve the DMFT equations on the imaginary axis. We note that closest to the $(U_c,T_c)$ point, up to 10,000 iterations were required to find converged solutions.
--- abstract: 'We introduce a new method for training deep Boltzmann machines jointly. Prior methods of training DBMs require an initial learning pass that trains the model greedily, one layer at a time, or do not perform well on classification tasks. In our approach, we train all layers of the DBM simultaneously, using a novel training procedure called [*multi-prediction training*]{}. The resulting model can either be interpreted as a single generative model trained to maximize a variational approximation to the generalized pseudolikelihood, or as a family of recurrent networks that share parameters and may be approximately averaged together using a novel technique we call the [*multi-inference trick*]{}. We show that our approach performs competitively for classification and outperforms previous methods in terms of accuracy of approximate inference and classification with missing inputs.' author: - | Ian J. Goodfellow Aaron Courville\ Université de Montréal Yoshua Bengio bibliography: - 'strings.bib' - 'strings-shorter.bib' - 'ml.bib' - 'aigaion-shorter.bib' - 'aigaion.bib' title: Joint Training of Deep Boltzmann Machines for Classification --- Deep Boltzmann machines ======================= A deep Boltzmann machine [@Salakhutdinov2009] is a probabilistic model consisting of many layers of random variables, most of which are latent. Typically, a DBM contains a set of $D$ input features $v$ that are called the [*visible units*]{} because they are always observed during both training and evaluation. The DBM is usually applied to classification problems and thus often represents the class label with a one-of-$k$ code in the form of a discrete-valued label unit $y$. $y$ is observed (on examples for which it is available) during training. The DBM also contains several latent variables that are never observed. These [*hidden units*]{} are usually organized into $L$ layers $h^{(i)}$ of size $N_i, i=1,\dots,L$, with each unit in a layer conditionally independent of the other units in the layer given the neighboring layers. These conditional independence properties allow fast Gibbs sampling because an entire layer of units can be sampled at a time. Likewise, mean field inference with fixed point equations is fast because each fixed point equation gives a solution to roughly half of the variational parameters. Inference proceeds by alternating between updating all of the even numbered layers and updating all of the odd numbered layers. A DBM defines a probability distribution by exponentiating and normalizing an energy function $$P(v,h,y) = \frac{1}{Z} \exp\left( -E(v,h,y) \right)$$ where $$Z = \sum_{v',h',y'} \exp \left( -E(v', h', y') \right).$$ $Z$, the partition function, is intractable, due to the summation over all possible states. Maximum likelihood learning requires computing the gradient of $\log Z$. Fortunately, the gradient can be estimated using an MCMC procedure [@Younes1999; @Tieleman08]. Block Gibbs sampling of the layers makes this procedure efficient. The structure of the interactions in $h$ determines whether further approximations are necessary. In the pathological case where every element of $h$ is conditionally independent of the others given the visible units, the DBM is simply an RBM and $\log Z$ is the only intractable term of the log likelihood. In the general case, interactions between different elements of $h$ render the posterior $P(h \mid v, y)$ intractable. @Salakhutdinov2009 overcome this by maximizing the lower bound on the log likelihood given by the mean field approximation to the posterior rather than maximizing the log likelihood itself. Again, block mean field inference over the layers makes this procedure efficient. An interesting property of the DBM is that the training procedure thus involves [*feedback connections*]{} between the layers. Consider the simple DBM consisting entirely of binary-valued units, with the energy function $$E(v,h) = - v^T W^{(1)} h^{(1)} -h^{(1)T} W^{(2)} h^{(2)}.$$ Approximate inference in this model involves repeatedly applying two fixed-point update equations to solve for the mean field approximation to the posterior. Essentially it involves running a recurrent net in order to obtain approximate expectations of the latent variables. Beyond their theoretical appeal as a deep model that admits simultaneous training of all components using a generative cost, DBMs have achieved excellent performance in practice. When they were first introduced, DBMs set the state of the art on the permutation-invariant[^1] version of the MNIST handwritten digit recognition task at 0.95%. Recently, new techniques were used in conjunction with DBM pretraining to set a new state of the art of 0.79% test error [@Hinton-et-al-arxiv2012]. The joint training problem ========================== ![The training procedure employed by @Salakhutdinov2009 on MNIST. a) An RBM comprising $v$ and $h^{(1)}$ is trained to maximize the log likelihood of $v$ using CD. Next, another RBM is trained with CD, using $y$ and samples of $h^{(1)}$ conditioned on $v$ as observed data, and $h^{(2)}$ as hidden units. b) The two RBMs are stitched together to form one DBM over $v$, $h^{(1)}$, $h^{(2)}$, and $y$. This DBM is trained to maximize the log likelihood of $v$ and $y$ using PCD. c) $y$ is deleted from the model. An extra MLP is built on top of $v$ and the mean field expectations of $h^(1)$ and $h^(2)$. The parameters of the DBM are frozen, and the parameters of the MLP are initialized based on the DBM parameters, then trained via nonlinear conjugate gradient descent to predict $y$ from $v$ and the mean field features. []{data-label="standard_dbm_training"}](standard_dbm){width="\textwidth"} Unfortunately, it is not possible to train a deep Boltzmann machine using only the variational bound and approximate gradient described above. See [@GoodfellowTPAMI_ToAppear] for an example of a DBM that has failed to learn using the naive training algorithm. @Salakhutdinov2009 found that for CD / PCD to work, the DBM must be initialized by training one layer at a time. After each layer is trained as an RBM, the RBMs can be modified slightly, assembled into a DBM, and the DBM may be trained with PCD learning rule described above. In order to achieve good classification results, an MLP designed specifically to predict $y$ from $v$ must be trained on top of the DBM model. Simply running mean field inference to predict $y$ given $v$ in the DBM model does not work nearly as well. See figure \[standard\_dbm\_training\] for a graphical description of the training procedure used by [@Salakhutdinov2009]. In this paper, we propose a method that enables the deep Boltzmann machine to be jointly trained, and to achieve excellent performance as a classifier without an additional classification-specific extension of the model. The standard approach requires training $L+2$ different models using $L+2$ different objective functions, and does not yield a single model that excels at answering all queries. Our approach requires training only one model with only one objective function, and the resulting model outperforms previous approaches at answering all kinds of queries (classification, classification with missing inputs, predicting arbitrary subsets of variables given arbitrary subsets of variables). Motivation ========== There are numerous reasons to prefer a single-model, single-training stage approach to deep Boltzmann machine learning: 1. [**Optimization**]{} As a greedy optimization procedure, layerwise training may be suboptimal. Small-scale experimental work has demonstrated this to be the case for deep belief networks [@Arnold+Ollivier-arxiv2012]. In general, for layerwise training to be optimal, the training procedure for each layer must take into account the influence that the deeper layers will provide. The standard training layerwise procedure simply does not attempt to be optimal. The procedures used by @LeRoux-Bengio-2008 [@Arnold+Ollivier-arxiv2012] make an optimistic assumption that the deeper layers will be able to implement the best possible prior on the current layer’s hidden units. This approach is not immediately applicable to Boltzmann machines because it is specified in terms of learning the parameters of $P(h^{(i-1)} | h^{(i)})$ assuming that the parameters of the $P(h^{(i)})$ will be set optimally later. In a DBM the symmetrical nature of the interactions between units means that these two distributions share parameters, so it is not possible to set the parameters of the one distribution, leave them fixed for the remainder of learning, and then set the parameters of the other distribution. Moreover, model architectures incorporating design features such as sparse connections, pooling, or factored multilinear interactions make it difficult to predict how best to structure one layer’s hidden units in order for the next layer to make good use of them. 2. [**Probabilistic modeling**]{} Using multiple models and having some models specialized for exactly one task (like predicting $y$ from $v$) loses some of the benefit of probabilistic modeling. If we have one model that excels at all tasks, we can use inference in this model to answer arbitrary queries, perform classification with missing inputs, and so on. 3. [**Simplicity** ]{} Needing to implement multiple models and training stages makes the cost of developing software with DBMs greater, and makes using them more cumbersome. Beyond the practical considerations, it can be difficult to monitor training and tell what kind of results during layerwise DBM pretraining will correspond to good classification accuracy later. Our joint training procedure allows the user to monitor the model’s ability of interest (usually ability to classify $y$ given $v$) from the very start of training. Multi-Prediction Training ========================= ![Mean field inference applied to MNIST digits. The first column shows the true digits. The second column shows pixels of the digits to be masked out, with red pixels indicating the region to be witheld from the input to the DBM. Yellow-boxed rows show input pixels. Green-boxed rows represent the class variables. The subsequent columns show the DBM incrementally predicting the missing variables, with each column being one iteration of mean field. On rows where the green-boxed class variable was masked out, the uncertainty over the class is represented by displaying a weighted average of templates for the 10 different classes.[]{data-label="animate_inpainting"}](inpainting.pdf){width="50.00000%"} Our proposed approach is to directly train the DBM to be good at solving all possible variational inference problems. We call this [*multi-prediction training*]{} because the procedure involves training the model to predict any subset of variables given the complement of that subset of variables. Specifically, we use stochastic gradient descent on the [*multi-prediction*]{} (MP) objective function $$J(v, \theta) = - \sum_i \log Q^*_{v_{-S_i}} ( v_{S_i} )$$ where $S$ is a sequence of subsets of the possible indices of $v$ and $$Q^*_{v_{-S_i}}(v_{S_i}, h) = \text{argmin}_Q D_{KL} \left( Q(v_{S_i}, h ) \Vert P( v_{S_i}, h \mid v_{-S_i} ) \right) .$$ In other words, the criterion for a single example $v$ is a sum of several terms, with term $i$ measuring the model’s ability to predict a subset of the inputs, $v_{S_i}$, given the remainder of the inputs, $v_{-S_i}$. During SGD training, we sample minibatches of values of $v$ and $S_i$. Sampling an $S_i$ uniformly simply requires sampling one bit (1 with probability 0.5) for each variable, to determine whether that variable should be an input to the inference procedure or a prediction target. In this paper, $Q$ is constrained to be factorial, though one could design model families for which it makes sense to use richer structure in $Q$. In order to accomplish the minimization, we instantiate a recurrent net that repreatedly runs the mean field fixed point equations, and backpropagrate the gradient of $J$ through the entire recurrent net. See Fig. \[mpt\] for a graphical description of this training procedure, and Fig. \[animate\_inpainting\] for an example of the inference procedure run on MNIST digits. The Multi-Inference Trick ========================= Mean field inference can be expensive due to needing to run the fixed point equations several times in order to reach convergence. In order to reduce this computational expense, it is possible to train using fewer mean field iterations than required to reach convergence. In this case, we are no longer necessarily minimizing $J$ as written, but rather doing partial training of a large number of fixed-iteration recurrent nets that solve related problems. We can approximately take the geometric mean over all predicted distributions $Q$ and renormalize in order to combine the predictions of all of these recurrent nets. This way, imperfections in the training procedure are averaged out, and we are able to solve inference tasks even if the corresponding recurrent net was never sampled during MP training. In order to approximate this average efficiently, we simply take the geometric mean at each step of inference, instead of attempting to take the correct geometric mean of the entire inference process. This is the same type of approximation used to take the average over several MLP predictions when using dropout [@Hinton-et-al-arxiv2012]. Here, the averaging rule is slightly different. In dropout, the different MLPs we average over either include or exclude diferent each variable. To take the geometric mean over a unit $h_j$ that receives input from $v_i$, we average together the contribution $v_i W_{ij}$ from the model that contains $v_i$ and the contribution $0$ from the model that does not. The final contribution from $v_i$ is $0.5 v_i W_{ij}$ so the dropout model averaging rule is to run an MLP with the weights divided by 2. For the multi-inference trick, each model we are averaging over solves a different inference problem. In half the problems, $v_i$ is observed, and constributes $v_i W_{ij}$ to $h_{j}$’s total input. In the other half of the problems, $v_i$ is inferred. If we represent the mean field estimate of $v_i$ with $r_i$, then in this case that unit contributes $r_i W_{ij}$ to $h_{j}$’s total input. To run multi-inference, we thus replace references to $v$ with $0.5 (v+r)$, where $r$ is updated at each mean field iteration. The main reason this approach is effective is that it gives a good way to incorporate information from many recurrent nets trained in slightly different ways. However, it can also be understand as including an input denoising step built into the inference. See Fig. \[mi\] for a graphical depiction of the method, and Fig. \[mi\_curve\] for an example of it in action. Justification and advantages ============================ In the case where we run the recurrent net for predicting $Q$ to convergence, the multi-prediction training algorithm follows the gradient of the objective function $J$. This can be viewed as a mean field approximation to the generalized pseudolikelihood. While both pseudolikelihood and likelihood are asymptotically consistent estimators, their behavior in the limited data case is different. Maximum likelihood should be better if the overall goal is to draw realistic samples from the model, but generalized pseudolikelihood can often be better for training a model to answer queries conditioning on sets similar to the $S_i$ used during training. Note that our variational approximation is not quite the same as the way variational approximations are usually applied. We use variational inference to ensure that the distributions we shape using backprop are as close as possible to the true conditionals. This is different from the usual approach to variational learning, where $Q$ is used to define a lower bound on the log likelihood and variational inference is used to make the bound as tight as possible. In the case where the recurrent net is not trained to convergence, there is an alternate way to justify MP training. Rather than doing variational learning on a single probabilistic model, the MP procedure trains a family of recurrent nets to solve related prediction problems by running for some fixed number of iterations. Each recurrent net is trained only a subset of the data (and most recurrent nets are never trained at all, but only work because they share parameters with the others). In this case, the multi-inference trick allows us to justify MP training as approximately training an ensemble of recurrent nets using bagging. @Stoyanov2011 have observed that a similar training strategy is useful because it trains the model to work well with the inference approximations it will be evaluated with at test time. We find these properties to be useful as well. The choice of this type of variational learning combined with the underlying generalized pseudolikelihood objective makes an MP-DBM very well suited for solving approximate inference problems but not very well suited for sampling. Our primary design consideration when developing multi-prediction training was ensuring that the learning rule was state-free. PCD training uses persistent Markov chains to estimate the gradient. These Markov chains are used to approximately sample from the model, and only sample from approximately the right distribution if the model parameters evolve slowly. The MP training rule does not make any reference to earlier training steps, and can be computed with no burn in. This means that the accuracy of the MP gradient is not dependent on properties of the training algorithm such as the learning rate–properties which can easily break PCD for many choices of the hyperparameters. Another benefit of MP is that it is easy to obtain an unbiased estimate of the MP objective from a small number of samples of $v$ and $i$. This is in contrast to the log likelihood, which requires estimating the log partition function. The best known method for doing so is AIS, which is relatively expensive [@Neal-2001]. Cheap estimates of the objective function enable early stopping based on the MP-objective (though we generally use early stopping based on classification accuracy) and optimization based on line searches (though we do not explore that possibility in this paper). Regularization ============== In order to obtain good generalization performance, @Salakhutdinov2009 regularized both the weights and the activations of the network. @Salakhutdinov2009 regularize the weights using an L2 penalty. We find that for joint training, it is critically important not to do this. When the second layer weights are not trained well enough for them to be useful for modeling the data, the weight decay term will drive them to become very small, and they will never have an opportunity to recover. It is much better to use constraints on the norms of the columns of the weight vectors, as advocated by @Hinton-et-al-arxiv2012. @Salakhutdinov2009 regularize the activities of the hidden units with a somewhat complicated sparsity penalty. See `http://www.mit.edu/~rsalakhu/DBM.html` for details. We use $\text{max}(|\mathbb{E}_{h\sim Q(h)} [h] - t| - \lambda, 0)$ and backpropagate this through the entire inference graph. $t$ and $\lambda$ are hyperparameters. Related work: centering ======================= @Montavon2012arxiv showed that reparameterizing the DBM to improve the condition number of the Hessian results in succesful generative training without a greedy layerwise pretraining step. However, this method has never been shown to have good classification performance, possibly because the reparameterization makes the features never be zero from the point of view of the final classifier. We evaluate its classification performance in more detail in this work. We consider two methods of PCD training. In one, we use Rao-Blackwellization of the negative phase particles to reduce the variance of the negative phase. In the other variant, we use a special negative phase that @Salakhutdinov2009 found useful. This negative phase uses a small amount of mean field, which reduces the variance further but introduces some bias, and has better symmetry with the positive phase. See `http://www.mit.edu/~rsalakhu/DBM.html` for details. MNIST experiments ================= In order to compare MP training and centering to standard DBM performance, we cross-validated each of the new methods by running 25 training experiments for each of three conditions: centered DBMs, centered DBMs with the special negative phase (“Centering+”), and MP training. For these experiments we did not use the multi-inference trick. All three conditions visited exactly the same set of 25 hyperparameter values for the momentum schedule, sparsity regularization hyperparameters, weight and bias initialization hyperparameters, weight norm constraint values, and number of mean field iterations. The centered DBMs also required one additional hyperparameter, the number of Gibbs steps to run for PCD. We used different values of the learning rate for the different conditions, because the different conditions require different ranges of learning rate to perform well. We use the same size of model, minibatch and negative chain collection as @Salakhutdinov2009, with 500 hidden units in the first layer, 1,000 hidden units in the second, 100 examples per minibatch, and 100 negative chains. See Fig. \[crossval\] for the results of cross-validation. On the validation set, MP training consistently performs better and is much less sensitive to hyperparameters than the other methods. This is likely because the state-free nature of the learning rule makes it perform better with settings of the learning rate and momentum schedule that result in the model distribution changing too fast for a method based on Markov chains to keep up. When we fine-tune the best model, the best “Centering+” DBM obtains a classification error of 1.22 % on the test set. The best MP-DBM obtains a classification error of 0.99 %. This compares to 0.95 % obtained by @Salakhutdinov2009. If instead of adding an MLP to the model to do fine tuning, we simply train a larger MP-DBM with twice as many hidden units in each layer, and apply the multi-inference trick, we obtain a slightly better classification error rate of 0.91 %. In other words, we are able to classify better using a single large DBM and a generic inference procedure, rather than using a DBM followed by an entirely separate MLP model specalized for classification. The original DBM was motivated primarily as a generative model with a high AIS score and as a classifier. Here we explore some more uses of the DBM as a generative model. First, we evaluate the use of the DBM to classify in the presence of missing inputs. See Fig. \[missing\_inputs\] for details. We find that for most amounts of missing inputs, the MP-DBM classifies better than the standard DBM or the best centering DBM. We also explored the ability of the DBM to resolve queries about random subsets of variables. See Fig. \[general\_queries\] for details. Again, we find that the MP-DBM outperforms the other DBMs. Conclusion ========== This paper has demonstrated that MP training and the multi-inference trick provide a means of training a single model, with a single stage of training, that matches the performance of standard DBMs but still works as a general probabilistic model, capable of handling missing inputs and answering general queries. In future work, we hope to obtain state of the art performance by combining MP training with dropout, and also to apply this method to other datasets. [^1]: By permutation-invariant, we mean that permuting all of the input pixels prior to learning the network should not cause a change in performance, so using synthetic image distortions or convolution to engineer knowledge about the structure of the images into the system is not allowed.
--- abstract: 'In this comment, we address a number of erroneous discussions and conclusions presented in a recent preprint by the HALQCD collaboration, arXiv:1703.07210 . In particular, we show that lattice QCD determinations of bound states at quark masses corresponding to a pion mass of $m_\pi=806$ MeV are robust, and that the extracted phases shifts for these systems pass all of the “sanity checks” introduced in arXiv:1703.07210 .' author: - 'Silas R. Beane' - Emmanuel Chang - Zohreh Davoudi - William Detmold - Kostas Orginos - Assumpta Parreño - 'Martin J. Savage' - 'Brian C. Tiburzi' - 'Phiala E. Shanahan' - 'Michael L. Wagman' - Frank Winter bibliography: - 'bibi.bib' title: 'Comment on “Are two nucleons bound in lattice QCD for heavy quark masses? - Sanity check with Lüscher’s finite volume formula -”' --- -1.1cm -0.5cm In the last decade, significant progress has been made in the study of multi-hadron systems using lattice QCD, with the first calculations of multi-baryon bound states and their electroweak properties and decays having been performed [@Fukugita:1994ve; @Beane:2006mx; @Ishii:2006ec; @Aoki:2008hh; @Nemura:2008sp; @Yamazaki:2009ua; @Aoki:2009ji; @Beane:2010hg; @Inoue:2010es; @Yamazaki:2011nd; @Beane:2011iw; @Beane:2012vq; @Beane:2013br; @Inoue:2011ai; @Yamazaki:2012hi; @HALQCD:2012aa; @Beane:2013kca; @Beane:2014ora; @Beane:2015yha; @Detmold:2015daa; @Berkowitz:2015eaa; @Yamazaki:2015asa; @Yamada:2015cra; @Chang:2015qxa; @Savage:2016kon; @Shanahan:2017bgi; @Tiburzi:2017iux]. It is imperative that the methods used in these calculations be robust; investigations such as those of the HALQCD collaboration in Ref. [@Iritani:2017rlk] are vital provided they are carried out correctly. However, as we show in detail, many of the conclusions reached in Ref. [@Iritani:2017rlk] (henceforth referred to as [HAL]{}), that cast doubt on the validity of multi-baryon calculations, are incorrect. Since we have recently refined one of the analyses that is criticized in [HAL]{}, we focus our attention on the conclusions drawn regarding this case in particular, see Ref. [@Wagman:2017tmp]. The central point addressed by [HAL]{} is whether there exist bound states in the $\si$ and $\siii$ two-nucleon channels at heavy quark masses. Three independent groups have analysed lattice QCD calculations at quark masses corresponding to a heavy pion mass of $\sim800$ MeV (one set of calculations used quenched QCD) and found that there are bound states in these channels. Each of these groups has concluded this by extracting energies from two-point correlation functions (with the quantum numbers of interest) at two or more lattice volumes and demonstrating, through extrapolations based on the finite-volume formalism of Lüscher [@Luscher:1986pf; @Luscher:1990ux], that these energies correspond to an infinite-volume state that is below the two-particle threshold and is hence a bound state. Each group has used different technical approaches, and all are in reasonable agreement given the uncertainties that are reported. The HALQCD collaboration has also investigated these two-particle channels using a method (also based on the work of Lüscher [@Luscher:1986pf; @Luscher:1990ux]) that involves constructing Bethe-Salpeter wavefunctions, but do not find evidence for bound states in these channels [@Ishii:2006ec; @Aoki:2008hh; @Aoki:2009ji; @HALQCD:2012aa; @Inoue:2011ai].[^1] We note, however, that the HALQCD method introduces unquantified systematic effects as discussed in, e.g., Refs. [@Detmold:2007wk; @Beane:2010em; @Detmold:2015jda] and the nuclear physics overview talks in recent proceedings of the International Symposium on Lattice Field Theory [@Walker-Loud:2014iea; @Yamazaki:2015nka; @Savage:2016egr]). Here, we focus our criticisms of [HAL]{} on several specific points. ### Misinterpretation of energies and source independence Figure 2 of [HAL]{} contains a compilation of results for the ground states of the $\si$ and $\siii$ two-nucleon channels. Unfortunately the figure includes a second state from Ref. [@Berkowitz:2015eaa] that the authors of Ref. [@Berkowitz:2015eaa] explicitly indicate is not the ground state, and reporting it as such is a significant error on which many of the invalid arguments of HAL are based.[^2] There is a small scatter in the remaining results that is due to statistical fluctuations, discretisation artifacts and exponentially-small residual finite-volume effects, but, taken as a whole, there is no inconsistency in these results. In addition, a further recent study of axial-current matrix elements using a different set of interpolators [@Savage:2016kon; @Shanahan:2017bgi; @Tiburzi:2017iux] (denoted in Fig. \[fig:binding\] by NPLQCD17) also finds a consistent negatively-shifted energy on the $32^3\times48$ ensemble used in this comparison. Figure 2 in [HAL]{} also fails to include the energies extracted in Ref. [@Beane:2012vq] on the largest volume, which dominate the extraction of the binding energy. Without the results from this large volume, the confidence in the binding energy in Ref. [@Beane:2012vq] would be significantly diminished. It is therefore vital that this information be included in any discussion of these results. Figure \[fig:binding\] below shows a (corrected) summary of the energy levels extracted for the ground states of the $\si$ and $\siii$ two-nucleon systems in different volumes that are published in the literature at this particular quark mass. No significant interpolator dependence is observed, as is indicated by simple fits to the reported results for each volume, with all these fits having acceptable values of $\chi^2$ per degree of freedom. ![Binding energies of the $\siii$ and $\si$ ground states at $m_\pi= 806$ MeV found in the literature: NPLQCD13 [@Beane:2012vq], Berkowitz16 [@Berkowitz:2015eaa], and NPLQCD17 [@Savage:2016kon; @Shanahan:2017bgi; @Tiburzi:2017iux] ($d=0$ and $d=2$ refer to the magnitude of the centre-of-mass momentum used in the calculations in units of $2\pi/L$). The three regions in each panel correspond to three different volumes: $L=24$, 32, and 48 from left to right. Uncertainties listed in the original references are combined in quadrature. The horizontal lines and shaded bands represent the central value and one standard deviation bands from uncorrelated fits, respectively.[]{data-label="fig:binding"}](figures/bindings.pdf){width="0.95\columnwidth"} Figure 13 of [HAL]{} is also erroneously described as indicating that scattering state results are not source independent. The results show three energy levels where different interpolating operators are consistent within one standard deviation, and one energy level that differs at two standard deviations. This indicates broad agreement within the reported uncertainties and, contrary to statements in HAL, does not provide a sound statistical basis for a claim of inconsistency. In summary, comparison of results from the different interpolators in Refs. [@Beane:2012vq; @Beane:2013br; @Berkowitz:2015eaa; @Tiburzi:2017iux] shows that both bound and scattering-state energy levels are source-independent within reported uncertainties. This is contrary to the claims in [HAL]{}. ### Volume scaling of energies The authors of HAL claim that the single-exponential behaviour found in our work, Refs. [@Beane:2012vq; @Beane:2013br], and in that of Ref. [@Berkowitz:2015eaa], is a “mirage” arising from the cancellation of two or more scattering eigenstates[^3] contributing to the correlation functions with opposite signs (see Ref. [@Iritani:2016jie] for elaborations on possible “mirage” plateaus). This interpretation of the negatively-shifted states in these works is exceedingly unlikely, however, as such cancellation would need to occur in an almost identical way for multiple different volumes. For each of the different analyses of the 806 MeV ensembles in Fig. \[fig:binding\] (NPLQCD2013 $d=0$, NPLQCD2013 $d=2$ and Berkowitz2016 $d=0$), identical sources and sinks were used in each of three volumes (two volumes in the case of Berkowitz2016). Scattering-state eigenenergies necessarily change significantly with volume, having power-law dependence as dictated by the Lüscher quantisation condition. While it is possible that, in a given volume, a correlator for a particular source-sink interpolator combination could exhibit a cancellation between contributions of two scattering states that produces an energy level below threshold, it is very unlikely that the cancellation would persist in different volumes as the scattering-state eigenenergies change significantly with volume. As shown in Fig. \[fig:eff\], for example, the volume-independent interpolators used in Ref. [@Beane:2012vq; @Beane:2013br] produce energy levels in the three different volumes that are statistically indistinguishable, and even the approach to single-exponential behaviour does not depend on volume. The figure shows the effective masses of the smeared-point correlation functions, but the same features are seen in all other source-sink interpolator combinations that are studied. This rules out the possibility that the negatively-shifted signals are caused by cancellations between scattering states. The largest volume used in our works [@Beane:2012vq; @Beane:2013br] makes this an extremely robust statement as the spatial volumes from which we draw these conclusions vary by a factor of eight. ![The effective mass plots associated with the $d=0$ smeared-point correlators in the $L=24$, 32, and 48 ensembles of Ref. [@Beane:2012vq; @Beane:2013br]. The left(right) panel shows the $\siii$($\si$) channel. Quantities are expressed in lattice units. The horizontal grey line marks the infinite-volume energy of two non-interacting nucleons.[]{data-label="fig:eff"}](figures/effSP.pdf){width="0.8\columnwidth"} ### Consistency of Effective Range Expansion (“[HAL]{} Sanity Check (i)”) If the effective range expansion (ERE) is a valid parametrization of the scattering amplitude at low energies, the analyticity of the amplitude as a function of the centre-of-mass energy implies that the ERE obtained from states with positively-shifted energies (${k^*}^2>0$, where $k^*$ is the centre-of-mass interaction momentum) must be consistent with that obtained from states with negatively-shifted energies (${k^*}^2<0$). Although [HAL]{} finds that the NPLQCD results pass this test, we demonstrate how robust the results in Refs. [@Beane:2012vq; @Beane:2013br] are in this regard through the plots presented in Fig. \[fig:ERE-n1-n2\]. This figure shows fits to the ERE using both ground states ($n=1$) and first excited states ($n=2$) (color-shaded bands). These are overlaid on ERE fits using only the ground states (hashed bands). The two sets of bands are fully consistent with each other, proving that this check is unambiguously passed. The same feature is seen for three-parameter ERE fits, with significantly larger uncertainty bands (see also Ref. [@Wagman:2017tmp]).[^4] The difference in the size of uncertainties in the phase shift between the fits with and without the $n=2$ data shows that conclusions about the behaviour and/or validity of the ERE for datasets only near the bound-state pole are likely subject to significant uncertainties. We note that scattering parameters extracted in the region near ${k^*}^2=0$ from a linear ERE will in general differ from those determined in the vicinity of a bound-state pole due to higher order terms in the ERE. Indeed, it is known that in nature, the ERE of the $\siii$ phase shift around ${k^*}^2=0$ and around the deuteron pole are different (albeit slightly) [@deSwart:1995ui]. ![$k^*\cot \delta$ vs. the square of the centre-of-mass momentum of two baryons, ${k^*}^2$, along with the bands representing fits to two-parameter EREs obtained from i) only the ground states ($n=1$) and ii) from both the ground states ($n=1$) and the first excited states ($n=2$). The plots show the consistency of the ERE between negative and positive ${k^*}^2$ regions in both the $\si$ and $\siii$ channels. These results are from our recent re-analysis of these ensembles [@Wagman:2017tmp], and are consistent with the initial analysis [@Beane:2012vq; @Beane:2013br], with the mean values in agreement within one standard deviation as defined by the combined (statistical and systematic) uncertainties of each result. Quantities are expressed in lattice units (l.u.).[]{data-label="fig:ERE-n1-n2"}](figures/sc1.pdf){width="0.8\columnwidth"} ### Residue of the S-matrix at the bound-state pole (“[HAL]{} Sanity check (iii)”) The sign of the residue of the S-matrix at the bound-state pole is fixed. This requirement leads to the following condition on $k^*\cot \delta$ : $$\begin{aligned} \left. \frac{d}{d{k^*}^2}(k^*\cot \delta+\sqrt{-{k^*}^2}) \right |_{{k^*}^2=-{\kappa^{(\infty)}}^2} < 0, \label{eq:slope}\end{aligned}$$ where $\kappa^{(\infty)}$ is the infinite-volume binding momentum. As is seen from Fig. \[fig:ERE-tangent\], which displays the results of The uncertainty in the tangent line to the $-\sqrt{-{k^*}^2}$ function at ${k^*}^2=-{\kappa^{(\infty)}}^2$ arises from the uncertainty in the values of $\kappa^{(\infty)}$ (see also Ref. [@Wagman:2017tmp]). A similar conclusion can be drawn from three-parameter ERE fits. ![ The two-parameter ERE is compared with the tangents to the $-\sqrt{-{k^*}^2}$ curve at values of ${k^*}^2=-{\kappa^{(\infty)}}^2$. The plots show that all the identified energy eigenstates in this work are consistent with the criterion in Eq. (\[eq:slope\]) within uncertainties. These results are from our recent re-analysis of these ensembles [@Wagman:2017tmp], and are consistent with the initial analysis [@Beane:2012vq; @Beane:2013br], with the mean values in agreement within one standard deviation as defined by the combined (statistical and systematic) uncertainties of each result. Quantities are expressed in lattice units (l.u.). []{data-label="fig:ERE-tangent"}](figures/sc3.pdf){width="0.8\columnwidth"} ![ []{data-label="fig:tangent13"}](figures/2013only.pdf){width="0.8\columnwidth"} ![ []{data-label="fig:tangent1317"}](figures/2013vs2017.pdf){width="0.8\columnwidth"} ### Discussion {#discussion .unnumbered} Given the discussion above, the NPLQCD results presented in the “NPL2013” row of Table IV of the published version of [HAL]{} [@Iritani:2017rlk], reproduced below, ------------------- -------------- ----- -------- ------- -------------- ----- ------ ------- Data Source Source independence (i) (ii) (iii) independence (i) (ii) (iii) NPL2013 \[28,29\] No \* \* No No \* \* ? ------------------- -------------- ----- -------- ------- -------------- ----- ------ ------- \ should be replaced by ------ -------------- -------- -------- -------- -------------- -------- -------- -------- Data Source Source independence (i) (ii) (iii) independence (i) (ii) (iii) Yes Passed Passed Passed Yes Passed Passed Passed Yes Passed Passed Passed Yes Passed Passed Passed ------ -------------- -------- -------- -------- -------------- -------- -------- -------- \ where we have taken the liberty of changing the notation (in their published version) used to indicate passing a “sanity check” in [HAL]{} from a “ \* ” entry to “Passed”. We are currently revisiting the other NPLQCD analyses discussed in [HAL]{}. Ref. [@Yamazaki:2017euu] refutes the [HAL]{} criticisms of source-dependence leveled at the works of the PACS-CS collaboration [@Yamazaki:2009ua]. Ref. [@Savage:2016egr] provides a summary of the evidence for the validity of ground-state identifications in two-nucleon systems. With the robust conclusion of the existence of bound states reached by independent groups, and argued in this Comment, the systematic uncertainties of the potential method used by the HALQCD collaboration requires further investigation to better understand the origin of its failure to identify two-nucleon bound states. SRB was partially supported by NSF continuing grant number PHY1206498 and by the U.S. Department of Energy through grant number DE-SC001347. EC was supported in part by the USQCD SciDAC project, the U.S. Department of Energy through grant number DE-SC00-10337, and by U.S. Department of Energy grant number DE-FG02-00ER41132. ZD, WD and PES were partly supported by U.S. Department of Energy Early Career Research Award DE-SC0010495 and grant number DE-SC0011090. KO was partially supported by the U.S. Department of Energy through grant number DE- FG02-04ER41302 and through contract number DE-AC05-06OR23177 under which JSA operates the Thomas Jefferson National Accelerator Facility. A.P. is partially supported by the Spanish Ministerio de Economia y Competitividad (MINECO) under the project MDM-2014-0369 of ICCUB (Unidad de Excelencia ’María de Maeztu’), and, with additional European FEDER funds, under the contract FIS2014-54762-P, by the Generalitat de Catalunya contract 2014SGR-401, and by the Spanish Excellence Network on Hadronic Physics FIS2014-57026-REDT. MJS was supported by DOE grant number DE-FG02-00ER41132, and in part by the USQCD SciDAC project, the U.S. Department of Energy through grant number DE-SC00-10337. BCT was supported in part by the U.S. National Science Foundation, under grant number PHY15-15738. MLW was supported in part by DOE grant number DE-FG02-00ER41132. FW was partially supported through the USQCD Scientific Discovery through Advanced Computing (SciDAC) project funded by U.S. Department of Energy, Office of Science, Offices of Advanced Scientific Computing Research, Nuclear Physics and High Energy Physics and by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under contract DE-AC05-06OR23177. [^1]: In the $\Lambda\Lambda$ channel, the HALQCD approach does indicate a bound state, but the binding energy is found to be significantly different from that determined by extrapolating finite-volume energy levels [@Beane:2012vq]. [^2]: Whether the quoted value for the second energy in Ref. [@Berkowitz:2015eaa] is a true estimate of an excited-state energy is a question for future discussion. However for the ground states, all results unambiguously agree. [^3]: The scattering states are loosely used here to denote states in a finite volume that correspond to the continuum states of infinite volume. [^4]: Our analysis of two-nucleon correlation functions generated from these ensembles of gauge-field configurations has been recently refined in a comprehensive re-analysis [@Wagman:2017tmp], including results at additional kinematic points. This new analysis has been used in obtaining the results shown in Figs. \[fig:ERE-n1-n2\] and \[fig:ERE-tangent\]. All of the energies extracted from the three lattice volumes, and the binding energies and ERE parameters subsequently obtained, are in agreement with our previous results; i.e., the differences in the mean values of the results from the previous and the new analyses are within one standard deviation as defined by the (statistical and systematic) uncertainties of the results combined in quadrature [@Beane:2012vq; @Beane:2013br].
--- abstract: | We present MMT spectroscopic observations of regions in 42 low luminosity galaxies in the *Spitzer* Local Volume Legacy (LVL) survey. For 31 of the 42 galaxies in our sample, we were able to measure the temperature sensitive \[\] $\lambda$4363 line at a strength of 4$\sigma$ or greater, and thus determine oxygen abundances using the “direct" method. Our results provide the first “direct" estimates of oxygen abundance for 19 of these galaxies. “Direct" oxygen abundances were compared to *B*-band luminosities, 4.5 $\mu$m luminosities, and stellar masses in order to characterize the luminosity-metallicity and mass-metallicity relationships at low-luminosity. We present and analyze a “Combined Select" sample composed of 38 objects (drawn from a sub-set of our parent sample and the literature) with “direct" oxygen abundances and reliable distance determinations (based on the tip of the red giant branch or Cepheid variables). Consistent with previous studies, the $B$-band and 4.5 $\mu$m luminosity-metallicity relationships for the 38 objects were found to be 12 + log(O/H) = (6.27$\pm0.21) + (-0.11\pm0.01) M_{B}$ and 12 + log(O/H) = (6.10$\pm0.21) + (-0.10\pm0.01) M_{[4.5]}$ with dispersions of $\sigma$ = 0.15 and 0.14 respectively. The slopes of the optical and near-IR L-Z relationships have been reported to be different for galaxies with luminosities greater than that of the LMC. However, the similarity of the slopes of the optical and near-IR L-Z relationships for our sample probably reflects little influence by dust extinction in the low luminosity galaxies. For this sample, we derive a mass-metallicity relationship of 12 + log(O/H) = (5.61$\pm0.24) + (0.29\pm0.03) \log(\mbox{M}_{\star})$, which agrees with previous studies; however, the dispersion ($\sigma$ = 0.15) is not significantly lower than that of the L-Z relationships. Because of the low dispersions in these relationships, if an accurate distance is available, the luminosity of a low luminosity galaxy is often a better indicator of metallicity than that derived using certain “strong-line” methods, so significant departures from the L-Z relationships may indicate that caution is prudent in such cases. With these new “direct" metallicities we also revisit the 70/160 $\mu$m color metallicity relationship. Additionally, we examine N/O abundance trends with respect to oxygen abundance and B-V color. We find a positive correlation between N/O ratio and B-V color for 0.05 $\lesssim B-V \lesssim$ 0.75: $\log(\mbox{N/O})$ = (1.18$\pm$0.9)$\times$(B-V) + ($-$1.92$\pm$0.08), with a dispersion of $\sigma$ = 0.14, that is in agreement with previous studies. author: - 'Danielle A. Berg, Evan D. Skillman, Andrew R. Marble, Liese van Zee, Charles W. Engelbracht, Janice C. Lee, Robert C. Kennicutt, Jr., Daniela Calzetti, Daniel A. Dale, and Benjamin D. Johnson' title: Direct Oxygen Abundances for Low Luminosity LVL Galaxies --- INTRODUCTION {#sec:intro} ============ There is a fundamental relationship between the mass of stars in a galaxy and its metallicity evolution [e.g., @tremonti04 hereafter, the M-Z relation]. Empirically, this has been observed as a luminosity-metallicity relationship (hereafter, the L-Z relation) for low redshift dwarf galaxies [e.g., @lequeux79; @skillman89; @lee06a and references therein] and spiral galaxies [e.g., @mccall85; @garnett87; @zaritsky94; @tremonti04 and references therein]. This relationship is observed over a range of 10 magnitudes in galaxy optical luminosity [e.g., @zaritsky94; @tremonti04; @lee06a], but the data are relatively sparse at the low luminosity end where the intrinsic faintness of these galaxies makes metallicity determinations more difficult. The physical driver of the M-Z relation remains under debate. One possibility is that low-mass galaxies are younger, in that they only recently started forming stars [@noeske2000; @leitner2011]. Another is that they have been less efficient at producing metals [@brooks07]. Many studies favor a different interpretation, where supernova driven winds preferentially expel metals from low-mass galaxies, resulting in a lower effective yield with decreasing mass [e.g., @dekel86]. However, @dalcanton07 emphasizes the importance of star formation efficiency as outflows are an insufficient regulator in the absence of depressed star formation. In addition, Dalcanton’s calculations show that low effective yields cannot be due to gas infall. Alternatively, @koppen07 showed that the M-Z relationship may be observed naturally if a SFR-dependent, and therefore mass-dependent, stellar initial mass function (IMF) is assumed. Clearly a better understanding of the mass-metallicity relationship at low-luminosity remains important to determine how galaxies evolve [e.g., see discussion in @moustakas12 and references therein]. In addition, a well defined low-luminosity M-Z relationship will provide clues to the source of its measurable scatter. While observational errors play a role, one or more physical processes may be responsible for the remainder. Suggestions for the scatter include variations in the star formation history [e.g., recent starbursts, @contini02], variations in stellar surface mass density [@ellison08], inflow of metal poor gas, perhaps triggered by interactions [@jlee2004], and variations in local galaxy density [e.g., @cooper08 and references therein]. As astronomers examine the interrelationship between chemical abundance measurements, star formation, gas accretion, and gas outflow by measuring the evolution of the M-Z relationship, a secure M-Z relationship for the current epoch is needed for comparison. Empirical and theoretical oxygen abundance calibrations often introduce bias, further limiting the M-Z relationship [e.g., @yin07; @perez-montero09; @moustakas10; @berg11]. Notably, for 53,000 SDSS galaxies, which span 10 orders in B-band magnitude, @tremonti04 found a dispersion of 0.16 for their L-Z relationship and 0.10 for their M-Z relationship. @lee06a [hereafter L06] were able to extend the mass-metallicity relation lower by 2.5 decades in stellar mass using 4.5 $\mu$m luminosities for 27 nearby dwarf irregular galaxies. Interestingly, L06 found the dispersion in the near-infrared L-Z relationship to be smaller than the corresponding dispersion in the B-band L-Z relationship and nearly identical to that of the M-Z relationship. The smaller dispersion in the near-infrared is not totally unexpected, as NIR luminosities are less sensitive to extinction from dust and variations in star formation rate. However, the significant but uncertain stochastic effects of asymptotic giant branch (AGB) stars on the total NIR luminosities of low luminosity galaxies must also be considered [see, e.g., @fouesneau10; @meidt12; @melbourne12]. To thoroughly examine the L-Z and M-Z relations, we need a robust sample of galaxies. The *Spitzer* Local Volume Legacy survey[^1] [LVL; @dale09] covers a volume-complete sample of 258 galaxies in the local universe with multiwavelength observations spanning the ultraviolet to the radio. The LVL is leveraged by ancillary data including H$\alpha$ [@kennicutt08] and UV [@jlee11] imaging from the 11 Mpc H$\alpha$ and Ultraviolet Galaxy Survey [11HUGS; @jlee11] and the Nearby Galaxy Survey [NGS; @gildepaz07]. A subsample of the LVL also contains stellar population mapping from the ACS Nearby Galaxy Survey Treasury [ANGST; @dalcanton09], HI mapping from the VLA and GMRT, and optical broad-band imaging [@cook12; @vanzee12] and spectroscopy. However, many of the faintest objects are missing the high-quality optical spectroscopy needed to determine “direct" oxygen-abundance metallicity estimates. As the L-Z relationship provides both a very strong constraint on theories of galaxy evolution and a tool to better understand galaxies at higher redshifts [@kobulnicky03], we are motivated to better characterize the low-luminosity end of the L-Z relationship. Thus, we obtained high-resolution MMT spectroscopy of 42 low luminosity star-forming galaxies in the Local Volume with the goal of detecting the \[\] $\lambda$4363 line in order to constrain electron temperature measurements. We present our low-luminosity sample in § \[sec:Sample\], with spectral observations obtained from the MMT in § \[sec:mmt\] and IRAC photometry in § \[sec:phot\]. Section \[sec:data\] describes the data reduction, followed by the description of the method used to determine “direct" oxygen abundances in § \[sec:metallicity\]. Our “Select" sample, compiled from objects with “direct" oxygen abundances and secure distance estimates, is defined in § \[sec:Gold\]. Using this sample, metallicity is compared to expected trends with *B*-band luminosity, 4.5 $\mu$m luminosity, and stellar mass in § \[sec:Lb-Z\], § \[sec:L-Z\], and § \[sec:M-Z\] respectively. N/O relative abundances are discussed in § \[sec:N/O\]. In § \[sec:discussion\] we discuss the results of the relationships found in § \[sec:Lb-Z\]-§ \[sec:M-Z\], the “young galaxy" hypothesis, and the quality of abundance estimators. Finally, we summarize our conclusions in § \[sec:conclusion\]. Appendix \[sec:strong\] presents the strong-line abundances for the low-luminosity LVL galaxies for which we were unable to determine “direct” abundances. and Appendix \[sec:outlier\] presents our new “direct" abundances in comparison to the color-temperature metallicity relationship of @engelbracht08. Sample Selection ================ *Spitzer* LVL Survey {#sec:LVL} -------------------- LVL is a *Spitzer Space Telescope* legacy program that combines IRAC (Infrared Array Camera) and MIPS (Multiband Imaging Photometer) infrared imaging for a complete sample of 258 galaxies for the nearest 11 Mpc of our local universe. These data build upon recent Local Volume galaxy surveys: narrowband H$\alpha$ [@kennicutt08], *GALEX* ultraviolet [@jlee11], and *Hubble Space Telescope* resolved stellar population imaging [@dalcanton09]. While previous surveys comprehensively cover high surface brightness systems in flux-limited samples, the LVL survey, although also biased toward high surface brightness galaxies, provides a multi-wavelength inventory of a statistically robust, approximately volume-limited sample, which is well-suited for studies of dwarf galaxies. By studying the nearby, low-luminosity galaxies, we can increase the dynamic range covered by the luminosity-metallicity and mass-metallicity relationships, which will help to better constrain the slopes. Low-Luminosity LVL Sample {#sec:Sample} ------------------------- We selected a sample of 42 low luminosity galaxies in the LVL survey in order to obtain new MMT high-resolution spectra. These low luminosity spirals and dwarf irregulars span a range in distance of 2.5 $\leq$ D $\leq$ 14.0 Mpc[^2]. The luminosities for this sample range in the near-IR (determined from IRAC [@fazio04] photometry) from $M_{[4.5]} = -13.1$ to $-21.7$, with *B*-band magnitudes of $-10.8 \geq M_B \geq -18.8$. Most of the objects were chosen because they lack “direct" oxygen abundances in the literature, their abundance estimates are dated, or were studied with instruments which were known to have problems. Although not LVL objects, two additional galaxies were added to the sample (increasing the sample total to 44 objects) because they played a role in motivating this project. Both UGC 4393 and UGC 10818 were identified by @engelbracht08 as low metallicity outliers from the global trend of 70/160 $\mu$m color temperature as a function of metallicity. These two galaxies affect the interpretation of the trend for aromatic emission to weaken below 12 + log(O/H) = 7.9 in the mid-IR [see e.g., @engelbracht08] and the far-IR [see e.g., @draine07; @engelbracht08]. Because of the possibility that these objects’ oxygen abundances were underestimated using the lower branch of the R$_{23}$ calibration [@pilyugin05], they were included in this sample to be re-examined (see discussion in Appendix \[sec:outlier\]). See Table \[tbl1\] for sample characteristics. DATA ==== MMT Spectra {#sec:mmt} ----------- ### Observations {#sec:mmtobs} New spectroscopy was acquired at the MMT in order to achieve high signal-to-noise (S$/$N) spectra with the goal of detecting the faint \[\] $\lambda$4363 auroral line at a strength of 4$\sigma$ or higher. The observations were obtained with the Blue Channel spectrograph [@schmidt89] on the UT dates of 2008 October 30-November 1, 2009 June 15-22, and 2010 January 11-12. Sky conditions varied, but contained minimal cloud coverage and approximately arcsecond seeing. A 500 line grating, $1\arcsec$ slit, and UV-36 blocking filter were used, yielding an approximate dispersion of 1.2 Å per pixel, a full width at half maximum resolution of $\lesssim3$ Å, and a wavelength coverage of 3690–6790 Å. The sensitivity, resolution, and wavelength coverage of the MMT and Blue Channel spectrograph combination allowed for the measurement of all emission lines relevant to oxygen abundance determinations. Bias frames, flat-field lamp images, and sky flats were taken each night. The latter were primarily necessary due to significant differences between the chip illumination patterns of the sky and the MMT Top Box that houses the “BC" incandescent flat-field lamp. On average, four standard stars from @oke90 with spectral energy distributions (SEDs) peaking in the blue and containing minimal absorption were observed throughout the night using a 5$\arcsec$ slit over a range of airmasses. This allows the flux calibration to be determined as a function of airmass. The large slit width mitigates the effects of atmospheric differential refraction and allows accurate measurements of relative fluxes across a large range in wavelength. Note that since we only care about relative abundances, an absolute flux calibration is not critical. All 44 galaxies had at least one strong H$\alpha$ brightness peak that was aligned with the $1\arcsec\times180\arcsec$ slit. Typically, three 900 second exposures[^3] were made with the slit at a fixed position angle which approximated the parallactic angle at the midpoint of the observation and laid across several H$\alpha$ bright regions when possible. This, in addition to observing the galaxies at airmasses less than 1.5, served to minimize the wavelength-dependent light loss due to differential refraction [@filippenko82]. A single slit position for each target was deemed sufficient to characterize the global oxygen abundance, as metallicity gradients are observed to be small or non-existent in low-mass galaxies [e.g., @skillman89; @ks96; @ks97; @lee06b; @croxall09]. Finally, combined helium, argon, and neon arc lamps were observed at each pointing for accurate wavelength calibration. A log of the observations is provided in Table \[tbl2\]. Figure \[fig1\] shows the R-band continuum and H$\alpha$ continuum-subtracted images for each galaxy, motivating our slit location choices. The brightest H$\alpha$ regions observed are ordered alphabetically by decreasing flux, and the slit positions on the galaxies are shown. The images scale as 60x60 arcseconds with North oriented up and East to the left. ### Spectra Reduction {#sec:mmtreduct} The MMT observations were processed using ISPEC2D [@moustakas06], a long-slit spectroscopy data reduction package written in IDL. A master bias frame was created from $\gtrsim 20$ zero second exposures by discarding the highest and lowest value at each pixel and taking the median. Master sky and dome flats were similarly constructed after normalizing the counts in the individual images. Those calibration files were then used to bias-subtract, flat-field, and illumination-correct the raw data frames. Dark current was measured to be an insignificant $\sim 1$ e$^{-}$ per pixel per hour and was not corrected for. Misalignment between the trace of the light in the dispersion direction and the orientation of the CCD detector was rectified via the mean trace of the standard stars for each night, providing alignment to within a pixel across the detector. A two-dimensional sky subtraction was performed using individually selected sky apertures, followed by a wavelength calibration applied from the HeArNe comparison lamps taken at the same telescope pointing. Airmass dependent atmospheric extinction and reddening were corrected for using the standard Kitt Peak extinction curve [@crawford70]. For each galaxy, the multiple sub-exposures were combined, eliminating cosmic rays in the process. The resulting images were then flux-calibrated using the sensitivity curve derived from the standard star observations taken throughout a given night. Finally, the trace fit to the strongest continuum source in the slit was used to extract the galaxy emission within apertures that encompassed $\gtrsim99\%$ of the light. Figure \[fig2\] shows a sample of four of the resulting one-dimensional spectra extracted for galaxies that had significant \[\] $\lambda$4363 detections. The inset windows display a narrower spectral range to emphasize the \[\] $\lambda$4363 strength. This sample does not feature the best spectra from our sample, but rather galaxies are ordered by ionizing radiation field strength from highest to lowest as given by the \[\] $\lambda5007$/\[\] $\lambda3727$ ratio, highlighting the variation within the sample. Photometry {#sec:phot} ---------- To better characterize our low-luminosity sample, absolute magnitudes in several different bands were obtained. Here we describe their origin and reference their subsequent use. $M_B$ values were determined by @vanzee12 using photometry from apertures matched to the infrared LVL photometry (unless otherwise noted). Optical photometry for the entire LVL sample is given in @cook12, whereas @vanzee12 focuses on the analysis of colors and EW gradients of dwarf galaxies. The data are used to examine the optical luminosity-metallicity relationship (see Section \[sec:Lb-Z\]). $M_{[4.5]}$ values from the 4.5 $\mu$m IRAC photometry presented in @dale09 were calculated using $$M_{[4.5]} = -2.5\log{\frac{F_{[4.5]} (d/10)^2}{179.7}},$$ where $F_{[4.5]}$ is the 4.5 $\mu$m flux in Janskys, d is the distance in parsecs, and 179.7 is the zero point flux in Janksys for the 4.5 $\mu$m IRAC band [@reach05]. Distances are taken from the literature, as described in Table \[tbl1\], and assumed to have $10\%$ uncertainty where none were provided. IRAC calibration uncertainties are $5-10\%$ for the 4.5 $\mu$m data. Later, in Section \[sec:M-Z\], we use these $M_{[4.5]}$ magnitudes to analyze the NIR luminosity-metallicity relationship. Similarly, $M_{K_S}$ values were determined by @dale09 from 2MASS imaging, where 666.7 is the zero point flux in Janksys for the 2MASS $K_S$ band. Although 2MASS $F_{K_S}$ values are available for those objects which [@dale09] don’t provide $K_S$ magnitudes, we choose not to use them. The small apertures used in the 2MASS extraction produce unexpectedly faint magnitudes for smaller galaxies when compared to similar extractions from IRAC 3.6 and 4.5 $\mu$m data , and so may not be terribly accurate for our sample. The $K_S$ magnitudes were used to determine stellar masses in Section \[sec:M-Z\]. Finally, *V*-band magnitudes were needed to calculate $B-V$ colors (see Table \[tbl1\]). When available, $M_{V}$ values were provided by @vanzee12, using the LVL elliptical aperture. In other cases, values are taken from [@devaucouleurs91] or are determined using *g*- and *r*-band photometry available from the Sloan Digital Sky Survey [SDSS; @york00]. The SDSS values are then used to estimate the $B-V$ color following @jester05: $$B-V = \frac{(g-r)+0.22}{1.02}.$$ The available $M_B$, $M_{[4.5]}$, and $B-V$ colors and references for this sample are listed in Table \[tbl1\]. Note that the main source of uncertainty in these magnitudes lies in the distance determinations. Eight of the objects in our sample have distance errors of approximately $10\%$. Furthermore, 20 of the 44 objects in our sample do not have uncertainties associated with their distance determinations. For these objects we used an uncertainty of $10\%$, which may be an underestimate for some of them. The distance uncertainties tend to dominate over the photometric uncertainties. NEBULAR ABUNDANCE ANALYSIS {#sec:data} ========================== Emission Line Measurements {#sec:iraf} -------------------------- Emission line strengths were measured using standard methods available within IRAF[^4]. In particular, the SPLOT routine was used to analyze the extracted one-dimensional spectra and to fit Gaussian profiles to emission lines to determine their integrated fluxes. Special attention was paid to the Balmer lines, which are sometimes located in troughs of significant underlying stellar absorption. The H$\alpha$ emission lines typically had equivalent widths of $\sim$ 350 Å, large enough that the underlying absorption was not a concern. Even for those H$\alpha$ emission lines with lower EWs, the underlying absorption was negligible. This was often not the case for H$\beta$ and the lower equivalent width Balmer lines. The H$\beta$ absorption EWs for our sample range from 1-8 Å. These values are typical of local low-luminosity galaxies, with the majority having H$\beta$ absorption EWs between 0 Å and 5 Å [see, e.g., Figure 6 in @berg11]. For the bluer Balmer lines, a multiple component fit was used in which the absorption was fit by a broad, negative Lorentzian profile and the emission was fit by a narrow, positive Gaussian profile. To ensure a proper fit of the \[\] $\lambda4363$ line, H$\gamma$ was first fit by a Gaussian profile, then \[\] $\lambda4363$ was forced to be fit to the same line profile with the assumption that the profile widths of these two neighboring lines should be the same. Note that we chose to fit the underlying Balmer absorption with Lorentzian profiles, as opposed to using stellar population synthesis continuum fitting common in many studies [e.g., @tremonti04]. Given the large equivalent widths of the Balmer emission lines, the differences between the two methods are negligible, and the Lorentzian profiles have the advantage of require no additional assumptions. Most importantly, for spectra dominated by young stars, at S/N values typical of our spectra, population synthesis models may not provide a unique solution. There are also very large variations in the population synthesis models for young ages, with large uncertainties in how the Wolfe-Rayet phase, stellar winds, rotation, and other parameters are treated. Since mass loss and mixing processes in stellar evolution are still poorly understood, stellar phases, like Wolf-Rayet stars or Red Super Giants, are particularly affected by such uncertainties [@leitherer2011]. Later phases, like AGB stars, are covered only crudely in models or not at all, pushing parameters into regimes that are not properly calibrated. When discrepancies between models are found, they can usually be attributed to different intrinsic input parameters and/or treatment of these aberrant stellar evolutionary phases [@vazquez2005; @conroy2010]. By not using the models to fit our continuum, we avoid the uncertainties associated with these implicit assumptions. The errors of the flux measurements were approximated using $$\sigma_{\lambda} \approx \sqrt{ {(2\times \sqrt{N}\times rms)}^2 + {(0.02\times F_{\lambda})}^2 } , \label{eq:uncertainty}$$ where N is the number of pixels spanning the Gaussian profile fit to the narrow emission lines. The rms noise in the continuum was taken to be the average of the rms on each side of an emission line. For weak lines, whose uncertainty is dominated by error from the continuum subtraction, the rms term determines the approximate uncertainty. For the lines with flux measurements much stronger than the rms noise of the continuum, (usually the H$\alpha$ lines and often the \[\] $\lambda\lambda$4959,5007 doublet) the error is dominated by flux calibration and de-reddening uncertainties. In this case, a minimum uncertainty of 2% was assumed, and the right hand term above dominates the uncertainty estimate. 31 of the 44 galaxies in our sample were measured to have \[\] $\lambda$4363 line strengths $> 4\sigma$. The measured \[\] $\lambda4959$/$\lambda5007$ ratios match theoretical expectations within the errors, supporting our error estimates and the assumption that the continuum subtraction dominates the uncertainties for the weak lines. For all the objects in the present sample, flux line strengths and corresponding errors are listed in Table \[tbl3\]. We concentrate the rest of our analysis on the objects for which direct electron temperature and chemical abundance determinations can be made. An analysis of the remaining spectra using strong-line methods is reported in Appendix \[sec:strong\]. Reddening Corrections {#sec:redcor} --------------------- The relative intensities of the Balmer lines are nearly independent of both density and temperature, so they can be used to solve for the reddening. The MMT spectra were de-reddened using the reddening law of @cardelli89, parameterized by $A_{V}=3.1\ E(B-V)$, where the extinction, $A_{1}(\lambda)$ was calculated using the York Extinction Solver [@mccall04][^5]. With these values, the reddening, $E(B-V)$, can be derived using $$\log{\frac{I(H\alpha)}{I(H\beta)}\ } = \log{\frac{F(H\alpha)}{F(H\beta)}\ } + 0.4\ E(B-V)\ [A_{1}(H\alpha)-A_{1}(H\beta)], \label{eq:dered}$$ where $F$(H$\alpha)/F$(H$\beta$) is the observed flux ratio and $I$(H$\alpha)/I$(H$\beta$) is the de-reddened line intensity ratio using case B from @storey87, assuming an electron temperature calculated from the \[\] line ratio and $n_{\rm e}=10^{2}$ cm$^{-3}$. For our sample, the electron temperature range is 9,500 K - 19,500 K, with an average of 13,300 K. This range agrees with the typical electron temperatures of 10,000 K - 20,000 K for metal-poor regions. This same process can be carried out for the H$\gamma$/H$\beta$ and H$\delta$/H$\beta$ ratios observed. When all the necessary Balmer lines were present, which is true of all of the objects in our “Select" sample, we used a minimized chi squared approach to find the best estimate of E(B-V) based on the H$\alpha$/H$\beta$, H$\gamma$/H$\beta$, and H$\delta$/H$\beta$ ratios. The resulting Balmer ratios are within errors of the @storey87 Case B values for all objects meeting the selection criteria of our “Select" sample (see § \[sec:Gold\]), with an average of $\chi^2=0.03$. Following @lee04, the reddening value can be converted to the logarithmic extinction at H$\beta$ as $$c(\mbox{H}\beta) = 1.43\ E(B-V). \label{eq:cHbeta}$$ Our reddening corrections are tabulated in Table \[tbl3\]. “Direct" OXYGEN ABUNDANCE DETERMINATIONS {#sec:metallicity} ======================================== Accurate “direct" oxygen abundance determinations from regions require a measurement of the electron temperature (typically via observation of the temperature sensitive auroral \[\] $\lambda$4363 line). For the 31 low-luminosity objects for which \[\] $\lambda$4363 strengths were measured to be $> 4\sigma$, we use the temperature sensitive ratio comparing “auroral" to “nebular" collisionally excited lines to determine electron temperatures. A simple, yet reasonable, approximation to the geometry of an region is to assume a two zone volume, where $t_2$ and $t_3$ are the electron temperatures (in units of $10^{4}$ K) in the low and high ionization zones respectively. For the high ionization zone, the \[\] I($\lambda\lambda$4959,5007)/I($\lambda$4363) ratio was used to derive a temperature using the IRAF task TEMDEN. This task computes the electron temperature of the ionized nebular gas within the 5-level atom approximation. The O$^{+}$ (low ionization) zone electron temperature can be related to the O$^{++}$ (high ionization) zone electron temperature [e.g., @campbell86; @pagel92]. We used the relation between $t_{2}$ and $t_{3}$ proposed by @pagel92, based on the photoionization modeling of @stasinska90 to determine the low ionization zone temperature: $${t_{2}}^{-1} = 0.5({t_{3}}^{-1} + 0.8).$$ The low and high ionization region temperatures are tabulated in Table \[tbl4\]. Typically regions are assumed to have electron temperatures within the range of 1 to 2 $\times10^4$ K. Temperatures for the present sample agree with this approximation, spanning 10,800 K - 15,200 K for the low ionization region, and 9,600 K - 19,400 K in the high ionization region. Since the MMT spectra include emission lines from both O$^+$ and O$^{++}$, we determine oxygen abundances based on our estimated two zone electron temperatures. Spectra which contained measurable \[\] $\lambda\lambda$6717,6731 were used to determine electron densities consistent with the low density limit. Thus, it is reasonable to simply assume $n_{\rm e}=10^2$ cm$^{-3}$ for this sample. Ionic abundances were calculated with: $${\frac{N(X^{i})}{N(H^{+})}\ } = {\frac{I_{\lambda(i)}}{I_{H\beta}}\ } {\frac{j_{H\beta}}{j_{\lambda(i)}}\ }. \label{eq:Nfrac}$$ The emissivity coefficients, which are functions of both temperature and density, were determined using the IONIC routine in IRAF. This routine applies the 5-level atom approximation, assuming the appropriate ionization zone electron temperature, as determined from the oxygen line ratios. Some abundance determinations require ionization correction factors to account for unobserved ionic species. Here we assume N/O = N$^{+}$/O$^{+}$ [@peimbert69]. [@nava06] have investigated the validity of this assumption. They concluded that although it could be improved upon with modern photoionization models, it is valid to within about 10%. Thus, we employ this assumption, mostly for the purposes of direct comparison with other studies in the literature. For the 9 objects with multiple regions containing strong \[\] $\lambda$4363, an error weighted average was used to determine a best estimate of relative abundances and oxygen abundances. The results from individual regions are tabulated in Table \[tbl4\] and the mean values, using a weight of $1/\sigma_i^2$ for each component, are listed in Table \[tbl5\]. The uncertainties for these mean values are represented by the standard deviation of the weighted mean or the weighted dispersion, which ever is greater. Calculated errors in this paper provide a statistical estimate only. Additional errors may be important, such as systematic errors due to temperature fluctuations or other imperfect assumptions. However, the purpose of this paper is to improve the L-Z and M-Z relationships with abundances from high quality spectra. The statistical errors allow such an assessment of the relative quality of the spectra used, which in turn are weighted higher in the regression fits. For 7 of the 9 dwarf galaxies with direct abundances from multiple regions, the derived oxygen abundances agree within the uncertainties. These support the interpretation that the ISM in typical dwarf galaxies is chemically well mixed, in agreement with past studies [e.g., @skillman89; @ks96; @ks97; @lee06b; @kehrig08; @croxall09; @perez-montero2011]. Various theoretical studies support this result [e.g., @roy95]. However, there are two galaxies for which the oxygen abundances don’t agree. For NGC 4449 the highest signal to noise spectrum is offset to higher log(O/H) values by 0.16 and 0.18 dex compared to the other two. This discrepancy may be due to the possible contamination of an embedded supernova remnant [e.g., @skillman85], or it may be truly offset. Additional spectra are needed to clarify this. NGC 2537 has two high quality optical spectra, but the derived values disagree by 0.26 dex. This factor of nearly two difference is intriguing, warranting further investigation of this object. We increased the error of the weighted mean to indicate the dispersion between the two values. Note that the lower value would be in better agreement with the L-Z relationships, but that the mean is not offset very far. Overall, the oxygen abundances determined in this paper are all relatively low (12 + log(O/H) $< 8.3$; average 12 + log(O/H) $= 7.84$) as we would expect for low-mass, low-luminosity galaxies. The abundances for the two additional objects outside of the LVL sample, UGC 4393 and UGC 10818, are discussed in Appendix \[sec:outlier\]. The L-Z and M-Z Relationships {#sec:relationships} ============================= The new “direct" oxygen abundances determined in this paper provide an opportunity to expand relationships previously limited by the reliability of empirical calibrations. In particular, these measurements allow us to re-examine the L-Z and M-Z relationships derived by L06, which are limited by small number statistics at the low luminosity end. The Total and “Select" Samples {#sec:Gold} ------------------------------ In the following, we analyze various samples based on both abundance measurement and distance measurement quality criteria. Specifically, we label the samples of galaxies with both direct oxygen abundance measurements and accurate distances as “Select.” We observed 31 objects with \[\] $\lambda4363$ detected at a strength greater than 4$\sigma$; this comprises our total sample. Our “direct" oxygen abundance measurements have relatively small errors, but comparisons to luminosity and stellar mass calculations require accurate distance determinations. This motivated further cuts from our sample to keep only objects with reliable distance determinations using the tip of the red giant branch (TRGB) or Cepheid variables (ceph), giving rise to our 13 object “Select" sample. In addition, the L06 data were updated with 4.5 $\mu$m photometry from @dale09 (to minimize the effects of aperture differences between the previous photometry and our own), distances from @dalcanton09, and “direct" oxygen abundances from @croxall09 when available. Those objects that passed the selection criteria were assembled into a similar “Select L06" sample of 14 objects. Other Local Volume objects presented in @vanzee06a and @marble10 were considered for an additional “Select" sample. Using the same criteria mentioned above, this provided 11 additional objects with “direct" abundances at a strength of $4\sigma$ or greater and accurate TRGB distances. The 13 “Select" objects from this paper are noted in Table \[tbl5\] and the properties of the additional objects taken from the literature are listed in Table \[tbl6\]. Together these data sets made the final “Combined Select" sample comprised of 38 objects with both secure distance (TRGB or ceph) and oxygen abundance determinations (\[\] $\lambda 4363 > 4\sigma$). Note that we have 18 objects with accurate oxygen abundances that require accurate distances from TRGB observations in order to be elevated to the “Select" caliber. Of these, 13 have distances less than 8 Mpc, so their TRGB distances could be obtained with a relatively small investment of *Hubble Space Telescope* time. Due to the wealth of *B*-band photometry available from previous studies, the majority of the sample has *B*-band absolute magnitude estimates. With the addition of *Spitzer* IRAC photometry, all members of the “Select" sample also have 4.5 $\mu$m absolute magnitudes as determined by @dale09. In the following sections we discuss the low-luminosity portion of both the optical and NIR L-Z relationships and the subsequently determined M$_{\star}$-Z relationship, for our whole sample of “direct" oxygen abundances and a comparison to the filtered “Combined Select" sample. *B*-band L-Z Relationship {#sec:Lb-Z} ------------------------- In the top panel of Figure \[fig:LbZ\] we compare “direct" metallicities to corresponding *B*-band luminosities. Taking into consideration the errors on both quantities [c.f., @press92], we determine the most likely linear fit to the data using the MPFITEXY routine [@williams10], which depends, in turn, on the MPFIT package [@markwardt09]. In this section, and those following, we provide the total scatter (intrinsic $+$ observational) output from the MPFITEXY routine, which is essentially a weighted mean of the scatter of the data about the linear fit. In each case, we compare our results to that of L06, who also use a weighted dispersion routine. The best fit to the 31 objects in the current sample with “direct" oxygen abundance measurements results in: $$12 + \log(\mbox{O/H}) = (6.59\pm0.32) + (-0.08\pm0.03) M{_B},$$ with a dispersion in log(O/H) of $\sigma=0.19$. Updated data for the L06 sample (see § \[sec:Gold\]) is also plotted, and compared to the original least-squares best fit of L06. The low-metallicity outlier at 12 + log(O/H) = 7.20 is the blue compact dwarf UGC 5340, supporting its classification by previous work as one of the most metal-deficient star-forming galaxies [e.g., @izotov07; @pustilnik08b]. However, @pustilnik08b note that its present distance could be significantly *underestimated* due to the large negative peculiar velocity in that region, which, if true, would result in an even larger discrepancy. @ekta08 and @pustilnik08a have discussed the HI observations of UGC 5340 and concluded that it is likely undergoing a merger, which could explain, at least in part, its discrepant position from the L-Z relationship. From HI observations of a sample of extremely metal poor galaxies, @ekta10 find that roughly half of these galaxies show evidence of interactions, and conclude that the very low metallicities in these galaxies are due to recent infall of metal poor gas [see also @jlee2004]). Thus, these galaxies do not lie on the L-Z relationship defined by the average low luminosity galaxy, and therefore, UGC 5340 has not been included in the relationships of the “Combined Select" sample.[^6] In the lower panel of Figure \[fig:LbZ\] we plot the 38 objects in the “Combined Select" sample. The best fit is given by: $$12 + \log(\mbox{O/H}) = (6.27\pm0.21) + (-0.11\pm0.01) M{_B}.$$ with a resultant dispersion in log(O/H) of $\sigma=0.15$[^7]. Note that the luminosity error bars represent the error propagated from the uncertainty in the photometry and distances. This relationship agrees with that of L06 within errors. Additionally, the MPFITEXY routine allows us to estimate the intrinsic scatter by ensuring that $\chi ^2$/(degrees or freedom) $\approx$ 1. Using this tool, the intrinsic scatter in log(O/H) for the *B*-band L-Z relationship for the “Combined Select" sample is 0.13 dex, i.e., most of the scatter in this relationship is intrinsic. 4.5 $\mu$m L-Z Relationship {#sec:L-Z} --------------------------- L06 found their L-Z slope to be smaller in the NIR than in the optical and to contain less scatter. This result might be expected since luminosities in redder bands are less sensitive to dust extinction and star formation rates than optical luminosities. However, these NIR luminosities are also vulnerable to stochastic effects from the high NIR luminosities of AGB stars. Following the motivation given in L06, we analyze the 4.5 $\mu$m L-Z relationship. In the top panel of Figure \[fig:LZ\], we plot the 4.5 $\mu$m L-Z relationship for our low-luminosity LVL sample. Our results are well matched to the luminosity-metallicity relationship for dwarf galaxies found by L06 [and corroborated by @marble10]. Using the MPFITEXY least-squares fit to our data, the resulting expression is: $$12+\log(\mbox{O/H})= (6.37\pm0.33) + (-0.08\pm0.02) M_{[4.5]},$$ with a standard deviation in log(O/H) of $\sigma=0.18$. The original L06 least-squares fit and the updated L06 data are also plotted in Figure \[fig:LZ\], displaying an equivalent slope, but with a notably smaller dispersion in log(O/H) of only 0.12. Note that while the two fits have the same slope, they are offset from one another by roughly 0.1 dex in log(O/H); this difference is within the error and can be attributed to the difference in samples and small sample size. In the bottom panel of Figure \[fig:LZ\] we have plotted the NIR L-Z relationship for the “Combined Select" sample. A least-squares fit results in: $$12+\log(\mbox{O/H})= (6.10\pm0.21) + (-0.10\pm0.01) M_{[4.5]}$$ and produces a standard deviation of $\sigma=0.14$[^8]. This is nearly identical to the standard deviation of $\sigma=0.15$ found for the “Combined Select" sample for the optical L-Z relationship, and the slopes are the same within the uncertainties. The intrinsic scatter in log(O/H) for the 4.5 $\mu$m L-Z relationship for the “Combined Select" sample is 0.11 dex. Since AGB stars can have significant impact on the NIR luminosities, we must consider the effect of stochastic sampling on the overall scatter of our relationship. However, since we find such a small scatter in the NIR L-Z relationship it is unlikely to be due to AGB stars, which would normally drive the data to a larger dispersion. L06 determined dispersions in the optical and NIR L-Z relationships of 0.161 and 0.122 respectively. In comparison, the present work does not find a significant difference between the dispersions of the NIR and optical L-Z relationships. However, the NIR intrinsic scatter in log(O/H) is slightly smaller than the intrinsic scatter for the *B*-band L-Z relationship for the “Combined Select" sample (0.11 versus 0.15 dex). M$_{\star}$-Z Relationship {#sec:M-Z} -------------------------- The underlying relationship between mass and luminosity and the relative ease of measuring luminosities has allowed a widespread use of the L-Z relationship. However, mass is thought to be more fundamentally related to metallicity [see, e.g. @tremonti04], and so, when possible, metallicity is also investigated as a function of stellar mass. In order to examine the M$_{\star}$-Z relationship, we need to estimate stellar masses in a self consistent way. Although SED fitting is commonly used to determine individual masses, the necessary spectral and/or photometric components were not available to us for our entire “Combined Select" sample. Stellar mass can also be inferred from luminosity, where optical colors have been widely used to estimate M/L ratios [e.g., @brinchmann00; @bell01]. It is important to note the uncertainties in M/L ratios that occur due to variations in the current star formation rate, which are most significant if galaxies have formed a substantial fraction ($>$10%) of their stars in a recent episode. Near IR magnitudes are often a better choice to characterize the galaxy luminosity because they are less sensitive than bluer bands to extinction and the age of the stellar population. The dominant emission in NIR wavelengths arises from the stellar populations (as opposed to dust) and is only marginally sensitive to recent star formation, but even so, NIR stellar M/L ratios can vary by up to a factor of $\sim$2 due to the star formation rate and stellar metallicity [@bell01]. Furthermore, @lee06a found that although individual stellar masses can vary by as much as $\sim$0.5 dex with M/L model, the subsequent M-Z relationship spanning four decades in stellar mass is nearly independent of the model chosen. We chose to estimate stellar mass in a uniform manner from 4.5 $\mu$m luminosity and $K-[4.5]$ and $B-K$ color following the method presented by L06: $$\log{\mbox{M}_{\star}}=\log(\mbox{M}_{\star}/L_{K}) + [\log{L_{[4.5]}} - 0.4\ (K-[4.5])]. \label{eq:Mstar}$$ L06 derived a mass-to-light ratio ($\mbox{M}_{\star}/L_{K}$) as a linear function of *B$-$K* color based on the Bruzual & Charlot model with a Salpeter IMF. Note that there is a systematic uncertainty in NIR M/L ratios of $\sim$0.2 dex due to uncertainties in AGB evolution [e.g., @conroy2010; @melbourne12]. Since $K_s$ photometry is available for the LVL sample [@dale09], unlike the procedure of L06, the $B-K$ color was calculated directly (we assume M$_K$ $\equiv$ M$_{K_{s}}$). Based on the direct relationship between the ratio of luminosities and ratio of absolute magnitudes for two objects, we calculated monochromatic luminosities, $L_{[4.5]}$, assuming $M_{[4.5]}\simeq3.3$ for the Sun (following the logic of L06). The M$_{\star}$ results are tabulated in Table \[tbl1\]. In principle, mass estimates can be improved using SED fitting to broad-band photometry which span from the UV to the IR. @johnson12 have determined masses for the LVL galaxies using this method. Unfortunately, the broad wavelength coverage and associated analysis is not available for the entire LVL survey, including objects in our sample. There are 41 LVL galaxies for which we have obtained new spectra or which have spectra in the literature with masses computed by @johnson12 to which we can compare our stellar masses determined from 4.5$\mu$m luminosities. We find an average difference of 0.23 dex in mass, or an offset of a factor of $\sim$2, in the sense that the SED derived masses are smaller and independent of luminosity or optical color. This difference can be accounted for by the use of different IMFs in the modeling (Salpeter IMF in [@bell01] and Chabrier IMF in [@johnson12]). Note that this average difference, as well as the dispersion of $\sigma$=0.24, is smaller than the typical uncertainty in our derived masses. Therefore, adopting these masses would not affect the slope of our derived M-Z relationship. Because we do not have SED derived masses for our entire “Combined Select" sample, we report the present relationship using the masses calculated here. M$_{\star}$-Z data are plotted in the top panel of Figure \[fig:MZ\] in comparison to the updated L06 data and original M$_{\star}$-Z relationship of L06. The best fit to our data, $$12+\log(\mbox{O/H})= (5.43\pm0.42) + (0.30\pm0.05) \log(\mbox{M}_{\star}),$$ with a dispersion of $\sigma=0.21$, agrees, within errors, with the fit to the L06 data set. This dispersion is notably larger than the 0.12 dispersion in log(O/H) found by L06. The mass error bars used here are the propagated errors from the 4.5 $\mu$m luminosity, K-\[4.5\] color, and mass-to-light ratio (where we substituted the uncertainty in B-K color). Note that the contrast in dispersion of the two data sets is largely due to the different errors. L06 assumed the same errors for their mass determinations as their 4.5 $\mu$m luminosities, whereas we incorporated the additional propagated error from the color terms. This difference accounts for the disparity in uncertainty. On the bottom of Figure \[fig:MZ\] we have plotted the “Combined Select" M-Z data. Fitting the combined data set produces the least-squares linear fit, $$12+\log(\mbox{O/H})= (5.61\pm0.24) + (0.29\pm0.03) \log(\mbox{M}_{\star}), \label{eqMZ}$$ with a standard deviation of $\sigma=0.15$[^9], which is essentially equivalent to the dispersions of the “Combined Select" L-Z data sets. The intrinsic scatter in log(O/H) for the M$_{\star}$-Z relationship for the “Combined Select" sample is 0.08 dex. This appears to be significantly smaller than the intrinsic scatter in log(O/H) for the 4.5 $\mu$m L-Z relationship for the “Combined Select" sample of 0.11 dex. The dual effects of increasing the number of objects observed and selecting only objects with both reliable oxygen abundances and distances has resulted in a better characterization of the L-Z and M-Z relationships. In this work, we assume that a galaxy with an region of sufficiently high surface brightness to allow a $\lambda$4363 measurement is a local property of the star forming region, and not related to a characteristic property of the host galaxy. Thus, we don’t believe our sample to be biased in terms of mass or galaxy type. Additionally, the observation that strong-line abundances of low-mass galaxies are consistent with the relationships derived here, albeit with increased scatter, supports this assumption. Therefore, the L-Z and M-Z relationships presented here should accurately represent low-mass galaxies in general. In high mass galaxies, @tremonti04 found a decrease in the dispersion in the L-Z relationships as one went from $\sigma=0.16$ for the optical B-band to $\sigma=0.13$ for the longer wavelength z-band, and then an even smaller dispersion of $\sigma=0.10$ for the M-Z relationship. The “Combined Select" data show a negligibly smaller dispersion for the NIR L-Z relationship compared to the B-band, and no similar decrease in dispersion for the M-Z relationship. N/O Relative Abundances {#sec:N/O} ======================= The N/O versus O/H trend is well studied in galaxies of varying types. @vce93 presented a thorough overview of theoretical expectations and observations available at the time. A salient point is that N can be produced as both a primary and a secondary element and that the secondary component is expected to be delayed relative to oxygen and to dominate at high abundances. A typical scenario might be described by oxygen production in Type II supernovae being released 10 Myr after star formation, whereas nitrogen forming in intermediate mass stars isn’t released until much later times [$> 10^{8}$ Myr; @ks96]. Initially, N/O is expected to rapidly decrease as oxygen is returned to the interstellar medium, but will gradually increase with time as nitrogen begins to be returned to the gas reservoir. Thus, in principle, the relative N/O abundance can be used as a clock [e.g., @henry00] to indicate the time since the most recent burst of star formation. Note that this effect is not expected if the star formation rate does not show significant variations [@molla06]. Table \[tbl5\] lists the error weighted average N/O values for our sample. The N/O errors were determined by first adding in quadrature the error in flux of both \[\] $\lambda3727$ and \[\] $\lambda6584$, then adding this value in quadrature with the error in temperature of the low ionization zone. The most extreme values extend from log(N/O) = $-1.77$ to $-1.00$, with an average of log(N/O) = $-1.47$; this is comparable to the isolated dwarf irregular sample examined by @vanzee06b [hereafter vZ06], with an average log(N/O) = $-1.41$. We tested for a correlation of N/O with reddening and found none, indicating an absence of bias in this regard. The 9 objects with multiple “direct" oxygen abundances provides the opportunity to study N/O variations in individual dwarf galaxies. The average N/O ratio dispersion of different regions in a given galaxy is only 0.08 dex, indicating that dwarf galaxies, despite appearing to be solid body rotators [@skillman88], are well mixed [see also e.g., @roy95]. Other studies, such as the *green pea* galaxies analyzed by @amorin10 and the nitrogen enriched dwarf galaxies analyzed by @perez-montero2011, find N/O abundance dispersions or small gradients hypothesized to be a combination of outflows of enriched gas and inflows of metal-poor gas. Note that errors in N/O account for the dispersion within four of the objects that have multiple N/O measurements (UGC 1056, UGC 4278, NGC 3738, and NGC 4449), but not for 5 others (NGC 784, NGC 2537, UGC 4393, UGC 5423, and UGC 8638). For two of these objects (NGC 784 and UGC 4393) the differences in N/O are significant (0.19 and 0.15). In these last two cases in could be that significant nitrogen enhancement has been detected, although not at the level of the well studied galaxy NGC 5253 [e.g., @ksrwr97; @ls12] or the more recently discovered N/O anomaly in MRK 996 [@james09]. vZ06 looked at several variables for their possible influence on N/O abundance. In particular, they found a correlation between N/O and color, in the sense that redder galaxies have higher N/O as one might expect from time delayed N release. In the top panel of Figure \[fig:NO\], log(N/O) is plotted vs. $B-V$ color for objects of our sample with “direct" abundances and measurable \[\]/\[\] abundances. Similar to vZ06, we find a fairly steep increase in N/O with redder color (demonstrated by the dotted least squares fit): $$\log(\mbox{N/O})= (-1.96\pm0.12) + (1.22\pm0.26)\times(B-V).$$ with a dispersion of $\sigma$ = 0.13. In fact, the two groupings of points are visually consistent with one another. When the additional objects from the literature are added to the plot, the least squares fit over 0.05 $\lesssim B-V \lesssim$ 0.75 to all of the data is $$\log(\mbox{N/O})= (-1.92\pm0.08) + (1.18\pm0.19)\times(B-V),$$ which agrees well with the relationship found by vZ06. Below $B-V$ = 0.10 there are two objects with discrepantly large N/O values. Therefore, we suggest this fit is most appropriate for the range of 0.20 $\lesssim B-V \lesssim$ 0.75. Note the appearance of significant scatter in this figure. We calculate a dispersion in log(N/O) of $\sigma$ = 0.14 dex, with an estimated intrinsic scatter of 0.10 dex. Additionally, the bottom panel of Figure \[fig:NO\] shows log(N/O) plotted vs. 12 + log(O/H) for the same sample. Above 12 + log(O/H) $\approx$ 7.7 a trend of N/O increasing with O/H is evident, despite the large scatter. For 12 + log(O/H) $\ge$ 7.7, the best fit to our data yields: $$\log(\mbox{N/O})= (-5.49\pm1.36) + (0.51\pm0.17)\times[12+\log(\mbox{O/H})],$$ with a dispersion of $\sigma$ = 0.16, where the estimated intrinsic scatter is 0.14 dex. With an increasing slope, this would be indicative of secondary N production in this region. @garnett90 proposed that much of the scatter in the 12 + log(O/H) vs log(N/O) relationship could be explained by the time delay between producing oxygen and secondary nitrogen. For the systems with 12 + log(O/H) $\le$ 7.7, in agreement with previous studies, there is little trend in N/O with O/H. We have calculated a weighted mean in N/O using the IDL routine MPFITEXY with the added constraint of setting the slope to zero for the points below 12 + log(O/H) $=$ 7.7. For our eight new observations, the weighted mean is log(N/O) = $-$1.56 with a standard deviation of 0.05. For the nine observations from the literature, the weighted mean is log(N/O) = $-$1.51 with a standard deviation of 0.04. For the two sets together we obtain log(N/O) = $-$1.56 with a standard deviation of 0.05. Of this dispersion, the intrinsic scatter is predicted to be 0.02, so observational scatter may play a large role in determining the observed scatter in this relationship. In most previous studies, no correlation is noted between 12 + log(O/H) and the relative N/O abundance at low oxygen abundances, where nitrogen is expected to behave like a primary nucleosynthesis element. Together the new observations are consistent with the trends in N/O with O/H observed by @vce93, @jlee2004, @vanzee06a, @molla06, and @liang06. Discussion {#sec:discussion} ========== The L-Z and M-Z Relations for Low Luminosity Galaxies {#sec:low} ----------------------------------------------------- The dual effects of increasing the sample size and selecting only objects with both reliable oxygen abundances and distances has resulted in an improved characterization of the L-Z and M-Z relationships. In high mass galaxies, @tremonti04 found a decrease in the dispersion in the L-Z relationship as one went from the optical B-band ($\sigma=0.16$) to the longer wavelength z-band ($\sigma=0.13$), and an even smaller dispersion for the M-Z relationship ($\sigma=0.10$). The present data show only a slightly smaller dispersion for the NIR L-Z relationship ($\sigma=0.14$) compared to the B-band ($\sigma=0.15$), but no similar decrease in dispersion for the M-Z relationship ($\sigma=0.15$). However, our estimates of the *intrinsic* scatter in the three relationships do show a decreasing trend in the sense that the *intrinsic* scatter of the B-band L-Z relationship is largest ($\sigma=0.13$), followed by the NIR L-Z relationship ($\sigma=0.12$), then the M-Z relationship ($\sigma=0.08$). While this trend could be an artifact of how the errors are estimated for the three different parameters, it is interesting that it follows the same pattern observed in the larger spiral galaxies. Perhaps what is most remarkable is the small *intrinsic* scatter in all three relationships. When averaging the light over an entire galaxy, as done in @tremonti04 one might expect relatively low dispersions. However, oxygen abundances derived from spectroscopic apertures only covering a fraction of the galaxy will be biased if radial gradients exist [e.g., @moustakas12]. Therefore, one might expect much larger dispersions when observing individual regions, yet this is not the case observed in most dwarf galaxies, as they have been shown to be relatively chemically homogeneous [e.g., @croxall09]. The L-Z and M-Z relationship slopes determined for the “Combined Select" sample are similar to those found in previous studies [e.g., @tremonti04; @lee06a]. For large galaxies, a different slope may apply as galaxies higher in mass and luminosity contain more metals and dust [e.g., @rosenberg06] causing them to appear under-luminous. For smaller, less luminous galaxies, even with the present sample included, the number of galaxies meeting our “Select" criteria is still relatively small. This limitation could affect our measurements of the scatter, but it appears that these relationships have intrinsically smaller dispersions. The evolutionary paths of dwarfs are still poorly understood, making the source of this inherent variation unclear. Some studies argue for the importance of gas infall and outflows [e.g., @garnett02], whereas others point to star formation efficiencies [e.g., @lequeux79; @brooks07], and variations in initial mass functions [e.g., @koppen07]. Still other studies have also seen significant scatter at low stellar masses [see for example @tremonti04; @amorin10]. [@amorin10] suggest that inherent variation in the L-Z and M-Z relations could result from these objects being relatively young and thus may still be converting large amounts of cold gas into stars. If these young galaxies have not had enough time for several generations of star formation to produce massive AGB stars, then we would expect very little absorption due to dust. The relative uniformity between the dispersions of the L-Z and M-Z relationships and between the slopes of the optical and near-IR L-Z relationships is consistent with this idea, suggesting no more absorption in the optical than in the near-IR, and thus very little dust is present in these low-luminosity galaxies. The fact that the scatter in the L-Z and M-Z relationships is small suggests that AGB stars do not play as significant of a role in determining the scatter in the NIR L-Z relationship for low-mass galaxies. In fact, in our sample it seems that AGB stars are balanced out by the effects of star formation histories. Whatever the actual source of the scatter may be, since we used the most reliable oxygen abundances and distance estimates possible in constructing the L-Z and M-Z relationships, it appears that the dispersion for this sample is real as it is larger than observational errors. However, the “young galaxy” hypothesis faces other observational challenges. N/O and the Young Galaxy Hypothesis {#sec:young} ----------------------------------- @garnett90 first showed that the N/O ratio in low metallicity star forming galaxies is relatively constant as a function of O/H (with a mean value of log(N/O) = $-$1.46$^{+0.10}_{-0.13}$) for these “plateau” objects. Later, @izotov99 drew attention to the plateau with small dispersion in log (N/O) ($-$1.60 $\pm$ 0.02) in extremely metal-poor (12 + log(O/H) $\le$ 7.6) blue compact dwarf galaxies. They proposed that the absence of time-delayed production of N (and C) is consistent with the scenario that extremely metal-poor galaxies are now undergoing their first burst of star formation, and that they are therefore young, with ages not exceeding 40 Myr. They further argued that if this were true, then this would argue against the commonly held belief that C and N are produced by intermediate-mass stars at very low metallicities (as these stars would not have yet completed their evolution in these lowest metallicity galaxies). [@nava06] revisited the observed N/O plateau with a large set of objects and determined a mean value for the N/O plateau of $-$1.43 with a standard deviation of $^{+0.071}_{-0.084}$. They further concluded from a $\chi^2$ analysis that only a small fraction of the observed scatter in N/O is intrinsic. From the bottom panel of Figure \[fig:NO\], we see that the sample assembled here also shows a plateau in N/O of log(N/O) = $-$1.56 $\pm$ 0.05. The level of the plateau in our data is slightly lover than found by @nava06, but agrees fairly well with that found by @izotov99. While the observed dispersion is larger than that found for the blue compact dwarfs by @izotov99, the *intrinsic* dispersion agrees well for the two samples. Clearly the relatively constant N/O value is a common characteristic of dwarf star forming galaxies, and not just those undergoing a current burst of star formation. @vanzee06b demonstrated that Leo A, with 12 + log(O/H) = 7.38 $\pm$ 0.10 and log(N/O) = $-$1.53 $\pm$ 0.09, and GR 8, with 12 + log(O/H) = 7.65 $\pm$ 0.06 and log(N/O) = $-$1.51 $\pm$ 0.07, which are *not* blue compact dwarf galaxies, are consistent with this plateau in log(N/O) at low values of O/H. However, both Leo A and GR 8 have detailed star formation histories derived from *Hubble Space Telescope* observations of their resolved stars which clearly show that the bulk of their star formation occurred well before the last 40 Myr [@tolstoy98; @cole07; @dohmpalmer98; @weisz11]. In fact, @weisz11 show, from a nearly volume limited sample, that the majority of dwarf galaxies formed the bulk of their stellar mass prior to z $\sim$ 1, regardless of current morphological type. Since the low mass, metal-poor galaxies in the present sample and works cited appear to have nearly the same value of N/O, regardless of whether they have a current burst of star formation, it would seem that the young galaxy hypothesis is not a valid explanation for the plateau in N/O at low metallicity. If the plateau in N/O is not due to young galaxy ages, what is its cause? Clearly nitrogen is behaving as a primary element at low metallicities. @henry06 considered various scenarios and concluded that a wide range were consistent with the observations. At this point, a definitive explanation for the N/O plateau appears elusive. Best Estimate of Abundances {#sec:best} --------------------------- Determining an accurate and reliable oxygen abundance for an individual region depends on measuring the combination of bright nebular and faint auroral emission lines (the “direct" method). Many studies have emphasized that a “direct” abundance is not without systematic uncertainties. Specifically, due to the high temperature sensitivity of the “direct” method, inhomogeneous temperature distributions will lead to abundance underestimates. The uncertainty in the absolute oxygen abundance determination by this method is $\sim$ 0.1 dex, but the error in relative metallicities is likely to be $<<$ 0.1 dex [@kewley08]. However, @bresolin2007 warns that T$_{e}$-based determinations only provide a lower limit if the temperature fluctuations are substantial. In the absence of a temperature-sensitive auroral line detection, a mix of strong emission lines are used as a proxy for metallicity (strong-line methods: empirical, semi-empirical, and theoretical calibrations). Strong-line calibrations are limited by sample selection effects, potentially making them appropriate for ranking objects on a single scale, but not useful for determining an absolute metallicity as the various methods do not converge [see e.g., @yin07; @kewley08; @bresolin09; @berg11]. If a strong-line method must be used, @stasinska10 recommends only using a strong-line method for nebulae having [*the same properties as those of the calibration sample*]{}. @oey00 [@vanzee06a; @vanzee06b; @yin07; @kewley08; @perez-montero09; @amorin10; @moustakas10], and others have investigated several strong-line calibrations including the O3N2 method, the N2 method, and the R$_{23}$ index, finding inconsistencies between methods that were largely related to variations in the hardness of the ionizing radiation field, nitrogen abundance, and/or age of the stellar cluster. There are several strong-line methods to chose from, but when compared they all have similar uncertainties of 0.1-0.2 dex and discrepancies between them as large as 0.6 dex [e.g., @liang06; @bresolin2007; @yin07; @kewley08]. Improvements have been made in strong-line calibrations by the introduction of photoionization models to simultaneously fit the most prominent emission lines [e.g., @tremonti04; @brinchmann04]. However, @yin07 found the MPA/JHU simultaneous line fitting SDSS abundances determined from the Charlot et al. (2006) photoionization models overestimate oxygen abundances by $\sim$0.34 dex compared to direct abundances. They postulate the difference to be due to the models treatment of the onset of secondary nitrogen production, and thus could be eliminated with improved modeling. One possible exception is the ONS calibration of @pilyugin10, for which they find deviations from T$_e$-based oxygen abundances of just $\sim$0.075 dex. Here we investigate a subset of strong line abundances for our objects with “direct" abundances. Following the methodology of @berg11, we calculated oxygen abundances from their strong lines for the 31 objects with “direct" abundances listed in Table \[tbl5\]. We determined abundances using the R$_{23}$ calibration of @mcgaugh91, the ONS calibration of @pilyugin10, and the N2 and O3N2 calibrations updated by @perez-montero09 [hereafter PMC09]. The R$_{23}$ calibration of @mcgaugh91 produces a bi-valued solution, so to discriminate between the two branches @mcgaugh94, @vanzee98, and others advised using the ratio of I(\[\] $\lambda6584$)/I(\[\] $\lambda3727$). @mcgaugh94 suggested that \[\]/\[\] is approximately $< 0.1$ for low abundances and $> 0.1$ for high abundances, giving a rough distinction between lower and upper branches. Using this distinction, we selected the appropriate branch calibration for each object. Note that for metal poor objects with enhanced nitrogen, \[\]/\[\] becomes a biased discriminator [e.g., @yin07; @berg11; @perez-montero2011]. In a similar fashion, the ONS method of @pilyugin10 requires two discriminators, \[\] and \[\]/\[\], to distinguish between three classes of regions. We followed @berg11 and assumed T$_{\rm e}=1.25\times10^4$ K to examine N/O ratios and calculate abundances with the N2 and O3N2 calibrations of PMC09. This correction may be important for NGC 2537 and UGC 4393, which appear to have somewhat discrepant nitrogen abundances (nitrogen enrichment for log(N/O) $>$ -1.0). The other objects in this sample have average N/O ratios for their masses [see e.g., @berg11]. The results are tabulated in Table \[tbl8\]. The mean offsets and dispersions relative to the direct abundances are calculated and given at the bottom of Table \[tbl8\]. Table \[tbl8\] shows that all four methods have significant dispersions, with the ONS method showing the smallest dispersion (although larger than anticipated) and the O3N2 method having the largest. The ONS method also has the smallest mean offset. Figure 7 presents a plot of differences between the R$_{23}$ and ONS method abundances and the direct abundances as a function of abundance. This illustrates the results of Table \[tbl8\], that the ONS method has a smaller dispersion and a smaller mean offset from the direct method. Thus, our data favor the ONS method, but do not support the claim of the very small error as found by @pilyugin10. In Figure 7 we find no clear trend exists between the “direct" method and the strong-line methods, implying that simple calibrations between methods are not possible. With the relatively precise M-Z and L-Z relationships in place, and their correspondingly low dispersions, oxygen abundances for normal (non-starburst) low luminosity galaxies can be inferred with relatively high confidence without a spectrum. In fact, given reliable distance and photometry measurements, the resulting luminosity and mass estimates can be used as more reliable predictors of oxygen abundance than some strong-line calibrations. As counter-intuitive as this idea may seem, it is a natural consequence of the inability of some strong-line methods to accurately predict the metallicity of individual regions. Studies of abundances in dwarfs which do not reproduce the L-Z and M-Z relationships, therefore, should raise suspicions concerning methodology. CONCLUSIONS {#sec:conclusion} =========== We have determined uniform oxygen abundance metallicities for 31 low luminosity galaxies in the *Spitzer* LVL survey. With high-resolution spectral observations taken at the MMT, we were able to measure the intrinsically faint \[\] $\lambda$4363 fluxes at strengths of 4$\sigma$ or greater and explicitly determine electron temperatures. Metallicity measurements are important for characterizing many other properties, especially when the more reliable “direct" method is used. However, metallicity relationships tend to suffer from small number statistics in the low luminosity regime. In particular, these measurements allowed us to better characterize the luminosity-metallicity and mass-metallicity relationships by doubling the number of reliable low-luminosity measurements. We created a “Combined Select" sample of objects that have both reliable “direct" oxygen abundance determinations and distances estimated from the tip of the red giant branch or Cepheid variables. With this sample, we find that both the luminosity-metallicity and the mass-metallicity relationships agree well with previous relationships defined for low luminosities. From the 38 objects making up the “Combined Select" sample, we found an optical L-Z relationship of $12+\log(\mbox{O/H})=(6.27\pm0.21)+(-0.11\pm0.01)M_{B}$, with a dispersion of $\sigma=0.15$. In comparison, the near-IR L-Z relationship for this data is $12+\log(\mbox{O/H})=(6.10\pm0.21)+(-0.10\pm0.01)M_{[4.5]}$, with a dispersion of $\sigma=0.14$. While the slopes of the two L-Z relationships agree, our findings confirm the work of L06 in that the near-IR relationship has lower scatter. By converting NIR luminosity to a stellar mass estimate, we determined the M-Z relationship for our data to be $12+\log(\mbox{O/H})=(5.61\pm0.24)+(0.29\pm0.03)M_{\star}$, with a dispersion of $\sigma=0.15$. In agreement with the idea that mass is more fundamentally related to metallicity than luminosity, we find that the intrinsic scatter of the optical L-Z, NIR L-Z, and M-Z relationships decreases from 0.13 to 0.12 to 0.08. However, the total dispersion of the M-Z relationship was measured to be no smaller than the L-Z relationships. This suggests, given a reliable distance measurement and appropriate photometry, luminosity is just as strong of a metallicity indicator as stellar mass. Furthermore, with the dispersions in luminosity and mass roughly equal, either may be used in combination with a reliable distance determination to estimate metallicity of a low luminosity dwarf with more confidence than when using strong-line calibrations. Our observations of N/O abundances are in agreement with previous studies. We find a positive correlation between N/O ratio and B-V color for 0.05 $\lesssim B-V \lesssim$ 0.75; $\log(\mbox{N/O}) = (-1.92\pm0.08)+(1.18\pm0.19)\times(B-V)$, with a dispersion of $\sigma$ = 0.14. Furthermore, in agreement with observations of blue compact galaxies, there are no objects with high N/O ratio (log(N/O) $>$ -1.4) below 12+log(O/H)=7.7. Since the typical low luminosity galaxy in the Local Volume displays roughly constant star formation over the age of the universe, the small dispersion in N/O at low values of O/H cannot be due to the very recent birth of the galaxy. DAB is grateful for support from a Penrose Fellowship and a NASA Space Grant Fellowship from the University of Minnesota. EDS is grateful for partial support from the University of Minnesota. We are grateful to the referee for a thorough analysis of this paper that greatly improved the analysis and presentation of this work. Special thanks to John Moustakas and L. Andrew Helton for many scientifically stimulating and helpful discussions. Observations reported here were obtained at the MMT Observatory, a joint facility of the Smithsonian Institution and the University of Arizona. MMT observations were obtained as part of the University of Minnesota’s guaranteed time on Steward Observatory facilities through membership in the Research Corporation and its support for the Large Binocular Telescope, and granted by NOAO, through the Telescope System Instrumentation Program (TSIP). TSIP is funded by the National Science Foundation. This research has made use of NASA’s Astrophysics Data System Bibliographic Services and the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This work was initiated as part of the Spitzer Space Telescope Legacy Science Program and was supported by National Aeronautics and Space Administration (NASA) through contract 1336000 issued by the Jet Propulsion Laboratory (JPL), California Institute of Technology (Caltech) under NASA contract 1407. Strong-Line Abundances for Galaxies Lacking Direct Abundances {#sec:strong} ============================================================= In Table \[tbl9\] we present strong-line abundances for the 12 objects in our sample without \[\] $\lambda4363$ detections. While these may not be as accurate as the direct abundances for the rest of our sample, they may be useful for studies of these individual galaxies. The O/H values derived using the ONS method for both these 12 objects (Table \[tbl9\]) and the objects with “direct abundances" (Table \[tbl8\]) are plotted in Figure \[fig:strong\] where they are compared to our “direct" abundances. The two methods display coincident trends in metallicity with mass, yet the O/H abundances derived via the ONS calibration have a larger dispersion. We have not conducted a statistical comparison, as not all galaxies have accurate distances, and the subset with accurate distance is quite small. 70/160 $\mu$m Color Temperature-Metallicity Outliers {#sec:outlier} ==================================================== As noted in § \[sec:Sample\], two objects were of particular interest to this study (UGC 10818 and UGC 4393) because they appear to be outliers from the global trend of 70/160 $\mu$m color temperature as a function of metallicity as determined by @engelbracht08. Specifically, based on *Spitzer* observations of 66 starburst galaxies, they showed that the far-infrared color temperature of large dust grains increases toward lower metallicity down to 12 + log(O/H) $\sim$ 8. However, the oxygen abundances found by @engelbracht08 for these two objects were based on the R$_{23}$ strong-line estimator. Our new spectroscopic results indicate that both UGC 4393 and UGC 10818 (SHOC 567) are near the transition region between the upper and lower branches based on their \[\]/\[\] ratios, and thus the R$_{23}$ method may not yield an accurate abundance for these systems. While our observations of UGC 10818 are still ambiguous due to the degeneracy in the strong-line metallicity calibrations, we derive an oxygen abundance of 12 + log(O/H) = 7.82 based on the @mcgaugh91 R$_{23}$ calibration. This increases the oxygen abundance of UGC 10818 by 0.51 dex compared to previous measurements and moves UGC 10818 (SHOC 567) closer to the original trend illustrated in @engelbracht08. Conversely, the “direct" oxygen abundance of UGC 4393 was determined in this paper to be 12 + log(O/H) = 8.02 +/- 0.05, in agreement with the strong-line estimate presented in @engelbracht08. Thus, at first glance, these new observations appear to only impact the location of one of the two most extreme outliers in the original plot. Perhaps more importantly, we have reproduced the 70/160 $\mu$m color temperature versus 12 + log(O/H) plot of @engelbracht08 with the addition of “direct" abundance objects from this work in Figure \[fig:temp\]. Note that the star-bursting objects from [@engelbracht08] tend to have higher dust temperatures than the low intensity objects studied in this paper. This may mean that the trend of increasing far-infrared dust temperature with decreasing metallicity was just a slice of a larger picture, where the selected samples were limited by star formation rates, which biased the view to a more narrow window. 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J. 1997b, , 114, 2479 , L., [Salzer]{}, J. J., [Haynes]{}, M. P., [O’Donoghue]{}, A. A., & [Balonek]{}, T. J. 1998, , 116, 2805 van Zee, L., & Haynes, M. P. 2006a, , 636, 214 van Zee, L., Skillman, E. D., & Haynes, M. P. 2006b, , 637, 269 van Zee, L., et al., 2012, in preparation V[á]{}zquez, G. A., & Leitherer, C. 2005, , 621, 695 Vila Costas, M. B., & Edmunds, M. G. 1993, , 265, 199 Weisz, D. R., Dalcanton, J. J., Williams, B. F., et al. 2011, , 739, 5 Williams, M. J., Bureau, M., & Cappellari, M. 2010, , 409, 1330 Yin, S. Y., Liang, Y. C., Hammer, F., Brinchmann, J., Zhang, B., Deng, L. C., & Flores, H. 2007, , 462, 535 York, D. G., et al. 2000, , 120, 1579 Zaritsky, D., Kennicutt, Jr., R. C., & Huchra, J. P. 1994, , 420, 87 [ccccccccccccc]{} UGC 521 & 00:51:12.1 & 12:01:26 & 2.24 & 5.43$\pm$2.88 & 10.9 & 13, v(flow) & -15.16$\pm$0.50 & -17.93$\pm$0.61 & -17.46$\pm$0.52 & 0.34$\pm$0.05 & 8.50$\pm$0.61 & 7.96$\pm$0.61\ UGC 695 & 01:07:46.4 & 01:03:52 & 3.02 & $< 8.45$ & 10.2 & 6, v(flow) & -15.13$\pm$0.50 & -18.11$\pm$0.61 & -17.80$\pm$0.52 & 0.45$\pm$0.06 & 8.57$\pm$0.61 & 8.08$\pm$0.61\ NGC 404 & 01:09:27.0 & 35:43:05 & 239 & 676$\pm$340 & 3.05$\pm$0.04 & 4, trgb & -16.39$\pm$0.07 & -20.23$\pm$0.35 & -19.94$\pm$0.14 & 0.83 & 9.42$\pm$0.35 & 9.20$\pm$0.35\ UGC 1056 & 01:28:47.6 & 16:41:21 & 4.14 & 29.1$\pm$4.0 & 10.32 & 6, v(flow) & -15.09$\pm$0.52 & -18.47$\pm$0.61 & -19.17$\pm$0.50 & 0.57$\pm$0.07 & 8.72$\pm$0.61 & 8.62$\pm$0.61\ UGC 1176 & 01:40:09.9 & 15:54:20 & 6.22 & $< 26.6$ & 9.04$\pm$1.66 & 9, bs & -15.48$\pm$0.93 & -18.63$\pm$0.98 & -18.78$\pm$0.92 & 0.31$\pm$0.10 & 8.78$\pm$0.98 & 8.48$\pm$0.98\ NGC 784 & 02:01:16.9 & 28:50:09 & 35.0 & 72.2$\pm$9.9 & 5.19$\pm$0.12 & 12, trgb & -16.50$\pm$0.12 & -19.30$\pm$0.36 & -18.66$\pm$0.20 & 0.40$\pm$0.04 & 9.05$\pm$0.36 & 8.48$\pm$0.36\ UGC 2716 & 03:24:08.1 & 17:45:15 & 7.58 & 22.2$\pm$4.7 & 6.2 & 6, v(flow) & -15.31$\pm$0.50 & -18.04$\pm$0.61 & -17.78$\pm$0.51 & 0.31$\pm$0.06 & 8.55$\pm$0.61 & 8.13$\pm$0.61\ KKH 37 & 06:47:45.4 & 80:07:26 & 1.58 & 6.43$\pm$2.81 & 3.39$\pm$0.12 & 4, trgb & -11.98$\pm$0.20 & -15.01$\pm$0.39 & -15.11$\pm$0.20 & 0.54$\pm$0.06 & 7.34$\pm$0.39 & 7.01$\pm$0.39\ NGC 2537 & 08:13:14.6 & 45.59:30 & 51.4 & 160$\pm$10 & 6.88 & 10, bs & -17.14$\pm$0.50 & -20.34$\pm$0.61 & -20.36$\pm$0.51 & 0.72 & 9.47$\pm$0.61 & 9.10$\pm$0.61\ UGC 4278 & 08:13:58.9 & 45:44:37 & 15.5 & 35.8$\pm$5.1 & 7.6 & 6, v(flow) & -16.36$\pm$0.50 & -19.24$\pm$0.60 & -18.73$\pm$0.52 & 0.35 & 9.03$\pm$0.60 & 8.50$\pm$0.60\ NGC 2552 & 08:19:19.2 & 50:00:37 & 22.7 & 54.9$\pm$8.1 & 7.7 & 6, v(flow) & -16.72$\pm$0.50 & -19.67$\pm$0.61 & -19.21$\pm$0.52 & 0.43$\pm$0.04 & 9.20$\pm$0.61 & 8.69$\pm$0.61\ UGC 4393 & 08:26:04.4 & 45:58:04 & 7.24 & & 16.8$\pm$2.9 & 15, TF & -17.67$\pm$0.85 & -21.65$\pm$0.86 & & 0.60 & 9.99$\pm$0.86 & 9.43$\pm$0.86\ CGCG 35-007 & 09:34:44.9 & 06:25:32 & 2.88 & 13.5$\pm$3.4 & 5.2 & 6, v(flow) & -13.38$\pm$0.51 & -16.58$\pm$0.61 & -16.83$\pm$0.51 & 0.54$\pm$0.10 & 7.97$\pm$0.61 & 7.69$\pm$0.61\ UGC 5139 & 09:40:30.0 & 71:11:05 & 5.98 & 16.0$\pm$7.3 & 3.90$\pm$0.05 & 4, trgb & -14.42$\pm$0.12 & -16.76$\pm$0.51 & -16.41$\pm$0.20 & 0.36$\pm$0.07 & 8.04$\pm$0.51 & 7.39$\pm$0.51\ IC 559 & 09:44:43.8 & 09:36:54 & 5.55 & 23.7$\pm$4.1 & 4.9 & 6, v(flow) & -14.12$\pm$0.50 & -17.19$\pm$0.61 & -17.34$\pm$0.51 & 0.48 & 8.21$\pm$0.61 & 7.86$\pm$0.61\ UGC 5272 & 09:50:22.4 & 31:29:16 & 5.12 & 15.8$\pm$4.2 & 7.11$\pm$0.77 & 8, bs & -14.98$\pm$0.55 & -17.90$\pm$0.64 & -17.70$\pm$0.55 & 0.37 & 8.49$\pm$0.64 & 8.00$\pm$0.64\ UGC 5340 & 09:56:45.8 & 28:49:32 & 1.94 & $<9.36$ & 12.1$\pm$0.7 & 16, trgb & -15.83$\pm$0.55 & -17.99$\pm$0.44 & -18.28$\pm$0.29 & 0.13$\pm$0.08 & 8.53$\pm$0.44 & 7.97$\pm$0.45\ UGC 5423 & 10:05:30.6 & 70:21:52 & 3.63 & 13.6$\pm$2.7 & 5.27$\pm$0.40 & 10, bs & -13.77$\pm$0.38 & -16.88$\pm$0.85 & -16.89$\pm$0.53 & 0.48$\pm$0.04 & 8.09$\pm$0.85 & 7.77$\pm$0.85\ UGC 5672 & 10:28:20.8 & 22:34:16 & 7.71 & 25.3$\pm$5.3 & 6.25 & 10, bs & -14.73$\pm$0.52 & -18.06$\pm$0.61 & -17.93$\pm$0.51 & 0.64$\pm$0.05 & 8.56$\pm$0.61 & 8.38$\pm$0.61\ UGC 5692 & 10:30:36.6 & 70:37:03 & 17.7 & 66.4$\pm$8.4 & 3.80$\pm$0.05 & 4, trgb & -14.68$\pm$0.08 & -17.88$\pm$0.35 & -17.89$\pm$0.11 & 0.68$\pm$0.04 & 8.48$\pm$0.35 & 8.16$\pm$0.35\ UGC 5797 & 10:39:25.2 & 01:43:05 & 4.09 & 11.7$\pm$4.2 & 6.8 & 6, v(flow) & -14.56$\pm$0.51 & -17.57$\pm$0.61 & -17.29$\pm$0.51 & 0.46$\pm$0.08 & 8.36$\pm$0.61 & 7.75$\pm$0.61\ UGC 5923 & 10:49:07.6 & 06:55:03 & 6.41 & 21.8$\pm$2.5 & 7.2 & 6, v(flow) & -14.70$\pm$0.50 & -18.16$\pm$0.61 & -18.06$\pm$0.51 & 0.66$\pm$0.02 & 8.59$\pm$0.61 & 8.29$\pm$0.61\ NGC 3738 & 11:35:48.6 & 54:31:29 & 39.9 & 104$\pm$8 & 4.9$\pm$0.6 & 3, trgb & -16.51$\pm$0.61 & -19.32$\pm$0.70 & -18.93$\pm$0.63 & 0.39 & 9.06$\pm$0.70 & 8.50$\pm$0.70\ NGC 3741 & 11:36:05.8 & 45:17:11 & 3.24 & 11.2$\pm$4.5 & 3.24$\pm$0.13 & 4, trgb & -13.18$\pm$0.22 & -15.69$\pm$0.40 & -15.62$\pm$0.22 & 0.31$\pm$0.07 & 7.61$\pm$0.40 & 7.05$\pm$0.40\ UGC 6782 & 11:48:57.0 & 23:50:17 & 2.30 & $< 8.88$ & 13.7 & 7, bs & -15.54$\pm$0.51 & -18.45$\pm$0.61 & -18.49$\pm$0.51 & 0.53 & 8.71$\pm$0.61 & 8.29$\pm$0.61\ UGC 6817 & 11:50:54.1 & 38:52:51 & 4.99 & 21.5$\pm$6.0 & 2.59$\pm$0.17 & 4, trgb & -13.70$\pm$0.34 & -15.68$\pm$0.48 & -15.84$\pm$0.34 & 0.30 & 7.60$\pm$0.48 & 6.97$\pm$0.48\ UGC 6900 & 11:55:39.4 & 31:31:10 & 5.41 & 18.1$\pm$5.2 & 7.5 & 6, v(flow) & -14.62$\pm$0.53 & -18.06$\pm$0.61 & -17.95$\pm$0.51 & 0.64 & 8.56$\pm$0.61 & 8.19$\pm$0.61\ NGC 4163 & 12:12:09.2 & 36:10:10 & 11.5 & 32.6$\pm$5.8 & 2.88$\pm$0.04 & 4, trgb & -13.65$\pm$0.12 & -16.81$\pm$0.35 & -16.52$\pm$0.14 & 0.44 & 8.06$\pm$0.35 & 7.61$\pm$0.35\ CGCG 269-049 & 12:15:47.2 & 52:23:17 & 1.24 & 3.31$\pm$1.99 & 1.60$\pm$0.04 & 4, trgb & -10.83$\pm$0.14 & -13.12$\pm$0.36 & -12.76$\pm$0.18 & 0.28$\pm$0.06 & 6.58$\pm$0.36 & 5.90$\pm$0.36\ UGC 7577 & 12:27:40.9 & 43:29:44 & 14.2 & 41.9$\pm$6.9 & 2.58$\pm$0.07 & 4, trgb & -14.12$\pm$0.14 & -16.80$\pm$0.36 & -16.55$\pm$0.18 & 0.48 & 8.05$\pm$0.36 & 7.50$\pm$0.36\ NGC 4449 & 12:28:10.1 & 44:05:31 & 315 & 893$\pm$46 & 3.82$\pm$0.26 & 14, trgb & -18.02$\pm$0.34 & -21.02$\pm$0.48 & -20.73$\pm$0.36 & 0.37$\pm$0.04 & 9.74$\pm$0.48 & 9.25$\pm$0.48\ UGC 7599 & 12:28:28.5 & 37:14:01 & 1.43 & 4.01$\pm$2.06 & 6.9 & 7, bs & -14.35$\pm$0.51 & -16.45$\pm$0.61 & -16.14$\pm$0.52 & 0.40 & 7.91$\pm$0.61 & 7.19$\pm$0.61\ UGC 7605 & 12:28:38.5 & 35:42.58 & 2.41 & 6.09$\pm$2.45 & 4.43$\pm$0.57 & 3, trgb & -13.49$\pm$0.66 & -16.05$\pm$0.73 & -15.63$\pm$0.66 & 0.29$\pm$0.04 & 7.75$\pm$0.73 & 7.12$\pm$0.73\ UGC 7639 & 12:29:53.4 & 47:31:52 & 7.68 & 26.4$\pm$5.5 & 7.1$\pm$0.5 & 11, sbf & -15.55$\pm$0.37 & -18.33$\pm$0.49 & -18.25$\pm$0.37 & & 8.67$\pm$0.49 & 8.25$\pm$0.49\ NGC 4656 & 12:43:57.7 & 32:10:05 & 70.5 & 135$\pm$14 & 8.6 & 6, v(flow) & -18.75$\pm$0.51 & -21.15$\pm$0.61 & -20.44$\pm$0.53 & 0.42 & 9.79$\pm$0.61 & 9.04$\pm$0.61\ UGC 8201 & 13:06:24.5 & 67:42:28 & 9.09 & 37.3$\pm$6.3 & 4.57$\pm$0.40 & 3, trgb & -15.17$\pm$0.44 & -17.56$\pm$0.60 & -17.67$\pm$0.45 & 0.24$\pm$0.04 & 8.36$\pm$0.60 & 7.82$\pm$0.60\ UGC 8245 & 13:08:35.2 & 78:56:14 & 5.98 & 20.0$\pm$4.3 & 3.64 & 6, v(flow) & -13.67$\pm$0.50 & -16.61$\pm$0.61 & -16.50$\pm$0.51 & 0.47$\pm$0.04 & 7.98$\pm$0.61 & 7.53$\pm$0.61\ UGC 8508 & 13:30:44.1 & 54:54:40 & 4.87 & 13.9$\pm$4.2 & 2.58$\pm$0.03 & 4, trgb & -13.03$\pm$0.07 & -15.64$\pm$0.35 & -15.36$\pm$0.13 & 0.37$\pm$0.03 & 7.59$\pm$0.35 & 7.00$\pm$0.35\ UGC 8638 & 13:39:19.2 & 24:46:36 & 5.01 & 15.2$\pm$4.6 & 4.27$\pm$0.34 & 5, trgb & -13.77$\pm$0.40 & -16.77$\pm$0.53 & -16.55$\pm$0.41 & 0.47$\pm$0.04 & 8.04$\pm$0.53 & 7.57$\pm$0.53\ UGC 8837 & 13:54:45.7 & 53:54:03 & 10.1 & 26.5$\pm$7.1 & 8.3 & 2, bs & -15.92$\pm$0.51 & -18.97$\pm$0.61 & -18.59$\pm$0.52 & 0.42 & 8.92$\pm$0.61 & 8.41$\pm$0.61\ NGC 5477 & 14:05:33.1 & 54:27:39 & 5.17 & 22.0$\pm$4.5 & 7.7 & 2, bs & -15.22$\pm$0.51 & -18.08$\pm$0.61 & -18.23$\pm$0.51 & 0.34 & 8.56$\pm$0.61 & 8.15$\pm$0.61\ UGC 9405 & 14:35:24.4 & 57:15:19 & 3.97 & 12.9$\pm$5.0 & 8.02$\pm$0.74 & 8, bs & -14.97$\pm$0.47 & -17.88$\pm$0.58 & -17.74$\pm$0.47 & 0.68 & 8.48$\pm$0.58 & 7.97$\pm$0.58\ UGC 10818 & 17:19:42 & 61:18:47 & 4.65 & & 56 & 1, h(flow) & -18.59$\pm$0.50 & -20.96$\pm$0.51 & & & 9.72$\pm$0.51 & 9.45$\pm$0.51\ KKH 98 & 23:45:34.3 & 38:43:00 & 1.56 & 7.56$\pm$3.09 & 2.45$\pm$0.04 & 4, trgb & -11.10$\pm$0.16 & -14.29$\pm$0.36 & -14.58$\pm$0.11 & 0.20$\pm$0.13 & 7.05$\pm$0.36 & 6.72$\pm$0.36\ \[tbl1\] [cccccc]{} UGC 521-A & 00:51:11.9 & 12:01:34 & -55.71 & Nov08 & 3 $\times$ 900\ UGC 521-B & 00:51:12.1 & 12:01:31 & -55.71 & Nov08 & 2 $\times$ 900\ UGC 695-E & 01:07:46.5 & 01:03:53 & 29.44 & Jan10 & 4 $\times$ 900\ NGC 0404-A & 01:09:26.0 & 35:43:00 & -76.70 & Jan10 & 3 $\times$ 600\ UGC 1056-A & 01:28:47.3 & 16:41:16 & 45.19 & Jan10 & 3 $\times$ 900\ UGC 1056-B & 01:28:47.5 & 16:41:21 & 45.19 & Jan10 & 4 $\times$ 900\ UGC 1176-A & 01:40:11.9 & 15:54:46 & 42.14 & Jan10 & 4 $\times$ 900\ UGC 784-B & 02:01:16.5 & 28:50:06 & 52.76 & Jan10 & 4 $\times$ 900\ UGC 784-A & 02:01:17.5 & 28:50:16 & 52.76 & Jan10 & 4 $\times$ 1200\ UGC 2716-A & 03:24:07.2 & 17:45:11 & 63.24 & Jan10 & 3 $\times$ 900\ KKH 037-A & 06:47:43.1 & 80:07:27 & -176.05 & Jan10 & 1 $\times$ 1800\ NGC 2537-A & 08:13:13.0 & 45:59:39 & -94.27 & Jan10 & 3 $\times$ 900\ NGC 2537-B & 08:13:13.3 & 45:59:39 & -94.27 & Jan10 & 3 $\times$ 900\ UGC 4278-B & 08:14:00.2 & 45:42:58 & -128.00 & Oct08 & 3 $\times$ 1800\ UGC 4278-A & 08:14:00.0 & 45:42.57 & -128.00 & Oct08 & 3 $\times$ 1800\ NGC 2552-A & 08:19:17.1 & 50:00:14 & -120.00 & Oct08 & 3 $\times$ 1200\ UGC 4393-B & 08:26:05.3 & 45:58:10 & -124.65 & Jan10 & 3 $\times$ 900\ UGC 4393-C & 08:26:01.5 & 45:47:43 & -124.65 & Jan10 & 3 $\times$ 900\ CGCG 035-007-A & 09:34:44.4 & 06:25:31 & 42.78 & Jan10 & 3 $\times$ 900\ UGC 5139-A & 09:40:16.0 & 71:10:06 & -140.00 & Nov08 & 4 $\times$ 1200\ IC 559-A & 09:44:42.9 & 09:36:54 & -64.15 & Jan10 & 4 $\times$ 900\ UGC 5272-A & 09:50:22.3 & 31:29:15 & -80.51 & Oct08 & 3 $\times$ 600\ UGC 5340-A & 09:56:46.8 & 28:50:10 & -75.61 & Jan10 & 4 $\times$ 900\ UGC 5423-A & 10:05:28.7 & 70:22:05 & 127.00 & Jan10 & 3 $\times$ 900\ UGC 5423-B & 10:05:32.1 & 70:21:52 & 127.00 & Jan10 & 3 $\times$ 900\ UGC 5672-A & 10:28:21.1 & 22:34:05 & -57.80 & Jan10 & 4 $\times$ 900\ UGC 5692-A & 10:30:34.8 & 70:37:11 & -147.53 & Jan10 & 4 $\times$ 900\ UGC 5797-A & 10:39:25.0 & 01:43:00 & -4.17 & Jan10 & 3 $\times$ 900\ UGC 5923-A & 10:49:07.5 & 06:55:08 & 20.00 & Jan10 & 5 $\times$ 600\ NGC 3738-A & 11:35:46.8 & 54:31:32 & 93.73 & Jun09 & 4 $\times$ 900\ NGC 3738-B & 11:35:48.2 & 54:31:31 & 93.73 & Jun09 & 4 $\times$ 900\ NGC 3741-A & 11:36:05.9 & 45:17:00 & 101.03 & Jun09 & 3 $\times$ 1200\ UGC 6782-A & 11:48:57.2 & 23:50:32 & 64.55 & Jun09 & 3 $\times$ 1200\ UGC 6817-A & 11:50:52.9 & 38:52:52 & 93.87 & Jun09 & 3 $\times$ 1200\ UGC 6900-A & 11:55:36.2 & 31:31:19 & 81.43 & Jun09 & 3 $\times$ 1200\ UGC 4163-A & 12:12:09.4 & 36:09:59 & 87.48 & Jun09 & 3 $\times$ 1200\ CGCG 269-049-A & 12:15:46.6 & 52:23:14 & -187.93 & Jan10 & 4 $\times$ 900\ UGC 7577-A & 12:27:42.8 & 43:29:06 & 100.00 & Jun09 & 3 $\times$ 1200\ NGC 4449- C & 12:28:14.5 & 44:07:13 & 75.00 & Jun09 & 3 $\times$ 600\ NGC 4449- B & 12:28:14.1 & 44:07:12 & 75.00 & Jun09 & 3 $\times$ 600\ NGC 4449-A & 12:28:13.9 & 44:07:10 & 75.00 & Jun09 & 3 $\times$ 600\ UGC 7599-A & 12:28:27.2 & 37:14:16 & 86.11 & Jun09 & 3 $\times$ 1500\ UGC 7605-A & 12:28:38.4 & 35:43:15 & 89.00 & Jun09 & 5 $\times$ 1200\ UGC 7639-A & 12:29:54.6 & 47:31:40 & 95.77 & Jun09 & 2 $\times$ 1200, 1 $\times$ 600\ NGC 4656-A & 12:43:56.6 & 32:10:12 & -79.80 & Jan10 & 3 $\times$ 900\ UGC 8201-A & 13:06:17.4 & 67:42:08 & 120.00 & Jun09 & 3 $\times$ 1200\ UGC 8245-A & 13:08:41.0 & 78:56:22 & 150.00 & Jun09 & 3 $\times$ 1200\ UGC 8508-A & 13:30:44.5 & 54:54:24 & 90.09 & Jun09 & 2 $\times$ 1200, 1$\times$ 900\ UGC 8638-A & 13:39:19.3 & 24:46:28 & 69.76 & Jun09 & 3 $\times$ 1200\ UGC 8638-B & 13:39:20.5 & 24:46:33 & 69.76 & Jun09 & 3 $\times$ 1200\ UGC 8837-A & 13:54:40.5 & 53:53:09 & 123.33 & Jun09 & 3 $\times$ 900\ NGC 5477-C & 14:05:32.9 & 54:27:41 & 99.00 & Jun09 & 3 $\times$ 900\ NGC 5477-A & 14:05:33.4 & 54:27:41 & 99.00 & Jun09 & 3 $\times$ 900\ UGC 9405-A & 14:35:25.9 & 57:15:29 & 125.00 & Jun09 & 4 $\times$ 1200\ UGC 10818-A & 17:19:41.1 & 61:18:31 & -180.00 & Jun09 & 3 $\times$ 900\ KKH 098-A & 23:45:33.5 & 38:43:15 & -110.00 & Jan10 & 3 $\times$ 1800\ \[tbl2\] [lcccccccccc]{} & [UGC]{} & [UGC]{} & [UGC]{} & [UGC]{} & [UGC]{} & [NGC]{} & [NGC]{} & [UGC]{} & [KKH]{} & [NGC]{}\ & [521 A]{} & [695 E]{} & [1056 A]{} & [1056 B]{} & [1176 A]{} & [784 A]{} & [784 B]{} & [2716 A]{} & [037 A]{} & [2537 A]{}\ & 1.59$\pm$0.03 & 3.25$\pm$0.07 & 3.32$\pm$0.13 & 2.80$\pm$0.06 & 2.27$\pm$0.05 & 2.19$\pm$0.06 & 2.61$\pm$0.07 & 2.08$\pm$0.04 & 6.51$\pm$0.08 & 3.42$\pm$0.07\ [He I $\lambda$3820]{} & & & & & & & & & &\ [H9 $\lambda$3835]{} & 0.07$\pm$0.02 & 0.11$\pm$0.01 & & & 0.07$\pm$0.01 & 0.04$\pm$0.01 & 0.07$\pm$0.01 & 0.08$\pm$0.02 & &\ [\[Ne III\] $\lambda$3868]{} & 0.33$\pm$0.02 & 0.19$\pm$0.01 & 0.27$\pm$0.04 & 0.30$\pm$0.02 & 0.28$\pm$0.01 & 0.33$\pm$0.01 & 0.29$\pm$0.01 & 0.38$\pm$0.02 & & 0.15$\pm$0.01\ [He I+H8 $\lambda$3889]{} & 0.17$\pm$0.02 & 0.25$\pm$0.01 & 0.17$\pm$0.03 & 0.24$\pm$0.02 & 0.19$\pm$0.01 & 0.18$\pm$0.01 & 0.23$\pm$ 0.01 & 0.26$\pm$0.01 & & 0.17$\pm$0.01\ [\[Ne III\]+H7 $\lambda$3968]{} & 0.34$\pm$0.02 & 0.36$\pm$0.01 & 0.37$\pm$ 0.02 & 0.54$\pm$0.02 & 0.59$\pm$0.01 & 0.20$\pm$0.01 & 0.51$\pm$ 0.01 & 0.32$\pm$0.01 & & 0.30$\pm$0.01\ [He I $\lambda$4026]{} & & & & & & & & & & 0.025$\pm$0.003\ [\[S II\] $\lambda$4068]{} & & 0.03$\pm$0.02 & & & & & & & &\ [H$\delta$ $\lambda$4101]{} & 0.26$\pm$0.02 & 0.29$\pm$0.02 & 0.27$\pm$0.02 & 0.26$\pm$0.01 & 0.26$\pm$0.01 & 0.264$\pm$0.006 & 0.59$\pm$0.01 & 0.27$\pm$0.01 & & 0.27$\pm$0.01\ [H$\gamma$ $\lambda$4340]{} & 0.48$\pm$0.01 & 0.53$\pm$0.01 & 0.49$\pm$0.01 & 0.50$\pm$0.01 & 0.46$\pm$0.01 & 0.460$\pm$0.009 & 0.48$\pm$0.01 & 0.51$\pm$0.01 & 0.47$\pm$0.20 & 0.47$\pm$0.01\ [\[O III\] $\lambda$4363]{} & 0.09$\pm$0.01 & 0.04$\pm$0.01 & 0.04$\pm$0.01 & 0.04$\pm$0.01 & 0.05$\pm$0.01 & 0.051$\pm$0.005 & 0.05$\pm$0.01 & 0.06$\pm$0.01 & & 0.012$\pm$0.002\ [He I $\lambda$4471]{} & & & & & & 0.036$\pm$0.005 & 0.03$\pm$0.01 & 0.038$\pm$0.005 & & 0.042$\pm$0.002\ [\[Fe III\] $\lambda$4658]{} & & & & & & & & & &\ [He II $\lambda$4686]{} & & & & & & & & & &\ [H$\beta$ $\lambda$4861]{} & 1.00$\pm$0.01 & 1.00$\pm$0.02 & 1.00$\pm$0.01 & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.04 & 1.00$\pm$0.02 & 1.00$\pm$0.17 & 1.00$\pm$0.02\ [\[O III\] $\lambda$4959]{} & 1.21$\pm$0.02 & 0.54$\pm$0.01 & 0.78$\pm$0.01 & 1.09$\pm$0.02 & 1.20$\pm$0.02 & 1.37$\pm$0.03 & 1.12$\pm$0.02 & 1.43$\pm$0.03 & & 0.71$\pm$0.01\ [\[O III\] $\lambda$5007]{} & 3.64$\pm$0.07 & 1.62$\pm$0.03 & 2.35$\pm$0.05 & 3.27$\pm$0.07 & 3.61$\pm$0.07 & 4.13$\pm$0.08 & 3.32$\pm$0.07 & 4.26$\pm$0.09 & 0.53$\pm$0.15 & 2.14$\pm$0.04\ [\[N I\] $\lambda$5199]{} & & & & & & & 0.013$\pm$0.003 & & &\ [He I $\lambda$5876]{} & 0.08$\pm$0.01 & 0.10$\pm$0.01 & 0.13$\pm$0.01 & 0.11$\pm$0.01 & 0.097$\pm$0.004 & 0.111$\pm$0.003 & 0.095$\pm$0.005 & 0.113$\pm$0.004 & & 0.12$\pm$0.03\ [\[O I\] $\lambda$6300]{} & 0.02$\pm$0.01 & 0.07$\pm$0.01 & 0.07$\pm$0.01 & 0.04$\pm$0.01 & 0.012$\pm$0.005 & 0.051$\pm$0.002 & 0.031$\pm$0.003 & 0.034$\pm$0.003 & & 0.045$\pm$0.002\ [\[S III\] $\lambda$6312]{} & 0.03$\pm$0.01 & & & 0.02$\pm$0.01 & 0.019$\pm$0.005 & 0.020$\pm$0.002 & 0.015$\pm$0.003 & 0.022$\pm$0.003 & & 0.016$\pm$0.002\ [\[O I\] $\lambda$6363]{} & 0.01$\pm$0.01 & 0.03$\pm$0.01 & 0.02$\pm$0.01 & 0.13$\pm$0.01 & & 0.022$\pm$0.002 & 0.016$\pm$0.003 & 0.011$\pm$0.003 & & 0.013$\pm$0.002\ [\[N II\] $\lambda$6548]{} & & 0.034$\pm$0.009 & 0.05$\pm$0.01 & 0.04$\pm$0.01 & 0.032$\pm$0.003 & 0.036$\pm$0.002 & 0.020$\pm$0.003 & 0.031$\pm$0.003 & & 0.15$\pm$0.01\ [H$\alpha$ $\lambda$6563]{} & 2.77$\pm$0.06 & 2.86$\pm$0.06 & 2.80$\pm$0.11 & 2.82$\pm$0.06 & 2.82$\pm$0.06 & 2.89$\pm$0.07 & 2.81$\pm$0.07 & 2.86$\pm$0.06 & 2.80$\pm$0.14 & 2.87$\pm$0.07\ [\[N II\] $\lambda$6584]{} & 0.05$\pm$0.01 & 0.125$\pm$0.009 & 0.14$\pm$0.01 & 0.12$\pm$0.01 & 0.122$\pm$0.003 & 0.109$\pm$0.003 & 0.080$\pm$0.003 & 0.092$\pm$0.003 & 0.14$\pm$0.12 & 0.48$\pm$0.01\ [He I $\lambda$6678]{} & 0.02$\pm$0.01 & & 0.04$\pm$0.01 & 0.029$\pm$0.004 & 0.028$\pm$0.004 & 0.027$\pm$0.002 & 0.025$\pm$0.003 & 0.034$\pm$0.003 & & 0.030$\pm$0.002\ [\[S II\] $\lambda$6717]{} & 0.11$\pm$0.01 & 0.434$\pm$0.009 & 0.40$\pm$0.02 & 0.26$\pm$0.01 & 0.17$\pm$0.01 & 0.203$\pm$0.004 & 0.144$\pm$0.004 & 0.196$\pm$0.004 & 0.38$\pm$0.10 & 0.36$\pm$0.01\ [\[S II\] $\lambda$6731]{} & 0.07$\pm$0.01 & 0.274$\pm$0.009 & 0.31$\pm$0.02 & 0.17$\pm$0.01 & 0.12$\pm$0.01 & 0.150$\pm$0.003 & 0.103$\pm$0.003 & 0.137$\pm$0.003 & 0.27$\pm$0.10 & 0.27$\pm$0.01\ & 0.00$\pm$0.01 & 0.17$\pm$0.01 & 0.23$\pm$0.05 & 0.28$\pm$0.01 & 0.22$\pm$0.01 & 0.43$\pm$0.02 & 0.41$\pm$0.02 & 0.29$\pm$0.02 & 0.63$\pm$0.04 & 0.43$\pm$0.02\ [$F$(H$\beta$)]{} & 8.69$\pm$0.07 & 13.5$\pm$0.3 & 9.75$\pm$0.11 & 20.8$\pm$0.42 & 19.3$\pm$0.4 & 22.6$\pm$0.5 & 29.1$\pm$1.2 & 24.2$\pm$0.5 & 0.51$\pm$0.09 & 117$\pm$2\ [EW(H$\beta$)]{} & 30.7 & 21.4 & 19.9 & 25.4 & 129 & 70.1 & 63.7 & 46.6 & 5.9 & 70.1\ [EW(H$\alpha$)]{} & 130. & 112 & 96.0 & 134 & 622 & 467 & 386 & 269 & 33.3 & 339\ \ [Ion]{} & [NGC]{} & [UGC]{} & [UGC]{} & [NGC]{} & [UGC]{} & [UGC]{} & [CGCG]{} & [UGC]{} & [IC]{} &\ & [2537 B]{} & [4278 B]{} & [4278 A]{} & [2552 A]{} & [4393 B]{} & [4393 C]{} & [035-007 A]{} & [5139 A]{} & [559 A]{} &\ & 2.79$\pm$0.06 & 2.03$\pm$0.04 & 1.68$\pm$0.04 & 2.39$\pm$0.05 & 2.60$\pm$0.06 & 4.06$\pm$0.09 & 4.00$\pm$0.09 & 1.88$\pm$0.04 & 3.12$\pm$0.06 &\ [He I $\lambda$3820]{} & & & & & & & & & &\ [H9 $\lambda$3835]{} & & 0.12$\pm$0.01 & 0.07$\pm$0.01 & & 0.08$\pm$0.01 & 0.05$\pm$0.01 & & 0.05$\pm$0.01 & &\ [\[Ne III\] $\lambda$3868]{} & 0.11$\pm$0.01 & 0.17$\pm$0.01 & 0.20$\pm$0.01 & 0.25$\pm$0.01 & 0.27$\pm$0.01 & 0.26$\pm$0.01 & & 0.30$\pm$0.01 & 0.25$\pm$0.02 &\ [He I+H8 $\lambda$3889]{} & 0.20$\pm$0.01 & 0.27$\pm$0.01 & 0.19$\pm$0.01 & 0.24$\pm$0.01 & 0.15$\pm$0.01 & 0.21$\pm$0.01 & & 0.19$\pm$0.01 & 0.16$\pm$0.01 &\ [\[Ne III\]+H7 $\lambda$3968]{} & 0.15$\pm$0.01 & 0.43$\pm$0.01 & 0.23$\pm$0.01 & 0.50$\pm$0.01 & 0.39$\pm$0.01 & 0.24$\pm$0.01 & & 0.24$\pm$0.01 & 0.16$\pm$0.01 &\ [He I $\lambda$4026]{} & 0.024$\pm$0.007 & & & & & 0.016$\pm$0.006 & & 0.03$\pm$0.01 & &\ [\[S II\] $\lambda$4068]{} & 0.018$\pm$0.007 & 0.06$\pm$0.01 & 0.022$\pm$0.005 & 0.02$\pm$0.01 & & 0.040$\pm$0.005 & & & &\ [H$\delta$ $\lambda$4101]{} & 0.27$\pm$0.01 & 0.25$\pm$0.01 & 0.24$\pm$0.01 & 0.25$\pm$0.01 & 0.26$\pm$0.01 & 0.27$\pm$0.01 & 0.24$\pm$0.02 & 0.25$\pm$0.01 & 0.270$\pm$0.008 &\ [H$\gamma$ $\lambda$4340]{} & 0.44$\pm$0.01 & 0.46$\pm$0.01 & 0.44$\pm$0.01 & 0.47$\pm$0.01 & 0.47$\pm$0.01 & 0.48$\pm$0.01 & 0.46$\pm$0.02 & 0.48$\pm$0.01 & 0.451$\pm$0.008 &\ [\[O III\] $\lambda$4363]{} & 0.016$\pm$0.004 & 0.033$\pm$0.005 & 0.046$\pm$0.004 & 0.021$\pm$0.003 & 0.034$\pm$0.008 & 0.036$\pm$0.005 & 0.06$\pm$0.02 & 0.051$\pm$0.008 & 0.028$\pm$0.008 &\ [He I $\lambda$4471]{} & 0.040$\pm$0.004 & 0.026$\pm$0.005 & 0.032$\pm$0.004 & 0.028$\pm$0.003 & 0.05$\pm$0.01 & 0.039$\pm$0.005 & & & 0.025$\pm$0.007 &\ [\[Fe III\] $\lambda$4658]{} & & & & & & & & & &\ [He II $\lambda$4686]{} & & & & & & & & & &\ [H$\beta$ $\lambda$4861]{} & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.02 &\ [\[O III\] $\lambda$4959]{} & 0.60$\pm$0.01 & 0.63$\pm$0.01 & 0.83$\pm$0.02 & 0.97$\pm$0.02 & 1.10$\pm$0.02 & 0.86$\pm$0.04 & 0.68$\pm$0.01 & 1.24$\pm$0.03 & 0.93$\pm$0.02 &\ [\[O III\] $\lambda$5007]{} & 1.78$\pm$0.04 & 1.91$\pm$0.04 & 2.53$\pm$0.05 & 2.91$\pm$0.06 & 3.28$\pm$0.07 & 2.59$\pm$0.05 & 2.03$\pm$0.04 & 3.66$\pm$0.07 & 2.80$\pm$0.06 &\ [\[N I\] $\lambda$5199]{} & & & 0.007$\pm$0.002 & 0.013$\pm$0.002 & & & & & &\ [He I $\lambda$5876]{} & 0.112$\pm$0.002 & 0.08$\pm$0.01 & 0.096$\pm$0.004 & 0.10$\pm$0.01 & 0.12$\pm$0.01 & 0.11$\pm$0.01 & 0.079$\pm$0.013 & & 0.09$\pm$0.01 &\ [\[O I\] $\lambda$6300]{} & 0.021$\pm$0.002 & 0.03$\pm$0.01 & 0.026$\pm$0.004 & 0.037$\pm$0.002 & 0.055$\pm$0.005 & 0.084$\pm$0.005 & & 0.04$\pm$0.01 & &\ [\[S III\] $\lambda$6312]{} & 0.013$\pm$0.002 & 0.013$\pm$0.006 & 0.013$\pm$0.004 & 0.015$\pm$0.002 & 0.011$\pm$0.005 & 0.017$\pm$0.005 & & 0.02$\pm$0.01 & &\ [\[O I\] $\lambda$6363]{} & 0.005$\pm$0.002 & 0.008$\pm$0.006 & 0.005$\pm$0.004 & 0.012$\pm$0.002 & 0.016$\pm$0.004 & 0.024$\pm$0.004 & & 0.06$\pm$0.01 & &\ [\[N II\] $\lambda$6548]{} & 0.128$\pm$0.003 & 0.017$\pm$0.004 & 0.017$\pm$0.003 & 0.08$\pm$0.01 & 0.092$\pm$0.004 & 0.096$\pm$0.003 & 0.06$\pm$0.01 & 0.02$\pm$0.01 & 0.034$\pm$0.014 &\ [H$\alpha$ $\lambda$6563]{} & 2.79$\pm$0.06 & 2.80$\pm$0.06 & 2.86$\pm$0.06 & 2.86$\pm$0.06 & 2.86$\pm$0.06 & 2.87$\pm$0.07 & 2.84$\pm$0.06 & 2.83$\pm$0.06 & 2.86$\pm$0.06 &\ [\[N II\] $\lambda$6584]{} & 0.41$\pm$0.01 & 0.060$\pm$0.004 & 0.054$\pm$0.003 & 0.25$\pm$0.01 & 0.30$\pm$0.01 & 0.31$\pm$0.01 & 0.21$\pm$0.01 & 0.07$\pm$0.01 & 0.15$\pm$0.01 &\ [He I $\lambda$6678]{} & 0.030$\pm$0.002 & 0.027$\pm$0.004 & 0.027$\pm$0.002 & 0.025$\pm$0.002 & 0.029$\pm$0.003 & 0.025$\pm$0.002 & & 0.029$\pm$0.006 & &\ [\[S II\] $\lambda$6717]{} & 0.26$\pm$0.01 & 0.18$\pm$0.01 & 0.140$\pm$0.003 & 0.34$\pm$0.01 & 0.28$\pm$0.01 & 0.39$\pm$0.01 & 0.49$\pm$0.01 & 0.136$\pm$0.006 & 0.35$\pm$0.02 &\ [\[S II\] $\lambda$6731]{} & 0.184$\pm$0.004 & 0.13$\pm$0.01 & 0.103$\pm$0.002 & 0.25$\pm$0.01 & 0.19$\pm$0.01 & 0.28$\pm$0.01 & 0.33$\pm$0.01 & 0.092$\pm$0.006 & 0.23$\pm$0.02 &\ & 0.28$\pm$0.01 & 0.10$\pm$0.01 & 0.31$\pm$0.02 & 0.14$\pm$0.01 & 0.27$\pm$0.02 & 0.34$\pm$0.02 & 0.16$\pm$0.01 & 0.085$\pm$0.010 & 0.30$\pm$0.02 &\ [$F$(H$\beta$)]{} & 80.0$\pm$1.6 & 8.93$\pm$0.17 & 15.9$\pm$0.32 & 37.0$\pm$0.7 & 24.2$\pm$0.5 & 19.3$\pm$0.4 & 11.3$\pm$0.2 & 6.40$\pm$0.13 & 21.3$\pm$0.4 &\ [EW(H$\beta$)]{} & 47.8 & 64.2 & 82.7 & 55.3 & 22.0 & 160 & 16.5 & 197 & 49.7 &\ [EW(H$\alpha$)]{} & 188 & 321 & 478 & 312 & 113 & 922 & 72.3 & 939 & 273 &\ \ [Ion]{} & [UGC]{} & [UGC]{} & [UGC]{} & [UGC]{} & [UGC]{} & [UGC]{} & [UGC]{} & [UGC]{} & [NGC]{} &\ & [5272 A]{} & [5340 A]{} & [5423 A]{} & [5423 B]{} & [5672 A]{} & [5692 A]{} & [5797 A]{} & [5923 A]{} & [3741 A]{} &\ & 1.06$\pm$0.02 & 0.58$\pm$0.02 & 2.08$\pm$0.04 & 1.78$\pm$0.04 & 3.28$\pm$0.04 & 2.47$\pm$0.06 & 1.59$\pm$0.03 & 4.06$\pm$0.11 & 1.60$\pm$0.03 &\ [He I $\lambda$3820]{} & 0.014$\pm$0.002 & & & & & & & & &\ [H9 $\lambda$3835]{} & 0.084$\pm$0.002 & 0.05$\pm$0.01 & 0.12$\pm$0.01 & & & & 0.27$\pm$0.04 & 0.18$\pm$0.03 & &\ [\[Ne III\] $\lambda$3868]{} & 0.375$\pm$0.007 & 0.16$\pm$0.01 & 0.33$\pm$0.01 & 0.28$\pm$0.02 & 0.20$\pm$0.04 & 0.21$\pm$0.05 & 0.52$\pm$0.04 & 0.24$\pm$0.03 & 0.24$\pm$0.01 &\ [He I+H8 $\lambda$3889]{} & 0.197$\pm$0.004 & 0.17$\pm$0.01 & 0.24$\pm$0.01 & 0.32$\pm$0.02 & 0.16$\pm$0.04 & 0.24$\pm$0.05 & 0.36$\pm$0.03 & 0.33$\pm$0.03 & 0.21$\pm$0.01 &\ [\[Ne III\]+H7 $\lambda$3968]{} & 0.194$\pm$0.004 & 0.47$\pm$ 0.01 & 0.28$\pm$0.01 & 0.50$\pm$0.02 & 0.19$\pm$0.04 & & 0.32$\pm$0.03 & 0.20$\pm$0.03 & 0.49$\pm$0.01 &\ [He I $\lambda$4026]{} & 0.015$\pm$0.003 & & & & & & & & &\ [\[S II\] $\lambda$4068]{} & & & 0.03$\pm$0.01 & & & & & & &\ [H$\delta$ $\lambda$4101]{} & 0.26$\pm$0.01 & 0.25$\pm$0.01 & 0.26$\pm$0.01 & 0.26$\pm$0.02 & 0.25$\pm$0.03 & 0.22$\pm$0.04 & 0.28$\pm$0.02 & 0.24$\pm$0.02 & 0.25$\pm$0.01 &\ [H$\gamma$ $\lambda$4340]{} & 0.46$\pm$0.01 & 0.47$\pm$0.01 & 0.46$\pm$0.01 & 0.46$\pm$0.01 & 0.42$\pm$0.03 & 0.54$\pm$0.03 & 0.47$\pm$0.02 & 0.51$\pm$0.02 & 0.46$\pm$0.01 &\ [\[O III\] $\lambda$4363]{} & 0.083$\pm$0.002 & 0.06$\pm$0.01 & 0.070$\pm$0.004 & 0.07$\pm$0.01 & 0.04$\pm$0.02 & 0.06$\pm$0.03 & 0.09$\pm$0.02 & 0.06$\pm$0.01 & 0.06$\pm$0.01 &\ [He I $\lambda$4471]{} & 0.036$\pm$0.002 & 0.04$\pm$0.01 & 0.033$\pm$0.004 & & & & & & 0.026$\pm$0.003 &\ [\[Fe III\] $\lambda$4658]{} & & & & & & & & & &\ [He II $\lambda$4686]{} & & 0.024$\pm$0.002 & 0.019$\pm$0.002 & & & & 0.036$\pm$0.006 & & &\ [H$\beta$ $\lambda$4861]{} & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.02 &\ [\[O III\] $\lambda$4959]{} & 1.66$\pm$0.03 & 0.64$\pm$0.01 & 1.16$\pm$0.02 & 1.24$\pm$0.02 & 0.85$\pm$0.02 & 0.57$\pm$0.02 & 1.85$\pm$0.04 & 0.84$\pm$0.02 & 0.94$\pm$0.02 &\ [\[O III\] $\lambda$5007]{} & 4.94$\pm$0.10 & 1.89$\pm$0.04 & 3.49$\pm$0.01 & 3.71$\pm$0.07 & 2.51$\pm$0.05 & 1.70$\pm$0.03 & 5.53$\pm$0.11 & 2.48$\pm$0.05 & 2.84$\pm$0.06 &\ [\[N I\] $\lambda$5199]{} & & & 0.012$\pm$0.004 & & & & & & &\ [He I $\lambda$5876]{} & 0.101$\pm$0.001 & 0.09$\pm$0.01 & 0.10$\pm$0.01 & 0.11$\pm$0.01 & 0.11$\pm$0.01 & 0.14$\pm$0.03 & 0.09$\pm$0.01 & & 0.097$\pm$0.003 &\ [\[O I\] $\lambda$6300]{} & 0.013$\pm$0.001 & & 0.080$\pm$0.004 & 0.02$\pm$0.01 & 0.04$\pm$0.01 & & 0.05$\pm$0.02 & 0.06$\pm$0.01 & 0.022$\pm$0.002 &\ [\[S III\] $\lambda$6312]{} & 0.020$\pm$0.001 & & 0.021$\pm$0.004 & 0.02$\pm$0.01 & & & 0.02$\pm$0.02 & 0.02$\pm$0.01 & 0.018$\pm$0.002 &\ [\[O I\] $\lambda$6363]{} & 0.004$\pm$0.001 & & 0.020$\pm$0.004 & 0.01$\pm$0.01 & & & 0.05$\pm$0.02 & 0.02$\pm$0.01 & 0.004$\pm$0.002 &\ [\[N II\] $\lambda$6548]{} & 0.010$\pm$0.001 & & 0.035$\pm$0.004 & 0.026$\pm$0.009 & 0.12$\pm$0.02 & 0.10$\pm$0.02 & 0.03$\pm$0.01 & 0.07$\pm$0.01 & 0.017$\pm$0.003 &\ [H$\alpha$ $\lambda$6563]{} & 2.83$\pm$0.06 & 2.81$\pm$0.06 & 2.86$\pm$0.06 & 2.86$\pm$0.06 & 2.87$\pm$0.07 & 2.79$\pm$0.06 & 2.84$\pm$0.06 & 2.78$\pm$0.07 & 2.83$\pm$0.06 &\ [\[N II\] $\lambda$6584]{} & 0.035$\pm$0.001 & 0.016$\pm$0.003 & 0.120$\pm$0.004 & 0.082$\pm$0.009 & 0.23$\pm$0.02 & 0.38$\pm$0.02 & 0.09$\pm$0.01 & 0.24$\pm$0.01 & 0.051$\pm$0.003 &\ [He I $\lambda$6678]{} & 0.029$\pm$0.002 & 0.023$\pm$0.007 & 0.027$\pm$0.003 & 0.029$\pm$0.007 & & & & 0.03$\pm$0.01 & 0.026$\pm$0.002 &\ [\[S II\] $\lambda$6717]{} & 0.079$\pm$0.002 & 0.05$\pm$0.007 & 0.31$\pm$0.006 & 0.153$\pm$0.007 & 0.31$\pm$0.01 & 0.61$\pm$0.02 & 0.22$\pm$0.01 & 0.31$\pm$0.01 & 0.122$\pm$0.002 &\ [\[S II\] $\lambda$6731]{} & 0.055$\pm$0.002 & 0.05$\pm$0.007 & 0.22$\pm$0.006 & 0.110$\pm$0.007 & 0.22$\pm$0.01 & 0.40$\pm$0.02 & 0.15$\pm$0.01 & 0.22$\pm$0.01 & 0.088$\pm$0.002 &\ & 0.08$\pm$0.01 & 0.00$\pm$0.01 & 0.24$\pm$0.01 & 0.29$\pm$0.02 & 0.40$\pm$0.02 & 0.25$\pm$0.01 & 0.15$\pm$0.01 & 0.51$\pm$0.02 & 0.10$\pm$0.01 &\ [$F$(H$\beta$)]{} & 101$\pm$2 & 13.6$\pm$0.27 & 22.8$\pm$0.46 & 9.38$\pm$0.19 & 5.80$\pm$0.12 & 4.71$\pm$0.10 & 8.40$\pm$0.17 & 20.2$\pm$0.40 & 45.0$\pm$0.9 &\ [EW(H$\beta$)]{} & 201 & 99.2 & 94.8 & 51.4 & 14.6 & 24.3 & 23.8 & 16.0 & 59.9 &\ [EW(H$\alpha$)]{} & 943 & 549 & 438 & 247 & 67.4 & 120 & 111 & 75.3 & 330 &\ \ [Ion]{} & [NGC]{} & [NGC]{} & [UGC]{} & [UGC]{} & [NGC]{} & [CGCG]{} & [CGCG]{} & [UGC]{} & [NGC]{} &\ & [3738 A]{} & [3738 B]{} & [6817 A]{} & [6900 A]{} & [4163 A]{} & [269-049 C]{} & [269-049 A]{} & [7577 A]{} & [4449 C]{} &\ & 2.91$\pm$0.06 & 3.45$\pm$0.07 & 0.94$\pm$0.02 & 4.13$\pm$0.27 & 3.72$\pm$0.07 & 1.76$\pm$0.04 & 1.02$\pm$0.02 & 1.65$\pm$0.03 & 3.26$\pm$0.07 &\ [He I $\lambda$3820]{} & & & & & & & & & &\ [H9 $\lambda$3835]{} & & & 0.08$\pm$0.01 & & & & 0.09$\pm$0.02 & 0.06$\pm$0.02 & 0.04$\pm$0.01 &\ [\[Ne III\] $\lambda$3868]{} & 0.28$\pm$0.01 & 0.31$\pm$0.04 & 0.25$\pm$0.01 & & & & 0.20$\pm$0.02 & 0.47$\pm$0.02 & 0.19$\pm$0.01 &\ [He I+H8 $\lambda$3889]{} & 0.11$\pm$0.01 & 0.32$\pm$0.04 & 0.19$\pm$0.01 & & & & 0.16$\pm$0.02 & 0.20$\pm$0.02 & 0.17$\pm$ 0.01 &\ [\[Ne III\]+H7 $\lambda$3968]{} & 0.39$\pm$0.01 & 0.24$\pm$0.04 & 0.53$\pm$0.01 & & & & 0.44$\pm$0.01 & 0.34$\pm$0.02 & 0.19$\pm$ 0.01 &\ [He I $\lambda$4026]{} & & & & & & & & & &\ [\[S II\] $\lambda$4068]{} & & & & & & & & & &\ [H$\delta$ $\lambda$4101]{} & 0.23$\pm$0.01 & 0.24$\pm$0.02 & 0.26$\pm$0.01 & 0.31$\pm$0.09 & 0.17$\pm$0.03 & 0.20$\pm$0.06 & 0.26$\pm$0.01 & 0.25$\pm$0.02 & 0.26$\pm$0.01 &\ [H$\gamma$ $\lambda$4340]{} & 0.47$\pm$0.01 & 0.45$\pm$0.01 & 0.46$\pm$0.01 & 0.55$\pm$0.06 & 0.31$\pm$0.04 & 0.29$\pm$0.06 & 0.46$\pm$0.01 & 0.47$\pm$0.01 & 0.45$\pm$0.01 &\ [\[O III\] $\lambda$4363]{} & 0.031$\pm$0.006 & 0.04$\pm$0.01 & 0.068$\pm$0.002 & & 0.014$\pm$0.003 & 0.05$\pm$0.01 & 0.062$\pm$0.003 & 0.08$\pm$0.01 & 0.018$\pm$0.004 &\ [He I $\lambda$4471]{} & 0.03$\pm$0.01 & & 0.033$\pm$0.002 & & & & 0.030$\pm$0.003 & 0.04$\pm$0.01 & &\ [\[Fe III\] $\lambda$4658]{} & & & & & & & & & &\ [He II $\lambda$4686]{} & & & & & & & & & &\ [H$\beta$ $\lambda$4861]{} & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.11 & 1.00$\pm$0.02 & 1.00$\pm$0.05 & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.02 &\ [\[O III\] $\lambda$4959]{} & 0.98$\pm$0.02 & 1.03$\pm$0.02 & 0.97$\pm$0.02 & 0.23$\pm$0.10 & 0.18$\pm$0.02 & 0.52$\pm$0.04 & 0.84$\pm$0.02 & 1.86$\pm$0.04 & 0.77$\pm$0.02 &\ [\[O III\] $\lambda$5007]{} & 2.96$\pm$0.06 & 3.11$\pm$0.06 & 2.89$\pm$0.06 & 0.60$\pm$0.10 & 0.49$\pm$0.02 & 1.53$\pm$0.04 & 2.51$\pm$0.05 & 5.25$\pm$0.11 & 2.33$\pm$0.05 &\ [\[N I\] $\lambda$5199]{} & 0.011$\pm$0.005 & & & & & & & & 0.017$\pm$0.003 &\ [He I $\lambda$5876]{} & 0.12$\pm$0.01 & 0.15$\pm$0.02 & 0.100$\pm$0.002 & & 0.10$\pm$0.02 & & 0.097$\pm$0.003 & 0.11$\pm$0.02 & 0.12$\pm$0.01 &\ [\[O I\] $\lambda$6300]{} & 0.05$\pm$0.01 & 0.06$\pm$0.01 & 0.015$\pm$0.002 & & 0.07$\pm$0.02 & & 0.008$\pm$0.003 & & 0.071$\pm$0.004 &\ [\[S III\] $\lambda$6312]{} & 0.02$\pm$0.01 & & 0.015$\pm$0.002 & & & & 0.012$\pm$0.003 & & 0.014$\pm$0.004 &\ [\[O I\] $\lambda$6363]{} & 0.02$\pm$0.01 & 0.02$\pm$0.01 & 0.004$\pm$0.002 & 0.29$\pm$0.11 & & & 0.003$\pm$0.001 & & 0.024$\pm$0.004 &\ [\[N II\] $\lambda$6548]{} & 0.06$\pm$0.01 & 0.06$\pm$0.01 & 0.008$\pm$0.001 & 0.16$\pm$0.10 & 0.04$\pm$0.02 & 0.02$\pm$0.04 & 0.010$\pm$0.001 & 0.03$\pm$0.01 & 0.074$\pm$0.003 &\ [H$\alpha$ $\lambda$6563]{} & 2.83$\pm$0.06 & 2.82$\pm$0.06 & 2.83$\pm$0.06 & 2.83$\pm$0.10 & 2.75$\pm$0.06 & 2.79$\pm$0.06 & 2.83$\pm$0.06 & 2.79$\pm$0.06 & 2.84$\pm$0.06 &\ [\[N II\] $\lambda$6584]{} & 0.19$\pm$0.01 & 0.20$\pm$0.01 & 0.03$\pm$0.01 & 0.52$\pm$0.10 & 0.13$\pm$0.02 & & 0.033$\pm$0.003 & 0.09$\pm$0.01 & 0.23$\pm$0.01 &\ [He I $\lambda$6678]{} & 0.03$\pm$0.01 & & 0.026$\pm$0.002 & & & & 0.024$\pm$0.002 & 0.03$\pm$0.01 & 0.020$\pm$0.003 &\ [\[S II\] $\lambda$6717]{} & 0.3 2$\pm$0.01 & 0.29$\pm$0.01 & 0.068$\pm$0.002 & 1.11$\pm$0.09 & 0.35$\pm$0.02 & & 0.064$\pm$0.002 & 0.20$\pm$0.01 & 0.35$\pm$0.01 &\ [\[S II\] $\lambda$6731]{} & 0.23$\pm$0.01 & 0.21$\pm$0.01 & 0.049$\pm$0.002 & 0.64$\pm$0.09 & 0.23$\pm$0.02 & & 0.046$\pm$0.002 & 0.15$\pm$0.01 & 0.25$\pm$0.01 &\ & 0.04$\pm$0.01 & 0.19$\pm$0.01 & 0.06$\pm$0.01 & 0.09$\pm$ 0.01 & 0.10$\pm$0.01 & 0.16$\pm$0.01 & 0.08$\pm$0.01 & 0.05$\pm$0.01 & 0.14$\pm$0.01 &\ [$F$(H$\beta$)]{} & 58.7$\pm$1.2 & 24.8$\pm$0.5 & 42.3$\pm$0.8 & 1.49$\pm$0.17 & 5.24$\pm$0.12 & 1.71$\pm$0.08 & 29.1$\pm$0.6 & 12.5$\pm$0.3 & 54.8$\pm$1.1 &\ [EW(H$\beta$)]{} & 35.2 & 23.4 & 146 & 20.0 & 9.21 & 6.8 & 81.3 & 216 & 119 &\ [EW(H$\alpha$)]{} & 183 & 121 & 834 & 75.5 & 40.9 & 37.2 & 434 & 854 & 437 &\ \ [Ion]{} & [NGC]{} & [NGC]{} & [UGC]{} & [UGC]{} & [NGC]{} & [UGC]{} & [UGC]{} & [UGC]{} & [UGC]{} &\ & [4449 B]{} & [4449 A]{} & [7605 A]{} & [7639 A]{} & [4656 A]{} & [8201 A]{} & [8245 A]{} & [8508 A]{} & [8638 A]{} &\ & 3.04$\pm$0.07 & 2.39$\pm$0.05 & 1.91$\pm$0.04 & 3.98$\pm$0.12 & 0.80$\pm$0.03 & 1.62$\pm$0.03 & 2.81$\pm$0.06 & 1.47$\pm$0.03 & 1.77$\pm$0.04 &\ [He I $\lambda$3820]{} & & 0.008$\pm$0.004 & & & & & & & &\ [H9 $\lambda$3835]{} & 0.05$\pm$0.01 & 0.062$\pm$0.004 & & & 0.08$\pm$0.02 & & & & &\ [\[Ne III\] $\lambda$3868]{} & 0.19$\pm$0.01 & 0.23$\pm$0.01 & 0.18$\pm$0.02 & & 0.50$\pm$0.02 & 0.25$\pm$0.01 & & 0.29$\pm$0.02 & 0.32$\pm$0.01 &\ [He I+H8 $\lambda$3889]{} & 0.19$\pm$0.01 & 0.167$\pm$0.004 & 0.15$\pm$0.02 & & 0.19$\pm$0.02 & 0.23$\pm$0.01 & & 0.26$\pm$0.02 & 0.22$\pm$0.01 &\ [\[Ne III\]+H7 $\lambda$3968]{} & 0.19$\pm$0.01 & 0.187$\pm$0.004 & 0.18$\pm$0.02 & & 0.29$\pm$0.02 & 0.19$\pm$0.01 & & 0.49$\pm$ 0.02 & 0.59$\pm$0.01 &\ [He I $\lambda$4026]{} & & 0.016$\pm$0.001 & & & & & & & &\ [\[S II\] $\lambda$4068]{} & & 0.014$\pm$0.001 & & & & & & & &\ [H$\delta$ $\lambda$4101]{} & 0.25$\pm$0.01 & 0.26$\pm$0.01 & 0.25$\pm$0.02 & 0.28$\pm$0.05 & 0.25$\pm$0.02 & 0.26$\pm$0.01 & 0.24$\pm$0.02 & 0.24$\pm$0.02 & 0.26$\pm$0.01 &\ [H$\gamma$ $\lambda$4340]{} & 0.48$\pm$0.01 & 0.47$\pm$0.01 & 0.46$\pm$0.01 & 0.44$\pm$0.07 & 0.45$\pm$0.01 & 0.47$\pm$0.01 & 0.41$\pm$0.02 & 0.45$\pm$0.01 & 0.45$\pm$0.01 &\ [\[O III\] $\lambda$4363]{} & 0.018$\pm$0.003 & 0.019$\pm$0.001 & 0.04$\pm$0.01 & 0.05$\pm$0.05 & 0.09$\pm$0.01 & 0.05$\pm$0.01 & 0.03$\pm$0.02 & 0.06$\pm$0.01 & 0.055$\pm$0.003 &\ [He I $\lambda$4471]{} & 0.032$\pm$0.003 & 0.039$\pm$0.001 & 0.02$\pm$0.01 & & & 0.05$\pm$0.01 & & 0.03$\pm$0.01 & 0.039$\pm$0.002 &\ [\[Fe III\] $\lambda$4658]{} & & & & & & & & & &\ [He II $\lambda$4686]{} & & & & & & & 0.05$\pm$0.01 & & &\ [H$\beta$ $\lambda$4861]{} & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.05 & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.02 &\ [\[O III\] $\lambda$4959]{} & 0.84$\pm$0.02 & 1.15$\pm$0.02 & 0.76$\pm$0.01 & 0.48$\pm$0.06 & 2.29$\pm$0.05 & 0.99$\pm$0.06 & 0.43$\pm$0.01 & 1.08$\pm$0.02 & 1.39$\pm$0.03 &\ [\[O III\] $\lambda$5007]{} & 2.54$\pm$0.05 & 3.46$\pm$0.07 & 2.33$\pm$0.05 & 1.33$\pm$0.06 & 6.71$\pm$0.13 & 2.94$\pm$0.06 & 1.25$\pm$0.03 & 3.25$\pm$0.06 & 4.17$\pm$0.08 &\ [\[N I\] $\lambda$5199]{} & 0.013$\pm$0.002 & 0.006$\pm$0.001 & & & & & & & &\ [He I $\lambda$5876]{} & 0.106$\pm$0.002 & 0.115$\pm$0.002 & 0.07$\pm$0.02 & & & 0.11$\pm$0.01 & 0.10$\pm$0.02 & 0.10$\pm$0.01 & 0.112$\pm$0.002 &\ [\[O I\] $\lambda$6300]{} & 0.050$\pm$0.002 & 0.028$\pm$0.001 & 0.05$\pm$0.02 & 0.09$\pm$0.06 & 0.02$\pm$0.01 & 0.06$\pm$0.01 & 0.15$\pm$0.02 & 0.04$\pm$0.01 & 0.016$\pm$0.001 &\ [\[S III\] $\lambda$6312]{} & 0.012$\pm$0.002 & 0.014$\pm$0.001 & 0.01$\pm$0.02 & & 0.03$\pm$0.01 & 0.02$\pm$0.01 & & 0.02$\pm$0.01 & 0.021$\pm$0.001 &\ [\[O I\] $\lambda$6363]{} & 0.015$\pm$0.002 & 0.009$\pm$0.001 & & & 0.01$\pm$0.01 & 0.04$\pm$0.01 & 0.08$\pm$0.02 & 0.02$\pm$0.01 & 0.008$\pm$0.001 &\ [\[N II\] $\lambda$6548]{} & 0.074$\pm$0.002 & 0.048$\pm$0.001 & 0.015$\pm$0.018 & 0.07$\pm$0.06 & 0.009$\pm$0.009 & & 0.04$\pm$0.02 & & 0.02$\pm$0.01 &\ [H$\alpha$ $\lambda$6563]{} & 2.86$\pm$0.06 & 2.83$\pm$0.06 & 2.83$\pm$0.06 & 2.83$\pm$0.06 & 2.86$\pm$0.06 & 2.81$\pm$0.06 & 2.83$\pm$0.06 & 2.83$\pm$0.06 & 2.82$\pm$0.06 &\ [\[N II\] $\lambda$6584]{} & 0.205$\pm$0.004 & 0.163$\pm$0.003 & 0.066$\pm$0.018 & 0.21$\pm$0.06 & 0.023$\pm$0.009 & 0.04$\pm$0.01 & 0.12$\pm$0.01 & 0.05$\pm$0.01 & 0.07$\pm$0.01 &\ [He I $\lambda$6678]{} & 0.025$\pm$0.002 & 0.029$\pm$0.001 & 0.05$\pm$0.01 & & & 0.03$\pm$0.01 & 0.02$\pm$0.01 & 0.029$\pm$0.001 & 0.028$\pm$0.002 &\ [\[S II\] $\lambda$6717]{} & 0.28$\pm$0.01 & 0.18$\pm$0.01 & 0.15$\pm$0.01 & 0.45$\pm$0.04 & 0.07$\pm$0.01 & 0.10$\pm$0.01 & 0.27$\pm$0.01 & 0.12$\pm$0.01 & 0.14$\pm$0.01 &\ [\[S II\] $\lambda$6731]{} & 0.20$\pm$0.01 & 0.13$\pm$0.01 & 0.09$\pm$0.01 & 0.28$\pm$0.04 & 0.04$\pm$0.01 & 0.07$\pm$0.01 & 0.18$\pm$0.01 & 0.08$\pm$0.01 & 0.10$\pm$0.01 &\ & 0.21$\pm$0.01 & 0.09$\pm$0.01 & 0.06$\pm$0.01 & 0.08$\pm$0.01 & 0.24$\pm$0.01 & 0.15$\pm$0.01 & 0.11$\pm$0.01 & 0.09$\pm$0.01 & 0.002$\pm$0.001 &\ [$F$(H$\beta$)]{} & 64.4$\pm$1.3 & 567$\pm$11 & 6.37$\pm$0.13 & 1.70$\pm$0.09 & 43.4$\pm$0.9 & 8.87$\pm$0.18 & 6.43$\pm$0.13 & 9.79$\pm$0.20 & 41.4$\pm$0.8 &\ [EW(H$\beta$)]{} & 119 & 239 & 75.8 & 8.25 & 171 & 120 & 20.8 & 74.0 & 129 &\ [EW(H$\alpha$)]{} & 587 & 851 & 611 & 37.8 & 1141 & 547 & 91.1 & 314 & 601 &\ \ [Ion]{} & [UGC]{} & [UGC]{} & [NGC]{} & [UGC]{} & [UGC]{} & [KKH]{} & & & &\ & [8638 B]{} & [8837 A]{} & [5477 A]{} & [9405 A]{} & [10818 A]{} & [098 A]{} & & & &\ & 1.75$\pm$0.04 & 3.46$\pm$0.01 & 1.31$\pm$0.05 & 3.66$\pm$0.37 & 2.73$\pm$0.06 & 1.85$\pm$0.05 & & & &\ [He I $\lambda$3820]{} & & & & & & & & & &\ [H9 $\lambda$3835]{} & 0.10$\pm$0.01 & 0.054$\pm$0.005 & 0.06$\pm$0.01 & & 0.16$\pm$0.02 & & & & &\ [\[Ne III\] $\lambda$3868]{} & 0.32$\pm$0.01 & 0.071$\pm$0.005 & 0.33$\pm$0.01 & & 0.20$\pm$0.02 & 0.21$\pm$0.04 & & & &\ [He I+H8 $\lambda$3889]{} & 0.24$\pm$0.01 & 0.21$\pm$0.01 & 0.23$\pm$0.01 & & 0.19$\pm$0.02 & 0.17$\pm$0.04 & & & &\ [\[Ne III\]+H7 $\lambda$3968]{} & 0.02$\pm$0.01 & 0.16$\pm$0.01 & 0.33$\pm$0.01 & & 0.28$\pm$0.01 & 0.09$\pm$0.04 & & & &\ [He I $\lambda$4026]{} & & & & & & & & & &\ [\[S II\] $\lambda$4068]{} & 0.02$\pm$0.01 & 0.02$\pm$0.01 & & & & & & & &\ [H$\delta$ $\lambda$4101]{} & 0.27$\pm$0.01 & 0.24$\pm$0.01 & 0.28$\pm$0.02 & & 0.23$\pm$0.01 & 0.36$\pm$0.03 & & & &\ [H$\gamma$ $\lambda$4340]{} & 0.47$\pm$0.01 & 0.47$\pm$0.01 & 0.47$\pm$0.02 & 0.35$\pm$0.22 & 0.45$\pm$0.01 & 0.58$\pm$0.02 & & & &\ [\[O III\] $\lambda$4363]{} & 0.06$\pm$0.01 & 0.019$\pm$0.003 & 0.06$\pm$0.01 & & 0.02$\pm$0.01 & & & & &\ [He I $\lambda$4471]{} & 0.03$\pm$0.01 & 0.030$\pm$0.003 & & & 0.03$\pm$0.01 & & & & &\ [\[Fe III\] $\lambda$4658]{} & & 0.009$\pm$0.003 & & & & & & & &\ [He II $\lambda$4686]{} & & & & & & & & & &\ [H$\beta$ $\lambda$4861]{} & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.02 & 1.00$\pm$0.13 & 1.00$\pm$0.02 & 1.00$\pm$0.02 & & & &\ [\[O III\] $\lambda$4959]{} & 1.39$\pm$0.03 & 0.42$\pm$0.01 & 1.55$\pm$0.03 & 0.46$\pm$0.14 & 0.74$\pm$0.02 & 0.63$\pm$0.01 & & & &\ [\[O III\] $\lambda$5007]{} & 4.15$\pm$0.08 & 1.26$\pm$0.03 & 4.64$\pm$0.10 & 1.44$\pm$0.14 & 2.22$\pm$0.04 & 1.91$\pm$0.01 & & & &\ [\[N I\] $\lambda$5199]{} & & 0.012$\pm$0.002 & & & 0.02$\pm$0.01 & & & & &\ [He I $\lambda$5876]{} & 0.107$\pm$0.004 & 0.091$\pm$0.002 & 0.11$\pm$0.01 & & 0.10$\pm$0.01 & 0.08$\pm$0.02 & & & &\ [\[O I\] $\lambda$6300]{} & 0.013$\pm$0.003 & 0.051$\pm$0.002 & 0.01$\pm$0.01 & & 0.06$\pm$0.01 & & & & &\ [\[S III\] $\lambda$6312]{} & 0.018$\pm$0.003 & 0.018$\pm$0.002 & 0.02$\pm$0.01 & & 0.01$\pm$0.01 & & & & &\ [\[O I\] $\lambda$6363]{} & 0.004$\pm$0.003 & 0.013$\pm$0.002 & 0.05$\pm$0.01 & & 0.02$\pm$0.01 & & & & &\ [\[N II\] $\lambda$6548]{} & 0.02$\pm$0.01 & 0.067$\pm$0.002 & 0.015$\pm$0.004 & & 0.08$\pm$0.01 & 0.04$\pm$0.02 & & & &\ [H$\alpha$ $\lambda$6563]{} & 2.81$\pm$0.06 & 2.83$\pm$0.06 & 2.84$\pm$0.11 & 2.79$\pm$0.17 & 2.79$\pm$0.06 & 2.79$\pm$0.06 & & & &\ [\[N II\] $\lambda$6584]{} & 0.06$\pm$0.01 & 0.202$\pm$0.004 & 0.047$\pm$0.001 & 0.40$\pm$0.16 & 0.25$\pm$0.01 & 0.08$\pm$0.02 & & & &\ [He I $\lambda$6678]{} & 0.026$\pm$0.003 & 0.021$\pm$0.002 & 0.028$\pm$0.001 & & & & & & &\ [\[S II\] $\lambda$6717]{} & 0.11$\pm$0.01 & 0.35$\pm$0.01 & 0.079$\pm$0.002 & 0.76$\pm$0.15 & & 0.15$\pm$0.02 & & & &\ [\[S II\] $\lambda$6731]{} & 0.09$\pm$0.01 & 0.25$\pm$0.01 & 0.059$\pm$0.001 & 0.55$\pm$0.14 & & 0.10$\pm$0.02 & & & &\ & 0.24$\pm$0.01 & 0.06$\pm$0.01 & 0.09$\pm$0.01 & 0.10$\pm$0.01 & 0.19$\pm$0.01 & 0.24$\pm$0.01 & & & &\ [$F$(H$\beta$)]{} & 16.7$\pm$0.3 & 35.4$\pm$0.71 & 78.0$\pm$1.6 & 0.48$\pm$0.06 & 11.3$\pm$0.2 & 3.07$\pm$0.06 & & & &\ [EW(H$\beta$)]{} & 57.1 & 114 & 177 & 43.6 & 46.3 & 50.6 & & & &\ [EW(H$\alpha$)]{} & 301 & 723 & 879 & 216 & 216 & 224 & & & &\ \[tbl3\] [cccccccccccc]{} UGC 521 & A & 14200$\pm$900 & 16500$\pm$1100 & 1.59$\pm$0.32 & 3.05$\pm$0.42 & 4.64$\pm$0.53 & 7.67$\pm$0.05 & 0.29$\pm$0.06 & -1.61$\pm$0.07 & 1.15$\pm$0.25\ UGC 695 & E & 14000$\pm$1900 & 15800$\pm$2200 & 3.45$\pm$1.48 & 1.50$\pm$0.45 & 4.95$\pm$1.55 & 7.69$\pm$0.12 & 1.07$\pm$0.29 & -1.49$\pm$0.04 & 1.61$\pm$0.53\ UGC 1056 & A & 13000$\pm$2500 & 13500$\pm$2600 & 4.50$\pm$2.89 & 3.26$\pm$1.56 & 7.75$\pm$3.29 & 7.89$\pm$0.15 & 1.48$\pm$0.59 & -1.48$\pm$0.05 & 2.57$\pm$1.14\ UGC 1056 & B & 12600$\pm$900 & 12700$\pm$900 & 4.23$\pm$1.06 & 5.43$\pm$1.04 & 9.66$\pm$1.49 & 7.98$\pm$0.06 & 1.34$\pm$0.21 & -1.49$\pm$0.03 & 3.16$\pm$0.52\ UGC 1176 & A & 12600$\pm$500 & 12800$\pm$500 & 3.40$\pm$0.47 & 5.91$\pm$0.59 & 9.31$\pm$0.75 & 7.97$\pm$0.03 & 1.29$\pm$0.11 & -1.40$\pm$0.02 & 3.73$\pm$0.34\ NGC 784 & B & 12900$\pm$600 & 13400$\pm$700 & 3.57$\pm$0.59 & 4.70$\pm$0.58 & 8.27$\pm$0.83 & 7.92$\pm$0.04 & 0.78$\pm$0.08 & -1.63$\pm$0.02 & 1.94$\pm$0.22\ NGC 784 & A & 12500$\pm$500 & 12400$\pm$500 & 3.41$\pm$0.48 & 7.20$\pm$0.78 & 10.61$\pm$0.91 & 8.03$\pm$0.04 & 1.24$\pm$0.11 & -1.44$\pm$0.02 & 3.87$\pm$0.36\ UGC 2716 & A & 12800$\pm$500 & 13100$\pm$500 & 2.96$\pm$0.42 & 6.41$\pm$0.70 & 9.37$\pm$0.82 & 7.97$\pm$0.04 & 0.99$\pm$0.09 & -1.47$\pm$0.02 & 3.14$\pm$0.31\ NGC 2537 & A & 10900$\pm$600 & 9700$\pm$500 & 8.85$\pm$1.94 & 8.56$\pm$1.60 & 17.4$\pm$2.5 & 8.24$\pm$0.06 & 7.40$\pm$0.99 & -1.07$\pm$0.02 & 14.8$\pm$2.25\ NGC 2537 & B & 11800$\pm$900 & 11200$\pm$900 & 5.26$\pm$1.48 & 4.23$\pm$0.96 & 9.49$\pm$1.76 & 7.98$\pm$0.07 & 5.13$\pm$0.88 & -1.00$\pm$0.02 & 9.41$\pm$1.82\ UGC 4278 & B & 13300$\pm$1000 & 14200$\pm$1000 & 2.53$\pm$0.61 & 2.32$\pm$0.41 & 4.85$\pm$0.73 & 7.69$\pm$0.06 & 0.57$\pm$0.09 & -1.63$\pm$0.03 & 1.11$\pm$0.20\ UGC 4278 & A & 13500$\pm$600 & 14600$\pm$600 & 2.01$\pm$0.27 & 2.86$\pm$0.28 & 4.87$\pm$0.38 & 7.69$\pm$0.03 & 0.52$\pm$0.05 & -1.58$\pm$0.03 & 1.27$\pm$0.13\ NGC 2552 & A & 11400$\pm$500 & 10400$\pm$500 & 5.27$\pm$0.88 & 8.93$\pm$1.23 & 14.19$\pm$1.52 & 8.15$\pm$0.04 & 3.57$\pm$0.37 & -1.16$\pm$0.02 & 9.68$\pm$1.10\ UGC 4393 & B & 12100$\pm$1000 & 11700$\pm$1000 & 4.55$\pm$1.36 & 6.92$\pm$1.64 & 11.5$\pm$0.2 & 8.06$\pm$0.07 & 3.63$\pm$0.66 & -1.09$\pm$0.02 & 9.39$\pm$1.82\ UGC 4393 & C & 12800$\pm$800 & 13100$\pm$800 & 5.80$\pm$1.22 & 3.92$\pm$0.63 & 9.72$\pm$1.37 & 7.99$\pm$0.06 & 3.25$\pm$0.42 & -1.24$\pm$0.02 & 5.55$\pm$0.82\ UGC 5139 & A & 12800$\pm$800 & 13000$\pm$800 & 2.70$\pm$0.60 & 5.65$\pm$0.95 & 8.35$\pm$1.13 & 7.92$\pm$0.05 & 0.72$\pm$0.12 & -1.56$\pm$0.05 & 2.30$\pm$0.41\ IC 559 & A & 12000$\pm$1400 & 11500$\pm$1300 & 5.63$\pm$2.31 & 6.18$\pm$2.02 & 11.81$\pm$3.07 & 8.07$\pm$0.10 & 1.74$\pm$0.46 & -1.47$\pm$0.05 & 3.97$\pm$1.13\ UGC 5272 & A & 13300$\pm$500 & 14100$\pm$200 & 1.34$\pm$0.17 & 6.12$\pm$0.22 & 7.46$\pm$0.27 & 7.87$\pm$0.02 & 0.34$\pm$0.03 & -1.59$\pm$0.02 & 1.92$\pm$0.12\ UGC 5340 & A & 15200$\pm$1200 & 19400$\pm$1500 & 0.47$\pm$0.11 & 1.12$\pm$0.16 & 1.59$\pm$0.19 & 7.20$\pm$0.05 & 0.09$\pm$0.02 & -1.60$\pm$0.08 & 0.40$\pm$0.09\ UGC 5423 & A & 13700$\pm$500 & 15300$\pm$500 & 2.33$\pm$0.27 & 3.55$\pm$0.24 & 5.88$\pm$0.36 & 7.77$\pm$0.03 & 1.08$\pm$0.08 & -1.32$\pm$0.02 & 2.82$\pm$0.21\ UGC 5423 & B & 13400$\pm$1100 & 14400$\pm$1200 & 2.17$\pm$0.59 & 4.37$\pm$0.87 & 6.54$\pm$1.05 & 7.82$\pm$0.06 & 0.79$\pm$0.16 & -1.43$\pm$0.05 & 2.43$\pm$0.48\ UGC 5797 & A & 13200$\pm$1000 & 14000$\pm$1000 & 2.04$\pm$0.51 & 7.06$\pm$1.29 & 9.11$\pm$1.39 & 7.96$\pm$0.06 & 0.95$\pm$0.19 & -1.35$\pm$0.06 & 4.11$\pm$0.84\ UGC 5923 & A & 14300$\pm$2500 & 16600$\pm$3000 & 4.04$\pm$2.22 & 2.08$\pm$0.78 & 6.12$\pm$2.35 & 7.79$\pm$0.14 & 1.97$\pm$0.67 & -1.30$\pm$0.04 & 3.10$\pm$1.23\ NGC 3738 & A & 12100$\pm$900 & 11800$\pm$800 & 4.98$\pm$1.27 & 6.00$\pm$1.21 & 10.98$\pm$1.76 & 8.04$\pm$0.06 & 2.31$\pm$0.36 & -1.33$\pm$0.02 & 5.14$\pm$0.86\ NGC 3738 & B & 12500$\pm$1400 & 12500$\pm$1400 & 5.36$\pm$2.15 & 5.39$\pm$1.67 & 10.75$\pm$2.72 & 8.03$\pm$0.10 & 2.23$\pm$0.55 & -1.37$\pm$0.03 & 4.57$\pm$1.21\ NGC 3741 & A & 13700$\pm$500 & 15200$\pm$400 & 1.81$\pm$0.21 & 2.93$\pm$0.20 & 4.74$\pm$0.29 & 7.68$\pm$0.03 & 0.42$\pm$0.04 & -1.61$\pm$0.03 & 1.15$\pm$0.10\ UGC 6817 & A & 14200$\pm$500 & 16500$\pm$300 & 0.94$\pm$0.10 & 2.47$\pm$0.11 & 3.41$\pm$0.15 & 7.53$\pm$0.02 & 0.26$\pm$0.02 & -1.53$\pm$0.03 & 1.01$\pm$0.07\ NGC 4163 & A & 14800$\pm$2000 & 18200$\pm$2500 & 3.28$\pm$1.34 & 0.34$\pm$0.09 & 3.62$\pm$1.34 & 7.56$\pm$0.14 & 1.00$\pm$0.30 & -1.49$\pm$0.06 & 1.17$\pm$0.47\ CGCG 269-049 & A & 14400$\pm$500 & 17100$\pm$500 & 1.00$\pm$0.11 & 1.95$\pm$0.11 & 2.96$\pm$0.16 & 7.47$\pm$0.02 & 0.20$\pm$0.02 & -1.57$\pm$0.03 & 0.80$\pm$0.08\ UGC 7577 & A & 13100$\pm$900 & 13700$\pm$900 & 2.17$\pm$0.49 & 7.10$\pm$1.18 & 9.27$\pm$1.28 & 7.97$\pm$0.06 & 0.91$\pm$0.15 & -1.37$\pm$0.04 & 3.96$\pm$0.67\ NGC 4449 & C & 11500$\pm$900 & 10600$\pm$800 & 6.92$\pm$1.99 & 6.69$\pm$1.59 & 13.62$\pm$2.54 & 8.13$\pm$0.07 & 3.16$\pm$0.55 & -1.33$\pm$0.02 & 6.30$\pm$1.22\ NGC 4449 & B & 11300$\pm$600 & 10400$\pm$600 & 6.73$\pm$1.44 & 7.79$\pm$1.39 & 14.52$\pm$2.00 & 8.16$\pm$0.06 & 2.96$\pm$0.39 & -1.36$\pm$0.02 & 6.30$\pm$0.91\ NGC 4449 & A & 10800$\pm$500 & 9600$\pm$200 & 6.36$\pm$1.16 & 14.50$\pm$1.08 & 20.87$\pm$1.59 & 8.32$\pm$0.03 & 2.50$\pm$0.28 & -1.39$\pm$0.02 & 8.50$\pm$0.73\ UGC 7605 & A & 13400$\pm$2000 & 15100$\pm$2200 & 2.18$\pm$1.01 & 2.44$\pm$0.80 & 4.61$\pm$1.29 & 7.66$\pm$0.11 & 0.57$\pm$0.23 & -1.54$\pm$0.11 & 1.32$\pm$0.53\ NGC 4656 & A & 12600$\pm$700 & 12700$\pm$700 & 1.20$\pm$0.22 & 11.03$\pm$1.53 & 12.23$\pm$1.55 & 8.09$\pm$0.05 & 0.27$\pm$0.11 & -1.66$\pm$0.14 & 2.66$\pm$1.10\ UGC 8201 & A & 12900$\pm$900 & 13600$\pm$900 & 2.18$\pm$0.50 & 4.07$\pm$0.69 & 6.25$\pm$0.85 & 7.80$\pm$0.06 & 0.28$\pm$0.06 & -1.77$\pm$0.07 & 1.06$\pm$0.30\ UGC 8508 & A & 13300$\pm$1100 & 14300$\pm$1200 & 1.82$\pm$0.50 & 3.90$\pm$0.78 & 5.72$\pm$0.92 & 7.76$\pm$0.07 & 0.34$\pm$0.08 & -1.60$\pm$0.07 & 1.44$\pm$0.36\ UGC 8638 & A & 12600$\pm$500 &12800$\pm$300 & 2.63$\pm$0.36 & 6.73$\pm$0.40 & 9.36$\pm$0.54 & 7.97$\pm$0.02 & 0.79$\pm$0.07 & -1.51$\pm$0.02 & 2.88$\pm$0.20\ UGC 8638 & B & 13000$\pm$500 & 13500$\pm$500 & 2.38$\pm$0.33 & 5.82$\pm$0.60 & 8.20$\pm$0.69 & 7.91$\pm$0.03 & 0.61$\pm$0.07 & -1.58$\pm$0.04 & 2.17$\pm$0.26\ UGC 8837 & A & 12900$\pm$900 & 13400$\pm$900 & 4.76$\pm$1.13 & 1.80$\pm$0.32 & 6.56$\pm$1.18 & 7.82$\pm$0.07 & 2.12$\pm$0.31 & -1.35$\pm$0.02 & 2.94$\pm$0.55\ NGC 5477 & A & 12800$\pm$500 & 13000$\pm$200 & 1.89$\pm$0.25 & 7.10$\pm$0.30 & 8.99$\pm$0.39 & 7.95$\pm$0.02 & 0.50$\pm$0.04 & -1.56$\pm$0.02 & 2.45$\pm$0.14\ [ccccccc]{} UGC 521 & & 7.67$\pm$0.05 & 2.29$\pm$0.06 & -1.61$\pm$0.07 & D: 12, 13\ UGC 695 & & 7.69$\pm$0.12 & 0.50$\pm$0.01 & -1.49$\pm$0.04 &\ UGC 1056 & & 7.97$\pm$0.06 & 0.91$\pm$0.02 & -1.49$\pm$0.02 &\ UGC 1176 & & 7.97$\pm$0.05 & 1.59$\pm$0.05 & -1.40$\pm$0.02 &\ NGC 784 & $\surd$ & 7.97$\pm$0.06 & 1.47$\pm$0.04 & -1.54$\pm$0.10 & S: 2,6\ UGC 2716 & & 7.97$\pm$0.05 & 2.05$\pm$0.06 & -1.47$\pm$0.02 &\ NGC 2537 & & 8.14$\pm$0.13 & 0.63$\pm$0.02 & -1.04$\pm$0.04 & S: 2, 21-23\ UGC 4278 & & 7.69$\pm$0.05 & 1.08$\pm$0.02 & -1.60$\pm$0.03 & D: 16, 17\ NGC 2552 & & 8.15$\pm$0.05 & 1.28$\pm$0.04 & -1.16$\pm$0.02 & S: 2, 17, 18\ UGC 4393 & & 8.02$\pm$0.05 & 0.59$\pm$0.01 & -1.15$\pm$0.08 & D: 16\ UGC 5139 & $\surd$ & 7.92$\pm$0.05 & 1.95$\pm$0.06 & -1.56$\pm$0.05 & D: 5, 8\ IC 559 & & 8.07$\pm$0.10 & 0.90$\pm$0.03 & -1.47$\pm$0.05 &\ UGC 5272 & & 7.87$\pm$0.05 & 4.66$\pm$0.13 & -1.59$\pm$0.02 & D: 6, 10, 11\ UGC 5340 & & 7.20$\pm$0.05 & 3.26$\pm$0.13 & -1.60$\pm$0.08 & D: 6, 7\ UGC 5423 & & 7.78$\pm$0.05 & 1.77$\pm$0.03 & -1.33$\pm$0.04 & D: 2, 5\ UGC 5797 & & 7.96$\pm$0.06 & 3.48$\pm$0.10 & -1.35$\pm$0.06 &\ UGC 5923 & & 7.79$\pm$0.14 & 0.61$\pm$0.02 & -1.30$\pm$0.04 & S: 9\ NGC 3738 & $\surd$ & 8.04$\pm$0.05 & 1.02$\pm$0.03 & -1.34$\pm$0.02 & D: 2, 18-20\ NGC 3741 & $\surd$ & 7.68$\pm$0.05 & 1.78$\pm$0.05 & -1.61$\pm$0.03 & S: 2, 3\ UGC 6817 & $\surd$ & 7.53$\pm$0.05 & 3.07$\pm$0.10 & -1.53$\pm$0.03 &\ NGC 4163 & $\surd$ & 7.56$\pm$0.14 & 0.15$\pm$0.01 & -1.49$\pm$0.06 & S: 2\ CGCG 269-049 & $\surd$ & 7.47$\pm$0.05 & 1.13$\pm$0.03 & -1.57$\pm$0.03 & D: 1\ UGC 7577 & $\surd$ & 7.97$\pm$0.06 & 3.18$\pm$0.09 & -1.37$\pm$0.04 &\ NGC 4449 & $\surd$ & 8.26$\pm$0.09 & 0.86$\pm$0.02 & -1.36$\pm$0.02 & D: 17, 19, 20, 24-26\ UGC 7605 & $\surd$ & 7.66$\pm$0.11 & 1.22$\pm$0.04 & -1.54$\pm$0.10 &\ NGC 4656 & & 8.09$\pm$0.05 & 8.39$\pm$0.35 & -1.66$\pm$0.14 & S: 2, 27\ UGC 8201 & $\surd$ & 7.80$\pm$0.06 & 1.82$\pm$0.05 & -1.77$\pm$0.07 & S: 8\ UGC 8508 & $\surd$ & 7.76$\pm$0.07 & 2.21$\pm$0.06 & -1.60$\pm$0.07 & S: 2, 4\ UGC 8638 & $\surd$ & 7.95$\pm$0.05 & 2.36$\pm$0.05 & -1.53$\pm$0.03 &\ UGC 8837 & & 7.87$\pm$0.07 & 0.36$\pm$0.01 & -1.43$\pm$0.03 & D:15\ NGC 5477 & & 7.95$\pm$0.02 & 0.54$\pm$0.01 & -1.56$\pm$0.02 & D: 14\ [ccccccccccccc]{}\ \ WLM & 00:01:58.6 & -15:27:12 & 62.9 & 117 & 0.97$\pm$0.02 & 1 & -13.50$\pm$0.05 & -16.29$\pm$0.34 & -15.54$\pm$0.34 & 0.46$\pm$0.03 & 7.85$\pm$0.34 & 7.19$\pm$0.34\ NGC 55 & 00:14:53.6 & -39:11:48 & 1390 & 2630 & 2.11$\pm$0.04 & 2 & -18.20$\pm$0.11 & -21.34$\pm$0.36 & -20.61$\pm$0.16 & 0.55$\pm$0.08 & 9.87$\pm$0.35 & 9.30$\pm$0.35\ UGC 00668 & 01:04:49.1 & 02:07:31 & 90.1 & 232 & 0.75$\pm$0.02 & 1 & -13.61$\pm$0.14 & -16.13$\pm$0.37 & -15.73$\pm$0.27 & 0.40$\pm$0.04 & 7.78$\pm$0.37 & 7.14$\pm$0.37\ NGC 1705 & 04:54:13.7 & -53:21:41 & 19.3 & 44.4 & 5.11$\pm$0.17 & 3 & -15.77$\pm$0.52 & -18.62$\pm$0.60 & -18.10$\pm$0.57 & 0.38$\pm$0.18 & 8.78$\pm$0.60 & 8.19$\pm$0.60\ NGC 2366 & 07:28:49.6 & 69:12:32 & 4.99 & 110 & 3.21$\pm$0.05 & 2 & -15.95$\pm$0.11 & -18.64$\pm$0.35 & -18.08$\pm$0.33 & 0.32$\pm$0.05 & 8.79$\pm$0.35 & 8.15$\pm$0.35\ UGC 4305 & 08:19:09.0 & 70:43:28 & 64.6 & 216 & 3.38$\pm$0.05 & 2 & -16.11$\pm$0.12 & -19.03$\pm$0.31 & -18.92$\pm$0.23 & 0.19$\pm$0.06 & 8.95$\pm$0.31 & 8.48$\pm$0.31\ UGC 4459 & 08:34:07.6 & 66:10:39 & 3.10 & 7.93 & 3.61$\pm$0.05 & 2 & -12.93$\pm$0.12 & -15.88$\pm$0.81 & -15.48$\pm$1.17 & 0.46$\pm$0.08 & 7.68$\pm$0.81 & 7.15$\pm$0.81\ Leo A & 09:59:24.8 & 30:44:49 & 13.6 & 34.8 & 0.81$\pm$0.04 & 4 & -10.91$\pm$0.26 & -14.24$\pm$0.43 & -13.82 & 0.24$\pm$0.06 & 7.03$\pm$0.43 & 6.58$\pm$0.43\ Sex B & 10:00:00.0 & 05:19:56 & 36.1 & 136 & 1.39$\pm$0.04 & 2 & -13.54$\pm$0.16 & -16.47$\pm$0.38 & -16.49$\pm$0.25 & 0.40$\pm$0.05 & 7.92$\pm$0.37 & 7.49$\pm$0.38\ Sex A & 10:11:00.7 & -04:41:37 & 24.9 & 62.2 & 1.38$\pm$0.05 & 2 & -13.62$\pm$0.19 & -16.05$\pm$0.39 & -15.62$\pm$0.49 & 0.24$\pm$0.06 & 7.75$\pm$0.39 & 7.08$\pm$0.39\ UGC 5666 & 10:28:35.3 & 68:25:53 & 111 & 165 & 3.79$\pm$0.05 & 2 & -16.81$\pm$0.13 & -19.87$\pm$0.30 & -18.88$\pm$0.33 & 0.34$\pm$0.06 & 9.28$\pm$0.30 & 8.62$\pm$0.30\ NGC 4214 & 12:15:39.0 & 36:19:35 & 224 & 488 & 3.03$\pm$0.05 & 2 & -17.15$\pm$0.10 & -20.15$\pm$0.36 & -19.58$\pm$0.17 & 0.40$\pm$0.03 & 9.39$\pm$0.36 & 8.83$\pm$0.36\ UGC 8091 & 12:58:39.8 & 14:13:06 & 2.29 & 8.40 & 2.08$\pm$0.02 & 2 & -11.76$\pm$0.07 & -14.35$\pm$0.35 & -14.34$\pm$0.97 & 0.27$\pm$0.04 & 7.07$\pm$0.35 & 6.55$\pm$0.36\ IC 5152 & 22:02:41.6 & -51:17:40 & 103 & 306 & 1.97$\pm$0.07 & 5 & -15.47$\pm$0.03 & -18.37$\pm$0.34 & -18.13$\pm$0.15 & 0.41$\pm$0.02 & 8.68$\pm$0.34 & 8.17$\pm$0.34\ \ \ \ SMC & 00:52:44.0 & -72:49:42 & 20.7 & & 0.056$\pm$0.002 & 6 & -16.04$\pm$0.20 & -8.89$\pm$0.34 & & 0.45$\pm$0.10 & 4.89$\pm$0.46 &\ UGC 00685 & 01:07:22.8 & 16:41:02 & 7.48 & 11.6 & 4.70$\pm$0.06 & 5 & -14.13$\pm$0.11 & -17.41$\pm$0.34 & -16.46 & 0.60$\pm$0.09 & 8.30$\pm$0.34 & 7.71$\pm$0.34\ NGC 625 & 01:35:03.9 & -41:26:14 & 88.3 & 242 & 3.89$\pm$0.13 & 7 & -16.28$\pm$0.04 & -19.68$\pm$0.34 & -19.34$\pm$0.17 & 0.59$\pm$0.02 & 9.20$\pm$0.34 & 8.80$\pm$0.34\ LMC & 05:23:34.6 & -69:45:22 & 1.41 & & 0.05$\pm$0.01 & 8 & -17.68$\pm$0.05 & -5.83$\pm$0.36 & & 0.51$\pm$0.08 & 3.66$\pm$0.35 &\ UGC 4483 & 08:37:03.3 & 69:46:34 & 0.92 & 4.19 & 3.41$\pm$0.12 & 2 & -12.71$\pm$0.19 & -14.44$\pm$0.40 & -14.66$\pm$1.21 & 0.15$\pm$0.05 & 7.11$\pm$0.39 & 6.42$\pm$0.40\ UGC 6541 & 11:33:28.8 & 49:14:23 & 3.59 & 12.6 & 3.89$\pm$0.52 & 9 & -13.51$\pm$0.06 & -16.20$\pm$0.34 & -16.14$\pm$0.64 & 0.42$\pm$0.04 & 7.81$\pm$0.34 & 7.30$\pm$0.34\ UGCA 292 & 12:38:40.7 & 32:45:41 & 0.54 & 6.57 & 3.60$\pm$0.08 & 2 & -11.52$\pm$0.23 & -13.94$\pm$0.40 & -15.27 & 0.07$\pm$0.14 & 6.91$\pm$0.40 & 6.68$\pm$0.40\ UGC 8651 & 13:39:53.9 & 40:44:26 & 3.90 & 15.9 & 3.14$\pm$0.05 & 2 & -13.13$\pm$0.11 & -15.83$\pm$0.36 & -15.93$\pm$0.79 & 0.36$\pm$0.06 & 7.66$\pm$0.36 & 7.19$\pm$0.36\ UGC 9128 & 14:15:56.8 & 23:03:22 & 2.43 & 9.25 & 2.21$\pm$0.07 & 2 & -12.12$\pm$0.18 & -14.55$\pm$0.38 & -14.58$\pm$0.88 & 0.31$\pm$0.07 & 7.15$\pm$0.38 & 6.59$\pm$0.39\ UGC 9240 & 14:24:43.1 & 44:31:37 & 10.7 & 33.2 & 2.79$\pm$0.04 & 2 & -13.89$\pm$0.09 & -16.67$\pm$0.36 & -16.47$\pm$0.47 & 0.43$\pm$0.06 & 8.00$\pm$0.36 & 7.47$\pm$0.36\ UGCA 442 & 23:43:46.3 & -31.57:25 & 7.51 & 13.4 & 4.27$\pm$0.53 & 9 & -14.34$\pm$0.62 & -17.21$\pm$0.71 & -16.41$\pm$1.14 & 0.38$\pm$0.03 & 8.21$\pm$0.71 & 7.56$\pm$0.71\ [cccc]{} WLM & 7.83$\pm$0.06 & -1.49$\pm$0.01 & 1\ NGC 55 & 8.05$\pm$0.10 & -1.26$\pm$0.05 & 2\ UGC 00668 & 7.62$\pm$0.05 & -1.51$\pm$0.10 & 3\ NGC 1705 & 8.21$\pm$0.05 & -1.75$\pm$0.06 & 4\ NGC 2366 & 7.91$\pm$0.05 & -1.17$\pm$0.26 & 5\ UGC 4305 & 7.92$\pm$0.10 & -1.52$\pm$0.11 & 6\ UGC 4459 & 7.82$\pm$0.09 & -1.32$\pm$0.17 & 7\ Leo A & 7.30$\pm$0.05 & -1.53$\pm$0.09 & 8\ Sex B & 7.53$\pm$0.05 & -1.49$\pm$0.06 & 9\ Sex A & 7.54$\pm$0.06 & -1.54$\pm$0.13 & 10\ UGC 5666 & 7.93$\pm$0.05 & -1.45$\pm$0.08 & 7\ NGC 4214 & 8.22$\pm$0.05 & -1.32$\pm$0.03 & 11\ UGC 8091 & 7.65$\pm$0.06 & -1.51$\pm$0.07 & 8\ IC 5152 & 7.92$\pm$0.07 & -1.05$\pm$0.12 & 6\ \ SMC & 7.96$\pm$0.15 & -1.55$\pm$0.15 & 12\ UGC 00685 & 8.00$\pm$0.03 & -1.45$\pm$0.08 & 13\ NGC 625 & 8.08$\pm$0.12 & -1.25$\pm$0.03 & 14\ LMC & 8.26$\pm$0.15 & -1.30$\pm$0.20 & 12\ UGC 4483 & 7.56$\pm$0.03 & -1.57$\pm$0.07 & 13\ UGC 6541 & 7.82$\pm$0.06 & -1.45$\pm$0.13 & 15\ UGCA 292 & 7.30$\pm$0.03 & -1.45$\pm$0.07 & 9\ UGC 8651 & 7.85$\pm$0.04 & -1.60$\pm$0.09 & 13\ UGC 9128 & 7.75$\pm$0.05 & -1.80$\pm$0.12 & 16\ UGC 9240 & 7.95$\pm$0.03 & -1.60$\pm$0.06 & 13\ UGCA 442 & 7.72$\pm$0.03 & -1.41$\pm$0.02 & 14\ [cccccc]{} UGC 521 A & 0.03 & 7.90 (L) & 7.65 (3) & 7.87 & 8.14\ UGC 695 E & 0.04 & 8.01 (L) & 7.93 (3) & 8.13 & 8.34\ UGC 1056 A & 0.04 & 8.08 (L) & 8.03 (3) & 8.15 & 8.29\ UGC 1176 A & 0.05 & 8.03 (L) & 7.90 (3) & 8.04 & 8.18\ NGC 784 A & 0.05 & 8.08 (L) & 7.95 (3) & 8.02 & 8.15\ UGC 2716 A & 0.04 & 8.08 (L) & 7.93 (3) & 7.99 & 8.15\ NGC 2537 A & 0.14 & 8.58 (U) & 8.41 (2) & 8.27 & 8.29\ UGC 4278 A & 0.03 & 7.78 (L) & 7.64 (3) & 7.88 & 8.19\ NGC 2552 A & 0.10 & 7.98 (L) & 8.06 (3) & 8.12 & 8.20\ UGC 4393 B & 0.12 & 8.56 (U) & 8.31 (2) & 8.16 & 8.19\ UGC 5139 A & 0.04 & 7.97 (L) & 7.78 (3) & 7.94 & 8.16\ IC 559 A & 0.05 & 8.09 (L) & 8.02 (3) & 8.14 & 8.26\ UGC 5272 A & 0.03 & 7.95 (L) & 7.71 (3) & 7.73 & 8.04\ UGC 5340 A & 0.03 & 7.34 (L) & 7.15 (3) & 7.51 & 8.09\ UGC 5423 A & 0.06 & 7.99 (L) & 7.97 (3) & 8.02 & 8.17\ UGC 5797 A & 0.06 & 8.13 (L) & 8.02 (3) & 7.92 & 8.07\ UGC 5923 A & 0.06 & 8.20 (L) & 8.23 (2) & 8.26 & 8.31\ NGC 3738 A & 0.07 & 8.07 (L) & 8.05 (3) & 8.15 & 8.24\ NGC 3741 A & 0.03 & 7.81 (L) & 7.64 (3) & 7.87 & 8.17\ UGC 6817 A & 0.03 & 7.64 (L) & 7.45 (3) & 7.69 & 8.10\ NGC 4163 A & 0.03 & 8.07 (L) & 7.55 (3) & 8.18 & 8.52\ CGCG 269-049 A & 0.03 & 7.60 (L) & 7.65 (3) & 7.72 & 8.05\ UGC 7577 A & 0.05 & 8.12 (L) & 8.07 (3) & 7.94 & 7.97\ NGC 4449 A & 0.07 & 8.04 (L) & 8.19 (2) & 8.08 & 8.19\ UGC 7605 A & 0.03 & 7.81 (L) & 7.66 (3) & 7.94 & 8.22\ NGC 4656 A & 0.03 & 8.10 (L) & 7.81 (3) & 7.62 & 7.96\ UGC 8201 A & 0.02 & 7.82 (L) & 7.55 (3) & 7.85 & 8.17\ UGC 8508 A & 0.03 & 7.83 (L) & 7.62 (3) & 7.85 & 8.14\ UGC 8638 A & 0.04 & 8.01 (L) & 7.82 (3) & 7.93 & 8.13\ UGC 8837 A & 0.06 & 8.02 (L) & 7.92 (3) & 8.20 & 8.38\ NGC 5477 A & 0.04 & 7.97 (L) & 7.72 (3) & 7.81 & 8.08\ \ Offset & & 0.14 & -0.001 & 0.12 & 0.32\ Dispersion & & 0.22 & 0.17 & 0.24 & 0.42\ [ccccccc]{} NGC 404 & 0.04 & 7.23 (L) & 7.55 (3) & 8.53 & 8.56\ KKH 037 & 0.06 & 8.32 (U) & 8.33 (1) & 8.53 & 8.56\ CGCG 035-007 & 0.05 & 8.00 (L) & 8.10 (3) & 8.57 & 8.37\ UGC 5672 & 0.06 & 8.00 (L) & 8.26 (3) & 8.37 & 8.29\ UGC 5692 & 0.15 & 8.40 (U) & 8.08 (3) & 8.53 & 8.31\ UGC 6782 & 0.08 & 7.85 (L) & & 8.18 & 8.27\ UGC 6900 & 0.12 & 8.00 (U) & 7.69 (3) & 8.08 & 8.17\ UGC 7599 & & 8.09 (U) & & &\ UGC 7639 & 0.05 & 7.77 (L) & 7.99 (3) & 8.50 & 8.42\ UGC 8245 & 0.04 & 7.59 (L) & 7.78 (3) & 8.40 & 8.39\ UGC 9405 & 0.11 & 7.77 (L) & 8.22 (3) & 8.63 & 8.40\ UGC 10818 & 0.09 & 7.82 (L) & & 8.45 & 8.29\ KKH 098 & 0.04 & 7.61 (L) & 7.66 (3) & 8.17 & 8.27\ \[fig:objects\] [^1]: http://www.ast.cam.ac.uk/research/lvls [^2]: Since the inception of the LVL Spitzer program, four galaxies included in the sample have updated distances which place them outside of 11 Mpc [see @dale09; @jlee11]. [^3]: Some galaxy observations were adjusted to shorter or longer exposures depending on the brightness of the \[\] $\lambda$4363 line strength, or included additional exposures when the observing program allowed for it; see Table \[tbl2\]. [^4]: IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. [^5]: http://www1.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/community/YorkExtinctionSolver/ [^6]: At this time the morphologies have not been analyzed for the LVL sample, so we cannot make predictions about the infall of unenriched gas for these galaxies. [^7]: Dispersion in log(O/H) of the “Combined Select" sample increases to $\sigma=0.18$ if UGC 5340 is included. [^8]: Dispersion in log(O/H) of the “Combined Select" sample increases to $\sigma=0.22$ if UGC 5340 is included. [^9]: Dispersion in log(O/H) of the “Combined Select" sample increases to $\sigma=0.21$ if UGC 5340 is included.
--- abstract: | One of the key approaches to save samples when learning a policy for a reinforcement learning problem is to use knowledge from an approximate model such as its simulator. However, *does knowledge transfer from approximate models always help to learn a better policy*? Despite numerous empirical studies of transfer reinforcement learning, an answer to this question is still elusive. In this paper, we provide a strong negative result, showing that even the *full knowledge* of an approximate model may *not* help reduce the number of samples for learning an accurate policy of the true model. We construct an example of reinforcement learning models and show that the complexity with or without knowledge transfer has the same order. On the bright side, effective knowledge transferring is still possible under additional assumptions. In particular, we demonstrate that knowing the (linear) *bases* of the true model significantly reduces the number of samples for learning an accurate policy. author: - 'Fei Feng[^1]' - 'Wotao Yin[^2]' - 'Lin F. Yang[^3]' bibliography: - 'references.bib' title: 'Does Knowledge Transfer Always Help to Learn a Better Policy?' --- Introduction ============ Reinforcement learning (RL) is the framework of learning to control an unknown system through trial and error. Recently, RL achieves phenomenal empirical successes, e.g, AlphaGo [@silver2016mastering] defeated the best human player in Go, and OpenAI used RL to precisely and robustly control a robotic arm [@NIPS2017_7090]. The RL framework is general enough such that it can capture a broad spectrum of topics, including health care, traffic control, and experimental design [@sutton1992reinforcement; @esteva2019guide; @si2001online; @wiering2000multi; @denil2016learning]. However, *successful* applications of RL in these domains are still rare. The major obstacle that prevents RL being widely used is its high sample complexity: both the AlphaGo and OpenAI arm took nearly a thousand years of human-equivalent experiences to achieve good performances. One way to reduce the number of training samples is to mimic how human beings learn – borrow knowledge from previous experiences. In robotics research, a robot may need to accomplish different tasks at different time. Instead of learning every task from scratch, a more ideal situation is that the robot can utilize the similarities between the underlying models of these tasks and adapt them to future new jobs quickly. Another example is that RL agents are often trained in simulators and then applied to real-world [@ng2006autonomous; @itsuki1995soccer; @dosovitskiy2017carla]. It is still desirable to have their performance improved after seeing samples collected from the real-world. One might hope that agents from simulators (approximate models) can adapt to the real world (true model) faster than knowing nothing. Both examples lead to a natural question: > *If models are similar, can we achieve fast adaptation through knowledge transferring?* This paper focuses on answering the above question. Suppose the true unknown model is a Markov Decision Process (MDP) $\cM$ and the RL agent is provided with an approximate model $\cM_0$ with $$\dist(\cM_0, \cM)\le \beta,$$ where $\dist(\cdot, \cdot)$ is a statistic distance and $\beta$ is a small scalar. We would like to study the sample complexity of learning a policy $\pi$ for $\cM$ such that its error[^4] is at most $\varepsilon$ with $\varepsilon\ll \beta$ (i.e., the *high-precision* regime). For a fixed $\varepsilon$, a common wisdom would suggest that better $\cM_0$ (e.g. smaller $\beta$) can help reduce the sample complexity. The most natural choice of $\dist(\cdot,\cdot)$ is the total-variation (TV) distance between the transition kernels of $\cM_0$ and $\cM$. It is well-known (see e.g. @puterman2014markov) that an optimal policy for $\cM_0$ has an error at most $\cO_{\cM}(\beta)$ in $\cM$, where $\cO_{\cM}$ hides the constants determined by the model. In this paper we show, however, to obtain an $\varepsilon$-optimal policy (the formal definition will be given in Sec. \[sec:pre\]), the number of samples is of the form, $$\Omega_{\cM}(\varepsilon^{-2})$$ when $\varepsilon \ll \beta$. Note that the sample complexity is *independent* of $\beta$. In particular, the complexity does not improve as $\beta$ becomes smaller (as long as $\varepsilon \ll \beta$). It renders the knowledge from a TV-distance ball of $\cM$ *useless* when pursing a high-precision control without further structural information of the model. In order to show the lower bound, we leverage techniques for proving hardness in the bandit literature (e.g. @mannor2004sample) and reinforcement learning (e.g. @azar2013minimax) to carefully show that the approximate model does not provide *critical information* that matters for high-precision control of the true model. Therefore, learning high-precision policy does not benefit from the approximate model. To complement the lower bound, we further investigate the possible structural information of a model that provably helps knowledge transferring. We show that if the unknown model is in the convex hull of a set of $K$ known base models, we are able to obtain a high-precision control with a number of samples significantly fewer than that of learning from scratch. Specifically, the number of samples is proportional to $$\cO(\poly(K))$$ rather than the much larger $|\cS|$, the number of states in the model. Related Work ------------ Reducing sample complexity is a core research goal in RL. Many related sub-branches of RL, e.g., multi-task RL [@brunskill2013sample; @ammar2014online; @calandriello2014sparse], lifelong RL [@abel2018policy; @brunskill2014pac], and meta-RL [@al2017continuous], provide different schemes to utilize experiences from previous tasks. Please also see a survey paper [@taylor2009transfer] for more related works. However, these results focus on different special cases of knowledge utilization rather than understanding the fundamental question of whether an approximate model is useful for policy learning and what guarantees we can have. In the area of Sim-to-Real[^5], some works point out that an imperfect approximate model may degrade the performance of learning and efforts have been made to address this issue, e.g [@kober2013reinforcement; @buckman2018sample; @kalweit2017uncertainty; @kurutach2018model]. There has been active empirical research, but little in theory is known. A more related work is @jiang2018pac, who shows that even if the approximate model differs from the real environment in a single state-action pair (but which one is unknown), such an approximate model could still be information-theoretically useless. This is another interesting direction to look at. However, the statistic distance from such a model to the true model can be arbitrarily large and hence the policy of the approximate model does not have a guarantee on the true model. The limitation of the benefit that an approximate model could bring can also be found in @jiang2015doubly, where the authors build a policy value estimator and use the approximate model to reduce variance. However, they demonstrate that, if no extra knowledge is provided, only the part of variance arising from the randomness in policy can be eliminated rather than the stochasticity in state transitions. In order to take more advantage of the previous experiences, additional structure information is needed. A number of structure settings have been studied in the literature. For instance, in @brunskill2013sample, all models are assumed to be drawn from a finite set of MDPs with identical state and action spaces, but different reward and/or transition probabilities; in @abel2018policy, one study case requires that all models share the same transition dynamics and only reward functions change with a hidden distribution; in @mann2012directed, a special mapping between the approximate model and the true model is assumed such that the approximate model can provide a good action-value initialization for the true model; in @calandriello2014sparse, all tasks can be accurately represented in a linear approximation space and the weight vectors are jointly sparse; in @modi2019sample, every model’s transition kernel and reward function lie in the linear span of $K$ known base models. To complement our lower bound, we study an MDP model that shares a similar information structure as in @modi2019sample. In contrast to @modi2019sample, our model is of infinite-horizon and the loss function is also different. Although not the main focus of this paper, our proposed model and algorithm provide another effective approach that supports knowledge transferring. It is worth mentioning that the structure information such as the existence of a lower-dimensional knowledge-sharing space is not exclusive to RL. One can also find their applications in supervised multi-task learning, e.g. @kumar2012learning [@ruvolo2013ella; @maurer2013sparse]. Preliminaries {#sec:pre} ------------- #### Notation We use small letters (e.g. $s,a,t, p(s'|s,a)$) for scalars, capital letters (e.g., $R, V, Q$) for vectors or functions and (e.g. $N, K, L$) for some specific scalars, capital boldface letters (e.g. $\vP$, $\vP^{\pi}, \vU$) for matrices, and calligraphic letters (e.g., $\cS,\cA$) for sets. The cardinality of a set is denoted by $|\cdot|$. We use $[K]$ to represent the set $\{1,2,\dots,K\}$. The simplex in $\RR^k$ is denoted by $\Delta^k:=\{(x_1,x_2,\dots,x_k)^T~|~\sum_{i=1}^k x_i=1, x_i\geq 0\}$. We abbreviate Kullback-Leibler divergence to KL and use $\cO, \Omega$ and $\Theta$ to denote leading orders in upper, lower, and minimax lower bounds, respectively; and we use $\widetilde{\cO}, \widetilde{\Omega}$ and $\widetilde{\Theta}$ to hide the polylog factors. #### Markov Decision Process In this paper, we focus on the discounted Markov Decision Process (MDP) with an infinite horizon, while the same analysis straightforwardly extends to other settings of MDP. We use $\cM$ to represent an MDP. Each MDP is described as a tuple $(\cS, \cA, \vP, R, \gamma)$, where $\mathcal{S}$ is a finite state space $\{s_1,s_2,\dots, s_{|\cS|}\}$, $\mathcal{A}$ is a finite action space $\{a_1,a_2,\dots, a_{|\cA|}\}$, $\vP$ is an $|\cS||\cA|\times |\cS|$ matrix with each row being a state transition distribution, $R: \cS\times \cA\times\cS\rightarrow [0,1]$ is the reward function, and $\gamma\in(0,1)$ is a discounted factor. We denote by $p(s'|s,a)$ the $\big((s,a),s'\big)$th entry in $\vP$ and $P(\cdot|s,a)^\top$ the $(s,a)-$th row in $\vP$. We use $R(s,a,\cdot)$ for the vector $[R(s,a,s_1);R(s,a,s_2);\cdots;R(s,a,s_{|\cS|})]\in\RR^{|\cS|}$ and $N$ for the total number of state-action pairs. At each time step, the controller observes a state $s \in \mathcal{S}$ and selects an action $a \in \mathcal{A}$ according to a *policy* $\pi$, where $\pi$ maps a state to an action. The action transitions the environment to a new state $s'$ with probability $p(s'|s,a)$. Meanwhile, the controller receives an instant reward $R(s,a,s')$. Given a policy $\pi: \mathcal{S} \rightarrow \mathcal{A}$, we define its value function $V^{\pi}:\cS\rightarrow \RR$ as: $$\label{eq:Vdef} V^{\pi}(s) := \mathbb{E}^\pi\bigg[\sum_{t=0}^\infty \gamma^{t} R(s_t ,a_t, s_{t+1})|s_0=s\bigg],$$ where the expectation is taken over the transition trajectory following $\pi$, i.e., $a_t = \pi(s_t)$ and $s_{t+1}\sim P(\cdot|s_t,a_t)$. The objective of RL is to learn an optimal policy $\pi^*$ such that its value (on any state $s\in \cS$) is maximized over all policies, i.e., $$\forall \pi, s:\quad V^{\pi}(s)\le V^{\pi^*}(s).$$ We also denote the *optimal value* function $ V^{\pi^*}$ as $V^*$. In practice, the optimal value/policy is in general not attainable. Therefore, it makes sense to study sub-optimal policies. Denote an *$\varepsilon$-optimal policy* $\pi$ for $\cM$ as such that $$\forall s\in \cS:\quad 0\le V^*(s) - V^{\pi}(s) \le \varepsilon.$$ We also denote the *action-value function* (or *Q-function*) for a policy $\pi$ as $Q^{\pi}:\cS\times\cA\rightarrow \RR$. Specifically, $$\label{eq:Qdef} Q^{\pi}(s,a) := P(\cdot|s,a)^\top (R(s,a,\cdot) +\gamma V^{\pi}) \quad \text{and}\quad Q^{*}(s,a) = Q^{\pi^*}(s,a).$$ We also adapt the notion of $\varepsilon$ sub-optimality for value function and Q-function, i.e., $V$ and $Q$ are $\varepsilon$-optimal if $\|V-V^*\|_{\infty}\le \varepsilon$ and $\|Q-Q^*\|_{\infty}\le \varepsilon$. Furthermore, if we let $R^{\pi}\in\RR^{|\cS|}$ with the $i$th coordinate being $P(\cdot|s_i,\pi(s_i))^\top R(s_i,\pi(s_i),\cdot)$ and $\vP^{\pi}\in\RR^{|\cS|\times|\cS|}$ with the $i$th row being $P(\cdot|s_i,\pi(s_i))^\top$, then by definition, it holds that $$\begin{aligned} \label{eq:Vpi} V^{\pi} = R^{\pi} + \gamma \vP^{\pi} V^{\pi}.\end{aligned}$$ #### TV-distance for MDPs To measure the closeness of two MDPs, $\cM^1:=(\cS,\cA,\vP^1,R^1,\gamma)$ and $\cM^2:=(\cS,\cA,\vP^2,R^2,\gamma)$, we introduce the following metric $d_{\mathrm{TV}}(\cdot, \cdot)$: $$d_{\mathrm{TV}}(\cM^1, \cM^2) = \max\Big\{ \|\vP^1-\vP^2\|_{\infty}, ~ \|R^1-R^2\|_{\infty}~\Big\}.$$ Note that the distance is only valid between MDPs with the same state and action spaces, and the discounted factor. The name TV comes from that $$\|\vP^1-\vP^2\|_{\infty} = \max_{(s,a)\in\cS\times\cA}\|P^1(\cdot|s,a)-P^2(\cdot|s,a)\|_1$$ and $\|P^1(\cdot|s,a)-P^2(\cdot|s,a)\|_1$ is also the *total variation* distance between $P^1(\cdot|s,a)$ and $P^2(\cdot|s,a)$. We denote by $\cM^2\in B_{\mathrm{TV}}(\cM^1,\beta)$ if $d_{\mathrm{TV}}(\cM^1,\cM^2)\leq \beta$ for some $\beta>0$. #### Generative Model Given an MDP, we define a special sample oracle – generative model. A generative model allows any $(s,a)\in\cS\times\cA$ as input and outputs $(s',R(s,a,s'))$ with $s'\sim P(\cdot|s,a)$. Problem Formulation and Illustration ==================================== We formalize our setting and the considered problem of knowledge transferring as below. Suppose the unknown true model is an MDP $\cM$ and an agent is provided with the full information of a prior model $\cM_0$ satisfying $\cM\in B_{\mathrm{TV}}(\cM_0, \beta)$, where $\beta >0$ is a known constant. How many samples does it take to learn an $\varepsilon$-optimal policy for $\cM$, where $\epsilon\ll \beta$ is an accuracy parameter? Due to the property of TV-distance, an optimal policy of $\cM_0$ is already a $\beta/(1-\gamma)$-optimal policy for $\cM$. It is natural to hope that a smaller $\beta$ leads to a smaller sample complexity, i.e., the knowledge from $\cM_0$ helps policy learning in $\cM$. However, when a higher precision is desired, i.e., $\varepsilon\ll \beta$, we show that the number of samples only depends on $\varepsilon$, rather than $\beta$, unless additional assumptions of the model are made. Such a conclusion is particularly striking since the knowledge of the approximate model from the vicinity of the true model is almost “useless.” A Basic Case ------------ We illustrate our point with a simple case. Define two MDPs $\cM_0$ and $\cM$ as shown in Figure \[fig:simple\]. Both of them have 5 states $\{x,y_1,y_2,z_1,z_2\}$, where $x$ has two actions $a_1$ and $a_2$ and the rest are all single-action states. After taking $a_i$, $x$ will transition to $y_i$ deterministically. In $\cM_0$, $y_1$ and $y_2$ transition to themselves with probabilities 0.5 and 0.4, respectively and to $z_1$ and $z_2$ with probability 0.5 and 0.6, respectively. In $\cM$, each $y_i$ transitions to itself with probability $p_i$ and to $z_i$ with probability $1-p_i$. $p_1$ and $p_2$ are unknown. In both models, $z_1$ and $z_2$ are absorbing states. They also have the same reward function: $R(s,a,s')=1$ if $s'\in\{y_1,y_2\}$ and $R(s,a,s')=0$ otherwise. Without loss of generality, we take $\beta=0.2$ (the reader can easily generate similar examples for other values of $\beta$ following the same principle). Since $\cM\in B_{\mathrm{TV}}(\cM_0, \beta)$, we have that $|p_1-0.5|\leq0.1$ and $|p_2-0.4|\leq 0.1$. ![The left MDP is $\cM_0$ and the right MDP is $\cM$, where $|p_1-0.5|\leq0.1$ and $|p_2-0.4|\leq0.1$.[]{data-label="fig:simple"}](toyexamples.png){width="220pt"} In $\cM$, if $p_1>p_2$, the optimal policy should have $\pi^*(x)=a_1$ and $V^*(x)=\frac{\gamma}{1-\gamma p_1}$. If a policy $\pi$ returns $a_2$ for $x$, then its state value at $x$ is $V^{\pi}(x) = \frac{\gamma}{1-\gamma p_2}$. Thus $\pi$ is $|\frac{\gamma}{1-\gamma p_1}- \frac{\gamma}{1-\gamma p_2}|$-optimal. When $\varepsilon<|\frac{\gamma}{1-\gamma p_1}- \frac{\gamma}{1-\gamma p_2}|$, to produce an $\varepsilon$-optimal policy, an algorithm must find out $p_1>p_2$ with high probability. For $p_2>p_1$, vice versa. Therefore, the problem of learning an $\varepsilon$-optimal policy is equivalent to identifying the larger value from $\{p_1, p_2$}. The knowledge we can use from $\cM_0$ is that $p_1\in [0.4,0.6]$ and $p_2\in [0.3,0.5]$, which does not help reduce the sample complexity due to the overlap. Empirical Verification {#sec:toyexample1} ---------------------- Besides the previous simple case, we also do a numerical demonstration on a sailing problem [@sailing]. In Figure \[fig:toytest\], we generate two MDPs $\cM_0$ and $\cM$ with $\cM\in B_{\mathrm{TV}}(\cM_0,0.3)$. We compare the performances of two algorithms: 1. direct Q-learning [@watkins1992q] with transition samples from $\cM$ (blue line); 2. use the full knowledge of $\cM_0$ to generate a nearly optimal Q-function for $\cM_0$, then use that Q-function to initialize the proceeding Q-learning algorithm with transition samples from $\cM$ (red line). Both algorithms use the same batch of transition samples from $\cM$. Since $\cM_0$ is close to $\cM$, the warm-start Q-learning is much better than the learning-from-scratch counterpart in the initial stage. However, these two curves overlap when they become closer to the optimal value, indicating similar sample complexities for both algorithms when pursuing a high-precision Q-value estimation. ![A toy test comparison between direct Q-learning and warm-start Q-learning with a nearly optimal Q-value of $\cM_0$ as initialization.[]{data-label="fig:toytest"}](toy.png "fig:"){width=".5\textwidth"}\ Lower Bound of Transfer Learning from a TV-distance Ball {#sec:lowerbdd} ======================================================== In this section, we formally prove that an approximate model does not help when learning a high-precision policy. In particular, we show the following lower bound. (Main Result) \[thm:pilowerbound\] Let $\cM$ be an unknown MDP. Suppose MDP $\cM_0$ is given and it satisfies $\cM\in B_{\mathrm{TV}}(\cM_0, \beta)$. There exists $\varepsilon_0$, $\delta_0\in(0,1)$ such that for all $\varepsilon\in(0,\varepsilon_0)$, $\delta\in(0, \delta_0)$, the sample complexity of learning an $\varepsilon$-optimal policy for $\cM$ with probability at least $1-\delta$ is $$\Omega\bigg(\frac{N}{(1-\gamma)^3\varepsilon^2}\log\big(\frac{1}{\delta}\big)\bigg).$$ As shown in @azar2013minimax and @sidford2018near [@agarwal2019optimality], the sample complexity of directly learning an $\varepsilon$-optimal policy for an MDP with high probability under a generative model is $$\widetilde{\Theta}\bigg(\frac{N}{(1-\gamma)^3\varepsilon^2}\log\Big(\frac{N}{\delta}\Big)\bigg).$$ We conclude that for any $\beta>0$, when $\varepsilon$ is small enough, the sample complexity of learning *with* prior knowledge is at least as hard as learning *without* prior knowledge, if we only know the true model lies in a small TV-distance ball of the approximate model. As any online algorithm can be applied in the generative model case, the lower bound automatically adapts to the online setting as well. Before starting the proof, we give the following definition about the correctness of RL algorithms. (($\cM_0, \beta, \varepsilon, \delta$)-correctness)\[def:epsilonalgorithm\] Given $\beta>0$ and a prior model $\cM_0$, we say that an RL algorithm $\sA$ is $(\cM_0, \beta, \varepsilon, \delta)$-correct if for every $\cM\in B_{\mathrm{TV}}(\cM_0, \beta)$, $\sA$ can output an $\varepsilon$-optimal policy with probability at least $1-\delta$. Next, we construct a class of MDPs. We are going to select one model $\cM^0$ from the class as prior knowledge. Then we show that if an RL algorithm $ \sA$ learns with samples significantly fewer than the lower bound, there would always exist an MDP $\cM\in B_{\mathrm{TV}}(\cM_0, \beta)$ such that $\sA$ cannot be $(\cM_0, \beta, \varepsilon, \delta)$-correct. Hence, the lower bound complexity is established. #### Construction of the Hard Case We define a family of MDPs $\mathbb{M}$. These MDPs have the structure as depicted in Figure \[fig:mdp\]. The state space $\cS$ consists of three disjoint subsets $\cX$ (gray nodes), $\cY_1$ (green nodes), and $\cY_2$ (blue nodes). The set $\cX$ includes $K$ states $\{x_1, x_2, \dots, x_K\}$ and each of them has $L$ available actions $\{a_1,a_2,\dots,a_L\}=:\cA$. States in $\cY_1$ and $\cY_2$ are all of single-action. In total, there are $N:=3KL$ state-action pairs. For state $x\in\cX$, by taking action $a\in\cA$, it transitions to a state $y_1(x,a)\in\cY_1$ with probability 1. Note that such a mapping is one-to-one from $\cX\times\cA$ to $\cY_1$. For state $y_1(x,a)\in\cY_1$, it transitions to itself with probability $p_{\cM}(x,a)\in(1/2,1)$ and to a corresponding state $y_2(y_1) \in \cY_2$ with probability $1-p_{\cM}(x,a)$. $p_{\cM}(x,a)$ can be different for different models. All states in $\cY_2$ are absorbing. The reward function is: $R(s,a,s')=1$, if $s'\in\cY_1$; $R(s,a,s')=0$, otherwise. ![The class of MDPs considered in the proof of Theorem \[thm:pilowerbound\]. Nodes represent states and arrows show transitions. $\cX$ consists of all grey nodes. $\cY_1$ comprises of all green nodes. Blue nodes form $\cY_2$.[]{data-label="fig:mdp"}](bigexample.png "fig:"){width=".98\textwidth"}\ $\mathbb{M}$ is a generalization of a multi-armed bandit problem used in @mannor2004sample to prove a lower bound on bandit learning. A similar example is also shown in @azar2013minimax to prove a lower bound on reinforcement learning without any prior knowledge. For an MDP $\cM\in\mathbb{M}$, it is fully determined by the parameter set $\{p_{\cM}(x_k,a_l), k\in[K], l\in[L]\}$. And its Q-function has the values: $$\begin{aligned} Q_{\cM}(x,a) = \frac{1}{1-\gamma p_{\cM}(x,a)}, \quad \forall~ (x,a)\in\cX\times\cA.\end{aligned}$$ #### Prior Model $\cM_0$ and Hypotheses of $\cM$ Now, we select a *prior* model $\cM_0\in\mathbb{M}$ with $$p_{\cM_0}(x,a)\equiv \frac{4\gamma-1}{3\gamma}=:p_0, \quad \forall~(x,a)\in\cX\times\cA,$$ where $\gamma$ is the discounted factor. We restrict $\gamma\in(0.4,1)$, then $p_0\in(1/2,1)$. Given $\beta>0$, let $\sA$ be an $(\cM_0, \beta, \varepsilon, \delta)$-correct algorithm. Denote by $\beta':=\min(\beta, 1-p_0)$. We consider $1+K(L-1)$ possibilities of $\cM$: $$\begin{aligned} \cM_{1}:& \begin{cases}p_{\cM_1}(x_k,a_1) = p_0+\alpha_1,\quad\forall~ k\in[K],\\ p_{\cM_1}(x_k,a_l)=p_0, \quad \forall~k\in[K], l\neq1;\end{cases}\\ \text {for every } k\in[K], l\neq1, \quad \cM_{k,l}:& \begin{cases}p_{\cM_{k,l}}(x_{k},a_{l}) = p_0+\alpha_2,\\ p_{\cM_{k,l}}(x_{k'},a_{l'}) = p_{\cM_1}(x_{k'},a_{l'}), \quad \forall (k',l')\neq(k,l),\end{cases} \end{aligned}$$ where $\alpha_1$ is selected such that $$\begin{aligned} Q_{\cM_{1}}(x_k,a_1)-Q_{\cM_1}(x_k,a_l)=\frac{1}{1-\gamma(p_0+\alpha_1)}-\frac{1}{1-\gamma p_0}=2\varepsilon\end{aligned}$$ and $\alpha_2 = \frac{4(1-\gamma p_0)^2\varepsilon}{\gamma}$ such that $$Q_{\cM_{k,l}}(x_{k},a_{l})-Q_{\cM_{k,l}}(x_{k},a_1) = \frac{1}{1-\gamma(p_0+\alpha_2)}-\frac{1}{1-\gamma (p_0+\alpha_1)}\geq2\varepsilon.$$ Note that the parameter set of $\cM_1$ differs from that of $\cM_0$ only on action $a_1$ and the parameter set of $\cM_{k,l}$ differs from that of $\cM_1$ only on pair $(x_{k},a_{l})$. When $\varepsilon\leq\frac{\beta'\gamma}{8(1-\gamma p_0)^2}$, $0<\alpha_1<\alpha_2\leq \beta'/2$. Thus, all models above lie in $B_{\mathrm{TV}}(\cM_0, \beta)$. We refer to them as *hypotheses* of $\cM$. Every hypothesis gives a probability measure over the same sample space. We denote by $\mathbb{E}_1$, $\mathbb{P}_1$ and $\mathbb{E}_{k,l}$, $\mathbb{P}_{k,l}$ the expectation and probability under hypothesis $\cM_1$ and $\cM_{k,l}$, respectively. These probability measures capture both the randomness in the corresponding MDP and the randomization carried out by the algorithm $\sA$, for example its sampling strategy. It is worth mentioning that in @azar2013minimax, the authors implicitly assume that the sampling numbers to different states are determined before the start of the algorithm and do not change during learning (this is due to their *conditionally independence* argument in Lemma 18). Such an assumption does not apply to adaptive sampling strategy. In our result, adaptive sampling is included. In the sequel, we fix $\varepsilon\in(0, \varepsilon_0)$ and $\delta\in(0, \delta_0)$, where $\varepsilon_0$ and $\delta_0$ will be determined later. Let $$t^* = \frac{c_1}{(1-\gamma)^3\varepsilon^2}\log\Big(\frac{1}{4\delta}\Big),$$ where $c_1>0$ is to be determined later. We also define $T_{k,l}$ the number of samples that algorithm $\sA$ calls from the generative model with input state $y_1(x_{k},a_{l})$ till $\sA$ stops (these sample calls are not necessarily consecutive). For every $k\in[K], l\neq1$, we define the following three events: $$\begin{aligned} A_{k,l} &= \{T_{k,l}\leq 4t^*\},\\ B_{k,l} &= \{ \sA \text{ outputs a policy } \pi \text{ with } \pi(x_k) = a_1\},\\ C_{k,l} &= \Big\{ S_{k,l}-p_0T_{k,l} \leq\sqrt{2p_0(1-p_0)T_{k,l}\log(1/4\delta)}\Big\},\end{aligned}$$ where $S_{k,l}$ is the sum of rewards (non-discounted) by calling the generative model $T_{k,l}$ times with input state $y_1(x_k,a_l)$. For these events, we have the following lemmas. For any $k\in[K], l\neq 1$, if  $\mathbb{E}_1[T_{k,l}]\leq t^*$, $\mathbb{P}_1(A_{k,l})> 3/4$. $$t^*\geq \mathbb{E}_1[T_{k,l}]> 4t^*\mathbb{P}_1(T_{k,l}>4t^*)=4t^*(1-\mathbb{P}_0(T_1\leq 4t^*)).$$ Thus, $\mathbb{P}_1(A_{k,l})> 3/4$. For any $k\in[K], l\neq1$, if  $\delta<1/16$, $\mathbb{P}_1(C_{k,l})\geq 3/4$. When $l\neq1$, under hypothesis $\cM_1$, $p_{\cM_1}(x_k,a_l)=p_0$. By definition, the instant rewards from state $y_1(x_k,a_l)$ are i.i.d. Bernoulli-$p_0$ random variables. Denote by $\epsilon:=\sqrt{2p_0(1-p_0)T_{k,l}\log(1/4\delta)}$. By Chernoff-Hoeffding bound and $p_0>1/2$, we have that $$\begin{aligned} &\mathbb{P}_1\bigg(S_{k,l}-p_0T_{k,l} \leq\sqrt{2p_0(1-p_0)T_{k,l}\log(1/4\delta)}\bigg)\\ \geq& ~1-\exp\bigg(-\text{KL}\bigg(p_0+\frac{\epsilon}{T_{k,l}}~||~p_0\bigg)\cdot T_{k,l}\bigg)\geq 1-\exp\bigg(-\frac{\epsilon^2}{2p_0(1-p_0)T_{k,l}}\bigg)=1-4\delta.\end{aligned}$$ Thus, when $\delta<1/16$, $\mathbb{P}_1(C_{k,l})\geq 3/4$. Now, we set $\delta_0$ as $1/16$, then for $\delta\in(0,\delta_0)$, $\sA$ should return a policy $\pi$ such that when $\cM=\cM_1$, $\pi(x_{k})=a_1$ for every $k\in[K]$ with probability at least $1-\delta$, i.e. $\mathbb{P}_1(B_{k,l})\geq 1-\delta\geq 1-1/4,$ for all $k\in[K]$ and $l\neq 1$. Define the event $\cE_{k,l}:=A_{k,l}\cap B_{k,l} \cap C_{k,l}$. Combining the results above, we have that $$\begin{aligned} \mathbb{P}_1(\cE_{k,l})>1-3/4=1/4, \quad \forall~ k\in[K], l\neq 1.\end{aligned}$$ Next, we show that if the expectation of number of samples in $\sA$ on any $y_1(x_k,a_l)$ is less than $t^*$, then $B_{k,l}$ occurs with probability greater than $\delta$ under the hypothesis $\cM_{k,l}$. \[lemma:newversion\] Let $\varepsilon_0 = \frac{\beta'\gamma}{8(1-\gamma p_0)^2}$. For any $k\in[K], l\neq1$, when $\varepsilon\in(0, \varepsilon_0)$, if  $\mathbb{E}_1[T_{k,l}]<t^*$, then $\mathbb{P}_{k,l}(B_{k,l})>\delta$. Given $k\in[K]$ and $l\neq 1$, we denote by $W$ the length-$T_{k,l}$ random sequence of the instant rewards by calling the generative model $T_{k,l}$ times with the input state $y_1(x_k,a_l)$. As one can see, if $\cM=\cM_1$, this is an i.i.d. Bernoulli-$p_0$ sequence; if $\cM=\cM_{k,l}$, this is an i.i.d Bernoulli-$(p_0+\alpha_2)$ sequence. We define the likelihood function $L_{k,l}$ by letting $$L_{k,l}(w) = \mathbb{P}_{k,l}(W=w)$$ for every possible realization $w$. This function can be used to define a random variable $L_{k,l}(W)$, where $W$ is the sample path of the random sequence. Following the previous notation, $S_{k,l}$ is the sum of rewards, i.e. the total number of getting 1s in $W$. Then we have the likelihood ratio $L_{k,l}(W)/L_1(W)$ as $$\begin{aligned} \frac{L_{k,l}(W)}{L_1(W)}=& \frac{(p_0+\alpha_2)^{S_{k,l}}(1-p_0-\alpha_2)^{T_{k,l}-S_{k,l}}}{(p_0)^{S_{k,l}}(1-p_0)^{T_{k,l}-S_{k,l}}}\\ =& \left(1+\frac{\alpha_2}{p_0}\right)^{S_{k,l}}\left(1-\frac{\alpha_2}{1-p_0}\right)^{T_{k,l}-S_{k,l}}\\ =& \left(1+\frac{\alpha_2}{p_0}\right)^{S_{k,l}}\left(1-\frac{\alpha_2}{1-p_0}\right)^{S_{k,l}\frac{1-p_0}{p_0}}\left(1-\frac{\alpha_2}{1-p_0}\right)^{T_{k,l}-S_{k,l}/p_0}.\end{aligned}$$ By our choice of $p_0$, $\alpha_2$, and $\varepsilon$, it holds that $\alpha_2/(1-p_0)\in(0, 1/2]$ and $\alpha_2/p_0\in(0,1/2)$. With the fact that $\log (1-u) \geq-u-u^{2}$ for $u\in[0,1/2]$ and $\exp (-u) \geq 1-u$ for $u\in[0,1]$, we have that $$\begin{aligned} \bigg(1-\frac{\alpha_2}{1-p_0}\bigg)^{\frac{1-p_0}{p_0}}& \geq \exp \left(\frac{1-p_0}{p_0}\left(-\frac{\alpha_2}{1-p_0}-\big(\frac{\alpha_2}{1-p_0}\big)^{2}\right)\right) \\ & \geq\left(1-\frac{\alpha_2}{p_0}\right)\left(1-\frac{\alpha_2^{2}}{p_0(1-p_0)}\right).\end{aligned}$$ Thus $$\begin{aligned} \frac{L_{k,l}(W)}{L_{1}(W)} & \geq\left(1-\frac{\alpha_2^2}{p_0^2}\right)^{S_{k,l}}\left(1-\frac{\alpha_2^{2}}{p_0(1-p_0)}\right)^{S_{k,l}}\bigg(1-\frac{\alpha_2}{1-p_0}\bigg)^{T_{k,l}-S_{k,l}/p_0} \\ & \geq\left(1-\frac{\alpha_2^2}{p_0^2}\right)^{T_{k,l}}\left(1-\frac{\alpha_2^{2}}{p_0(1-p_0)}\right)^{T_{k,l}}\bigg(1-\frac{\alpha_2}{1-p_0}\bigg)^{T_{k,l}-S_{k,l}/p_0}\end{aligned}$$ due to $S_{k,l}\leq T_{k,l}$. Next, we proceed on the event $\cE_{k,l}$. By definition, if $\cE_{k,l}$ occurs, event $A_{k,l}$ has occurred. Using $\log(1-u)\geq -2u$ for $u\in[0,1/2]$, it follows that $$\left(1-\frac{\alpha_2^2}{p_0^2}\right)^{T_{k,l}} \geq \left(1-\frac{\alpha_2^2}{p_0^2}\right)^{4t^*} \geq \exp \left(-8t^* \frac{\alpha_2^2}{p_0^2}\right) \geq\left(4\delta\right)^{7000c_1}.$$ Using $\log(1-u)\geq -2u$ for $u\in[0,1/2]$, we have that $$\left(1-\frac{\alpha_2^{2}}{p_0(1-p_0)}\right)^{T_{k,l}} \geq \left(1-\frac{\alpha_2^{2}}{p_0(1-p_0)}\right)^{4t^*}\geq \exp \left(-8t^* \frac{\alpha_2^{2}}{p_0(1-p_0)}\right) \geq\left(4\delta\right)^{7000c_1}.$$ Further, we have that when $\cE_{k,l}$ occurs, $C_{k,l}$ also occurs. Therefore, $$\begin{aligned} \left(1-\frac{\alpha_2}{1-p_0}\right)^{T_{k,l}-S_{k,l}/{p_0}} & \geq\left(1-\frac{\alpha_2}{1-p_0}\right)^{\sqrt{\frac{1-p_0}{p_0}T_{k,l}\log(1/4\delta)}} \geq\left(1-\frac{\alpha_2}{1-p_0}\right)^{\sqrt{\frac{1-p_0}{p_0}4t^*\log(1/4\delta)}} \\ & \geq \exp \left(-\sqrt{16\frac{\alpha_2^2}{p_0(1-p_0)}t^*\log(1/4\delta)}\right) \\ & \geq\left(4\delta \right)^{\sqrt{13000c_1}}.\end{aligned}$$ By taking $c_1$ small enough, e.g. $c_1=10^{-5}$, we have $$\frac{L_{k,l}(W)}{L_{1}(W)} \geq (4\delta).$$ By a change of measure, we deduce that $$\label{eq:p2} \mathbb{P}_{k,l}(B_{k,l})\geq \mathbb{P}_{k,l}(\cE_{k,l})=\mathbb{E}_{k,l}[\mathbf{1}_{\cE_{k,l}}]=\mathbb{E}_1\left[\frac{L_{k,l}(W)}{L_1(W)}\mathbf{1}_{\cE_{k,l}}\right]\geq 4\delta*1/4=\delta.$$ If $\sA$ is $(\cM^0,\beta,\varepsilon, \delta)$-correct, under hypothesis $\cM_{k,l}$, $\sA$ should produce a policy $\pi$ such that $\pi(x_k)=a_l$ with probability greater than $1-\delta$. Thus, we should have $\mathbb{P}_{k,l}(B_{k,l})<\delta$ for all $k\in[K], l\neq 1$. From Lemma \[lemma:newversion\], it requires $\mathbb{E}_1[T_{k,l}]>t^*$ for all $k\in[K], l\neq 1$. In total, we need $\Omega\left(\frac{N}{(1-\gamma)^3\varepsilon^2}\log(1/\delta)\right)$ samples, which concludes our proof of Theorem \[thm:pilowerbound\]. A Case Study for Knowledge Transfer in Reinforcement Learning ============================================================= In this section, we impose a new assumption on *similarities* among models such that transferring knowledge achieves fast adaptation. We consider a sequence of MDPs, where they have the same state and action spaces, and the discounted factor, but different transition dynamics and/or reward functions. At each time step $t$, we want to learn an $\varepsilon$-optimal policy for $\cM_t:=(\cS,\cA,\vP_t, R_t, \gamma).$ The assumption we propose is a convex hull structure as stated below. \[ass:convex\]Given a finite set of MDPs $\mathbb{M}:=\{\cM^1, \cM^2, \dots, \cM^K\}$ where $\cM^k=(\cS,\cA,\vP^k, R^k, \gamma)$, we have $\cM_t\in \text{conv}(\mathbb{M})$[^6] for all $t>0$. We have full knowledge of all MDPs in $\mathbb{M}$ and access to a generative model of each $\cM_t$. We define a set of matrices $\cV:=\{\vV_{s,a}\in\RR^{|\cS|\times K}, s\in \cS, a\in\cA\}$, where the $k$th column of $\vV_{s,a}$ is $P^k(\cdot|s,a)$. Since $\cM_t\in\text{conv}(\mathbb{M})$, there exists a vector $C_t\in\Delta^K$ such that $$\begin{aligned} \label{eq:convex} P_t(\cdot|s,a)=\vV_{s,a}~ C_t, \quad \forall~(s,a)\in\cS\times\cA.\end{aligned}$$ We define a matrix $\vU\in\RR^{|\cS|^2|\cA|\times K}$ by stacking all $\vV_{s,a}$ vertically, i.e. $$\vU:=\begin{bmatrix} \vV_{s_1,a_1}\\ \vV_{s_1,a_2}\\ \vdots\\ \vV_{s_{|\cS|}, a_{|\cA|}}\end{bmatrix}.$$ We make the following assumption about $\vU$. \[ass:fullrank\] $\vU$ has full column rank. Since $K$ is much smaller than $|\cS|^2|\cA|$, the assumption is easy to be satisfied in real applications. Then a direct result is: \[lemma:kpair\] There exists a set $\{(s'_k,a'_k)\}_{k=1}^K$ such that the matrix formed by stacking all $\vV_{s'_k,a'_k}$ vertically has column rank $K$. The proof is easy with basic linear algebra. Let $\{(s'_k,a'_k)\}_{k=1}^K$ be the set in Lemma \[lemma:kpair\]. We define $$\begin{aligned} \label{eq:Utrun} &\vU_{\text{trun}}\in\RR^{K|\cS|\times K}:=\frac{1}{K}\begin{bmatrix}\vV_{s'_1,a'_1}\\ \vV_{s'_2,a'_2}\\ \vdots\\ \vV_{s'_K,a'_K}\end{bmatrix}, \quad P_t\in\RR^{K|\cS|} :=\frac{1}{K}\begin{bmatrix}P_t(\cdot|s'_1,a'_1)\\ P_t(\cdot|s'_2,a'_2)\\ \vdots\\ P_t(\cdot|s'_K,a'_K)\end{bmatrix}. \end{aligned}$$ Basically, we shrink the size of $\vU$ and normalize it to $\vU_{\text{trun}}$. Equation reduces to $$\label{eq:realC} P_t = \vU_{\text{trun}} C_t.$$ Let $\lambda_{\text{max}}$ and $\lambda_{\text{min}}$ be the largest and smallest eigenvalues of $\vU_{\text{trun}}^\top\vU_{\text{trun}}$, respectively. Note that $\lambda_{\text{min}}>0$ due to the full column rank property of $\vU_{\text{trun}}$. We give an algorithm to learn an $\varepsilon$-optimal policy for $\cM_t$. The algorithm is presented in Algorithm \[alg:case2\]. For every $t>0$, we first take samples to construct $P_t$’s empirical estimation $\widehat{P}_t$. Then we find a vector $\widehat{C}_t\in\Delta^K$ such that $\vU_{\text{trun}}\widehat{C}_t\approx \widehat{P}_t$. Next, we use $\widehat{C}_t$ to form a model $\widetilde{\cM}_t\in\text{conv}(\mathbb{M})$ as an approximation to the true $\cM_t$. Finally, the algorithm returns a policy which is $\varepsilon/2$-optimal policy for $\widetilde{\cM_t}$. We will show in the proceeding that the output policy is $\varepsilon$-optimal for $\cM_t$. **Input:** $\mathbb{M}:=\{\cM^1,\cM^2,\dots,\cM^K\}, \{(s'_k,a'_k)\}_{k=1}^K, \text{ a generative model } $`GM`$ , \varepsilon>0, \delta>0,$ a zero vector $\widehat{P}_t\in\RR^{K|\cS|}, L=\lceil\frac{432K\lambda_{\text{max}}}{\varepsilon^2(1-\gamma)^4\lambda_{\text{min}}^2}\log(\frac{1+K|\cS|}{\delta})\rceil$. $i\gets 1$, $\widehat{P}_t\gets 0$; Sample $j\in[K]$ uniformly; Sample $(s,r)\gets \texttt{GM}(s'_j,a'_j)$; Increment the $n$th coordinate in $\widehat{P}_t$ by 1, where $n=(j-1)|\cS|+s$; $i\gets i+1;$ Normalize $\widehat{P}_t$: $\widehat{P_t}\gets \frac{1}{L}\widehat{P_t}$; Calculate $\widehat{C}_t:=\Proj_{\Delta^K}(\vU_{\text{trun}}^\top\vU_{\text{trun}})^{-1}\vU_{\text{trun}}^\top\widehat{P}_t$ ($\vU_{\text{trun}}$ as defined in Equation ); Formulate $\widetilde{\cM}_t:=(\cS,\cA,\sum_{i=1}^K c_t^K \vP^k, \sum_{k=1}^K c_t^KR^k, \gamma)$, where $[c_t^1, c_t^2, \dots, c_t^K]^\top=\widehat{C}_t;$ Run any planning algorithm and get an $\varepsilon/2$-optimal policy $\pi_t$ for $\widetilde{\cM}_t$. **Output:** $\pi_t$. Our proof consists of two steps: 1. we show in Lemma \[lemma:closemdp\] that if the convex coefficients of two MDPs $\cM, \widehat{\cM} \in \text{conv}(\mathbb{M})$ are close under $\|\cdot\|_2$, then an $\varepsilon$-optimal policy $\pi$ for $\widehat{\cM}$ is also nearly optimal for $\cM$; 2. we show in Lemma \[lemma:cbound\] that the convex coefficients of $\widetilde{\cM}_t$ and $\cM_t$ are close under $\|\cdot\|_2$. \[lemma:closemdp\] Suppose for MDPs $\cM:=(\cS,\cA,\vP,R,\gamma)$ and $\widehat{\cM}:=(\cS,\cA,\widehat{\vP},\widehat{R},\gamma)$, there exists two vectors $C:=[c_1,c_2,\dots,c_K]\in\Delta^K$ and $D:=[d_1,d_2,\dots,d_K]\in\Delta^K$ such that $$\begin{aligned} \vP&=\sum_{k=1}^K c_k \vP^k,\quad R=\sum_{k=1}^K c_k R^k;\\ \widehat{\vP} &=\sum_{k=1}^K d_k \vP^k, \quad \widehat{R} = \sum_{k=1}^K d_k R^k. \end{aligned}$$ If $\|C-D\|_2\leq \alpha$, then an $\varepsilon'$-optimal policy for $\widehat{\cM}$ is $\bigg(\varepsilon'+6\alpha\sqrt{K}/(1-\gamma)^2\bigg)$-optimal for $\cM$. Denote by $V^*$ and $\widehat{V}^*$ the optimal value vectors for $\cM$ and $\widehat{\cM}$, respectively. Given an $\varepsilon'$-optimal policy $\pi$ for $\widehat{\cM}$, we denote by $V^{\pi}$ and $\widehat{V}^{\pi}$ the value vectors following $\pi$ in $\cM$ and $\widehat{\cM}$, respectively. Then, by triangle inequality, we first have that $$\begin{aligned} \|V^{\pi}-V^*\|_{\infty} &\leq \|V^{\pi}-\widehat{V}^{\pi}\|_{\infty}+\|\widehat{V}^{\pi}-\widehat{V}^*\|_{\infty}+\|\widehat{V}^*-V^*\|_{\infty}\\ &\leq \|V^{\pi}-\widehat{V}^{\pi}\|_{\infty}+\varepsilon'+\|\widehat{V}^*-V^*\|_{\infty}\label{eq:valuebound}\end{aligned}$$ To bound $\|V^{\pi}-\widehat{V}^{\pi}\|_{\infty}$, we notice that from Equation , $$\begin{aligned} V^{\pi}=(\vI-\gamma \vP^{\pi})^{-1} R^{\pi}, \quad \widehat{V}^{\pi}=(\vI-\gamma \widehat{\vP}^{\pi})^{-1} \widehat{R}^{\pi}.\end{aligned}$$ By Gershgorin Circle Theorem [@gershgorin1931uber], the absolute values of all eigenvalues of $\gamma \vP^{\pi}$ are strictly smaller than 1. So $(\vI-\gamma \vP^{\pi})^{-1}=\vI+\sum_{n=1}^\infty \big(\gamma\vP^{\pi}\big)^n$. Based on this, it holds that $$\begin{aligned} \|V^{\pi}-\widehat{V}^{\pi}\|_{\infty}&= \left\|(\vI-\gamma \vP^{\pi})^{-1} R^{\pi}-(\vI-\gamma \widehat{\vP}^{\pi})^{-1} \widehat{R}^{\pi}\right\|_{\infty} \\ &= \left\|R^{\pi}-\widehat{R}^{\pi} + \sum_{n=1}^{\infty} \gamma^n \Big[(\vP^{\pi})^n R^{\pi}-(\widehat{\vP}^{\pi})^n \widehat{R}^{\pi}\Big]\right\|_{\infty}\\ &\leq \|R^{\pi}-\widehat{R}^{\pi}\|_{\infty}+\sum_{n=1}^{\infty} \gamma^n \Big[\left\|\big((\vP^{\pi})^n-(\widehat{\vP}^{\pi})^n\big) R^{\pi}\right\|_{\infty} + \left\|(\widehat{\vP}^{\pi})^n(R^{\pi}-\widehat{R}^{\pi})\right\|_{\infty}\Big]\\ &\leq \|R^{\pi}-\widehat{R}^{\pi}\|_{\infty}+\sum_{n=1}^{\infty} \gamma^n \left[\left\|(\vP^{\pi}-\widehat{\vP}^{\pi})\Big(\sum_{i=0}^{n-1} (\vP^{\pi})^i(\widehat{\vP}^{\pi})^{n-1-i}\Big) R^{\pi}\right\|_{\infty} + \|R^{\pi}-\widehat{R}^{\pi}\|_{\infty}\right]\\ &= \frac{\|R^{\pi}-\widehat{R}^{\pi}\|_{\infty}}{1-\gamma}+\sum_{n=1}^{\infty} \gamma^n \Big[\|(\vP^{\pi}-\widehat{\vP}^{\pi})R_n\|_{\infty}\Big]\label{eq:ineq}\end{aligned}$$ where $R_n:=\sum_{i=0}^{n-1} (\vP^{\pi})^i(\widehat{\vP}^{\pi})^{n-1-i}R^{\pi}\in[0,n]^{|\cS|}$. The third line is by triangle inequality and the fourth line is due to that $(\widehat{\vP}^{\pi})^n$ is a transition matrix. For the first term in Equation , we have that $$\begin{aligned} \|R^{\pi}-\widehat{R}^{\pi}\|_{\infty}&=\max_{s\in\cS}\left|P(\cdot|s,\pi(s))^\top R(s,\pi(s),\cdot)-\widehat{P}(\cdot|s,\pi(s))^\top \widehat{R}(s,\pi(s),\cdot)\right|\\ &\leq \max_{s\in\cS}\left| P(\cdot|s,\pi(s))^\top R(s,\pi(s),\cdot)-\widehat{P}(\cdot|s,\pi(s))^\top R(s,\pi(s),\cdot)\right|\\ &\quad+\max_{s\in\cS}\left|\widehat{P}(\cdot|s,\pi(s))^\top R(s,\pi(s),\cdot) - \widehat{P}(\cdot|s,\pi(s))^\top \widehat{R}(s,\pi(s),\cdot)\right|\\ &=\max_{s\in\cS}\left|\sum_{k=1}^K (c_k-d_k) P^k(\cdot|s,\pi(s))^\top R(s,\pi(s), \cdot)\right|+\max_{s\in\cS}\left|\sum_{k=1}^K(c_k-d_k) \widehat{P}(\cdot|s,\pi(s))^\top R^k(s,\pi(s), \cdot)\right|\\ &\leq 2\|C-D\|_1\leq 2\sqrt{K}\|C-D\|_2,\label{eq:Rpi}\end{aligned}$$ where the second line is by triangle inequality and the last line is due to $R^k(s,\pi(s),\cdot)\in[0,1]^{|\cS|}$. For the second term in Equation , observe that $$\begin{aligned} \|(\vP^{\pi}-\widehat{\vP}^{\pi})R_n\|_{\infty}&=\max_{s\in\cS} \left|\sum_{k=1}^K (c_k-d_k)P^k(\cdot|s,\pi(s))^\top R_n\right|\\ &\leq \|C-D\|_1 \|R_n\|_{\infty}\leq n\sqrt{K}\|C-D\|_2.\label{eq:pn}\end{aligned}$$ Combining and , we have the following result: $$\begin{aligned} \eqref{eq:ineq} &\leq 2\alpha\sqrt{K}/(1-\gamma) + \alpha \sqrt{K} \gamma/(1-\gamma)^2\leq 3\alpha\sqrt{K}/(1-\gamma)^2.\end{aligned}$$ To bound $\|\widehat{V}^*-V^*\|_{\infty}$, let $\pi^*$ and $\widehat{\pi}^*$ be optimal policies for $\cM$ and $\widehat{\cM}$, respectively. Then it holds that $$\begin{aligned} \widehat{V}^*-V^* \leq \|\widehat{V}^{\widehat{\pi}^*}-V^{\widehat{\pi}^*}\|_{\infty}, \quad V^* -\widehat{V}^* \leq \|V^{\pi^*}-\widehat{V}^{\pi^*}\|_{\infty}.\end{aligned}$$ Following the same steps as before, we have $\|\widehat{V}^{\widehat{\pi}^*}-V^{\widehat{\pi}^*}\|_{\infty}\leq 3\alpha\sqrt{K}/(1-\gamma)^2$ and $ \|V^{\pi^*}-\widehat{V}^{\pi^*}\|_{\infty}\leq 3\alpha\sqrt{K}/(1-\gamma)^2$. Therefore, $\|\widehat{V}^*-V^*\|_{\infty}\leq 3\alpha\sqrt{K}/(1-\gamma)^2$. The desired result is obtained. \[lemma:cbound\] Let $C_t$ be the solution of Equation . Then in Algorithm \[alg:case2\], for each $t>0$, $\|\widehat{C}_t-C_t\|_2\leq (1-\gamma)^2\varepsilon/(12\sqrt{K})$ with probability at least $1-\delta$. As defined in Equation , $\|P_t\|_1=1$. Therefore, $P_t$ can be taken as a distribution. The samples we take in Algorithm \[alg:case2\] are indeed all i.i.d. samples following $P_t$. Therefore, $\widehat{P}_t$ is an empirical approximation of $P_t$ with $L$ samples. By Bernstein inequality for matrices [@tropp2015introduction Theorem 6.1.1], we have that $$\mathbb{P}\Big(\|\widehat{P}_t-P_t\|_2\leq \epsilon \Big)\geq 1-(1+K|\cS|)\exp \Big(\frac{-L^2\epsilon^2}{2L + 2L/3}\Big)\geq 1-(1+K|\cS|)\exp\Big(\frac{-L\epsilon^2}{3}\Big).$$ Taking $\epsilon=\frac{\lambda_{\text{min}}\varepsilon(1-\gamma)^2}{12\sqrt{\lambda_{\text{max}}}\sqrt{K}}$, when $L=\lceil\frac{432K\lambda_{\text{max}}}{\varepsilon^2(1-\gamma)^4\lambda_{\text{min}}^2}\log(\frac{1+K|\cS|}{\delta})\rceil$, we have that $$\mathbb{P}\Big(\|\widehat{P}_t-P_t\|_2\leq \frac{\lambda_{\text{min}}\varepsilon(1-\gamma)^2}{12\sqrt{\lambda_{\text{max}}}\sqrt{K}}\Big)\geq 1- \delta.$$ Based on this, it holds that with probability at least $1-\delta$, $$\begin{aligned} \|\widehat{C}_t-C_t\|_2 &= \| \Proj_{\Delta^K}\big((U_{\text{trun}}^\top U_{\text{trun}})^{-1}U_{\text{trun}}^\top \widehat{P}_t\big)- (U_{\text{trun}}^\top U_{\text{trun}})^{-1}U_{\text{trun}}^\top P_t\|_2\\ &\leq \|(U_{\text{trun}}^\top U_{\text{trun}})^{-1}U_{\text{trun}}^\top \widehat{P}_t- (U_{\text{trun}}^\top U_{\text{trun}})^{-1}U_{\text{trun}}^\top P_t\|_2\\ & \leq \frac{\sqrt{\lambda_{\text{max}}}}{\lambda_{\text{min}}}\|\widehat{P}_t-P_t\|_2,\end{aligned}$$ where the second line is due to that $\Delta^K$ is a convex set and $C_t\in\Delta^K$. Therefore, $\|\widehat{C}_t-C_t\|_2\leq \frac{\varepsilon(1-\gamma)^2}{12\sqrt{K}}$ with probability at least $1-\delta$. Combining Lemma \[lemma:closemdp\] and \[lemma:cbound\], and the fact that $\pi_t$ is $\varepsilon/2$-optimal for $\widetilde{\cM}_t$, we have the following result. \[prop:scforconv\] Let $\varepsilon>0$ and $\delta>0$. Under Assumption \[ass:convex\] and \[ass:fullrank\], for any $t>0$, with probability at least $1-\delta$, Algorithm \[alg:case2\] returns an $\varepsilon$-optimal policy for $\cM_t$ with samples $$\begin{aligned} \cO\bigg(\frac{K}{\varepsilon^2(1-\gamma)^4}\log(\frac{1+K|\cS|}{\delta})\bigg).\end{aligned}$$ The sample complexity in Proposition \[prop:scforconv\] is irrelevant with $|\cS|$ which makes it significantly smaller than the lower bound $\Omega\big(\frac{N}{\varepsilon^2(1-\gamma)^3}\log(1/\delta)\big).$ Therefore, fast adaptation is achieved. Conclusion ========== In this paper, we show that transfer learning from a TV-distance neighbourhood cannot reduce the sample complexity of learning a high precision policy of an MDP, unless extra structural information is provided. We study the case where the new unknown model can be represented as a convex combination of a finite set of base models. In this setting, transfer learning achieves significantly lower sample complexity compared with learning from scratch. [^1]: fei.feng@math.ucla.edu [^2]: wotaoyin@math.ucla.edu [^3]: linyang@ee.ucla.edu, corresponding author. [^4]: The error of a policy is the difference between the values of the policy and the optimal policy. [^5]: It stands for simulator-to-real-environment [^6]: $\cM\in \text{conv}(\mathbb{M})$ if there exists a vector $C:=[c_1,c_2,\dots,c_k]^\top\in\Delta^k$ such that for any $(s,a)\in\cS\times\cA$, $P(\cdot|s,a)= \sum_{k=1}^K c_k P^k(\cdot|s,a)$ and $R=\sum_{k=1}^K c_k R^k$.
--- abstract: 'Inductive and coinductive structures are everywhere in mathematics and computer science. The induction principle is well known and fully exploited to reason about inductive structures like natural numbers and finite lists. To prove theorems about coinductive structures such as infinite streams and infinite trees we can appeal to bisimulation or the coinduction principle. Pure inductive and coinductive types however are not the only data structures we are interested to reason about. In this paper we present a calculus to prove theorems about mutually defined inductive and coinductive data types. Derivations are carried out in an infinitary sequent calculus for first order intuitionistic multiplicative additive linear logic with fixed points. We enforce a condition on these derivations to ensure their cut elimination property and thus validity. Our calculus is designed to reason about linear properties but we also allow appealing to first order theories such as arithmetic, by adding an adjoint downgrade modality. We show the strength of our calculus by proving several theorems on (mutual) inductive and coinductive data types.' author: - Farzaneh Derakhshan - Frank Pfenning title: 'Circular Proofs in First-Order Linear Logic with Least and Greatest Fixed Points' --- &lt;ccs2012&gt; &lt;concept&gt; &lt;concept\_id&gt;10011007.10011006.10011008&lt;/concept\_id&gt; &lt;concept\_desc&gt;Software and its engineering General programming languages&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10003456.10003457.10003521.10003525&lt;/concept\_id&gt; &lt;concept\_desc&gt;Social and professional topics History of programming languages&lt;/concept\_desc&gt; &lt;concept\_significance&gt;300&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;/ccs2012&gt; Introduction ============ The induction principle is well known and presented in the literature in many different contexts. Computer scientists use this principle to reason about inductive data types such as natural numbers and finite lists. To show properties of coinductive data types, e.g. streams and infinite trees, a dual principle of coinduction is needed. In the literature bisimulation has been used effectively to prove equality of structures defined as greatest fixed points. To prove properties other than equality for coinductive data types one needs to use the somewhat less familiar coinduction principle[@brandt1998coinductive; @hermida1998structural; @niqui2009coinductive]. Kozen and Silva established a practical proof principle to produce sound proofs by coinduction [@kozen2017practical]. However for data types mutually defined by induction and coinduction these separate principles are insufficient. One recent approach in type theory integrates induction and coinduction by pattern and copattern matching and explicit well-founded induction on ordinals[@abel2016well], following a number of earlier representations of induction and coinduction in type theory [@abel2013wellfounded]. Here, we pursue a different line of research in *linear logic* with fixed points. In this paper we introduce a sequent calculus to reason about linear predicates defined as nested least and greatest fixed points. Instead of applying induction and coinduction principles directly, we follow the approach of Brotherston et al. [@brotherston2005cyclic] to allow circularity in derivations. We use cyclic reasoning in the context of first order intuitionistic multiplicative additive linear logic extended with least and greatest fixed points. To ensure soundness of the proofs we impose a validity condition on our derivations. Fortier et al. introduced an infinitary sequent calculus for propositional singleton logic with fixed points, where antecedent and succedent consist of exactly one formula [@Fortier13csl; @santocanale2002calculus]. Adding circularity comes with the cost of losing the cut elimination property. To recover this property they introduced a guard condition that ensures soundness of possibly infinite derivations. They provide a cut elimination algorithm and show its productivity on derivations satisfying their guard condition. Fortier and Santocanale’s result has been generalized by Baelde et al. [@baelde2016infinitary; @doumane2017infinitary] for propositional MALL with fixed points. In this paper, we extend Fortier et al.’s results to *first order* multiplicative additive linear logic with fixed points. Our notion of validity is adapted from its counterpart in their system. We introduce a similar cut elimination algorithm and prove its local termination on valid derivations with a dual approach. It is worth mentioning that our calculus is essentially different from the finitary one introduced by Baelde for the first order MALL with fixed points [@baelde2007least] since we allow for circularity. We will show with several examples that our calculus is strong enough to prove many (mutual) inductive and coinductive theorems. To make the examples concise we may use pattern matching for defining inductive predicates [@brotherston2005cyclic; @rosu2017matching]. Our underlying system is designed to reason about linear structures. However, some properties of linear structures rely on first order non-linear theories such as theory of arithmetic or order theory. To be able to prove these properties as well we extend our calculus by mixing linear and structural formulas. Our approach is to use a restricted version of the adjoint logic presented by Pfenning et al. [@benton1994mixed; @pfenning2015polarized]. The restriction is that only linear formulas can depend on non-linear ones and not vice versa. Thus we only add the *downgrade* operator ($\downarrow$) that embeds a nonlinear formula into a linear one to our language. In this way we can isolate the reasoning about nonlinear properties to the pure structural part, and use any sound nonlinear theory in a modular way. In summary, the main contributions of this paper is to introduce an infinitary sequent calculus for first order multiplicative additive linear logic with (mutual) least and greatest fixed points. We provide a validity condition on derivations that ensures the cut elimination property. Our calculus is a tool to reason about a rich signature of mutually defined inductive and coinductive predicates and also allows using nonlinear first order theories. We show its strength by providing several examples including properties defined as nested least and greatest fixed points. First order intuitionistic linear logic with fixed points ========================================================= The syntax of formulas in the first order intuitionistic multiplicative additive linear logic with fixed points ($\mathit{FIMALL}^{\infty}_{\mu,\nu}$) follows the grammar [$$\begin{array}{lcl} A & ::= & 1 \mid 0 \mid \top \mid A \otimes A \mid A \multimap A \mid A \oplus A \mid A\,\&\, A \\ & & \mid \exists x.\, A(x) \mid \forall x.\, A(x) \mid s=t \mid T(\overline{t}) \end{array}$$]{}where $s,t$ stand for terms[^1] and $x,y$ for term variables. $T(\overline{t})$ is a predicate variable defined using least and greatest fixed points in a *signature* $\Sigma$. [$$\Sigma ::= \cdot \mid \Sigma, T(\overline{x})=^{i}_{\mu} A \mid \Sigma, T(\overline{x})=^{i}_\nu A$$]{} The subscript $a$ of a fixed point $T(\overline{x})=^i_{a}$ determines its polarity. If $a=\mu$, then predicate $T(\overline{x})$ is of positive polarity and if $a=\nu$ it is of negative polarity. We represent inductively defined predicates (e.g., the property of being a natural number) as fixed points with positive polarity and coinductively defined predicates (e.g., the lexicographic order on streams) as fixed points with negative polarity. Here we restrict $\Sigma$ to the definitions in which each predicate occurs only in positive (variant) or negative(contravariant) positions, i.e. we do not allow mixed positions[@REYNOLDS81A; @pierce2002types]. The superscript $i\in \mathbb{N}$ is the relative priority of $T(\overline{x})$ in the signature $\Sigma$ with the condition that if $ T_1(\overline{x})=^{i}_{a} A, T_2(\overline{x})=^{i}_b B \in \Sigma$, then $a=b$. Similar to prior work ([@Fortier13csl; @derakhshan2019circular]) we use priority on predicates to define the validity condition on infinite derivations. \[Nat-pred\] Let signature $\Sigma_1$ be [$$\begin{array}{lcl} \mathtt{Nat}(x)&=^1_{\mu} &(\exists y. (x=\mathsf{s} y)\, \otimes \, \mathtt{Nat}(y))\, \oplus\, ((x=\mathsf{z})\, \otimes\, 1)\\ \mathtt{Even}(x)&=^2_{\mu}& (\exists y. (x=\mathsf{s}y)\, \otimes\, \mathtt{Odd}(y)) \oplus ((x=\mathsf{z})\, \otimes\, 1)\\ \mathtt{Odd}(x)&=^2_{\mu}& (\exists y. (x=\mathsf{s}y)\, \otimes\, \mathtt{Even}(y)) \end{array}$$]{}where positive predicates $\mathtt{Nat}$, $\mathtt{Even}$, and $\mathtt{Odd}$ refer to the properties of being natural, even, and odd numbers respectively. We interpret it as $\mathtt{Nat}$ having a higher priority relative to $\mathtt{Even}$ and $\mathtt{Odd}$. A judgment in $\mathit{FIMALL}^{\infty}_{\mu,\nu}$ is of the form $\Gamma \vdash_{\Sigma} A$ where $\Gamma$ is a set of formulas and $\Sigma$ is the signature. We omit $\Sigma$ from the judgments, since it never changes throughout a proof. The infinitary sequent calculus for this logic is given in Figure \[fig:rules-1\], in which we generalize $\oplus$ and $\&$ to be $n$-ary connectives $\oplus\{l_j:A_j\}_{j \in I}$ and $\&\{l_i:A_i\}_{j \in I}$. The binary disjunction and conjunction are defined as $A\oplus B=\oplus \{\pi_1:A,\pi_2:B\}$ and $A\& B=\& \{\pi_1:A,\pi_2:B\}$. Constants $0$ and $\top$ defined as the nullary version of these connectives: $0=\oplus\{\}$ and $\top=\&\{\}$. $$\begin{array}{cc} \infer[\mathtt{fwd}]{A \vdash A}{} & \infer[\mathtt{Cut}]{\Gamma, \Gamma' \vdash C}{\Gamma \vdash A & \Gamma' , A \vdash C} \\[1em] \infer[1R]{\cdot \vdash 1}{} & \infer[1L]{\Gamma, 1 \vdash C}{ \Gamma \vdash C} \\[1em] \infer[\otimes R]{\Gamma, \Gamma' \vdash A_1 \otimes A_2}{\Gamma \vdash A_1 & \Gamma' \vdash A_2}&\infer[\otimes L]{\Gamma, A_1 \otimes A_2 \vdash B}{\Gamma, A_1, A_2\vdash B }\\[1em] \infer[\multimap R]{\Gamma \vdash A_1 \multimap A_2}{\Gamma , A_1 \vdash A_2}&\infer[\multimap L]{\Gamma, \Gamma', A_1 \multimap A_2 \vdash B}{\Gamma \vdash A_1 & \Gamma', A_2 \vdash B}\\[1em] \infer[\oplus R]{\Gamma \vdash \oplus \{l_i: A_i\}_{i \in I}}{\Gamma \vdash A_k & k \in I}&\infer[\oplus L]{\Gamma, \oplus\{l_i:A_i\}_{i\in I} \vdash B}{\Gamma, A_i\vdash B & \forall i \in I}\\[1em] \infer[\& R]{\Gamma \vdash \& \{l_i: A_i\}_{i \in I}}{\Gamma \vdash A_i & \forall i \in I}&\infer[\& L]{\Gamma, \&\{l_i:A_i\}_{i\in I} \vdash B}{\Gamma, A_k\vdash B & k \in I}\\[1em] \infer[\exists R]{\Gamma \vdash \exists x. P(x)}{\Gamma \vdash P(t)}&\infer[\exists L]{\Gamma, \exists x. P(x) \vdash B}{\Gamma, P(x)\vdash B }\\[1em] \infer[\forall R]{\Gamma \vdash \forall x. P(x)}{\Gamma \vdash P(x)}&\infer[\forall L]{\Gamma, \forall x. P(x) \vdash B}{\Gamma, P(t)\vdash B }\\[1em] \infer[\mu_{T} R]{\Gamma \vdash T(\overline{t})}{\Gamma \vdash [\overline{t}/\overline{x}]A & T(\overline{x})=_{\mu} A } &\infer[\mu_{T} L]{\Gamma, T(\overline{t}) \vdash B}{\Gamma, [\overline{t}/\overline{x}]A\vdash B & T(\overline{x})=_{\mu} A }\\[1em] \infer[\nu_{T} R]{\Gamma \vdash T(\overline{t})}{\Gamma \vdash [\overline{t}/\overline{x}]A & T(\overline{x})=_{\nu}A } &\infer[\nu_{T} L]{\Gamma, T(\overline{t}) \vdash B}{\Gamma, [\overline{t}/\overline{x}]A\vdash B & T(\overline{x})=_{\nu} A }\\[1em] \infer[= R]{\cdot \vdash s=s}{}&\infer[= L]{\Gamma, s=t \vdash B}{\Gamma[\theta] \vdash B[\theta] & \theta \in \mathtt{mgu}(t,s) } \end{array}$$ \[Odd1\] Consider signature $\Sigma_1$ and predicates $\mathtt{Even}$ and $\mathtt{Odd}$ defined in Example \[Nat-pred\]. The following derivation is a finite proof of one ($\mathsf{s}\mathsf{z}$) being an odd number. [$$\infer[\mu_{\mathtt{Odd}}R]{\cdot \vdash \mathtt{Odd}(\mathsf{s}\mathsf{z})}{\infer[\exists R]{ \cdot \vdash \exists y. (\mathsf{s}\mathsf{z}=\mathsf{s}y)\, \otimes\, \mathtt{Even}(y) }{ \infer[\otimes R]{\cdot \vdash (\mathsf{s}\mathsf{z}=\mathsf{s}\mathsf{z})\, \otimes\, \mathtt{Even}(\mathsf{z})}{\infer[=R]{\cdot \vdash \mathsf{s}\mathsf{z}=\mathsf{s}\mathsf{z}}{} & \infer[\mu_{\mathtt{Even}R}]{\cdot \vdash \mathtt{Even}(\mathsf{z})}{\infer[\oplus R]{\cdot \vdash(\exists y. (x=\mathsf{s}y)\, \otimes\, \mathtt{Odd}(y)) \oplus ((\mathsf{z}=\mathsf{z})\, \otimes\, 1)}{\infer[\otimes R]{\cdot \vdash (\mathsf{z}=\mathsf{z})\, \otimes\, 1}{\infer[=R]{\cdot \vdash \mathsf{z}=\mathsf{z}}{} & \infer[1R]{\cdot \vdash 1}{}}}}}}}$$]{} The calculus in Figure \[fig:rules-1\] is infinitary, meaning that it allows building infinite derivations as well. The infinite derivations we are interested in, are those we can represent in a finite way. A *circular derivation* is the finite representation of an infinite one in which we can identify each open subgoal with an identical interior judgement. In the first order context we may need to use a substitution rule right before a circular edge to make the subgoal and interior judgment exactly identical [@brotherston2005cyclic]: [$$\qquad \qquad \infer[\mathtt{subst}_{\theta}]{\Gamma[\theta] \vdash B[\theta]}{\Gamma\vdash B}$$]{}We can transform a circular derivation to its underlying infinite derivation in a productive way by deleting the $\mathtt{subst}_{\theta}$ rule and the circular edge. We need to instantiate the derivation to which the circular edge pointed with substitution $\theta$. This instantiation exists and does not change the structure of derivation by Lemma \[lem:subst\] in the Appendix. \[Oddsx\] Consider Signature $\Sigma_1$ and predicates $\mathtt{Nat}$, $\mathtt{Even}$, and $\mathtt{Odd}$ defined in Example \[Nat-pred\]. Figure \[fig:ex-evenodd\] represents a circular derivation for $\mathtt{Even}(x) \vdash \mathtt{Odd}(\mathsf{s}\,x)$. $\Pi$ is the finite derivation given in Example \[Odd1\]. We can interpret the proof in Example \[Oddsx\] as an inductive proof where its circular edge corresponds to applying the induction hypothesis. In the next two examples we represent two coinductive proofs in our circular calculus. Both examples are adapted from @kozen2017practical. \[bisim\] Define $\Sigma_2$ to consist of a single predicate with negative polarity [$\sim(x,y)=^1_{\nu} (\mathsf{hd}\, x = \mathsf{hd}\, y) \, \& \sim(\mathsf{tl}\, x, \mathsf{tl}\, y). $]{}\ Predicate $\sim(x,y)$ can be read as a bisimulation between streams $x$ and $y$. We present a circular derivation for $\sim$ being symmetric in Figure \[fig:bisim\]. We can reason about the properties of stream operations in our calculus as well. Consider three operations $\mathsf{merge}$, $\mathsf{split}_1$ and $\mathsf{split}_2$. Operation $\mathsf{merge}$ receives two streams and merge them into a single stream by alternatively outputting an element of each. Operations $\mathsf{split}_1$ and $\mathsf{split}_2$ receive a stream $x$ as an input and return the odd and even elements of it, respectively. We define these operations as negative predicates in our language. Define signature $\Sigma_3$ as [$$\begin{array}{lcl} \mathtt{Merge}(x,y,z)&=^1_{\nu}& (\mathsf{hd} \, z = \mathsf{hd} \, x\, \&\, \mathtt{Merge}\, (y , \mathsf{tl}\, x, \mathsf{tl}\, z))\\ \mathtt{Split}_1(x,y)&=^1_{\nu}& (\mathsf{hd} \, y = \mathsf{hd} \, x\, \&\, \mathtt{Split}_2 ( \mathsf{tl} \, x, \mathsf{tl}\, y))\\ \mathtt{Split}_2(x,y)&=^1_{\nu}& (1\, \&\, \mathtt{Split}_1 ( \mathsf{tl} \, x, y))\\ \end{array}$$]{}The derivation given in Figure \[fig:streamrev\] shows that operations $\mathsf{merge}$ and $\mathsf{split}_i$ are inverses: Split a stream $x$ into two streams $y_1$ and $y_2$ using $\mathsf{split}_1$ and $\mathsf{split}_2$, respectively, then merge $y_1$ and $y_2$. The result is $x$. Pattern Matching ================ It may not be feasible to present a large piece of derivation fully in the calculus of Figure \[fig:rules-1\]. For the sake of brevity, we may represent predicates of positive polarity in the signature using pattern matching and build equivalent derivations based on that signature [@rosu2017matching; @brotherston2005cyclic]. In all the examples we use pattern matching for it should be clear how to transform the signature and derivations into our main logical system. \[pattern\] Redefine predicates $\mathtt{Even}$, $\mathtt{Odd}$, and $\mathtt{Nat}$ in Example \[Nat-pred\] by pattern matching in Signature $\Sigma'_1$ as: [$$\begin{array}{lcl lcl} \mathtt{Nat}(\mathsf{z}) & =^1_{\mu} & 1 & \qquad \mathtt{Nat}(sy) & =^1_{\mu} & \mathtt{Nat}(y)\\ \mathtt{Odd}(\mathsf{z}) & =^1_{\mu} & 0& \qquad \mathtt{Odd}(sy) & =^1_{\mu} & \mathtt{Even}(y) \\ \mathtt{Even}(\mathsf{z})& =^1_{\mu} & 1 & \qquad \mathtt{Even}(sy) &=^1_{\mu} & \mathtt{Odd}(y) \end{array}$$]{}The circular derivation in Example \[Oddsx\] can be simplified in the following way: [$$\begin{array}{cc} \textit{[1]}\quad \infer[\mu L]{\dagger \, \mathtt{Even}(\mathsf{z}) \vdash \mathtt{Odd(s\,\mathsf{z})}}{\infer[1 L]{1 \vdash \mathtt{Odd}(s\,\mathsf{z})}{\infer[\mu R]{\cdot \vdash \mathtt{Odd}(s \, \mathsf{z})}{\infer[\mu R]{\cdot \vdash \mathtt{Even}(\mathsf{z})}{\infer[1 R]{\cdot \vdash 1}{}}}}} & \textit{[2]}\quad \infer[\mu L]{\dagger \, \mathtt{Even}(s \, x) \vdash \mathtt{Odd(s\,s \, x)}}{\infer[\mu R]{\mathtt{Odd}(x) \vdash \mathtt{Odd}(s\, s\,x)}{\deduce{\mathtt{Odd}(x) \vdash \mathtt{Even}(s \, x)}{\star}}} \\ \textit{[3]}\quad \infer[\mu L]{\star\, \mathtt{Odd}(z) \vdash \mathtt{Even(s\,z)}}{\infer[0 L]{0 \vdash \mathtt{Odd}(z)}{}} & \textit{[4]}\quad \infer[\mu L]{\star\, \mathtt{Odd}(s \, x) \vdash \mathtt{Even(s\,s \, x)}}{\infer[\mu R]{\mathtt{Even}(x) \vdash \mathtt{Even}(s\, s\,x)}{\deduce{\mathtt{Even}(x) \vdash \mathtt{Odd}(s \, x)}{\dagger}}} \end{array}$$]{} By the definition of signature $\Sigma'_1$, the pattern of $x$ in $\mathtt{Odd}(x)$ is either of the form $\mathsf{s}\, y$ or $\mathsf{z}$. At the subgoal marked with $\star$ in subderivation $2$, we form a branch similar to the $\oplus \, L$ rule to cover all possible patterns of $x$; we continue with subderivations $3$ and $4$. With the same reasoning at the subgoal marked with $\dagger$ in the subderivation $4$ we form a branch with subderivations $1$ and $2$. A major contribution of this paper is to give a criterion for validity of theorems proved by simultaneous induction and coinduction. In the next example we see an interplay between positive and negative fixed points in the derivation. Define predicate $\mathtt{run}(x,t)$ to represent computation of a stream processor, where $x$ is the list of operations we want to compute. Operations in $x$ can be either a $\mathit{skip}$ or a $\mathit{put}\langle x \rangle$. Operation $\mathit{skip}$ simply skips one step and does not contribute to the output stream $t$. Operation $\mathit{put}\langle x \rangle$ puts element $\mathsf{z}$ as the head of the output stream $t$ and inserts a new list of operations $x$ to the original list of operations. After computing $\mathit{skip}$ the length of remaining operations in $x$ goes down by one. So we can define $\mathtt{run}(\mathit{skip};x, t)$ inductively as a positive predicate. $\mathit{put}\langle x \rangle$ increases the length of the operations, but produces an element of the output stream. So $\mathtt{run}(\mathit{put}\langle x \rangle;y,t)$ needs to be defined as a negative predicate rather than a positive one. Define the signature $\Sigma_4$ to be [$$\begin{array}{lcl} \mathtt{run}(\cdot \, , t) &=^1_{\mu}& 1 \\ \mathtt{run}(skip;x, t) &=^1_{\mu}& \mathtt{run} (x, t) \\ \mathtt{run}( put \langle x \rangle;y, t ) &=^1_{\mu}& \, \mathtt{nrun}\, (x, y, t) \\ \mathtt{nrun}( x, y, t) &=^2_{\nu}& \mathtt{hd}\, t= \mathsf{z}\, \&\, \mathtt{run} (x;y, \mathtt{tl}\, t) \end{array}$$]{}The equivalent signature without pattern matching is [$$\begin{array}{lll} \mathtt{run}(x,t)& =^1_{\mu} & \oplus\{\mathsf{e}:x=\cdot\, \otimes 1,\\ && \quad \mathsf{s}: \exists x'. x=skip;x' \otimes \mathtt{run}(x',t),\\ && \quad \mathsf{p}: \exists x'. \exists y. x= \mathit{put}(x');y \otimes \mathtt{nrun}(x',y,t) \}\\ \mathtt{nrun}( x, y, t) &=^2_{\nu}& \mathtt{hd}\, t= \mathsf{z}\, \&\, \mathtt{run} (x;y, \mathtt{tl}\, t) \end{array}$$]{}Here we define $\mathtt{run}( put \langle x \rangle;y, t )$ in two steps to follow the rules of definition by pattern matching. We can abbreviate this definition to one step as: [$$\mathtt{run}( put \langle x \rangle;y, t)=^2_{\nu}\mathtt{hd}\, t= \mathsf{z}\, \&\, \mathtt{run} (x;y, \mathtt{tl}\, t)$$]{} We want to prove that a run of any list of operations $x$ produces a (possibly infinite) list of elements $\mathsf{z}$. [$$\begin{array}{lcl} \mathtt{zlist} (t) &=^1_{\mu}& 1 \oplus \mathtt{ztream}(t) \\ \mathtt{ztream}(t) &=^2_{\nu}& \mathtt{hd}\, t= \mathsf{z}\, \&\, \mathtt{zlist}\, (\mathtt{tl}\, t) \\ \end{array}$$]{}We give circular derivations for both $(\dagger)\, \mathtt{run}(x, t) \vdash \mathtt{zlist}(t)$ and $(\star)\, \mathtt{nrun}(x,y, t) \vdash \mathtt{ztream}(t)$ in Figure \[fig:run\] to show the interplay between coinductive and inductive predicates. [$$\infer[\nu_{\mathtt{ztream}} R]{\star\, \mathtt{nrun}\, (x, y, t)\vdash \mathtt{ztream}(t)}{\infer[\& R]{\mathtt{nrun}\, (x, y, t)\vdash \mathsf{hd}\,t= \mathsf{z}\, \& \, \mathtt{zlist}(\mathsf{tl}\, t)}{\infer[\nu_{\mathtt{nrun} }L]{\mathtt{nrun}\, (x, y, t)\vdash \mathsf{hd}\,t= \mathsf{z}}{\infer[\&L]{\mathtt{hd}\, t= \mathsf{z}\, \&\, \mathtt{run} (x;y, \mathtt{tl}\, t)\vdash \mathsf{hd}\,t= \mathsf{z}}{\infer[\mathtt{ID}]{\mathtt{hd}\, t= \mathsf{z} \vdash \mathsf{hd}\,t= \mathsf{z}}{}}} & \infer[\nu_{\mathtt{nrun}} L]{\mathtt{nrun}\, (x, y, t)\vdash \mathtt{zlist}(\mathsf{tl}\, t)}{\infer[\& L]{\mathtt{hd}\, t= \mathsf{z}\, \&\, \mathtt{run} (x;y, \mathtt{tl}\, t) \vdash \mathtt{zlist}(\mathsf{tl}\, t)}{\deduce{ \mathtt{run} (x;y, \mathtt{tl}\, t) \vdash \mathtt{zlist}(\mathsf{tl}\, t)}{\dagger}} }}}$$]{} A Validity Condition ==================== Adding fixed point rules to the calculus comes with the price of losing the cut elimination property. Infinite derivations in this calculus do not necessarily enjoy the cut elimination property and thus are called *pre-proofs* instead of *proofs*. We introduce a validity condition on derivations such that the cut elimination property holds for the derivations satisfying it. Our condition is adapted from the Guard condition introduced by @Fortier13csl for singleton logic. We annotate formulas with position variables $\mathbf{x},\mathbf{y},\mathbf{z}$ and track their generations $\alpha, \beta$ to capture evolution of a formula in a derivation. With this annotation we can keep track of behaviour of any particular formula throughout the whole derivation. Our validity condition requires that at least one formula in every infinite branch behaves in a way that justifies validity of that branch. A basic judgment in the annotated calculus is of the form $\Delta \vdash_{\Omega} \mathbf{z}^\beta:C$ where $\Delta= \cdot \mid \mathbf{x}^\alpha:A, \Delta$. The set $\Omega$ keeps the relation between different generation of position variables in a derivation. We will use the set $\Omega$ to define our validity condition. Figure \[fig:rules-2\] shows the calculus annotated with position variable generations and their relations. A new generation of a position variable is introduced when a fixed point rule applies on it. The relation of a new generation to its priors is determined by the role of the rule that introduces it in (co)induction. $\mu L$ rule breaks down an inductive antecedent and $\nu R$ produces a coinductive information. They both take a step toward termination/productivity of the proof: we put the new generation of the position variable they introduce to be less than the prior ones in the given priority. Their counterpart rules $\mu R$ and $\nu L$, however, do not contribute to termination/productivity. They break the relation between the new generation and its prior ones for the given priority. In the $\mathit{Cut}$ rule we introduce a fresh position variable of generation zero, $\mathbf{w}^0$. Since it refers to appearance of a new formula, we put it to be incomparable to other position variables. We do not consider $\mathbf{w}^0$ as a continuation of $\mathbf{z}^\beta$ in the rule $\otimes R$ either; we need to restrict left branching on succedent position variables to prove Theorem \[thm:main\].[^2] The fresh position variable $\mathbf{w}^0$ introduced in $\multimap R$ (resp. $\multimap L$) rule switches its polarity from right to left (resp. left to right) so it cannot be equal to $\mathbf{z}^\beta$ (resp. $\mathbf{y}^\alpha$).[^3] As none of the above reasons hold for $\mathbf{w}^0$ in $\otimes L$, we keep its relation with $\mathbf{y}^\alpha$ in $\Omega$. For a given signature $\Sigma$, define *snapshot* of an annotated position variable $\mathbf{x}^\alpha$ as a list $\mathsf{snap}(\mathbf{x}^\alpha)=[\mathbf{x}^\alpha_i]_{i <n}$, where $n$ is the maximum priority in $\Sigma$. The list $\mathsf{snap}(\mathbf{x}^\alpha)$ stores the information of the fixed point unfolding rules applied on previous generations of position variable $\mathbf{x}^\alpha:A$ in a derivation. \[snapex\] For signature $\Sigma_1$ defined in Example \[Nat-pred\]: [$$\begin{array}{lcl} \mathtt{Nat}(x)&=^1_{\mu} &(\exists y. (x=\mathsf{s} y)\, \otimes \, \mathtt{Nat}(y))\, \oplus\, ((x=\mathsf{z})\, \otimes\, 1)\\ \mathtt{Even}(x)&=^2_{\mu}& (\exists y. (x=\mathsf{s}y)\, \otimes\, \mathtt{Odd}(y)) \oplus ((x=\mathsf{z})\, \otimes\, 1)\\ \mathtt{Odd}(x)&=^2_{\mu}& (\exists y. (x=\mathsf{s}y)\, \otimes\, \mathtt{Even}(y)) \end{array}$$]{}and position variables $\mathbf{x}^\alpha$ and $\mathbf{z}^\beta$ in the judgment [$\mathbf{x}^\alpha: \mathtt{Odd}(x) \vdash \mathbf{z}^\beta : \mathtt{Even}(\mathsf{s}x)$]{} we have [$\mathsf{snap}(\mathbf{x}^\alpha)=[\mathbf{x}^\alpha_i]_{i <2}= [\mathbf{x}_1^\alpha, \mathbf{x}_2^\alpha]$]{} and [$\mathsf{snap}(\mathbf{z}^\beta)=[\mathbf{z}^\beta_i]_{i <2}= [\mathbf{z}_1^\beta, \mathbf{z}_2^\beta]$.]{} Having the relation between annotated position variables in $\Omega$, we can define a partial order on snapshots of annotated position variables. We write [$$\mathsf{snap}(\mathbf{x}^\alpha)=[\mathbf{x}^\alpha_1\cdots \mathbf{x}^\alpha_n]<_{\Omega}[\mathbf{z}^\beta_1\cdots \mathbf{z}^\beta_n]= \mathsf{snap}(\mathbf{z}^\beta)$$]{}if the list [$[\mathbf{x}^\alpha_1\cdots \mathbf{x}^\alpha_n]$]{} is less than [$[\mathbf{z}^\beta_1\cdots \mathbf{z}^\beta_n]$]{} by the lexicographic order defined by the transitive closure of the relations in $\Omega$. Let [$\Omega=\{\mathbf{x}_1^\alpha = \mathbf{z}_1^{\beta}, \mathbf{x}_2^\alpha < \mathbf{z}_2^\gamma, \mathbf{z}_2^\gamma <\mathbf{z}_2^\beta\}.$]{} For $\mathsf{snap}(\mathbf{x}^\alpha)$ and $\mathsf{snap}(\mathbf{z}^\beta)$ defined over signature $\Sigma_1$ in Example \[snapex\], we have [$\mathsf{snap}(\mathbf{x}^\alpha)=[\mathbf{x}^\alpha_1, \mathbf{x}^\alpha_2]<_{\Omega}[\mathbf{z}^\beta_1, \mathbf{z}^\beta_2]= \mathsf{snap}(\mathbf{z}^\beta).$]{} We adapt the definitions of left $\mu$-trace and right $\nu$-trace from Fortier and Santocanale to our own settings. \[def:mu\] An infinite branch of a derivation is a *left $\mu$-trace* if for infinitely many position variables $\mathbf{x1}^{\alpha_1}, \mathbf{x2}^{\alpha_2}, \cdots$ appearing as antecedents of judgments in the branch we can form an infinite chain of inequalities [$$\mathsf{snap}(\mathbf{x1}^{\alpha_1})>_{\Omega_1}\mathsf{snap}(\mathbf{x2}^{\alpha_2})>_{\Omega_2}\cdots.$$]{}Dually, an infinite branch of a derivation is a *right $\nu$-trace* if for infinitely many position variables $\mathbf{y1}^{\beta_1}, \mathbf{y2}^{\beta_2}, \cdots$ appearing as the succedents of judgments in the branch, we can form an infinite chain of inequalities [$$\mathsf{snap}(\mathbf{y1}^{\beta_1})>_{\Omega_1}\mathsf{snap}(\mathbf{y2}^{\beta_2})>_{\Omega_2}\cdots$$]{}. An infinite derivation is a *valid proof* if each of its infinite branches is either a left $\mu$-trace or a right $\nu$-trace. A circular *proof* has a valid underlying infinite derivation. We can rewrite derivation of Example \[Oddsx\] in the annotated calculus as in Figure \[fig:ex-evenoddannot\]. To check the validity of this derivation, it is enough to observe that [$$\mathsf{snap}(\mathbf{x}^2)=[\mathbf{x}^2_1,\mathbf{x}^2_2] <_{\Omega_6}[\mathbf{x}^0_1,\mathbf{x}^0_2]=\mathsf{snap}(\mathbf{x}^0).$$]{} (2)[ [ $ \infer[\mu_{\mathtt{Even}}]{\mathbf{x}^0:\mathtt{Even}(x) \vdash_{\emptyset} \mathbf{y}^0:\mathtt{Odd}(s(x)) }{\infer[\oplus L]{ \mathbf{x}^1:(\exists y. (x=\mathtt{s}y)\, \otimes\, \mathtt{Odd}(y)) \oplus ((x=0)\, \otimes\, 1) \vdash_{\Omega_1} \mathbf{y}^0:\mathtt{Odd}(s(x))}{\infer[\exists L]{\mathbf{x}^1:\exists y. (x=\mathtt{s}y)\, \otimes\, \mathtt{Odd}(y) \vdash_{\Omega_1} \mathbf{y}^0:\mathtt{Odd}(s(x)) }{\infer[\otimes L]{\mathbf{x}^1:(x=\mathtt{s}y)\, \otimes\, \mathtt{Odd}(y) \vdash_{\Omega_1} \mathbf{y}^0:\mathtt{Odd}(s(x))}{\infer[= L]{\mathbf{w}^0:(x=\mathtt{s}y), \mathbf{x}^1:\mathtt{Odd}(y) \vdash_{\Omega_2} \mathbf{y}^0:\mathtt{Odd}(s(x))}{\infer[\mu_{\mathtt{Odd}R}]{\mathbf{x}^1:\mathtt{Odd}(y) \vdash_{\Omega_2} \mathbf{y}^0:\mathtt{Odd}(s(sy))}{\infer[\exists R]{\mathbf{x}^1:\mathtt{Odd}(y) \vdash_{\Omega_3} \mathbf{y}^1:\exists z. (ssy=\mathtt{s}z)\, \otimes\, \mathtt{Even}(z)}{\infer[\otimes R]{\mathbf{x}^1:\mathtt{Odd}(y) \vdash_{\Omega_3} \mathbf{y}^1: (ssy=\mathtt{s}sy)\, \otimes\, \mathtt{Even}(sy)}{\infer[=R]{\cdot \vdash_{\Omega_3} \mathbf{z}^0:(ssy=\mathtt{s}sy)}{} & \infer[\mu_{\mathtt{Odd}L}]{\mathbf{x}^1:\mathtt{Odd}(y) \vdash_{\Omega_3} \mathbf{y}^1:\mathtt{Even}(sy)}{ \infer[\exists L]{\mathbf{x}^2:\exists z. (y=\mathtt{s}z)\, \otimes\, \mathtt{Even}(z) \vdash_{\Omega_4} \mathbf{y}^1:\mathtt{Even}(sy)}{\infer[\otimes L]{\mathbf{x}^2:(y=\mathtt{s}z)\, \otimes\, \mathtt{Even}(z) \vdash_{\Omega_4} \mathbf{y}^1:\mathtt{Even}(sy)}{\infer[=L]{v^0:(y=\mathtt{s}z), \mathbf{x}^2:\mathtt{Even}(z) \vdash_{\Omega_5} \mathbf{y}^1:\mathtt{Even}(sy)}{\infer[\mu_{\mathtt{Even} R}]{\mathbf{x}^2:\mathtt{Even}(z) \vdash_{\Omega_5} \mathbf{y}^1:\mathtt{Even}(ssz)}{\infer[\oplus R ]{\mathbf{x}^2:\mathtt{Even}(z) \vdash_{\Omega_6} \mathbf{y}^2:(\exists y. (ssz=\mathtt{s}y)\, \otimes\, \mathtt{Odd}(y)) \oplus ((ssz=0)\, \otimes\, 1)}{\infer[\exists R]{\mathbf{x}^2:\mathtt{Even}(z) \vdash_{\Omega_6} \mathbf{y}^2:(\exists y. (ssz=\mathtt{s}y)\, \otimes\, \mathtt{Odd}(y))}{\infer[\otimes R]{\mathbf{x}^2:\mathtt{Even}(z) \vdash_{\Omega_6} \mathbf{y}^2:(ssz=\mathtt{s}sz)\, \otimes\, \mathtt{Odd}(sz)}{\infer[=R]{\cdot \vdash_{\Omega_6} \mathbf{u}^0:ssz=ssz}{} & \infer[\mathtt{Subst_{[z/x]}}]{\mathbf{x}^2:\mathtt{Even}(z) \vdash_{\Omega_6} \mathbf{y}^2:\mathtt{Odd}(sz)}{\mathbf{x}^2:\mathtt{Even}(x) \vdash_{\Omega_6} \mathbf{y}^2:\mathtt{Odd}(sx)} }}} }}}}}}} }}} } & \cdots } } $]{} ]{}; (4.8,4.55).. controls (7,7) and (15,-6) .. (2.9,-4.6); [$\Omega_1=\{\mathbf{x}^1_{2}<\mathbf{x}^0_2, \mathbf{x}^1_1=\mathbf{x}^0_1\},$ $\Omega_2=\Omega_1 \cup \{\mathbf{w}^0_{2}=\mathbf{x}^1_2, \mathbf{w}^0_1=\mathbf{x}^1_1\},$ $\Omega_3=\Omega_2 \cup \{\mathbf{y}^1_1=\mathbf{y}^0_1\},$ $\Omega_4=\Omega_3 \cup \{\mathbf{x}^2_{2}<\mathbf{x}^1_2, \mathbf{x}^2_1=\mathbf{x}^1_1\},$ $\Omega_5=\Omega_4 \cup \{v^0_{2}=\mathbf{x}^2_2, v^0_1=\mathbf{x}^2_1\},$ and $\Omega_6=\Omega_5 \cup \{\mathbf{y}^2_1=\mathbf{y}^1_1\}.$]{} Since the annotation of position variables is straightforward, for the sake of conciseness, we present future examples as circular derivations in the calculus of Figure \[fig:rules-1\]. We also use pattern matching whenever possible. All derivations presented in this paper are valid by this definition. We leave it to the reader to check their validity. A productive cut elimination algorithm ====================================== @Fortier13csl introduced a cut elimination algorithm for infinite pre-proofs in singleton logic with fixed points. They proved that for infinite proofs satisfying their guard condition the algorithm is productive. In this section we adapt their cut elimination algorithm to $\mathit{FIMALL}^{\infty}_{\mu,\nu}$ and prove its productivity for valid derivations. The algorithm receives an infinite proof as an input and outputs a cut-free infinite proof. Since we are dealing with infinite derivations, to make the algorithm productive we need to push every cut away from the root with a lazy strategy (BFS). With this strategy we may need to permute two consecutive cuts which results into a loop. To overcome this problem, similar to Fortier and Santocanale and also @baelde2007least we generalize binary cuts to $n$-ary cuts using the notion of a *branching tape* the prior notion of *tape*. \[def:tape\] A *branching tape* $\mathcal{C}$ is a finite list of sequents $\Delta \vdash \mathbf{w}^\beta: A$[^4], such that - Every two judgments $\Delta \vdash \mathbf{w}^\beta: A$ and $\Delta' \vdash \mathbf{w}'^{\beta'}: A'$ on the tape share at most one position variable $\mathbf{z}^\alpha:B$. If they share such position variable, we call them connected. Moreover, assuming that $\Delta \vdash \mathbf{w}^\beta: A$ appears before $\Delta' \vdash \mathbf{w}'^{\beta'}: A'$ on the list, we have $\mathbf{z}^\alpha:B\in \Delta'$ and $\mathbf{z}^\alpha:B= \mathbf{w}^\beta:A$. - Each position variable $\mathbf{z}^\beta$ appears at most twice in a tape and if it appears more than once it connects two judgments. - Every tape is connected and acyclic. The *conclusion* $\mathsf{conc}_{\mathcal{M}}$ of a branching tape $\mathcal{M}$ is a sequent $\Delta \vdash \mathbf{x}^\alpha:A$ such that - there is a sequent $\Delta' \vdash \mathbf{x}^\alpha:A$ in the tape that $\mathbf{x}^\alpha:A$ does not connect it to any other sequent in the tape. - For every $\mathbf{y}^\beta:B\in \Delta$ there is a sequent $\Delta', \mathbf{y}^\beta:B \vdash \mathbf{z}^\gamma:C$ on the tape such that $\mathbf{y}^\beta:B$ does not connect it to any other sequent in the tape. We call $\Delta$ the set of *leftmost formulas* of $\mathcal{M}$: $\mathsf{lft}(\mathcal{M})$. And $x^\alpha:A$ is the *rightmost formula* of tape $\mathcal{M}$: $\mathsf{rgt}(\mathcal{M})$. The conclusion of a branching tape always exists and is unique. An $n$-ary cut is a rule formed from a tape $\mathcal{M}$ and its conclusion $\mathsf{conc}_{\mathcal{M}}$:$\infer[nCut]{\mathsf{conc}_{\mathcal{M}}}{\mathcal{M}}$ We generalize Fortier and Santaconale’s set of primitive operations to account for $\mathit{FIMALL}^{\infty}_{\mu,\nu}$. They closely resemble the reduction rules given by @doumane2017infinitary. Figure \[fig:Reductions\] depicts a few interesting Internal and External reductions [^5]. [ $$\begin{array}{ccc} \infer[\mathit{nCut}]{\Delta \vdash \mathbf{v}:C}{\mathcal{C}_1& \infer[\multimap R]{ \Delta' \vdash \mathbf{z}^{\beta}:A_1 \multimap A_2}{\Delta', \mathbf{u}^0: A_1 \vdash \mathbf{z}^{\beta}:A_2}& \mathcal{C}_2 & \infer[\multimap L]{\Delta'', z^{\beta}:A_1 \multimap A_2 \vdash \mathbf{w}^\alpha:B}{\Delta_1''\vdash \mathbf{u}^{0}:A_1 & \Delta_2'', \mathbf{z}^{\beta}: A_2 \vdash \mathbf{w}^\alpha:B } & \mathcal{C}_3 } & \xRightarrow{\mathsf{Reduce}}& \hspace*{10em} \\ \multicolumn{3}{r}{ \infer[\mathit{nCut}]{\Delta \vdash \mathbf{v}:C}{\mathcal{C}_1& \mathcal{C}_2 & \Delta_1''\vdash \mathbf{u}^{0}:A_1 & \Delta', \mathbf{u}^0: A_1 \vdash \mathbf{z}^{\beta}:A_2 & \Delta_2'', \mathbf{z}^{\beta}: A_2 \vdash \mathbf{w}^\alpha:B & \mathcal{C}_3 }}\\ \end{array}$$]{} $${\small \begin{array}{lcl} \infer[\mathit{nCut}]{\Delta \vdash \mathbf{v}:C}{\mathcal{C}_1 & \infer[\mu R]{ \Delta' \vdash \mathbf{z}^{\beta}:T(\overline{t})}{ \Delta'\vdash \mathbf{z}^{\beta+1}:[\overline{t}/\overline{x}]A & T(\overline{x})=_{\mu}A }& \mathcal{C}_2 &\infer[\mu L]{\Delta'', \mathbf{z}^{\beta}:T(\overline{t}) \vdash \mathbf{w}^\alpha:B}{\Delta'', \mathbf{z}^{\beta+1}: [\overline{t}/\overline{x}]A \vdash \mathbf{w}^\alpha:B & T(\overline{x})=_{\mu}A} & \mathcal{C}_3 }& \xRightarrow{\mathsf{Reduce}}& \hspace*{10em} \\ \multicolumn{3}{r}{ \infer[\mathit{nCut}]{\Delta \vdash \mathbf{v}:C}{\mathcal{C}_1 & \Delta'\vdash \mathbf{z}^{\beta+1}:[\overline{t}/\overline{x}]A &\mathcal{C}_2 &\Delta'', \mathbf{z}^{\beta+1}: [\overline{t}/\overline{x}]A \vdash \mathbf{w}^\alpha:B& \mathcal{C}_3 }}\\ \end{array} }$$ $${\small \begin{array}{lcl} \infer[\mathit{nCut}]{\Delta \vdash \mathbf{v}:C}{\mathcal{C}_1 & \infer[\exists R]{ \Delta' \vdash \mathbf{z}^{\beta}:\exists x. P(x)}{ \Delta'\vdash \mathbf{z}^{\beta}:P(t) }& \mathcal{C}_2 &\infer[\exists L]{\Delta'', \mathbf{z}^{\beta}:\exists x. P(x) \vdash \mathbf{w}^\alpha:B}{\deduce{\Delta'', \mathbf{z}^{\beta}: P(x) \vdash \mathbf{w}^\alpha:B}{\Pi'} } & \mathcal{C}_3 }& \xRightarrow{\mathsf{Reduce}}& \hspace*{20em} \\ \multicolumn{3}{r}{ \infer[\mathit{nCut}]{\Delta \vdash \mathbf{v}:C}{\mathcal{C}_1 & \Delta'\vdash \mathbf{z}^{\beta}:P(t) &\mathcal{C}_2 &\deduce{\Delta'', \mathbf{z}^{\beta}: P(t) \vdash \mathbf{w}^\alpha:B}{\Pi'[t/x]}& \mathcal{C}_3 }}\\ \infer[\mathit{nCut}]{\Delta \vdash \mathbf{v}:C}{\mathcal{C}_1 & \infer[=R]{ \cdot \vdash \mathbf{z}^{\beta}:s=s}{}&\mathcal{C}_2 &\infer[=L]{\Delta'', \mathbf{z}^{\beta}:s=s \vdash \mathbf{w}^\alpha:B }{\Delta'' \vdash \mathbf{w}^\alpha:B} & \mathcal{C}_3 }& \xRightarrow{\mathsf{Reduce}} & \infer[\mathit{nCut}]{\Delta \vdash \mathbf{v}:C}{\mathcal{C}_1 & \mathcal{C}_2 & \Delta'' \vdash \mathbf{w}^\alpha:B & \mathcal{C}_3 }\\ \end{array} }$$ $${ \begin{array}{lcl} \infer[\mathit{nCut}]{\Delta_1,\Delta_2 \vdash \mathbf{z}^\beta:A_1 \otimes A_2}{\mathcal{C}& \infer[\otimes R]{ \Delta_1', \Delta'_2 \vdash \mathbf{z}^{\beta}:A_1 \otimes A_2}{\Delta'_1\vdash \mathbf{u}^{0}:A_1 & \Delta'_2\vdash \mathbf{z}^{\beta}:A_2} }& \xRightarrow{\mathsf{RFLIP}} & \infer[\otimes R]{\Delta_1, \Delta_2 \vdash \mathbf{z}^\beta:A_1 \otimes A_2}{\infer[nCut]{\Delta_1\vdash \mathbf{u}^{0}:A_1}{ \mathcal{C}_{\Delta'_1} & \Delta'_1\vdash \mathbf{u}^{0}:A_1} & \infer[\mathit{nCut}]{\Delta_2\vdash \mathbf{z}^{\beta}:A_2}{\mathcal{C}_{\Delta'_2} & \Delta'_2\vdash \mathbf{z}^{\beta}:A_2 } }\\\\ \infer[\mathit{nCut}]{\Delta, \mathbf{z}^\beta: A_1 \otimes A_2 \vdash \mathbf{v}:C}{\mathcal{C}_1 & \infer[\otimes L]{ \Delta',\mathbf{z}^\beta: A_1 \otimes A_2 \vdash \mathbf{w}^{\alpha}:B}{\Delta', \mathbf{u}^0: A_1, \mathbf{z}^\beta: A_2 \vdash \mathbf{w}^{\alpha}:B} & \mathcal{C}_2 }& \xRightarrow{\mathsf{LFLIP}} & \infer[\otimes L]{\Delta, \mathbf{z}^\beta: A_1 \otimes A_2 \vdash \mathbf{v}:C}{ \infer[\mathit{nCut}]{\Delta, \mathbf{u}^0: A_1 , \mathbf{z}^\beta:A_2 \vdash \mathbf{u}^{0}:A_1}{ \mathcal{C}_1 & \Delta', \mathbf{u}^0: A_1, \mathbf{z}^\beta: A_2 \vdash \mathbf{w}^{\alpha}:B & \mathcal{C}_2}}\\\\ \infer[\mathit{nCut}]{\Delta, \mathbf{z}^\beta:s=t \vdash \mathbf{w}^\alpha:B}{ \mathcal{C}_1 & \infer[=L]{\Delta',\mathbf{z}^\beta:s=t \vdash \mathbf{w}^\alpha:B}{\Delta'[\theta] \vdash \mathbf{w}^\alpha:B'[\theta] & \theta \in \mathtt{mgu}(t,s)}& \mathcal{C}_2} & \xRightarrow{\mathsf{LFLIP}} & \infer[=L]{\Delta, \mathbf{z}^\beta:s=t \vdash \mathbf{w}^\alpha:B}{ \infer[\mathit{nCut}]{\Delta[\theta] \vdash \mathbf{w}^\alpha:B[\theta] }{\mathcal{C}_1[\theta] & \Delta'[\theta] \vdash \mathbf{w}^\alpha:B'[\theta] & \mathcal{C}_2[\theta] } & \theta \in \mathtt{mgu}(t,s) }\\\\ \infer[\mathit{nCut}]{\Delta \vdash \mathbf{z}^\beta:C }{\mathcal{C}_1 & \infer[ID]{\mathbf{x}^\alpha:A \vdash \mathbf{w}^\gamma:A }{} & \mathcal{C}_2 } & \xRightarrow{\mathsf{ID-Elim}} & \infer[\mathit{nCut}]{\Delta \vdash \mathbf{z}^\beta:C}{\mathcal{C}_1 & \mathcal{C}_2[\mathbf{x}^\alpha/\mathbf{w}^\gamma] } \end{array} }$$ Our cut elimination algorithm is given as Algorithms \[algorithm\] and \[alg:treat\]. The output of the algorithm is a tree labelled by $\{0,1\}$. For each node $w \in \{0,1\}^*$ of the tree it also identifies the corresponding sequent, $\mathit{s}(w)$, and the rule applied on the node, $\mathit{r}(w)$. The algorithm *Treat* reduces the sequence in a branching tape with internal reductions until either a left rule is applied on one of its leftmost formulas or a right rule is applied on its rightmost formula. While this condition holds, the algorithm applies a *flip* rule on a leftmost/rightmost formula of the tape. The flipping step is always productive since it pushes a cut one step up. It suffices to show that the treating part is terminating to prove productivity of the algorithm. Initialization: $\Lambda \leftarrow \emptyset; Q \leftarrow [(\epsilon, [v])]$; $v$ is the root sequent. $\rho(s)$ is the rule applied on formula annotated with position variable $s$, it can either be an $\mathsf{ID}$, $\mathsf{Cut}$, a $L$ rule, or a $R$ rule. $\mathsf{lft}(M)$ and $\mathsf{rgt}(M)$ are defined in Definition \[def:tape\]. The $\mathsf{FLip}$ rules will return a rule that they permuted down, the sequent corresponding to that, and a list $\mathit{List}$ of one or two tapes.\ Initialization: $M$ is a branching tape. $i$ and $j$ in $\mathsf{Reduce}(M,i,j)$ are the index of the two sequents in tape on which the reduction rules are applied. Similarly $i$ in $\mathsf{Merge}(M,i)$ and $\mathsf{idElim}$ is the index of the sequent in the tape on which the corresponding rule is applied. $\rho'(i)$ is the rule applied on the $i$-th sequent of the tape, it can either be an $\mathsf{ID}$, $\mathsf{Cut}$, a $L$ rule, or a $R$ rule.\ \[thm:main\] For every input tape $M$, computation of $\mathit{Treat}(M)$ halts. By a proof similar to FS, except that we use $\mu$-threads instead of $\nu$-threads to show that $\mathit{Treat}(M)$ does not have an infinite computation tree. Assume for the sake of contradiction that $\mathit{Treat}(M)$ has an infinite computation tree $\Psi$. We prove the following three contradictory statements, where Here $<$ and $\bigwedge$ are defined according to the lexicographic order on the tree $\Psi$. 1. The greatest infinite branch of $\Psi$ is a $\mu$-branch. 2. Let $E$ be a nonempty collection of $\mu$-branches and let $\gamma=\bigwedge E$. Then $\gamma$ is a $\mu$-branch. 3. If $\beta$ is a $\mu$-branch, then there exists another $\mu$-branch $\beta' < \beta$. The complete proof is given in the Appendix. Adding finite structural derivations ==================================== In the proof of theorems about linear structures, we may want to appeal to some first order (structural) theories such as arithmetic. To allow a restricted form of such structural (that is, nonlinear) reasoning in our system we add nonlinear formulas $A_s$ to the linear system as [$$A_l::= \cdots \mid {\downarrow }A_s$$]{}labeling all linear propositions as $A_l$. We do not prescribe the exact syntax for nonlinear proposition since our development is parametric in those (subject to a few assumptions) and may vary with the particular application. Nonlinear judgments are of the form $\Psi \Vdash A_s$. The only restriction we put on the calculus for pure structural judgments is to have the cut elimination property. We generalize linear judgments to $\Psi; \Delta \vdash_{\Omega} x^\alpha:A_l$ where $\Psi$ is a structural context (i.e., it allows weakening and contraction) and $\Delta$ a linear context. Judgments of the rules in Figure \[fig:rules-2\] are refined by adding an extra structural context $\Psi$ to them. This change is straightforward since none of these rules affect the structural context $\Psi$. We add the following three rules to Figure \[fig:rules-2\] to connect linear and structural reasoning. [$$\begin{array}{ccc} \infer[\downarrow L]{\Psi; \Delta, x^\alpha: \downarrow A_s \vdash_{\Omega} \mathbf{z}^\beta: B_l}{\Psi, A_s; \Delta \vdash_{\Omega} \mathbf{z}^\beta:B_l} & \infer[\downarrow R]{\Psi; \cdot \vdash_{\Omega} x^\alpha: \downarrow B_s}{\Psi \Vdash B_s}\end{array}$$ $$\infer[\mathsf{Cut_{sl}}]{\Psi; \Gamma \vdash_{\Omega} B_l }{\Psi \Vdash A_s & \Psi, A_s; \Gamma \vdash_{\Omega} B_l}$$]{}In both $\downarrow L$ and $\downarrow R$ rules, when position variable $x^\alpha$ becomes structural we simply delete it. The $\mathsf{Cut}_{sl}$ rule does not create any fresh position variable either. So our validity condition remains intact after adding the structural component. There are three different cut rules in this system: [$$\begin{array}{cc} \infer[\mathsf{Cut}_{ss}]{\Psi \Vdash A_s}{ \Psi \Vdash B_s & \Psi, B_s \Vdash A_s} & \infer[\mathsf{Cut_{sl}}]{\Psi; \Gamma \vdash_{\Omega} B_l }{\Psi \Vdash A_s & \Psi, A_s; \Gamma \vdash_{\Omega} B_l} \end{array}$$ $$\infer[\mathsf{Cut}_{ll}]{\Psi;\Delta, \Delta' \vdash_{\Omega} \mathbf{z}^\beta:C}{\Psi;\Delta \vdash_{\Omega} \mathbf{w}^0:A & \Psi;\Delta', \mathbf{w}^0:A \vdash_{\Omega} \mathbf{z}^\beta:C}$$]{}By assumption, $\mathsf{Cut}_{ss}$ can be eliminated. It is enough to eliminate $\mathsf{Cut}_{ll}$ and $\mathsf{Cut}_{sl}$ rules in a productive way. We define a generalized $n$-ary cut to account for the two latter two cut rules. A *generalized branching tape* $\mathcal{M}_{\Psi}$ is of the form $\mathcal{S}_{\Psi} \mid \mathcal{C}_{\Psi}$ where $\mathcal{C}_{\Psi}$ is a branching tape by Definition \[def:tape\] and $\mathcal{S}_{\Psi}$ is a set of structural judgments $\Psi \Vdash A_s$. For each structural judgment $\Psi \Vdash A_s \in \mathcal{S}$, formula $A_s$ appears in the structural context of at least one judgment in $\mathcal{C}_{\Psi}$. For a judgment $\Psi'; \Delta \vdash B_l$ in $\mathcal{C}_{\Psi}$, we have $\Psi'=\Psi, \Phi$, where all formulas in $\Phi$ are the succedent of a structural judgment in $\mathcal{S}_{\Psi}$. The generalized n-ary cut rule is $$\infer[\mathit{nCut}]{\Psi; \Delta \vdash \mathbf{x}^\alpha:B_l}{\mathcal{M}_{\Psi}}$$ Where $\Delta \vdash \mathbf{x}^\alpha: B_l$ is the conclusion of the linear part of the tape as defined in Definition \[def:tape\]. We add one reduction step, a rule to merge a $\mathsf{Cut}_{sl}$ rule to the tape, and two flips to the set of primitive operations (Figure \[fig:str-Reductions\]). We keep all the primitive operations for the purely linear system. To preserve the invariants of the (generalized) branching tape in some of the operations we silently remove the structural sequents in which their succedent is not used in any linear judgment. In the $\mathsf{Reduce}$, $\mathsf{Merge{-}Cut}$, and $\mathsf{RFlip}$ rules we know that $\Psi_1=\Psi, \Phi$, where all elements in $\Phi$ appear as a succedent in $S_{\Psi}$. By cut elimination for pure linear judgments we get a proof for $\Psi \Vdash A_s$. In $\mathsf{LFlip}$ rule we use admissibility of Weakening for the structural context: We can prove coinductively that if there is a derivation for $\Psi; \Delta \vdash \mathbf{x}^\alpha: A_l$ in our calculus, there is also a derivation for $\Psi, B_s; \Delta \vdash \mathbf{x}^\alpha:A_1$ with the same structure. $${\small \begin{array}{lcl} \infer[\mathit{nCut}]{\Psi; \Delta\vdash \mathbf{v}:C}{ \mathcal{S}_{\Psi} \mid {\mathcal{C}_1}_{\Psi} & \infer[\mathsf{Cut}_{sl}]{ \Psi_1; \Delta', \mathbf{z}^\beta: B \vdash \mathbf{w}^\alpha:B }{\Psi_1 \Vdash A_s & \Psi_1, A_s; \Delta' \vdash \mathbf{w}^\alpha:B } & {\mathcal{C}_2}_{\Psi} } & \xRightarrow{\mathsf{Merge{-}Cut}} & \infer[\mathit{nCut}]{\Psi; \Delta \vdash \mathbf{v}:C}{ \mathcal{S}_{\Psi}, \Psi \Vdash A_s \mid {\mathcal{C}_1}_{\Psi} & \Psi_1, A_s; \Delta' \vdash \mathbf{w}^\alpha:B & {\mathcal{C}_2}_{\Psi} }\\\\ \infer[\mathit{nCut}]{\Psi; \cdot \vdash \mathbf{z}^\beta: \downarrow A_s}{\mathcal{S}_{\Psi} &\mid \cdot & \infer[\downarrow R]{ \Psi_1; \cdot \vdash \mathbf{z}^{\beta}:\downarrow A_s}{\Psi_1\Vdash A_s} }& \xRightarrow{\mathsf{RFLip}} & \infer[\downarrow R]{\Psi; \cdot \vdash \mathbf{z}^\beta:\downarrow A_s }{ {\Psi\Vdash A_s}}\\\\ \infer[\mathit{nCut}]{\Psi; \Delta, \mathbf{z}^\beta:\downarrow A_s \vdash \mathbf{v}:C}{ \mathcal{S}_{\Psi} \mid {\mathcal{C}_1}_{\Psi} & \infer[\downarrow L]{\Psi_1; \Delta', \mathbf{z}^\beta:\downarrow A_s \vdash \mathbf{w}^\alpha:B }{\Psi_1, A_s; \Delta' \vdash \mathbf{w}^\alpha:B } & {\mathcal{C}_2}_{\Psi} } & \xRightarrow{\mathsf{LFlip}} & \infer[\downarrow L]{\Psi; \Delta, \mathbf{z}^\beta:\downarrow A_s \vdash \mathbf{v}:C}{\infer[\mathit{nCut}]{\Psi, A_s; \Delta \vdash \mathbf{v}:C}{\mathcal{S}_{\Psi, A_s} \mid {\mathcal{C}_1}_{\Psi, A_s} & \Psi_1, A_s; \Delta' \vdash \mathbf{w}^\alpha:B &{\mathcal{C}_2}_{\Psi, A_s}}} \end{array} }$$ The next example shows how to use a structural context to prove a property of infinite streams. We define the lexicographic order on streams [@kozen2017practical] in signature $\Sigma_5$ as a negative predicate [$$\begin{array}{lcl} x \preceq y & =^1_{\nu} & \downarrow(\mathsf{hd} x < \mathsf{hd} y) \oplus (\downarrow (\mathsf{hd}x = \mathsf{hd} y)\, \& \, \mathsf{tl} x \preceq \mathsf{tl} y). \\ \end{array}$$]{}where the relation $<$ is a transitive partial order on the elements of streams. Our goal is to show that relation $\preceq$ is transitive by using the (structural) first order theory of orders ($\mathbb{O}$). In Figure \[fig:transt\] we show two branches of this proof, the rest of the proof can be completed in a similar way. We can define even and odd predicates alternatively using structural arithmetic formulas. In the next example we show how these alternative definitions can be deduced from the ones we introduced in Example \[Nat-pred\]. \[ex:evenalt\] Define Signature $\Sigma_6$ to be [$$\begin{array}{lcl lcl} \mathtt{Odd}(\mathsf{z}) & =^1_{\mu} & 0 & \qquad \mathtt{Odd}(sy) & =^1_{\mu} & \mathtt{Even}(y) \\ \mathtt{Even}(\mathsf{z})& =^1_{\mu} & 1 & \qquad \mathtt{Even}(sy) &=^1_{\mu} & \mathtt{Odd}(y) \end{array}$$]{}[$$\begin{array}{lcllcl} \mathtt{E}(x)& =^2_{\mu}& \exists y.\, \downarrow (x=2y) & \quad \mathtt{O}(x)&=^2_{\mu}& \exists y.\, \downarrow (x=2y+1) \end{array}$$]{} Put $\mathbb{P}$ to be the rules of arithmetic. We present circular derivations for $\star \; \mathbb{P};\mathtt{Even}(x) \vdash \mathtt{E}(x)$ and $\dagger \; \mathbb{P};\mathtt{Odd}(x) \vdash \mathtt{O}(x)$ in Figure \[fig:evenalt\]. These derivations satisfy the validity condition since in every infinite branch infinitely many $\mu_{\mathtt{Odd}}L$ and $\mu_{\mathtt{Even}}L$ rules are applied on the antecedent. Conclusion ========== In this paper we introduced an infinitary sequent calculus for first order intuitionistic multiplicative additive linear logic with fixed points. This system is mainly designed for linear reasoning but we also allow appealing to first order theories such as arithmetic, by adding an adjoint downgrade modality. Inspired by the work of @Fortier13csl we provide an algorithm to identify valid proofs among all infinite derivations. We have provided several examples to show the strength of calculus in proving theorems about mutually inductive and coinductive data types. One of our main motivations for introducing this calculus was to have a system for reasoning about programs behaviour. In particular we want to use this calculus to give a direct proof for the strong progress property of locally valid binary session typed processes [@derakhshan2019circular]. The importance of a direct proof other than its elegance is that it can be adapted for a more general validity condition on processes without the need to prove cut elimination productivity for their underlying derivations. The connection to the type theoretic approach by Abel et al [@abel2016well] is an interesting item for future research. A first step in this general direction was taken by Sprenger and Dam [@sprenger2003structure] who justify cyclic inductive proofs using inflationary iteration. Appendix ======== \[lem:subst\] For a valid derivation [$$\infer{\Delta \vdash \mathbf{w}^\alpha:A}{\Pi}$$]{} in the infinite system and substitution $\theta$, there is a valid derivation for [$$\infer{\Delta[\theta] \vdash \mathbf{w}^\alpha:A[\theta]}{\Pi[\theta]}$$]{} Where $\Pi[\theta]$ is the whole derivation $\Pi$ or a prefix of it instantiated by $\theta$. The proof is by coinduction on the structure of [$$\Delta \vdash \mathbf{w}^\alpha:A.$$]{} The only interesting case is where we get to the $= L$ rule. [$$\infer[=L]{\Gamma, s=t \vdash B}{ \deduce{\Gamma[\theta']\vdash B[\theta']}{\Pi'}& \theta' \in \mathsf{mgu}(t,s)}$$]{} If the set [$\mathsf{mgu}(t[\theta],s[\theta])$]{}is empty then so is [$\Pi[\theta]$]{}. Otherwise if $\eta$ is the single element of [$ \mathsf{mgu}(t[\theta],s[\theta])$]{}, then for some substitution $\lambda$ we have [$\theta\eta=\theta'\lambda,$]{} and we can form the rest of derivation for substitution $\lambda$ as [$\Pi'[\lambda]$]{} coinductively. [\[thm:main\]]{} For every input tape $M$, computation of $\mathit{Treat}(M)$ halts. We show that $\mathit{Treat}(M)$ does not have an infinite computation tree. Assume for the sake of contradiction that $\mathit{Treat}(M)$ has an infinite computation tree and gets into an infinite loop. We follow the proof by @Fortier13csl(FS) closely. Put $M_i$ for $i \ge 1$ to be the branching tape in memory before the $i$-th turn of the loop, with $M_1=M$. We build the full trace $T$ of the algorithm with essentially the same transition rules as in FS. In our algorithm the sequents subject to reduction may not be next to each other. The $\mathsf{Reduce}$ function needs to receive two indices in the tape to find the sequents for reduction. All reductions except those corresponding to $\otimes$ and $\multimap$ are non-branching($\mathit{nb}$) and their transition rules are quite similar to the one introduced by FS. - If $M_{n+1}= \mathsf{Reduce}_{\mathit{nb}}(M_{n},i,j)$ then - $(n,k)\rightarrow^\bot (n+1,k)$ for $k\not\in\{i,j\}$, - $(n,i) \rightarrow^0 (n+1, i)$, - $(n,j)\rightarrow^{0} (n+1,j)$. . The reductions corresponding to $\otimes$ and $\multimap$, however, produce a branch and need to be defined separately: - If $M_{n+1}= \mathsf{Reduce}_{\otimes}(M_{n},i,j)$ then - $(n,k)\rightarrow^\bot (n+1,k)$ for $k<i$, - $(n,i) \rightarrow^1 (n+1, i)$ and $(n,i) \rightarrow^2 (n+1, i+1)$, - $(n,j)\rightarrow^{0} (n+1,j+1)$, - $(n,k)\rightarrow^{\bot} (n+1,k+1)$ for $i<k<j$ or $k>j$. . - If $M_{n+1}= \mathsf{Reduce}_{\multimap}(M_{n},i,j)$ then - $(n,k)\rightarrow^\bot (n+1,k)$ for $k<i$, - $(n,i) \rightarrow^0 (n+1, j)$, - $(n,k)\rightarrow^\bot (n+1,k-1)$ for $i<k<j$, - $(n,j) \rightarrow^1 (n+1, j-1)$ and $(n,j) \rightarrow^2 (n+1, j+1)$, - $(n,k)\rightarrow^{\bot} (n+1,k+1)$ for $k<j$. Transitions labelled by $\bot$ mean that the sequent has not evolved by a reduction rule, while other labels show that the sequent is evolved into one or two (in the case of branching rules) new sequents in the next tape. We get the real trace $\Psi$ by collapsing the transitions labelled by $\bot$. $\Psi$ is an infinite, finitely branching labelled tree with prefix order $\sqsubseteq$ and lexicographical order $<$. A branch in $\Psi$ is a maximal path. The set of all branches of $\Psi$ ordered lexicographically forms a complete lattice. An infinite branch is a $\mu$- branch (resp. $\nu$-branch) if its corresponding derivation is a $\mu$- trace (resp. $\nu$-trace). By our validity condition $\Psi$ satisfies the property that a $\nu$-branch can only admit finitely many branches on its right side (it may include cuts, $\otimes$, or $\multimap$ reductions). We prove the following three contradictory statements dual to FS: 1. The greatest infinite branch of $\Psi$ is a $\mu$-branch: The greatest infinite branch of $\Psi$ exists by Konig’s lemma and is either a $\mu$- or a $\nu$- branch. Assume it is a $\nu$- branch. Then either it forms infinitely many branches on its right or there is an infinite branch greater than it. In both cases we can form a contradiction. 2. Let $E$ be a nonempty collection of $\mu$-branches. Then $\gamma=\bigwedge E$ is a $\mu$-branch: If $\gamma \in E$ then it is trivially true. Otherwise, by the way we constructed $\Psi$, it means that $\gamma$ has infinitely many branches on its right and thus cannot be a $\nu$ branch. 3. If $\beta$ is a $\mu$-branch, then there exists another $\mu$-branch $\beta' < \beta$: $\beta$ is a $\mu$-branch so for infinitely many position variables $\mathbf{x1}^{\alpha_1}, \mathbf{x2}^{\alpha_2}, \cdots$ on the antecedents of $\beta$ we can form an infinite chain of inequalities [$$\mathsf{snap}(\mathbf{x1}^{\alpha_1})>_{\Omega_1^\beta}\mathsf{snap}(\mathbf{x2}^{\alpha_2})>_{\Omega_2^\beta}\cdots.$$]{} There are two possibilities here: 1. There is an infinite branch $\beta'<\beta$ with infinitely many position variables $\mathbf{xi}^{\alpha_{i}}, \mathbf{x\{i+1\}}^{\alpha_{i+1}}, \cdots$ as its succedents. Note that these position variables connect sequents in $\beta$ to the sequents in $\beta'$ infinitely many times. So every $\mu/\nu L$ rule in $\beta$ reduces with a $\mu/\nu R$ rule in $\beta'$. This means that a $\mu R$ rule with priority $i$ is applied on the succedent of $\beta'$ infinitely often but no priority $j<i$ has an infinitely many $\nu R$ rule in $\beta'$. 2. There is an infinite branch $\beta'<\beta$ with infinitely many branches on its right. In both cases $\beta'$ cannot be a $\nu$-branch and thus is a $\mu$-branch. Items (i)-(iii) form a contradiction. We can form the nonempty collection $E$ of all $\mu$- branches in $\Psi$ by (i). By (ii) we get $(\gamma= \bigwedge E)\in E$, which forms a contradiction with (iii). [19]{} \#1 \#1[\#1]{}\#1 \#1 \#1 \#1 \#1[\#1]{} \#1[\#1]{} [abel2016well]{} . . (). [abel2013wellfounded]{} . . In , Vol. . ACM, . [baelde2016infinitary]{} . . (). [baelde2007least]{} . . In . Springer, . [benton1994mixed]{} . . In . Springer, . [brandt1998coinductive]{} . . , (), . [brotherston2005cyclic]{} . . In . Springer, . [derakhshan2019circular]{} . . (). [doumane2017infinitary]{} . **. . [Fortier13csl]{} . . In , (Ed.). , , . [hermida1998structural]{} . . , (), . [kozen2017practical]{} . . , (), . [niqui2009coinductive]{} . . In . . [pfenning2015polarized]{} . . In . Springer, . [pierce2002types]{} . . . [REYNOLDS81A]{} . . In , (Eds.). , , . [rosu2017matching]{} . . (). [santocanale2002calculus]{} . . In . Springer, . [sprenger2003structure]{} . . In . Springer, . [^1]: We do not specify a grammar for terms; all terms are of the only type $U$. [^2]: This restriction aligns with the computational interpretation of linear logic as session types. [^3]: One can maintain their relation despite the polarity change by introducing shifts in the language. We reserve this for a further work. [^4]: For brevity we elide the set $\Omega$ in the judgments. [^5]: $\mathcal{C}_{\Delta_1'}$ in the fourth operation of Figure \[fig:Reductions\] is a subset of the tape $\mathcal{C}$ connected to $\Delta'_1$. By definition of tape, two sets $\mathcal{C}_{\Delta_1'}$ and $\mathcal{C}_{\Delta_2'}$ partition $\mathcal{C}$.
--- address: - | Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218\ [()]{} -   author: - 'M. Franz and Z. Tešanović' title: Vortex state in a doped Mott insulator --- Introduction ============ Nature of the ground state as a function of doping remains one of the recurring unresolved issues in the theory of high-$T_c$ cuprate superconductors. The problem is partly due to formidable difficulties related to the theoretical description of doped Mott insulators and partly due to experimental hurdles in accessing the normal state properties in the $T\to 0$ limit because of the intervening superconducting order. Probes that suppress superconductivity and reveal the properties of the underlying ground state are therefore of considerable value. So far only pulsed magnetic fields[@ando1] in excess of $H_{c2}$ and impurity doping beyond the critical concentration[@lemberger1] have been used towards this goal. Here we argue that the vortex core spectroscopy performed using scanning tunneling microscope (STM) can provide new insights into the nature of the ground state in cuprates. We analyze the existing experimental data[@maggio1; @renner1; @pan1; @pan2] and conclude that they imply strongly correlated “normal” ground state, presumably derivable from a doped Mott insulator. We then develop a theoretical framework for the problem of tunneling in the vortex state of such a doped Mott insulator. In the vortex core the superconducting order parameter is locally suppressed to zero and the region within a coherence length $\xi$ from its center can be to the first approximation thought of as normal. Spectroscopy of the vortex core therefore provides information on the normal state electronic excitation spectrum in the $T\to 0$ limit. More accurately, the core spectroscopy reflects the spectrum in the spatially non-uniform situation where the order parameter amplitude rapidly varies in response to the singularity in the phase imposed by the external magnetic field. In order to extract useful information regarding the underlying ground state from such measurements a detailed understanding of the vortex core physics is necessary. So far the problem has been addressed using the weak coupling approach based on the Bogoliubov-de Gennes theory generalized to the $d$-wave symmetry of the order parameter[@soininen1; @wang1; @franz1; @kita1], and semiclassical calculations[@volovik1; @maki1; @ichioka1]. The early theoretical debate focused on the existence or absence of the vortex core bound states [@maki1; @franz2; @himeda1]. This debate, now resolved in favor of absence of any bound states in pure $d_{x^-y^2}$ state [@franz1; @kita1; @resende1], has somewhat eclipsed the possibly more important issues related to the nature of the ground state in cuprates. The body of work based on mean field, weak coupling calculations [@wang1; @franz1; @kita1; @ichioka1] yields results for the local density of states in the vortex core which exhibit two generic features: (i) the coherence peaks (occurring at $E=\pm\Delta_0$ in the bulk) are suppressed, with the spectral weight transferred to a (ii) broad featureless peak centered around the zero energy. Here we wish to emphasize the heretofore little appreciated fact that these features are [*qualitatively inconsistent*]{} with the existing experimental data on cuprate superconductors. STM spectroscopy on Bi$_2$Sr$_2$CaCu$_2$O$_8$ (BSCCO) at 4.2K indicates a “pseudogap” spectrum in the vortex core with the spectral weight from the coherence peaks at $\pm\Delta_0\simeq 40$meV transferred to [*high energies*]{}, and no peak whatsoever around $E=0$[@renner1]. Recent high resolution data on the same compound[@pan2] confirmed these findings down to 200mK and found evidence for weak bound states at $\pm7$meV. Experiments on YBa$_2$Cu$_3$O$_7$ (YBCO)[@maggio1] also indicate low energy bound states, but are somewhat more difficult to interpret because of the high zero-bias conductance of unknown origin appearing even in the absence of magnetic field. The fundamental discrepancy between the theoretical predictions and the experimental findings strongly suggests that models based on a simple weak coupling theory break down in the vortex core. The pseudogap observed in the core hints that the underlying ground state revealed by local suppression of the superconducting order parameter is a doped Mott insulator and not a conventional metal. Taking into account the effects of strong correlations appears to be necessary to consistently describe the physics of the vortex core. Conversely, studying the vortex core physics could provide information essential for understanding the nature of the underlying ground state in cuprates. The first step in this direction was taken by Arovas [*et al.*]{} [@arovas1] who proposed that within the framework of the SO(5) theory [@zhang1] vortex cores could become antiferromagnetic (AF). They found that such AF cores can be stabilized at low $T$ but only in the close vicinity of the bulk AF phase. In contrast, experimentally the pseudogap in the core is found to persist into the overdoped region[@renner1]. More recently microscopic calculations within the same model[@andersen1] revealed electronic excitations in such AF cores with behavior roughly resembling the experimental data. Quantitatively, however, these spectra exhibit asymmetric shifts in the coherence peaks (related to the fact that spin gap in the AF core is no longer tied to the Fermi level) not observed experimentally. These discrepancies suggest that generically cores will not exhibit the true AF order. Finally, these previous approaches are still of the Hartree-Fock-Bogoliubov type and cannot be expected to properly capture the effects of strong correlations. Here we consider a model for the vortex core based on a version of the U(1) gauge field slave boson theory formulated recently by Lee[@dhlee1]. Originally proposed by Anderson[@anderson1] the slave boson theory was formulated to describe strongly correlated electrons in the CuO$_2$ planes of the high-$T_c$ cuprates. Various versions of this theory have been extensively discussed in the literature [@baskaran1; @ruckenstein1; @kotliar1; @affleck1; @lee1]. Interest in spin-charge separated systems revived recently[@wen1; @lee2; @balents1; @dhlee1] due to the realization that it provides a natural description of the pseudogap phenomenon observed in the underdoped cuprates. The common ingredient in these theories is “splintering” of the electron into quasiparticles carrying its spin and charge degrees of freedom. Within the theories based on Hubbard and $t$-$J$ models this splintering is formally implemented by the decomposition of the electron creation operator $$c^\dagger_{i\sigma}=f^\dagger_{i\sigma} b_i \label{}$$ into a fermionic spinon $f_{i\sigma}$ and bosonic holon $b_i$. The local constraint of the single occupancy $b^\dagger_ib_i+ f^\dagger_{i\sigma}f_{i\sigma}=1$ is enforced by a fluctuating U(1) gauge field ${\bf a}$. The mean field phase diagram is known to contain four phases distinguished by the formation of spinon pairs, $\Delta_{ij}=\langle\epsilon_{\sigma\sigma'} f^\dagger_{i\sigma}f^\dagger_{j\sigma'}\rangle$, and Bose-Einstein condensation of the individual holons $b=\langle b_i\rangle$[@lee1], and is illustrated in Figure (\[fig1\]). =8.5cm The effects of magnetic field on such spin-charge separated system is most conveniently studied in the framework of an effective Ginzburg-Landau (GL) theory for the condensate fields $\Delta$ and $b$. The corresponding effective action can be constructed[@sachdev1; @lee3] based on the requirements of local gauge invariance with respect to the physical electromagnetic vector potential ${\bf A}$ and the internal gauge field ${\bf a}$: $$\begin{aligned} f_{\rm GL}&=&|(\nabla-2i{\bf a})\Delta|^2 +{r_\Delta}|\Delta|^2 + {1\over 2}{u_\Delta}|\Delta|^4\nonumber \\ &+&|(\nabla-i{\bf a}-ie{\bf A})b|^2 +r_b|b|^2 +{1\over 2} u_b|b|^4 +v|\Delta|^2|b|^2 \nonumber \\ &+& {1\over 8\pi}(\nabla\times{\bf A})^2 +f_{\rm gauge}. \label{fgl1}\end{aligned}$$ The factor of 2 in the spinon gradient term reflects the fact that [*pairs*]{} of spinons were assumed to condense. $f_{\rm gauge}$ describes the dynamics of the internal gauge field ${\bf a}$. We note that unlike the physical electromagnetic field ${\bf A}$ the gauge field ${\bf a}$ has no independent dynamics in the underlying microscopic model since it serves only to enforce a constraint. Sachdev[@sachdev1] and Nagaosa and Lee[@lee3] assumed that upon integrating out the microscopic degrees of freedom a term $$f_{\rm gauge}={\sigma\over2}(\nabla\times{\bf a})^2 \label{fgauge}$$ is generated in the free energy. They then analyzed vortex solutions of the free energy (\[fgl1\]) and came to the conclusion that two types of vortices are permissible: a “holon vortex” with the singularity in the $b$ field and a “spinon vortex” with the singularity in the $\Delta$ field. Because holons carry electric charge $e$ the holon vortex is threaded by electronic flux quantum $hc/e$, i.e. twice the conventional superconducting flux quantum $\Phi_0=hc/2e$. Spinons on the other hand condense in pairs, and the spinon vortex therefore carries flux $\Phi_0$. Stability analysis then implies that spinon vortex will be stable over the most of the superconducting phase diagram, while the $hc/e$ holon vortex can be stabilized only in the close vicinity of the phase boundary on the underdoped side [@sachdev1; @lee3]. This is a direct consequence of the fact that singly quantized vortices are always energetically favorable[@abrikosov1; @fetter1]. As far as the electronic excitations are concerned, the spinon vortex is virtually indistinguishable from the vortex in a conventional weak coupling mean field theory: the spin gap $\Delta$, which gives rise to the gap in the electron spectrum, vanishes in the core. Consequently, the vortex state based on the results of Sachdev-Nagaosa-Lee (SNL) theory[@sachdev1; @lee3] does not exhibit the pseudogap in the core and suffers from the same discrepancy with the experimental data as the weak coupling theories[@soininen1; @wang1; @franz1; @kita1] based on the conventional Fermi liquid description. Moreover, no evidence exists at present for stable doubly quantized holon vortices predicted by SNL. What is needed to account for the experimental data is a [*singly quantized holon vortex*]{} stable over the large portion of the superconducting phase in the phase diagram of Figure \[fig1\]. In the core of such a holon vortex the spin gap $\Delta$ remains finite and leads naturally to the pseudogap excitation spectrum. In what follows we show that under certain conditions the free energy (\[fgl1\]) permits precisely such solution. The results of the SNL theory are predicated upon the assumption that the “stiffness” $\sigma$ of the gauge field is relatively large and that singular configurations in which $\nabla\times{\bf a}$ contains a full flux quantum through an elementary plaquette are prohibited. Consider now a precisely opposite physical situation, allowing unconstrained fluctuations in ${\bf a}$. This amounts to the assumption that the $f_{\rm gauge}$ term (\[fgauge\]) can be neglected in (\[fgl1\]), i.e. $\sigma\to 0$. Physically this corresponds to the “extreme type-I” limit of the GL “superconductor” (\[fgl1\]) with respect to fluctuations in ${\bf a}$. Based on Elitzur’s theorem[@elitzur1] Nayak[@nayak1] recently argued that the exact local U(1) symmetry of the model cannot be broken, implying absence of the phase stiffness term (\[fgauge\]) at all energy scales. Our assumption therefore appears reasonable and in Section III. we shall give a more thorough discussion of the significance of the $f_{\rm gauge}$ term for the vortex solutions of interest here. For the time being we shall assume that $f_{\rm gauge}$ can be neglected and explore physical consequences of the resulting theory. $f_{\rm GL}$ given by Eq. (\[fgl1\]) is quadratic in ${\bf a}$ and with the $\nabla\times{\bf a}$ term absent the gauge fluctuations can be trivially integrated out. Within the closely related microscopic model this procedure has been recently implemented by Lee[@dhlee1]. The resulting effective free energy density reads $$\begin{aligned} f&=& f_{\rm amp} +{\rho_\Delta^2\rho_b^2\over 4\rho_\Delta^2+\rho_b^2} (\nabla\phi-2\nabla\theta+2e{\bf A})^2 \nonumber \\ &+&{1\over 8\pi}(\nabla\times{\bf A})^2, \label{feff}\end{aligned}$$ where we have set $\Delta=\rho_\Delta e^{i\phi}$, $b=\rho_b e^{i\theta}$, and $$\begin{aligned} f_{\rm amp}&=&(\nabla\rho_\Delta)^2 +{r_\Delta}\rho_\Delta^2 + {1\over 2}{u_\Delta}\rho_\Delta^4\nonumber \\ &+&(\nabla\rho_b)^2 +r_b\rho_b^2 +{1\over 2} u_b\rho_b^4 +v\rho_\Delta^2\rho_b^2 \label{famp}\end{aligned}$$ is the amplitude piece. The most important feature of the effective free energy (\[feff\]) is that it no longer depends on the individual phases $\phi$ and $\theta$ but only on their particular combination $$\Omega=\phi-2\theta. \label{omega}$$ Since the physical superconducting order parameter $\Psi=\Delta^*b^2= \rho_\Delta\rho_b^2 e^{-i(\phi-2\theta)}$ it is reasonable to identify $\Omega$ with the phase of a [*Cooper pair*]{}. Physically, the unconstrained fluctuations of the gauge field in Eq. (\[fgl1\]) resulted in partial restoration of the original electronic degrees of freedom in Eq.  (\[feff\]). In the underlying microscopic model this means that on long length scales spinons and holons are always confined, in agreement with Elitzur’s theorem [@elitzur1; @nayak1]. On lengthscales shorter than the confinement length, such as inside the vortex core, spinons and holons can still appear locally decoupled. In the present effective theory this aspect is reflected by two amplitude degrees of freedom present in (\[feff\]). More detailed discussion of these issues is given in Refs. [@dhlee1; @nayak1]. We have thus arrived at an effective theory of a spin-charge separated system containing one phase degree of freedom $\Omega$ and two amplitudes, $\rho_\Delta$ and $\rho_b$. Deep in the superconducting phase, where both amplitudes are finite, the physics of (\[feff\]) will be very similar to that of a conventional GL theory. In the situations where the superconducting order parameter $\Psi$ is strongly suppressed, such as in the vortex core, near an impurity or a wall, the new theory has an extra degree of richness, associated with the fact that it is sufficient (and generally preferred by the energetics) when only [*one*]{} of the two amplitudes is suppressed. Since the two amplitudes play very different roles in the electronic excitation spectrum, the effective theory (\[feff\]) will lead to a number of nontrivial effects. To illustrate this consider what will happen in the core of a superconducting vortex. Under the influence of the magnetic field the phase $\Omega$ will develop a singularity such that $\nabla\Omega \sim 1/r$ close to the vortex center. For the free energy to remain finite the amplitude prefactor in the second term of Eq. (\[feff\]) must vanish for $r\to 0$. This is analogous to $|\Psi|$ vanishing in the core of a conventional vortex. In the present case, however, it is sufficient when the product $\rho_\Delta\rho_b$ vanishes. Since suppressing any of the two amplitudes costs condensation energy, in general only one amplitude will be driven to zero. Which of the two is suppressed will be determined by the energetics of the amplitude term (\[famp\]). On general grounds we expect that the state in the vortex core will be the same as the corresponding bulk “normal” state obtained by raising temperature above $T_c$. Thus, very crudely, we expect that holon vortex will be stable in the underdoped while the spinon vortex will be stable in the overdoped region of the phase diagram Figure \[fig1\]. An important point by which our approach differs from the SNL theory is that in the present theory [*both*]{} types of vortices carry the [*same*]{} superconducting flux quantum $\Phi_0$ and thus compete on equal footing. This is a direct consequence of our assumption of the vanishing phase stiffness $\sigma$. In what follows we study in detail the vortex solutions of the free energy (\[feff\]). Our main objective is to obtain the precise estimates for the energy of the two types of vortices as a function of temperature and doping and deduce the corresponding phase diagram for the state inside the vortex core. We show that for generic parameters in (\[feff\]) the singly quantized holon vortex with a pseudogap spectrum in the core can be stabilized over a large portion of the superconducting phase, as required by the experimental constraints discussed above. Solution for a single vortex ============================ General considerations ---------------------- In order to provide a more quantitative discussion we now adopt some assumptions about the coefficients entering the free energy (\[feff\]). We assume that $$r_i=\alpha_i(T-T_i), \ \ i=b,\Delta, \label{ri}$$ where $T_i$ are corresponding “bare” critical temperatures, which we assume depend on doping concentration $x$ in the following way: $$T_\Delta = T_0(2x_m-x), \ \ \ T_b = T_0x. \label{tc}$$ Here $x_m$ denotes the optimal doping and $T_0$ sets the overall temperature scale. We furthermore assume that $u_i$ and $v$ are all positive and independent of doping and temperature. It is easy to see that such choice of parameters qualitatively reproduces the bulk phase diagram of cuprates in the $x$-$T$ plane shown in Figure \[fig1\]. The effect of the $v$-term is to suppress $T_c$ from its bare value away from the optimal doping. In real systems fluctuations will lead to additional suppression of $T_c$ which we do not consider here. In the absence of perturbations the bulk values of the amplitudes are given by $$\begin{aligned} \bar{\rho}_\Delta^2 &=& -({r_\Delta} u_b-r_bv)/D, \nonumber \\ \bar{\rho}_b^2 &=& -(r_b{u_\Delta} -{r_\Delta} v)/D, \label{rho}\end{aligned}$$ with $D=u_b{u_\Delta}-v^2$. In analogy with conventional GL theories we may define coherence lengths for the two amplitudes[@sachdev1] $$\begin{aligned} \xi_\Delta^{-2} &=& -({r_\Delta}-r_bv/u_b), \nonumber \\ \xi_b^{-2} &=& -(r_b-{r_\Delta} v/{u_\Delta}), \label{xi}\end{aligned}$$ one of which always diverges at $T_c$ as $(T-T_c)^{-1/2}$. Minimization of the free energy (\[feff\]) with respect to the vector potential ${\bf A}$ yields an equation $$\nabla\times\nabla\times{\bf A}=e\rho_s(\nabla\Omega-2e{\bf A}), \label{lona}$$ where $$\rho_s={4\rho_\Delta^2\rho_b^2\over 4\rho_\Delta^2+\rho_b^2} \label{rhos}$$ is the effective superfluid density. The term in brackets can be identified as twice the conventional superfluid velocity $${\bf v}_s={1\over 2}\nabla\Omega-e{\bf A}.$$ Making use of the Ampere’s law $4\pi{\bf j}=\nabla\times{\bf B}$ we see that Eq. (\[lona\]) specifies the supercurrent in terms superfluid density and velocity: ${\bf j}=2e\rho_s{\bf v}_s$. Minimization of (\[feff\]) with respect to $\Omega$ then implies $\nabla\cdot{\bf j}=0$; the supercurrent is conserved. Minimizing the free energy (\[feff\]) with respect to the amplitudes results in the pair of coupled GL equations: \[gl:all\] $$-\nabla^2\rho_\Delta + {r_\Delta}\rho_\Delta + {u_\Delta}\rho_\Delta^3 + v\rho_b^2\rho_\Delta +{4\rho_\Delta^2\rho_b^2\over (4\rho_\Delta^2+\rho_b^2)^2}{\bf v}_s^2 = 0, \label{gl:a}$$ $$-\nabla^2\rho_b + r_b\rho_b + u_b\rho_b^3 + v\rho_\Delta^2\rho_b +{16\rho_\Delta^2\rho_b^2\over (4\rho_\Delta^2+\rho_b^2)^2}{\bf v}_s^2 = 0. \label{gl:b}$$ We are interested in the behavior of the amplitudes in the vicinity of the vortex center. In this region, for a strongly type-II superconductor, we may neglect the vector potential [[**A**]{}]{} in the superfluid velocity ${\bf v}_s$. In a singly quantized vortex $\Omega$ winds by $2\pi$ around the origin leading to a singularity of the form ${\bf v}_s\simeq{1\over 2}\nabla\Omega =\hat\varphi/2r$. First, for the [*holon*]{} vortex we assume that $\rho_b$ vanishes in the core as some power $\rho_b(r)\sim r^\nu$ and $\rho_\Delta(r)\approx\bar\rho_\Delta$ remains approximately constant. Eq. (\[gl:b\]) then becomes $$({1\over 4}-\nu^2)r^{\nu-2} + (r_b+v\bar\rho_\Delta^2)r^\nu + u_b\bar\rho_b^2r^{3\nu}=0, \label{glcore}$$ where we have neglected $\rho_b^2(r)$ compared to $4\bar\rho_\Delta^2$ in the denominator of the last term in Eq. (\[gl:b\]). The most singular term in Eq. (\[glcore\]) is the first one and we must demand that the coefficient of $r^{\nu-2}$ vanishes. This implies $\nu={1\over 2}$. The asymptotic short distance behavior of the holon amplitude therefore can be written as $$\rho_b(r)\simeq c_b\bar\rho_b \left(r\over\xi_b\right)^{1/2}, \label{rhobcore}$$ where $c_b$ is a constant of order unity which may be determined by the full integration of Eqs. (\[gl:all\]). Similar analysis of Eq. (\[gl:a\]) in the vicinity of the [*spinon*]{} vortex yields $$\rho_\Delta(r)\simeq c_\Delta\bar\rho_\Delta \left(r\over\xi_d\right), \label{rhodcore}$$ with $\rho_b$ approximately constant. We notice the different power laws in the holon and spinon results. Operationally this difference arises from different numerical prefactors of the respective superfluid velocity terms in Eqs. (\[gl:all\]). Physically, the unusual $r$ dependence of the holon amplitude in the core reflects the fact that the field $b$ describes a condensate of single holons, each carrying charge $e$. Superconducting vortex with the flux quantum $\Phi_0$ represents a magnetic “half-flux” for the holon field which results in non-analytic behavior of $\rho_b(r)$ at the origin. Singly quantized holon vortex is therefore a peculiar object and we shall discuss it more fully in Section III. Here we note that the physical superconducting order parameter amplitude $|\Psi|=\rho_\Delta\rho_b^2$ remains analytic in the core of both the spinon and the holon vortex. Holon vs. spinon vortex: the phase diagram ------------------------------------------ We are now in the position to estimate the energies of the two types of vortices and deduce the phase diagram for the “normal” state in the vortex core. To this end we consider a single isolated vortex centered at the origin. The total vortex line energy can be divided into electromagnetic and core contributions[@fetter1]. The electromagnetic contribution consists of the energy of the supercurrents and the magnetic field outside the core region. It may be estimated by assuming that the amplitudes $\rho_\Delta$ and $\rho_b$ have reached their bulk values $\bar\rho_\Delta$ and $\bar\rho_b$ respectively. Taking curl of Eq. (\[lona\]) and noting that $\nabla\times\nabla\Omega=2\pi\delta({\bf r})$ for a singly quantized vortex we obtain the London equation for the magnetic field ${\bf B}=\nabla\times{\bf A}$ of the form $$B-\lambda^2\nabla^2B=\Phi_0\delta({\bf r}) \label{lon}$$ where $$\lambda^{-2}=8\pi e^2 {4\bar\rho_\Delta^2\bar\rho_b^2\over 4\bar\rho_\Delta^2+\bar\rho_b^2}. \label{lam}$$ has the meaning of the London penetration depth for the effective GL theory (\[feff\]). Aside from the unusual form of $\lambda$, Eq. (\[lon\]) is identical to the conventional London equation. The corresponding electromagnetic energy is therefore the same for both types of vortices and can be calculated in the usual manner[@abrikosov1; @fetter1; @sachdev1] obtaining $$E_{\rm EM}\simeq\left({\Phi_0\over 4\pi\lambda}\right)^2 \ln\kappa, \label{em}$$ with $\kappa=\lambda/\max(\xi_\Delta,\xi_b)$ being the generalized GL ratio. To estimate the core contribution to the vortex line energy we assume that one of the amplitudes is suppressed to zero in the core $$\rho_i(r)=0, \ \ \ r<\xi_i, \label{rhor}$$ while the other one stays constant and equal to its bulk value. This is a very crude approximation which we justify below by an exact numerical computation. With these assumptions, the core energy is $$E_{\rm core}^{(i)}\simeq\left({\Phi_0\over 4\pi\lambda_i}\right)^2, \label{ecore}$$ where $i=\Delta,b$ for spinon and holon vortex respectively and $$\lambda^{-2}_i=8\pi e^2 \bar\rho_i^2. \label{lami}$$ Such a crude approximation overestimates the core energy. A more accurate analysis[@abrikosov1; @fetter1], which we do not pursue here, allows for a more realistic variation of $\rho_i(r)$ in the core and indicates that the value of $E_{\rm core}^{(i)}$ has the same form as Eq. (\[ecore\]) multiplied by a numerical factor $c_1\approx 0.5$[@hu1; @alama1]. Thus, the total energy of the vortex line can be written as $$E^{(i)}=\left({\Phi_0\over 4\pi\lambda}\right)^2\ln\kappa +c_1\left({\Phi_0\over 4\pi\lambda_i}\right)^2, \label{evortex}$$ where again $i=\Delta,b$ for spinon and holon vortex respectively. Eq. (\[evortex\]) parallels the Abrikosov expression for the vortex line energy in a conventional GL theory[@abrikosov1] where $\lambda$ and $\lambda_i$ are identical and equal to the ordinary London penetration depth. =8.5cm In the vortex state described by the free energy (\[feff\]) the vortex with lower energy $E^{(i)}$ will be stabilized. Eq. (\[evortex\]) implies that the difference in energy between the two types of vortices comes primarily from the core contribution, as expected on the basis of the physical argument presented above. Condition $\lambda_\Delta=\lambda_b$ marks the transition point between the two solutions. For fixed GL parameters $T_0$, $x_m$, $\alpha_i$, $u_i$ and $v$ this defines a transition line in the $x$-$T$ plane. According to (\[lami\]) the equation for this line is $$\bar\rho_\Delta(x,T)=\bar\rho_b(x,T). \label{tran}$$ Using Eqs. (\[ri\]-\[rho\]) one can obtain an explicit expression for the transition temperature $T_g$ between two types of vortices as a function of doping $$T_g(x)=T_0\left[{2x_m-x\over 1-\beta}+{x\over 1-\beta^{-1}}\right], \label{tg}$$ with $$\beta={\alpha_b(u_\Delta+v)\over \alpha_\Delta(u_b+v)}. \label{beta}$$ Eq. (\[tg\]) describes a straight line in the $x$-$T$ plane, originating at $[x_m,T_0x_m]$, i.e. maximal $T_c$ at optimum doping, and terminating at $[2x_m/(1+\beta),0]$. Generically, we expect that parameters $\alpha_i$ and $u_i$ will be comparable in magnitude for the holon and spinon channels. Parameter $\beta$ defined in Eq. (\[beta\]) will therefore be of order unity. The typical situation for $\beta=0.77$ is illustrated in Figure \[fig2\]. More generally the quartic coefficients $u_i$ and $v$ could exhibit weak doping and temperature dependences leading to a curvature in the phase boundary. The appealing feature of the present theory is that parameter $\beta$ may vary from compound to compound. Thus, the experimental fact that in BSCCO the pseudogap in the core persists into the overdoped region is easily accounted for in the present theory. It would be interesting to see if the transition from holon to spinon vortex as a function of doping could be experimentally observed. A good candidate for such observation would be LSCO, where the transport measurements in pulsed magnetic fields[@ando1] established a metal-insulator transition around optimal doping, i.e. $\beta\approx 1$. The current theory predicts a holon vortex with the pseudogap spectrum in the underdoped (insulating) region and spinon vortex with conventional metallic spectrum on the overdoped side. Numerical results ----------------- In order to put the above analytical estimates on firmer ground we now pursue numerical computation of the vortex line energy. For simplicity we consider the strongly type-II situation $(\kappa\gg1)$ where the vector potential term in ${\bf v}_s$ can be neglected to an excellent approximation, as long as we focus on the behavior close to the core. We are then faced with the task of numerically minimizing the free energy (\[feff\]) with respect to the two cylindrically symmetric amplitudes $\rho_\Delta(r)$ and $\rho_b(r)$. As noted by Sachdev [@sachdev1] direct numerical minimization of the free energy (\[feff\]) provides a more robust solution than the numerical integration of the coupled differential equations (\[gl:all\]). We discretize the free energy functional (\[feff\]) on a disk of a radius $R\gg\xi_i$ in the radial coordinate $r$ with up to $N= 2000$ spatial points. We then employ the Polak-Ribiere variant of the Conjugate Gradient Method [@numrec] to minimize this discretized functional with respect to $\rho_\Delta(r_j)$ and $\rho_b(r_j)$, initialized to suitable single vortex trial functions. The procedure converges very rapidly and the results are insensitive to the detailed shape of the trial functions as long as they saturate to the correct bulk values outside the vortex core. Typical results of our numerical computations are displayed in Figure (\[fig3\]) and are in complete agreement with the analytical considerations of the preceding subsections. Note in particular that $\rho_b(r)$ in the holon vortex vanishes with infinite slope, consistent with Eq. (\[rhobcore\]). Plotting $\rho_b^2(r)$ confirms that the exponent is indeed $1/2$. In the spinon vortex $\rho_\Delta(r)$ is seen to vanish linearly as expected on the basis of Eq. (\[rhodcore\]). The nonvanishing order parameter is slightly elevated in the core reflecting the effective “repulsion” between the two amplitudes contained in the $v$-term of the free energy. The results for the spinon vortex are consistent with those of Ref. [@sachdev1]. =8.5cm We explored a number of other parameter configurations and obtained similar results. We find that Eq. (\[tran\]) is a good predictor of the transition line between the holon and spinon vortex, although the precise numerical value of the transition temperature $T_g$ for given $x$ tends to deviate slightly from the value predicted by Eq. (\[tg\]). This is illustrated in Figure (\[fig2\]) where we compare the vortex core phase diagrams obtained numerically and from Eq. (\[tg\]). Interestingly, the deviation always tends to enlarge the holon vortex sector of the phase diagram at the expense of the spinon vortex sector. This is presumably because the sharper $\sim\sqrt{r}$ suppression of the holon order parameter in the core costs less condensation energy. Gauge fluctuations and the spectral properties in the core ========================================================== Theory of the vortex core based on the effective action (\[feff\]) appears to yield results consistent with the STM data on cuprates [@renner1; @pan2] in that it implies stable holon vortex solution over the large portion of the superconducting phase diagram. The state inside the core of such a holon vortex is characterized by vanishing amplitude of the holon condensate field, $|b|=0$, and a finite spin gap $|\Delta|\approx\Delta_{\rm bulk}$. This is the same state as in the pseudogap region above $T_c$. One would thus expect the electronic spectrum in the core to be similar to that found in the normal state of the underdoped cuprates, in agreement with the data[@renner1; @pan2]. The holon vortex with this property carries conventional superconducting flux quantum $\Phi_0$, in accord with experiment. This general agreement between theory and experiment would suggest that the effective action (\[feff\]) provides the sought for phenomenological description of the vortex core physics in cuprates. In what follows we amplify our argumentation that it is also tenable in a broader theoretical context in that it naturally follows from the U(1) slave boson models extensively studied in the classic and more recent high-$T_c$ literature. We then provide a more detailed discussion of the vortex core spectra and propose an explanation for the experimentally observed core bound states. Significance of the $f_{\rm gauge}$ term ---------------------------------------- Derivation of the effective action (\[feff\]) from the more general U(1) action (\[fgl1\]) hinges on our assumption that the stiffness $\sigma$ of the gauge field ${\bf a}$ is low and that the $f_{\rm gauge}$ term (\[fgauge\]) can be neglected. Assumption of large $\sigma$ by SNL leads to very different vortex solutions[@sachdev1; @lee3] which appear inconsistent with the recent experimental data. We first expand on our discussion as to why is $f_{\rm gauge}$ term important and then we argue why it may be permissible to neglect it in the realistic models of cuprates. To facilitate the discussion let us rewrite Eq. (\[fgl1\]) by resolving the complex matter fields into amplitude and phase components: $$\begin{aligned} f_{\rm GL}&=& f_{\rm amp} +\rho_\Delta^2(\nabla\phi -2{\bf a})^2 +\rho_b^2(\nabla\theta-{\bf a}-e{\bf A})^2 \nonumber \\ &+&{1\over 8\pi}(\nabla\times{\bf A})^2 + {\sigma\over2}(\nabla\times{\bf a})^2, \label{fgl2}\end{aligned}$$ with $f_{\rm amp}$ specified by Eq. (\[famp\]). Now consider situation in which the sample is subjected to uniform magnetic field ${\bf B}=\nabla\times{\bf A}$. Two scenarios (discussed previously by SNL) appear possible. In the first, the internal gauge field develops no net flux, $\langle\nabla\times{\bf a}\rangle=0$, and the holon phase $\theta$ develops singularities in response to ${\bf A}$ such that $$\nabla\times\nabla\theta=2\pi\sum_j\delta({\bf r}-{\bf r}_j),$$ where ${\bf r}_j$ denotes the vortex positions. The holon amplitude $\rho_b$ is driven to zero at ${\bf r}_j$, essentially to prevent the free energy from diverging due to the singularity in the phase gradient. Since holons carry charge $e$, each vortex is threaded by flux $hc/e$, i.e. twice the superconducting flux quantum $\Phi_0=hc/2e$. This solution represents the doubly quantized holon vortex lattice, considered by SNL. In the second scenario ${\bf a}$ develops a net flux such that ${\bf a}\approx -e{\bf A}$, which screens out the ${\bf A}$ field in the holon term but produces a net flux $-2e{\bf A}$ in the spinon term. In response to this flux, spinon phase $\phi$ develops singularities such that $$\nabla\times\nabla\phi=2\pi\sum_j\delta ({\bf r}-\tilde{\bf r}_j),$$ corresponding to the spinon vortex lattice. $\tilde{\bf r}_j$ denotes vortex positions which will be different from ${\bf r}_j$ since at the fixed field $B$ there will be twice as many spinon vortices as holon vortices. (Spinon vortices carry conventional superconducting quantum of flux $\Phi_0$.) In this case $\rho_\Delta$ is driven to zero at the vortex centers. In this scenario one pays a penalty for nucleating the net flux in $\nabla\times{\bf a}$ due to last term in Eq. (\[fgl2\]). This energy cost can be estimated as $$E_\sigma\simeq 8\pi\sigma e^2 \left({\Phi_0\over 4\pi\lambda}\right)^2 \label{esig}$$ per vortex. Stiffness $\sigma$ must be small enough so that $E_\sigma$ is small compared to the vortex energy (\[evortex\]). Taking the dominant $E_{\rm EM}$ term and neglecting $\ln\kappa$ this implies that $$\sigma\ll {1\over 8\pi e^2}, \label{sig}$$ which is the same condition as considered in Ref. [@sachdev1]. Now consider a [*third*]{} scenario in which a [*singly quantized*]{} holon vortex emerges. As a starting point consider the spinon vortex solution just described. In the underdoped regime the amplitude piece $f_{\rm amp}$ would favor suppressing the holon amplitude in the core instead of the spinon amplitude but according to our previous considerations this would ordinarily require formation of a doubly quantized vortex whose magnetic energy is too large. However, if the gauge field stiffness $\sigma$ is sufficiently small, the system could lower its free energy by setting up singularities in ${\bf a}$ which would precisely cancel the singularities in $\nabla\phi$ and shift them to the holon term. To arrive at this situation imagine contracting the initially uniform flux $\nabla\times{\bf a}$ so that it becomes localized in the individual vortex core regions. Taking this procedure to the extreme, i.e. taking the limit $\sigma\to 0$, the gauge field will form “flux spikes” of the form $$2(\nabla\times{\bf a}) =-\nabla\times\nabla\phi=-2\pi\sum_j\delta({\bf r}-\tilde{\bf r}_j), \label{sing}$$ completely localized at the vortex centers. Gauge field of this form indeed completely cancels the singularities in the spinon phase gradient in Eq. (\[fgl2\]) and $\rho_\Delta$ is no longer forced to vanish in the core. The singularities now appear in the holon term, but they stem from ${\bf a}$ rather that $\nabla\theta$ which remains nonsingular. Consequently, $\rho_b$ is forced to vanish in the vortex cores. By construction the vortices are located at $\tilde{\bf r}_j$ and are therefore singly quantized. This is the singly quantized holon vortex discussed in the framework of the free energy (\[feff\]). Based on the above discussion the singly quantized holon vortex can be thought of as a composite object formed by attaching half quantum $(h/2)$ of the fictitious gauge flux $\nabla\times{\bf a}$ to the spinon vortex. Within the full compact U(1) theory this is essentially equivalent to the Z$_2$ vortex discussed by Wen[@wen2] in the framework of topological orders in spin liquids. In the framework of the free energy (\[fgl2\]) one pays a penalty for such a singular solution due to the gauge stiffness term. In the present continuum model this penalty per single vortex is actually infinite, since according to Eq. (\[sing\]) it involves a spatial integral over $[\delta({\bf r}-\tilde{\bf r}_j)]^2$. Thus, in the continuum model the singular solutions of this type are prohibited. In reality, however, we have to recall that our effective action (\[fgl1\]) descended from a microscopic lattice model for spinons and holons in which the gauge field ${\bf a}$ lives on the nearest neighbor bonds of the ionic lattice. The ionic lattice constant $d$ therefore provides a natural short distance cutoff and the delta function in Eq. (\[sing\]) should be interpreted as a flux quantum $\Phi_0$ piercing an elementary plaquette of the lattice. The energy cost per vortex thus becomes finite and is given by $$E'_\sigma\simeq {\sigma e^2\over 2} \left({\Phi_0\over d}\right)^2. \label{esigp}$$ Again, for the solution to be stable, $E'_\sigma$ must be negligible compared to the vortex energy (\[evortex\]). This implies $$\sigma\ll {1\over 8\pi^2e^2} \left({d\over \lambda}\right)^2, \label{sigp}$$ which is a much more stringent condition than (\[sig\]) since in cuprates $d\ll\lambda$. When condition (\[sigp\]) is satisfied it is permissible to neglect the $f_{\rm gauge}$ term in the effective action (\[fgl1\]) and it becomes fully equivalent to (\[feff\]) as far as the vortex solutions are concerned. Eq. (\[sigp\]) gives the precise meaning to the requirement of the weak stiffness of the gauge field loosely stated when deriving the effective action (\[feff\]). Microscopic considerations -------------------------- As mentioned in the introduction, the gauge field ${\bf a}$ has no dynamics in the original U(1) microscopic model, as it only serves to enforce a constraint on spinons and holons. The stiffness term (\[fgauge\]) in the effective theory was assumed to arise in the process of integrating out the microscopic degrees of freedom[@sachdev1; @lee3]. While such term is certainly permitted by symmetry, assessing its strength $\sigma$ is a nontrivial issue since even deep in the superconducting phase neither holons nor spinons are truly gapped. Thus, in general, integrating out these degrees of freedom may lead to singular and nonlocal interactions between the condensate and the gauge fields. To our knowledge the procedure has not been explicitly performed for the U(1) model and the precise form or magnitude of the gauge stiffness term is unknown. General considerations[@nayak1] suggest that the gauge stiffness term is negligible in the class of models with exact local U(1) symmetry connecting the phases of holons and spinons. Consider now an intermediate representation of the problem where only high energy microscopic degrees of freedom have been integrated out. In the presence of a cutoff this is a well defined procedure even for gapless excitations, as explicitly shown by Kwon and Dorsey[@kwon1] for a simple BCS model. The corresponding effective Lagrangian density of the present U(1) model can be written as $$\begin{aligned} {\cal L}_{\rm eff}&=& {\kappa_\Delta^\mu\over 2}(\partial_\mu\phi-2a_\mu)^2 +{\kappa_b^\mu\over 2}(\partial_\mu\theta-a_\mu-eA_\mu)^2 -f_{\rm amp} \nonumber \\ &+&(\partial_\mu\phi-2a_\mu)J_{\rm sp}^\mu +(\partial_\mu\theta-a_\mu-eA_\mu)J_h^\mu \nonumber \\ &+&{\cal L}_{\rm sp}[\psi_{\rm sp},\psi_{\rm sp}^\dagger;\rho_\Delta] +{\cal L}_h[\psi_h,\psi_h^\dagger;\rho_b] +{\cal L}_{\rm EM}[A_\mu]. \label{leff}\end{aligned}$$ The Greek index $\mu$ runs over time and two spatial dimensions, $\kappa_i^0$ are compressibilities of the holon and spinon condensates, while $$\kappa_i^j=-2(\rho_i)^2,\ \ \ i=\Delta, b, \ \ \ j=1,2, \label{kappa}$$ are the respective phase stiffnesses. $J_{\rm sp}^\mu$ and $J_h^\mu$ are spinon and holon three currents respectively and ${\cal L}_{\rm sp}$ and ${\cal L}_h$ are the low energy effective Lagrangians for the fermionic spinon field $\psi_{\rm sp}$ and bosonic holon field $\psi_h$. ${\cal L}_{\rm EM}$ is the Maxwell Lagrangian for the physical electromagnetic field. Thus, ${\cal L}_{\rm eff}$ describes an effective low energy theory of spinons and holons coupled to their respective collective modes and a fluctuating U(1) gauge field. Similar theory has been recently considered by Lee[@dhlee1]. The precise form of the microscopic Lagrangians ${\cal L}_{\rm sp}$ and ${\cal L}_h$ is not important for our discussion. The salient feature which we exploit here is that only the amplitude of the respective condensate field enters into ${\cal L}_{\rm sp}$ and ${\cal L}_h$. Coupling to the phases and the gauge field is contained entirely in the Doppler shift terms \[second line of Eq. (\[leff\])\]. Such form of the coupling is largely dictated by the requirements of the gauge invariance and the particular form Eq. (\[leff\]) can be explicitly derived by gauging away the respective phase factors from the $\psi$ fields[@balents1; @kwon1]. The gauge field $a_\mu$ enters the effective Lagrangian (\[leff\]) only via two gauge invariant terms: $(\partial_\mu\phi-2a_\mu)$ and $(\partial_\mu\theta-a_\mu-eA_\mu)$, which may be interpreted as the three velocities of the spinon and holon condensates respectively. Furthermore, the only coupling between holons and spinons arises from $a_\mu$. Therefore, if we now proceed to integrate out the remaining microscopic degrees of freedom from ${\cal L}_{\rm eff}$, the two velocity terms will not mix. This consideration suggests that upon integrating out all of the microscopic degrees of freedom, the resulting gauge stiffness term will be of the form $$\begin{aligned} f'_{\rm gauge}&=&{\sigma_\Delta\over2}[\nabla\times(2{\bf a}-\nabla\phi)]^2 \nonumber \\ &+&{\sigma_b\over2}[\nabla\times({\bf a}+e{\bf A}-\nabla\theta)]^2. \label{fgaugep}\end{aligned}$$ Clearly, such term is permitted by the gauge symmetry. Furthermore, we note that for smooth (i.e. vortex free) configurations of phases the gradient terms will contribute nothing and we recover the gauge term considered in Ref. [@lee3]. In the presence of a vortex in $\phi$ or $\theta$ the $f'_{\rm gauge}$ term will contribute formally divergent energy. Regularizing this on the lattice, as discussed above Eq. (\[esigp\]), this energy will become finite and can be interpreted simply as the energy of the spinon or holon vortex core states, which have been integrated out. In the microscopic theory (\[leff\]) such energy would arise upon solving the relevant fermionic or bosonic vortex problem. We stress that, as concluded in the preceding subsection, the main theoretical obstacle to the formation of a singly quantized holon vortex in the original SNL theory was the appearance of a formally divergent contribution in the $f_{\rm gauge}$ term (\[fgauge\]). The argument above suggests that $f_{\rm gauge}$ in Eq. (\[fgl1\]) should be replaced by Eq.  (\[fgaugep\]), in which such formally divergent contribution appears for [*arbitrary*]{} vortex configuration and upon regularization has a simple physical interpretation in terms of the energy of the vortex core states. Usage of the physically motivated term (\[fgaugep\]) in place of (\[fgauge\]) therefore removes the bias against the singly quantized holon vortex solution, which appears to be realized in real materials. With (\[fgaugep\]) any bias between the holon and spinon vortex solutions can result only from the difference between the two stiffness constants $\sigma_\Delta$ and $\sigma_b$. It is reasonable on physical grounds to assume that constants $\sigma_\Delta$ and $\sigma_b$ are of the similar magnitudes. Furthermore, on the basis of Ref. [@nayak1] we expect these constants to be negligibly small in the physically relevant models. Consequently we expect that neglecting the $f_{\rm gauge}$ term as in our derivation of effective action (\[feff\]) will result in accurate determination of the phase diagram for the state in the vortex core. Vortex core states ------------------ The phenomenological theory based on the effective action (\[feff\]) does not allow us to address the interesting question of the nature of the fermionic states in the vortex core. To do this we need to consider the microscopic Lagrangian density (\[leff\]). While the fully self consistent calculation is likely to be prohibitively difficult, one can obtain qualitative insights by first solving the GL theory (\[feff\]) as described in Sec. II, and then using the order parameters $\rho_\Delta$ and $\rho_b$ as an input to the fermionic and bosonic sectors of the theory specified by Eq. (\[leff\]). The work on a detailed solution of this type is in progress. Here we wish to point out some interesting features of such a theory and argue that it may indeed exhibit structure in the low energy spectral density similar to that found experimentally[@maggio1; @pan2]. It is instructive to integrate out the gauge fluctuations from the Lagrangian (\[leff\]) as first discussed by Lee[@dhlee1]. Since ${\cal L}_{\rm eff}$ is quadratic in $a_\mu$ the integration can be explicitly performed resulting in the Lagrangian of the form $$\begin{aligned} {\cal L}'_{\rm eff}&=& {1\over 2}K_\mu(v_s^\mu)^2 -f_{\rm amp} + {\cal L}_{\rm EM}\nonumber \\ &-& {2\kappa_b^\mu\over 4\kappa_\Delta^\mu+\kappa_b^\mu} (v_s^\mu J_{\rm sp}^\mu) +{4\kappa_\Delta^\mu\over 4\kappa_\Delta^\mu+\kappa_b^\mu} (v_s^\mu J_h^\mu)\nonumber \\ &+&{\cal L}_{\rm sp} +{\cal L}_h -{1\over 2}{1\over 4\kappa_\Delta^\mu+\kappa_b^\mu}(2J_{\rm sp}^\mu+J_h^\mu)^2, \label{leffi}\end{aligned}$$ where $K_\mu=4\kappa_\Delta^\mu\kappa_b^\mu/(4\kappa_\Delta^\mu+\kappa_b^\mu)$ and $$v_s^\mu=(\partial_\mu\theta-{1\over2}\partial_\mu\phi-eA_\mu) \label{vs}$$ is the physical superfluid velocity. The first line reproduces the GL effective action (\[feff\]) for the condensate fields, the second line describes the Doppler shift coupling of the superfluid velocity to the microscopic currents, and the third line contains spinon and holon pieces with additional current-current interactions generated by the gauge fluctuations[@dhlee1]. We now discuss the physical implications of Eq. (\[leffi\]) for the two types of vortices. We focus on the static solutions (i.e. we ignore the time dependences of various quantities, e.g. taking $v_s^0=0$) of ${\cal L}'_{\rm eff}$ in the presence of a single isolated vortex. We are interested in the local spectral function of a physical electron. This is given by a convolution in the energy variable of the spinon and holon spectral functions. According to the analysis presented in Ref. [@lee2], at low temperatures the electron spectral function will be essentially equal to the spinon spectral function. Convolution with the holon spectral function which is dominated by the sharp coherent peak due to the condensate merely leads to a small broadening of the order $T$. In the following we therefore focus on the behavior of spinons in the vicinity of the two types of vortices. By inspecting Eq. (\[leffi\]) it is easy to see that the excitations inside the [*spinon vortex*]{} will be qualitatively very similar to those found in the conventional vortex described by the weak coupling $d$-wave BCS theory[@wang1; @franz1; @kita1]. In particular according to Eq. (\[rhodcore\]) we have $\kappa_\Delta\sim r^2$, and $\kappa_b\sim$ const in the core. Recalling furthermore that $|{\bf v}_s|\sim 1/r$ we observe that the spinon current ${\bf J}_{\rm sp}$ is coupled to a term that diverges as $1/r$ in the core (just as in a conventional vortex), while the holon current ${\bf J}_h$ is coupled to a nonsingular term. Thus, one may conclude that holons remain essentially unperturbed by the phase singularity in the spinon vortex while the spinons obey the essentially conventional Bogoliubov-de Gennes equations for a $d$-wave vortex. In the [*holon vortex*]{} the situation is quite different. According to Eq. (\[rhobcore\]) we have $\kappa_b\sim r$ and $\kappa_\Delta\sim$ const in the core. The spinon current ${\bf J}_{\rm sp}$ is now coupled to a nonsingular term ($1/r$ divergence in $v_s$ is canceled by $\kappa_b\sim r$). Therefore, there will be no topological perturbation in the spinon sector and we expect the spinon wavefunctions to be essentially unperturbed by the diverging superfluid velocity. Spinon spectral density in the core should be qualitatively similar to that far outside the core. This is our basis for expecting a pseudogap-like spectrum in the core of a holon vortex. We now address the possible origin of the experimentally observed vortex core states[@maggio1; @pan2] within the present scenario for a holon vortex. To this end consider the effect of the last term in Eq. (\[leffi\]) which we ignored so far. Upon expanding the binomial the temporal component is seen to contain a density-density interaction of the form $J_{\rm sp}^0 J_h^0$ where $J_h^0$ is the local density of [*uncondensed*]{} holons. Since the holon order parameter vanishes in the core and the electric neutrality dictates that the total density of holons must be approximately constant in space, we expect that uncondesed holon density will behave roughly as $$J_h^0(r)=\bar\rho_b-\rho_b(r);$$ $J_h^0(r)$ will have a spike in the core of a holon vortex. Insofar as $J_h^0(r)$ can be viewed as a static potential acting on spinons, the uncondensed holons in the vortex core can be thought of as creating a scattering potential, akin to an impurity embedded in a $d$-wave superconductor. In fact, formally the spinon problem is identical to the problem of a fermionic quasiparticle in a $d$-wave superconductor in zero field in the presence of a localized impurity potential. It is known that such problem exhibits a pair of marginally bound impurity states[@balatsky1] at low energies which result in sharp resonances in the spectral density inside the gap. Such states have been extensively studied theoretically [@balatsky2; @flatte1; @shnirman1; @atkinson1] and their existence was recently confirmed experimentally by Pan [*et al.*]{} [@pan1]. We propose here that, within the formalism of Eq. (\[leffi\]), the same mechanism could give rise to the low energy quasiparticle states in the core of a holon vortex. Such structure, if indeed confirmed by a microscopic calculation, could explain the spectral features observed experimentally in the vortex cores of cuprate superconductors [@maggio1; @pan2]. Conclusions =========== Scanning tunneling spectroscopy of the vortex cores affords a unique opportunity for probing the underlying “normal” ground state in cuprate superconductors. The existing experimental data on YBCO and BSCCO strongly suggest that conventional mean field weak coupling theories [@soininen1; @wang1; @franz1; @kita1; @volovik1; @maki1; @ichioka1] fail to describe the physics of the vortex core. Our main objective was to develop a theoretical framework for understanding these spectra and the nature of the strongly correlated electronic system which emerges once the superconducting order is suppressed. We have shown that phenomenological model (\[fgl1\]) based on a variant of the U(1) gauge field slave boson theory [@dhlee1] contains the right physics, provided that the gauge field stiffness is vanishingly small. The latter assumption is consistent with the general arguments involving local gauge symmetry[@nayak1]. In such a theory the gauge field can be explicitly integrated out, resulting in the effective action (\[feff\]) which contains one phase degree of freedom representing the phase of a Cooper pair and two amplitude degrees of freedom representing the holon and spinon condensates. Analysis of the effective theory (\[feff\]) in the presence of a magnetic field establishes existence of two types of vortices, spinon and holon, with contrasting spectral properties in their core regions. Our holon vortex is singly quantized and therefore differs in a profound way from the doubly quantized holon vortex discussed by SNL[@sachdev1; @lee3]. As indicated in Figure \[fig2\] such a singly quantized holon vortex is expected to be stable over the large portion of the phase diagram on the underdoped side. Quasiparticle spectrum in the core of a holon vortex is predicted to exhibit a “pseudogap”, similar to that found in the underdoped normal region above $T_c$. This is consistent with the data of Renner [*et al.*]{} [@renner1] who pointed out a remarkable similarity between the vortex core and the normal state spectra in BSCCO. Spinon vortex, on the other hand, should be virtually indistinguishable from the conventional $d$-wave BCS vortex and is expected to occur on the overdoped side of the phase diagram. Transition from the insulating holon vortex to the metallic spinon vortex as a function of doping is a concrete testable prediction of the present theory. Phenomenological theory based on the effective action (\[feff\]) does not permit explicit evaluation of the electronic spectral function. To this end we have considered the corresponding microscopic theory (\[leffi\]) and concluded that holon vortex will indeed exhibit a pseudogap like spectrum. Such qualitative analysis furthermore suggests a plausible mechanism for the sharp vortex core states observed in YBCO[@maggio1] and BSCCO [@pan2]. We stress that conventional mean field weak coupling theories yield neither pseudogap nor the core states. In the core of a holon vortex such states will arise as a result of spinons scattering off of the locally uncondensed holons, in a manner analogous to the quasiparticle resonant states in the vicinity of an impurity in a $d$-wave superconductor [@balatsky1; @balatsky2; @flatte1; @shnirman1; @atkinson1]. The latter conclusion is somewhat speculative and must be confirmed by explicitly solving the fermionic sector of the microscopic theory (\[leffi\]). On a broader theoretical front the importance of the vortex core spectroscopy as a window to the normal state in the $T\to 0$ limit lies in its potential to discriminate between various microscopic theories of cuprates. It is reasonable to assume that the observed pseudogap in the vortex core reflects the same physics as the pseudogap observed in the normal state. This means that the mechanism responsible for the pseudogap must be operative on extremely short lengthscales, of order of several lattice spacings. The U(1) slave boson theory considered in this work apparently satisfies this requirement. Obtaining the correct vortex core spectral functions could serve as an interesting test for other theoretical approaches describing the physics of the underdoped cuprates[@lee2; @pines1; @levin1]. It will be of interest to explore the implications of the effective theories (\[feff\]) and (\[leffi\]) in other physical situations. Of special interest are situations where the holon condensate amplitude is suppressed, locally or globally, giving rise to “normal” transport properties (vanishing superfluid density) but quasiparticle excitations that are characteristic of a superconducting state. These include the spectra in the vicinity of an impurity, twin boundary or a sample edge. In the latter case one might hope to observe a signature of the zero bias tunneling peak anomaly (normally seen for certain geometries deep in the superconducting phase in the optimally doped cuprates) even above $T_c$ in the underdoped samples. The authors are indebted to A. J. Berlinsky, J. C. Davis, Ø. Fischer, C. Kallin, D.-H. Lee, P. A. Lee, S.-H. Pan, C. Renner, J. Ye and S.C. Zhang for helpful discussions. This research was supported in part by NSF grant DMR-9415549 and by Aspen Center for Physics where part of the work was done. After submission of this manuscript we learned about complementary microscopic treatments of the spin-charge separated state in the vortex core within U(1) [@han1] and SU(2) [@wen1] slave boson theories. The former agrees qualitatively with our phenomenological theory. Ref. [@wen1] proposes a new type of vortex which takes advantage of the larger symmetry group SU(2). 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--- author: - 'Alexander C. McLain,$^{*1}$ Rajeshwari Sundaram,$^2$ Marie Thoma,$^3$ and Germaine M. Buck Louis$^4$' bibliography: - 'Reference.bib' title: 'Cautionary note on “Semiparametric modeling of grouped current duration data with preferential reporting”' --- Introduction ============ This report is designed to clarify a few points about the article “Semiparametric modeling of grouped current duration data with preferential reporting” by McLain, Sundaram, Thoma and Louis in *Statistics in Medicine* [@McLetal14 hereafter MSTL] regarding using the methods under right censoring. In simulation studies, it has been found that bias can occur when right censoring is present. Current duration data normally does not have censored values, but censoring can be induced at a value, say $\tau$, after which the data values are thought to be unreliable. As noted in MSTL, some right censored data require an assumption on the parametric form of the data beyond $\tau$. While this assumption was given in MSTL, the implications of the assumption were not sufficiently explored. Here we present simulations and evaluate the methods of MSTL under type I censoring, give some settings under which the method works well even in presence of censoring, state when the model is correctly specified and discuss the reasons of the bias. Tail Assumptions Under Right Censoring ====================================== The bias observed under censoring is a result of model misspecification under censoring. To see this, we note the following form of the current duration probability mass function for the semi-parametric model $$\begin{aligned} \label{eq.discrete.CD} g(y|{{\mbox{\boldmath $Z$}}}) = \frac{\exp\left\{-\exp({{\mbox{\boldmath${\beta}$}}}^\top {{\mbox{\boldmath $Z$}}})\sum_{j=0}^y \alpha_j\right\} }{\sum_{y=0}^\infty \exp\left\{-\exp({{\mbox{\boldmath${\beta}$}}}^\top {{\mbox{\boldmath $Z$}}})\sum_{j=0}^y \alpha_j\right\}},\end{aligned}$$ where $\alpha_j \geq 0$ for all $j$ with $\alpha_0\equiv 0$. When there is no censoring, the denominator in (\[eq.discrete.CD\]) is calculated by setting $\alpha_y = \infty$ for $y>Y_{(m)}$ where $Y_{(m)}$ is the maximum observed current duration. The infinite sum in (\[eq.discrete.CD\]) then stops at $Y_{(m)}$. However, such an approach cannot be taken under right censoring. As noted in the Estimation section of MSTL, page 3966, > Let $\tilde Y_{(1)}, \tilde Y_{(2)}, \ldots, \tilde Y_{(m)} \leq \tau$ denote the ordered and distinctly observed uncensored current durations, and $\bar{G}(y|{{\mbox{\boldmath $Z$}}}) = 1-\sum_{j=0}^y g(j|{{\mbox{\boldmath $Z$}}})$. When censoring is present, we cannot set $\alpha_y = \infty$ for $y>\tilde Y_{(m)}$ because the likelihood for those censored at $\tau$ would be $\bar{G}(\tau|{{\mbox{\boldmath $Z$}}})=0$. To allow for $\bar{G}(\tau|{{\mbox{\boldmath $Z$}}})>0$, we introduce an additional parameter $\alpha_{\tau}$ and set $\alpha_y = \alpha_{\tau}$ for all $y >\tilde Y_{(m)}$. That is, under type I censoring the model assumes that $\alpha_j$ are equal for all $j\geq \tau$. The tail assumption is needed because a semiparametric model cannot estimate the mean under type I censoring without making a parametric assumption on the distribution beyond the value of $\tau$. Recall that the relationship between $Y$ and $T$ is $\bar{F}_T(y) = g(y)\mu_T$, thus $\mu_T = E(T)$ is required to specify the model. Under type I censoring at $\tau$ we can only estimate $E(T|T\leq \tau)$ with a semiparametric model. This is similar to the fact that $\mu_T$ cannot be estimated from a Kaplan-Meier curve if the maximum value is censored. To estimate $\mu_T$ the above tail assumption is used, which implies that the discrete hazard probability of $T$ takes the parametric form $$\lambda(y|{{\mbox{\boldmath $Z$}}}) = P(T = y|T \geq y, {{\mbox{\boldmath $Z$}}}) = 1 - \exp\{-\alpha_{\tau^+} \exp({{\mbox{\boldmath${\beta}$}}}^\top {{\mbox{\boldmath $Z$}}})\} \mbox{ for all }y\geq \tau.$$ Notice that this implies that the discrete hazard probabilities are constant in $y$, thus $T$ follows a geometric distribution in the tail, i.e., $\lambda(y|{{\mbox{\boldmath $Z$}}})$ is constant in $y$ for $y\geq \tau$. When this assumption is misspecified biases can occur. For example, if $\lambda(y|{{\mbox{\boldmath $Z$}}})$ is non-constant in $y$ for $y\geq \tau$, the denominator in (\[eq.discrete.CD\]) is misspecified since it is a function of $\lambda(y|{{\mbox{\boldmath $Z$}}})$ for $y\geq \tau$. The misspecification in the denominator cannot be absorbed in any way, and results in model misspecification. This same phenomena happens with the piecewise constant model of MSTL, where $\alpha_y$ is constant beyond the largest knot. If the values of $T$ were observed the tail behavior of the $\alpha_j$’s would not impact the estimation since they would not enter the likelihood. However, since we observe the $Y$ values with probability mass function given in (\[eq.discrete.CD\]), the tail values of $\alpha_j$ impact the estimation. This explains why this problem is unique to current duration analysis. Another issue with censoring is how to truncate the upper limit of the infinite sum in the denominator in (\[eq.discrete.CD\]), which we denote by $Y^+$. In theory this value should be set at a point where negligible probability mass occurs thereafter. For cases when there is no known upper boundary to the distribution, we have observed in simulation studies that when $Y^+$ is too large it causes instability in the estimates, especially for the piecewise constant model, and having $Y^+$ too small results in biased estimates. Whether a value is “too small” or “too large” will depend on the distribution of the data. A strategy we found effective in simulation studies was to set $Y^+$ to twice the largest value before the administrative censoring was implemented. MSLT set $Y^+=1000$, which we found could be too large based on some of the new simulation settings tested. Simulation Studies ================== To test the properties of the models in MSLT, numerous simulation studies were performed. The current duration for the $i$th subject was simulated by generating the unobserved total durations as $T_{ij} {\sim} F$ for $j = 1,2,\ldots,K$, where $K = \min(k; \sum_{j=1}^k T_{ij}>M)$ and $M$ is a fixed large integer then setting $Y_{ij} = T_{iK} - M$. This setting replicates a renewal process in equilibrium with renewal distribution [see @Fel66 for details]. All of the simulation scenarios used data that was discretely distributed with a simple binary covariate $X$ with 0.5 success probability. The underlying distribution of the survival times is $P(T=t|T\geq t) = 1 - \exp\{-\alpha_t\exp(\beta_1 X)\}$ where $\beta_1=0.5$. The value of $\alpha_t$ was set to (a) $\alpha_t=\theta$, (b) $\alpha_t=\theta \alpha_0 t_k^{\alpha_0-1}$ for $t \in (t_{k-1},t_k]$ or (c) $\alpha_t=\theta\{t^{\alpha_0} - (t-1)^{\alpha_0}\}$. Here, (a) corresponds to a geometric setting, (b) corresponds to a piecewise geometric distribution, and the survival function for (c) is equal to $\bar F(t|X)=\exp\{-\alpha_0^t\exp(\beta_1 X)\}$ with we refer to as the discrete Weibull setting (note that (c) is equivalent to (a) when $\alpha_0=1$). For (a) we set $\theta=1/5$, for (b) and (c) $\alpha_0=4/5$ and $\theta$ was varied to alter the proportion of censored values. For (b) $\theta=3/16$ or $\theta=3/8$, while for (c) $\theta=1/4$ or $\theta=1/8$. The lower $\theta$ values induce more censoring. For (b) we set $\{t_1,\ldots,t_7\}=\{1, 2, 4, 5, 7, 10,18\}$ and $t_0=0$, which match the knots used for the piecewise constant model. For each setting, type I censoring at $\tau = \{3,6,12,24,36\}$ along with no censoring was applied. All simulations used $n=1000$ subjects. The above distributions were fitted with the semiparametric and piecewise constant models from MSLT where the piecewise constant model had knots at $\{1, 2, 4, 5, 7, 10,18\}$, equal to those used for simulating the data. For the geometric setting in (a) the tail assumption is correctly specified regardless of the value of $\tau$. The tail assumption is also correctly specified in (b) when $\tau \geq 18$ since $\alpha_j=\alpha_{18}$ for all $j \geq 18$. The misspecified scenarios include (b) when $\tau < 18$, and setting (c). Programs to simulate and fit all models are available from the first authors website (see the ‘Programs’ Section below). -------- ---------------------------------------------------- ---------------------------------------------------- ---------------------------------------------------- -------------------------------------------------- --------------------------------------------------- -- ---------------------------------------------------- ---------------------------------------------------- -------------------------------------------------- --------------------------------------------------- -------------------------------------------------------- $\tau$ <span style="font-variant:small-caps;">true</span> <span style="font-variant:small-caps;">mean</span> <span style="font-variant:small-caps;">bias</span> <span style="font-variant:small-caps;">sd</span> <span style="font-variant:small-caps;">ecp</span> <span style="font-variant:small-caps;">mean</span> <span style="font-variant:small-caps;">bias</span> <span style="font-variant:small-caps;">sd</span> <span style="font-variant:small-caps;">ecp</span> <span style="font-variant:small-caps;">prop cen</span> 3 0.5 0.496 -0.004 0.082 0.959 0.498 -0.002 0.087 0.945 0.365 6 0.5 0.505 0.005 0.077 0.951 0.498 -0.003 0.081 0.949 0.178 12 0.5 0.503 0.003 0.075 0.946 0.499 -0.001 0.075 0.949 0.039 24 0.5 0.504 0.004 0.074 0.947 0.509 0.009 0.073 0.954 0.003 36 0.5 0.507 0.007 0.073 0.947 0.509 0.009 0.073 0.950 0.001 None 0.5 0.507 0.007 0.073 0.958 0.509 0.009 0.073 0.950 0.000 $\tau$ <span style="font-variant:small-caps;">true</span> <span style="font-variant:small-caps;">mean</span> <span style="font-variant:small-caps;">bias</span> <span style="font-variant:small-caps;">sd</span> <span style="font-variant:small-caps;">ecp</span> <span style="font-variant:small-caps;">mean</span> <span style="font-variant:small-caps;">bias</span> <span style="font-variant:small-caps;">sd</span> <span style="font-variant:small-caps;">ecp</span> <span style="font-variant:small-caps;">prop cen</span> 3 0.5 0.591 0.091 0.077 0.809 0.580 0.080 0.079 0.864 0.453 6 0.5 0.567 0.067 0.071 0.869 0.561 0.061 0.074 0.906 0.281 12 0.5 0.538 0.038 0.062 0.926 0.527 0.027 0.065 0.944 0.134 24 0.5 0.508 0.008 0.058 0.957 0.509 0.009 0.059 0.956 0.055 36 0.5 0.509 0.009 0.057 0.960 0.514 0.014 0.059 0.950 0.025 None 0.5 0.509 0.009 0.057 0.959 0.516 0.016 0.059 0.951 0.000 $\tau$ <span style="font-variant:small-caps;">true</span> <span style="font-variant:small-caps;">mean</span> <span style="font-variant:small-caps;">bias</span> <span style="font-variant:small-caps;">sd</span> <span style="font-variant:small-caps;">ecp</span> <span style="font-variant:small-caps;">mean</span> <span style="font-variant:small-caps;">bias</span> <span style="font-variant:small-caps;">sd</span> <span style="font-variant:small-caps;">ecp</span> <span style="font-variant:small-caps;">prop cen</span> 3 0.5 0.705 0.205 0.117 0.573 0.689 0.189 0.120 0.672 0.752 6 0.5 0.654 0.154 0.098 0.659 0.636 0.136 0.102 0.764 0.631 12 0.5 0.587 0.087 0.079 0.808 0.555 0.055 0.081 0.930 0.472 24 0.5 0.506 0.006 0.060 0.947 0.484 -0.016 0.067 0.928 0.311 36 0.5 0.506 0.006 0.058 0.941 0.488 -0.012 0.064 0.924 0.209 None 0.5 0.506 0.006 0.057 0.938 0.510 0.010 0.056 0.950 0.000 $\tau$ <span style="font-variant:small-caps;">true</span> <span style="font-variant:small-caps;">mean</span> <span style="font-variant:small-caps;">bias</span> <span style="font-variant:small-caps;">sd</span> <span style="font-variant:small-caps;">ecp</span> <span style="font-variant:small-caps;">mean</span> <span style="font-variant:small-caps;">bias</span> <span style="font-variant:small-caps;">sd</span> <span style="font-variant:small-caps;">ecp</span> <span style="font-variant:small-caps;">prop cen</span> 3 0.5 0.548 0.048 0.088 0.930 0.540 0.040 0.092 0.946 0.571 6 0.5 0.523 0.023 0.080 0.946 0.520 0.020 0.084 0.938 0.395 12 0.5 0.511 0.011 0.070 0.948 0.504 0.004 0.074 0.942 0.200 24 0.5 0.503 0.003 0.066 0.937 0.500 0.000 0.068 0.934 0.058 36 0.5 0.503 0.003 0.065 0.943 0.508 0.008 0.068 0.937 0.018 None 0.5 0.502 0.002 0.065 0.942 0.510 0.010 0.067 0.938 0.000 $\tau$ <span style="font-variant:small-caps;">true</span> <span style="font-variant:small-caps;">mean</span> <span style="font-variant:small-caps;">bias</span> <span style="font-variant:small-caps;">sd</span> <span style="font-variant:small-caps;">ecp</span> <span style="font-variant:small-caps;">mean</span> <span style="font-variant:small-caps;">bias</span> <span style="font-variant:small-caps;">sd</span> <span style="font-variant:small-caps;">ecp</span> <span style="font-variant:small-caps;">prop cen</span> 3 0.5 0.570 0.070 0.123 0.932 0.555 0.055 0.125 0.943 0.783 6 0.5 0.551 0.051 0.101 0.930 0.538 0.038 0.103 0.949 0.661 12 0.5 0.536 0.036 0.084 0.936 0.517 0.017 0.091 0.943 0.481 24 0.5 0.523 0.023 0.070 0.952 0.497 -0.003 0.077 0.941 0.267 36 0.5 0.519 0.019 0.066 0.955 0.493 -0.007 0.072 0.949 0.154 None 0.5 0.518 0.018 0.064 0.954 0.515 0.015 0.066 0.949 0.000 -------- ---------------------------------------------------- ---------------------------------------------------- ---------------------------------------------------- -------------------------------------------------- --------------------------------------------------- -- ---------------------------------------------------- ---------------------------------------------------- -------------------------------------------------- --------------------------------------------------- -------------------------------------------------------- : Summary of $1,000$ simulated samples with $n=1000$ for the piecewise constant and semi-parametric models under various discrete distributional assumptions with fixed type I censoring at $\tau$. Displayed is the true coefficient (<span style="font-variant:small-caps;">true</span>), the average estimated coefficient (<span style="font-variant:small-caps;">mean</span>), the empirical bias (<span style="font-variant:small-caps;">bias</span>), the empirical standard deviation (<span style="font-variant:small-caps;">sd</span>), the empirical coverage probability (<span style="font-variant:small-caps;">ecp</span>) and the censoring proportion (<span style="font-variant:small-caps;">prop cen</span>). \[SimComp1\] In Table \[SimComp1\] we present bias, standard deviation and empirical coverage probabilities for various distributional assumptions corresponding to the distributions discussed above, which were varied by the fixed censoring value and the $\theta$ and $\alpha_0$ parameters. As expected, the effect of the varying censoring value on the geometric setting is relatively small. There does appear to be a decrease in the overall parameter estimate as the censoring value decreases, but overall the estimates are relatively unbiased. For the piecewise geometric setting the parameters are relatively unbiased for $\tau \geq 24$. This is as hypothesized since when $\tau \geq 24$ the tail assumption is correctly specified. When $\tau < 24$ the tail assumption is misspecified and we see increasing bias as $\tau$ gets closer to zero. Further, when the proportion censored increases the results remain consistent. This suggests that the value of $\tau$, not the overall censoring proportion, is what is driving the bias. Thus, when the tail assumption is correctly specified the results appear to be relatively unbiased regardless of the proportion censored. The Weibull setting shows noticeable bias in the estimates when the censoring percentage is larger than 10%. It should be noted that the piecewise constant model is misspecified under the Weibull, so some bias is expected. This misspecification appears to have a larger impact on the bias for the ‘high censoring’ distribution. For the semi-parametric setting the results have small bias when the censoring proportion is less than 30%. Discussion ========== The purpose of this paper was to investigate the properties of the MSTL model when all data are censored at a fixed value (i.e., type I censoring at $\tau$). The impact of censoring is that a parametric assumption on the tail behavior of the data must be assumed. Specifically, under censoring the model assumes that the hazard probability is constant for all $y\geq \tau$ where $\tau$ is the censoring value. The simulation studies show that when the tail behavior is correctly specified both models have relatively unbiased results regardless of the amount of censoring. This can be seen in the relatively unbiased results for the geometric setting for both models (another setting with higher censoring showed similar results), and the results for both piecewise geometric settings when $\tau >18$. Recall that the last knot of the piecewise scenario was $18$ so the true $\alpha_j$ values are constant beyond this value. Thus, when $\tau >18$ the distribution is geometric beyond the censoring value. The discrete Weibull setting is misspecified for all values of $\tau$. Further, the piecewise constant model is misspecified when there is no censoring. Our simulation results show that under misspecification the degree of bias depends on the amount of censoring. The analysis included in MSTL censored all values at $\tau=36$. The simulation studies suggest that $\tau=36$ will not have large impact on the results, however, this could be sensitive to the true distribution. The analysis was repeated without censoring and the results were largely unchanged. The previous analysis with the piecewise model found significant associations for both age ($\beta = -0.035$ with 95% CI $[-0.054, -0.015]$) and parity ($\beta = -0.492$ with 95% CI $[0.257, 0.728]$). This analysis also found significant associations for both age ($\beta = -0.037$ with 95% CI $[-0.053, -0.021]$) and parity ($\beta = 0.470$ with 95% CI $[0.272, 0.667]$). For the semi-parametric model the effect of age changed from $\beta = -0.036 \ [-0.074, 0.003]$ in the old analysis to $\beta = -0.040 \ [-0.057, -0.022]$ with no censoring. The effect of parity showed attenuation with $\beta = 0.747 \ [0.206, 1.289]$ in the old analysis and $\beta = 0.472 \ [0.272, 0.672]$ with the new analysis. The “geometric in the tail” assumption allows calculation of the necessary quantities needed to implement maximum likelihood estimation under censoring. Specifically, it assures that $\bar{G}(\tau|{{\mbox{\boldmath $Z$}}})>0$ for all ${{\mbox{\boldmath $Z$}}}$ which is required for likelihood calculation. When the “geometric in the tail” assumption is misspecified it will lead to biased results of varying degrees (as explored Section 3). When the tail assumption is misspecified, one option is to impose different tail behavior. Some examples include (i) $\alpha_t = \alpha_{\tau^+}^{\gamma(t-\tau-1)}$, (ii) $\alpha_t =\theta\{t^{\alpha_0} - (t-1)^{\alpha_0}\}$, or (iii) $\alpha_t = (t-\tau-1)^\gamma$ for $t\geq \tau$. It is important to keep in mind that sparse data are available to determine the tail behavior. We implemented different tail assumptions in simulations studies and found unstable results when two parameters were included in the calculation of the tail behavior of $\alpha_j$. So if (i) or (ii) were used one of the parameters should be fixed. In summary, the simulations in the paper show that censoring should be employed with caution when using the MSTL method. Further, if censoring is required multiple values of $\tau$ should be used to test the sensitivity of the results. Unlike the situation found in standard survival analysis, the model assumptions extend beyond the censoring value. The main reason for censoring in current duration data is due to concerns of measurement errors associated with large responses. Censoring is an attractive option when measurement error is likely, but we recommend that it be used cautiously in keeping with the specified parametric assumptions. One solution in this case is to use the piecewise model, which as shown in MSTL can correct for random digit preference in the outcome. Software {#software .unnumbered} ======== A zip file containing all the programs to implement the MSTL model can be found at through the following link [ https://sites.google.com/site/alexmclain/research](https://sites.google.com/site/alexmclain/research). See the link under the reference for MSTL “Zip file with R code to run the programs.” This file contains all of the programs to run the semiparametric and piecewise models, along with a nonparametric method. It also contains sample data, along with two programs that will generate current duration data for the discrete Weibull and piecewise constant distributions used in Section 3. The geometric distribution can be generated as a special case of the discrete Weibull distribution when $\alpha=1$. Acknowledgements {#acknowledgements .unnumbered} ================ We would like to thank Professor Niels Keiding and his group for alerting us to this issue.
--- abstract: 'Edge sampling is an important topic in network analysis. It provides a natural way to reduce network size while retaining desired features of the original network. Sampling methods that only use local information are common in practice as they do not require access to the entire network and can be parallelized easily. Despite promising empirical performance, most of these methods are derived from heuristic considerations and therefore still lack theoretical justification. To address this issue, we study in this paper a simple edge sampling scheme that uses network local information. We show that when local connectivity is sufficiently strong, the sampled network satisfies a strong spectral property. We quantify the strength of local connectivity by a global parameter and relate it to more common network statistics such as clustering coefficient and Ricci curvature. Based on this result, we also derive a condition under which a hypergraph can be sampled and reduced to a weighted network.' address: 'University of California, Davis' author: - 'Can M. Le' bibliography: - 'C:/Users/CML/Dropbox/all-ref/tex/latex/allref.bib' title: Edge sampling using network local information --- [^1] Introduction ============ Network analysis has become an important area in many research domains. It provides a natural way to model and analyze data in the presence of complex interdependence among entities. A network typically consists of a set of nodes representing the entities of interest and a set of edges between nodes encoding the relations between the nodes. For example, in a social network such as Facebook or Twitter, nodes are users and there is an edge between two users if they are friends. Studying the structure of a network provides valuable information about how entities interact and may help predict the formmation of different groups [@Goldenberg2010; @Fortunato2010]. As real-world networks are often very large, it is difficult and often impossible to store or even get access to the entire data set. It is therefore desirable to preprocess the data to reduce the network size before performing any analysis. A natural method that has been studied in network literature is to reduce the number of edges, also known as graph sparsification . For a network of $n$ nodes, instead of storing possibly $n^2$ edges, one can sample and store just $O(n\log n)$ edges and still retain some features of the original network, depending on the sampling scheme . One of the simplest sampling schemes is to independently delete edges with probability $1-\varepsilon$, where $\varepsilon\in (0,1)$ is a tuning parameter; this is also known as bond percolation . For a network $G=(V,E)$ with the set of nodes $V$ and the set of edges $E$, the adjacency matrix $A$ is a symmetric matrix with entries $A_{ij}=1$ if there is an edge between node $i$ and node $j$, and $A_{ij}=0$ otherwise. If $\varepsilon$ is sufficiently large so that $\Omega(n\log n)$ edges are retained, then by a concentration inequality [@Oliveira2010], $$\|A_\varepsilon - \varepsilon A\| = O(\sqrt{\varepsilon\|A\|\log n}).$$ Here $A_\varepsilon$ is the adjacency matrix of the sparsified network and hereafter $\|.\|$ denotes the operator norm. The advantage of this sampling scheme is that it can be done separately for every edge without additional information from other edges. However, this method only preserves cuts of large sets of nodes ; here, the cut of a set of nodes is the number of edges between that set and its complement in $V$. Another sampling scheme that approximately preserves all cuts is proposed by . Instead of using the same probability $\varepsilon$ as in bond percolation model, each edge is now sampled with a probability proportional to its effective resistance. Under this sampling scheme, the (weighted) sparsified network $H$ of $G$, also known as the spectral sparsifier, satisfies $$\label{eq: spectral property intro} (1-\e) x^\tran L_G x \le x^\tran L_H x \le (1+\e) x^\tran L_G x$$ for all vectors $x$. Here $L_G = D - A$ denotes the Laplacian of $G$, where $D$ is the diagonal matrix with degrees $d_i = \sum_{j\in V} A_{ij}$ on the diagonal; similarly, $L_H=D_H-W_H$ denotes the Laplacian of $H$, where $W_H=(w_{ij})$ is the weighted adjacency matrix of $H$ and $D_H$ is the diagonal matrix with weighted degrees $\sum_{j=1}^n w_{ij}$ on the diagonal. Despite the strong spectral property, this method requires access to the entire network for computing effective resistances of edges, which may be prohibitive in practice. Also, the computation involves a complicated linear system solver of Spielman and Teng that is not easy to implement in practice. Although some improvements of are now available, they still rely on complicated linear system solvers . It has been observed that many real networks have very strong non-Euclidian local structure . This reflects the belief that incident nodes exhibit the transitivity property: if $i$ and $j$ are connected and $j$ and $k$ are connected then it is likely that $j$ and $k$ are also connected. One way to quantify the transitivity is via clustering coefficient . For each node $i\in V$, the local clustering coefficient of $i$ is defined as the ratio between the number of triangles containing $i$ and the maximum number of triangles it can form with incident nodes $$c_i = \frac{|\{ (j,k)\in E: (i,j)\in E, (i,k)\in E \}|}{d_i(d_i-1)/2}.$$ The clustering coefficient of a network $G$ is the average of all local clustering coefficients $$c = \frac{1}{|V|} \sum_{i\in V} c_i.$$ Another measure of network transitivity that has recently attracted much attention is the Ricci curvature . In Riemanian geometry, Ricci curvature is a fundamental quantity that measures the degree to which the local geometry of a manifold deviates from Euclidian geometry. It is well known that a manifold has positive Ricci curvature if and only if the geodesic distance between any two close points is larger than the optimal transportation distance between two small balls around these points . Based on this property, the notion of Ricci curvature has been extended to metric spaces by [@Ollivier2009]. In particular, when the metric space is a network equipped with the geodesic distance, Ricci curvature is closely related to the local clustering coefficient . Network local information has been used by several edge sampling methods that aim at preserving certain features of networks such as number of connected components, diameter, homophily, node centrality measures or community structure [@Newman2010]. These methods sample each edge of a network according to certain edge scores that depend for instance on Jaccard similarity score , the number of triangles [@Hamann.et.al.2016] or the number of quadrangles containing the edge ; see also [@Hamann.et.al.2016] for methods based on other local measures. Although these methods have been empirically shown to perform well and can be parallelized easily, to our best knowledge, there is still no theoretical guarantee of their performance. It is also unclear if other features of networks (besides the targeting features considered) are preserved. To address this issue, we study in this paper a simple edge sampling scheme similar to methods that use Jaccard similarity or number of triangles . Specifically, we sample each edge $(i,j)\in E$ of a network $G=(V,E)$ with probability proportional to the number of common neighbors that $i$ and $j$ have. For simplicity, we assume that numbers of common neighbors are known for all edges. In practice, these numbers can be computed exactly in parallel fashion or approximated by hashing . We show that when the local connectivity of a network is sufficiently strong, our sampling method satisfies the spectral property ; this provides theoretical evidence supporting edge sampling methods based on local information . Intuitively, as numbers of common neighbors increase, the network transitivity becomes stronger and our sampling scheme becomes more similar to the sampling scheme using effective resistances of edges. In contrast, as numbers of common neighbors decreases, the sampling scheme become more similar to bond percolation. We quantify this intuition using a global measure and relate it to the network transitivity and Ricci curvature. Based on this result, we also derive a condition under which a hypergraph can be sampled and reduced to a weighted graph. While the sparsified network under our sampling scheme approximately preserves all cuts, it is interesting that the approximation accuracy only depends on the average of local features. This result also confirms the usefulness of network local information observed for example in the context of community detection . Edge sampling using common neighbors ==================================== We are interested in finding a sparsifier of $G=(V,E)$ that satifies the spectral notion of similarity introduced by Spielman and Teng . A sparsifier of $G$ is a weighted sparse network $H=(V,E_H,W_H)$ such that $E_H\subseteq E_G$ and holds for any vector $x\in \R^n$. Note that is equivalent to $(1-\e)L_G \preceq L_H \preceq (1+\e) L_G$ and we write $X\preceq Y$ if $Y-X$ is a semi-positive definite matrix. For each edge $(i,j)\in E$, let $T_{ij}$ be the number of common neighbors of $i$ and $j$. To form a sparsifier $H=(V,E_H,W_H)$, we sample $m$ edges of $G$ independently according to a multinomial distribution with probabilities $$\label{eq: pij} p_{ij} = \frac{\frac{2}{T_{ij}+2}}{\sum_{(i,j)\in E_G}\frac{2}{T_{ij}+2}}.$$ If an edge $(i,j)\in E_G$ of the original graph $G$ is selected, we add an edge to $H$ with the weight $(mp_{ij})^{-1}$. Weights of edges of $H$ are summed if edges are selected more than once. \[thm: main theorem\] Consider an undirected connected graph $G=(V,E)$. For each edge $(i,j)\in E$ denote by $T_{ij}$ the number of common neighbors of $i$ and $j$. Let $\e\in(0,1)$ and denote $$\label{eq: main assumption} \a = \frac{1}{n}\sum_{(i,j)\in E_G} \frac{2}{T_{ij}+2}.$$ Form a weighted graph $H$ by sampling $8\a n\log n/\e^2$ edges of $G$ as described above. Then with probability at least $1-1/n$ the following holds: $$\label{eq: spectral property} (1-\e)L_G \preceq L_H \preceq (1+\e) L_G.$$ Parameter $\a$ in measures the average strength of network local connectivity. When the local connectivity is strong, i.e. $\alpha = O(1)$, Theorem \[thm: main theorem\] shows that we can preserve network topology if we locally sample and retain $O(n\log n)$ edges. In contrast, if the local connectivity is weak (for example when $T_{ij}=O(1)$) then $p_{ij}$ are similar, therefore the sampling scheme is similar to bond percolation. Table \[tb: stats\] shows the value of $\a$ and clustering coefficient $c$ for some well-known real-world networks. Note that while these networks are relatively sparse, their values of $\a$ are quite small, which suggests that real-world networks have strong local connectivity. The proof of Theorem \[thm: main theorem\] is given in Appendix \[sec: appendix\]. $n$ Average degree Clustering coefficient $\alpha$ ----------------------------- --------- ---------------- ------------------------ ---------- Karate Club [@Zachary1977] $34$ $4.59$ $0.59$ $1.40$ Dolphins [@Lusseau2003] $62$ $5.13$ $0.30$ $1.58$ Political Blogs [@Adamic05] $1222$ $27.36$ $0.36$ $3.04$ Ego-Facebook $4039$ $43.69$ $0.62$ $1.96$ Astro Physics Collaboration $18772$ $21.10$ $0.68$ $1.94$ Enron Email $36692$ $10.02$ $0.72$ $1.56$ : Statistics of some real-world networks.[]{data-label="tb: stats"} Lower bound on parameter $\a$ ============================= The following lemma provides a lower bound for parameter $\a$ in term of clustering coefficient $c$ and degrees $d_i$. \[lem: lower bound resistance sum\] For any undirected connected graph the following holds: $$\label{eq: lower bound rs} \a \ge\frac{1}{4c+ \frac{2}{n}\sum_{i\in V}\frac{1}{d_i}}.$$ For each node $i$, denote by $N_i$ and $t_i$ the set of neighbors of $i$ and the number of triangles that contain $i$, respectively. Using Lemma \[lem: simple inequality\], we have $$\sum_{j\in N_i}\frac{2}{T_{ij}+2} \ge \frac{2|N_i|^2}{\sum_{j\in N_i} (T_{ij}+2)} = \frac{d_i^2}{t_i + d_i} \ge \frac{1}{2c_i+\frac{1}{d_i}}.$$ Summing over all nodes $i$ and applying Lemma \[lem: simple inequality\] again, we obtain $$\sum_{(i,j)\in E}\frac{4}{T_{ij}+2} \ge \sum_{i\in V} \frac{1}{2c_i+\frac{1}{d_i}} \ge \frac{|V|^2}{\sum_{i\in V}\left(2c_i+\frac{1}{d_i}\right)} = \frac{n}{2c+ \frac{1}{n}\sum_{i\in V}\frac{1}{d_i}}.$$ The proof is complete by dividing both sides of this inequality by $2n$. \[rm: order of alpha\] Lemma \[lem: lower bound resistance sum\] provides a lower bound for $\a$ in . If $c\gtrsim 1/n\sum_{i\in V} 1/d_i$ then the constant $\a$ in Theorem \[thm: main theorem\] satisfies $\a \gtrsim 1/c$. The proof of Lemma \[lem: lower bound resistance sum\] suggests that the upper bound $\a \lesssim 1/c$ also holds if $T_{ij}$ are similar for most of edges of $G$. Table \[tb: stats\] provides a numerical evidence supporting this heuristic argument. Clearly, the bound $\a \lesssim 1/c$ does not hold for all networks. Below is an example of a network for which two sides of are of different orders. Let $G = R_n\cup K_n$ be the union of a random $d$-regular graph $R_n$ ($c=d/n$) of size $n$ and a complete graph $K_n$ ($c=1$), also of size $n$ (we can connect $R_n$ and $K_n$ by an arbitrary edge to make $G$ a connected graph). If $d = o(n)$ then an easy calculation shows that the left hand-side of is of order $n^2/d$ while the right hand-side is of order $n$. Upper bound on parameter $\a$ {#sec: Ricci curvature} ============================= In this section we recall the definition of Ricci curvature for graphs and show that if the Ricci curvature of a graph is bounded from below by some constant $\kappa_0>0$ then $\a\le 1/\kappa_0$. For a graph $G$, denote by $d(i,j)$ the length of a shortest path connecting nodes $i$ and $j$. We attach to each node $i$ of $G$ a uniform probability measure $m_i$ with support being the set of neighbors of $i$: $$m_i(k) = \left\{ \begin{array}{ll} \frac{1}{d_i}, & \hbox{if } k\in N_i\\ 0, & \hbox{otherwise.} \end{array} \right.$$ The optimal transportation distance between $m_i$ and $m_j$ is defined as follows: $$W_1(m_i,m_j) = \inf_{\xi\in\Pi (m_i,m_j)}\sum_{(k,k')\in V\times V} d(k,k')\xi(k,k'),$$ where $\Pi(m_i,m_j)$ is the set of all probability measures on $V\times V$ with marginals $m_i$ and $m_j$. Intuitively, $\xi(k,k')$ represents the mass transported from $k$ to $k'$, and $W_1(m_i,m_j)$ is the optimal cost for moving a unit mass distributed evenly among neighbors of $i$ to neighbors of $j$. With this notion of distance between probability measures on $G$, the Ricci curvature of two nodes $i$ and $j$ is defined by $$\kappa(i,j) = 1 - \frac{W_1(m_i,m_j)}{d(i,j)}.$$ Figure \[fig: karate\] shows Zachary’s karate club network [@Zachary1977] together with information of Ricci curvatures of incident nodes (edges). Edges with negative curvatures are in blue, positive curvatures – in red and zero curvatures – in black; widths of edges are proportional to magnitudes of curvatures. ![Zachary’s karate club network [@Zachary1977]. Edges with negative Ricci curvatures are in blue, positive Ricci curvatures – in red and zero Ricci curvatures – in black; widths of edges are proportional to magnitudes of Ricci curvatures. []{data-label="fig: karate"}](karate "fig:"){width="40.00000%"}\ A lower bound $\kappa\ge \kappa_0>0$ on the curvature implies that $W_1(m_i,m_j) \le (1-\kappa_0) d(i,j)$ for each pair of nodes $i$ and $j$. In particular, if $i$ and $j$ are neighbors then $W_1(m_i,m_j) \le 1-\kappa_0$. Note that if $G$ is a connected graph then the inverse is also true: If $W_1(m_i,m_j) \le 1-\kappa_0$ holds for all pairs of neighbors $i$ and $j$ then $W_1(m_i,m_j) \le (1-\kappa_0) d(i,j)$ holds for all $(i,j)\in V\times V$ by a triangle inequality. For social networks, positive Ricci curvature reflects the idea that people are better off with the help of friends. Imagine that person $i$ needs to transfer money to person $j$ who is not his friend. Without knowing the best route to reach $j$, $i$ divides the money and asks his friends to help him transfer the money to $j$. Similarly, $j$ asks his friends to accept the transferred money on his behalf. Positive Ricci curvature ensures that the cost of transferring money in this way is smaller than the cost of sending money directly along a shortest path. Ricci curvature is also closely related to a simple random walk on a graph. If $\kappa\ge \kappa_0>0$ then [@Ollivier2009] shows that the spectral gap between the two largest eigenvalues of the transition probability matrix $D^{-1}A$ is bounded from below by $\kappa_0$ (see also for an improvement of the bound). Thus, Ricci curvature of a graph controls how fast a simple random walk on that graph mixes. \[lem: upper bound er\] If the Ricci curvature $\kappa$ of a graph $G$ satisfies $\kappa\ge \kappa_0$ for some constant $\kappa_0>0$ then $\alpha \le 1/\kappa_0$. Consider an edge $(i,j)$ of $G$. The masses of $m_i$ and $m_j$ are evenly distributed among neighbors of $i$ and $j$, respectively. To transport $m_i$ to $m_j$, except those masses at common neighbors of $i$ and $j$ that may not have to be moved, we need to move the rest along routes of distances at least one. Since the masses that do not require transportation is at most $c_{ij}\cdot\min\{1/d_i,1/d_j\}$ and $d(i,j)=1$, we have $$1-\kappa_0 \ge W_1(m_i,m_j) \ge 1-c_{ij}\cdot \min\{1/d_i,1/d_j\}.$$ This implies $c_{ij}\ge \kappa_0/\min\{1/d_i,1/d_j\}$. Summing over all edges $(i,j)$ of $G$, we obtain $$\sum_{(i,j)\in E} \frac{2}{2+c_{ij}} \le \frac{2}{\kappa_0}\sum_{(i,j)\in E} \min\{1/d_i,1/d_j\} = \frac{1}{\kappa_0}\sum_{i\in V}\sum_{j\in N_i} \min\{1/d_i,1/d_j\} \le \frac{n}{\kappa_0}.$$ For the last inequality we use the fact that $\sum_{j\in N_i} \min\{1/d_i,1/d_j\} \le 1$. The proof is complete. Lemma \[lem: upper bound er\] requires that Ricci curvatures of all edges of graph $G$ are bounded from below by $\kappa_0$. This assumption can be relaxed by a weaker assumption that the number of edges with Ricci curvatures less than $\kappa_0$ is of order $O(n)$. With this assumption, the upper bound of $\alpha$ becomes $\alpha\le 1/\kappa_0 + O(1)$. Sparsifying Hypergraphs ======================= Strong local conectivity of a network is often caused by the fact that each node belongs to several tightly connected small groups . To simplify the analysis, we assume in this section that within each small group, all nodes are connected. Under this assumption, a network can be modeled by a hypergraph $\mathcal{G}=(V,\mathcal{E})$ which consists of a set of nodes $V$ and a set of hyperedges $\mathcal{E}$ where each hyperedge is a subset of $V$. We derive in this section a condition under which a hypergraph can be sampled and reduced to a weighted network. This provides an example when our sampling scheme works well and may be useful in practice as a computational acceleration technique. The Laplacian previously defined for networks can be naturally extended to hypergraphs through clique expansion . For a hypergraph $\mathcal{G}=(V,\mathcal{E})$, the evaluation of the Laplacian $L_\mathcal{G}$ at a vector $x$ is defined by $$L_{\mathcal{G}}(x) = \sum_{e\in\mathcal{E}}\sum_{i,j\in e} (x_i-x_j)^2.$$ If we view $x$ as a function on $V$ then $L_{\mathcal{G}}(x)$ measures the smoothness of $x$ and it occurs naturally in many problems of estimating smooth functions . Let $G=(V,E,W)$ be a weighted graph such that $(i,j)\in E$ if and only if both $i$ and $j$ belong to at least one hyperedge of $\mathcal{G}$; the weight $w_{ij}$ of edge $(i,j)$ is the number of hyperedges that both $i$ and $j$ belong to. It is easy to see that $L_{\mathcal{G}}(x) = x^\tran L_{G}x$ for every $x$, where $L_G$ is the Laplacian of the weighted graph $G$ defined by $$x^\tran L_G x = \sum_{(i,j)\in E} w_{ij}(x_i-x_j)^2.$$ Thus, if we are only interested in smoothness induced by $\mathcal{G}$ of functions on $V$ then we can replace $\mathcal{G}$ with $G$. We call $G$ the weighted graph induced by $\mathcal{G}$. To form a sparsifier $H = (V,E_H,W_H)$ of $G$, we sample with replacement $m$ edges of $G$ with probability $$\label{eq: pij weighted graph} \mathcal{P}_{ij} = \frac{\mathcal{C}_{ij}^{-1}}{\sum_{(i,j)\in E_G} \mathcal{C}_{ij}^{-1}}, \quad \text{where } \ \mathcal{C}_{ij} = \sum_{e\in\mathcal{E}: \ \{i,j\}\subseteq e} |e|.$$ If an edge $(i,j)\in E_G$ is selected, we add $(i,j)$ to $E_H$ with weight $(m\mathcal{P}_{ij})^{-1}$. Again, weights are summed up if edges are sampled more than once. Next lemma shows that a condition similar to holds if each node of $\mathcal{G}$ belongs to a finite number of hyperedges. Let $\mathcal{G}=(V,\mathcal{E})$ be a hypergraph and $G=(V,E,W)$ be the weighted graph induced by $\mathcal{G}$. If each node of $\mathcal{G}$ belongs to at most $d$ hyperedges of $\mathcal{G}$ then $$\label{eq: alpha hypergraph} \sum_{(i,j)\in E_G} \mathcal{C}_{ij}^{-1} \le dn/2.$$ By definition of $E_G$ we have $$\sum_{(i,j)\in E_G} \mathcal{C}_{ij}^{-1} \le \sum_{e\in E_\mathcal{G}}\sum_{\{i,j\}\subseteq e} \mathcal{C}_{ij}^{-1}.$$ Since $\mathcal{C}_{ij} \ge |e|$ for each $e\in E_\mathcal{G}$ that contains $\{i,j\}$ and there are $|e|(|e|-1)/2$ pairs $\{i,j\}\in e$, it follows from above inequality that $$\sum_{(i,j)\in E_G} \mathcal{C}_{ij}^{-1} \le \sum_{e\in E_\mathcal{G}}\frac{|e|-1}{2} \le \frac{1}{2} \sum_{e\in E_\mathcal{G}} |e| \le \frac{dn}{2}.$$ For the last inequality we use the assumption that each node belongs to at most $d$ hyperedges. Without further assumptions on $\mathcal{G}$, the dependence of the right hand-side of on $d$ is optimal. To see this, consider the following example. Let $k>0$ be an integer, $n = k^2$ and $V_1,...,V_k$ be a partition of $[n]:=\{1,...,n\}$ such that each $V_i$ contains exactly $k$ elements $V_{i1},...,V_{ik}$. For each $i\in[k]$, let $\sigma_i$ be a permutation of $[k]$ given by $\sigma_i(j) = i + j$ (mode $k$). Define the set of hyperedges of $\mathcal{G}$ as a collection of subsets of the form $$\left\{V_{1j},V_{2\sigma_i(j)},...,V_{k\sigma_i^{k-1}(j)}\right\}, \quad 1\le i,j\le k.$$ It is easy to see that every node of $\mathcal{G}$ is contained in exactly $d = k$ hyperedges and every pair of nodes of $\mathcal{G}$ is contained in at most one hyperedge. A simple calculation shows that $\sum_{(i,j)\in E_G} 1/\mathcal{C}_{ij} = (d-1)n/2$. \[lem: alpha bound hypergraph\] Let $\mathcal{G}=(V,\mathcal{E})$ be a hypergraph and $G=(V,E,W)$ be the weighted graph induced by $\mathcal{G}$. Let $\varepsilon\in (0,1)$ and assume that each node of $\mathcal{G}$ belongs to at most $d$ hyperedges of $\mathcal{G}$. Form a weighted graph $H$ by sampling $4dn\log n/\varepsilon^2$ edges of $G$ with probability $\mathcal{P}_{ij}$. Then with probability at least $1-1/n$ the following holds: $$\label{eq: hypergraph spectral property} (1-\e)L_G \preceq L_H \preceq (1+\e) L_G.$$ The proof of this lemma is similar to the proof of Theorem \[thm: main theorem\] with one exception that we use the bound in Lemma \[lem: alpha bound hypergraph\] instead of condition . Discussion ========== We study in this paper an edge sampling algorithm that uses only the numbers of common neighbors of incident nodes. These simple statistics provide an easy way to measure the strength of network local connectivity through parameter $\alpha$. However, in practice we often have access to not only the numbers of common neighbors but also local networks around edges. In that case, we should use the information of these local networks, provided that it is available or easily computed. Measuring the strength of local connectivity through local networks is more challenging and we leave it for future work. Proof of Theorem \[thm: main theorem\] {#sec: appendix} ====================================== To prove Theorem \[thm: main theorem\], we will use the following result about concentration of the sum of random matrices [@VershyninNote2009]. \[thm: concentration of sum of matrices\] Let $Y_k$ be independent $n\times n$ random matrices such that $Y_k \succeq 0$ and $\|Y_k\|\le M$ for all $1\le k \le m$. Let $S_m=\sum_{k=1}^m Y_k$ and $E = \sum_{k=1}^m \| \operatorname{\mathbb{E}}Y_k\|$. Then for every $\varepsilon\in (0,1)$ we have $${\mathbb{P}_{} \left\{ \|S_m-\operatorname{\mathbb{E}}S_m\| > \varepsilon E \rule{0mm}{3mm}\right\}} \le n\cdot \exp\left(\frac{-\varepsilon^2 E}{4M}\right).$$ Let $X$ be a random matrix that takes one of the $|E_G|$ matrix values: $$X = \frac{1}{p_{ij}}(e_i-e_j)(e_i-e_j)^\tran \quad \text{with probability } \ p_{ij},$$ where $(i,j)\in E_G$ and $p_{ij}$ is defined by . Then $$\label{eq: E X} \operatorname{\mathbb{E}}X = \sum_{(i,j)\in E_G} p_{ij} \times \frac{1}{p_{ij}}(e_i-e_j)(e_i-e_j)^\tran = L_G.$$ Let $X_k$ be $m$ independent copies of $X$. By the sampling scheme we have $$L_H = \frac{1}{m}\sum_{k=1}^m X_k, \quad \operatorname{\mathbb{E}}L_H = L_G.$$ Denote by $L_G^{-1}$ the Moore-Penrose pseudoinverse of $L_G$ and by $L_G^{-1/2}$ the squared root of $L_G^{-1}$. Note that the kernel of the map $L_G$ is an one-dimensional vector space spanned by the all-one vector $\onevector$ and it is contained in the kernel of $L_H$. Therefore inequality is equivalent to $$\label{eq: spectral property 1} (1-\e)I_\onevector \preceq \frac{1}{m}\sum_{k=1}^m L_G^{-1/2} X_k L_G^{-1/2} \preceq (1+\e) I_\onevector,$$ where $I_\onevector = I-(1/n)\onevector\onevector^\tran$ is the identity map on the $(n-1)$-dimensional subspace orthogonal to the all-one vector $\onevector$. To prove , we apply Theorem \[thm: concentration of sum of matrices\] to $Y_k := L_G^{-1/2} X_k L_G^{-1/2}$. Since $X_k\succeq 0$ and $\operatorname{\mathbb{E}}X_k= L_G$ by , it follows that $Y_k \succeq 0$ and $\|\operatorname{\mathbb{E}}Y_k\|= \|I_\onevector\| = 1$. To bound $\|Y_k\|$, note that $Y_k$ takes one of the following matrix values $$\frac{1}{p_{ij}} \left(L_G^{-1/2} (e_i - e_j)\right) \left(L_G^{-1/2} (e_i - e_j)\right)^\tran, \quad (i,j)\in E_G.$$ By and assumption we have $1/p_{ij}\le n\a(c_{ij}+2)/2$. Therefore $$\label{eq: bound Yk 1} \|Y_k\| \le \max_{(i,j)\in E_G} \frac{n\a(c_{ij}+2)}{2} \cdot(e_i-e_j)^\tran L_G^{-1}(e_i-e_j).$$ For each $(i,j)\in E_G$, let $N_{ij}$ be the set of common neighbors of $i$ and $j$. Denote by $G_{ij}=(V_{ij},E_{ij})$ the subgraph of $G$ such that $$V_{ij}=\{i,j\}\cup N_{ij}, \qquad E_{ij} = \{(i,j),(i,k),(j,k): k\in N_{ij}\}.$$ Thus, $G_{ij}$ contains $c_{ij}+2$ vertices and $2c_{ij}+1$ edges. Since $G_{ij}$ is a subgraph of $G$, it follows that $L_{G_{ij}} \preceq L_G$. On the $(c_{ij}+1)$-dimensional subspace spanned by $\{e_k:k\in V_{ij}\}$ and orthogonal to all-one vector $\onevector$, both $L_G$ and $L_{G_{ij}}$ are nonsingular, therefore $L_{G_{ij}}^{-1} \succeq L_G^{-1}$ (see e.g. ). In particular, $$\label{eq: bound Yk midstep} (e_i-e_j)^\tran L_G^{-1}(e_i-e_j) \le (e_i-e_j)^\tran L_{G_{ij}}^{-1}(e_i-e_j).$$ We claim that the right hand side of is equal to $2/(c_{ij}+2)$. Let $x = L^{-1}_{G_{ij}}(e_i-e_j)$. Then $L_{G_{ij}} x = e_i-e_j$ and by comparing the $i$-th and $j$-th components of $L_{G_{ij}} x$ and $e_i-e_j$, we have $$(c_{ij}+1)x_i - x_j - \sum_{k\in N_{ij}} x_k = 1, \qquad x_i- (c_{ij}+1)x_j+\sum_{k\in N_{ij}} x_k = 1.$$ Adding these equalities, we obtain $x_i-x_j= 2/(c_{ij}+2)$. Since the right hand side of is $(e_i-e_j)^\tran x=x_i-x_j$, the claim is proved. Together with and this implies $\|Y_k\| \le n\a$. Therefore by Theorem \[thm: concentration of sum of matrices\] we have $${\mathbb{P}_{} \left\{ \left\|\frac{1}{m}\sum_{k=1}^m Y_k-I_{\onevector}\right\| >\varepsilon \rule{0mm}{3mm}\right\}} \le n\cdot \exp\left(\frac{-\varepsilon^2 m}{4\a n}\right).$$ Inequality then follows by choosing $m=8\a n \log n /\varepsilon^2$. \[lem: simple inequality\] For positive numbers $x_1,x_2,...,x_k$ the following inequality holds $$\label{eq: simple inequality} \left(x_1+x_2+\cdots+x_k\right)\left(\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_k}\right)\ge k^2.$$ The two sides are equal if and only if $x_1=x_2=\cdots = x_n$. Using the inequality of arithmetic and geometric means, we have $$x_1+x_2+\cdots+x_k\ge k(x_1x_2\cdots x_k)^{1/k}, \quad \frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_k} \ge k(x_1x_2\cdots x_k)^{-1/k}.$$ Lemma \[lem: simple inequality\] follows directly from these inequalities. [^1]:
--- abstract: 'Confocal laser endomicroscopy (CLE) is a novel imaging modality that provides *in vivo* histological cross-sections of examined tissue. Recently, attempts have been made to develop miniaturized *in vivo* imaging devices, specifically confocal laser microscopes, for both clinical and research applications. However, current implementations of miniature CLE components, such as confocal lenses, compromise image resolution, signal-to-noise ratio, or both, which negatively impacts the utility of *in vivo* imaging. In this work, we demonstrate that software-based techniques can be used to recover lost information due to endomicroscopy hardware miniaturization and reconstruct images of higher resolution. Particularly, a densely connected convolutional neural network is used to reconstruct a high-resolution CLE image from a low-resolution input. In the proposed network, each layer is directly connected to all subsequent layers, which results in an effective combination of low-level and high-level features and efficient information flow throughout the network. To train and evaluate our network, we use a dataset of 181 high-resolution CLE images. Both quantitative and qualitative results indicate superiority of the proposed network compared to traditional interpolation techniques and competing learning-based methods. This work demonstrates that software-based super-resolution is a viable approach to compensate for loss of resolution due to endoscopic hardware miniaturization.' author: - 'Saeed Izadi, Kathleen P. Moriarty, and Ghassan Hamarneh' bibliography: - 'refs.bib' title: 'Can Deep Learning Relax Endomicroscopy Hardware Miniaturization Requirements?' --- =1 Introduction ============ Last year, colorectal cancer caused an estimated 50,260 deaths in the United States alone and another 140,030 people are expected to be diagnosed with this disease during 2018 [@siegel2017colorectal; @siegel2018cancer]. Accordingly, it is the third most commonly diagnosed cancer among both men and women [@siegel2018cancer]. Early diagnosis and treatment of colorectal cancer is crucial for reducing the mortality rate. Gastroenterologists screen and monitor the status of their patients’ digestive systems through specialized endoscopy procedures such as colonoscopy and sigmoidoscopy. During colonoscopy, a flexible video endoscope is guided through the large intestine, capturing images used to differentiate between neoplastic (intraepithelial neoplasia, cancer) and non-neoplastic (e.g., hyperplastic polyps) tissues. Since the introduction of endoscopy to gastroenterology, many significant advances have been made toward improving the diagnostic and therapeutic yield of endoscopy. Confocal laser endomicroscopy (CLE), first introduced to the endoscopy field in 2004 [@kiesslich2004confocal], is an emerging imaging modality that allows histological analysis at cellular and subcellular resolutions during ongoing endoscopy. An endomicroscope is integrated into the distal tip of a conventional video colonoscope, providing an *in vivo* microscopic visualization of tissue architecture and cellular morphology in real-time. Endomicroscopes offer a magnification and resolution comparable to that obtained from *ex vivo* histology imaging techniques, without the need for biopsy (i.e., tissue removal, sectioning and staining). Despite the promise of confocal laser endomicroscopy, both clinicians and researchers prefer compact instruments with relatively large penetration depth to recognize tissue structures such as the mucosa, the submucosa, and the muscular layers. Compact instruments can also directly benefit the patients, as smaller devices improve early diagnostic procedures by offering greater flexibility during hand-held use, for a quicker and less invasive endoscopy [@helmchen_2002]. In this regard, further attempts have been made to design miniaturized confocal scanning lasers capable of capturing images from the tissue subsurface with micron resolution *in vivo*, once installed on top of a flexible fiber bundle. However, miniaturization implies using smaller optical elements, which introduces pixelation artifacts in images. Therefore, there exists a trade-off between miniaturizing the CLE components and the resultant image resolution. Image super-resolution, transforms an image from low-resolution (LR) to high-resolution (HR) by recovering the high-frequency cues and reconstructing textural information. In the past decade, various learning-based approaches have been proposed to learn the desired LR-to-HR mapping, including dictionary learning [@5466111; @Yang2012], linear regression [@6751179; @timofte2014a+], and random decision forests [@7299003]. In recent years, deep learning models have been applied to various image interpretation tasks. Among such efforts, convolutional neural networks (CNN) have been utilized to resolve the ill-posed inverse problem of super-resolution. Dong et al. [@Dong2016srcnn] demonstrated that a fully convolutional network trained end-to-end can be used to perform the LR-to-HR nonlinear mapping. The same authors extended their previous work by introducing deconvolutional layers at the end of the architecture, such that the mapping between LR and HR images is learned directly without image interpolation [@dong2016accelerating]. They also slightly increased the depth of the network and adopted smaller kernels for better performance. Instead of HR images, Kim et al. [@Kim_2016_VDSR] suggested to train deeper neural networks through predicting the residual images, which when summed with an interpolated image gives the desired output. Increasing the network depth by adding weighted layers introduces more parameters, which can lead to overfitting. Kim et al. [@Kim2016_rCNN] tackled overfitting by using a deeply-recursive convolutional network. In their work, the same convolutional layers are used recursively without the need for extra parameters. To simplify the training of the network, they suggested recursive supervision and skip connections to avoid the problem of vanishing/exploding gradients. Given the constraints imposed by CLE hardware miniaturization, we propose to leverage state-of-the-art deep learning super-resolution methods to mitigate the unwanted trade-off between miniaturization and image resolution. In other words, we show that the pixelation artifact, which is a consequence of hardware miniaturization, can be significantly remedied through an efficient and practical use of software-based techniques, particularly machine learning methods. To this end, we employ a densely connected CNN in which extensive usage of skip connections is exploited [@Tong2017_DLSR]. Dense connections help information flow in backpropagation algorithms and alleviate the vanishing gradient problem. Furthermore, the low-level features from early layers are efficiently combined with those of later layers. In addition, we use sub-pixel convolutional layers [@Shi2016_subpixCNN] to render the upsampling operation learnable and expedite the reconstruction process. Method ====== Our main goal in this work is to super-resolve an LR image by passing it through a set of nonlinear transformations to recover high-frequency details and reconstruct the HR image, effectively increasing the number of pixels from $N_{LR}\times N_{LR} \text{ to } N_{HR}\times N_{HR}$, where $\frac{N_{HR}}{N_{LR}}$ is the scale factor. The proposed architecture consists of dense blocks and upsampling layers which are efficiently designed to combine the features from earlier layers with those of later layers and improve information flow throughout the model. Fig. \[fig:overall\_arch\] depicts the architecture of the employed model. **Low-Level Features**. A series of low-level features are extracted from small regions of the LR input image using two successive convolutional layers with kernel size $3\times3$ and ReLU non-linearity. The number of feature channels for the first and second layer is 64 and 128, respectively. The learned low-level features are used to efficiently represent the intrinsic textural differences between LR and HR images. ![image](Fig1.png){width="\textwidth"} **High-Level Features**. The resultant low-level feature maps are used as the input to a fully convolutional DenseNet architecture to provide high-level features. DenseNet, which was first introduced by Huang et al. [@Huang2017_denseCNN], consists of a set of dense blocks in which any layer is connected to every other layer in a feed-forward fashion. Alternatively stated, the $i^{th}$ layer in a dense block receives the concatenation of outputs by all preceding layers as the input: $$L_i = relu(\psi_{\theta^i}(L_1 {\mathbin{+\mkern-10mu+}}L_2 {\mathbin{+\mkern-10mu+}}... {\mathbin{+\mkern-10mu+}}L_{i-1}))$$ where $\psi_{\theta^i}$ denotes the transformation of the $i^{th}$ layer parameterized by $\theta^i$ and ${\mathbin{+\mkern-10mu+}}$ denotes the concatenation operation. Dense skip connections help alleviate the vanishing-gradient problem and improve information flow throughout the network. Counter-intuitively, the number of parameters is also reduced since the previously-generated feature maps are re-used in the subsequent layers, thus minimizing the need for learning redundant features. As depicted in Fig. \[fig:overall\_arch\], a single dense block consists of $m$ convolutional layers, each producing $k$ feature maps, referred to as the *growth rate*. Accordingly, the final output of each dense block has $m \times k$ features maps. The growth rate regulates how much new information each layer contributes to achieving the final performance. In this study, we set $m$ and $k$ to be 8 and 16, respectively. Thus, each dense block receives and produces 128 feature maps as input and output. We stack 12 dense blocks in a feed-forward fashion to construct the DenseNet part of our proposed architecture. **Upsampling Layers**. In some SR methods [@7410407; @Dong2016srcnn; @Kim_2016_VDSR], the LR image is first resized to match the HR spatial dimensions using bicubic interpolation. Thereafter, several convolution layers are employed to enhance the interpolated input in the HR space. In addition to having a considerable increase in memory usage and computational complexity, these interpolation methods are categorized as non-learnable upsampling techniques, which do not leverage data statistics to bring new information for more accurate reconstruction. As an alternative, deconvolutional layers, which are learnable operations, are utilized to enlarge the spatial dimensions of the LR image. However, the most prominent problem associated with deconvolutional layers is the presence of checkerboard artifacts in the output image. To overcome this, extra post-processing steps or smoothness constraints are required. In this work, we use sub-pixel convolutional layers [@Shi2016_subpixCNN], to upsample the spatial size of the feature maps within the network. Suppose that we desire to spatially upsample $c$ feature maps of size $h \times w \times c$ to size $H \times W \times c$, by a scale factor $r=H/h=W/w$. The LR feature maps would be fed into a convolution layer that increases the number of channels by a factor of $r^2$, resulting in a volume of size $h \times w \times (c \times r^2)$. Next, the resultant volume is simply re-arranged to be of shape $(h\times r) \times (w \times r) \times c$, which is equal to $H \times W \times c$. Here, we use successive $\times 2$ upsampling layers to gradually increase the spatial dimensionality. Each upsampling block contains a single convolutional layer with $3 \times 3$ kernel size and ReLU non-linearity. **Integration Layer**. Once the features maps match the spatial dimension in the HR space, an integration layer is used to consolidate the features across the channels into a single channel. The integration layer is a convolutional layer with $3 \times 3$ kernel size and a single output channel. Finally, a *sigmoid* activation function is employed to produce the super-resolved image. Experiments =========== **Data**. We evaluate our study on the dataset provided by Leong et al. [@LEONG20081870]. The dataset contains 181 gray scale confocal images of size $1024 \times 1024$ from 31 patients and 50 different anatomical sites. Each patient has undergone a confocal gastroscopy (Pentax EC-3870FK, Pentax, Tokyo, Japan) under conscious sedation. CLE images and forceps biopsies of the same sites were taken sequentially at standardized locations (i.e., sites of the small intestine). Each forceps biopsy was then assessed by 2 experienced blinded histopathologists. Despite our application of interest being colorectal cancer, we used the publicly available CLE celiac dataset as a proof-of-concept. Colorectal cancer images are assessed primarily in the large intestine as opposed to the small intestine used in celiac assessment, however the imaging procedure (CLE) remains the same. This dataset was made publicly available as part of an International Symposium on Biomedical Imaging (ISBI) challenge and we used the provided training and test sets, consisting of 108 and 73 images, respectively. **Implementation Details**. We partition the HR images into $64\times64$ non-overlapping patches. Then, the HR patches are downsampled by bicubic interpolation to construct $<$LR, HR$>$ pairs for training the model. The network is optimized with Adam [@kingma2014adam] optimizer with default parameters, i.e. $\beta_1=0.9$, $\beta_2=0.999$ and $\epsilon=10^{-4}$. We set the mini-batch size to 128. The learning rate is first initialized with 0.001 and is multiplied by $\gamma=10$ at epochs 50 and 200. The network is trained for 300 epochs using L1 loss. For data augmentation, we use random horizontal and vertical flips. The proposed method is implemented in PyTorch and is trained using two Nvidia Titan X (Pascal) GPUs. It takes 2 days to train the networks for each upsampling factor. All hyper-parameters (optimizer, learning rate, batch size, and distance metric) are found via grid search on 20 images from the training set. ![image](Fig2_new.png){width="\textwidth"} **Qualitative Results**. In Fig. \[fig:qualitative\], we visually compare our proposed super-resolution method to three traditional interpolation techniques and two learning-based approaches with scale factors of $\times 2$, $\times 4$ and $\times 8$. Evidently, DenseNet produces output images of higher quality by reconstructing high-frequency cues and removing visual artifacts, e.g. over-smoothness and pixelation. Specifically for a $\times 8$ scale factor, the densely connected network can accurately recover high-level textural patterns such as grids and granular patterns. Moreover, a more rigorous examination of smaller regions for $\times4$ scale factor clearly reveals the superiority of DenseNet model in producing sharper edges and improved contrast for lines and shapes. From a clinician’s point of view, the reconstruction power of the method offers a clear advantage over others. In Fig. \[fig:reconstruct\] we illustrate the trade-off between the amount of lost information after downsampling and the quality of the reconstructed image. As can be seen, a large portion of pixels is discarded in downsampling, restricting the networks to a small fraction of the original image pixels for reconstruction. However, deep learning approaches are clearly capable of generating a sharp image from only 1.6% of pixels (for a scale factor of $\times 8$) with very small L1 distance values which indicates a minimal loss of information. **Quantitative Results**. Table. \[tab:quant\] compares our proposed method with three interpolation methods and two learning-based techniques in terms of PSNR (Peak Signal to Noise Ratio) and SSIM (Structural Similarity). PSNR is a well-known metric for image quality assessment which is inversely proportional to Mean Square Error. SSIM also measures the the similarity between two images and is correlated with quality perception in human visual system. In terms of PSNR, DenseNet yields $2.08$, $1.93$ and $1.14$ average improvements over Nearest, Bilinear and Bicubic interpolation methods across all scale factors, respectively. For learning-based approaches, DenseNet outperforms A+ [@timofte2014a+] and SRCNN [@Dong2016srcnn] in terms of average SSIM by $0.020$ and $0.019$ over all scale factors, respectively. ![image](Fig3_new.png){width="\textwidth"} [c @ccccccccccccc]{} & & & & & &\ & [PSNR]{} & [SSIM]{} & [PSNR]{} & [SSIM]{} & [PSNR]{} & [SSIM]{} & [PSNR]{} & [SSIM]{} & [PSNR]{} & [SSIM]{} & [PSNR]{} & [SSIM]{}\ **$\times$2** & 35.32 & 0.881 & 34.21 & 0.849 & 35.80 & 0.908 & 36.21 & 0.925 & 35.54 & 0.930 & **38.57** & **0.950**\ **$\times$4** & 31.64 & 0.658 & 32.38 & 0.707 & 32.87 & 0.755 & 33.00 & 0.781 & 33.01 & 0.778 & **33.32** & **0.801**\ **$\times$8** & 30.59 & 0.528 & 31.40 & 0.586 & 31.70 & 0.615& 31.74 & 0.636 & 31.80 & 0.636 & **31.90** & **0.651**\ Conclusion ========== Developing smaller hardware for medical imaging devices has several advantages such as increased portability and reduced patient discomfort. However, hardware miniaturization comes at the expense of reduced image quality. In this preliminary study, we obtained encouraging results to support that software-based methods can be used to counteract the loss of image quality due to miniaturized device components. Compared to common interpolation methods, our qualitative and quantitative results indicate that a densely connected convolutional neural network can significantly yield higher PSNR and SSIM scores, resulting in super-resolved images of higher quality. In future work, we will focus on how super-resolved images, compared to low-resolution images, can be advantageous to clinical and research applications. For example, super-resolution images may be used as input to automated machine-learning based disease classification. **Acknowledgments**. Thanks to the NVIDIA Corporation for the donation of Titan X GPUs used in this research and to the Collaborative Health Research Projects (CHRP) for funding.
--- abstract: 'The purpose of this article is to extend the earliest results of A.A. Brudno, connecting the topological entropy of a subshift $\bfX$ over $\N$ to the Kolmogorov complexity of words in $\bfX$, to subshifts over computable groups that posses computable F[ø]{}lner monotilings, which we introduce in this work as a computable version of the notion of a F[ø]{}lner monotiling originally due to B. Weiss. For every $d \in \N$, the groups $\Z^d$ and ${\mathsf{UT}_{d}(\Z)}$ posses particularly nice computable F[ø]{}lner monotilings for which we can provide the required computing algorithms ‘explicitly’. Following the work of Weiss further, we show that the class of computable groups admitting computable F[ø]{}lner monotilings is closed under group extensions.' address: 'Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands' author: - Nikita Moriakov bibliography: - 'zdbrudnobib.bib' date: - - title: | Computable F[ø]{}lner monotilings\ and a theorem of Brudno I. --- [^1] Introduction ============ It was proved by A.A. Brudno in [@brudno1974] that the topological entropy of a subshift $\bfX$ over $\N$ equals the limit of the average maximum Kolmogorov complexities of words on $[1,2,\dots,n] \subset \N$, i.e. $$h(\bfX) = \lim\limits_{n \to \infty} \frac 1 n \max\limits_{\omega \in \prX} \frac{\uK(\omega|_{[1,\dots,n]})}{n},$$ where $h(\bfX)$ is the topological entropy of $\bfX$ and $\uK(\omega|_{[1,\dots,n]})$ is the *Kolmogorov complexity* of the word $\omega|_{[1,\dots,n]}$ of length $n$. Roughly speaking, $\uK(\omega|_{[1,\dots,n]})$ is the length of the shortest description of $\omega|_{[1,\dots,n]}$ for a fixed ‘optimal decompressor’ that takes finite binary words as an input and produces finite words as an output. So, for instance, the Kolmogorov complexity of long periodic words will be small compared to their length, while ‘random’ words would be most complex. The notion of a topological entropy has been since then extended to actions of discrete amenable groups, so one can wonder if the results in [@brudno1974] can be extended as well. In this article we give such an extension for a large class of amenable groups, including both classical examples, such as the groups $\Z^d$ for $d \in \N$, and interesting new ones, such as the nilpotent groups ${\mathsf{UT}_{d}(\Z)}$ of upper-triangular matrices with integer entries, as well. The paper is structured as follows. We provide a basic background on amenable groups and topological entropy for amenable group actions in Section \[ss.amengr\]. Section \[ss.regfoln\] is devoted to the notion of a F[ø]{}lner monotiling, which was suggested by B. Weiss in [@weiss2001]. The classical concepts of a computable function, a computable set and Kolmogorov complexity are explained in Section \[ss.compcompl\]. In Section \[ss.compspaces\] we introduce new notions: computable spaces, morphisms between computable spaces, word presheaves and Kolmogorov complexity of sections of word presheaves. Asymptotic Kolmogorov complexity of word presheaves is defined at the end of this section, too. Using this language we define computable groups in Section \[ss.compgrp\]. ‘Computable version’ of the notion of a F[ø]{}lner monotiling is developed in Section \[ss.compfoln\], at the end of which we prove that the class of computable groups admitting computable F[ø]{}lner monotilings is closed under ‘computable’ group extensions. The main theorem of this article is Theorem \[thm.brudno\] in Section \[s.brudnocfm\], where we prove that the topological entropy of a subshift over a computable group admitting computable normal F[ø]{}lner monotiling equals the asymptotic Kolmogorov complexity of the associated word presheaf. The requirement of normality is nonrestrictive: we show in Lemma \[l.normalization\] that each computable F[ø]{}lner monotiling gives rise to a computable normal F[ø]{}lner monotiling. The groups $\Z^d$ and ${\mathsf{UT}_{d}(\Z)}$ do admit computable F[ø]{}lner monotilings, thus Theorem \[thm.brudno\] gives a nontrivial abstract generalization of the results of A.A.Brudno in [@brudno1974] and, partially, the results of S. G. Simpson in [@simpson2015] for $\Z$ and $\Z^d$ cases respectively. I would like to thank my advisor Markus Haase for reading the draft and providing corrections. I would also like to thank Stephen G. Simpson for explaining certain details in [@simpson2015]. Preliminaries ============= Amenable groups and entropy theory {#ss.amengr} ---------------------------------- In this section we will remind the reader of the classical notion of amenability, and state some results from entropy theory of amenable group actions. We stress that all the groups that we consider in this paper are discrete and countably infinite. Let $\Gamma$ be a group with the counting measure ${ \left| \cdot \right|}$. A sequence of finite sets $(F_n)_{n \geq 1}$ is called a **weak F[ø]{}lner sequence** if for every finite set $K \subseteq \Gamma$ one has $$\frac{{ \left| F_n \sdif K F_n \right|}}{{ \left| F_n \right|}} \to 0 ;$$ a **strong F[ø]{}lner sequence** if for every finite set $K \subseteq \Gamma$ one has $$\frac{{ \left| {\partial_{K}(F_n)} \right|}}{{ \left| F_n \right|}} \to 0 ,$$ where $${\partial_{K}(F)}:=K^{-1}F \cap K^{-1} \comp{F}$$ is the **$K$-boundary** of $F$; One can show that a sequence of sets $(F_n)_{n \geq 1}$ is a weak F[ø]{}lner sequence if and only if it is a strong F[ø]{}lner sequence (see [@ceccherini2010], Section 5.4), hence we will simply call it a **F[ø]{}lner sequence**. A group $\Gamma$ is called **amenable** if it admits a F[ø]{}lner sequence. Since $\Gamma$ is infinite, for every F[ø]{}lner sequence $(F_n)_{n \geq 1}$ we have ${ \left| F_n \right|} \to \infty$ as $n \to \infty$. For finite sets $F, K \subseteq \Gamma$ the set $${\mathrm{int}_{K}(F)}:=F \setminus {\partial_{K}(F)}$$ is called the **$K$-interior** of $F$. It is clear that if a sequence of finite sets $(F_n)_{n \geq 1}$ is a F[ø]{}lner sequence, then for every finite $K \subseteq \Gamma$ one has $${ \left| {\mathrm{int}_{K}(F_n)} \right|}/{ \left| F_n \right|} \to 1 \text{ as } n \to \infty.$$ We will now briefly remind the reader of the notion of topological entropy for amenable group actions. Let $\alpha=\{ A_1,\dots,A_n\}$ be a finite open cover of a topological space $\prX$. The **topological entropy of a cover** $\alpha$ is defined by $$H(\alpha):=\log \min\{ \card \beta : \beta \subseteq \alpha \text{ a subcover}\}.$$ The entropy of a cover is always a nonnegative real number. We say that a finite open cover $\alpha$ is **finer** than a finite open cover $\beta$ if for every $B \in \beta$ there exists $A \in \alpha$ such that $A \subseteq B$. If $\alpha,\beta$ are two finite open covers, then $$\alpha \vee \beta:=\{ A \cap B: A \in \alpha, B \in \beta \}$$ is a finite open cover as well. It is clear that $\alpha \vee \beta$ is finer than $\alpha$ and $\beta$. Given a topological dynamical system $\bfX=(\prX,\Gamma)$, where the discrete amenable group $\Gamma$ acts on the topological space $\prX$ on the left by homeomorphisms, we can also define (dynamical) entropy of a cover. For every element $g \in \Gamma$ and every finite open cover $\alpha$ we define a finite open cover $g^{-1} \alpha$ by $$g^{-1} \alpha := \{ g^{-1} A: A \in \alpha\}.$$ Next, for every finite subset $F \subseteq \Gamma$ and every finite open cover $\alpha$ we define a finite open cover $$\alpha^F:=\bigvee\limits_{g \in F} g^{-1} \alpha.$$ Let $(F_n)_{n \geq 1}$ be a F[ø]{}lner sequence in $\Gamma$ and $\alpha$ be a finite open cover. The limit $$h(\alpha,\Gamma):=\lim\limits_{n \to \infty} \frac{H(\alpha^{F_n})}{{ \left| F_n \right|}}$$ exists and it is a nonnegative real number independent of the choice of a F[ø]{}lner sequence due to the lemma of D.S. Ornstein and B.Weiss (see [@gromov1999],[@krieger2007]). The limit $h(\alpha,\Gamma)$ is called the **dynamical entropy of $\alpha$**. Finally, the **topological entropy** of a topological system $\bfX=(\prX,\Gamma)$ is defined by $$h(\bfX):=\sup\{ h(\alpha,\Gamma): \alpha \text{ a finite open cover of } \prX\}.$$ F[ø]{}lner monotilings {#ss.regfoln} ---------------------- The purpose of this section is to discuss notion of a F[ø]{}lner monotiling, that was introduced by B.Weiss in [@weiss2001]. The (adapted) notion of a *normal* F[ø]{}lner monotiling, central to the results of this paper, will also be suggested below. A **monotiling** $[F, \calZ]$ in a discrete group $\Gamma$ is a pair of a finite set $F\subseteq \Gamma$, which we call a **tile**, and a set $\calZ \subseteq \Gamma$, which we call a set of **centers**, such that $\{ F z: z \in \calZ \}$ is a covering of $\Gamma$ by disjoint translates of $F$. A **F[ø]{}lner monotiling** is a sequence of monotilings $([F_n, \calZ_n])_{n \geq 1}$ s.t. $(F_n)_{n \geq 1}$ is a F[ø]{}lner sequence in $\Gamma$. We call a F[ø]{}lner monotiling $([F_n,\calZ_n])_{n \geq 1}$ **normal** if $\frac{{ \left| F_n \right|}}{\log n} \to \infty$ as $n \to \infty$; $\ue \in F_n$ for every n. Consider the group $\Z^d$ for some $d \geq 1$ and the F[ø]{}lner sequence $(F_n)_{n\geq 1}$ in $\Z^d$ given by $$F_n:= [0,1,2,\dots,n-1]^d.$$ Furthermore, for every $n$ let $$\calZ_n:= n \Z^d.$$ It is easy to see that $([F_n,\calZ_n])_{n \geq 1}$ is a normal F[ø]{}lner monotiling. Later we will see that the requirement of normality is not essentially restrictive for our purposes. We will need the following simple \[prop.fmonot\] Let $([F_n, \calZ_n])_{n\geq 1}$ be a F[ø]{}lner monotiling of $\Gamma$ s.t. $\ue \in F_n$ for every $n$. Then for every fixed $k$ $$\frac{{ \left| {\mathrm{int}_{F_k}(F_n)} \cap \calZ_k \right|}}{{ \left| F_n \right|}} \to \frac 1 {{ \left| F_k \right|}}$$ and $$\frac{{ \left| F_n \cap \calZ_k \right|}}{{ \left| F_n \right|}} \to \frac 1 {{ \left| F_k \right|}}$$ as $n \to \infty$. Observe first that for every set $A \subseteq \Gamma$ we have $$g \in {\mathrm{int}_{F_k}(A)} \Leftrightarrow F_k g \subseteq A.$$ For every $n \in \N$, consider the finite set $A_{n,k}:=\{ z \in \calZ_k: F_k z \cap {\mathrm{int}_{F_k}(F_n)} \neq \varnothing \}$. Then the translates $\{ F_k z: z \in A_{n,k} \}$ form a disjoint cover of the set ${\mathrm{int}_{F_k}(F_n)}$. It is easy to see that $$\Gamma = {\mathrm{int}_{F_k}(F_n)} \sqcup {\partial_{F_k}(F_n)} \sqcup {\mathrm{int}_{F_k}(\comp{F_n})}.$$ Since $A_{n,k} \cap {\mathrm{int}_{F_k}(\comp{F_n})} = \varnothing$, we can decompose the set of centers $A_{n,k}$ as follows $$A_{n,k} = (A_{n,k} \cap {\mathrm{int}_{F_k}(F_n)}) \sqcup (A_{n,k} \cap {\partial_{F_k}(F_n)}).$$ Since $(F_n)_{n \geq 1}$ is a F[ø]{}lner sequence, $$\frac{{ \left| F_k (A_{n,k} \cap {\partial_{F_k}(F_n)}) \right|}}{{ \left| F_n \right|}}=\frac{{ \left| F_k \right|} \cdot { \left| A_{n,k} \cap {\partial_{F_k}(F_n)} \right|}}{{ \left| F_n \right|}} \to 0$$ and ${ \left| {\mathrm{int}_{F_k}(F_n)} \right|}/{ \left| F_n \right|} \to 1$ as $n \to \infty$. Then from the inequalities $$\begin{aligned} \frac{{ \left| {\mathrm{int}_{F_k}(F_n)} \right|}}{{ \left| F_n \right|}} &\leq \frac{{ \left| F_k (A_{n,k} \cap {\partial_{F_k}(F_n)}) \right|}}{{ \left| F_n \right|}} + \frac{{ \left| F_k (A_{n,k} \cap {\mathrm{int}_{F_k}(F_n)}) \right|}}{{ \left| F_n \right|}} \\ &\leq \frac{{ \left| F_k (A_{n,k} \cap {\partial_{F_k}(F_n)}) \right|}}{{ \left| F_n \right|}} + 1\end{aligned}$$ we deduce that $$\frac{{ \left| F_k \right|} \cdot { \left| A_{n,k} \cap {\mathrm{int}_{F_k}(F_n)} \right|}}{{ \left| F_n \right|}} \to 1$$ as $n \to \infty$. It remains to note that $A_{n,k} \cap {\mathrm{int}_{F_k}(F_n)} = \calZ_k \cap {\mathrm{int}_{F_k}(F_n)}$ and the first statement follows. The second statement follows from the first one and the fact that $(F_n)_{n \geq 1}$ is a strong F[ø]{}lner sequence. Later, in the Section \[ss.compfoln\], we will add a *computability* requirement to the notion of a normal F[ø]{}lner monotiling. The central result of this paper says that the Brudno’s theorem holds for groups admitting a computable normal F[ø]{}lner monotiling. Computability and Kolmogorov complexity {#ss.compcompl} --------------------------------------- In this section we will discuss the standard notions of computability and Kolmogorov complexity that we will use in this work. We refer to Chapter 7 in [@hedman2004] for details, more definitions and proofs. For a natural number $k$ a $k$-ary **partial function** is any function of the form $f: D \to \N \cup \{ 0 \}$, where $D$, **domain of definition**, is a subset of $(\N \cup \{ 0 \})^k$ for some natural $k$. A $k$-ary partial function is called **computable** if there exists an algorithm which takes a $k$-tuple of nonnegative integers $(a_1,a_2,\dots,a_k)$, prints $f((a_1,a_2,\dots,a_k))$ and terminates if $(a_1,a_2,\dots,a_k)$ is in the domain of $f$, while yielding no output otherwise. A function is called **total**, if it is defined everywhere. The term *algorithm* above stands, informally speaking, for a computer program. One way to formalize it is through introducing the class of *recursive functions*, and the resulting notion coincides with the class of functions computable on *Turing machines*. We do not focus on these question in this work, and we will think about computability in an ‘informal’ way. A set $A \subseteq \N$ is called **recursive** (or **computable**) if the indicator function ${\mathbf{1}_{A}}$ of $A$ is computable. It is easy to see that finite and co-finite subsets of $\N$ are computable. Furthermore, for computable sets $A,B \subseteq \N$ their union and intersection are also computable. If a total function $f: \N \to \N$ is computable and $A \subseteq \N$ is a computable set, then $f^{-1}(A)$, the full preimage of $A$, is computable. The image of a computable set via a total computable bijection is computable, and the inverse of such a bijection is a computable function. A sequence of subsets $(F_n)_{n \geq 1}$ of $\N$ is called **computable** if the total function ${\mathbf{1}_{F_{\cdot}}}: (n,x) \mapsto {\mathbf{1}_{F_n}}(x)$ is computable. It is easy to see that a total function $f: \N \to \N$ is computable if and only if the sequence of singletons $(f(n))_{n \geq 1}$ is computable in the sense above. It is very often important to have a numeration of elements of a set by natural numbers. A set $A \subseteq \N$ is called **enumerable** if there exist a total computable surjective function $f: \N \to A$. If the set $A$ is infinite, we can also require $f$ to be injective. This leads to an equivalent definition because an algorithm computing the function $f$ can be modified so that no repetitions occur in its output. Finite and cofinite sets are enumerable. It can be shown (Proposition 7.44 in [@hedman2004]) that a set $A$ is computable if and only if both $A$ and $\N \setminus A$ are enumerable. Furthermore, for a set $A \subsetneq \N$ the following are equivalent: $A$ is enumerable; $A$ is the domain of definition of a partial recursive function. Finally, we can introduce the Kolmogorov complexity for finite words. Let $A$ be a computable partial function defined on a domain $D$ of finite binary words with values in the set of all finite words over finite alphabet $\Lambda$. Of course, we have defined computable functions on subsets of $(\N \cup \{ 0 \})^k$ with values in $\N \cup \{ 0 \}$ above, but this can be easily extended to (co)domains of finite words over finite alphabets. We can think of $A$ as a ‘decompressor’ that takes compressed binary descriptions (or ‘programs’) in its domain, and decompresses them to finite words over alphabet $\Lambda$. Then we define the **Kolmogorov complexity** of a finite word $\omega$ with respect to $A$ as follows: $${\uK_{A}^{0}(\omega)}:=\inf\{ l(p): A(p)=w \},$$ where $l(p)$ denotes the length of the description. If some word $\omega_0$ does not admit a compressed version, then we let ${\uK_{A}^{0}(\omega_0)} = \infty$. The **average Kolmogorov complexity** with respect to $A$ is defined by $${\overline{\uK}_{A}^0(\omega)}:=\frac{{\uK_{A}^{0}(\omega)}}{l(\omega)},$$ where $l(\omega)$ is the length of the word $\omega$. Intuitively speaking, this quantity tells how effective the compressor $A$ is when describing the word $\omega$. Of course, some decompressors are intuitively better than some others. This is formalized by saying that $A_1$ is **not worse** than $A_2$ if there is a constant $c$ s.t. for all words $\omega$ $$\label{eq.optdecomp} {\uK_{A_1}^{0}(\omega)} \leq {\uK_{A_2}^{0}(\omega)}+c.$$ A theorem of Kolmogorov says that there exist a decompressor $A_1$ that is optimal, i.e. for every decompressor $A_2$ there is a constant $c$ s.t. for all words $\omega$ the Equation \[eq.optdecomp\] holds. The notion of Kolmogorov complexity can be extended to words defined on finite subsets of $\N$, and this will be essential in the following sections. More precisely, let $X \subset \N$ be a finite subset, $\imath_X: X \to \{ 1,2,\dots, \card X\} $ an increasing bijection, $\Lambda$ a finite alphabet, $A$ a decompressor and $\omega \in \Lambda^Y$ a word defined on some set $Y \supseteq X$. Then we let $$\label{eq.kcnsubs} \uK_A(\omega,X):={\uK_{A}^{0}(\omega \circ \imath_X^{-1})}.$$ and $$\label{eq.kcansubs} \overline{\uK}_A(\omega,X):=\frac{{\uK_{A}^{0}(\omega \circ \imath_X^{-1})}}{\card X}.$$ We call $\uK_A(\omega,X)$ the **Kolmogorov complexity** of $\omega$ over $X$ with respect to $A$, and $\overline{\uK}_A(\omega,X)$ is called the **mean Kolmogorov complexity** of $\omega$ over $X$ with respect to $A$. If a decompressor $A_1$ is not worse than a decompressor $A_2$ with some constant $c$, then for all $X, \omega$ above $$\uK_{A_1}(\omega,X)\leq\uK_{A_2}(\omega,X)+c.$$ From now on, we will (mostly) use a fixed optimal decompressor ${A^{\ast}}$ and write $\uK(\omega,X)$, $\overline{\uK}(\omega,X)$ omitting explicit reference to ${A^{\ast}}$. When estimating the Kolmogorov complexity of words we will often have to encode nonnegative integers using binary words. We will now fix some notation that will be used later. When $n$ is a nonnegative integer, we write $\underline \un$ for the **binary encoding** of $n$ and $\overline \un$ for the **doubling encoding** of $n$, i.e. if $b_l b_{l-1} \dots b_0$ is the binary expansion of $n$, then $\underline \un$ is the binary word $\ub_l \ub_{l-1} \dots \ub_0$ of length $l+1$ and $\overline \un$ is the binary word $\ub_l \ub_l \ub_{l-1} \ub_{l-1} \dots \ub_0 \ub_0$ of length $2l+2$. We denote the length of the binary word $\uw$ by $l(\uw)$, and is clear that $l(\underline \un) \leq \lfloor \log n\rfloor+1$ and $l(\overline \un) \leq 2 \lfloor \log n\rfloor+2$. We write $\widehat \un$ for the encoding $\overline{l(\underline \un)} {\mathrm{01}}\underline \un$ of $n$, i.e. the encoding begins with the length of the binary word $\underline \un$ encoded using doubling encoding, then the delimiter ${\mathrm{01}}$ follows, then the word $\underline n$. It is clear that $l(\widehat \un) \leq 2 \lfloor \log(\lfloor \log n \rfloor + 1) \rfloor + \lfloor \log n \rfloor + 5$. This encoding enjoys the following property: given a binary string $$\widehat x_1 \widehat x_2 \dots \widehat x_l,$$ the integers $x_1,\dots,x_l$ are unambiguously restored. We will call such an encoding a **simple prefix-free encoding**. Computable Spaces and Sheaves {#ss.compspaces} ----------------------------- The goal of this section is to introduce the notions of *computable space*, *computable function* between computable spaces and *word sheaves* over computable spaces. The complexity of sections of word presheaves and asymptotic complexity of word presheaves is introduced in this section as well. An **indexing** of a set $X$ is an injective mapping $\imath: X \to \N$ such that $\imath(X)$ is a computable subset. Given an element $x \in X$, we call $\imath(x)$ the **index** of $x$. If $i \in \imath(X)$, we denote by $x_i$ the element of $X$ having index $i$. A **computable space** is a pair $(X, \imath)$ of a set $X$ and an indexing $\imath$. Preimages of computable subsets of $\N$ under $\imath$ are called **computable subsets** of $(X,\imath)$. Each computable subset $Y \subseteq X$ can be seen as a computable space $(Y, \imath|_Y)$, where $\imath|_Y$ is the restriction of the indexing function. Of course, the set $\N$ with identity as an indexing function is a computable space, and the computable subsets of $(\N, \id)$ are precisely the computable sets of $\N$ in the sense of Section \[ss.compcompl\]. Let $(X_1, \imath_1), (X_2, \imath_2), \dots, (X_k, \imath_k),(Y,\imath)$ be computable spaces. A (total) function $f: X_1\times X_2 \times \dots \times X_k \to Y$ is called **computable** if the function $\widetilde f: \imath_1(X_1) \times \imath_2(X_2) \times \dots \times \imath_k(X_k) \to \imath(Y)$ determined by the condition $$\widetilde f(\imath_1(x_1),\imath_2(x_2),\dots,\imath_k(x_k)) = \imath(f(x_1,x_2,\dots,x_k))$$ for all $(x_1,x_2,\dots,x_k) \in X_1 \times X_2 \times \dots \times X_k$ is computable. This definition extends the standard definition of computability from Section \[ss.compcompl\] when the computable spaces under consideration are $(\N,\id)$. A computable function $f: (X,\imath_1) \to (Y,\imath_2)$ is called a **morphism** between computable spaces. This yields the definition of the **category of computable spaces**. Let $(X, \imath_1)$, $(X, \imath_2)$ be computable spaces. The indexing functions $\imath_1$ and $\imath_2$ of $X$ are called **equivalent** if $\id: (X,\imath_1) \to (X,\imath_2)$ is an isomorphism. It is clear that the classes of computable functions and computable sets do not change if we pass to equivalent indexing functions. Given a computable space $(X,\imath)$, we call a sequence of subsets $(F_n)_{n \geq 1}$ of $X$ **computable** if the function ${\mathbf{1}_{F_{\cdot}}}: \N \times X \to \{ 0,1\}, (n,x) \mapsto {\mathbf{1}_{F_n}}(x)$ is computable. We will also need a special notion of computability for sequences of *finite* subsets of $(X,\imath)$. A sequence of finite subsets $(F_n)_{n \geq 1}$ of $X$ is called **canonically computable** if there is an algorithm that, given $n$, prints the set $\imath(F_n)$ and halts. One way to make this more precise is by introducing the canonical index of a finite set. Given a finite set $A=\{ x_1,x_2,\dots,x_k\} \subset \N$, we call the number ${\mathrm{I}(A)}:=\sum\limits_{i=1}^k 2^{x_i}$ the **canonical index** of $A$. Hence a sequence of finite subsets $(F_n)_{n \geq 1}$ of $X$ is canonically computable if and only if the total function $n \mapsto {\mathrm{I}(\imath(F_n))}$ is computable. Of course, a canonically computable sequence of finite sets is computable, but the converse is not true due to the fact that there is no effective way of determining how large a finite set with a given computable indicator function is. It is easy to see that the class of canonically computable sequences of finite sets does not change if we pass to an equivalent indexing. The proof of the following proposition is straightforward: Let $(X,\imath)$ be a computable space. Then If $(F_n)_{n \geq 1}, (G_n)_{n \geq 1}$ are (canonically) computable sequences of sets, then the sequences of sets $(F_n \cup G_n)_{n \geq 1}$, $(F_n \cap G_n)_{n \geq 1}$ and $(F_n \setminus G_n)_{n \geq 1}$ are (canonically) computable. If $(F_n)_{n \geq 1}$ is a canonically computable sequence of sets and $(G_n)_{n \geq 1}$ is a computable sequence of sets, then the sequence of sets $(F_n \cap G_n)_{n \geq 1}$ is canonically computable. Let $(X,\imath)$ be a computable space and $\Lambda$ be a finite alphabet. A **word presheaf** $\calF_{\Lambda}$ on $X$ consists of A set $\calF_{\Lambda}(U)$ of $\Lambda$-valued functions defined on the set $U$ for every computable subset $U \subseteq X$; A restriction mapping $\rho_{U,V}: \calF_{\Lambda}(U) \to \calF_{\Lambda}(V)$ for each pair $U,V$ of computable subsets s.t. $V \subseteq U$, that takes functions in $\calF_{\Lambda}(U)$ and restricts them to the subset $V$. It is easy to see that the standard ‘presheaf axioms’ are satisfied: $\rho_{U,U}$ is identity on $\calF_{\Lambda}(U)$ for every computable $U \subseteq X$, and for every triple $V \subseteq U \subseteq W$ we have that $\rho_{W,V} = \rho_{U,V} \circ \rho_{W,U}$. Elements of $\calF_{\Lambda}(U)$ are called **sections** over $U$, or **words** over $U$. We will often write $s|_V$ for $\rho_{U,V}s$, where $s \in \calF_{\Lambda}(U)$ is a section. We have introduced Kolmogorov complexity of words supported on subsets of $\N$ in the previous section, now we want to extend this by introducing complexity of sections. Let $(X,\imath)$ be a computable space and let $\calF_{\Lambda}$ be a word presheaf over $(X,\imath)$. Let $U \subseteq X$ be a finite set and $\omega \in \calF_{\Lambda}(U)$. Then we define the **Kolmogorov complexity** of $\omega \in \calF_{\Lambda}(U)$ by $${\uK(\omega,U)}:={\uK(\omega \circ \imath^{-1},\imath(U))}$$ and the **mean Kolmogorov complexity** of $\omega \in \calF_{\Lambda}(U)$ by $${\overline{\uK}(\omega,U)}:={\overline{\uK}(\omega \circ \imath^{-1},\imath(U))}.$$ The quantities on the right hand side here are defined in the Equations \[eq.kcnsubs\] and \[eq.kcansubs\] respectively (which are special cases of the more general definition when the computable space $X$ is $(\N,\id)$). Let $(F_n)_{n \geq 1}$ be a sequence of finite subsets of $X$ s.t. $\card F_n \to \infty$ as $n \to \infty$. Then we define **asymptotic Kolmogorov complexity** of the word presheaf $\calF_{\Lambda}$ along the sequence $(F_n)_{n \geq 1}$ by $${\widetilde{\uK}(\calF_{\Lambda})}:= \limsup\limits_{n \to \infty} \max\limits_{\omega \in \calF_{\Lambda}(F_n)} {\overline{\uK}(\omega,F_n)}.$$ Dependence on the sequence is omitted in the notation, but it will always be clear from the context which sequence we take. If $A'$ is some decompressor, then it follows from the optimality of ${A^{\ast}}$ that $${\widetilde{\uK}(\calF_{\Lambda})} \leq \widetilde{\uK}_{A'}(\calF_{\Lambda}).$$ To simplify the notation, we adopt the following convention. We say explicitly what indexing function we use when introducing a computable space, but later, when the indexing is fixed, we shall often omit the indexing function from the notation and think about computable spaces as computable subsets of $\N$, endowed with induced ordering. Words defined on subsets of a computable space become words defined on subsets of $\N$. This will help to simplify the notation without introducing much ambiguity. Computable Groups {#ss.compgrp} ----------------- In this section we provide the definitions of a computable group and a few related notions, connecting results from algebra with computability. This section is based on [@rabin1960]. Let $\Gamma$ be a group with respect to the multiplication operation $\ast$. An indexing $\imath$ of $\Gamma$ is called **admissible** if the function $\ast: (\Gamma,\imath) \times (\Gamma,\imath) \to (\Gamma,\imath)$ is a computable function in the sense of Section \[ss.compspaces\]. A **computable group** is a pair $(\Gamma,\imath)$ of a group $\Gamma$ and an admissible indexing $\imath$. Of course, the groups $\Z^d$ and ${\mathsf{UT}_{d}(\Z)}$ possess ‘natural’ admissible indexings. For the group $\Z$ we fix the indexing $$\imath: n \mapsto 2|n| + {\mathbf{1}_{n \geq 0}}.$$ It is clear that the groups $\Z^d$ for $d>1$ possess admissible indexing functions such that all coordinate projections onto $\Z$, endowed with the indexing function $\imath$ above, are computable. We leave the details to the reader. The following lemma from [@rabin1960] shows that in a computable group taking the inverse is a computable operation. Let $(\Gamma, \imath)$ be a computable group. Then the function ${\mathrm{inv}}: (\Gamma,\imath) \to (\Gamma,\imath), g \mapsto g^{-1}$ is computable. $(\Gamma, \imath)$ is a computable space, and we can talk about computable subsets of $(\Gamma, \imath)$. A subgroup of $\Gamma$ which is a computable subset will be called a **computable subgroup**. A homomorphism between computable groups that is computable as a map between computable spaces will be called a **computable homomorphism**. The proof of the proposition below is straightforward. Let $(\Gamma,\imath)$ be a computable group. Then the following assertions hold: Given a computable set $A \subseteq \Gamma$ and $g \in \Gamma$, the sets $A^{-1}, gA$ and $Ag$ are computable; Given a (canonically) computable sequence $(F_n)_{n \geq 1}$ of (finite) subsets of $\Gamma$ and $g \in \Gamma$, the sequences $(g F_n)_{n \geq 1}, (F_n g)_{n \geq 1}$ are (canonically) computable. It is interesting to see that a computable version of the ‘First Isomorphism Theorem’ also holds. Let $(G,\imath)$ be a computable group and let $(H,\imath|_H)$ be a computable normal subgroup, where $\imath|_H$ is the restriction of the indexing function $\imath$ to $H$. Then there is a compatible indexing function $\imath'$ on the factor group $\fact G H$ such that the quotient map $\pi: (G,\imath) \to (\fact G H, \imath')$ is a computable homomorphism. For the proof we refer the reader to the Theorem 1 in [@rabin1960]. Computable F[ø]{}lner sequences and computable monotilings {#ss.compfoln} ---------------------------------------------------------- The notions of an amenable group and a F[ø]{}lner sequence are well-known, but, since we are working with computable groups, we need to develop their ‘computable’ versions. Let $(\Gamma,\imath)$ be a computable group. A F[ø]{}lner monotiling $([F_n, \calZ_n])_{n \geq 1}$ of $\Gamma$ is called **computable** if $(F_n)_{n \geq 1}$ is a canonically computable sequence of finite subsets of $\Gamma$; $(\calZ_n)_{n \geq 1}$ is a computable sequence of subsets of $\Gamma$. \[ex.zdmonot\] Consider the group $\Z^d$ for some $d \geq 1$. Let $\imath$ be any admissible indexing such that all the coordinate projections $\Z^d \to \Z$ are computable. Then the F[ø]{}lner sequence $F_n = [0,1,2,\dots,n-1]^d$ is canonically computable. Furthermore, the set $\calZ_n$ of centers equals $n \Z^d$ for every $n$, hence $([F_n,\calZ_n])_{n \geq 1}$ is a computable normal F[ø]{}lner monotiling. In general, the procedures computing F[ø]{}lner monotilings might be difficult to write out explicitly and, given a F[ø]{}lner sequence, there might be different corresponding sequences of tilings. However, for the groups ${\mathsf{UT}_{d}(\Z)}$ of upper-triangular matrices with integer entries there exist very ‘natural’ F[ø]{}lner monotilings that can be easily computed. For details we refer the reader to the work [@golodets2002] of Ya. Golodets and S. D. Sinelshchikov. The main results of this paper are proved for groups that admit a computable *normal* F[ø]{}lner monotiling. In the lemma below we show that this requirement is not restrictive. \[l.normalization\] Let $(\Gamma,\imath)$ be a computable group and $([F_n,\calZ_n])_{n \geq 1}$ be a computable F[ø]{}lner monotiling. Then there is a computable function $r_{\cdot}: \N \to \Gamma$ and a computable function $n_{\cdot}: \N \to \N$ such that $([F_{n_i} r_{n_i}^{-1}, r_{n_i} \calZ_{n_i}])_{i\geq 1}$ is a computable normal F[ø]{}lner monotiling. Let $r_i$ be the first element of the set $F_i$ for every $i$ when we view $F_i$ as a subset of $\N$ via the indexing mapping $\imath$. Then $(F_n r_n^{-1})_{n \geq 1}$ is a F[ø]{}lner sequence such that $\ue \in F_n r_n^{-1}$ for every $n$, and $([F_n r_n^{-1}, r_n \calZ_n])_{n \geq 1}$ is a F[ø]{}lner monotiling. It is clear that we can pick the function $n_{\cdot}$ such that the growth condition is satisfied as well. The following lemma shows that we can ‘computably’ determine which elements of canonically computable F[ø]{}lner sequences are ‘invariant enough’ in computable groups. \[prop.indexing\] Let $(F_n)_{n \geq 1}$ be a canonically computable F[ø]{}lner sequence in a computable group $(\Gamma, \imath)$. Then there is a computable function $i \mapsto n_i$ such that $$\label{eq.indexing} \frac{{ \left| F_{n_i} \sdif g F_{n_i} \right|}}{{ \left| F_{n_i} \right|}} < \frac 1 {2 i}$$ for all $g \in K_i$ and all $i$, where $K_i$ is the set of the first $i$ elements of $\Gamma$ with respect to the indexing $\imath$. For every $i$ we let $K_i$ be the finite set defined above. We let $n_i$ be the first index such that (\[eq.indexing\]) holds for all $g \in K_i$ (such $n_i$ exists because $(F_n)_{n \geq 1}$ is a F[ø]{}lner sequence). The following lemma simplifies checking the computability of $(\calZ_n)_{n \geq 1}$. \[prop.monoteq\] Let $(\Gamma, \imath)$ be a computable group. Let $([F_n,\calZ_n])_{n \geq 1}$ be a F[ø]{}lner monotiling of $\Gamma$ such that $(F_n)_{n \geq 1}$ is a canonically computable sequence of finite sets and $\ue \in F_n$ for all $n \geq 1$. Then the following assertions are equivalent: There is a total computable function $\phi: \N^2 \to \Gamma$ such that for every $n \in \N$ $$\calZ_n = \{ \phi(n,1),\phi(n,2), \dots\}.$$ The sequence of sets $(\calZ_n)_{n \geq 1}$ is computable. One implication is clear. For the converse, note that to prove computability of the function ${\mathbf{1}_{\calZ_{\cdot}}}$ we have to devise an algorithm that, given $n\in \N$ and $g \in \Gamma$, decides whether $g \in \calZ_n$ or not. Let $\phi: \N^2 \to \Gamma$ be the function from assertion (i). Then the following algorithm answers the question. Start with $i:=1$ and compute $\ue \phi(n,i), h_{1,n} \phi(n,i),\dots, h_{k,n} \phi(n,i)$, where $F_n=\{ \ue, h_{1,n},\dots,h_{k,n}\}$. This is possible since $(F_n)_{n \geq 1}$ is a canonically computable sequence of finite sets. If $g = \ue \phi(n,i) $, then the answer is ‘Yes’ and we stop the program. If $g = h_{j,n} \phi(n,i)$ for some $j$, then the answer is ‘No’ and we stop the program. If neither is true, then we set $i:=i+1$ and go to the beginning. Since $\Gamma = F_n \calZ_n$, the algorithm terminates for every input. The following theorem, whose proof is essentially due to B. Weiss [@weiss2001], shows that the class of groups admitting computable normal F[ø]{}lner monotilings is closed under group extensions. \[thm.weiss\] Let $$1 \to (E,\imath_E) \overset{\id}{\to} (F,\imath_F) \overset{\psi}{\to} (G,\imath_G) \to 1$$ be an exact sequence of computable groups such that $\id, \psi$ are computable homomorphisms. Suppose that $([E_k,\calQ_k])_{k \geq 1}, ([G_m,\calS_m])_{m \geq 1}$ are computable normal F[ø]{}lner monotilings of the groups $(E,\imath_E)$ and $(G,\imath_G)$ respectively. Then there is a computable normal F[ø]{}lner monotiling $([F_l,\calR_l])_{l \geq 1}$ in the group $(F,\imath_F)$. We begin by describing an auxiliary construction that provides us with ‘computable’ sections of computable sets in $F$ over $G$. Let $T \subseteq F$ be a computable set. Let ${\mathbf{1}_{T}}$ be the characteristic function of $T$. Now we construct a characteristic function of a computable section $T'$ of $T$ as follows: $${\mathbf{1}_{T'}}(n) := \begin{cases} 1 & \text{if } {\mathbf{1}_{T}}(n)=1 \text{ and } ((\forall \ l < n \ \ \psi(n)\neq \psi(l)) \vee (n=\ue));\\ 0 & \text{otherwise.} \end{cases}$$ That is, $T'$ is the set of the first members of each $E$-coset in $T$ except for the coset $\ue E$, on which we pick $\ue$ instead if $\ue \in T$. Since the functions ${\mathbf{1}_{T}}, \psi$ are computable, it is easy to see that ${\mathbf{1}_{T'}}$ is computable as well. In particular, the function $x \mapsto \psi^{-1}(x)'$ from $G$ to $F$ is computable. Let $l \in \N$ be fixed. Let $K_l=\{f_1,f_2,\dots,f_l \} \subset F$ be the set of the first $l$ elements of $F$ with respect to the indexing $\imath_F$. We will describe an algorithm that yields a tile $F_l \subset F$ such that for all $g \in K_l$ we have $$\frac{{ \left| F_l \sdif g F_l \right|}}{{ \left| F_l \right|}} \leq \frac 1 l.$$ It is easy to see that such a sequence $(F_l)_{l \geq 1}$ is a canonically computable F[ø]{}lner sequence. We will use Proposition \[prop.monoteq\] to show that the corresponding sequence of centers $(\calR_l)_{l \geq 1}$ is computable, and it will be clear from the proof why the monotiling $([F_l,\calR_l])_{l \geq 1}$ is normal as well. Consider the finite set $\psi(K_l) \subset G$. Then the maximum $I_l \geq l$ of the indices of elements of $\psi(K_l)$ is a computable function of $l$. We let $Q_l:=\{ g_1,g_2,\dots,g_{I_l} \}$ be the finite set of the first $I_l$ elements of $G$. Let $m_{\cdot}: i \mapsto m_i$ be the computable function from the Proposition \[prop.indexing\] applied to $(G,\imath_G)$ and $(G_m)_{m \geq 1}$, then the function $m_{\cdot}^{\ast}: l \mapsto \max(m_{I_l},2l)$ is a computable function. Let $G_{{m_{l}^{\ast}}}$ be the corresponding F[ø]{}lner tile in $G$. Consider its preimage $\psi^{-1}(G_{{m_{l}^{\ast}}})$, which is a computable subset of $F$. We note that the sequence of sets $l \mapsto \psi^{-1} (G_{{m_{l}^{\ast}}})$ is computable because the sequence of sets $(G_m)_{m \geq 1}$ is computable and the functions $\psi, {m_{\cdot}^{\ast}}$ are computable. Let $T_l \subset \psi^{-1}(G_{{m_{l}^{\ast}}})$ be the computable section of $\psi^{-1}(G_{{m_{l}^{\ast}}})$ over $G$ as defined above, then $\psi$ is bijective as a map from $T_l$ to $G_{{m_{l}^{\ast}}}$. Observe further that the sequence of *finite* sets $l \mapsto T_l$ is *canonically* computable. B. Weiss proved that if $U$ is a sufficiently invariant monotile in $E$, then $T_l U$ is a sufficiently invariant monotile in $F$. Below we examine his construction closely. Let ${G_{{m_{l}^{\ast}}}^{\circ}}:=\{ t \in G_{{m_{l}^{\ast}}}: Q_l t \subset G_{{m_{l}^{\ast}}} \} \subset G$ be the part of $G_{{m_{l}^{\ast}}}$ that stays within $G_{{m_{l}^{\ast}}}$ when shifted by elements of $Q_l$. It is clear that $T_l^{\circ}:=\psi^{-1}({G_{{m_{l}^{\ast}}}^{\circ}})' \subset T_l$, and that the sequence of finite sets $l \mapsto {T_{l}^{\circ}}$ is canonically computable. For all $x \in K_l$ and $t \in {T_{l}^{\circ}}$ we deduce that $$x t = \lambda_l(x,t) \rho_l(x,t),$$ where $\lambda_l(x,t) \in T_l$ and $\rho_l(x,t) \in E$. The functions $\lambda_{\cdot}(\cdot,\cdot)$ and $\rho_{\cdot}(\cdot,\cdot)$ are uniquely determined by this condition and are partial computable. Consider the finite subset $$P_l:=\{ \rho_l(x,t): x \in K_l, t\in {T_{l}^{\circ}} \}.$$ The maximum index $J_l$ of elements of this set is a computable function of $l$. Let $k_{\cdot}: i\mapsto k_i$ be the computable function from the Proposition \[prop.indexing\] applied to $(E,\imath_E)$ and $(E_k)_{k \geq 1}$, then the function ${k_{\cdot}^{\ast}}: l \mapsto \max(k_{J_l}, {m_{l}^{\ast}})$ is computable. Consider the F[ø]{}lner tile $E_{{k_{l}^{\ast}}}$, and let $E_{{k_{l}^{\ast}}}^{\circ}:=\{ s \in E_{{k_{l}^{\ast}}}: P_l s \subset E_{{k_{l}^{\ast}}}\}$ be the part of $E_{{k_{l}^{\ast}}}$ that stays in $E_{{k_{l}^{\ast}}}$ when shifted by elements of $P_l$. We claim that the tile $$F_l:=T_l E_{{k_{l}^{\ast}}}$$ is ‘invariant enough’. Observe that $$K_l {T_{l}^{\circ}} E_{{k_{l}^{\ast}}}^{\circ} \subset T_l E_{{k_{l}^{\ast}}}.$$ By definition, the set ${G_{{m_{l}^{\ast}}}^{\circ}}$ is large enough: $${ \left| {G_{{m_{l}^{\ast}}}^{\circ}} \right|} \geq \left(1-\frac{1}{2 l}\right){ \left| G_{{m_{l}^{\ast}}} \right|},$$ hence $${ \left| {T_{l}^{\circ}} \right|} \geq \left(1-\frac{1}{2 l}\right){ \left| T_l \right|}.$$ Similarly, $${ \left| E_{{k_{l}^{\ast}}}^{\circ} \right|} \geq \left(1-\frac{1}{2 l}\right){ \left| E_{{k_{l}^{\ast}}} \right|},$$ and it follows that $${ \left| T_l^{\circ} E_{{k_{l}^{\ast}}}^{\circ} \right|} \geq \left(1-\frac 1 {2 l}\right){ \left| T_l E_{{k_{l}^{\ast}}} \right|}.$$ We deduce that for every $g \in K_l$ $$\begin{aligned} \frac{{ \left| F_l \sdif g F_l \right|}}{{ \left| F_l \right|}} = \frac{{ \left| F_l \sdif g^{-1} F_l \right|}}{{ \left| F_l \right|}} \leq \frac{1}{l},\end{aligned}$$ and this shows that $F_l$ is ‘invariant enough’. We have obtained a canonically computable F[ø]{}lner sequence $l \mapsto F_l$. It remains to prove that for each $l$ the set $F_l$ is a tile and that the sequence of centers $(\calR_l)_{l \geq 1}$ is computable. Let $\phi_E, \phi_G$ be computable functions from the Proposition \[prop.monoteq\] applied to groups $(E,\imath_E),(G,\imath_G)$ respectively. Let $\theta: \N^2 \to F$ be the total computable function $(n,i) \mapsto \psi^{-1}(\phi_G(n,i))'$, i.e. we compute $\phi_G(n,i)$ first and then pick an element in its fiber in $F$. It is clear that $$\{ \phi_E({k_{l}^{\ast}},i) \theta({m_{l}^{\ast}},j )\}_{i,j \geq 1}$$ is a set of centers for the tile $F_l$. If $\nu: \N \to \N^2$ is any computable bijection, then the total computable function $\phi_F: (n,i) \mapsto \phi_E({k_{n}^{\ast}}, \nu_1(i)) \theta({m_{n}^{\ast}}, \nu_2(i))$ satisfies the conditions of Proposition \[prop.monoteq\], and the proof is complete. Brudno’s theorem {#s.brudnocfm} ================ We are now ready to prove the main theorem of this article. First, we will explain the definitions and introduce some notation that will be used in the proofs. By a **subshift** $\bfX=(\prX,\Gamma)$ we mean a closed $\Gamma$-invariant subset $\prX$ of $\Lambda^{\Gamma}$, where $\Lambda$ is the finite **alphabet** of $\bfX$. The words consisting of letters from the alphabet $\Lambda$ will be often called **$\Lambda$-words**. Of course, we can assume without loss of generality that $\Lambda = \{ 1,2,\dots,k\}$ for some $k$. The left action of the group $\Gamma$ on $\prX$ is given by $$(g\cdot \omega)(x) := \omega(x g) \ \ \ \forall x,g \in \Gamma, \omega \in \prX.$$ We can associate a word presheaf $\calF_{\Lambda}$ to the subshift $\bfX$ by setting $$\calF_{\Lambda}(F):=\{ \omega|_F: \omega \in \prX\}.$$ That is, $\calF_{\Lambda}(F)$ is the set of all restrictions of words in $\prX$ to the set $F$ for every computable $F$. The goal of this section is to prove the following: \[thm.brudno\] Let $(\Gamma,\imath)$ be a computable group with a fixed computable normal F[ø]{}lner monotiling $([F_n,\calZ_n])_{n \geq 1}$. Let $\bfX=(\prX,\Gamma)$ be a subshift on $\Gamma$ and $\calF_{\Lambda}$ be the associated word presheaf on $\Gamma$. Then $${\widetilde{\uK}(\calF_{\Lambda})} = h(\bfX),$$ where the asymptotic complexity of the word presheaf $\calF_{\Lambda}$ is computed along the sequence $(F_n)_{n \geq 1}$. The proof is split into two parts, establishing respective inequalities in Theorems \[thm.brudnogeq\] and \[thm.brudnoleq\]. Given a subshift $\bfX$ on the alphabet $\Lambda$, we define the cover $$\alpha_{\Lambda}:=\{A_1,\dots,A_k\}, \ A_i:=\{ \omega \in \prX: \omega(\ue) = i\} \text{ for } i = 1,\dots,k.$$ Then $\alpha_{\Lambda}$ is, clearly, a generating cover: for every finite open cover $\beta$ of $\prX$ there exists a finite subset $F \subseteq \Gamma$ such that the cover $\alpha_{\Lambda}^F$ is finer than $\beta$. We will use the following well-known Let $\bfX$ be a subshift of $\Lambda^{\Gamma}$ and $\alpha_{\Lambda}$ be the cover defined above. Then $$h(\alpha_{\Lambda},\Gamma) = \lim\limits_{n\to \infty} \frac{\log \card \calF_{\Lambda}(F_n)}{{ \left| F_n \right|}}= h(\bfX).$$ We will now establish the first inequality. The proof is essentially the same as the original one from [@brudno1974]. \[thm.brudnoleq\] In the setting of Theorem \[thm.brudno\] we have $$h(\bfX) \leq {\widetilde{\uK}(\calF_{\Lambda})}$$ By the definition, we have to show that $$\lim\limits_{n\to \infty} \frac{\log \card \calF_{\Lambda}(F_n)}{{ \left| F_n \right|}} \leq \limsup\limits_{n \to \infty} \max\limits_{\omega \in \calF_{\Lambda}(F_n)} {\overline{\uK}(\omega,F_n)}.$$ Suppose that ${\widetilde{\uK}(\calF_{\Lambda})} < t$ for some $t \geq 0$. Then there exists $n_0$ such that for all $n \geq n_0$ and all $\omega \in \calF_{\Lambda}(F_n)$ the inequality ${\uK(\omega,F_n)} \leq t { \left| F_n \right|}$ holds. There are at most $2^{t { \left| F_n \right|} + 1}$ valid binary programs for the decompressor ${A^{\ast}}$ of length at most $t { \left| F_n \right|}$, hence $\card \calF_{\Lambda}(F_n) \leq 2^{t { \left| F_n \right|}+1}$. Taking the logarithm shows that for all $n \geq n_0$ we have $$\frac{\log \card \calF_{\Lambda}(F_n)}{{ \left| F_n \right|}} \leq t+ \frac{1}{|F_n|},$$ and this completes the proof. The proof of the second inequality requires more work. The proof we provide is based on the idea of the proof of Lemma 5.1 from [@simpson2015]. \[thm.brudnogeq\] In the setting of Theorem \[thm.brudno\] we have $$h(\bfX) \geq {\widetilde{\uK}(\calF_{\Lambda})}$$ By the definition, we have to show that $$\limsup\limits_{n \to \infty} \max\limits_{\omega \in \calF_{\Lambda}(F_n)} {\overline{\uK}(\omega,F_n)} \leq \lim\limits_{n\to \infty} \frac{\log \card \calF_{\Lambda}(F_n)}{{ \left| F_n \right|}}.$$ The alphabet $\Lambda$ is finite, so we encode each letter of $\Lambda$ using precisely $\lfloor \log \card \Lambda \rfloor+1$ bits. We fix this encoding. Then binary words of length $N\left(\lfloor \log \card \Lambda \rfloor+1 \right)$ are unambiguously interpreted as $\Lambda$-words of length $N$. We will now describe a decompressor ${A^!}$ that will be used to prove the theorem. The decompressor is defined on the domain of binary programs of the form $$\label{eq.prog} \up = \widehat \uk \widehat \un \widehat \uN \widehat \uL \widehat \ul \uw_1 \uw_2 \dots \uw_N \uv \widehat{\ui}_1 \widehat{\ui}_2 \dots \widehat{\ui}_s.$$ Here $\widehat \uk, \widehat \un, \widehat \uN, \widehat \uL, \widehat \ul$ are simple prefix-free encodings of the natural numbers $k,n,N,L,l$. Binary words $\uw_1, \uw_2, \dots, \uw_N$ have all length $L$. Words $\widehat{\ui}_1, \widehat{\ui}_2, \dots, \widehat{\ui}_s$ encode some natural numbers $i_1,i_2,\dots,i_s$ that are required to be less or equal to $N$. Finally, $\uv$ is a binary word of length $l$. Observe that programs of the form \[eq.prog\] are indeed unambiguously interpreted. The decompressor ${A^!}$ works as follows. First, given $k,n$ above, the finite sets $$F_k, F_n, {\mathrm{int}_{F_k}(F_n)}, {\mathrm{int}_{F_k}(F_n)} \cap \calZ_k \subseteq \N$$ are computed. We let $I_{k,n}:={\mathrm{int}_{F_k}(F_n)} \cap \calZ_k$ and compute the set $$\Delta_{k,n}:=F_k \left( {\mathrm{int}_{F_k}(F_n)} \cap \calZ_k \right) \subseteq F_n.$$ We treat $N$ binary words $\uw_1, \uw_2, \dots, \uw_N$ of length $L$ as encodings of $\Lambda$-words $w_1,w_2,\dots,w_N$ of length $F_k$, if $L \neq { \left| F_k \right|} \left( \lfloor \log \card \Lambda \rfloor+1 \right)$ the algorithm terminates without producing output. Words $w_1,w_2,\dots,w_N$ form the *dictionary* that we will use to encode parts of the words. We require that $s = { \left| {\mathrm{int}_{F_k}(F_n)} \cap \calZ_k \right|}$ and $$l = { \left| F_n \setminus \Delta_{k,n} \right|} \left( \lfloor \log \card \Lambda \rfloor+1 \right),$$ the algorithm terminates without producing output if this does not hold. Otherwise, the binary word $\uv$ of length $l$ is seen as a binary encoding of the $\Lambda$-word $v$ of length ${ \left| F_n \setminus \Delta_{k,n} \right|}$. We will now compute a $\Lambda$-word $\omega$ defined on $F_n$. The set ${\mathrm{int}_{F_k}(F_n)} \cap \calZ_k$ is ordered as a subset of $\N$. For $j$-th element $g_j \in {\mathrm{int}_{F_k}(F_n)} \cap \calZ_k$ we require that $\omega|_{F_k g_j} \circ \imath_{F_k g_j}^{-1} = w_{i_j}$, where $j=1,2,\dots,s$. That is, we require that the restriction of $\omega$ to the subset $F_k g_j$ coincides with $i_j$-th element of the dictionary for every $j$. It is clear that this determines the restriction $\omega|_{\Delta_{k,n}}$, and it remains to describe $\omega|_{F_n \setminus \Delta_{k,n}}$. We require that $\omega|_{F_n \setminus \Delta_{k,n}} \circ \imath_{F_n \setminus \Delta_{k,n}}^{-1} = v$. The decompressor ${A^!}$ prints the $\Lambda$-word $\omega \circ \imath_{F_n}^{-1}$. Fix $k \geq 1$ and $\varepsilon>0$. Let $n_0$ be such that for all $n \geq n_0$ we have $$\frac{{ \left| F_n \setminus \Delta_{k,n} \right|}}{{ \left| F_n \right|}} \leq \varepsilon.$$ Let $\omega \in \calF_{\Lambda}(F_n)$. We use the following program to encode $\omega$. We let $N := \card \calF_{\Lambda}(F_k)$, $L:={ \left| F_k \right|} \left( \lfloor \log \card \Lambda \rfloor+1 \right)$ and $w_1,w_2,\dots,w_N$ be the list of words $\upsilon \circ \imath_{F_k}^{-1}$ for $\upsilon \in \calF_{\Lambda}(F_k)$ (say, in lexicographic order). For every $g_j \in {\mathrm{int}_{F_k}(F_n)} \cap \calZ_k$ and every $x \in F_k$ note that $$\omega|_{F_k g_j} (x g_j) = (g_j \cdot \omega)|_{F_k}(x),$$ where $g_j \cdot \omega \in \prX$ by invariance. Hence we let $i_j$ be the index of the word $(g_j \cdot \omega)|_{F_k} \circ \imath_{F_k}^{-1}$ in the dictionary $w_1,w_2,\dots,w_N$ for every $j=1,2,\dots, { \left| {\mathrm{int}_{F_k}(F_n)} \cap \calZ_k \right|}$. Finally, we let $v$ be the remainder $\omega|_{F_n \setminus \Delta_{k,n}} \circ \imath_{F_n \setminus \Delta_{k,n}}^{-1}$ and $l$ be the length of the binary encoding of the word $v$. It is clear that the program \[eq.prog\] with the parameters determined above does describe $\omega|_{F_n}$. It remains to estimate the length of $\up$. It is easy to see that $$\begin{aligned} l(\up) &\leq l(\widehat \uk) + l(\widehat \un)+l(\widehat{\uN})+ l(\widehat{\uL})+l(\widehat{\ul})+\card{\calF_{\Lambda}(F_k)} { \left| F_k \right|} \left( \lfloor \log \card \Lambda \rfloor+1 \right)+\\ &+{ \left| F_n \setminus \Delta_{k,n} \right|}\left( \lfloor \log \card \Lambda \rfloor+1 \right)+{ \left| I_{k,n} \right|}l(\widehat{\uN}),\end{aligned}$$ and taking the limit as $n \to \infty$ we see (using Proposition \[prop.fmonot\]) that $$\begin{aligned} \limsup\limits_{n \to \infty} & \max\limits_{\omega \in \calF_{\Lambda}(F_n)} {\overline{\uK}(\omega,F_n)} \leq \varepsilon \left( \lfloor \log \card \Lambda \rfloor+1 \right)+ \\ &+\frac{2 \lfloor \log(\lfloor \log \card \calF_{\Lambda}(F_k) \rfloor + 1) \rfloor + \lfloor \log \card \calF_{\Lambda}(F_k) \rfloor + 5}{{ \left| F_k \right|}}.\end{aligned}$$ Since $k, \varepsilon$ are arbitrary the conclusion follows. It is clear that the proof of Theorem \[thm.brudno\] now follows from Theorem \[thm.brudnogeq\] and Theorem \[thm.brudnoleq\]. [^1]: The author kindly acknowledges the support from ESA CICAT of TU Delft
--- author: - | [ Lili Hu , Chunhui Lai]{}\ [Department of Mathematics, Zhangzhou Teachers College,]{}\ [Zhangzhou, Fujian 363000, P. R. of CHINA.]{}\ [jackey2591924@163.com ( Lili Hu)]{}\ [zjlaichu@public.zzptt.fj.cn(Chunhui Lai, Corresponding author)]{} title: ' [**On Potentially $K_5-E_3$-graphic Sequences**]{} [^1]' --- 0.1in [**Abstract**]{} [ Let $K_m-H$ be the graph obtained from $K_m$ by removing the edges set $E(H)$ of $H$ where $H$ is a subgraph of $K_m$. In this paper, we characterize the potentially $K_5-P_3$, $K_5-A_3$, $K_5-K_3$ and $K_5-K_{1,3}$-graphic sequences where $A_3$ is $P_2\cup K_2$. Moreover, we also characterize the potentially $K_5-2K_2$-graphic sequences where $pK_2$ is the matching consisted of $p$ edges.]{} [**Key words:**]{} graph; degree sequence; potentially $K_5-H$-graphic sequences [**AMS Subject Classifications:**]{} 05C07 Introduction ============ We consider finite simple graphs. Any undefined notation follows that of Bondy and Murty $[1]$. The set of all non-increasing nonnegative integer sequence $\pi=(d_1,d_2,\cdots,d_n)$ is denoted by $NS_n$. A sequence $\pi\epsilon NS_n$ is said to be graphic if it is the degree sequence of a simple graph $G$ of order $n$; such a graph $G$ is referred as a realization of $\pi$. The set of all graphic sequence in $NS_n$ is denoted by $GS_n$. A graphic sequence $\pi$ is potentially $H$-graphic if there is a realization of $\pi$ containing $H$ as a subgraph. Let $C_k$ and $P_k$ denote a cycle on $k$ vertices and a path on $k+1$ vertices, respectively. Let $\sigma(\pi)$ the sum of all the terms of $\pi$ and let $A_3$ and $Z_4$ denote $P_2\cup K_2$ and $K_4-P_2$, respectively. We use the symbol $E_3$ to denote graphs on 5 vertices and 3 edges. A graphic sequence $\pi$ is said to be potentially $H$-graphic if it has a realization $G$ containing $H$ as a subgraph. Let $G-H$ denote the graph obtained from $G$ by removing the edges set $E(H)$ where $H$ is a subgraph of $G$. In the degree sequence, $r^t$ means $r$ repeats $t$ times, that is, in the realization of the sequence there are $t$ vertices of degree $r$. Given a graph $H$, what is the maximum number of edges of a graph with $n$ vertices not containing $H$ as a subgraph? This number is denoted $ex(n,H)$, and is known as the Turán number. In terms of graphic sequences, the number $2ex(n,H)+2$ is the minimum even integer $l$ such that every $n$-term graphical sequence $\pi$ with $\sigma (\pi)\geq l $ is forcibly $H$-graphical. Gould, Jacobson and Lehel \[4\] considered the following variation of the classical Turán-type extremal problems: determine the smallest even integer $\sigma(H,n)$ such that every n-term positive graphic sequence $\pi=(d_1,d_2,\cdots,d_n)$ with $\sigma(\pi)\geq \sigma(H,n)$ has a realization $G$ containing $H$ as a subgraph. They proved that $\sigma(pK_2, n)=(p-1)(2n-p)+2$ for $p\ge 2$; $\sigma(C_4, n)=2[{{3n-1}\over 2}]$ for $n\ge 4$. Erdös,Jacobson and Lehel \[3\] showed that $\sigma(K_k, n)\ge (k-2)(2n-k+1)+2$ and conjectured that the equality holds. In the same paper, they proved the conjecture is true for $k=3$ and $n\geq6$. The cases $k=4$ and 5 were proved separately (see \[4\] and \[17\], and \[18\]). For $k\geq6$ and $n\geq {k \choose 2} $+3, Li, Song and Luo \[19\] proved the conjecture true via linear algebraic techniques. Recently, Ferrara, Gould and Schmitt proved the conjecture $[5]$ and they also determined in $[6]$ $\sigma(F_k,n)$ where $F_k$ denotes the graph of $k$ triangles intersecting at exactly one common vertex. Yin, Li, and Mao \[25\] determined $\sigma(K_{r+1}-e,n)$ for $r\geq3$ and $r+1\leq n\leq 2r$ and $\sigma(K_5-e,n)$ for $n\geq5$, and Yin and Li \[24\] further determined $\sigma(K_{r+1}-e,n)$ for $r\geq2$ and $n\geq3r^2-r-1$. Moreover, Yin and Li in \[24\] also gave two sufficient conditions for a sequence $\pi\epsilon GS_n$ to be potentially $K_{r+1}-e$-graphic. Yin \[27\] determined $\sigma(K_{r+1}-K_3,n)$ for $r\geq3$ and $n\geq 3r+5$. Lai \[12-15\] determined $\sigma(K_4-e,n)$ for $n\geq4$ and $\sigma(K_5-C_4,n)$, $\sigma(K_5-P_3,n)$, $\sigma(K_5-P_4,n)$, $\sigma(K_5-K_3,n)$ for $n\geq5$. Lai \[10-11\] proved that $\sigma(C_{2m+1}, n)=m(2n-m-1)+2$, for $m\geq 2, n\geq 3m$; $\sigma(C_{2m+2} , n)=m(2n-m-1)+4$, for $ m\geq 2, n\geq 5m-2$. Lai and Hu \[16\] determined $\sigma(K_{r+1}-H,n)$ for $n\geq4r+10$, $r\geq3$, $r+1\geq k\geq4$ and $H$ be a graph on $k$ vertices which containing a tree on 4 vertices but not contain a cycle on 3 vertices and $\sigma(K_{r+1}-P_2,n)$ for $n\geq4r+8$, $r\geq3$. A harder question is to characterize the potentially $H$-graphic sequences without zero terms. Luo \[21\] characterized the potentially $C_k$-graphic sequences for each $k=3,4,5$. Recently, Luo and Warner \[22\] characterized the potentially $K_4$-graphic sequences. Eschen and Niu \[23\] characterized the potentially $K_4-e$-graphic sequences. Yin and Chen \[26\] characterized the potentially $K_{r,s}$-graphic sequences for $r=2,s=3$ and $r=2,s=4$. Chen \[2\] characterized the potentially $K_5-2K_2$-graphic sequences for $5\leq n\leq8$. Hu and Lai \[7-8\] characterized the potentially $K_5-C_4$ and $K_5-Z_4$-graphic sequences. In this paper, we completely characterize the potentially $K_5-E_3$ - graphic sequences, that is potentially $K_5-P_3$, $K_5-A_3$, $K_5-K_3$ and $K_5-K_{1,3}$-graphic sequences. Moreover, we also characterize the potentially $K_5-2K_2$-graphic sequences. Preparations ============ Let $\pi=(d_1,\cdots,d_n)\epsilon NS_n,1\leq k\leq n$. Let $$\pi_k^{\prime\prime}=\left\{ \begin{array}{ll}(d_1-1,\cdots,d_{k-1}-1,d_{k+1}-1, \cdots,d_{d_k+1}-1,d_{d_k+2},\cdots,d_n), \\ \mbox{ if $d_k\geq k,$}\\ (d_1-1,\cdots,d_{d_k}-1,d_{d_k+1},\cdots,d_{k-1},d_{k+1},\cdots,d_n), \\ \mbox{if $d_k < k.$} \end{array} \right.$$ Denote $\pi_k^\prime=(d_1^\prime,d_2^\prime,\cdots,d_{n-1}^\prime)$, where $d_1^\prime\geq d_2^\prime\geq\cdots\geq d_{n-1}^\prime$ is a rearrangement of the $n-1$ terms of $\pi_k^{\prime\prime}$. Then $\pi_k^{\prime}$ is called the residual sequence obtained by laying off $d_k$ from $\pi$. In this paper, we denote $\pi_n^\prime$ by $\pi^\prime$. For a nonincreasing positive integer sequence $\pi=(d_1,d_2,\cdots,d_n)$, we write $m(\pi)$ and $h(\pi)$ to denote the largest positive terms of $\pi$ and the smallest positive terms of $\pi$, respectively. We need the following results. [**Theorem 2.1 \[4\]**]{} If $\pi=(d_1,d_2,\cdots,d_n)$ is a graphic sequence with a realization $G$ containing $H$ as a subgraph, then there exists a realization $G^\prime$ of $\pi$ containing $H$ as a subgraph so that the vertices of $H$ have the largest degrees of $\pi$. [**Theorem 2.2 \[20\]**]{} If $\pi=(d_1,d_2,\cdots,d_n)$ is a sequence of nonnegative integers with $1\leq m(\pi)\leq2$, $h(\pi)=1$ and even $\sigma(\pi)$, then $\pi$ is graphic. [**Theorem 2.3 \[21\]**]{} Let $\pi=(d_1,d_2,\cdots,d_n)$ be a graphic sequence. Then $\pi$ is potentially $C_4$-graphic if and only if the following conditions hold: (1) $d_4\geq2$; (2) $d_1=n-1$ implies $d_2\geq3$; (3) If $n=5,6$, then $\pi\neq(2^n)$. [**Lemma 2.4 \[2\]**]{} Let $\pi=(d_1,d_2,\cdots,d_n)\epsilon NS_n$, $1\leq j\leq n-5$, $0\leq k\leq [{{n-j-i-4}\over 2}]$. Let $$\pi = \left\{ \begin{array}{ll}(n-i,n-j,3^{n-i-j-2k}, 2^{2k},1^{i+j-2}) \\ \mbox{ $n-i-j$ is even;}\\ (n-i,n-j,3^{n-i-j-2k-1}, 2^{2k+1},1^{i+j-2}) \\ \mbox{ $n-i-j$ is odd.} \end{array} \right.$$ Let $S_1$ be the set consisting of the above sequences and let $S_2$ be the set of the following sequences: $(n-1,3^5,1^{n-6})$ and $(n-1,3^6,1^{n-7})$. If $\pi \epsilon S_1$ or $\pi \epsilon S_2$, then $\pi$ is not potentially $K_{1,2,2}$-graphic. [**Lemma 2.5 \[8\]**]{} If $\pi=(d_1,d_2,\cdots,d_n)$ is a nonincreasing sequence of positive integers with even $\sigma(\pi)$, $n\geq4$, $d_1\leq3$ and $\pi\neq(3^3,1),(3^2,1^2)$, then $\pi$ is graphic. [**Lemma 2.6 (Kleitman and Wang \[9\])**]{}   $\pi$ is graphic if and only if $\pi^\prime$ is graphic. The following corollary is obvious. [**Corollary 2.7**]{}   Let $H$ be a simple graph. If $\pi^\prime$ is potentially $H$-graphic, then $\pi$ is potentially $H$-graphic. Main Theorems ============== **Theorem 3.1** Let $\pi=(d_1,d_2,\cdots,d_n)$ be a graphic sequence with $n\geq5$. Then $\pi$ is potentially $K_5-P_3$-graphic if and only if the following conditions hold: $(1)$ $d_1\geq4$, $d_3\geq3$ and $d_5\geq2$. $(2)$ $\pi\neq (4,3^2,2^3)$, $(4,3^2,2^4)$ and $(4,3^6)$. [**Proof:**]{} Assume that $\pi$ is potentially $K_5-P_3$-graphic. $(1)$ and $(2)$ are obvious. To prove the sufficiency, we use induction on $n$. Suppose the graphic sequence $\pi$ satisfies the conditions (1) and (2). We first prove the base case where $n=5$. In this case, $\pi$ is one of the following: $(4^5)$, $(4^3,3^2)$, $(4^2,3^2,2)$, $(4,3^4)$, $(4,3^2,2^2)$. It is easy to check that all of these are potentially $K_5-P_3$-graphic. Now we assume that the sufficiency holds for $n-1(n\geq6)$, we will show that $\pi$ is potentially $K_5-P_3$-graphic in terms of the following cases: **Case 1:** $d_n\geq4$. Clearly, $\pi^\prime$ satisfies $(1)$ and $(2)$, then by the induction hypothesis, $\pi^\prime$ is potentially $K_5-P_3$-graphic, and hence so is $\pi$. **Case 2:** $d_n=3$. Consider $\pi^\prime=(d_1^\prime,d_2^\prime,\cdots,d_{n-1}^\prime)$ where $d_{n-3}^\prime\geq3$ and $d_{n-1}^\prime\geq2$. If $\pi^\prime$ satisfies $(1)$ and $(2)$, then by the induction hypothesis, $\pi^\prime$ is potentially $K_5-P_3$-graphic, and hence so is $\pi$. If $\pi^\prime$ does not satisfy $(1)$, i.e., $d_1^\prime=3$, then $\pi^\prime=(3^k,2^{n-1-k})$ where $n-3\leq k \leq n-1$. Since $\sigma(\pi^\prime)$ is even, $k$ must be even. If $k=n-3$, then $\pi=(4,3^{n-1})$ where $n$ is odd. Since $\pi\neq(4,3^6)$, we have $n\geq9$. By Lemma 2.5, $\pi_1=(3^{n-5})$ is graphic. Let $G_1$ be a realization of $\pi_1$, then $K_{1,2,2}\cup G_1$ is a realization of $\pi=(4,3^{n-1})$. Thus, $\pi=(4,3^{n-1})$ is potentially $K_5-P_3$-graphic since $K_5-P_3\subseteq K_{1,2,2}$. If $k=n-2$, then $\pi=(4^2,3^{n-2})$ where $n$ is even. It is easy to see that $\pi=(4^2,3^4)$ and $\pi=(4^2,3^6)$ are potentially $K_5-P_3$-graphic. Let $G_2$ be a realization of $(4^2,3^4)$, which contains $K_5-P_3$. If $n\geq10$, then $\pi_2=(3^{n-6})$ is graphic by Lemma 2.5. Let $G_3$ be a realization of $\pi_2$, then $G_2\cup G_3$ is a realization of $\pi=(4^2,3^{n-2})$. In other words, $\pi=(4^2,3^{n-2})$ is potentially $K_5-P_3$-graphic. If $k=n-1$, then $\pi=(4^3,3^{n-3})$ where $n$ is odd. It is easy to see that $\pi=(4^3,3^4)$ is potentially $K_5-P_3$-graphic. If $n\geq9$, then $K_5-e\cup G_1$ is a realization of $\pi=(4^3,3^{n-3})$. Thus, $\pi=(4^3,3^{n-3})$ is potentially $K_5-P_3$-graphic since $K_5-P_3\subseteq K_5-e$. If $\pi^\prime$ does not satisfy $(2)$, then $\pi^\prime$ is just $(4,3^6)$, and hence $\pi=(5,4^2,3^5)$ or $(4^4,3^4)$. It is easy to see that these sequences are potentially $K_5-P_3$-graphic. **Case 3:** $d_n=2$. Consider $\pi^\prime=(d_1^\prime,d_2^\prime,\cdots,d_{n-1}^\prime)$ where $d_2^\prime\geq3$ and $d_{n-1}^\prime\geq2$. If $\pi^\prime$ satisfies $(1)$ and $(2)$, then by the induction hypothesis, $\pi^\prime$ is potentially $K_5-P_3$-graphic, and hence so is $\pi$. If $\pi^\prime$ does not satisfy $(1)$, there are two subcases: **Subcase 1:** $d_1^\prime\geq4$ and $d_3^\prime=2$. Then $\pi=(d_1,3^2,2^{n-3})$ where $d_1\geq 5$. Since $\sigma(\pi)$ is even, $d_1$ must be even. We will show that $\pi$ is potentially $K_5-P_3$-graphic. It is enough to show $\pi_1=(d_1-4,2^{n-5})$ is graphic. It clearly suffices to show $\pi_2=(2^{n-1-d_1},1^{d_1-4})$ is graphic. By $\sigma(\pi_2)$ being even and Theorem 2.2, $\pi_2$ is graphic. **Subcase 2:** $d_1^\prime=3$. Then $d_1=4$, $d_3=3$, $d_2=4$ or $d_2=3$. If $d_2=4$, then $\pi=(4^2,3^k,2^{n-2-k})$ where $k\geq1$ and $n-2-k\geq1$. Since $\sigma(\pi)$ is even, $k$ must be even. We will show that $\pi$ is potentially $K_5-P_3$-graphic. First, we consider $\pi=(4^2,3^2,2^{n-4})$. It is enough to show $\pi_1=(2^{n-5},1^2)$ is graphic. By $\sigma(\pi_1)$ being even and Theorem 2.2, $\pi_1$ is graphic. Then we consider $\pi=(4^2,3^k,2^{n-2-k})$ where $k\geq4$. It is easy to see that $(4^2,3^4)$ is potentially $K_5-P_3$-graphic. Let $G_1$ be a realization of $(4^2,3^4)$, which contains $K_5-P_3$. If $n\geq10$, then $\pi_2=(3^{k-4},2^{n-2-k})$ is graphic by Lemma 2.5. Let $G_2$ be a realization of $\pi_2$, then $G_1\cup G_2$ is a realization of $\pi=(4^2,3^k,2^{n-2-k})$. If $n\leq9$, then $\pi$ is one of the following: $(4^2,3^4,2)$, $(4^2,3^4,2^2)$, $(4^2,3^4,2^3)$, $(4^2,3^6,2)$. It is easy to check that all of these are potentially $K_5-P_3$-graphic. In other words, $\pi=(4^2,3^k,2^{n-2-k})$ is potentially $K_5-P_3$-graphic. If $d_2=3$, then $\pi=(4,3^k,2^{n-1-k})$ where $k\geq2$ and $n-1-k\geq1$. Since $\sigma(\pi)$ is even, $k$ must be even. We will show that $\pi$ is potentially $K_5-P_3$-graphic. First, we consider $\pi=(4,3^2,2^{n-3})$. Since $\pi\neq(4,3^2,2^3)$ and $(4,3^2,2^4)$, we have $n\geq8$. It is enough to show $\pi_1=(2^{n-5})$ is graphic. Clearly, $C_{n-5}$ is a realization of $\pi_1$. Second, we consider $\pi=(4,3^4,2^{n-5})$. It is enough to show $\pi_2=(2^{n-5},1^2)$ is graphic. By $\sigma(\pi_2)$ being even and Theorem 2.2, $\pi_2$ is graphic. Then we consider $\pi=(4,3^k,2^{n-1-k})$ where $k\geq6$. If $n\geq9$, then $\pi_3=(3^{k-4},2^{n-1-k})$ is graphic by Lemma 2.5. Let $G_1$ be a realization of $\pi_3$, then $K_{1,2,2}\cup G_1$ is a realization of $\pi=(4,3^k,2^{n-1-k})$. Hence, $\pi=(4,3^k,2^{n-1-k})$ is potentially $K_5-P_3$-graphic since $K_5-P_3\subseteq K_{1,2,2}$. If $n\leq8$, then $\pi=(4,3^6,2)$. It is easy to see that $\pi$ is potentially $K_5-P_3$-graphic. In other words, $\pi=(4,3^k,2^{n-1-k})$ is potentially $K_5-P_3$-graphic. If $\pi^\prime$ does not satisfy $(2)$, then $\pi^\prime$ is one of the following: $(4,3^2,2^3)$, $(4,3^2,2^4)$, $(4,3^6)$. Hence $\pi$ is one of the following: $(5,4,3,2^4)$, $(5,3^3,2^3)$, $(4^3,2^4)$, $(5,4,3,2^5)$, $(5,3^3,2^4)$, $(4^3,2^5)$, $(5,4,3^5,2)$, $(4^3,3^4,2)$. It is easy to check that all of these are potentially $K_5-P_3$-graphic. **Case 4:** $d_n=1$. Consider $\pi^\prime=(d_1^\prime,d_2^\prime,\cdots,d_{n-1}^\prime)$ where $d_3^\prime\geq3$ and $d_5^\prime\geq2$. If $\pi^\prime$ satisfies $(1)$ and $(2)$, then by the induction hypothesis, $\pi^\prime$ is potentially $K_5-P_3$-graphic, and hence so is $\pi$. If $\pi^\prime$ does not satisfy $(1)$, i.e., $d_1^\prime=3$, then $\pi=(4,3^k,2^t,1^{n-1-k-t})$ where $k\geq2$, $k+t\geq4$ and $n-1-k-t\geq1$. Since $\sigma(\pi)$ is even, $n-1-t$ must be even. We will show that $\pi$ is potentially $K_5-P_3$-graphic. First, we consider $\pi=(4,3^2,2^t,1^{n-3-t})$. It is enough to show $\pi_1=(2^{t-2},1^{n-3-t})$ is graphic. By $\sigma(\pi_1)$ being even and Theorem 2.2, $\pi_1$ is graphic. Second, we consider $\pi=(4,3^3,2^t,1^{n-4-t})$. It is enough to show $\pi_2=(2^{t-1},1^{n-3-t})$ is graphic. By $\sigma(\pi_2)$ being even and Theorem 2.2, $\pi_2$ is graphic. Third, we consider $\pi=(4,3^4,2^t,1^{n-5-t})$. It is enough to show $\pi_3=(2^t,1^{n-3-t})$ is graphic. By $\sigma(\pi_3)$ being even and Theorem 2.2, $\pi_3$ is graphic. Then we consider $\pi=(4,3^k,2^t,1^{n-1-k-t})$ where $k\geq5$. Let $\pi_4=(3^{k-4},2^t,1^{n-1-k-t})$. If $n\geq9$ and $\pi_4\neq(3^3,1)$ or $(3^2,1^2)$, then $\pi_4$ is graphic by Lemma 2.5. Let $G_1$ be a realization of $\pi_4$, then $K_{1,2,2}\cup G_1$ is a realization of $\pi=(4,3^k,2^t,1^{n-1-k-t})$. Hence, $\pi=(4,3^k,2^t,1^{n-1-k-t})$ is potentially $K_5-P_3$-graphic since $K_5-P_3\subseteq K_{1,2,2}$. If $n=9$ and $\pi_4=(3^3,1)$ or $(3^2,1^2)$, then $\pi=(4,3^7,1)$ or $(4,3^6,1^2)$. If $n\leq8$, then $\pi=(4,3^5,1)$ or $(4,3^5,2,1)$. It is easy to check that all of these are potentially $K_5-P_3$-graphic. In other words, $\pi=(4,3^k,2^t,1^{n-1-k-t})$ is potentially $K_5-P_3$-graphic. If $\pi^\prime$ does not satisfy $(2)$, then $\pi^\prime$ is one of the following: $(4,3^2,2^3)$, $(4,3^2,2^4)$, $(4,3^6)$. Hence $\pi$ is one of the following: $(5,3^2,2^3,1)$, $(4^2,3,2^3,1)$, $(5,3^2,2^4,1)$, $(4^2,3,2^4,1)$, $(5,3^6,1)$, $(4^2,3^5,1)$. It is easy to check that all of these are potentially $K_5-P_3$-graphic. **Theorem 3.2** Let $\pi=(d_1,d_2,\cdots,d_n)$ be a graphic sequence with $n\geq5$. Then $\pi$ is potentially $K_5-A_3$-graphic if and only if the following conditions hold: $(1)$ $d_4\geq3$ and $d_5\geq2$. $(2)$ $\pi\neq(n-1,3^3,2^{n-k},1^{k-4})$ where $n\geq6$ and $k=4,5,\cdots,n-2$, $n$ and $k$ have the same parity. $(3)$ $\pi\neq (3^4,2^2),(3^6),(3^4,2^3),(3^6,2),(4,3^6),(3^7,1),(3^8),(n-1,3^5,1^{n-6})$ and $(n-1,3^6,1^{n-7})$. [**Proof:**]{} First we show the conditions (1)-(3) are necessary conditions for $\pi$ to be potentially $K_5-A_3$-graphic. Assume that $\pi$ is potentially $K_5-A_3$-graphic. $(1)$ is obvious. If $\pi=(n-1,3^3,2^{n-k},1^{k-4})$ is potentially $K_5-A_3$-graphic, then according to Theorem 2.1, there exists a realization $G$ of $\pi$ containing $K_5-A_3$ as a subgraph so that the vertices of $K_5-A_3$ have the largest degrees of $\pi$. Therefore, the sequence $\pi^*=(n-4,2^{n-1-k},1^{k-4})$ obtained from $G-(K_5-A_3)$ must be graphic, which is impossible since $G-(K_5-A_3)$ has only $n-4$ vertices, $\triangle(G-(K_5-A_3))\leq n-5$. Hence, $(2)$ holds. Now it is easy to check that $(3^4,2^2),(3^6),(3^4,2^3),(3^6,2),(4,3^6),(3^7,1)$ and $(3^8)$ are not potentially $K_5-A_3$-graphic. If $\pi=(n-1,3^5,1^{n-6})$ is potentially $K_5-A_3$-graphic, then according to Theorem 2.1, there exists a realization $G$ of $\pi$ containing $K_5-A_3$ as a subgraph so that the vertices of $K_5-A_3$ have the largest degrees of $\pi$. Therefore, the sequence $\pi^*=(n-4,3,1^{n-5})$ obtained from $G-(K_5-A_3)$ must be graphic. It follows that the sequence $\pi_1=(2)$ must be graphic, a contradiction. Hence, $\pi\neq(n-1,3^5,1^{n-6})$. If $\pi=(n-1,3^6,1^{n-7})$ is potentially $K_5-A_3$-graphic, then according to Theorem 2.1, there exists a realization $G$ of $\pi$ containing $K_5-A_3$ as a subgraph so that the vertices of $K_5-A_3$ have the largest degrees of $\pi$. Therefore, the sequence $\pi^*=(n-4,3^2,1^{n-6})$ obtained from $G-(K_5-A_3)$ must be graphic. It follows that the sequence $\pi_2=(2^2)$ must be graphic, a contradiction. Hence, $\pi\neq(n-1,3^6,1^{n-7})$. In other words, $(3)$ holds. Now we turn to show the conditions (1)-(3) are sufficient conditions for $\pi$ to be potentially $K_5-A_3$-graphic. Suppose the graphic sequence $\pi$ satisfies the conditions (1)-(3). Our proof is by induction on $n$. We first prove the base case where $n=5$. In this case, $\pi$ is one of the following: $(4^5)$, $(4^3,3^2)$, $(4^2,3^2,2)$, $(4,3^4)$, $(3^4,2)$. It is easy to check that all of these are potentially $K_5-A_3$-graphic. Now suppose that the sufficiency holds for $n-1(n\geq6)$, we will show that $\pi$ is potentially $K_5-A_3$-graphic in terms of the following cases: **Case 1:** $d_n\geq3$. Clearly, $\pi^\prime$ satisfies $(1)$. If $\pi^\prime$ also satisfies $(2)$ and $(3)$, then by the induction hypothesis, $\pi^\prime$ is potentially $K_5-A_3$-graphic, and hence so is $\pi$. If $\pi^\prime$ does not satisfy $(2)$, then $\pi^\prime$ is just $(5,3^3,2^2)$, and hence $\pi=(6,3^6)$ which is impossible by $(3)$. If $\pi^\prime$ does not satisfy $(3)$, since $\pi\neq(4,3^6)$ and $(3^8)$, then $\pi^\prime$ is only one of the following: $(3^6),(3^6,2),(4,3^6)$, $(3^8)$, $(5,3^5)$, $(6,3^6)$. Hence, $\pi$ is one of the following: $(4^3,3^4),(4^2,3^6),(5,4^2,3^5),(4^4,3^4), (4^3,3^6)$, $(6,4^2,3^4)$, $(7,4^2,3^5)$. It is easy to check that all of these are potentially $K_5-A_3$-graphic. **Case 2:** $d_n=2$. Consider $\pi^\prime=(d_1^\prime,d_2^\prime,\cdots,d_{n-1}^\prime)$ where $d_2^\prime\geq3$ and $d_{n-1}^\prime\geq2$. If $\pi^\prime$ satisfies $(1)$-$(3)$, then by the induction hypothesis, $\pi^\prime$ is potentially $K_5-A_3$-graphic, and hence so is $\pi$. If $\pi^\prime$ does not satisfy $(1)$, then $d_4^\prime=2$. Hence $\pi=(d_1,3^3,2^{n-4})$. Since $\sigma(\pi)$ is even, $d_1$ must be odd. We will show that $\pi$ is potentially $K_5-A_3$-graphic. If $d_1=3$, then $\pi=(3^4,2^{n-4})$. Since $\pi\neq(3^4,2^2)$ and $(3^4,2^3)$, we have $n\geq8$. It is enough to show $\pi_1=(2^{n-5})$ is graphic. Clearly, $C_{n-5}$ is a realization of $\pi_1$. If $d_1\geq5$, since $\pi\neq(n-1,3^3,2^{n-4})$, we have $d_1\leq n-2$. It is enough to show $\pi_2=(d_1-3,2^{n-5})$ is graphic. It clearly suffices to show $\pi_3=(2^{n-2-d_1},1^{d_1-3})$ is graphic. By $\sigma(\pi_3)$ being even and Theorem 2.2, $\pi_3$ is graphic. Thus, $\pi=(d_1,3^3,2^{n-4})$ is potentially $K_5-A_3$-graphic. If $\pi^\prime$ does not satisfy $(2)$, i.e., $\pi^\prime=(n-2,3^3,2^{n-5})$. Since $\sigma(\pi^\prime)$ is even, $n$ must be odd. Hence $\pi=(n-1,4,3^2,2^{n-4})$ or $(n-1,3^4,2^{n-5})$. We will show that both of them are potentially $K_5-A_3$-graphic. It is enough to show $\pi_1=(n-4,2^{n-5},1)$ is graphic. It clearly suffices to show $\pi_2=(1^{n-5})$ is graphic. By $\sigma(\pi_2)$ being even and Theorem 2.2, $\pi_2$ is graphic. If $\pi^\prime$ does not satisfy $(3)$, then $\pi^\prime$ is one of the following: $(3^4,2^2)$, $(3^6)$, $(3^4,2^3)$, $(3^6,2)$, $(4,3^6)$, $(3^8)$, $(5,3^5)$, $(6,3^6)$. Since $\pi\neq(3^6,2)$, then $\pi$ is one of the following: $(4^2,3^2,2^3)$, $(4,3^4,2^2)$, $(4^2,3^4,2)$, $(4^2,3^2,2^4)$, $(4,3^4,2^3)$, $(3^6,2^2)$, $(4^2,3^4,2^2)$, $(4,3^6,2)$, $(5,4,3^5,2)$, $(4^3,3^4,2)$, $(4^2,3^6,2)$, $(6,4,3^4,2)$, $(7,4,3^5,2)$. It is easy to check that all of these are potentially $K_5-A_3$-graphic. **Case 3:** $d_n=1$. Consider $\pi^\prime=(d_1^\prime,d_2^\prime,\cdots,d_{n-1}^\prime)$ where $d_3^\prime\geq3$ and $d_5^\prime\geq2$. If $\pi^\prime$ satisfies $(1)$-$(3)$, then by the induction hypothesis, $\pi^\prime$ is potentially $K_5-A_3$-graphic, and hence so is $\pi$. If $\pi^\prime$ does not satisfy $(1)$, then $d_4^\prime=2$. Hence $\pi=(3^4,2^k,1^{n-4-k})$ where $k\geq1$ and $n-4-k\geq1$. Since $\sigma(\pi)$ is even, $n-4-k$ must be even. We will show that $\pi$ is potentially $K_5-A_3$-graphic. It is enough to show $\pi_1=(2^{k-1},1^{n-4-k})$ is graphic. By $\sigma(\pi_1)$ being even and Theorem 2.2, $\pi_1$ is graphic. If $\pi^\prime$ does not satisfy $(2)$, i.e., $\pi^\prime=(n-2,3^3,2^{n-1-k},1^{k-4})$. Hence $\pi=(n-1,3^3,2^{n-1-k},1^{k-3})$ which contradicts condition $(2)$. If $\pi^\prime$ does not satisfy $(3)$, then by $\pi\neq(n-1,3^5,1^{n-6})$ and $(n-1,3^6,1^{n-7})$, $\pi^\prime$ is only one of the following: $(3^4,2^2)$, $(3^6)$, $(3^4,2^3)$, $(3^6,2)$, $(4,3^6)$, $(3^7,1)$, $(3^8)$. Since $\pi\neq(3^7,1)$, then $\pi$ is one of the following: $(4,3^3,2^2,1)$, $(3^5,2,1)$, $(4,3^5,1)$, $(4,3^3,2^3,1)$, $(3^5,2^2,1)$, $(4,3^5,2,1)$, $(5,3^6,1)$, $(4^2,3^5,1)$, $(4,3^6,1^2)$, $(4,3^7,1)$. It is easy to check that all of these are potentially $K_5-A_3$-graphic. **Theorem 3.3** Let $\pi=(d_1,d_2,\cdots,d_n)$ be a graphic sequence with $n\geq5$. Then $\pi$ is potentially $K_5-K_3$-graphic if and only if the following conditions hold: $(1)$ $d_2\geq4$ and $d_5\geq2$. $(2)$ $\pi\neq (4^2,2^4)$, $(4^2,2^5)$, $(4^3,2^3)$ and $(4^6)$. [**Proof:**]{} Assume that $\pi$ is potentially $K_5-K_3$-graphic. $(1)$ and $(2)$ are obvious. To prove the sufficiency, we use induction on $n$. Suppose the graphic sequence $\pi$ satisfies the conditions (1) and (2). We first prove the base case where $n=5$. In this case, $\pi$ is one of the following: $(4^5)$, $(4^3,3^2)$, $(4^2,3^2,2)$, $(4^2,2^3)$. It is easy to check that all of these are potentially $K_5-K_3$-graphic. Now suppose that the sufficiency holds for $n-1(n\geq6)$, we will show that $\pi$ is potentially $K_5-K_3$-graphic in terms of the following cases: **Case 1:** $d_n\geq4$. Clearly, $\pi^\prime$ satisfies $(1)$. If $\pi^\prime$ also satisfies $(2)$, then by the induction hypothesis, $\pi^\prime$ is potentially $K_5-K_3$-graphic, and hence so is $\pi$. If $\pi^\prime$ does not satisfy $(2)$, then $\pi^\prime$ is just $(4^6)$, and hence $\pi=(5^4,4^3)$. It is easy to see that $\pi$ is potentially $K_5-K_3$-graphic. **Case 2:** $d_n=3$. Consider $\pi^\prime=(d_1^\prime,d_2^\prime,\cdots,d_{n-1}^\prime)$ where $d_{n-2}^\prime\geq3$ and $d_{n-1}^\prime\geq2$. If $\pi^\prime$ satisfies $(1)$ and $(2)$, then by the induction hypothesis, $\pi^\prime$ is potentially $K_5-K_3$-graphic, and hence so is $\pi$. If $\pi^\prime$ does not satisfy $(1)$, i.e., $d_2^\prime=3$, then $d_2=4$ and $3\leq d_4\leq d_3\leq4$. There are three subcases: **Subcase 1:** $d_4=4$. Then $\pi=(4^4,3^{n-4})$. Since $\sigma(\pi)$ is even, $n$ must be even. We will show that $\pi$ is potentially $K_5-K_3$-graphic. It is easy to see that $(4^4,3^2)$ and $(4^4,3^4)$ are potentially $K_5-K_3$-graphic. Let $G_1$ be a realization of $(4^4,3^2)$, which contains $K_5-K_3$. If $n\geq10$, then $\pi_1=(3^{n-6})$ is graphic by Lemma 2.5. Let $G_2$ be a realization of $\pi_1$, then $G_1\cup G_2$ is a realization of $\pi=(4^4,3^{n-4})$. In other words, $\pi=(4^4,3^{n-4})$ is potentially $K_5-K_3$-graphic. **Subcase 2:** $d_4=3$ and $d_3=4$. Then $\pi=(d_1,4^2,3^{n-3})$. Since $\sigma(\pi)$ is even, $d_1$ and $n$ have different parities. We will show that $\pi$ is potentially $K_5-K_3$-graphic. It is enough to show $\pi_1=(d_1-4,3^{n-5},2,1^2)$ is graphic and the vertex with degree $d_1-4$ is not adjacent to the vertices with degree 2 or 1 in the realization of $\pi_1$. Hence, it suffices to show $\pi_2=(3^{n-1-d_1},2^{d_1-3},1^2)$ is graphic. By Lemma 2.5, $\pi_2$ is graphic. Thus, $\pi=(d_1,4^2,3^{n-3})$ is potentially $K_5-K_3$-graphic. **Subcase 3:** $d_3=3$. then $\pi=(d_1,4,3^{n-2})$. Since $\sigma(\pi)$ is even, $d_1$ and $n$ have the same parity. We will show that $\pi$ is potentially $K_5-K_3$-graphic. It is enough to show $\pi_1=(d_1-4,3^{n-5},1^3)$ is graphic and the vertex with degree $d_1-4$ is not adjacent to the vertices with degree 1 in the realization of $\pi_1$. Hence, it suffices to show $\pi_2=(3^{n-1-d_1},2^{d_1-4},1^3)$ is graphic. By Lemma 2.5, $\pi_2$ is graphic. If $\pi^\prime$ does not satisfy $(2)$, then $\pi^\prime$ is just $(4^6)$, and hence $\pi=(5^3,4^3,3)$. It is easy to check that $\pi$ is potentially $K_5-K_3$-graphic. **Case 3:** $d_n=2$. Consider $\pi^\prime=(d_1^\prime,d_2^\prime,\cdots,d_{n-1}^\prime)$ where $d_2^\prime\geq3$ and $d_{n-1}^\prime\geq2$. If $\pi^\prime$ satisfies $(1)$ and $(2)$, then by the induction hypothesis, $\pi^\prime$ is potentially $K_5-K_3$-graphic, and hence so is $\pi$. If $\pi^\prime$ does not satisfy $(1)$, i.e., $d_2^\prime=3$, then $d_2=4$. There are two subcases: $d_1=4$ and $d_1\geq5$. **Subcase 1:** $d_1=4$. If $d_3=4$, then $\pi=(4^3,3^k,2^{n-3-k})$ where $n-3-k\geq1$. Since $\sigma(\pi)$ is even, $k$ must be even. We will show that $\pi$ is potentially $K_5-K_3$-graphic. First, we consider $\pi=(4^3,2^{n-3})$. Since $\pi\neq(4^3,2^3)$, we have $n\geq7$. It is enough to show $\pi_1=(2^{n-4})$ is graphic. Clearly, $C_{n-4}$ is a realization of $\pi_1$. Second, we consider $\pi=(4^3,3^2,2^{n-5})$. It is easy to see that $\pi=(4^3,3^2,2)$ and $\pi=(4^3,3^2,2^2)$ are potentially $K_5-K_3$-graphic. If $n\geq8$, then $K_5-e \cup C_{n-5}$ is a realization of $\pi=(4^3,3^2,2^{n-5})$. Thus, $\pi=(4^3,3^2,2^{n-5})$ is potentially $K_5-K_3$-graphic since $K_5-K_3\subseteq K_5-e$. Then we consider $\pi=(4^3,3^k,2^{n-3-k})$ where $k\geq4$. If $n\geq9$, then $\pi_1=(3^{k-2},2^{n-3-k})$ is graphic by Lemma 2.5. Let $G_1$ be a realization of $\pi_1$, then $K_5-e \cup G_1$ is a realization of $\pi=(4^3,3^k,2^{n-3-k})$. Thus, $\pi$ is potentially $K_5-K_3$-graphic since $K_5-K_3\subseteq K_5-e$. If $n\leq8$, then $\pi=(4^3,3^4,2)$. It is easy to see that $(4^3,3^4,2)$ is potentially $K_5-K_3$-graphic. In other words, $\pi=(4^3,3^k,2^{n-3-k})$ is potentially $K_5-K_3$-graphic. If $d_3\leq3$, then $\pi=(4^2,3^k,2^{n-2-k})$ where $n-2-k\geq1$. Since $\sigma(\pi)$ is even, $k$ must be even. We will show that $\pi$ is potentially $K_5-K_3$-graphic. First, we consider $\pi=(4^2,2^{n-2})$. Since $\pi\neq(4^2,2^4)$ and $(4^2,2^5)$, we have $n\geq8$. It is enough to show $\pi_1=(2^{n-5})$ is graphic. Clearly, $C_{n-5}$ is a realization of $\pi_1$. Second, we consider $\pi=(4^2,3^2,2^{n-4})$. It is enough to show $\pi_2=(2^{n-5},1^2)$ is graphic. By $\sigma(\pi_2)$ being even and Theorem 2.2, $\pi_2$ is graphic. Then we consider $\pi=(4^2,3^k,2^{n-2-k})$ where $k\geq4$. It is easy to check that $\pi=(4^2,3^4)$ is potentially $K_5-K_3$-graphic. Let $G_1$ be a realization of $(4^2,3^4)$, which contains $K_5-K_3$. If $n\geq10$, then $\pi_3=(3^{k-4},2^{n-2-k})$ is graphic by Lemma 2.5. Let $G_2$ be a realization of $\pi_3$, then $G_1\cup G_2$ is a realization of $\pi=(4^2,3^k,2^{n-2-k})$. If $n\leq9$, then $\pi$ is one of the following: $(4^2,3^4,2)$, $(4^2,3^4,2^2)$, $(4^2,3^4,2^3)$, $(4^2,3^6,2)$. It is easy to check that all of these are potentially $K_5-K_3$-graphic. In other words, $\pi=(4^2,3^k,2^{n-2-k})$ is potentially $K_5-K_3$-graphic. **Subcase 2:** $d_1\geq5$. Then $\pi=(d_1,4,3^k,2^{n-2-k})$ where $n-2-k\geq1$. Since $\sigma(\pi)$ is even, $d_1$ and $k$ have the same parity. We will show that $\pi$ is potentially $K_5-K_3$-graphic. First, we consider $\pi=(d_1,4,2^{n-2})$. It is enough to show $\pi_1=(d_1-4,2^{n-5})$ is graphic. It clearly suffices to show $\pi_2=(2^{n-1-d_1},1^{d_1-4})$ is graphic. By $\sigma(\pi_2)$ being even and Theorem 2.2, $\pi_2$ is graphic. Second, we consider $\pi=(d_1,4,3,2^{n-3})$. It is enough to show $\pi_1=(d_1-4,2^{n-5},1)$ is graphic and there exists no edge between two vertices with degree $d_1-4$ and $1$ in the realization of $\pi_1$. Hence, it suffices to show $\pi_2=(2^{n-1-d_1},1^{d_1-3})$ is graphic. By $\sigma(\pi_2)$ being even and Theorem 2.2, $\pi_2$ is graphic. Third, we consider $\pi=(d_1,4,3^2,2^{n-4})$. It is enough to show $\pi_1=(d_1-4,2^{n-5},1^2)$ is graphic and the vertex with degree $d_1-4$ is not adjacent to the vertices with degree $1$ in the realization of $\pi_1$. Hence, it suffices to show $\pi_2=(2^{n-1-d_1},1^{d_1-2})$ is graphic. By $\sigma(\pi_2)$ being even and Theorem 2.2, $\pi_2$ is graphic. Fourth, we consider $\pi=(d_1,4,3^3,2^{n-5})$. It is enough to show $\pi_1=(d_1-4,2^{n-5},1^3)$ is graphic and the vertex with degree $d_1-4$ is not adjacent to the vertices with degree $1$ in the realization of $\pi_1$. Hence, it suffices to show $\pi_2=(2^{n-1-d_1},1^{d_1-1})$ is graphic. By $\sigma(\pi_2)$ being even and Theorem 2.2, $\pi_2$ is graphic. Then we consider $\pi=(d_1,4,3^k,2^{n-2-k})$ where $k\geq4$. It is enough to show $\pi_1=(d_1-4,3^{k-3},2^{n-2-k},1^3)$ is graphic and the vertex with degree $d_1-4$ is not adjacent to the vertices with degree $1$ in the realization of $\pi_1$. Assume that the vertex with degree $d_1-4$ is adjacent to $t$$(t\leq k-3)$ vertices with degree $3$ and $d_1-4-t$ vertices with degree $2$ in the realization of $\pi_1$. Hence, it suffices to show $\pi_2=(3^{k-3-t},2^{n+2-d_1-k+2t},1^{d_1-1-t})$ is graphic. By Lemma 2.5, $\pi_2$ is graphic. Thus, $\pi=(d_1,4,3^k,2^{n-2-k})$ is potentially $K_5-K_3$-graphic. If $\pi^\prime$ does not satisfy $(2)$, then $\pi^\prime$ is one of the following: $(4^2,2^4)$, $(4^2,2^5)$, $(4^3,2^3)$, $(4^6)$. Hence $\pi$ is one of the following: $(5^2,2^5)$, $(5^2,2^6)$, $(5^2,4,2^4)$, $(5^2,4^4,2)$. It is easy to check that all of these are potentially $K_5-K_3$-graphic. **Case 4:** $d_n=1$. Consider $\pi^\prime=(d_1^\prime,d_2^\prime,\cdots,d_{n-1}^\prime)$ where $d_1^\prime\geq4$, $d_2^\prime\geq3$ and $d_5^\prime\geq2$. If $\pi^\prime$ satisfies $(1)$ and $(2)$, then by the induction hypothesis, $\pi^\prime$ is potentially $K_5-K_3$-graphic, and hence so is $\pi$. If $\pi^\prime$ does not satisfy $(1)$, i.e., $d_2^\prime=3$, then $\pi=(4^2,3^k,2^t,1^{n-2-k-t})$ where $k+t\geq3$ and $n-2-k-t\geq1$. Since $\sigma(\pi)$ is even, $n-2-t$ must be even. We will show that $\pi$ is potentially $K_5-K_3$-graphic. First, we consider $\pi=(4^2,2^t,1^{n-2-t})$. It is enough to show $\pi_1=(2^{t-3},1^{n-2-t})$ is graphic. By $\sigma(\pi_1)$ being even and Theorem 2.2, $\pi_1$ is graphic. Second, we consider $\pi=(4^2,3,2^t,1^{n-3-t})$. It is enough to show $\pi_1=(2^{t-2},1^{n-2-t})$ is graphic. By $\sigma(\pi_1)$ being even and Theorem 2.2, $\pi_1$ is graphic. Third, we consider $\pi=(4^2,3^2,2^t,1^{n-4-t})$. It is enough to show $\pi_1=(2^{t-1},1^{n-2-t})$ is graphic. By $\sigma(\pi_1)$ being even and Theorem 2.2, $\pi_1$ is graphic. Fourth, we consider $\pi=(4^2,3^3,2^t,1^{n-5-t})$. It is enough to show $\pi_1=(2^t,1^{n-2-t})$ is graphic. By $\sigma(\pi_1)$ being even and Theorem 2.2, $\pi_1$ is graphic. Then we consider $\pi=(4^2,3^k,2^t,1^{n-2-k-t})$ where $k\geq4$ and $n-2-k-t\geq1$. It is easy to see that $\pi=(4^2,3^4)$ is potentially $K_5-K_3$-graphic. Let $G_1$ be a realization of $(4^2,3^4)$, which contains $K_5-K_3$. Let $\pi_1=(3^{k-4},2^t,1^{n-2-k-t})$. If $n\geq10$ and $\pi_1\neq(3^3,1)$, $(3^2,1^2)$, then $\pi_1$ is graphic by Lemma 2.5. Let $G_2$ be a realization of $\pi_1$, then $G_1 \cup G_2$ is a realization of $\pi=(4^2,3^k,2^t,1^{n-2-k-t})$. If $n=10$ and $\pi_1=(3^3,1)$ or $(3^2,1^2)$, then $\pi=(4^2,3^7,1)$ or $(4^2,3^6,1^2)$. If $n\leq9$, then $\pi=(4^2,3^4,1^2)$, $(4^2,3^4,2,1^2)$, $(4^2,3^5,1)$ or $(4^2,3^5,2,1)$. It is easy to check that all of these are potentially $K_5-K_3$-graphic. In other words, $\pi=(4^2,3^k,2^t,1^{n-2-k-t})$ is potentially $K_5-K_3$-graphic. If $\pi^\prime$ does not satisfy $(2)$, then $\pi^\prime$ is one of the following: $(4^2,2^4)$, $(4^2,2^5)$, $(4^3,2^3)$, $(4^6)$. Hence $\pi$ is one of the following: $(5,4,2^4,1)$, $(5,4,2^5,1)$, $(5,4^2,2^3,1)$, $(5,4^5,1)$. It is easy to check that all of these are potentially $K_5-K_3$-graphic. **Theorem 3.4** Let $\pi=(d_1,d_2,\cdots,d_n)$ be a graphic sequence with $n\geq5$. Then $\pi$ is potentially $K_5-K_{1,3}$-graphic if and only if the following conditions hold: $(1)$ $d_1\geq4$ and $d_4\geq3$. $(2)$ $\pi\neq (4,3^4,2)$, $(4^6)$, $(4^2,3^4)$, $(4,3^6)$, $(4^7)$, $(4,3^5,1)$, $(n-1,3^4,1^{n-5})$ and $(n-1,3^5,1^{n-6})$. [**Proof:**]{} Assume that $\pi$ is potentially $K_5-K_{1,3}$-graphic. $(1)$ is obvious. Now it is easy to check that $(4,3^4,2)$, $(4^6)$, $(4^2,3^4)$, $(4,3^6)$, $(4^7)$, $(4,3^5,1)$ are not potentially $K_5-K_{1,3}$-graphic. If $\pi=(n-1,3^4,1^{n-5})$ is potentially $K_5-K_{1,3}$-graphic, then according to Theorem 2.1, there exists a realization $G$ of $\pi$ containing $K_5-K_{1,3}$ as a subgraph so that the vertices of $K_5-K_{1,3}$ have the largest degrees of $\pi$. Therefore, the sequence $\pi^*=(n-5,2,1^{n-5})$ obtained from $G-(K_5-K_{1,3})$ must be graphic and there must be no edge between two vertices with degree $n-5$ and $2$ in the realization of $\pi^*$. Thus, $\pi^*$ satisfies: $(n-5)+2\leq n-5$, a contradiction. Hence, $\pi\neq(n-1,3^4,1^{n-5})$. If $\pi=(n-1,3^5,1^{n-6})$ is potentially $K_5-K_{1,3}$-graphic, then according to Theorem 2.1, there exists a realization $G$ of $\pi$ containing $K_5-K_{1,3}$ as a subgraph so that the vertices of $K_5-K_{1,3}$ have the largest degrees of $\pi$. Therefore, the sequence $\pi^*=(n-5,3,2,1^{n-6})$ obtained from $G-(K_5-K_{1,3})$ must be graphic and there must be no edge between two vertices with degree $n-5$ and $2$ in the realization of $\pi^*$. It follows that the sequence $\pi_1=(2^2)$ must be graphic, a contradiction. Hence, $\pi\neq(n-1,3^5,1^{n-6})$. In other words, $(2)$ holds. Now we prove the sufficient conditions. Suppose the graphic sequence $\pi$ satisfies the conditions $(1)$ and $(2)$. Our proof is by induction on $n$. We first prove the base case where $n=5$. Since $\pi\neq(4,3^4)$, then $\pi$ is one of the following: $(4^5)$, $(4^3,3^2)$, $(4^2,3^2,2)$, $(4,3^3,1)$. It is easy to check that all of these are potentially $K_5-K_{1,3}$-graphic. Now suppose that the sufficiency holds for $n-1(n\geq6)$, we will show that $\pi$ is potentially $K_5-K_{1,3}$-graphic in terms of the following cases: **Case 1:** $d_n\geq4$. Clearly, $\pi^\prime$ satisfies $(1)$. If $\pi^\prime$ also satisfies $(2)$, then by the induction hypothesis, $\pi^\prime$ is potentially $K_5-K_{1,3}$-graphic, and hence so is $\pi$. If $\pi^\prime$ does not satisfy $(2)$, since $\pi\neq(4^6)$ and $(4^7)$, then $\pi^\prime$ is just $(4^6)$ or $(4^7)$, and hence $\pi=(5^4,4^3)$ or $(5^4,4^4)$. It is easy to check that these sequences are potentially $K_5-K_{1,3}$-graphic. **Case 2:** $d_n=3$. Consider $\pi^\prime=(d_1^\prime,d_2^\prime,\cdots,d_{n-1}^\prime)$ where $d_{n-3}^\prime\geq3$ and $d_{n-1}^\prime\geq2$. If $\pi^\prime$ satisfies $(1)$ and $(2)$, then by the induction hypothesis, $\pi^\prime$ is potentially $K_5-K_{1,3}$-graphic, and hence so is $\pi$. If $\pi^\prime$ does not satisfy $(1)$, there are two subcases: **Subcase 1:** $d_1^\prime\geq4$ and $d_4^\prime=2$. Then $\pi^\prime=(4,3^2,2^2)$, and hence $\pi=(5,3^5)$ which contradicts condition $(2)$. **Subcase 2:** $d_1^\prime=3$. Then $\pi^\prime=(3^k,2^{n-1-k})$ where $n-3\leq k\leq n-1$. Since $\sigma(\pi^\prime)$ is even, $k$ must be even. If $n$ is odd, then $k=n-3$ or $n-1$. If $k=n-3$, then $\pi=(4,3^{n-1})$. Since $\pi\neq(4,3^6)$, we have $n\geq9$. It is easy to check that $(4,3^8)$ and $(4,3^{10})$ are potentially $K_5-K_{1,3}$-graphic. Let $G_1$ be a realization of $(4,3^8)$, which contains $K_5-K_{1,3}$. If $n\geq13$, then $\pi_1=(3^{n-9})$ is graphic by Lemma 2.5. Let $G_2$ be a realization of $\pi_1$, then $G_1\cup G_2$ is a realization of $\pi=(4,3^{n-1})$. In other words, $\pi=(4,3^{n-1})$ is potentially $K_5-K_{1,3}$-graphic. If $k=n-1$, then $\pi=(4^3,3^{n-3})$. It is easy to see that $\pi=(4^3,3^4)$ is potentially $K_5-K_{1,3}$-graphic. If $n\geq9$, then $\pi_2=(3^{n-5})$ is graphic by Lemma 2.5. Let $G_3$ be a realization of $\pi_2$, then $K_5-e\cup G_3$ is a realization of $\pi=(4^3,3^{n-3})$. Hence, $\pi=(4^3,3^{n-3})$ is potentially $K_5-K_{1,3}$-graphic since $K_5-K_{1,3}\subseteq K_5-e$. If $n$ is even, then $k=n-2$, thus $\pi=(4^2,3^{n-2})$. Since $\pi\neq (4^2,3^4)$, we have $n\geq8$. It is easy to see that $(4^2,3^6)$ and $(4^2,3^8)$ are potentially $K_5-K_{1,3}$-graphic. Let $G_4$ be a realization of $(4^2,3^6)$, which contains $K_5-K_{1,3}$. If $n\geq12$, then $\pi_3=(3^{n-8})$ is graphic by Lemma 2.5. Let $G_5$ be a realization of $\pi_3$, then $G_4\cup G_5$ is a realization of $\pi=(4^2,3^{n-2})$. In other words, $\pi=(4^2,3^{n-2})$ is potentially $K_5-K_{1,3}$-graphic. If $\pi^\prime$ does not satisfy $(2)$, then $\pi^\prime$ is one of the following: $(4,3^4,2)$, $(4^6)$, $(4^2,3^4)$, $(4,3^6)$, $(4^7)$, $(4,3^4)$, $(5,3^5)$. Hence $\pi$ is one of the following: $(5,4,3^5)$, $(5^3,4^3,3)$, $(5^2,4,3^4)$, $(5,4^3,3^3)$, $(4^5,3^2)$, $(5,4^2,3^5)$, $(4^4,3^4)$, $(5^3,4^4,3)$, $(5,4^2,3^3)$, $(4^4,3^2)$, $(6,4^2,3^4)$. It is easy to check that all of these are potentially $K_5-K_{1,3}$-graphic. **Case 3:** $d_n=2$. Consider $\pi^\prime=(d_1^\prime,d_2^\prime,\cdots,d_{n-1}^\prime)$ where $d_3^\prime\geq3$ and $d_{n-1}^\prime\geq2$. If $\pi^\prime$ satisfies $(1)$ and $(2)$, then by the induction hypothesis, $\pi^\prime$ is potentially $K_5-K_{1,3}$-graphic, and hence so is $\pi$. If $\pi^\prime$ does not satisfy $(1)$, there are two subcases: **Subcase 1:** $d_1^\prime\geq4$ and $d_4^\prime=2$. Then $\pi=(d_1,3^3,2^{n-4})$ where $d_1\geq5$. Since $\sigma(\pi)$ is even, $d_1$ must be odd. We will show that $\pi$ is potentially $K_5-K_{1,3}$-graphic. It is enough to show $\pi_1=(d_1-4,2^{n-5},1)$ is graphic and there exists no edge between two vertices with degree $d_1-4$ and $1$ in the realization of $\pi_1$. Hence, it suffices to show $\pi_2=(2^{n-1-d_1},1^{d_1-3})$ is graphic. By $\sigma(\pi_2)$ being even and Theorem 2.2, $\pi_2$ is graphic. **Subcase 2:** $d_1^\prime=3$. Then $d_1=4$, $d_3=d_4=3$, $d_2=4$ or $d_2=3$. If $d_2=4$, then $\pi=(4^2,3^k,2^{n-2-k})$ where $k\geq2$ and $n-2-k\geq1$. Since $\sigma(\pi)$ is even, $k$ must be even. We will show that $\pi$ is potentially $K_5-K_{1,3}$-graphic. First, we consider $\pi=(4^2,3^2,2^{n-4})$. It is enough to show $\pi_1=(2^{n-5},1^2)$ is graphic. By $\sigma(\pi_1)$ being even and Theorem 2.2, $\pi_1$ is graphic. Second, we consider $\pi=(4^2,3^4,2^{n-6})$. It is easy to see that $(4^2,3^4,2)$, $(4^2,3^4,2^2)$ and $(4^2,3^4,2^3)$ are potentially $K_5-K_{1,3}$-graphic. Let $G_1$ be a realization of $(4^2,3^4,2)$, which contains $K_5-K_{1,3}$. If $n\geq10$, then $G_1\cup C_{n-7}$ is a realization of $\pi=(4^2,3^4,2^{n-6})$. In other words, $\pi=(4^2,3^4,2^{n-6})$ is potentially $K_5-K_{1,3}$-graphic. Then we consider $\pi=(4^2,3^k,2^{n-2-k})$ where $k\geq6$. It is easy to see that $\pi=(4^2,3^6)$ is potentially $K_5-K_{1,3}$-graphic. Let $G_2$ be a realization of $(4^2,3^6)$, which contains $K_5-K_{1,3}$. If $n\geq12$, then $\pi_2=(3^{k-6},2^{n-2-k})$ is graphic by Lemma 2.5. Let $G_3$ be a realization of $\pi_2$, then $G_2\cup G_3$ is a realization of $\pi=(4^2,3^k,2^{n-2-k})$. If $n\leq11$, then $\pi$ is one of the following: $(4^2,3^6,2)$, $(4^2,3^6,2^2)$, $(4^2,3^6,2^3)$, $(4^2,3^8,2)$. It is easy to check that all of these are potentially $K_5-K_{1,3}$-graphic. In other words, $\pi=(4^2,3^k,2^{n-2-k})$ is potentially $K_5-K_{1,3}$-graphic. If $d_2=3$, then $\pi=(4,3^k,2^{n-1-k})$ where $k\geq3$ and $n-1-k\geq1$. Since $\sigma(\pi)$ is even, $k$ must be even. We will show that $\pi$ is potentially $K_5-K_{1,3}$-graphic. First, we consider $\pi=(4,3^4,2^{n-5})$. Since $\pi\neq(4,3^4,2)$, we have $n\geq7$. It is enough to show $\pi_1=(2^{n-4})$ is graphic. Clearly, $C_{n-4}$ is a realization of $\pi_1$. Second, we consider $\pi=(4,3^6,2^{n-7})$. It is easy to see that $(4,3^6,2)$, $(4,3^6,2^2)$ and $(4,3^6,2^3)$ are potentially $K_5-K_{1,3}$-graphic. Let $G_1$ be a realization of $(4,3^6,2)$, which contains $K_5-K_{1,3}$. If $n\geq11$, then $G_1\cup C_{n-8}$ is a realization of $\pi=(4,3^6,2^{n-7})$. In other words, $\pi=(4,3^6,2^{n-7})$ is potentially $K_5-K_{1,3}$-graphic. Then we consider $\pi=(4,3^k,2^{n-1-k})$ where $k\geq8$. It is easy to see that $\pi=(4^,3^8)$ is potentially $K_5-K_{1,3}$-graphic. Let $G_2$ be a realization of $(4,3^8)$, which contains $K_5-K_{1,3}$. If $n\geq13$, then $\pi_2=(3^{k-8},2^{n-1-k})$ is graphic by Lemma 2.5. Let $G_3$ be a realization of $\pi_2$, then $G_2\cup G_3$ is a realization of $\pi=(4,3^k,2^{n-1-k})$. If $n\leq12$, then $\pi$ is one of the following: $(4,3^8,2)$, $(4,3^8,2^2)$, $(4,3^8,2^3)$, $(4,3^{10},2)$. It is easy to check that all of these are potentially $K_5-K_{1,3}$-graphic. In other words, $\pi=(4,3^k,2^{n-1-k})$ is potentially $K_5-K_{1,3}$-graphic. If $\pi^\prime$ does not satisfy $(2)$, then $\pi^\prime$ is one of the following: $(4,3^4,2)$, $(4^6)$, $(4^2,3^4)$, $(4,3^6)$, $(4^7)$, $(4,3^4)$, $(5,3^5)$. Hence $\pi$ is one of the following: $(5,4,3^3,2^2)$, $(5,3^5,2)$, $(4^3,3^2,2^2)$, $(5^2,4^4,2)$, $(5^2,3^4,2)$, $(5,4^2,3^3,2)$, $(4^4,3^2,2)$, $(5,4,3^5,2)$, $(4^3,3^4,2)$, $(5^2,4^5,2)$, $(5,4,3^3,2)$, $(4^3,3^2,2)$, $(6,4,3^4,2)$. It is easy to check that all of these are potentially $K_5-K_{1,3}$-graphic. **Case 4:** $d_n=1$. Consider $\pi^\prime=(d_1^\prime,d_2^\prime,\cdots,d_{n-1}^\prime)$ where $d_4^\prime\geq3$. If $\pi^\prime$ satisfies $(1)$ and $(2)$, then by the induction hypothesis, $\pi^\prime$ is potentially $K_5-K_{1,3}$-graphic, and hence so is $\pi$. If $\pi^\prime$ does not satisfy $(1)$, i.e., $d_1^\prime=3$, then $\pi=(4,3^k,2^t,1^{n-1-k-t})$ where $k\geq3$ and $n-1-k-t\geq1$. Since $\sigma(\pi)$ is even, $n-1-t$ must be even. We will show that $\pi$ is potentially $K_5-K_{1,3}$-graphic. First, we consider $\pi=(4,3^3,2^t,1^{n-4-t})$. If $t=0$, it is enough to show $\pi_1=(1^{n-5})$ is graphic. By $\sigma(\pi_1)$ being even and Theorem 2.2, $\pi_1$ is graphic. If $t\geq1$, it is enough to show $\pi_2=(2^{t-1},1^{n-3-t})$ is graphic. By $\sigma(\pi_2)$ being even and Theorem 2.2, $\pi_2$ is graphic. Second, we consider $\pi=(4,3^4,2^t,1^{n-5-t})$. It is enough to show $\pi_1=(2^{t+1},1^{n-5-t})$ is graphic. By $\sigma(\pi_1)$ being even and Theorem 2.2, $\pi_1$ is graphic. Then we consider $\pi=(4,3^k,2^t,1^{n-1-k-t})$ where $k\geq5$. Since $\pi\neq(4,3^5,1)$, we have $n\geq8$. It is enough to show $\pi_1=(3^{k-4},2^{t+1},1^{n-1-k-t})$ is graphic. By Lemma 2.5, $\pi_1$ is graphic. If $\pi^\prime$ does not satisfy $(2)$, since $\pi\neq(n-1,3^4,1^{n-5})$ and $(n-1,3^5,1^{n-6})$, then $\pi^\prime$ is one of the following: $(4,3^4,2)$, $(4^6)$, $(4^2,3^4)$, $(4,3^6)$, $(4^7)$, $(4,3^5,1)$. Hence, $\pi$ is one of the following: $(5,3^4,2,1)$, $(4^2,3^3,2,1)$, $(5,4^5,1)$, $(5,4,3^4,1)$, $(4^3,3^3,1)$, $(5,3^6,1)$, $(4^2,3^5,1)$, $(5,4^6,1)$, $(5,3^5,1^2)$, $(4^2,3^4,1^2)$. It is easy to check that all of these are potentially $K_5-K_{1,3}$-graphic. **Theorem 3.5** Let $\pi=(d_1,d_2,\cdots,d_n)$ be a graphic sequence with $n\geq5$. Then $\pi$ is potentially $K_5-2K_2$-graphic if and only if the following conditions hold: $(1)$ $d_1\geq4$ and $d_5\geq3$; $(2)$ $$\pi \neq \left\{ \begin{array}{ll}(n-i,n-j,3^{n-i-j-2k}, 2^{2k},1^{i+j-2})\\ \mbox{ $n-i-j$ is even;}\\ (n-i,n-j,3^{n-i-j-2k-1}, 2^{2k+1},1^{i+j-2}) \\ \mbox{$n-i-j$ is odd.} \end{array} \right.$$      where $1\leq j\leq n-5$ and $0\leq k\leq [{{n-j-i-4}\over 2}]$. $(3)$ $\pi\neq (4^2,3^4)$, $(4,3^4,2)$, $(5,4,3^5)$, $(5,3^5,2)$, $(4^7)$, $(4^3,3^4)$, $(4^2,3^4,2)$, $(4,3^6)$, $(4,3^5,1)$,$(4,3^4,2^2)$, $(5,3^7)$, $(5,3^6,1)$, $(4^8)$, $(4^2,3^6)$, $(4^2,3^5,1)$, $(4,3^6,2)$, $(4,3^5,2,1)$, $(4,3^7,1)$, $(4,3^6,1^2)$, $(n-1,3^5,1^{n-6})$ and $(n-1,3^6,1^{n-7})$. [**Proof:**]{} Assume that $\pi$ is potentially $K_5-2K_2$-graphic. $(1)$ is obvious. According to Lemma 2.4, (2) holds. Now it is easy to check that $(4^2,3^4)$, $(4,3^4,2)$,   $(5,4,3^5)$, $(5,3^5,2)$,   $(4^7)$,   $(4^3,3^4)$,   $(4^2,3^4,2)$, $(4,3^6)$,   $(4,3^5,1)$,  $(4,3^4,2^2)$, $(5,3^7)$, $(5,3^6,1)$, $(4^8)$, $(4^2,3^6)$, $(4^2,3^5,1)$, $(4,3^6,2)$, $(4,3^5,2,1)$, $(4,3^7,1)$, $(4,3^6,1^2)$ are not potentially $K_5-2K_2$-graphic and by Lemma 2.4, $\pi\neq(n-1,3^5,1^{n-6})$ and $(n-1,3^6,1^{n-7})$. Hence, $(3)$ holds. Now we prove the sufficient conditions. Suppose the graphic sequence $\pi$ satisfies the conditions $(1)$-$(3)$. Our proof is by induction on $n$. We first prove the base case where $n=5$. In this case, $\pi$ is one of the following: $(4^5)$, $(4^3,3^2)$, $(4,3^4)$. It is easy to check that all of these are potentially $K_5-2K_2$-graphic. Now suppose that the sufficiency holds for $n-1(n\geq6)$, we will show that $\pi$ is potentially $K_5-2K_2$-graphic in terms of the following cases: **Case 1:** $d_n\geq4$. Clearly, $\pi^\prime=(d_1^\prime,d_2^\prime,\cdots,d_n^\prime)$ satisfies $(1)$ and (2). If $\pi^\prime$ also satisfies $(3)$, then by the induction hypothesis, $\pi^\prime$ is potentially $K_5-2K_2$-graphic, and hence so is $\pi$. If $\pi^\prime$ does not satisfy $(3)$, since $\pi\neq(4^7)$ and $(4^8)$, then $\pi^\prime$ is just $(4^7)$ or $(4^8)$, and hence $\pi=(5^4,4^4)$ or $(5^4,4^5)$. It is easy to check that these sequences are potentially $K_5-2K_2$-graphic. **Case 2:** $d_n=3$. Consider $\pi^\prime=(d_1^\prime,d_2^\prime,\cdots,d_{n-1}^\prime)$ where $d_{n-3}^\prime\geq3$ and $d_{n-1}^\prime\geq2$. If $\pi^\prime$ satisfies $(1)$-$(3)$, then by the induction hypothesis, $\pi^\prime$ is potentially $K_5-2K_2$-graphic, and hence so is $\pi$. If $\pi^\prime$ does not satisfy $(1)$, there are three subcases: **Subcase 1:** $d_1^\prime=d_5^\prime=3$. Then $\pi^\prime=(3^k,2^{n-1-k})$ where $n-3\leq k\leq n-1$. Since $\sigma(\pi^\prime)$ is even, $k$ must be even. If $k=n-3$, then $\pi=(4,3^{n-1})$ where $n$ is odd. Since $\pi\neq(4,3^6)$, we have $n\geq9$. By Lemma 2.5, $\pi_1=(3^{n-5})$ is graphic. Let $G_1$ be a realization of $\pi_1$, then $K_{1,2,2}\cup G_1$ is a realization of $\pi=(4,3^{n-1})$. In other words, $\pi=(4,3^{n-1})$ is potentially $K_5-2K_2$-graphic. If $k=n-2$, then $\pi=(4^2,3^{n-2})$ where $n$ is even. Since $\pi\neq(4^2,3^4)$ and $(4^2,3^6)$, we have $n\geq 10$. It is easy to see that $(4^2,3^8)$ and $(4^2,3^{10})$ are potentially $K_5-2K_2$-graphic. Let $G_2$ be a realization of $(4^2,3^8)$, which contains $K_5-2K_2$. If $n\geq14$, then $\pi_2=(3^{n-10})$ is graphic by Lemma 2.5. Let $G_3$ be a realization of $\pi_2$, then $G_2\cup G_3$ is a realization of $\pi=(4^2,3^{n-2})$. In other words, $\pi=(4^2,3^{n-2})$ is potentially $K_5-2K_2$-graphic. If $k=n-1$, then $\pi=(4^3,3^{n-3})$ where $n$ is odd. Since $\pi\neq(4^3,3^4)$, we have $n\geq9$. Clearly, $K_5-e\cup G_1$ is a realization of $\pi=(4^3,3^{n-3})$. Thus, $\pi=(4^3,3^{n-3})$ is potentially $K_5-2K_2$-graphic since $K_5-2K_2\subseteq K_5-e$. **Subcase 2:** $d_1^\prime\geq 4$ and $d_5^\prime=2$. Since $d_{n-3}^\prime \geq3$, we have $n=6$ or $n=7$. Then $\pi$ is $(5^2,3^4)$, $(5,3^5)$ or $(6,3^6)$, which is impossible by condition (2) and (3). **Subcase 3:** $d_1^\prime=3$ and $d_5^\prime=2$. Then $\pi=(4^2,3^4)$ or $(4,3^6)$, which is impossible by condition (3). If $\pi^\prime$ does not satisfy $(2)$, then $\pi^\prime=((n-2)^2,3^{n-3})$ or $((n-2)^2,3^{n-4},2)$. Hence, $\pi=((n-1)^2,4,3^{n-3})$ or $((n-1)^2,3^{n-2})$. But $\pi=((n-1)^2,3^{n-2})$ contradicts condition (2), thus $\pi=((n-1)^2,4,3^{n-3})$. Since $\pi_1^\prime=(n-2,3,2^{n-3})$  is   potentially   $C_4$-graphic   by   Theorem 2.3, thus  $\pi=((n-1)^2,4,3^{n-3})$ is potentially $K_5-2K_2$-graphic. If $\pi^\prime$ does not satisfy $(3)$, since $\pi\neq(5,4,3^5)$ and $(5,3^7)$, then $\pi^\prime$ is one of the following: $(4^2,3^4)$, $(5,4,3^5)$, $(5,3^5,2)$, $(4^7)$, $(4^3,3^4)$, $(4^2,3^4,2)$, $(4,3^6)$, $(5,3^7)$, $(4^8)$, $(4^2,3^6)$, $(4,3^6,2)$, $(5,3^5)$, $(6,3^6)$. Hence, $\pi$ is one of the following: $(5^2,4,3^4)$, $(5,4^3,3^3)$, $(4^5,3^2)$, $(6,5,4,3^5)$, $(6,4^3,3^4)$, $(6,4,3^6)$, $(5^3,4^4,3)$, $(5^3,3^5)$, $(5^2,4^2,3^4)$, $(5,4^4,3^3)$, $(4^6,3^2)$, $(5^2,3^6)$, $(5,4^2,3^5)$, $(4^4,3^4)$, $(6,4^2,3^6)$, $(5^3,4^5,3)$, $(5^2,4,3^6)$, $(5,4^3,3^5),$ $(4^5,3^4)$, $(5,4,3^7)$, $(6,4^2,3^4)$, $(7,4^2,3^5)$. It is easy to check that all of these are potentially $K_5-2K_2$-graphic. **Case 3:** $d_n=2$. Consider $\pi^\prime=(d_1^\prime,d_2^\prime,\cdots,d_{n-1}^\prime)$ where $d_4^\prime\geq3$ and $d_{n-1}^\prime\geq2$. If $\pi^\prime$ satisfies $(1)$-$(3)$, then by the induction hypothesis, $\pi^\prime$ is potentially $K_5-2K_2$-graphic, and hence so is $\pi$. If $\pi^\prime$ does not satisfy $(1)$, there are three subcases: **Subcase 1:** $d_1^\prime=d_5^\prime=3$. Then $d_1=4$, $d_3=d_4=d_5=3$ and $3\leq d_2\leq4$. If $d_2=4$, then $\pi=(4^2,3^k,2^{n-2-k})$ where $k\geq3$ and $n-2-k\geq1$. Since $\sigma(\pi)$ is even, $k$ must be even. We will show that $\pi$ is potentially $K_5-2K_2$-graphic. It is enough to show $\pi_1=(3^{k-3},2^{n-2-k},1)$ is graphic. If $n\geq8$, then $\pi_1$ is graphic by Lemma 2.5. If $n\leq7$, then $\pi=(4^2,3^4,2)$, which is impossible by (3). If $d_2=3$, then $\pi=(4,3^k,2^{n-1-k})$ where $k\geq6$, $n-1-k\geq1$ and $k$ is even. Since $\pi\neq(4,3^6,2)$, we have $n\geq9$. We will show that $\pi$ is potentially $K_5-2K_2$-graphic. It is enough to show $\pi_2=(3^{k-4},2^{n-1-k})$ is graphic. By Lemma 2.5, $\pi_2$ is graphic. **Subcase 2:** $d_1^\prime\geq 4$ and $d_5^\prime=2$. Then $d_1\geq5$, $d_2=d_3=d_4=d_5=3$ and $d_6=\cdots=d_{n-1}=2$. Hence, $\pi=(d_1,3^4,2^{n-5})$. Since $\sigma(\pi)$ is even, $d_1$ must be even. We will show that $\pi$ is potentially $K_5-2K_2$-graphic. It is enough to show $\pi_1=(d_1-4,2^{n-5})$ is graphic. It clearly suffices to show $\pi_2=(2^{n-1-d_1},1^{d_1-4})$ is graphic. By $\sigma(\pi_2)$ being even and Theorem 2.2, $\pi_2$ is graphic. **Subcase 3:** $d_1^\prime=3$ and $d_5^\prime=2$. Then $\pi=(4,3^4,2^{n-5})$. Since $\pi\neq(4,3^4,2)$ and $(4,3^4,2^2)$, we have $n\geq8$. Clearly, $K_{1,2,2}\cup C_{n-5}$ is a realization of $\pi$. In other words, $\pi$ is potentially $K_5-2K_2$-graphic. If $\pi^\prime$ does not satisfy $(2)$, i.e., $$\pi^\prime= \left\{ \begin{array}{ll}((n-2)^2,3^{n-3-2k}, 2^{2k}), \ \ \ \ \mbox{ $n$ is odd;}\\ ((n-2)^2,3^{n-4-2k}, 2^{2k+1}),\ \ \mbox{$n$ is even.} \end{array} \right.$$ If $n\geq7$, then $$\pi= \left\{ \begin{array}{ll}((n-1)^2,3^{n-3-2k}, 2^{2k+1}), \ \ \ \ \mbox{ $n$ is odd;}\\ ((n-1)^2,3^{n-4-2k}, 2^{2k+2}),\ \ \ \ \ \ \mbox{$n$ is even.} \end{array} \right.$$ which contradicts condition (2). If $n=6$, then $\pi^\prime=(4^2,3^2,2)$ and hence $\pi=(5^2,3^2,2^2)$ or $(4^4,2^2)$, which is impossible by (1). If $\pi^\prime$ does not satisfy $(3)$, then $\pi^\prime$ is one of the following: $(4^2,3^4)$, $(4,3^4,2)$, $(5,4,3^5)$, $(5,3^5,2)$, $(4^7)$, $(4^3,3^4)$, $(4^2,3^4,2)$, $(4,3^6)$, $(4,3^4,2^2)$, $(5,3^7)$, $(4^8)$, $(4^2,3^6)$, $(4,3^6,2)$, $(5,3^5)$, $(6,3^6)$. Since $\pi\neq(5,3^5,2)$, then $\pi$ is one of the following: $(5^2,3^4,2)$, $(5,4^2,3^3,2)$, $(4^4,3^2,2)$, $(5,4,3^3,2^2)$,   $(4^3,3^2,2)$,   $(6,5,3^5,2)$,  $(6,4^2,3^4,2)$, $(6,4,3^4,2^2)$,   $(6,3^6,2)$,   $(5^2,4^5,2)$, $(5^2,4,3^4,2)$, $(5,4^3,3^3,2)$, $(4^5,3^2,2)$, $(5^2,3^4,2^2)$, $(5,4^2,3^3,2^2)$, $(4^4,3^2,2^2)$, $(5,4,3^5,2)$, $(4^3,3^4,2)$,  $(5,4,3^3,2^3)$,   $(5,3^5,2^2)$, $(4^3,3^2,2^3)$,  $(6,4,3^6,2)$,  $(5^2,4^6,2)$, $(5^2,3^6,2)$,   $(5,4^2,3^5,2)$,   $(4^4,3^4,2)$, $(5,4,3^5,2^2)$,  $(5,3^7,2)$,  $(4^3,3^4,2^2)$, $(6,4,3^4,2)$, $(7,4,3^5,2)$. It is easy to check that all of these are potentially $K_5-2K_2$-graphic. **Case 4:** $d_n=1$. Consider $\pi^\prime=(d_1^\prime,d_2^\prime,\cdots,d_{n-1}^\prime)$ where $d_5^\prime\geq3$. If $\pi^\prime$ satisfies $(1)$-$(3)$, then by the induction hypothesis, $\pi^\prime$ is potentially $K_5-2K_2$-graphic, and hence so is $\pi$. If $\pi^\prime$ does not satisfy $(1)$, i.e., $d_1^\prime=3$, then $d_1=4$ and $d_2=\cdots=d_5=3$. Hence, $\pi=(4,3^k,2^t,1^{n-1-k-t})$ where $k\geq4$ and $n-1-k-t\geq1$. Since $\sigma(\pi)$ is even, $n-1-t$ must be even. We will show that $\pi$ is potentially $K_5-2K_2$-graphic. It is enough to show $\pi_1=(3^{k-4},2^t,1^{n-1-k-t})$ is graphic. Since $\pi\neq(4,3^7,1)$ and $(4,3^6,1^2)$, we have $\pi_1\neq(3^3,1)$ and $(3^2,1^2)$. If $n\geq9$, then $\pi_1$ is graphic by Lemma 2.5. If $n\leq8$, since $\pi\neq(4,3^5,1)$ and $(4,3^5,2,1)$, then $\pi=(4,3^4,1^2)$ or $(4,3^4,2,1^2)$. It is easy to see that $\pi$ is potentially $K_5-2K_2$-graphic. If $\pi^\prime$ does not satisfy $(2)$, i.e., $$\pi^\prime= \left\{ \begin{array}{ll}(n-1-i,n-1-j,3^{(n-1)-i-j-2k}, 2^{2k},1^{i+j-2}), \\ \mbox{ $n-1-i-j$ is even;}\\ (n-1-i,n-1-j,3^{(n-1)-i-j-2k-1}, 2^{2k+1},,1^{i+j-2}),\ \\ \mbox{$n-1-i-j$ is odd.} \end{array} \right.$$ where $1\leq j\leq (n-1)-5$ and $0\leq k\leq [{{(n-1)-j-i-4}\over 2}]$. If $n-i>n-j+1$ or $n-i=n-j$, then $$\pi= \left\{ \begin{array}{ll}(n-i,n-(j+1),3^{n-i-(j+1)-2k}, 2^{2k},1^{i+(j+1)-2}),\\ \mbox{ $n-i-(j+1)$ is even;}\\ (n-i,n-(j+1),3^{n-i-(j+1)-2k-1}, 2^{2k+1},1^{i+(j+1)-2}), \\\mbox{$n-i-(j+1)$ is odd.} \end{array} \right.$$ which contradicts condition (2). If $n-i=n-j+1$, i.e., $$\pi^\prime= \left\{ \begin{array}{ll}(n-1-i,n-2-i,3^{n-2i-2k-2}, 2^{2k},1^{2i-1}),\\ \mbox{ $n$ is even;}\\ (n-1-i,n-2-i,3^{n-2i-2k-3}, 2^{2k+1},1^{2i-1}),\ \ \\\mbox{$n$ is odd.} \end{array} \right.$$ Then $$\pi= \left\{ \begin{array}{ll}(n-i,n-i-2,3^{n-2i-2k-2}, 2^{2k},1^{2i}), \\ \mbox{ $n$ is even;}\\ (n-i,n-i-2,3^{n-2i-2k-3}, 2^{2k+1},1^{2i}),\ \\\mbox{$n$ is odd.} \end{array} \right.$$ or $$\pi= \left\{ \begin{array}{ll}((n-1-i)^2,3^{n-2i-2k-2}, 2^{2k},1^{2i}), \\ \mbox{ $n$ is even;}\\ ((n-1-i)^2,3^{n-2i-2k-3}, 2^{2k+1},1^{2i}),\\ \mbox{$n$ is odd.} \end{array} \right.$$ which contradicts condition (2). If $\pi^\prime$ does not satisfy $(3)$, since $\pi\neq(5,3^6,1)$, $(4^2,3^5,1)$, $(n-1,3^5,1^{n-6})$ and $(n-1,3^6,1^{n-7})$, then $\pi^\prime$ is one of the following: $(4^2,3^4)$, $(4,3^4,2)$, $(5,4,3^5)$, $(5,3^5,2)$, $(4^7)$, $(4^3,3^4)$, $(4^2,3^4,2)$, $(4,3^5,1)$, $(4,3^4,2^2)$, $(5,3^7)$, $(5,3^6,1)$, $(4^8)$, $(4^2,3^6)$, $(4^2,3^5,1)$, $(4,3^6,2)$, $(4,3^5,2,1)$, $(4,3^7,1)$, $(4,3^6,1^2)$. Hence, $\pi$ is one of the following: $(5,4,3^4,1)$, $(4^3,3^3,1)$, $(5,3^4,2,1)$, $(4^2,3^3,2,1)$, $(6,4,3^5,1)$, $(5^2,3^5,1)$, $(6,3^5,2,1)$, $(5,4^6,1)$, $(5,4^2,3^4,1)$, $(4^4,3^3,1)$, $(5,4,3^4,2,1)$, $(4^3,3^3,2,1)$, $(5,3^5,1^2)$, $(4^2,3^4,1^2)$, $(5,3^4,2^2,1)$, $(4^2,3^3,2^2,1)$, $(6,3^7,1)$, $(6,3^6,1^2)$, $(5,4^7,1)$, $(5,4,3^6,1)$, $(4^3,3^5,1)$, $(5,4,3^5,1^2)$, $(4^3,3^4,1^2)$, $(5,3^6,2,1)$, $(4^2,3^5,2,1)$, $(5,3^5,2,1^2)$, $(4^2,3^4,2,1^2)$, $(5,3^7,1^2)$, $(4^2,3^6,1^2)$, $(5,3^6,1^3)$, $(4^2,3^5,1^3)$. It is easy to check that all of these are potentially $K_5-2K_2$-graphic. Application ============= Using Theorem 3.1 and Theorem 3.3, we give simple proofs of the following theorems due to Lai: **Theorem 4.1** (Lai \[14\]) For $n\geq5$, $\sigma(K_5-P_3,n)=4n-4$. **Proof:** First we claim that for $n\geq5, \sigma(K_5-P_3,n)\geq4n-4$. It is enough to show that there exists $\pi_1$ with $\sigma(\pi_1)=4n-6$, such that $\pi_1$ is not potentially $K_5-P_3$-graphic. Take $\pi_1=((n-1)^2,2^{n-2})$, then $\sigma(\pi_1)=4n-6$, and it is easy to see that $\pi_1$ is not potentially $K_5-P_3$-graphic by Theorem 3.1. Now we show that if $\pi$ is an $n$-term $(n\geq5)$ graphical sequence with $\sigma(\pi)\geq4n-4$, then there exists a realization of $\pi$ containing $K_5-P_3$. Hence, it suffices to show that $\pi$ is potentially $K_5-P_3$-graphic. If $d_5=1$, then $\sigma(\pi)=d_1+d_2+d_3+d_4+(n-4)$ and $d_1+d_2+d_3+d_4\leq12+(n-4)=n+8$. Therefore, $\sigma(\pi)\leq2n+4<4n-4$, a contradiction. Thus, $d_5\geq2$. If $d_3\leq2$, then $\sigma(\pi)\leq d_1+d_2+2(n-2)\leq2(n-1)+2(n-2)=4n-6<4n-4$, a contradiction. Thus, $d_3\geq3$. If $d_1\leq3$, then $\sigma(\pi)\leq3n<4n-4$, a contradiction. Thus, $d_1\geq4$. Since $\sigma(\pi)\geq4n-4$, then $\pi$ is not one of the following: $(4,3^2,2^3)$, $(4,3^2,2^4)$, $(4,3^6)$. Thus, $\pi$ satisfies the conditions (1) and (2) in Theorem 3.1. Therefore, $\pi$ is potentially $K_5-P_3$-graphic. **Theorem 4.2** (Lai \[13\]) For $n\geq5$, $\sigma(K_5-C_4,n)=4n-4$. **Proof:** Obviously, for $n\geq5$, $\sigma(K_5-C_4,n)\leq \sigma(K_5-P_3,n)=4n-4$. Now we claim $\sigma(K_5-C_4,n) \geq 4n-4$ for $n \geq5$. We would like to show there exists $\pi_1$ with $\sigma(\pi_1)=4n-6$, such that $\pi_1$ is not potentially $K_5-C_4$-graphic. Let $\pi_1=((n-1)^2,2^{n-2})$. It is easy to see that $\sigma(\pi_1)=4n-6$ and the only realization of $\pi_1$ does not contain $K_5-C_4$. Thus, $\sigma(K_5-C_4,n)=4n-4$. **Theorem 4.3** (Lai \[10\], Luo\[21\]) $\sigma(C_5,n)=4n-4$ for $n\geq5$. **Proof:** Obviously, for $n\geq5$, $\sigma(K_5-C_5,n)\leq \sigma(K_5-P_3,n)=4n-4$$(K_5-C_5=C_5)$. Now we claim $\sigma(C_5,n) \geq 4n-4$ for $n \geq5$. We would like to show there exists $\pi_1$ with $\sigma(\pi_1)=4n-6$, such that $\pi_1$ is not potentially $C_5$-graphic. Let $\pi_1=((n-1)^2,2^{n-2})$. It is easy to see that $\sigma(\pi_1)=4n-6$ and the only realization of $\pi_1$ does not contain $C_5$. Thus, $\sigma(C_5,n)=4n-4$. **Theorem 4.4** (Lai \[15\]) For $n=5$ and $n\geq7$, $$\sigma(K_{3,1,1} ,n)=4n-2.$$ For $n=6$, if $\pi$ is a 6-term graphical sequence with $\sigma(\pi) \geq 22$, then either there is a realization of $\pi$ containing $K_{3,1,1}$ or $\pi=(4^{6})$. (Thus $\sigma(K_{3,1,1} ,6)=26$.) **Proof:** First we claim that for $n\geq5, \sigma(K_5-K_3,n)\geq4n-2(K_{3,1,1}=K_5-K_3)$. It is enough to show that there exists $\pi_1$ with $\sigma(\pi_1)=4n-4$, such that $\pi_1$ is not potentially $K_5-K_3$-graphic. Take $\pi_1=(n-1,3^{n-1})$, then $\sigma(\pi_1)=4n-4$, and it is easy to see that $\pi_1$ is not potentially $K_5-K_3$-graphic by Theorem 3.3. Now we show that if $\pi$ is an $n$-term $(n\geq5)$ graphical sequence with $\sigma(\pi)\geq4n-2$, then there exists a realization of $\pi$ containing $K_5-K_3$(unless $\pi=(4^6)$). Hence, it suffices to show that $\pi$ is potentially $K_5-K_3$-graphic. If $d_5=1$, then $\sigma(\pi)=d_1+d_2+d_3+d_4+(n-4)$ and $d_1+d_2+d_3+d_4\leq12+(n-4)=n+8$. Therefore, $\sigma(\pi)\leq2n+4<4n-2$, a contradiction. Thus, $d_5\geq2$. If $d_2\leq3$, then $\sigma(\pi)\leq d_1+3(n-1)\leq n-1+3(n-1)=4n-4<4n-2$, a contradiction. Thus, $d_2\geq4$. Since $\sigma(\pi)\geq4n-2$, then $\pi\neq(4^2,2^5)$. Hence, for $n=5$ and $n\geq7$, $\pi$ satisfies the conditions (1) and (2) in Theorem 3.3. Therefore, $\pi$ is potentially $K_5-K_3$-graphic. For $n=6$, since $\sigma(\pi)\geq4\times6-2=22$, then $\pi$ is not one of the following: $(4^2,2^4)$, $(4^3,2^3)$. Thus, by Theorem 3.3, either there is a realization of $\pi$ containing $K_{3,1,1}$ or $\pi=(4^{6})$. Acknowledgments {#acknowledgments .unnumbered} =============== The authors are grateful to the referee for his valuable comments and suggestions.  J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, The Macmillan Press Ltd., 1976.  Gang Chen, The characterization on potentially $K_{1,2,2}$-graphic sequences, Journal of Qingdao University of Science and Technology, 27(2006), 86-88.  P.Erdös, M.S. Jacobson and J. Lehel, Graphs realizing the same degree sequences and their respective clique numbers, in Graph Theory, Combinatorics and Application, Vol. 1(Y. Alavi et al., eds.), John Wiley and Sons, Inc., New York, 1991, 439-449.  R.J. Gould, M.S. Jacobson and J. Lehel, Potentially $G$-graphic degree sequences,in Combinatorics, Graph Theory and Algorithms,Vol. 2 (Y. 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Graph Theory, 29(1998), 63-72.  Jiong-Sheng Li and Zi-Xia Song, An extremal problem on the potentially $P_k$-graphic sequences, The International Symposium on Combinatorics and Applications, June 28-30, 1996 (W.Y.C. Chen et. al., eds.) Tianjin, Nankai University 1996, 269-276.  Jiong-sheng Li, Zi-Xia Song and Rong Luo, The Erdös-Jacobson-Lehel conjecture on potentially $P_k$-graphic sequence is true, Science in China(Series A), 41(5)(1998), 510-520.  Jiong-sheng Li and Jianhua Yin, A variation of an extremal theorem due to Woodall, Southeast Asian Bulletin of Math., 25(2001), 427-434.  Rong Luo, On potentially $C_k$-graphic sequences, Ars Combinatoria 64(2002), 301-318.  Rong Luo, Morgan Warner, On potentially $K_k$-graphic sequences, Ars Combin. 75(2005), 233-239.  Elaine M. Eschen and Jianbing Niu, On potentially $K_4-e$-graphic sequences, Australasian Journal of Combinatorics, 29(2004), 59-65.  Jianhua Yin and Jiongsheng Li, Two sufficient conditions for a graphic sequence to have a realization with prescribed clique size, Discrete Math.,301(2005) 218-227.  Jianhua Yin, Jiongsheng Li and Rui Mao, An extremal problem on the potentially $K_{r+1}-e$-graphic sequences, Ars Combinatoria 74(2005), 151-159.  Jianhua Yin and Gang Chen, On potentially $K_{r_1,r_2,\cdots,r_m}$-graphic sequences, Utilitas Mathematica 72(2007), 149-161.  Mengxiao Yin, The smallest degree sum that yields potentially $K_{r+1}-K_3$-graphic sequences, Acta Math. Appl. Sin. Engl. Ser. 22(2006), no. 3, 451-456. [^1]: Project Supported by NNSF of China(10271105), NSF of Fujian(Z0511034), Science and Technology Project of Fujian, Fujian Provincial Training Foundation for “Bai-Quan-Wan Talents Engineering” , Project of Fujian Education Department and Project of Zhangzhou Teachers College.
--- abstract: 'In this article, we present a discrete time modeling framework, in which the shape and dynamics of a Limit Order Book (LOB) arise endogenously from an equilibrium between multiple market participants (agents). We use the proposed modeling framework to analyze the effects of trading frequency on market liquidity in a very general setting. In particular, we demonstrate the dual effect of high trading frequency. On the one hand, the higher frequency increases market efficiency, if the agents choose to provide liquidity in equilibrium. On the other hand, it also makes markets more fragile, in the sense that the agents choose to provide liquidity in equilibrium only if they are market-neutral (i.e., their beliefs satisfy certain martingale property). Even a very small deviation from market-neutrality may cause the agents to stop providing liquidity, if the trading frequency is sufficiently high, which represents an endogenous liquidity crisis (aka flash crash) in the market. This framework enables us to provide more insight into how such a liquidity crisis unfolds, connecting it to the so-called adverse selection effect.' author: - | Roman Gayduk and Sergey Nadtochiy[^1] [^2] [^3]\ $\,\,\,\,$\ *University of Michigan* bibliography: - 'MFGLOB\_refs.bib' title: 'Liquidity Effects of Trading Frequency [^4]' --- [**Key words**]{}: liquidity, trading frequency, Limit Order Book, continuum-player games, conditional tails of Itô processes. Introduction ============ This paper is concerned with liquidity effects of trading frequency on an auction-style exchange, in which the participating agents can post limit or market orders. On the one hand, higher trading frequency provides more opportunities for the market participants to trade, hence, improving the liquidity of the market and increasing its efficiency. On the other hand, higher trading frequency also provides more opportunities for some participants to manipulate the price and disrupt the market liquidity. Such a manipulation creates a new type of risk, which reveals itself in unusually high price deviations, which cannot be explained by changes in the fundamental value of the asset. The most famous example of this phenomenon is the “flash crash” of $2010$. This example motivates the need for a comprehensive study of the tradeoff between the liquidity providing role of strategic players and the liquidity risk they generate, and its relation to trading frequency. The collective liquidity of the market is captured by the *Limit Order Book (LOB)*, which contains all the limit buy and sell orders. The goal of the present paper is two-fold. First, we develop a new framework for modeling market microstructure, in which the shape of the LOB, and its dynamics, arise *endogenously* from the interactions between the agents. Among the many advantages of such an approach is the possibility of modeling the market reaction to changes in the rules of the exchange: e.g., limited trading frequency, transaction tax, etc. The second, and most important, goal of the present work is to investigate the *liquidity effects* of *trading frequency*, using the proposed modeling framework. In particular, the main results of this paper (cf. the discussion in Section \[se:examples\], as well as Theorems \[le:main.zeroTermSpread\], \[thm:main.necessary\] and Corollary \[prop:main.smallspread\], in Section \[se:main\]) describe the dual effect of high trading frequency. On the one hand, if the agents choose to provide liquidity in equilibrium, higher trading frequency decreases the bid-ask spread and makes the expected profits of all market participants converge to the same (fundamental) value, thus, improving the market efficiency. On the other hand, higher trading frequency also makes the LOB more sensitive to the deviations of the agents’ attitudes from market-neutrality. It is, of course, clear that a strong bullish or bearish signal induces market participants to trade at a higher or lower price. However, the novelty of our observation is in the role that the trading frequency plays in amplifying this effect. Namely, we show that, if the trading frequency is high, even if agents have plenty of inventory, a very small deviation from market-neutrality may cause them to stop providing liquidity, by either withdrawing from the market completely, or by posting limit orders far away from the fundamental price. Such actions cause disproportional deviations in the LOB, which cannot be explained by any fundamental reasons: they are much higher than the trading signal (i.e., the expected change in the fundamental price), and they occur without any shortage of supply or demand for the asset. We refer to such a deviation as an *endogenous* liquidity crisis, because it is due to the trading mechanism (i.e., the rules by which the market participants interact), rather than any fundamental reasons (note the similarity with the flash crash). Our framework provides insights into how such a liquidity crisis unfolds, connecting it to the so-called *adverse selection* effect. In particular, Section \[se:examples\] constructs an equilibrium in which an endogenous liquidity crisis does not occur because of an abnormally large market order, wiping out the liquidity on one side of the LOB, but because the optimal strategies of the agents require them to stop providing liquidity on one side of the LOB. On the mathematical side, our analysis uses the properties of conditional tails of the increments of a general It[ô]{} process. The main result in that regard, in Lemma \[le:necessary.marginal.maximum\], provides a uniform exponential bound on the conditional tails of the increments of a general It[ô]{} process. We believe that this result is useful in its own right, and, to the best of our knowledge, it is not available in the existing literature. In recent years, we observed an explosion in the amount of literature devoted to the study of market microstructure. In addition to various empirical studies, a large part of the existing theoretical work focuses on the problem of optimal execution: see, among others, [@MMS1], [@MMS2], [@MMS6], [@MMS7], [@MMS11], [@MMS13], [@MMS15], [@MMS16], [@MMS20], [@MMS22], [@MMS23], [@MMS26], [@MMS25], and references therein. In these articles, the dynamics and shape of the LOB are modeled exogenously, or, equivalently, the arrival processes of the limit and market orders are specified exogenously. In particular, none of these articles attempts to explain the shape and dynamics of the LOB, arising directly from the interaction between the market participants. A different approach to the analysis of market microstructure has its roots in the economic literature. For example, [@MMS.g1], [@MMS.g2], [@MMS.g3], [@MMS.g4], [@MMS.g5], [@MMS.g6], [@DA.DuZhu], [@Bressan1], [@Bressan2], [@Bressan4] consider equilibrium models of market microstructure, and they are more closely related to the present work. However, the models proposed in the aforementioned papers do not aim to represent the mechanics of an auction-style exchange with sufficient precision, and, in particular, they are not well suited for analyzing the liquidity effects of trading frequency, which is the main focus of the present paper. A somewhat related strand of literature focuses on the endogenous formation of LOB in markets with a designated market maker: see e.g., [@MMS.gmm1], [@MMS.gmm2], [@MMS.gmm3], [@MMS.gmm4], [@MMS.gmm5]. In these papers, the LOB is not an outcome of a multi-agent equilibrium: instead, it is controlled by a single agent, the market maker. In the present paper, we model the entire LOB as an output of an equilibrium between a large number of agents, each of whom is allowed to both consume and provide liquidity (in particular, we have no designated market maker). Our setting is related to the literature on *double auctions* (cf. [@DA.Vayanos], [@DA.DuZhu]), with the crucial difference that the participants of each auction are allowed to choose two “asymmetric" types of strategies: market or limit orders. In addition, the present framework assumes that, ex ante, all agents have access to the same information, and, in this sense, it is similar to [@MMS.g1], [@MMS.g3], [@MMS.g6]. In particular, the *adverse selection* effect, herein, does not arise from any a priori information asymmetry between agents, instead, it is caused by the *mechanics* of the exchange. We formulate the problem as a *continuum-player game* – this abstraction allows us to obtain computationally tractable results (cf. [@Aumann], [@Schmeidler], [@GCarmona] for the concept of a continuum-player game, and [@MFG1], [@MFG2], [@MFG3], [@MFG4] for the subclass of mean field games). The paper is organized as follows. Subsection \[se:setup\] describes the probabilistic setting, along with the execution rules of the exchange and the resulting state processes of the agents. Subsection \[se:equil.def\] defines the equilibrium and introduces the notion of *degeneracy* of the market (which represents an endogenous liquidity crisis). In Section \[se:examples\], we construct an equilibrium in a simple model, illustrating how an endogenous liquidity crisis unfolds, and how it is connected to the adverse selection effect. Theorems \[le:main.zeroTermSpread\], \[thm:main.necessary\], and Corollary \[prop:main.smallspread\], in Section \[se:main\], are the main results of the paper: they formalize and generalize the conclusions of Section \[se:examples\]. In Section \[se:tails\], we prove the key technical results on the (conditional) tails of marginal distributions of Itô processes. Sections \[se:pf.1\], \[se:pf.2\] contain the proofs of the main results. We conclude in Section \[se:conclusion\]. Modeling framework for a finite-frequency auction-style exchange {#se:setup} ================================================================ Mechanics of the exchange {#se:setup} ------------------------- We consider an exchange in which trading can only occur at discrete times $n=0,1,\ldots,N$. We assume that the market participants are split into two groups: the *external investors*, who are “impatient", in the sense that they only submit market orders, and the *strategic players*, who can submit both market and limit orders, and who are willing to optimize their actions over a given (short) time horizon, in order to get a better execution price.[^5] In our study, we focus on the strategic players, who are referred to as *agents*, and we model the behavior of the external investors exogenously, via an *exogenous demand*. The interpretation of the external investors is clear: these are the investors who either have a longer-term view on the market, or who simply need to buy or sell the asset for reasons other than short-term profits. The strategic players (i.e., agents), on the contrary, are short-term traders, who attempt to maximize their objective at a shorter time horizon $N$. During every time period $[n,n+1)$, all the orders coming to the exchange are split into *limit* and *market* orders. The limit orders are collected in the so-called *Limit Order Book (LOB)*, and the market orders form the *demand curve*. At time $n+1$, the market orders in the demand curve are executed against the limit orders in the LOB. Then, this process is repeated in the next time interval. In particular, during a time period $[n,n+1)$ (for simplicity, we say “at time $n$"), an agent is allowed to submit a market order, post a limit buy or sell order, or wait (i.e., do nothing). If a limit order is not executed in a given time period, it costs nothing to cancel or re-position it for the next time period. Notice that our framework does not model the time-priority of limit orders. However, introducing a time-priority would not change the agents’ maximum objective value, as the “tick size" is assumed to be zero (i.e., the set of possible price levels is ${\mathbb R}$), and, hence, an agent can always achieve a priority by posting her order “infinitesimally" above or below a given competing order. Further details on modeling the formation of an LOB and the execution rules are presented below. The demand curves are modeled exogenously by a random field $D=\left(D_n(p)\right)_{p\in{\mathbb R},n=1,\ldots,N}$ on a filtered probability space $\left(\Omega,\mathbb{F}=\left(\mathcal{F}_n\right)_{n=0}^N,{\mathbb P}\right)$, such that $\mathcal{F}_0$ is a trivial sigma-algebra, completed w.r.t. ${\mathbb P}$. The random variable $D^+_n(p) = \max(D_n(p),0)$ denotes the number of shares of the asset that the external investors and the agents submitting market orders are willing to purchase at or below the price $p$, accumulated over the time period $[n-1,n)$, and $D^-_n(p) = -\min(D_n(p),0)$ denotes the number of shares of the asset that the external investors and the agents submitting market orders are willing to sell at or above the price $p$, in the same time period. We assume that $D_n(\cdot)$ is a.s. nonincreasing and measurable w.r.t. $\mathcal{F}_n\otimes\mathcal{B}({\mathbb R})$. We denote by $\mathbb{A}$ a Borel space of *beliefs*, and, for each $\alpha\in\mathbb{A}$, there exists a *subjective probability measure* ${\mathbb P}^{\alpha}$ on $\left(\Omega,\mathcal{F}_N\right)$, which is absolutely continuous with resect to ${\mathbb P}$. We assume that, for any $n=0,\ldots,N$ and any $\alpha\in\mathbb{A}$, there exists a regular version of the conditional probability ${\mathbb P}^{\alpha}$ given $\mathcal{F}_n$, denoted ${\mathbb P}^{\alpha}_n$.[^6] We denote the associated conditional expectations by ${\mathbb E}^{\alpha}_n$. We also need to assume that, for any $\alpha\in\mathbb{A}$, there exists a modification of the family $\left\{{\mathbb P}^{\alpha}_n\right\}_{n=0}^N$, which satisfies the *tower property with respect to ${\mathbb P}$*, in the following sense: for any $n\leq m$ and any r.v. $\xi$, such that ${\mathbb E}^{\alpha} \xi^+ < \infty$, we have $${\mathbb E}^{\alpha}_n {\mathbb E}^{\alpha}_m \xi = {\mathbb E}^{\alpha}_n \xi,\,\,\,\,\,\,\,\,\,{\mathbb P}\text{-a.s.}$$ There exists such a modification, for example, if ${\mathbb P}^{\alpha}\sim{\mathbb P}$. In any market model, for every $\alpha$, we fix such a modification of conditional probabilities (up to a set of ${\mathbb P}$-measure zero) and assume that all conditional expectations $\left\{{\mathbb E}^{\alpha}_n\right\}$ are taken under this family of measures. The *Limit Order Book (LOB)* is given by a pair of adapted processes $\nu=(\nu^+_n,\nu^-_n)_{n=0}^{N}$, such that every $\nu^+_n$ and $\nu^-_n$ is a finite sigma-additive random measure on ${\mathbb R}$ (w.r.t. $\mathcal{F}_n\otimes\mathcal{B}({\mathbb R})$). Herein, $\nu^+_n$ corresponds to the cumulative limit sell orders, and $\nu^-_n$ corresponds to the cumulative limit buy orders, posted at time $n$.The bid and ask prices at any time $n=0,\ldots,N$ are given by the random variables $$p^b_n = \sup \text{supp}(\nu^-_n), \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,p^a_n = \inf \text{supp}(\nu^+_n),$$ respectively. Notice that these extended random variables are always well defined but may take infinite values. We define the *state space* of an agent as $\mathbb{S}={\mathbb R}\times\mathbb{A}$, where the first component denotes the *inventory* of an agent, and the second component denotes her *beliefs*. Every agent in state $(s,\alpha)$ models the future outcomes using the subjective probability measure ${\mathbb P}^{\alpha}$. There are infinitely many agents, and their distribution over the state space is given by the *empirical distribution* process $\mu=(\mu_n)_{n=0}^{N}$, such that every $\mu$ is a finite sigma-additive random measure on $\mathbb{S}$ (w.r.t. $\mathcal{F}_n\otimes\mathcal{B}(\mathbb{S})$). In particular, the total mass of agents in the set $S\subset\mathbb{S}$ at time $n$ is given by $\mu_n(S)$. The inventory level $s$ represents the *number of shares per agent*, held in state $(s,\alpha)$. In particular, the total number of shares held by all agents in the set $S\subset\mathbb{S}$ is given by $\int_{S} s\mu_n(ds,d\alpha)$. The interpretation of this definition in a finite-player game is discussed in Remark \[rem:finplayer\] below. We refer the reader to [@GCarmona] for the general concept of a continuum-player game. \[rem:finplayer\] The continuum-player game defined in this section can be related to a finite-player game as follows. Denote by $\mu_0$ the empirical distribution of the agents’ states at a given time. Recall that $\mu_0$ is a measure on $\mathbb{S}={\mathbb R}\times\mathbb{A}$, and assume that it is a finite linear combination of Dirac measures: $\mu_0 = \frac{1}{M} \sum_{i=1}^M \delta_{(s^{i},\alpha^{i})}$. In this case, we interpret $s^i$ as the [**number of shares per agent**]{} held by the agents in the $i$th group. Let us explain how this notion is related to the actual inventory levels (i.e., the actual numbers of shares held by the agents) in the associated finite-player game. To this end, consider a collection of $M$ agents, whose states are given by their (actual) inventories and beliefs, $(s,\alpha)$, with the current states being $\{(\tilde{s}^i = s^i/M,\alpha^i)\}$. Define the “unit mass" of agents to be $M$. In this finite-player collection, the mass of agents (measured relative to the unit mass, $M$) at any state $(Ms,\alpha)$ is precisely $\mu_0(\{(s,\alpha)\})$, and their total inventory is $Ms\mu_0(\{(s,\alpha)\})$. The number of shares per agent is, then, defined as the total inventory held by these agents divided by their mass, and it is equal to $Ms$. Choosing $s=\tilde{s}^i$, we conclude that, in the finite-player collection, the number of shares per agent held by the agents at state $(\tilde{s}^i,\alpha^i)$ is given by $M\tilde{s}^i = s^i$, which coincides with our interpretation of $s^i$ in the continuum-player game. It is also easy to show that an equilibrium in the proposed continuum-player game (defined in the next subsection) produces an approximate equilibrium in the associated finite-player game, when the inventory levels $\{\tilde{s}^i\}$ are small (cf. Subsection 2.3 in the extended version of this paper, [@GaydukNadtochiy1]) As the parameter $\alpha$ does not change over time, the state process of an agent, denoted $(S_n)$, is an adapted ${\mathbb R}$-valued process, representing her inventory.[^7] The control of every agent is given by a triplet of adapted processes $(p,q,r) = (p_n,q_n,r_n)_{n=0}^{N-1}$ on $\left(\Omega,\mathbb{F}\right)$, with values in ${\mathbb R}^2\times\left\{0,1\right\}$. The first coordinate, $p_n$, indicates the location of a limit order placed at time $n$, and $q_n$ indicates the size of the order (measured in shares per agent, and with negative values corresponding to buy orders).[^8] The last coordinate $r_n$ shows whether the agent submits a market order (if $r_n = 1$) or a limit order (if $r_n=0$). Assume that an agent posts a limit sell order at a price level $p_n$. If the demand to buy the asset at this price level, $D^+_{n+1}(p_n)$, exceeds the amount of all limit sell orders posted below $p_n$ at time $n$, then (and only then) the limit sell order of the agent is executed. Market orders of the agents are always executed at the bid or ask prices available at the time when the order is submitted. We interpret an internal market order (i.e., the one submitted by an agent) as the decision of an agent to join the external investors, in the given time period. Summing up the above, we obtain the following dynamics for the state process of an agent, starting with initial inventory $s\in{\mathbb R}$ at time $m=0,\ldots,N-1$: $$S^{(p,q,r)}_m(m,s,\nu) = s,\,\,\,\,\,\, \Delta S^{(p,q,r)}_{n+1}(m,s,\nu) = S^{(p,q,r)}_{n+1}(m,s,\nu) - S^{(p,q,r)}_n(m,s,\nu) = -q_n \bone_{\left\{ r_n=1 \right\}}$$ $$\label{eq.stateProc.def} - \bone_{\left\{ r_n=0 \right\}} \left( q^+_n \bone_{\left\{D^+_{n+1}(p_n) > \nu^+_n((-\infty,p_n))\right\}} - q^-_n \bone_{\left\{D^-_{n+1}(p_n) > \nu^-_n((p_n,\infty))\right\}}\right), \,\,\,\,n=m,\ldots,N-1.$$ The above dynamics represent an optimistic view on the execution by the agents. In particular, they imply that all limit orders at the same price level are executed in full, once the demand reaches them: i.e., each agent believes that her limit order will be executed first among all orders at a given price level. In addition, all agents’ market orders are executed at the bid and ask prices: i.e., each agent believes that her market order will be executed first, when the demand curve is cleared against the LOB, at the end of a given time period. These assumptions can be partially justified by the fact that the agents’ orders are infinitesimal: $q_n$ is measured in shares per agent, and an individual agent has zero mass. However, if a non-zero mass of agents submit limit orders at the same price level, or execute market orders, at the same time, then, the above state dynamics may violate the market clearing condition: the total size of executed market orders (both in shares and in dollars) may not coincide with the total size of executed limit orders (at least, as viewed by the agents). Nevertheless, this issue is resolved if, at any time, the mass of agents posting limit orders at the same price level or posting market orders is zero. In other words, $(\nu,p,q,r)$ satisfy, ${\mathbb P}$-a.s.: $\nu_n$ is continuous, as a measure on ${\mathbb R}$ (i.e., it has no atoms), and $r_n=0$. Such an equilibrium is constructed in Section 8 of the extended version of this paper, [@GaydukNadtochiy1]. The general definition of a continuum-player game and its connection to a finite-player game can be found, e.g., in [@GCarmona] and in the references therein (see also Subsection 2.3 in the extended version of this paper, [@GaydukNadtochiy1]). The modeling framework proposed herein has a close connection to the models of *double auctions*, in the economic literature (cf. [@DA.DuZhu], [@DA.Vayanos]). The main difference is in the non-standard design of the auction. Namely, in the proposed setting, the auction participants may choose different styles of trading, i.e., market or limit orders, which generates an ex-post information asymmetry between participants: the limit orders have to be submitted before the demand curve is observed, while the market orders are submitted using complete information about the LOB. This difference is not coincidental – it is, in fact, crucial for a realistic representation of the risks associated with each order type, and it is at the core of the results established herein. A more detailed discussion of the information structure is provided in the next subsection. Equilibrium {#se:equil.def} ----------- The objective function of an agent, starting at the initial state $(s,\alpha)\in\mathbb{S}$, at any time $m=0,\ldots,N$, and using the control $(p,q,r)$, is given by the $\mathcal{F}_m$-measurable random variable $$\label{eq.intro.Jm.def} J^{(p,q,r)}(m,s,\alpha,\nu) = {\mathbb E}^{\alpha}_m \left[ \left(S^{(p,q,r)}_N(m,s,\nu)\right)^+ p^b_N - \left(S^{(p,q,r)}_N(m,s,\nu)\right)^- p^a_N \right.$$ $$\left. - \sum_{n=m}^{N-1} \left(p_n\bone_{\left\{ r_n = 0\right\}} + p^a_n\bone_{\left\{ r_n = 1, q_n <0\right\}} + p^b_n\bone_{\left\{ r_n = 1, q_n >0\right\}} \right) \Delta S^{(p,q,r)}_{n+1}(m,s,\nu) \right],$$ where we assume that $0\cdot\infty = 0$. In the above expression, we assume that, at the final time $n=N$, each agent is forced to liquidate her position at the bid or ask prices available at that time. Alternatively, one can think of it as *marking to market* the residual inventory, right after the last external market order is executed. \[def:admis\] For a given LOB $\nu$, integer $m=0,\ldots,N-1$, and state $(s,\alpha)\in\mathbb{S}$, the triplet of adapted processes $(p,q,r)$ is an [**admissible control**]{} if the positive part of the expression inside the expectation in (\[eq.intro.Jm.def\]) has a finite expectation under ${\mathbb P}^{\alpha}$. For a given LOB $\nu$, an initial condition $(m,s,\alpha)$, and a triplet of $\mathbb{F}\times\mathcal{B}(\mathbb{S})$-adapted random fields $(p,q,r)$, we identify the latter (whenever it causes no confusion) with stochastic processes $(p,q,r)$ via: $$p_n=p_n\left(S^{(p,q,r)}_n(m,s,\nu),\alpha\right),\,\,\, q_n=q_n\left(S^{(p,q,r)}_n(m,s,\nu),\alpha\right),\,\,\, r_n=r_n\left(S^{(p,q,r)}_n(m,s,\nu),\alpha\right),$$ and the state dynamics (\[eq.stateProc.def\]), for $n=m,\ldots,N$. This system determines $(p,q,r)$ and $S^{(p,q,r)}$ recursively. \[def:optControl\] For a given LOB $\nu$, we call the triplet of progressively measurable random fields $(p,q,r)$ an [**optimal control**]{} if, for any $m=0,\ldots,N$ and any $(s,\alpha)\in\mathbb{S}$, we have: - $(p,q,r)$ is admissible, - $J^{(p,q,r)}(m,s,\alpha,\nu) \geq J^{(p',q',r')}(m,s,\alpha,\nu)$, ${\mathbb P}$-a.s., for any admissible control $(p',q',r')$. In the above, we make the standard simplifying assumptions of continuum-player games: each agent is too small to affect the empirical distribution of cumulative controls (reflected in $\nu$) when she changes her control (cf. [@GCarmona]). Note also that our definition of the optimal control implies that it is time consistent: re-evaluation of the optimality at any future step, using the same terminal criteria, must lead to the same optimal strategy. Next, we discuss the notion of equilibrium in the proposed game. First, we notice that, if $p^b_N$ or $p^a_N$ becomes infinite, the agents with positive or negative inventory may face the objective value of “$-\infty$", for any control they use. In such a case, their optimal controls may be chosen in an arbitrary way, resulting in unrealistic equilibria. To avoid this, we impose the additional regularity condition on $\nu$. \[def:admis.LOB\] A given LOB $\nu$ is admissible if, for any $m=0,\ldots,N-1$ and any $\alpha\in\mathbb{A}$, we have, ${\mathbb P}$-a.s.: $${\mathbb E}^{\alpha}_m |p^a_N|\vee|p^b_N| < \infty.$$ Let us consider the (stochastic) value function of an agent for a fixed $(m,s,\alpha,\nu)$: $$\label{eq.gen.Val.randField} V^{\nu}_m(s,\alpha) = \text{esssup}_{p,q,r} J^{(p,q,r)}\left(m,s,\alpha,\nu\right),$$ where the essential supremum is taken under ${\mathbb P}$, over all admissible controls $(p,q,r)$, and $J^{(p,q,r)}$ is given by (\[eq.intro.Jm.def\]). Appendix A shows that, for any admissible $\nu$, $V^{\nu}_m(\cdot,\alpha)$ has a continuous modification under ${\mathbb P}$, which we refer to as the value function of an agent with beliefs $\alpha$. Using the Dynamic Programming Principle, Appendix A provides an explicit system of recursive equations that characterize optimal strategies and the value function. In particular, the results of Appendix A (cf. Corollary \[cor:piecewiseLin\]) yield the following proposition. \[cor:piecewiseLin.new\] Assume that, for an admissible LOB $\nu$, there exists an optimal control $(\hat{p},\hat{q},\hat{r})$. Then, for any $(s,\alpha)\in\mathbb{S}$, the following holds ${\mathbb P}$-a.s., for all $n=0,\ldots,N-1$: $$V^{\nu}_n(s,\alpha) = s^+ \lambda^a_n(\alpha) - s^- \lambda^b_n(\alpha),$$ with some adapted processes $\lambda^a(\alpha)$ and $\lambda^b(\alpha)$, such that $\lambda^a_N(\alpha) = p^b_N$ and $\lambda^b_N(\alpha) = p^a_N$. The values of $\lambda^a(\alpha)$ and $\lambda^b(\alpha)$ can be interpreted as the *expected execution prices* of the agents with beliefs $\alpha$, who are long and short the asset, respectively. \[def:equil.def\] Consider an empirical distribution process $\mu=(\mu_n)_{n=0}^N$ and a market model, as described in Subsection \[se:setup\]. We say that a given LOB process $\nu$ and a control $(p,q,r)$ form an [**equilibrium**]{}, if there exists a Borel set $\tilde{\mathbb{A}}\subset \mathbb{A}$, called the [**support**]{} of the equilibrium, such that: 1. $\mu_n\left(\mathbb{{\mathbb R}}\times\left(\mathbb{A}\setminus \tilde{\mathbb{A}}\right)\right)=0$, ${\mathbb P}$-a.s., for all $n$, 2. $\nu$ is admissible, and $(p,q,r)$ is an optimal control for $\nu$, on the state space $\tilde{\mathbb{S}}={\mathbb R}\times\tilde{\mathbb{A}}$, 3. and, for any $n=0,\ldots,N-1$, we have, ${\mathbb P}$-a.s., $$\label{eq.nuplus.fixedpoint.def} \nu^+_n((-\infty,x]) = \int_{\tilde{\mathbb{S}}} \bone_{\left\{p_n(s,\alpha)\leq x, r_n(s,\alpha)=0\right\}}\, q^+_n(s,\alpha) \mu_n(ds,d\alpha), \,\,\,\,\,\,\forall\, x\in{\mathbb R},$$ $$\label{eq.numinus.fixedpoint.def} \nu^-_n((-\infty,x]) = \int_{\tilde{\mathbb{S}}} \bone_{\left\{p_n(s,\alpha)\leq x, r_n(s,\alpha)=0\right\}}\, q^-_n(s,\alpha) \mu_n(ds,d\alpha), \,\,\,\,\,\,\forall\, x\in{\mathbb R}.$$ It follows from Proposition \[cor:piecewiseLin.new\] that, in equilibrium, it is optimal for an agent with zero initial inventory to do nothing. Hence, in equilibrium, roundtrip strategies are impossible. To allow for roundtrip strategies in equilibrium, one can, e.g., introduce an upper bound on $|q|$ or on the total inventory of an agent (as it is done, e.g., in [@MMS.gliq1]). However, we do not believe that such a modification would change the qualitative behavior of market liquidity as a function of trading frequency, which is the main focus of the present paper. Notice that, because the optimal controls are required to be time consistent under ${\mathbb P}$, the above definition, in fact, defines a *sub-game perfect equilibrium*. It is also worth mentioning that Definition \[def:equil.def\] defines a *partial equilibrium*, as the empirical distribution process $\mu$ is given exogenously. A more traditional version of Nash equilibrium would require $\mu$ to be determined by the initial distribution and the values of the state processes: $$\label{eq.endog.mu} \mu_n = \mu_0 \circ \left( (s,\alpha)\mapsto \left(S_n^{(p,q,r)}(0,s,\nu),\alpha\right) \right)^{-1},$$ which must hold ${\mathbb P}$-a.s., for all $n=0,\ldots,N$, with $S_n^{(p,q,r)}(0,s,\nu)$ defined via (\[eq.stateProc.def\]), in addition to the other conditions in Definition \[def:equil.def\]. Nevertheless, we choose not to enforce the condition (\[eq.endog.mu\]) in the definition of equilibrium, in order to allow new agents to enter the game, which, in effect, amounts to modeling $\mu$ exogenously. If one assumes that no new agent arrives to the market, then, the fixed-point condition (\[eq.endog.mu\]) has to be enforced. Note also that our interpretation of the demand curve $D_n(\cdot)$ implies that it consists of both the external (i.e., due external investors) and internal (i.e., due to the agents) market orders. Therefore, it may be reasonable to consider an additional consistency condition for an equilibrium. A part of this condition is to ensure that a non-zero mass of agents submit market buy orders only if the fundamental price rises above the ask price (i.e., only if a market buy order is actually executed), and, similarly, a non-zero mass of agents submit market sell orders only if the fundamental price falls below the bid price. We assume that the agents’ market orders enter into the demand curve with the highest level of priority: e.g., their market buy orders enter the demand curve at the price level infinitesimally close to, but below, the fundamental price, in order to guarantee that they are the first ones to be executed. Thus, another part of the aforementioned consistency condition is to ensure that the absolute value of the demand curve to the left or to the right of the fundamental price is sufficiently large to account for all internal market orders. Mathematically, such consistency condition can be formulated as follows: $$\label{eq.endog.D.1} d^b_n:=\mu_n\left( \left\{(s,\alpha)\,:\,q_n(s,\alpha)<0,\,r_n(s,\alpha)=1\right\}\right)>0\,\, \Rightarrow\,\,p^0_{n+1}>p^a_{n},\,\,\lim_{p\uparrow p^0_{n+1}}D^+_{n+1}(p)\geq d^b_n,$$ $$\label{eq.endog.D.2} d^a_n:=\mu_n\left( \left\{(s,\alpha)\,:\,q_n(s,\alpha)>0,\,r_n(s,\alpha)=1\right\}\right)>0\,\, \Rightarrow\,\,p^0_{n+1}<p^b_{n},\,\,\lim_{p\downarrow p^0_{n+1}}D^-_{n+1}(p)\geq d^a_n.$$ The above conditions become redundant if the agents never submit market orders in equilibrium. Section 8 of the extended version of this paper, [@GaydukNadtochiy1], shows how to construct an equilibrium which satisfies condition (\[eq.endog.mu\]), and in which the agents never submit market orders (hence, (\[eq.endog.D.1\]) and (\[eq.endog.D.2\]) are also satisfied). However, it is important to emphasize that the main results of the present work (cf. Section \[se:main\]) provide necessary conditions for [**all**]{} equilibria: for those satisfying the conditions (\[eq.endog.mu\]), (\[eq.endog.D.1\]), (\[eq.endog.D.2\]) and for the ones that do not. Let us comment on the information structure of the game. In the present setting, all agents observe the same information, given by the filtration ${\mathbb F}$. We consider an open-loop Nash equilibrium, in which the agent’s strategy is viewed as an adapted stochastic process (rather than a function of the states and controls of other players), and the definition of optimality is chosen accordingly. In addition, as $\mu$ is adapted to ${\mathbb F}$, each agent has complete information about the present and past states of other agents, and their beliefs. However, as the agents use different (subjective) measures $\{{\mathbb P}^{\alpha}\}$, their views on the future values of $\mu$ may be different. Of course, it would be more realistic to assume that the agents do not have complete information about each other’s current states, but this would make the problem significantly more complicated. In the present setting, the agents also have complete information about the current location of the fundamental price. In our follow-up paper, [@GaydukNadtochiy2], we relax this assumption, which allows us to develop a more realistic model for the “local" behavior of an individual agent. However, such a relaxation does not seem necessary for the questions analyzed herein. As all agents use the same information, the present article belongs to the strand of literature that attempts to explain microstructure phenomena without information asymmetry (cf. [@MMS.g3], [@MMS.g6], [@MMS.g1], [@MMS.g2]). Nevertheless, it is important to mention that information asymmetry arises ex-post, between the market participants submitting market and limit orders. This asymmetry is not due to superior information a priori available to any of the agents. Instead, it stems from the very nature of limit orders, which are “passive" by design (cf. the discussion on the last paragraph of Subsection \[se:setup\]). Similar observation is made in [@MMS.g3]. Next, we need to add another condition to the notion of equilibrium. Notice that equations (\[eq.nuplus.fixedpoint.def\])–(\[eq.numinus.fixedpoint.def\]) should serve as the fixed-point constraints that enable one to obtain the optimal controls $(p,q,r)$, along with the LOB $\nu$. However, these equations only hold for $n=0,\ldots,N-1$: indeed, the agents do not need to choose their controls at time $n=N$, as the game is over and their residual inventory is marked to the bid and ask prices. However, the terminal bid and ask prices are determined by the LOB $\nu_N$, which, in turn, can be chosen arbitrarily. To avoid such ambiguity, we impose an additional constraint on the equilibria studied herein. First, we introduce the notion of a *fundamental price*. \[def:het.p0\] Assume that ${\mathbb P}$-a.s., for any $n=1,\ldots,N$, there exists a unique $p^0_n$ satisfying $D_n\left(p^0_n\right)=0$. Then, the adapted process $(p^0_n)_{n=1}^N$ is called the [**fundamental price process**]{}. Whenever the notion of a fundamental price is invoked, we assume that it is well defined. The intuition behind $p^0$ is clear: it is a price level at which the immediate demand is balanced. However, it is important to stress that we do not assume that the asset can be traded at the fundamental price level. Rather, $p^0$ is a feature of the abstract current demand curve, whereas all actual trading happens on the exchange, against the current LOB. This aspect of our setting differs from many other approaches in the literature. \[def:het.\] Assume that the fundamental price is well defined and denote $\xi_N = p^0_N - p^0_{N-1}$. Then, an equilibrium with LOB $\nu$ is [**linear at terminal crossing (LTC)**]{} if $$\label{eq.LTC.def} \nu_N = \nu_{N-1}\circ (x\mapsto x+\xi_N)^{-1},\,\,\,\,\,\,\,\,{\mathbb P}\text{-a.s.}$$ The above definition assumes that the terminal LOB $\nu_N$ is obtained from $\nu_{N-1}$ by a simple shift, with the size of the shift equal to the increment in the fundamental price. This definition connects the LOB at the terminal time with the demand process, ruling out many unnatural equilibria. In particular, the question of existence of an equilibrium becomes non-trivial. However, the mere existence of an equilibrium is not the main focus of the present work: the existence results, established herein, are limited to Section \[se:examples\], which constructs an LTC equilibrium in a specific Gaussian random walk model (a slightly more general existence result is given in Section 8 of the extended version of this paper, [@GaydukNadtochiy1]). What is central to the present investigation is the observation that the agents may reach an equilibrium in which one side of the LOB becomes empty (as demonstrated by the example of Section \[se:examples\]). We call such LOB, and the associated equilibrium, *degenerate*. We say that an equilibrium with LOB $\nu$ is [**non-degenerate**]{} if $\nu^{+}_n({\mathbb R})>0$ and $\nu^{-}_n({\mathbb R})>0$, for all $n=0,\ldots,N-1$, ${\mathbb P}$-a.s.. Intuitively, the degeneracy of the LOB refers to a situation where, with positive probability, one side of the LOB disappears from the market: i.e., $\nu^+_n({\mathbb R})$ or $\nu^-_n({\mathbb R})$ becomes zero. Clearly, this happens when the agents who are supposed to provide liquidity choose to post market orders (i.e. consume liquidity) or wait (neither provide nor consume liquidity). Such a degeneracy can be interpreted as the *endogenous liquidity crisis* – the one that arises purely from the interaction between the agents, and cannot be justified by any fundamental economic reasons (e.g., the external demand for the asset may still be high, on both sides). Taking an optimistic point of view, we assume that the agents choose a non-degenerate equilibrium, whenever one is available. However, if a non-degenerate equilibrium does not exist, an endogenous liquidity crisis may occur with positive probability. One of the main goals of this paper is to provide insights into the occurrence of an endogenous liquidity crisis and its relation to trading frequency. Example: a Gaussian random walk model {#se:examples} ===================================== In this section, we consider a specific market model for the external demand $D$, to construct a non-degenerate LTC equilibrium. More importantly, using this model, we illustrate the liquidity effects of trading frequency. The present example, albeit very simplistic, enables us to identify the important changes in the optimal strategies of the agents (and, hence, to the LOB) as the trading frequency increases. In particular, we demonstrate how the *adverse selection* effect may be amplified disproportionally by the high trading frequency and may cause a liquidity crisis. Note that the adverse selection phenomenon, in the present setting, is not a consequence of any ex-ante information asymmetry but is due to the mechanics of the exchange (i.e., the nature of limit orders), which is similar to the phenomena documented in [@MMS.g3], [@MMS.g2]. In the rest of the paper, we show that the conclusions of this section are not due to the particular choice of a model made in the present section and, in fact, persist in a much more general setting. On a complete stochastic basis $(\Omega,\tilde{{\mathbb F}}=(\tilde{\mathcal{F}}_t)_{t\in[0,T]},{\mathbb P})$, we consider a continuous time process $\tilde{p}_0$: $$\label{eq.p0.BM} \tilde{p}^0_t = p^0_0 + \alpha t + \sigma W_t,\quad p^0_0 \in {\mathbb R}, \quad t\in[0,T],$$ where $\alpha\in{\mathbb R}$ and $\sigma>0$ are constants, and $W$ is a Brownian motion. We also consider an arbitrary progressively measurable random field $(\tilde{D}_t(p))$, s.t., ${\mathbb P}$-a.s., the function $\tilde{D}_t(\cdot)-\tilde{D}_s(\cdot)$ is strictly decreasing and vanishing at zero, for any $0\leq s < t\leq T$. Finally, we introduce the empirical distribution process $(\tilde{\mu}_t)$, with values in the space of finite sigma-additive measures on $\mathbb{S}$. We partition the time interval $[0,T]$ into $N$ subintervals of size $\Delta t=T/N$. A discrete time model is obtained by discretizing the continuous time one[^9] $$\mathcal{F}_n = \tilde{\mathcal{F}}_{n\Delta t},\quad p^0_n = \tilde{p}^0_{n\Delta t}, \quad D_n(p) = (\tilde{D}_{n\Delta t}-\tilde{D}_{(n-1)\Delta t})(p-p^0_n),\quad \mu_n = \tilde{\mu}_{n\Delta t}.$$ In this section, for simplicity, we assume that the set of agents’ beliefs is a singleton: $\mathbb{A}=\left\{\alpha\right\}$ and ${\mathbb P}^{\alpha}={\mathbb P}$. We also assume that (at least, from the agents’ point of view) there are always some long and short agents present in the market: $\mu_n\left((0,\infty)\times\mathbb{A}\right),\mu_n\left((-\infty,0)\times\mathbb{A}\right)>0$, ${\mathbb P}$-a.s., for all $n$. Clearly, $N$ represents the trading frequency, and the continuous time model represents the “limiting model," which the agents use as a benchmark, in order to make consistent predictions in the markets with different trading frequencies. We assume that the benchmark model is fixed, and $N$ is allowed to vary. In the remainder of this section, we propose a method for constructing a non-degenerate LTC equilibrium in the above discrete time model. We show that the method succeeds for any $(N,\sigma)$ if $\alpha=0$. However, for $\alpha\neq 0$, we demonstrate numerically that the method fails as $N$ becomes large enough. We show why, precisely, the proposed construction fails, providing an economic interpretation of this phenomenon. Moreover, we analyze the market close to the moment when a non-degenerate equilibrium fails to exist and demonstrate that the agents’ behavior at this time follows the pattern typical for an endogenous liquidity crisis. In view of Proposition \[cor:piecewiseLin.new\], in order to construct a non-degenerate LTC equilibrium, we need to find a control $(\hat{p},\hat{q},\hat{r})$, and the expected execution prices $(\hat{\lambda}^a,\hat{\lambda}^b)$, s.t. the value function of an agent with inventory $s$ is given by $V_n(s) = s^+\hat{\lambda}^a_n - s^-\hat{\lambda}^b_n$, and it is attained by the strategy $(\hat{p},\hat{q},\hat{r})$. In addition, we need to find a non-degenerate LOB $\nu$, s.t. (\[eq.nuplus.fixedpoint.def\]), (\[eq.numinus.fixedpoint.def\]) and (\[eq.LTC.def\]) hold. Our ansatz is as follows $$\nu_n = \left(h^a_n \delta_{p^a_n}, h^b_n \delta_{p^b_n}\right), \quad p^a_n = \hat{p}^a_n + p^0_n, \quad p^b_n = \hat{p}^b_n + p^0_n, \quad -\infty < \hat{p}^b_n,\, \hat{p}^a_n<\infty,$$ $$\hat{p}_n(s) = p^a_n\bone_{\{s>0\}} + p^b_n\bone_{\{s<0\}}, \quad \hat{q}_n(s) = s, \quad \hat{r}_n(s) = 0, \quad \lambda^a_n = \hat{\lambda}^a_n + p^0_n, \quad \lambda^b_n = \hat{\lambda}^b_n + p^0_n,$$ where $\delta$ is the Dirac measure, $(\hat{p}^a,\hat{p}^b,\hat{\lambda}^a,\hat{\lambda}^b)$ are deterministic processes, and $h^a_n=\int_0^{\infty}s\mu_n(ds)>0$, $h^b_n=\int_{-\infty}^0|s|\mu_n(ds)>0$. With such an ansatz, the conditions (\[eq.nuplus.fixedpoint.def\]), (\[eq.numinus.fixedpoint.def\]) are satisfied automatically. Thus, we only need to choose finite deterministic processes $(\hat{p}^a,\hat{p}^b,\hat{\lambda}^a,\hat{\lambda}^b)$ s.t.: $\hat{p}^a_N = \hat{p}^a_{N-1}$, $\hat{p}^b_N = \hat{p}^b_{N-1}$ (so that the equilibrium is LTC) and the associated $(\hat{p},\hat{q},0)$ form an optimal control, producing the value function $V_n(s) = s^+\lambda^a_n - s^-\lambda^b_n$. Appendix A contains necessary and sufficient conditions for characterizing such families $(p^a,p^b,\lambda^a,\lambda^b)$. In particular, we deduce from Corollaries \[cor:piecewiseLin\] and \[cor:piecewiseLin.verif\] that $(\hat{p}^a_{N-1},\hat{p}^b_{N-1},\hat{\lambda}^a_{N-1},\hat{\lambda}^b_{N-1})$ form a suitable family in a single-period case, $[N-1,N]$, if they solve the following system: $$\label{eq.ex.singleStep.1} \left\{ \begin{array}{l} {\hat{p}^a_{N-1} \in \text{arg}\max_{p\in{\mathbb R}} {\mathbb E}\left((p - \hat{p}^b_{N-1} - \xi) \bone_{\left\{ \xi > p \right\}}\right), \quad \hat{p}^b_{N-1}<0,\phantom{\frac{\frac{1}{2}}{2}}}\\ {\hat{p}^b_{N-1} \in \text{arg}\max_{p\in{\mathbb R}} {\mathbb E}\left((\hat{p}^a_{N-1} - p + \xi) \bone_{\left\{ \xi < p \right\}}\right), \quad \hat{p}^a_{N-1}>0,\phantom{\frac{\frac{1}{2}}{2}}}\\ {\hat{\lambda}^a_{N-1} = \hat{p}^b_{N-1} + \alpha\Delta t + {\mathbb E}\left((\hat{p}^a_{N-1} - \hat{p}^b_{N-1} - \xi) \bone_{\left\{ \xi > \hat{p}^a_{N-1} \right\}}\right), \phantom{\frac{\frac{\frac{1}{2}}{2}}{2}}}\\ {\hat{\lambda}^b_{N-1} = \hat{p}^a_{N-1} + \alpha\Delta t - {\mathbb E}\left((\hat{p}^a_{N-1} - \hat{p}^b_{N-1} + \xi) \bone_{\left\{ \xi < \hat{p}^b_{N-1} \right\}}\right), \phantom{\frac{\frac{\frac{1}{2}}{2}}{2}}}\\ {\hat{p}^b_{N-1}\leq \hat{\lambda}^a_{N-1}, \quad \hat{\lambda}^b_{N-1} \leq \hat{p}^a_{N-1}, \quad \hat{p}^a_{N-1} \geq \hat{p}^b_{N-1} + |\alpha|\Delta t, \phantom{\frac{\frac{1}{2}}{2}}} \end{array} \right.$$ where $\xi=\Delta p^0_N\sim \mathcal{N}(\alpha\Delta t, \sigma^2 \Delta t)$. Let us comment on the economic meaning of the equations in (\[eq.ex.singleStep.1\]). The expectations in the first two lines represent the *relative expected profit* from executing a limit order at time $N$, at the chosen price level $p+p^0_{N-1}$, versus marking the inventory to market at time $N$, at the best price available on the other side of the book: i.e., $p^b_N = \hat{p}^b_{N-1} + \xi + p^0_{N-1}$ or $p^a_N = \hat{p}^a_{N-1} + \xi + p^0_{N-1}$. Notice that a limit order is executed if and only if the fundamental price at time $N$ is above or below the chosen limit order: i.e., if $p^0_{N-1} + \xi > p + p^0_{N-1}$ or $p^0_{N-1} + \xi < p + p^0_{N-1}$.[^10] Clearly, it is only optimal for an agent to post a limit order if the relative expected profit is nonnegative, which is the case if and only if $\hat{p}^b_{N-1}<0<\hat{p}^a_{N-1}$. The third and fourth lines in (\[eq.ex.singleStep.1\]) represent the expected execution prices of the agents at time $N-1$, assuming they use the controls given by $(\hat{p}^a_{N-1},\hat{p}^b_{N-1})$. Each of the right hand sides is a sum of two components: the relative expected profit from posting a limit order and the expected value of marking to market at time $N$, measured relative to $p^0_{N-1}$. Let us analyze the inequalities in the last line of (\[eq.ex.singleStep.1\]). If the bid price at time $N-1$ exceeds the expected execution price of a long agent, i.e., $\hat{p}^b_{N-1} + p^0_{N-1}> \hat{\lambda}^a_{N-1}+ p^0_{N-1}$, then every agent with positive inventory prefers to submit a market order, rather than a limit order, at time $N-1$, which causes the ask side of the LOB to degenerate. Similarly, we establish $\hat{\lambda}^b_{N-1} \leq \hat{p}^a_{N-1}$. Finally, if $\alpha>0$ and $\hat{p}^a_{N-1} < \hat{p}^b_{N-1} + \alpha\Delta t$, an agent may buy the asset using a market order at time $N-1$, at the price $\hat{p}^a_{N-1}+p^0_{N-1}$, and sell it at time $N$, at the expected price $\hat{p}^b_{N-1} + p^0_{N-1} + \alpha \Delta t > \hat{p}^a_{N-1}+p^0_{N-1}$ (a reverse strategy works for $\alpha<0$). This strategy can be scaled to generate infinite expected profit and, hence, is excluded by the last inequality in the last line of (\[eq.ex.singleStep.1\]). We construct a solution to (\[eq.ex.singleStep.1\]) by solving a fixed-point problem given by the first two lines of (\[eq.ex.singleStep.1\]) and verifying that the desired inequalities hold.[^11] We implement this computation in MatLab, and the results can be seen as the right-most points on the graphs in Figure \[fig:2\]. From the numerical solution, we see that, whenever $\Delta t$ is small enough, the conditions $\hat{p}^b_{N-1}\leq \hat{\lambda}^a_{N-1}$ and $\hat{\lambda}^b_{N-1} \leq \hat{p}^a_{N-1}$ are satisfied (cf. the right part of Figure \[fig:2\]).[^12] In addition, for $\alpha\geq0$, we have $$0 < {\mathbb E}\left(\hat{p}^a_{N-1} - \hat{p}^b_{N-1} - \xi\,\vert\,\xi > \hat{p}^a_{N-1} \right) = \hat{p}^a_{N-1} - \hat{p}^b_{N-1} - {\mathbb E}\left(\xi\,\vert\,\xi > \hat{p}^a_{N-1} \right) \leq \hat{p}^a_{N-1} - \hat{p}^b_{N-1} - \alpha\Delta t,$$ which yields the last inequality in (\[eq.ex.singleStep.1\]). The case of $\alpha<0$ is treated similarly. Notice that $\hat{\lambda}^a_N = \hat{p}^b_N = \hat{p}^b_{N-1}$ and $\hat{p}^a_{N-1} = \hat{p}^a_N = \hat{\lambda}^b_N$. Thus, the single-period equilibrium we have constructed satisfies: $$\label{eq.example.1period.ineq.1} \hat{p}^b_{n}\leq \hat{\lambda}^a_{n},\quad \hat{\lambda}^b_{n} \leq \hat{p}^a_{n}, \quad \hat{\lambda}^a_{n+1} < 0,\quad \hat{\lambda}^b_{n+1}>0,$$ for $n=N-1$. If one of the first two inequalities in (\[eq.example.1period.ineq.1\]) fails, the agents choose to submit market orders, as opposed to limit orders, which leads to *degeneracy* of the LOB – one side of it disappears. If one of the last two inequalities fails, the execution of a limit order, at any price level, yields a negative relative expected profit for the agents on one side of the book (given by the expectation in the first or second line of (\[eq.ex.singleStep.1\])). As a result, it becomes optimal for all such agents to stop posting any limit orders, and the LOB degenerates. The latter is interpreted as the *adverse selection* effect. For example, if the third inequality in (\[eq.example.1period.ineq.1\]) fails, then, every long agent believes that, no matter the price at which her limit order is posted, if it is executed in the next time period, her expected execution price at the next time step will be higher than the price at which the limit order is executed. Hence, it suboptimal to post a limit order at all. In a single period $[N-1,N]$, by choosing small enough $\Delta t$, we can ensure that the inequalities in (\[eq.example.1period.ineq.1\]) are satisfied. However, it turns out that, as we progress recursively, constructing an equilibrium, we may encounter a time step at which one of the inequalities in (\[eq.example.1period.ineq.1\]) fails, implying that a non-degenerate LTC equilibrium cannot be constructed for the given time period (at least, using the proposed method). To see this, consider the recursive equations for $(\hat{p}^a,\hat{\lambda}^a)$ (which are chosen to satisfy the conditions of Corollary \[cor:piecewiseLin\], in Appendix A, given our ansatz): $$\label{eq.RW.pan} \left\{ \begin{array}{l} {\hat{p}^a_n \in \text{arg}\max_{p\in{\mathbb R}} {\mathbb E}\left(\left(p-\hat{\lambda}^a_{n+1} - \xi\right) \bone_{\left\{ \xi> p \right\}}\right), \phantom{\frac{\frac{1}{2}}{2}}}\\ {\hat{\lambda}^a_n = \hat{\lambda}^a_{n+1} + \alpha\Delta t + {\mathbb E}\left( \left(\hat{p}^a_n - \hat{\lambda}^a_{n+1} - \xi\right) \bone_{\left\{\xi> \hat{p}^a_n ) \right\}} \right) <0,\phantom{\frac{\frac{1}{2}}{2}}} \end{array} \right.$$ and similarly for $(\hat{p}^b,\hat{\lambda}^b)$. Using the properties of the Gaussian distribution, it is easy to see that, if $\hat{\lambda}^a_{n+1}<0$, we have $\hat{p}^a_n>0$. Similar conclusion holds for $(\hat{\lambda}^b,\hat{p}^b)$. Thus, if $\hat{\lambda}^a_k< 0 < \hat{\lambda}^b_k$, for $k=n+1,\ldots,N$, our method allows us to construct a non-degenerate LTC equilibrium on the time interval $[n,N]$, with $\hat{p}^b<0<\hat{p}^a$. Such a construction always succeeds if the agents are market-neutral: i.e., $\alpha=0$. Indeed, in this case, assuming $\hat{\lambda}^a_{n+1} < 0 < \hat{\lambda}^b_{n+1}$, we have $\hat{p}^b_n < 0 < \hat{p}^a_n$ and $$\hat{\lambda}^a_{n+1} + \left({\mathbb E}\left( \left(\hat{p}^a_n - \hat{\lambda}^a_{n+1} - \xi\right) \bone_{\left\{\xi> \hat{p}^a_n ) \right\}} \right)\right)^+ = {\mathbb E}\left( \hat{\lambda}^a_{n+1} \bone_{\left\{\xi> \hat{p}^a_n ) \right\}} \right) + {\mathbb E}\left( \left(\hat{p}^a_n - \xi\right) \bone_{\left\{\xi> \hat{p}^a_n ) \right\}} \right) < 0.$$ Hence, $\hat{\lambda}^a_n < 0$, and, similarly, we deduce that $\hat{\lambda}^b_n>0$. By induction, we obtain a non-degenerate LTC equilibrium on $[0,N]$, for any $(N,\sigma)$, as long as $\alpha=0$. Corollary \[prop:main.smallspread\] shows that, as $N\rightarrow\infty$, the processes $(\hat{\lambda}^a,\hat{\lambda}^b)$ converge to zero, which means that the expected execution prices converge to the fundamental price. The latter is interpreted as *market efficiency* in the high-frequency trading regime: any market participant expects to buy or sell the asset at the fundamental price. The left hand side of Figure \[fig:3\] shows that the bid and ask prices also converge to the fundamental price if $\alpha=0$. This can be interpreted as a *positive liquidity effect* of increasing the trading frequency. However, the situation is quite different if $\alpha\neq 0$. Assume, for example, that $\alpha>0$. Then, the second line of (\[eq.RW.pan\]) implies that $\hat{\lambda}^a$ increases by, at least, $\alpha\Delta t$ at each step of the (backward) recursion. Recall that the number of steps is $N=T/\Delta t$, hence, $\hat{\lambda}^a_0 \geq \hat{\lambda}^a_N + \alpha T$. If $|\hat{\lambda}^a_N|$ is small (which is typically the case if $N$ is large), then, we may obtain $\hat{\lambda}^a_{n+1}\geq 0$, at some time $n$, which violates the third inequality in (\[eq.example.1period.ineq.1\]), or, equivalently, implies that the objective in the first line of (\[eq.RW.pan\]) is strictly negative for all $p$. The latter implies that it is suboptimal for the agents with positive inventory to post limit orders, and the proposed method fails to produce a non-degenerate LTC equilibrium in the interval $[n,N]$. Figure \[fig:2\] shows that this does, indeed, occur. Figures \[fig:2\] and \[fig:3\] also show that, for a given (finite) frequency $N$, if $|\alpha|$ is small enough, a non-degenerate equilibrium may still be constructed. Nevertheless, for any $|\alpha|\neq0$, however small it is, there exists a large enough $N$, s.t. the non-degenerate LTC equilibrium fails to exist (at least, within the class defined by the proposed method). This is illustrated in Figure \[fig:3\]. It is important to provide an economic interpretation of why such degeneracy occurs. A careful examination of Figure \[fig:2\] reveals that, around the time when $\hat{\lambda}^a$ becomes nonnegative, the ask price $\hat{p}^a$ explodes. This means that the agents who want to sell the asset are only willing to sell it at a very high price. Notice also that this price is several magnitudes larger than the expected change in the fundamental price (represented by the black dashed line in the left hand side of Figure \[fig:2\]). Hence, such a behavior cannot be justified by the behavior of the fundamental. Indeed, this is precisely what is called an *endogenous liquidity crisis*. So, what causes such a liquidity crisis? Recall that there are two potential reasons for the market to degenerate: agents may choose to submit market orders (if $\hat{p}^b_n>\hat{\lambda}^a_n$ or $\hat{p}^a_n<\hat{\lambda}^b_n$), or they may choose to wait and do nothing (if $\hat{\lambda}^a_{n+1}\geq 0$ or $\hat{\lambda}^b_{n+1}\leq 0$). The right hand side of Figure \[fig:2\] shows that the degeneracy is caused by the second scenario. This means that the naive explanation of an endogenous liquidity crisis, based on the claim that, in a bullish market, those who need to buy the asset will submit market orders wiping out liquidity on the sell side of the book, is wrong. Instead, if the agents on the sell side of the book have the same beliefs, they will increase the ask price so that it is no longer profitable for the agents who want to buy the asset to submit market buy orders. In fact, the ask price may increase disproportionally to the expected change in the fundamental price (i.e., the signal), and this is what causes an endogenous liquidity crisis. The size of the resulting change in the bid or ask price depends not only on the signal, but also on the trading frequency, which demonstrates the *negative liquidity effect* of increasing the trading frequency: it fragilizes the market with respect to deviations of the agents from market-neutrality. The latter, in turn, is explained by the fact that higher trading frequency exacerbates the *adverse selection* effect. To see this, consider, e.g., an agent who is trying to sell one share of the asset. Increasing the trading frequency increases the expected execution value of this agent, bringing it closer to the fundamental price: this corresponds to $\hat{\lambda}^a$ approaching zero (from below). Assume that the agent posts a limit sell order at a price level $p$. If this order is executed in the next period, then, the agent receives $p$, but, for this to happen, the fundamental price value at the next time step, $p^0_{n+1}$, has to be above $p$. On the other hand, the expected execution price of the agent at the next time step is $p^0_{n+1} + \hat{\lambda}^a_{n+1}$. Thus, the expected relative profit, given the execution of her limit order, is ${\mathbb E}_n (p - p^0_{n+1} - \hat{\lambda}^a_{n+1} \, |\,p^0_{n+1}>p )$. The latter expression cannot be positive, unless $\hat{\lambda}^a_{n+1}<0$ and $|\hat{\lambda}^a_{n+1}|$ is sufficiently large. Therefore, if $|\hat{\lambda}^a_{n+1}|$ is small relative to ${\mathbb E}_n (p^0_{n+1} - p\, |\,p^0_{n+1}>p)$, the agent is reluctant to post a limit order at the price level $p$. Hence, $p$ needs to be sufficiently large, to ensure that ${\mathbb E}_n (p^0_{n+1} - p\, |\,p^0_{n+1}>p )$ is smaller than $|\hat{\lambda}^a_{n+1}|$ (in the Gaussian model of this section, the latter expectation vanishes as $p\rightarrow\infty$) – and the extent to which $p$ needs to increase determines the effect of adverse selection. It turns out that, if the agents are market-neutral (i.e. $\alpha=0$), as the frequency $N$ increases, the quantity ${\mathbb E}_n (p^0_{n+1} - p\, |\,p^0_{n+1}>p )$, for any fixed $p$, converges to zero at the same rate as $|\hat{\lambda}^a_{n+1}|$, hence, the above adverse selection effect does not get amplified. On the contrary, if the agents are not market-neutral, $\hat{\lambda}^a_{n+1}$ reaches zero at some high enough (but finite) frequency, while ${\mathbb E}_n (p^0_{n+1} - p\, |\,p^0_{n+1}>p )$ remains strictly positive, for any finite $p$, which produces an “infinite" adverse selection effect and causes the market to degenerate. Of course, so far, these conclusions are based on a very specific example and on a particular method of constructing an equilibrium. The next section shows that they remain valid in any model (with, possibly, heterogeneous beliefs) in which the fundamental price is given by an It[ô]{} process. It is worth mentioning that a similar adverse selection effect arises in [@MMS.g3], and it is referred to as the “winner’s curse" in [@MMS.g2]. However, the latter papers do not investigate the nature of this phenomenon and focus on other questions instead. In the literature on double auctions (cf. [@DA.DuZhu], [@DA.Vayanos]), a similar effect arises when the auction participants choose to decrease their trading activity in a given auction, because they expect many more opportunities to trade in the future. The latter is similar to the agents choosing to forgo limit orders and wait, in the present example. Main results {#se:main} ============ In this section, we generalize the previous conclusions, so that they hold in a general model and for any equilibrium. As before, we begin with the “limiting" continuous time model. Consider a terminal time horizon $T>0$ and a complete stochastic basis $(\Omega,\tilde{{\mathbb F}}=(\tilde{\mathcal{F}}_t)_{t\in[0,T]},{\mathbb P})$, with a Brownian motion $W$ on it.[^13] We define the adapted process $\tilde{p}^0$ as a continuous modification of $$\label{eq.p0.cont} \tilde{p}^0_t = p^0_0 + \int_0^t \sigma_s dW_s,\,\,\,\,\,\,\,\,\,\,\,p^0_0 \in {\mathbb R},$$ where $\sigma$ is a progressively measurable locally square integrable process. \[ass:sigma\] There exists a constant $C>1$, such that, $1/C\leq \sigma_t\leq C$, for all $t\in[0,T]$, ${\mathbb P}$-a.s.. Consider a Borel set of beliefs $\mathbb{A}$ and the associated family of measures $\left\{{\mathbb P}^{\alpha}\right\}_{\alpha\in\mathbb{A}}$ on $(\Omega,\tilde{\mathcal{F}}_T)$, absolutely continuous with respect to ${\mathbb P}$. Then, for any $\alpha\in\mathbb{A}$, we have $$\label{eq.p0.cont.a} \tilde{p}^0_t = p^0_0 + A^{\alpha}_t + \int_0^t \sigma_s dW^{\alpha}_s,\quad p^0_0 \in {\mathbb R}, \quad {\mathbb P}^{\alpha}\text{-a.s.},\,\,\forall t\in[0,T],$$ where $W^{\alpha}$ is a Brownian motion under ${\mathbb P}^{\alpha}$, and $A^{\alpha}$ is a process of finite variation. We assume that $A^{\alpha}$ is absolutely continuous: i.e., for any $\alpha\in\mathbb{A}$, there exists a locally integrable process $\mu^{\alpha}$, such that $$A^{\alpha}_t = \int_0^t \mu^{\alpha}_s ds,\quad {\mathbb P}^{\alpha}\text{-a.s.},\,\,\forall t\in[0,T].$$ \[ass:A.alpha\] For any $\alpha\in\mathbb{A}$, the process $\mu^{\alpha}$ is ${\mathbb P}$-a.s. right-continuous, and there exists a constant $C>0$, such that $|\mu^{\alpha}_t| \leq C$, for all $t\in[0,T]$, ${\mathbb P}$-a.s.. Thus, we can rewrite the dynamics of $\tilde{p}^0$, under each ${\mathbb P}^{\alpha}$, as follows: ${\mathbb P}^{\alpha}$-a.s., the following holds for all $t\in[0,T]$ $$\label{eq.p0.cont.a.alpha} \tilde{p}^0_t = p^0_0 + \int_0^t \mu^{\alpha}_s ds + \int_0^t \sigma_s dW^{\alpha}_s,\quad p^0_0 \in {\mathbb R}.$$ In addition, we modify the above stochastic integral on a set of ${\mathbb P}^{\alpha}$-measure zero, so that (\[eq.p0.cont.a.alpha\]) holds for *all* $(t,\omega)$. In what follows, we often need to analyze the future dynamics of $\tilde{p}^0$ under ${\mathbb P}^{\alpha}$, conditional on $\tilde{\mathcal{F}}_t$, for various $(t,\alpha)$ simultaneously. This is why we need the following joint regularity assumption. \[ass:joint.cond.reg\] There exists a modification of regular conditional probabilities $$\left\{\tilde{{\mathbb P}}^{\alpha}_t={\mathbb P}^{\alpha}\left(\cdot\,|\,\tilde{\mathcal{F}}_t\right) \right\}_{t\in[0,T],\,\alpha\in\mathbb{A},}$$ such that it satisfies the tower property with respect to ${\mathbb P}$ (as described in Section \[se:setup\]). Assumption \[ass:joint.cond.reg\] is satisfied, for example, if ${\mathbb P}^{\alpha}\sim {\mathbb P}$, for all $\alpha\in\mathbb{A}$, or if the set $\mathbb{A}$ is countable. Throughout the rest of the paper, $\tilde{{\mathbb P}}^{\alpha}_t$ refers to a member of the family appearing in Assumption \[ass:joint.cond.reg\]. All conditional expectations $\tilde{{\mathbb E}}^{\alpha}_t$ are taken under such $\tilde{{\mathbb P}}^{\alpha}_t$. The main results of this section require additional continuity assumptions on $\sigma$ and $\mu^{\alpha}$. The following assumption can be viewed as a stronger version of $\mathbb{L}^2$-continuity of $\sigma$. \[ass:main.L2.strong\] There exists a function $\varepsilon(\cdot)\geq0$, such that $\varepsilon(\Delta t)\rightarrow0$, as $\Delta t\rightarrow0$, and, ${\mathbb P}$-a.s., $$\tilde{{\mathbb P}}^{\alpha}_{t} \left({\mathbb E}^{\alpha}\left(\left(\sigma_{s\vee\tau} - \sigma_{\tau} \right)^2 \,|\,\mathcal{F}_{\tau} \right) \leq \varepsilon(\Delta t) \right) = 1$$ holds for all $t\in[0,T-\Delta t]$, all $s\in[t,t+\Delta t]$, all stopping times $t\leq\tau\leq s$, and all $\alpha\in\mathbb{A}$. The above assumption is satisfied, for example, if $\sigma$ is an Itô process with bounded drift and diffusion coefficients. Next, we state a continuity assumption on the drift, which can be interpreted as a uniform right-continuity in probability of the martingale $\tilde{{\mathbb E}}^{\alpha}_{t} \mu^{\alpha}_s$. \[ass:main.mu.cont.strong\] For any $\alpha\in\mathbb{A}$ and any $t\in[0,T)$, there exists a deterministic function $\varepsilon(\cdot)\geq0$, such that $\varepsilon(\Delta t)\rightarrow0$, as $\Delta t\rightarrow0$, and, ${\mathbb P}^{\alpha}$-a.s., $$\tilde{{\mathbb P}}^{\alpha}_{t'} \left( \left| \int_{t}^T\left(\tilde{{\mathbb E}}^{\alpha}_{t''} \mu^{\alpha}_s - \tilde{{\mathbb E}}^{\alpha}_{t'} \mu^{\alpha}_s\right) ds\right| \geq \varepsilon(\Delta t)\right) \leq \varepsilon(\Delta t)$$ holds for all $t\leq t' \leq t'' \leq t+\Delta t\leq T$. Notice that Assumptions \[ass:joint.cond.reg\], \[ass:main.L2.strong\], and \[ass:main.mu.cont.strong\] are not quite standard. Therefore, below, we describe a more specific (although, still, rather general) diffusion-based framework, in which the Assumptions \[ass:sigma\]–\[ass:main.mu.cont.strong\] reduce to standard regularity conditions on the diffusion coefficients, and are easily verified. To this end, consider a model in which $\mu^{\alpha}_t = \bar{\mu}^{\alpha}(t,Y_t)$, $\sigma_t = \bar{\sigma}(t,Y_t)$, and, under ${\mathbb P}$, the process $Y$ is a diffusion taking values in ${\mathbb R}^d$ $$dY_t = \Gamma(t,Y_t)dt + \Sigma(t,Y_t) d\bar{B}_t,$$ where $\Gamma:[0,T]\times{\mathbb R}^d\rightarrow{\mathbb R}^d$, $\Sigma=(\Sigma^{i,j})$ is a mapping on $[0,T]\times{\mathbb R}^d$ with values in the space of $d\times m$ matrices, and $\bar{B}$ is $m$-dimensional Brownian motion under ${\mathbb P}$ (on the original stochastic basis). We assume that $\Gamma$ and $\Sigma$ possess enough regularity to conclude that $Y$ is a strongly Markov process. Notice that Assumptions \[ass:sigma\] and \[ass:A.alpha\] reduce to the upper and lower bounds on the functions $\bar{\mu}^{\alpha}$ and $\bar{\sigma}$. Assumption \[ass:joint.cond.reg\] is satisfied if we assume that ${\mathbb P}^{\alpha}\sim{\mathbb P}$, for all $\alpha\in\mathbb{A}$. Let us further assume that the Radon-Nikodym derivative of each measure is in Girsanov form: $$\frac{d{\mathbb P}^{\alpha}}{d{\mathbb P}} = \exp\left(-\frac{1}{2} \int_0^t \|\gamma^{\alpha}(s,Y_s)\|^2 ds + \int_0^t \gamma^{\alpha}(s,Y_s) d\bar{B}_s \right),$$ with an ${\mathbb R}^d$-valued function $\gamma^{\alpha}$, for each $\alpha\in\mathbb{A}$. Let us assume that all entries of $\Gamma$, $\gamma^{\alpha}$ and $\Sigma$ are absolutely bounded by a constant (uniformly over $\alpha\in\mathbb{A}$). Assuming, in addition, that $\bar{\sigma}$ is globally Lipschitz, we easily verify Assumption \[ass:main.L2.strong\]. In order to verify Assumption \[ass:main.mu.cont.strong\], we assume that the quadratic form generated by $A(t,y):=\Sigma(t,y) \Sigma^T(t,y)$ is bounded away from zero, uniformly over all $(t,y)$, and that the entries of $\Gamma$, $\gamma^{\alpha}$ and $\Sigma$ are continuously differentiable with absolutely bounded derivatives (uniformly over $\alpha\in\mathbb{A}$). Then, the Feynman-Kac formula implies that, for any $t\leq s$, $$\tilde{{\mathbb E}}^{\alpha}_t \mu^{\alpha}_s = u^{s,\alpha}(t,Y_t),$$ where $u^{s,\alpha}$ is the unique solution to the associated partial differential equation (PDE) $$\partial_t u^{s,\alpha} + \sum_{i=1}^d \Gamma^{\alpha,i} \partial_{y_i} u^{s,\alpha} + \frac{1}{2}\sum_{i,j=1}^d A^{i,j} \partial^2_{y_i y_j} u^{s,\alpha} = 0,\,\,\,\,(t,y)\in (0,s)\times{\mathbb R}^d, \quad u^{s,\alpha}(s,y)=\bar{\mu}^{\alpha}(s,y),$$ and $\Gamma^{\alpha}=\Gamma + \Sigma \gamma^{\alpha}$. Assume that, for each $s\in[0,T]$, the function $\bar{\mu}^{\alpha}(s,\cdot)$ is continuously differentiable with absolutely bounded derivatives, uniformly over all $(s,\alpha)$. Then, the standard Gaussian estimates for derivatives of the fundamental solution to the above PDE (cf. Theorem 9.4.2 in [@Friedman.book]) imply that every $\partial_{y_i}u^{s,\alpha}$ is absolutely bounded, uniformly over all $(s,\alpha)$. Then, Itô’s formula and Itô’s isometry yield, for all $t'\leq t''$ and $s\geq t'$: $$\tilde{{\mathbb E}}^{\alpha}_{t'}\left(\tilde{{\mathbb E}}^{\alpha}_{t''} \mu^{\alpha}_s - \tilde{{\mathbb E}}^{\alpha}_{t'} \mu^{\alpha}_s\right)^2 = \sum_{j=1}^m \int_{t'}^{t''\wedge s} \tilde{{\mathbb E}}^{\alpha}_{t'}\left(\sum_{i=1}^d\partial_{y_i} u^{s,\alpha}(v,Y_v)\Sigma^{i,j}(v,Y_v) \right)^2 dv \leq C_1 (t''\wedge s\,-\,t'),$$ with some constant $C_1>0$. The above estimate and Jensen’s inequality imply the statement of Assumption \[ass:main.mu.cont.strong\] and complete the description of the diffusion-based setting. As in Section \[se:examples\], we also consider a progressively measurable random field $\tilde{D}$, s.t. ${\mathbb P}$-a.s. the function $\tilde{D}_t(\cdot)-\tilde{D}_s(\cdot)$ is strictly decreasing and vanishing at zero, for any $0\leq s < t \leq T$. We assume that the demand curve, $\tilde{D}_t(\cdot)-\tilde{D}_s(\cdot)$, cannot be “too flat". \[ass:main.demandInv.unif\] There exists $\varepsilon>0$, s.t., for any $0\leq t - \varepsilon \leq s < t \leq T$, there exists a $\tilde{\mathcal{F}}_{s}\otimes \mathcal{B}({\mathbb R})$-measurable random function $\kappa_s(\cdot)$, s.t., ${\mathbb P}$-a.s., $\kappa_{s}(\cdot)$ is strictly decreasing and $\left|\tilde{D}_t(p)-\tilde{D}_s(p)\right| \geq \left| \kappa_{s}(p) \right|$, for all $p\in{\mathbb R}$. Finally, we introduce the empirical distribution process $(\tilde{\mu}_t)$, with values in the space of finite sigma-additive measures on $\mathbb{S}$. The next assumption states that every $\tilde{\mu}_t$ is dominated by a deterministic measure. \[ass:dom.mu\] For any $t\in[0,T]$, there exists a finite sigma-additive measure $\mu^{0}_t$ on $\left(\mathbb{S}, \mathcal{B}\left(\mathbb{S} \right)\right)$, s.t., ${\mathbb P}$-a.s., $\tilde{\mu}_t$ is absolutely continuous w.r.t. $\mu^0_t$. We partition the time interval $[0,T]$ into $N$ subintervals of size $\Delta t=T/N$. A discrete time model is obtained by discretizing the continuous time one $$\mathcal{F}_n = \tilde{\mathcal{F}}_{n\Delta t},\quad p^0_n = \tilde{p}^0_{n\Delta t},\quad D_n(p) = (\tilde{D}_{n\Delta t}-\tilde{D}_{(n-1)\Delta t})(p-p^0_n),\quad \mu_n = \tilde{\mu}_{n\Delta t}. $$ Before we present the main results, let us comment on the above assumptions. These assumptions are important from a technical point of view, however, some of them have economic interpretation that may provide (partial) intuitive explanations of the results that follow. In particular, Assumption \[ass:sigma\] ensures that the fundamental price remains “noisy," which implies that an agent can execute a limit order very quickly by posting it close to the present value of $p^0$, if there are no other orders posted there. In combination with Assumption \[ass:main.demandInv.unif\], the latter implies that, when the frequency, $N$, is high, an agent has a lot of opportunities to execute her limit order at a price close to the fundamental price (at least, if no other orders are posted too close to the fundamental price). Intuitively, this means that the agent’s execution value should improve as the frequency increases. Assumption \[ass:main.mu.cont.strong\] ensures that, if an agent has a signal about the direction of the fundamental price, this signal is persistent – i.e., it is continuous in the appropriate sense. When the trading frequency $N$ is large, such persistency means that an agent has a large number of opportunities to exploit the signal, implying that she is in no rush to have her order executed immediately. The main results of this work, presented below, along with their proofs, confirm that these heuristic conclusions are, indeed, correct. As mentioned in the preceding sections, our main goal is to analyze the liquidity effects of increasing the trading frequency. Therefore, we fix a limiting continuous time model, and consider a sequence of discrete time models, obtained from the limiting one as described above, for $N\rightarrow\infty$. This can be interpreted as observing the same population of agents, each of whom has a fixed continuous time model for future demand, in various exchanges that allow for different trading frequencies. We begin with the following theorem, which shows that, if every market model in a given sequence admits a non-degenerate equilibrium, then, the terminal bid and ask prices converge to the fundamental price, as the trading frequency goes to infinity. \[le:main.zeroTermSpread\] Let Assumptions \[ass:sigma\], \[ass:A.alpha\], \[ass:joint.cond.reg\], \[ass:main.L2.strong\], \[ass:main.demandInv.unif\], \[ass:dom.mu\] hold. Consider a family of uniform partitions of a given time interval $[0,T]$, with diameters $\left\{\Delta t=T/N>0\right\}$ and with the associated family of discrete time models, and denote the associated fundamental price process by $p^{0,\Delta t}$. Assume that every such model admits a non-degenerate LTC equilibrium, and denote the associated bid and ask prices by $p^{b,\Delta t}$ and $p^{a,\Delta t}$ respectively. Then, there exists a deterministic function $\varepsilon(\cdot)$, s.t. $\varepsilon(\Delta t)\rightarrow0$, as $\Delta t\rightarrow0$, and, for all small enough $\Delta t>0$, the following holds ${\mathbb P}$-a.s.: $$\left|p^{a,\Delta t}_{N} - p^{0,\Delta t}_{N}\right| + \left|p^{b,\Delta t}_{N} - p^{0,\Delta t}_{N}\right| \leq \varepsilon(\Delta t)$$ The above theorem has a useful corollary, which can be interpreted as follows: *if the market does not degenerate as the frequency increases, then, such an increase improves market efficiency*. Here, we understand the “improving efficiency" in the sense that the expected execution price (i.e., the price per share that an agent expects to receive or pay by the end of the game) of every agent converges to the fundamental price. \[prop:main.smallspread\] Under the assumptions of Theorem \[le:main.zeroTermSpread\], denote the support of every equilibrium by $\tilde{\mathbb{A}}^{\Delta t}$ and the associated expected execution prices by $\lambda^{a,\Delta t}$ and $\lambda^{b,\Delta t}$. Then, there exists a deterministic function $\varepsilon(\cdot)$, such that $\varepsilon(\Delta t)\rightarrow0$, as $\Delta t\rightarrow0$, and, ${\mathbb P}$-a.s., $$\sup_{n=0,\ldots,N,\,\alpha\in\tilde{\mathbb{A}}^{\Delta t}}\left(\left|\lambda^{a,\Delta t}_n(\alpha) - p^{0,\Delta t}_n\right| + \left|\lambda^{b,\Delta t}_n(\alpha) - p^{0,\Delta t}_n\right|\right) \leq \varepsilon(\Delta t),$$ for all small enough $\Delta t>0$. Denote ${\mathbb E}^{\alpha}_n = \tilde{{\mathbb E}}^{\alpha}_{n\Delta t}$. It follows from Corollary \[cor:piecewiseLin\], in Appendix A, and the definition of LTC equilibrium that $\lambda^{a,\Delta t}_{N}(\alpha) = p^{b,\Delta t}_N$ and $\lambda^{b,\Delta t}_{N}(\alpha)=p^{a,\Delta t}_N$. It also follows from Corollary \[cor:piecewiseLin\] (or, more generally, from the definition of a value function) that $\lambda^{a,\Delta t}(\alpha)$ is a supermartingale, and $\lambda^{b,\Delta t}(\alpha)$ is a submartingale, under ${\mathbb P}^{\alpha}$. Thus, we have: $\lambda^{a,\Delta t}_{n}(\alpha) \geq {\mathbb E}^{\alpha}_{n} p^{b,\Delta t}_{N}$ and $\lambda^{b,\Delta t}_{n}(\alpha) \leq {\mathbb E}^{\alpha}_{n} p^{a,\Delta t}_{N}$. On the other hand, notice that we must have: $\lambda^{a,\Delta t}_{n}(\alpha) \leq {\mathbb E}^{\alpha}_{n} p^{a,\Delta t}_{N}$ and $\lambda^{b,\Delta t}_{n}(\alpha) \geq {\mathbb E}^{\alpha}_{n} p^{b,\Delta t}_{N}$. Assume, for example, that $\lambda^{a,\Delta t}_{n}(\alpha) > {\mathbb E}^{\alpha}_{n} p^{a,\Delta t}_{N}$ on the event $\Omega'$ of positive ${\mathbb P}^{\alpha}$-probability. Consider an agent at state $(0,\alpha)$, who follows the optimal strategy of an agent at state $(1,\alpha)$, starting from time $n$ and onward, on the event $\Omega'$ (otherwise, she does not do anything). It is easy to see that the objective value of this strategy is $${\mathbb E}^{\alpha}\left( \bone_{\Omega'} \left( \lambda^{a,\Delta t}_{n}(\alpha) - {\mathbb E}^{\alpha}_{n} p^{a,\Delta t}_{N} \right)\right) > 0,$$ which contradicts Corollary \[cor:piecewiseLin\]. The second inequality is shown similarly. Thus, we conclude that, for any $n=0,\ldots,N-1$, both $\lambda^{a,\Delta}_{n}(\alpha)$ and $\lambda^{b,\Delta}_{n}(\alpha)$ belong to the interval $$\left[{\mathbb E}^{\alpha}_{n} p^{b,\Delta t}_{N},\,{\mathbb E}^{\alpha}_{n} p^{a,\Delta t}_{N}\right],$$ which, in turn, converges to zero, as $\Delta t\rightarrow0$, due to the deterministic bounds obtained in the proof of Proposition \[le:main.zeroTermSpread\]. The results of Theorem \[le:main.zeroTermSpread\] and Corollary \[prop:main.smallspread\] can be viewed as a specific case of a more general observation: markets become more efficient as the frictions become smaller. In the present setting, the limited trading frequency is viewed as friction, and the market efficiency is measured by the difference between the bid and ask prices, or between the expected execution prices. Many more instances of analogous results can be found in the literature, depending on the choice of a friction type. For example, the markets become efficient in [@MMS.gmm1] and [@MMS.gmm2] as the number of insiders vanishes. Similarly, the markets become efficient in [@DA.DuZhu] as the trading frequency increases and the size of private signals vanishes. It is also mentioned in [@MMS.gliq1] that the market would become efficient if there was no restriction on the size of agents’ inventories therein. The above results demonstrate the positive role of high trading frequency. However, they are based on the assumption that the market does not degenerate as frequency increases. In the context of Section \[se:examples\], we saw that the markets do not degenerate only if the agents are market-neutral (i.e. $\alpha=0$). If this condition is violated and the frequency $N$ is sufficiently high, the market does not admit any non-degenerate equilibrium (i.e., there exists no safe regime, in which the liquidity crisis would never occur). It turns out that this conclusion still holds in the general setting considered herein. \[thm:main.necessary\] Let Assumptions \[ass:sigma\], \[ass:A.alpha\], \[ass:joint.cond.reg\], \[ass:main.L2.strong\], \[ass:main.mu.cont.strong\], \[ass:main.demandInv.unif\], \[ass:dom.mu\] hold. Consider a family of uniform partitions of a given time interval $[0,T]$, with diameters $\left\{\Delta t=T/N>0\right\}$, containing arbitrarily small $\Delta t$, and with the associated family of discrete time models. Assume that every such model admits a non-degenerate LTC equilibrium, with the same support $\tilde{\mathbb{A}}$. Then, for all $\alpha\in\tilde{\mathbb{A}}$, we have: $\tilde{p}^{0}$ is a [**martingale**]{} under ${\mathbb P}^{\alpha}$. The above theorem shows that the market degenerates even if the signal $\mu^{\alpha}$ is very small (but non-zero), provided the trading frequency $N$ is large enough. Therefore, as discussed at the end of Section \[se:examples\], such degeneracy cannot be attributed to any fundamental reasons, and we refer to it as the *endogenous liquidity crisis*. Let us provide an intuitive (heuristic) argument for why the statement of Theorem \[thm:main.necessary\] holds. Assume, first, that all long agents (i.e., those having positive inventory) are bullish about the asset (i.e., have a positive drift $\mu^{\alpha}$). Then, similar to Section \[se:examples\], the higher trading frequency amplifies the *adverse selection effect*, forcing the long agents to withdraw liquidity from the market (i.e., they prefer to do nothing and wait for a higher fundamental price level). Note that, in the present setting, the agents may have different beliefs, the LOB may have a complicated shape and dynamics, and the expected execution prices are no longer deterministic. All this makes it difficult to provide a simple description of how the high frequency amplifies the adverse selection. Nevertheless, the general analysis of this case is still based on the idea discussed at the end of Section \[se:examples\]: it has to do with how fast $\tilde{{\mathbb E}}^{\alpha}_{n\Delta t} (p^0_{n+1} - p\, |\,p^0_{n+1}>p )$ vanishes (as the frequency increases), relative to the rate at which the expected execution prices approach the fundamental price. Thus, to avoid market degeneracy, there must be a non-zero mass of long agents who are market-neutral or bearish. As the trading frequency grows, these agents will post their limit orders at lower levels. Next, assume that there exists a bullish agent (long, short, or with zero inventory). Then, at a sufficiently high trading frequency, the agent’s expected value of a long position in a single share of the asset will exceed the ask prices posted by the market-neutral and bearish long agents. In this case, the bullish agent prefers to buy more shares at the posted ask price, in order to sell them later. As the agents are small and their objectives are linear, the bullish agent can scale up her strategy to generate infinite expected profits. This contradicts the definition of optimality and implies that an equilibrium fails to exist. Thus, all agents have to be either market-neutral or bearish. Applying a symmetric argument, we conclude that all agents must be market-neutral.[^14] A rigorous formulation of the above arguments, which constitutes the proof of Theorem \[thm:main.necessary\], is given in Section \[se:pf.2\]. It is worth mentioning that the possible degeneracy of the LOB is also documented in [@MMS.gmm1], and is referred to as a “market shut down". The setting used in the latter paper is very different: it analyzes a quote-driven exchange (i.e., the one with a designated market maker) and assumes the existence of insiders with superior information. Nevertheless, it is possible to draw a parallel with the LOB degeneracy in the present setting. Namely, the degeneracy in [@MMS.gmm1] occurs when the number of insiders increases, which implies that the signal, generated by the insiders’ trading, becomes sufficiently large. The latter is similar to the deviation from martingality of the fundamental price in the present setting. However, an increase in the number of insiders in [@MMS.gmm1] also implies an increase in frictions (since the insiders can be interpreted as friction in [@MMS.gmm1]). Theorem \[thm:main.necessary\], on the other hand, states that a market degeneracy will occur when the frictions are sufficiently small. Perhaps, this dual role of the number of insiders did not allow for a detailed analysis of market shut downs in [@MMS.gmm1]. Many other models of market microstructure (cf. [@MMS.g3], [@MMS.g6], [@MMS.g1], [@MMS.g2], [@DA.DuZhu]) are not well suited for the analysis of market degeneracy, either because the agents in these models pursue “one-shot" strategies (i.e., they cannot choose to wait and post a limit order later) or because the fundamental price (or its analogue) is restricted to be a martingale. Conditional tails of the marginal distributions of Itô processes {#se:tails} ================================================================ As follows from the discussion in the preceding sections, in order to prove the main results of the paper, we need to investigate the properties of marginal distributions of the fundamental price $\tilde{p}^0$ (more precisely, the distributions of its increments). In order to prove Theorem \[le:main.zeroTermSpread\], we need to show that the difference between the fundamental price and the bid or ask prices converges to zero, as the frequency $N$ increases to infinity. It turns out that, for this purpose, it suffices to show that the distribution of a normalized increment of $\tilde{p}^0$ converges to the standard normal distribution. The following lemma summarizes these results. It is rather simple, but technical, hence, its proof is postponed to Appendix B. In order to formulate the result (and to facilitate the derivations in subsequent sections), we introduce addiitonal notation. For notational convenience, we drop the superscript $\Delta t$ for some variables (we only emphasize this dependence when it is important). For any market model on the time interval $[0,T]$, associated with a uniform partition with diameter $\Delta t=T/N>0$, and having a fundamental price process $p^0$, we define $$\label{eq.xi.not} \xi_n = p^{0}_n - p^{0}_{n-1} = \tilde{p}^0_{t_n} - \tilde{p}^0_{t_{n-1}}, \quad {\mathbb E}^{\alpha}_n = \tilde{{\mathbb E}}^{\alpha}_{t_n}, \quad {\mathbb P}^{\alpha}_n = \tilde{{\mathbb P}}^{\alpha}_{t_n}, \quad t_n = n \Delta t, \quad n=1,\ldots,NT/\Delta t.$$ We denote by $\eta_0$ a standard normal random variable (on a, possibly, extended probability space), which is independent of $\mathcal{F}_N$ under every ${\mathbb P}^{\alpha}$. \[gapproxapplied\] Let Assumptions \[ass:sigma\], \[ass:A.alpha\], \[ass:joint.cond.reg\], \[ass:main.L2.strong\] hold. Then, there exists a function $\varepsilon(\cdot)\ge0$, s.t. $\varepsilon(\Delta t)\to0$, as $\Delta t\to0$, and the following holds ${\mathbb P}$-a.s., for all $p\in{\mathbb R}$, all $\alpha\in\mathbb{A}$, and all $n=1,\ldots,N$, - $(|p|\vee 1)\left|{\mathbb P}^{\alpha}_{n-1}\left(\frac{\xi_n}{\sqrt{\Delta t}} >p\right) - {\mathbb P}^{\alpha}_{n-1}\left(\sigma_{t_{n-1}}\eta_0>p\right)\right| \le\varepsilon(\Delta t)$, - $\left|{\mathbb E}^{\alpha}_{n-1}\left( \frac{\xi_n}{\sqrt{\Delta t}}\bone_{\left\{\xi_n/\sqrt{\Delta t}>p\right\}}\right) - {\mathbb E}^{\alpha}_{n-1}\left(\sigma_{t_{n-1}}\eta_0 \bone_{\left\{\sigma_{t_{n-1}}\eta_0>p\right\}}\right) \right|\le\varepsilon(\Delta t)$. In addition, the above estimates hold if we replace $(\xi_n,\eta_0,p)$ by $(-\xi_n,-\eta_0,-p)$. In order to prove Theorem \[thm:main.necessary\] we need to compare the rates at which the conditional expectations ${\mathbb E}^{\alpha}_n (p^0_{n+1} - p\, |\,p^0_{n+1}>p )$ vanish (as the frequency $N$ goes to infinity) to the rate at which the expected execution prices converge to the fundamental price. This requires a more delicate analysis – in particular, the mere proximity of the distribution of a (normalized) fundamental price increment to the Gaussian distribution is no longer sufficient. In fact, what we need is a precise uniform estimate of the conditional tail of the distribution of a fundamental price increment. The desired property is formulated in the following lemma, which, we believe, is valuable in its own right. This result enables us to estimate the tails of the conditional marginal distribution of an It[ô]{} process $X$ uniformly by an exponential. To the best of our knowledge, this result is new. The main difficulties in establishing the desired estimates are: (a) the fact that we estimate the *conditional*, as opposed to the regular, tail, and (b) the fact that the estimates need to be uniform over the values of the argument. Note that, even in the case of a diffusion process $X$, the classical Gaussian-type bounds for the tails of the marginal distributions of $X$ are not sufficient to establish the desired estimates. The reason is that, in general, the Gaussian estimates of the regular tails from above and from below have different orders of decay, for the large values of the argument, which makes them useless for estimating the conditional tail (which is a ratio of two regular tails). \[le:necessary.marginal.maximum\] Consider the following continuous semimartingale on a stochastic basis $(\hat{\Omega},(\hat{\mathcal{F}}_t)_{t\in[0,1]},\hat{{\mathbb P}})$: $$X_t = \int_0^t \hat{\mu}_u du + \int_{0}^t \hat{\sigma}_u dB_u,\,\,\,\,\,\,\,\,\,\,\,\,t\in[0,1],$$ where $B$ is a Brownian motion (with respect to the given stochastic basis), $\hat{\mu}$ and $\hat{\sigma}$ are progressively measurable processes, such that the above integrals are well defined. Assume that, for any stopping time $\tau$ with values in $[0,1]$, $c\leq |\hat{\sigma}_{\tau}| \leq C$ holds a.s. with some constants $c,C>0$. Then, there exists $\varepsilon>0$, depending only on $(c,C)$, s.t., if $$\hat{\mu}^2_{\tau}\leq \varepsilon,\quad \hat{{\mathbb E}}\left( (\hat{\sigma}_{s\vee\tau} - \hat{\sigma}_{\tau})^2 \,|\,\hat{\mathcal{F}}_{\tau} \right) \leq \varepsilon\,\,\,\, \text{a.s.},$$ for all $s\in[0,1]$ and all stopping time $\tau$, with values in $[0,1]$, then, for any $c_1>0$, there exists $C_1>0$, depending only on $(c,C,\varepsilon,c_1)$, s.t. the following holds: $$\hat{{\mathbb P}}(X_1 > x+z\,\vert\, X_1>x) \leq C_1 e^{-c_1 z},\quad\forall x,z\geq0.$$ In the course of this proof, we will use the shorthand notation, $\hat{{\mathbb E}}_{\tau}$ and $\hat{{\mathbb P}}_{\tau}$, to denote the conditional expectation and the conditional probability w.r.t $\hat{\mathcal{F}}_{\tau}$. We also denote $$A_t = \int_0^t \hat{\mu}_u du, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,G_t = \int_{0}^t \hat{\sigma}_u dB_u.$$ For any $x\geq0$, let us introduce $\tau_x = 1\wedge\inf\left\{t\in[0,1]\,:\, X_t = x \right\}$. Then $$\hat{{\mathbb P}}(X_1 > x+z) \leq \hat{{\mathbb P}}(\sup_{t\in[0,1]} X_t > x+z) = \hat{{\mathbb E}} \left( \bone_{\left\{ \tau_x < 1 \right\}} \hat{{\mathbb P}}_{\tau_x} \left(\sup_{s\in[\tau_x,1]} (X_s - x) > z \right) \right)$$ Notice that, on $\left\{\tau_x\leq s \right\}$, we have: $X_s - x = A_{s\vee\tau_x} - A_{\tau_x} + G_{s\vee\tau_x} - G_{\tau_x}$. In addition, the process $(Y)_{s\in[0,1]}$, with $Y_s = A_{s\vee\tau_x} - A_{\tau_x}$, is adapted to the filtration $(\hat{\mathcal{F}}_{\tau_x\vee s})$, while the process $(Z)_{s\in[0,1]}$, with $Z_s = G_{s\vee\tau_x} - G_{\tau_x}$, is a martingale with respect to it. Next, on $\left\{\tau_x < 1 \right\}$, we have: $$\hat{{\mathbb P}}_{\tau_x} \left(\sup_{s\in[\tau_x,1]} (X_s - x) > z \right) = \hat{{\mathbb P}}_{\tau_x} \left(\sup_{s\in[0,1]} (Y_s + Z_s) > z\right)$$ $$\leq \hat{{\mathbb P}}_{\tau_x} \left(\sup_{s\in[0,1]} \exp\left(c_1Z_s - \frac{1}{2}c_1^2\langle Z\rangle_s \right) > \exp\left( c_1z - c_1\sqrt{\varepsilon} - \frac{1}{2} c_1^2C^2\right)\right),$$ where we used the fact that $\langle Z\rangle_s \leq \langle X\rangle_1 \leq C^2$, for all $s\in[0,1]$. Using the Novikov’s condition, it is easy to check that $$M_s = \exp\left(c_1Z_s - \frac{1}{2}c_1^2\langle Z\rangle_s \right),\,\,\,\,\,\,\,\,\,s\in[0,1],$$ is a true martingale, and, hence, we can apply the Doob’s martingale inequality to obtain, on $\left\{\tau_x < 1 \right\}$: $$\hat{{\mathbb P}}_{\tau_x} \left(\sup_{s\in[0,1]} \exp\left(c_1Z_s - \frac{1}{2}c_1^2\langle Z\rangle_s \right) > \exp\left( c_1z - c_1\sqrt{\varepsilon} - \frac{1}{2}c_1^2 C^2\right)\right) \leq \exp\left( -c_1z + c_1\sqrt{\varepsilon} + \frac{1}{2} c_1^2 C^2\right).$$ Collecting the above inequalities, we obtain $$\label{eq.necessary.biglemma.step1.res} \hat{{\mathbb P}}(X_1 > x+z) \leq \hat{{\mathbb P}}(\sup_{t\in[0,1]} X_t > x+z) \leq C_2(\varepsilon) e^{-c_1z} \hat{{\mathbb P}}(\tau_x < 1) = C_2(\varepsilon) e^{-c_1z} \hat{{\mathbb P}}(\sup_{t\in[0,1]} X_t > x).$$ The next step is to estimate the distribution tails of a running maximum via the tails of the distribution of $X_1$. To do this, we proceed as before: $$\label{eq.necessary.biglemma.step2.1} \hat{{\mathbb P}}(X_1 > x) = \hat{{\mathbb E}} \left( \bone_{\left\{ \tau_x < 1 \right\}} \hat{{\mathbb P}}_{\tau_x}\left(Y_1 + Z_1 > 0\right) \right),$$ with $Y$ and $Z$ defined above. Notice that, on $\left\{\tau_x < 1 \right\}$, $$\hat{{\mathbb P}}_{\tau_x}\left(Y_1 + Z_1 > 0\right) = \hat{{\mathbb P}}_{\tau_x}\left(\hat{\sigma}_{\tau_x} \frac{B_1-B_{\tau_x}}{\sqrt{1-\tau_x}} + \frac{1}{\sqrt{1-\tau_x}} \int_{\tau_x}^{1} \hat{\mu}_{u} du + \frac{1}{\sqrt{1-\tau_x}} \int_{0}^{1} (\hat{\sigma}_{u\vee\tau_x} - \hat{\sigma}_{\tau_x}) dB^x_u > 0\right),$$ where $B^x_s = B_{s\vee\tau_x}$ is a continuous square-integrable martingale with respect to $(\hat{\mathcal{F}}_{s\vee\tau_x})$. Denote $$R_s = \int_{0}^{s} (\hat{\sigma}_{u\vee\tau_x} - \hat{\sigma}_{\tau_x}) dB^x_u, \quad s\in[0,1],$$ and notice that it is a square-integrable martingale with respect to $(\hat{\mathcal{F}}_{s\vee\tau_x})$. Then, on $\left\{\tau_x<1\right\}$ (possibly, without a set of measure zero), we have: $$\hat{{\mathbb E}}_{\tau_x} \left(\frac{1}{\sqrt{1-\tau_x}} R_1 \right)^2 =\frac{1}{1-\tau_x} \hat{{\mathbb E}}_{\tau_x} R^2_1 \leq \frac{1}{1-\tau_x}\int_{\tau_x}^{1} \hat{{\mathbb E}}_{\tau_x}(\hat{\sigma}_{u\vee\tau_x} - \hat{\sigma}_{\tau_x})^2 du \leq \varepsilon.$$ In addition, $$\hat{{\mathbb E}}_{\tau_x} \left( \frac{1}{\sqrt{1-\tau_x}} \int_{\tau_x}^{1} \hat{\mu}_{u} du \right)^2 \leq \varepsilon.$$ Collecting the above and using Chebyshev’s inequality, we obtain, on $\left\{\tau_x < 1 \right\}$: $$\left|\hat{{\mathbb P}}_{\tau_x}\left(Y_1 + Z_1 > 0\right) - \hat{{\mathbb P}}_{\tau_x}\left(\hat{\sigma}_{\tau_x} \frac{B_1-B_{\tau_x}}{\sqrt{1-\tau_x}} \leq -\varepsilon^{1/3} \right)\right| \leq 2\varepsilon^{1/6}.$$ On the other hand, due to the strong Markov property of Brownian motion, on $\left\{\tau_x<1\right\}$, we have, a.s.: $$\hat{{\mathbb P}}_{\tau_x}\left(\hat{\sigma}_{\tau_x} \frac{B_1-B_{\tau_x}}{\sqrt{1-\tau_x}} \leq -\varepsilon^{1/3} \right) = \left.\hat{{\mathbb P}} \left(\xi \leq -\frac{\varepsilon^{1/3}}{\sigma} \right)\right|_{\sigma=\hat{\sigma}_{\tau_x}},$$ where $\xi$ is a standard normal. As $\hat{\sigma}_{\tau_x}\in[c,C]$, we conclude that the right hand side of the above converges to $1/2$, as $\varepsilon\rightarrow0$, uniformly over almost all random outcomes in $\left\{\tau_x<1\right\}$. In particular, for all small enough $\varepsilon>0$, we have: $$\bone_{\left\{\tau_x<1 \right\}} \left|\hat{{\mathbb P}}_{\tau_x}\left(Y_1 + Z_1 \leq 0\right) - \hat{{\mathbb P}}_{\tau_x}\left(Y_1 + Z_1 > 0\right) \right| \leq \bone_{\left\{\tau_x<1 \right\}} \delta(\varepsilon)<1,$$ and, in view of (\[eq.necessary.biglemma.step2.1\]), $$\hat{{\mathbb P}}(X_1>x) \geq \hat{{\mathbb E}} \left( \bone_{\left\{ \tau_x < 1 \right\}} \hat{{\mathbb P}}_{\tau_x}\left(Y_1 + Z_1 \leq 0\right) \right) - \delta(\varepsilon) \hat{{\mathbb P}}(\tau_x<1)$$ Summing up the above inequality and (\[eq.necessary.biglemma.step2.1\]), we obtain $$2\hat{{\mathbb P}}(X_1>x) \geq (1-\delta(\varepsilon))\hat{{\mathbb P}}(\tau_x<1) = (1-\delta(\varepsilon))\hat{{\mathbb P}}(\sup_{t\in[0,1]}X_t > x),$$ which, along with (\[eq.necessary.biglemma.step1.res\]), yields the statement of the lemma. Proof of Theorem \[le:main.zeroTermSpread\] {#se:pf.1} =========================================== Within the scope of this proof, we adopt the notation introduced in (\[eq.xi.not\]) and use the following convention. \[not:shift\] The LOB, the bid and ask prices, the expected execution prices, and the demand, are all measured relative to $p^0$. Namely, we use $\nu_n$ to denote $\nu_n\circ (x\mapsto x+p^0_n)^{-1}$, $p^a_n$ to denote $p^a_n-p^0_n$, $p^b_n$ to denote $p^b_n-p^0_n$, $\lambda^a_n$ to denote $\lambda^a_n-p^0_n$, $\lambda^b_n$ to denote $\lambda^b_n-p^0_n$, and $D_n(p)$ to denote $D_n(p^0_n+p)$. Herein, we are only concerned with what happens in the last trading period – at time $(N-1)$, where $N=T/\Delta t$. Hence, we omit the subscript $N-1$ whenever it is clear from the context. In particular, we write $p^a$ and $p^b$ for $p^a_{N-1}$ and $p^b_{N-1}$, $\nu$ for $\nu_{N-1}$, and $\xi$ for $\xi_N$. Note also that, in an LTC equilibrium, we have: $p^a=p^a_N=p^a_{N-1}$, with similar equalities for $p^b$ and $\nu$. For convenience, we also drop the superscript $\Delta t$ in the LOB and the associated bid and ask prices. Finally, we denote by $\tilde{\mathbb{A}}$ the support of a given equilibrium. As the roles of $p^a$ and $p^b$ in our model are symmetric, we will only prove the statement of the proposition for $p^b$. We are going to show that, under the assumptions of the theorem, there exists a constant $C_0>0$, depending only on the constant $C$ in Assumptions \[ass:sigma\] and \[ass:A.alpha\], such that, for all small enough $\Delta t$, we have, ${\mathbb P}$-a.s.: $$\label{eq.prop1.target} -C_0\leq p^b/\sqrt{\Delta t} < 0$$ First, we introduce $\hat{A}^\alpha(p;x)$, which we refer to as the simplified objective: $$\label{eq.simp.obj.def} \hat{A}^\alpha(p;x)={\mathbb E}^\alpha_{N-1}\left((p-x-\xi)\bone_{\{\xi>p\}}\right).$$ Recall that the expected relative profit from posting a limit sell order at price level $p$, in the last time period,[^15] is given by $A^\alpha(p;p^b_{N})$, where $$\label{eq.true.obj.def} A^\alpha(p;x)={\mathbb E}^\alpha_{N-1}\left((p-x-\xi)\bone_{\{D^+_N(p-\xi)>\nu^+((-\infty,p))\}}\right).$$ The simplified objective is similar to $A^\alpha$, but it assumes that there are no orders posted at better prices than the one posted by the agent. In particular, $\hat{A}^\alpha(p;x)=A^\alpha(p;x)$ for $p\le p^a$. Corollary \[cor:piecewiseLin\], in Appendix A, states that, in equilibrium, ${\mathbb P}$-a.s., if the agents in the state $(s,\alpha)$ post limit sell orders, then they post them at a price level $p$ that maximizes the true objective $A^\alpha(p;p^b)$. The following lemma shows that the value of the modified objective becomes close to the value of the true objective, for the agents posting limit sell orders close to the ask price. \[le:simp.to.true.val\] ${\mathbb P}$-a.s., either $\nu^+(\{p^a\})>0$ or we have: $$\left\vert A^{\alpha}(p;p^b) - \hat{A}^{\alpha}(p^a;p^b)\right\vert\to0,$$ as $p\downarrow p^a$, uniformly over all $\alpha\in\tilde{\mathbb{A}}$. If $\nu^+(\{p^a\})=0$, then $\nu^+$ is continuous at $p^a$, and $\nu^+((-\infty,p])\rightarrow0$, as $p\downarrow p^a$. Then, we have $$\left| A^{\alpha}(p;p^b) - \hat{A}^{\alpha}(p^a;p^b)\right|$$ $$=\left| {\mathbb E}^{\alpha}_{N-1}\left((p - p^b-\xi)\bone_{\{D^+_N(p-\xi)>\nu^+((-\infty,p))\}}\right) - {\mathbb E}^{\alpha}_{N-1}\left((p^a-p^b-\xi)\bone_{\{\xi>p^a\}}\right)\right|$$ $$\le|p-p^a|+\left\Vert p^a-p^b-\xi\right\Vert_{\mathbb{L}^2\left({\mathbb P}^{\alpha}_{N-1}\right)}{\mathbb P}^{\alpha}_{N-1}\left(\xi>p^a,\,D^+_N(p-\xi)\le\nu^+((-\infty,p))\right)$$ Thus, it suffices to show that: (i) $\left\Vert p^a-p^b-\xi \right\Vert_{\mathbb{L}^2({\mathbb P}^{\alpha}_{N-1})}$ is bounded by a finite random variable independent of $\alpha$, and (ii) $${\mathbb P}^{\alpha}_{N-1}\left(\xi_N>p^a,\, D^+_N(p-\xi)\le\nu^+((-\infty, p))\right) \to0, \quad {\mathbb P}\text{-a.s.},$$ as $p\downarrow p^a$, uniformly over $\alpha$. For (i), we have: $$\left\Vert p^a-p^b-\xi \right\Vert_{\mathbb{L}^2({\mathbb P}^{\alpha}_{N-1})}\le |p^a-p^b|+\left\Vert\xi\right\Vert_{\mathbb{L}^2({\mathbb P}^{\alpha}_{N-1})} \leq |p^a-p^b| + 2C \sqrt{\Delta t},$$ where the constant $C$ appears in Assumptions \[ass:sigma\] and \[ass:A.alpha\]. For (ii), we note that $$\{\xi_N>p^a,\, D^+_N(p-\xi) \le \nu^+((-\infty,p))\} = \{\xi_N>p^a,\, \xi \le p - D^{-1}_N\left(\nu^+((-\infty,p))\right)\},$$ as $D_N(\cdot)$ is strictly decreasing, with $D_N(0)=0$. Assumption \[ass:main.demandInv.unif\] implies that $$\kappa^{-1}(\nu^+((-\infty,p)))\leq D^{-1}_N(\nu^+((-\infty,p))) < 0,$$ where $\kappa$ is known at time $N-1$. Therefore, $${\mathbb P}^{\alpha}_{N-1}\left(\xi>p^a,\, D^+_N(p-\xi)\le\nu^+((-\infty, p))\right) \leq {\mathbb P}^{\alpha}_{N-1} \left( \xi \in \left(p^a, p - \kappa^{-1}(\nu^+((-\infty,p))) \right] \right).$$ It remains to show that, ${\mathbb P}$-a.s., the right hand side of the above converges to zero, uniformly over all $\alpha$. Assume that it does not hold. Then, with positive probability ${\mathbb P}$, there exists $\varepsilon>0$ and a sequence of $(p_k,\alpha_k)$, such that $p_k\downarrow p^a$ and $${\mathbb P}^{\alpha_k}_{N-1} \left( \xi \in (p^a, p_k - \kappa^{-1}(\nu^+((-\infty,p_k))) ] \right) \geq \varepsilon.$$ Notice that, ${\mathbb P}$-a.s., the family of measures $\left\{ \hat{\mu}_k = {\mathbb P}^{\alpha_k}_{N-1}\circ \xi^{-1} \right\}_k$ is tight. The latter follows, for example, from the fact that, ${\mathbb P}$-a.s., the conditional second moments of $\xi$ are bounded uniformly over all $\alpha$ (which, in turn, is a standard exercise in stochastic calculus). Prokhorov’s theorem, then, implies that there is a subsequence of these measures that converges weakly to some measure $\hat{\mu}$ on ${\mathbb R}$. Next, notice that, for any fixed $k$ in the chosen subsequence, there exists a large enough $k'$, such that $$\left|\hat{\mu} \left( \left(p^a, p_k - \kappa^{-1}(\nu^+((-\infty,p_k))) \right] \right) - \mu_{k'}\left( \left(p^a, p_k - \kappa^{-1}(\nu^+((-\infty,p_k))) \right] \right)\right| \leq \varepsilon/2.$$ Thus, for any $k$ in the subsequence, we have $$\hat{\mu} \left( \left(p^a, p_k - \kappa^{-1}(\nu^+((-\infty,p_k))) \right] \right) \geq \varepsilon/2.$$ The above is a contradiction, as the intersection of the corresponding intervals, $(p^a, p_k - \kappa^{-1}(\nu^+((-\infty,p_k))) ]$, over all $k$ is empty. Now we are ready to prove the upper bound in (\[eq.prop1.target\]). \[bidasksigns\] In any non-degenerate LTC equilibrium, $p^b<0<p^a$, ${\mathbb P}$-a.s.. We only show that $p^b<0$ holds, the other inequality being very similar. Assume that $p^b\ge0$ on some positive ${\mathbb P}$-probability set $\Omega'\in\mathcal{F}_{N-1}$. We are going to show that this results in a contradiction. First, Corollary \[cor:piecewiseLin\], in Appendix A, implies that, ${\mathbb P}$-a.s., if the agents in state $(s,\alpha)$ post a limit sell order, then we must have: $\sup\limits_{p\in{\mathbb R}} A^{\alpha}(p;p^b) \geq0$. In addition, on $\Omega'$, we have: $\hat{A}^{\alpha}(p^a;p^b)<0$ for all $\alpha\in\tilde{\mathbb{A}}$, as $\xi$ has full support in ${\mathbb R}$ under every ${\mathbb P}^{\alpha}_{N-1}$ (which, in turn, follows from the fact that $\sigma$ is bounded uniformly away from zero). Then, Lemma \[le:simp.to.true.val\] implies that there exists a $\mathcal{F}_{N-1}$-measurable $\bar{p}\geq p^a$, such that, on $\Omega'$, the following holds a.s.: if $\nu^+(\{p^a\})=0$ then $\bar{p}>p^a$, and, in all cases, $$\label{eq.bidasksigns.ubopt} A^{\alpha}(p;p^b) < 0,\,\,\,\,\,\,\,\,\,\forall p\in[p^a, \bar{p}], \,\,\,\,\forall \alpha\in\tilde{\mathbb{A}}$$ Clearly, it is suboptimal for an agent to post a limit sell order below $\bar{p}$. However, an agent’s strategy only needs to be optimal up to a set of ${\mathbb P}$-measure zero, and these sets can be different for different $(s,\alpha)$. Therefore, a little more work is required to obtain the desired contradiction. Consider the set $B\subset \Omega'\times\mathbb{{\mathbb R}}\times\tilde{\mathbb{A}}$: $$B = \left\{(\omega,s,\alpha)\,|\, \hat{q}(s,\alpha)>0,\,\,\hat{p}(s,\alpha)\leq \bar{p} \right\}.$$ This set is measurable with respect to $\mathcal{F}_{N-1}\otimes \mathcal{B}\left(\mathbb{{\mathbb R}}\times\tilde{\mathbb{A}}\right)$, due to the measurability properties of $\hat{q}$ and $\hat{p}$. Notice that, due to the above discussion and the optimality of agents’ actions (cf. Corollary \[cor:piecewiseLin\], in Appendix A), for any $(s,\alpha)\in\mathbb{{\mathbb R}}\times\tilde{\mathbb{A}}$, we have: $${\mathbb P}(\left\{\omega\,|\, (\omega,s,\alpha)\in B \right\}) = 0,$$ and hence $${\mathbb E}_{N-1} \int_{\mathbb{{\mathbb R}}\times\tilde{\mathbb{A}}} \bone_{B}(\omega,s,\alpha) \mu_{N-1}(ds,d\alpha) = \int_{\mathbb{{\mathbb R}}\times\tilde{\mathbb{A}}} {\mathbb E}_{N-1}\left(\bone_{B}(\omega,s,\alpha) \rho_{N-1}(\omega,s,\alpha)\right) \mu^0_{N-1}(ds,d\alpha) = 0,$$ where $\rho_{N-1}$ is the Radon-Nikodym density of $\mu_{N-1}$ w.r.t. to the deterministic measure $\mu^0_{N-1}$ (cf. Assumption \[ass:dom.mu\]). The above implies that, ${\mathbb P}_{N-1}$-a.s., $\bone_{B}(\omega,s,\alpha)\rho_{N-1}(\omega,s,\alpha)=0$, for $\mu^0_{N-1}$-a.e. $(s,\alpha)$. Notice also that, for all $(\omega,s,\alpha)\in \Omega'\times\mathbb{{\mathbb R}}\times\tilde{\mathbb{A}}$, $$\bone_{\left\{\hat{p}(s,\alpha)\leq \bar{p} \right\}} \hat{q}^+(s,\alpha) \bone_{B^c} = 0.$$ From the above observations and the condition (\[eq.nuplus.fixedpoint.def\]) in the definition of equilibrium (cf. Definition \[def:equil.def\]), we conclude that, on $\Omega'$, the following holds a.s.: $$\nu^+([p^a,\bar{p}])=0,$$ where $\bar{p}\geq p^a$, and, if $\nu^+(\{p^a\})=0$, then $\bar{p}> p^a$. This contradicts the definition of $p^a$ (recall that $p^a$ is ${\mathbb P}$-a.s. finite, due to non-degeneracy of the LOB). It only remains to prove the lower bound on $p^b$ in (\[eq.prop1.target\]). Assume that it does not hold. That is, assume that there exists a family of equilibria, with arbitrary small $\Delta t$, and positive ${\mathbb P}$-probability $\mathcal{F}_{N-1}$-measurable sets $\Omega^{\Delta t}$, such that $p^b<-C_0\sqrt{\Delta t}$ on $\Omega^{\Delta t}$. We are going to show that this leads to a contradiction with $p^a>0$. To this end, assume that the agents maximize the simplified objective function, $\hat{A}^{\alpha}$, instead of the true one, $A^{\alpha}$. Then, if $p^b$ is negative enough, the optimal price levels become negative for all $\alpha$. The precise formulation of this is given by the following lemma. \[gap\] There exists a constant $C_0>0$, s.t., for any small enough $\Delta t$, there exist constants $\epsilon,\delta>0$, s.t., ${\mathbb P}$-a.s., we have $$\hat{A}^{\alpha}(-\delta;x)\ge\epsilon+\sup\limits_{y\ge0}\hat{A}^\alpha(y;x),$$ for all $\alpha\in\tilde{\mathbb{A}}$ and all $x\le-C_0\sqrt{\Delta t}$. Denote $\bar{\xi}=\xi/\sqrt{\Delta t}$ and consider the random function $$\bar{A}^\alpha(p;x)={{\mathbb E}^\alpha_{N-1}}\left((p-x-\bar{\xi})\bone_{\{\bar{\xi}>p\}}\right).$$ Notice that $$\hat{A}^\alpha(p;x)=\sqrt{\Delta t}\bar{A}^\alpha\left(p/\sqrt{\Delta t}; x/\sqrt{\Delta t}\right),$$ and, hence, we can reformulate the statement of the lemma as follows: there exists a constant $C_0>0$, s.t., for any small enough $\Delta t$, there exist constants $\epsilon,\delta>0$, s.t., ${\mathbb P}$-a.s., we have $$\bar{A}^{\alpha}(-\delta;x)\ge\epsilon+\sup\limits_{y\ge0} \bar{A}^\alpha(y;x),$$ for all $\alpha\in\tilde{\mathbb{A}}$ and all $x\le-C_0$. Notice that $$\begin{gathered} \bar{A}^\alpha(-\delta;x)-\bar{A}^\alpha(y;x) = -x{{\mathbb E}^\alpha_{N-1}}\left(\bone_{\{-\delta<\bar{\xi}\le y\}}\right) - {{\mathbb E}^\alpha_{N-1}}\left(\xi\bone_{\{-\delta<\bar{\xi}\le y\}}\right) - \delta{{\mathbb E}^\alpha_{N-1}}\left(\bone_{\{\bar{\xi}>-\delta\}}\right) - y{{\mathbb E}^\alpha_{N-1}}\left(\bone_{\{\bar{\xi}>y\}}\right)\end{gathered}$$ is non-increasing in $x$, and, hence, such is $\bar{A}^{\alpha}(-\delta;x)-\sup\limits_{y\ge0}\bar{A}^\alpha(y;x)$. Hence, it suffices to prove the above statement for $x=-C_0$. Next, consider the deterministic function $A_\sigma(p;x)$, defined via $$\label{eq.Asigma.def} A_\sigma(p;x)=\hat{{\mathbb E}}\left((p-x-\sigma\eta_0)\bone_{\{\sigma\eta_0>p\}}\right),$$ where $\eta_0$ is a standard normal random variable on some auxiliary probability space $(\hat{\Omega},\hat{{\mathbb P}})$. It follows from Lemma \[gapproxapplied\] that there exists a function $\varepsilon_2(\cdot)\ge0$, s.t. $\varepsilon_2(\Delta t)\to0$, as $\Delta t\to0$, and, ${\mathbb P}$-a.s., we have $$\left|\bar{A}^\alpha(p;-C_0)-A_{\sigma_{t_{N-1}}}(p;-C_0)\right|\le\varepsilon_2(\Delta t),$$ for all $\alpha\in\tilde{\mathbb{A}}$ and all $p\in{\mathbb R}$. Then, as we can always choose $\Delta t$ small enough, so that $\varepsilon_2(\Delta t)<\epsilon$, the statements of the lemma would follow if we can show that there exist constants $\epsilon,\delta,C_0>0$, s.t., ${\mathbb P}$-a.s., $$A_{\sigma_{t_{N-1}}}(-\delta;-C_0)\ge3\epsilon+\sup\limits_{y\ge0}A_{\sigma_{t_{N-1}}}(y;-C_0)$$ As $\sigma_{t_{N-1}}(\omega)\in[1/C,C]$, ${\mathbb P}$-a.s., it suffices to find $\epsilon,\delta,C_0>0$, s.t. $$A_\sigma(-\delta;-C_0)\ge3\epsilon+\sup\limits_{y\ge0}A_\sigma(y;-C_0), \quad\forall\,\sigma\in[1/C,C].$$ Note that the above inequality does not involve $\omega$ or $\xi$, and it is simply a property of a deterministic function. Notice also that $A_\sigma(p;x)=\sigma A_1\left(p/\sigma;x/\sigma\right)$, with $A_1$ given in (\[eq.Asigma.def\]). Then, if we denote by $F(x)$ and $f(x)$, respectively, the cdf and pdf of a standard normal, we obtain $$A_1(p;x)=(p-x)(1-F(p))-\int_p^{\infty} t f(t)\text{d}t.$$ A straightforward calculation gives us the following useful properties of $A_1$ and $A_\sigma$ \(i) For any $\sigma>0$ and any $x<0$, the function $p\mapsto A_\sigma(p;x)$ has a unique maximizer $p_\sigma(x)$, in particular, it is increasing in $p\le p_\sigma(x)$ and decreasing in $p\ge p_\sigma(x)$. \(ii) The function $$x\mapsto p_\sigma(x)=\sigma p_1(x/\sigma)=\sigma\left((1-F)/f\right)^{-1}(-x/\sigma)$$ is increasing in $x<0$ and converges to $-\infty$, as $x\to-\infty$. Then, choosing $C_0$ large enough, so that $p_1(-C_0/C)<0$, ensures $p_\sigma(-C_0)<0$, for all $\sigma\in[1/C,C]$. Setting $\delta=-p_1(-C_0/C)/C$ guarantees that $p_\sigma(-C_0)\le-\delta$, for all $\sigma\in[1/C,C]$. Then, by property (i) above, we have, for all $\sigma\in[1/C,C]$ $$A_\sigma(-\delta;-C_0)>A_\sigma(0;-C_0)=\sup\limits_{y\ge0}A_\sigma(y;-C_0).$$ Finally, as $A_\sigma(-\delta;-C_0)-A_\sigma(0;-C_0)$ is a continuous function of $\sigma\in[1/C,C]$, we can find $\epsilon$, such that $$A_\sigma(-\delta;-C_0)\ge3\epsilon+\sup\limits_{y\ge0}A_\sigma(y;-C_0), \quad\forall\,\sigma\in[1/C,C].$$ Recall that our assumption is that $p^b<-C_0\sqrt{\Delta t}$ holds on a set $\Omega^{\Delta t}$ of positive ${\mathbb P}$-measure. Recall also that $p^a>0$, ${\mathbb P}$-a.s., due to Lemma \[bidasksigns\]. Then, Lemmas \[le:simp.to.true.val\] and \[gap\] imply that there exists $\mathcal{F}_{N-1}$-measurable $\bar{p}\geq p^a$, s.t., on $\Omega^{\Delta t}$, we have a.s.: if $\nu^+(\{p^a\})=0$ then $\bar{p}>p^a$, and, in all cases, $$A^{\alpha}(p;p^b) < \sup_{p'\in{\mathbb R}} A^{\alpha}(p';p^b),\,\,\,\,\,\,\,\,\forall p\in[p^a,\bar{p}],\,\,\,\,\forall \alpha\in\tilde{\mathbb{A}}.$$ It is intuitively clear that posting limit sell orders at the above price levels $p$ must be suboptimal for the agents. However, the above inequality, on its own, does not yield a contradiction, as the agents’ strategies are only optimal up to a set of ${\mathbb P}$-probability zero, and these sets may be different for different states $(s,\alpha)$. To obtain a contradiction with the definition of $p^a$, we simply repeat the last part of the proof of Lemma \[bidasksigns\] (following equation (\[eq.bidasksigns.ubopt\])). This ensures that (\[eq.prop1.target\]) holds and completes the proof of the theorem. Proof of Theorem \[thm:main.necessary\] {#se:pf.2} ======================================= Within the scope of this proof, we adopt the notation introduced in (\[eq.xi.not\]) and use Notational Convention \[not:shift\] (i.e. we measure the LOB, the expected execution prices, and the demand, relative to $p^0$, but keep the same variables’ names). Assume that the statement of the theorem does not hold: i.e., there exists $\alpha_0\in\tilde{\mathbb{A}}$, such that $\tilde{p}^0$ is not a martingale under ${\mathbb P}^{\alpha_0}$. Then, there exists $s\in[0,T)$, s.t., with positive probability ${\mathbb P}^{\alpha_0}$, we have $$\tilde{{\mathbb E}}^{\alpha_0}_{s}\tilde{p}^0_T \neq \tilde{p}^0_{s}.$$ Without loss of generality, we assume that there exists a constant $\delta>0$ and a set $\Omega'\in\mathcal{F}_{s}$, having positive probability ${\mathbb P}^{\alpha_0}$ (and hence ${\mathbb P}$), s.t., for all random outcomes in $\Omega'$, we have $$\label{eq.thm1.delta.def} \tilde{{\mathbb E}}^{\alpha_0}_{s}(\tilde{p}^0_T - \tilde{p}^0_{s})\geq \delta$$ (the case of negative values is analogous). Next, we fix an arbitrary $\Delta t$ from a given family and consider the associated non-degenerate LTC equilibrium. \[le:necessary.1\] There exists a deterministic function $\varepsilon(\cdot)\geq0$, s.t. $\varepsilon(\Delta t)\rightarrow0$, as $\Delta t\rightarrow0$, and, for any small enough $\Delta t>0$, there exists $n=0,\ldots,N - 3$ and $\Omega''\in\mathcal{F}_n$, s.t. ${\mathbb P}^{\alpha_0}_n(\Omega'')>0$ and the following holds on $\Omega''$ $${\mathbb P}^{\alpha_0}_{n+2} \left( {\mathbb E}^{\alpha_0}_{n+3} \left(p^0_N - p^0_{n+3} \right) \leq \delta/2\right) \leq \varepsilon(\Delta t).$$ The proof follows from Assumption \[ass:main.mu.cont.strong\]. Consider $t=t'=s$ and $t'' = t_{n+2}$. Then, Assumption \[ass:main.mu.cont.strong\] implies $$\tilde{{\mathbb P}}^{\alpha_0}_{s} \left( \left|\tilde{{\mathbb E}}^{\alpha_0}_{t_{n+2}} \int_{s}^T \mu^{\alpha_0}_u du - \tilde{{\mathbb E}}^{\alpha_0}_{s} \int_{s}^T \mu^{\alpha_0}_u du\right| \geq \varepsilon(\Delta t)\right) \leq \varepsilon(\Delta t)$$ on $\Omega'$, a.s.. Notice also that $$\tilde{{\mathbb E}}^{\alpha_0}_{s} (\tilde{p}^0_T - \tilde{p}^0_{s}) = \tilde{{\mathbb E}}^{\alpha_0}_{s} \int \limits_{s}^T \mu^{\alpha_0}_u \text{d}u.$$ Then, assuming that $\varepsilon(\Delta t)$ is small enough and recalling (\[eq.thm1.delta.def\]), we obtain $$\tilde{{\mathbb P}}^{\alpha_0}_{s} \left( \tilde{{\mathbb E}}^{\alpha_0}_{t_{n+2}} \int_{s}^T \mu^{\alpha_0}_u du \leq 3\delta/4 \right) \leq \varepsilon(\Delta t),$$ on $\Omega'$. Therefore, there exists a set $\Omega''\in\mathcal{F}_{s}\subset\mathcal{F}_{t_{n}}$, s.t. $\tilde{{\mathbb P}}^{\alpha_0}_{t_{n}}(\Omega'')>0$ and $$\tilde{{\mathbb E}}^{\alpha_0}_{t_{n+2}} \int_{s}^T \mu^{\alpha_0}_u du \geq 3\delta/4,$$ on $\Omega''$. Next, we choose $t=s$, $t'=t_{n+2}$, $t'' = t_{n+3}$, and use Assumption \[ass:main.mu.cont.strong\], to obtain $$\tilde{{\mathbb P}}^{\alpha_0}_{t_{n+2}} \left( \left|\tilde{{\mathbb E}}^{\alpha_0}_{t_{n+3}} \int_{s}^T \mu^{\alpha_0}_u du - \tilde{{\mathbb E}}^{\alpha_0}_{t_{n+2}} \int_{s}^T \mu^{\alpha_0}_u du\right| \geq \varepsilon(\Delta t)\right) \leq \varepsilon(\Delta t),$$ on $\Omega''$, a.s.. Assuming that $\varepsilon(\Delta t)$ is small enough and using the last two inequalities, we obtain $$\tilde{{\mathbb P}}^{\alpha_0}_{t_{n+2}} \left( \tilde{{\mathbb E}}^{\alpha_0}_{t_{n+3}} \int_{s}^T \mu^{\alpha_0}_u du \leq \delta/2 \right) \leq \varepsilon(\Delta t).$$ Finally, due to Assumption \[ass:A.alpha\], and as $\Delta t$ is small, we can replace $\int_{s}^T \mu^{\alpha_0}_u du$ by $\int_{t_{n+3}}^T \mu^{\alpha_0}_u du$, and $\delta/2$ by $\delta/4$, in the above equation. This completes the proof of the lemma. Using the strategy at which the agent in state $(1,\alpha_0)$ waits until the last moment $n=N$, we conclude that the process $(\lambda^a_n(\alpha_0) + p^0_n)$ must be a supermartingale under ${\mathbb P}^{\alpha_0}$. More precisely, due to the definition of an optimal strategy, we have, ${\mathbb P}$-a.s. $$\lambda^a_{n+2}(\alpha_0) \geq {\mathbb E}^{\alpha_0}_{n+2} \lambda^a_N(\alpha_0) + {\mathbb E}^{\alpha_0}_{n+2}\left( {\mathbb E}^{\alpha_0}_{n+3}(p^0_N - p^0_{n+3}) + \xi_{n+3} \right).$$ Recall that $\lambda^a_N(\alpha_0) = p^b_N$ and, due to Theorem \[le:main.zeroTermSpread\] (more precisely, it follows from the proof of the theorem), there exists a constant $C_0>0$, s.t., for all small enough $\Delta t>0$, the following holds ${\mathbb P}$-a.s. $$-C_0\sqrt{\Delta t}\le p^{b}_N <0<p^{a}_N\le C_0\sqrt{\Delta t}.$$ Thus, we have, ${\mathbb P}$-a.s. $$\label{eq.necessary.lambdab.est.1} \lambda^a_{n+2}(\alpha_0) \geq -C_0\sqrt{\Delta t} + {\mathbb E}^{\alpha_0}_{n+2}\left( {\mathbb E}^{\alpha_0}_{n+3}(p^0_N - p^0_{n+3}) \right) + {\mathbb E}^{\alpha_0}_{n+2} \xi_{n+3}.$$ Due to Assumption \[ass:A.alpha\], we have, ${\mathbb P}$-a.s. $${\mathbb E}^{\alpha_0}_{n+2} \xi_{n+3} \leq C\Delta t, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \left|{\mathbb E}^{\alpha_0}_{n+3}(p^0_N - p^0_{n+3})\right| \leq C T,$$ and, hence, $$\lambda^a_{n+2}(\alpha_0) \geq -C_0\sqrt{\Delta t} + CT + C\Delta t.$$ In addition, making use of Lemma \[le:necessary.1\], we conclude that, for any small enough $\Delta t$, there exist $n=0,\ldots,N - 2$ and $\Omega''\in\mathcal{F}_n$, s.t. ${\mathbb P}^{\alpha_0}_n(\Omega'')>0$ and $${\mathbb P}^{\alpha}_{n+2} \left( {\mathbb E}^{\alpha}_{n+3} \left(p^0_N - p^0_{n+3} \right) \leq \delta/2\right) \leq \varepsilon(\Delta t),\,\,\,\,\,\text{on}\,\,\Omega''.$$ Using (\[eq.necessary.lambdab.est.1\]) and assuming that $\Delta t$ is small enough, we obtain $$\lambda^a_{n+2}(\alpha_0) \geq \delta/4,\,\,\,\,\,\,\,\text{on}\,\,\Omega''.$$ Next, Corollary \[cor:piecewiseLin\], in Appendix A, implies that, ${\mathbb P}$-a.s., $$p^b_{n+1}\geq {\mathbb E}^{\alpha_0}_{n+1}\left(\lambda^a_{n+2}(\alpha_0)+\xi_{n+2}\big\vert \xi_{n+2}<p^b_{n+1}\right).$$ Thus, on $\Omega''$, we obtain $$\label{eq.necessary.pa.est.1} p^b_{n+1} - {\mathbb E}^{\alpha_0}_{n+1}\left(\xi_{n+2}\big\vert\xi_{n+2}<p^b_{n+1}\right) \geq \delta /4.$$ The following lemma shows that, for any number $p$, the conditional expectation of the fundamental price increment, ${\mathbb E}^{\alpha_0}_{n+1}(\xi_{n+2}|\xi_{n+2}<p)$, approaches $p$ as the size of the time interval vanishes. This result follows from Lemma \[le:necessary.marginal.maximum\]. \[le:necessary.2\] There exists a constant $C_3>0$, s.t., for all small enough $\Delta t>0$, and for any $t\in[0,T-\Delta t]$, the following holds ${\mathbb P}$-a.s. $$\sup\limits_{p\le0}\left| p - \tilde{{\mathbb E}}^{\alpha_0}_t\left(\tilde{p}^0_{t+\Delta t} - \tilde{p}^0_t\,\big\vert\, \tilde{p}^0_{t+\Delta t}-\tilde{p}^0_t<p\right) \right| \leq C_3 \sqrt{\Delta t}.$$ Fix $t$ and $\Delta t>0$ and consider the evolution of $\tilde{p}^0_s$, for $s\in[t,t+\Delta t]$, under ${\mathbb P}^{\alpha_0}_t$ $$\tilde{p}^0_{s} - \tilde{p}^0_t = \int_t^s \mu^{\alpha_0}_u du + \int_t^s \sigma_u dW^{\alpha_0}_u,$$ where $W^{\alpha_0}$ is a Brownian motion under ${\mathbb P}^{\alpha_0}$. Rescaling by $\sqrt{\Delta t}$, we obtain $$(\tilde{p}^0_{s} - \tilde{p}^0_t)/\sqrt{\Delta t} = X_{(s-t)/\Delta t}, \quad X_s = \int_0^s \hat{\mu}_u du + \int_0^s \hat{\sigma}_u d\hat{W}_u, \quad s\in[0,1],$$ with $$\hat{\mu}_s = \sqrt{\Delta t} \, \mu^{\alpha_0}_{t+s\Delta t}, \quad \hat{\sigma}_s = \sigma_{t+s\Delta t}, \quad \hat{W}_s = \frac{1}{\sqrt{\Delta t}} \left(W^{\alpha_0}_{t+s\Delta t} - W^{\alpha_0}_t\right), \quad s\in[0,1].$$ Notice that the above processes are adapted to the new filtration $\hat{\mathbb{F}}$, with $\hat{\mathcal{F}}_s = \tilde{\mathcal{F}}_{t+s\Delta t}$, and, ${\mathbb P}$-a.s., under $\tilde{{\mathbb P}}^{\alpha_0}_t$, $\hat{W}$ is a Brownian motion with respect to $\hat{\mathbb{F}}$. Next, due to Assumptions \[ass:sigma\] and \[ass:main.L2.strong\], for any small enough $\Delta t>0$, ${\mathbb P}$-a.s., the dynamics of $(-X_s)$, under $\tilde{{\mathbb P}}^{\alpha_0}_t$, satisfy all the assumptions of Lemma \[le:necessary.marginal.maximum\]. As a result, we obtain $$\tilde{{\mathbb P}}^{\alpha_0}_t(X_1 < -x-z) \leq C_1 e^{-z} \tilde{{\mathbb P}}^{\alpha_0}_t(X_1 < -x), \,\,\,\,\,\,\,\,\,\,\,\,\,\forall x,z\geq0.$$ Finally, we notice that $$\sup\limits_{p\le0}\left| p - \tilde{{\mathbb E}}^{\alpha_0}_t\left(\tilde{p}^0_{t+\Delta t} - \tilde{p}^0_t\big\vert \tilde{p}^0_{t+\Delta t}-\tilde{p}^0_t<p\right)\right| = \sqrt{\Delta t} \sup\limits_{p\le0}\left| p - \tilde{{\mathbb E}}^{\alpha_0}_t\left(X_1\big\vert X_1<p\right)\right|$$ $$= \sqrt{\Delta t} \sup\limits_{p\le0}\left| p - \frac{\int_{-p}^{\infty} x \,d\,\tilde{{\mathbb P}}^{\alpha_0}_t(X_1 < -x) }{\tilde{{\mathbb P}}^{\alpha_0}_t(X_1 < p)} \right| = \sqrt{\Delta t} \sup\limits_{p\le0}\left| \frac{\int_{0}^{\infty} \tilde{{\mathbb P}}^{\alpha_0}_t(X_1 < p - z) dz}{\tilde{{\mathbb P}}^{\alpha_0}_t(X_1 < p)} \right| \leq C_1 \sqrt{\Delta t},$$ which completes the proof of the lemma. Using (\[eq.necessary.pa.est.1\]) and Lemma \[le:necessary.2\], we conclude that, for all small enough $\Delta t$, we have: $p^b_{n+1} > 0$ on $\Omega''$, ${\mathbb P}$-a.s.. In addition, Corollary \[cor:piecewiseLin\], in Appendix A, implies that, for any $\alpha\in\tilde{\mathbb{A}}$, the following holds ${\mathbb P}$-a.s. $$\lambda^a_{n+1}(\alpha) \geq p^b_{n+1}.$$ Next, with a slight abuse of notation (similar notation was introduced in the proof of Proposition \[le:main.zeroTermSpread\]), we consider the simplified objective of an agent who posts a limit sell order at the ask price $p^a_n$ $$\hat{A}^{\alpha}(p^a_n;\lambda^a_{n+1}) = {\mathbb E}^{\alpha}_n\left( p^a_n - \lambda^a_{n+1} - \xi_{n+1} \,|\, \xi_{n+1} > p^a_n \right).$$ The above estimates imply that, on $\Omega''$, we have, ${\mathbb P}$-a.s. $$\label{eq.necessary.simpObj.neg} \hat{A}^{\alpha}(p^a_n;\lambda^a_{n+1}) \leq {\mathbb E}^{\alpha}_n\left( p^a_n - \xi_{n+1} \,|\, \xi_{n+1} > p^a_n \right) - {\mathbb E}^{\alpha}_n\left( p^b_{n+1}\bone_{\Omega''} \,|\, \xi_{n+1} > p^a_n \right) < 0,\,\,\,\,\,\,\,\,\,\,\,\forall \alpha\in\tilde{\mathbb{A}}.$$ To obtain the last inequality in the above, we recall that $\Omega''\in\mathcal{F}_n$ and, ${\mathbb P}$-a.s., $\bone_{\Omega''}{\mathbb P}_n(\Omega\setminus\Omega'')=0$, $p^b_{n+1} > 0$ on $\Omega''$, and ${\mathbb P}^{\alpha}_n(\xi_{n+1} > p^a_n)>0$, for all $\alpha\in\tilde{\mathbb{A}}$. Next, repeating the proof of Lemma \[le:simp.to.true.val\] (and using the fact that $\lambda^a_{n+1}$ is absolutely bounded, as shown in Corollary \[prop:main.smallspread\]), we conclude that, ${\mathbb P}$-a.s., either $\nu^+_n(\{p^a_n\})>0$, or we have $$\left\vert A^{\alpha}(p;\lambda^a_{n+1}) - \hat{A}^{\alpha}(p^a_n;\lambda^a_{n+1})\right\vert\to0,$$ as $p\downarrow p^a$, uniformly over all $\alpha\in\tilde{\mathbb{A}}$, where we introduce the true objective, $$A^\alpha(p;\lambda^a_{n+1})={\mathbb E}^\alpha_{n}\left(\left(p-\lambda^a_{n+1}-\xi_{n+1}\right)\bone_{\{D^+_{n+1}(p-\xi_{n+1})>\nu^+_n((-\infty,p))\}}\right).$$ This convergence, along with (\[eq.necessary.simpObj.neg\]), implies that there exists a $\mathcal{F}_{n}$-measurable $\bar{p}\geq p^a_n$, such that, on $\Omega''$, the following holds ${\mathbb P}$-a.s.: if $\nu^+_n(\{p^a_n\})=0$ then $\bar{p}>p^a_n$, and, in all cases, $$A^{\alpha}(p;\lambda^a_{n+1}) < 0,\,\,\,\,\,\,\,\,\,\forall p\in[p^a_n, \bar{p}], \,\,\,\,\forall \alpha\in\tilde{\mathbb{A}}.$$ Finally, we repeat the last part of the proof of Lemma \[bidasksigns\] (following equation (\[eq.bidasksigns.ubopt\])), to obtain a contradiction with the definition of $p^a_n$, and complete the proof of the theorem. The last argument also shows that, when $\Delta t$ is small enough, it becomes suboptimal for the agents to post limit sell orders, as the expected relative profit from this action becomes negative, causing the market to degenerate. Summary and future work {#se:conclusion} ======================= In this paper, we present a new framework for modeling market microstructure, which does not require the existence of a designate market maker, and in which the LOB arises endogenously, as a result of equilibrium between multiple strategic players (aka agents). This framework is based on a continuum-player game. It closely approximates the mechanics of an auction-style exchange, so that, in particular, it can be used to analyze the liquidity effects of changes in the rules of the exchange. We use the proposed framework to study the liquidity effects of high trading frequency. In particular, we demonstrate the dual nature of high trading frequency. On the one hand, in the absence of a bullish or bearish signal about the asset, the higher trading frequency improves the efficiency of the market. On the other hand, at a sufficiently high trading frequency, even a very small trading signal may amplify the adverse selection effect, creating a disproportionally large change in the LOB, which is interpreted as an endogenous liquidity crisis. The present article raises many questions for further research. Notice that our main results are of a qualitative nature: they demonstrate the general behavior of the LOB, as a function of trading frequency, but do not immediately allow for computations. It would also be interesting to establish quantitative results. In particular, we would like to construct an equilibrium in a more realistic, and more concrete, model than the one used in Section \[se:examples\]. Such a model would allow for heterogeneous beliefs, and it would prescribe the specific sources of information (i.e., relevant market factors) used by the agents to form their beliefs. A model of this type could be calibrated to market data and used to study the effects of changes in relevant market parameters on the LOB. Finally, it would be interesting to develop a continuous time version of the proposed framework, in order to better capture the present state of the markets, where the trading frequency is not restricted. All these questions are the subject of our follow-up paper [@GaydukNadtochiy2]. Appendix A ========== This section contains several useful technical results on the representation of the value function of an agent in the proposed game. Notice that (\[eq.stateProc.def\]) and (\[eq.intro.Jm.def\]) imply that, if $\nu$ is admissible, then, for any $(\alpha,m,p,q,r)$, we have, ${\mathbb P}$-a.s. $$\left|J^{(p,q,r)}\left(m,s,\alpha,\nu\right) - J^{(p,q,r)}\left(m,s',\alpha,\nu\right)\right| \leq |s-s'|\, {\mathbb E}^{\alpha}_m |p^a_N| \vee |p^b_N|, \,\,\,\,\,\,\,\,\,\,\forall s,s'\in\mathbb{{\mathbb R}}$$ This implies that every $J^{(p,q,r)}\left(m,\cdot,\alpha,\nu\right)$ and $V^{\nu}_m(\cdot,\alpha)$ has a continuous modification under ${\mathbb P}$. Thus, whenever $\nu$ is admissible, we define the value function of an agent as the aforementioned continuous modification of the left hand side of (\[eq.gen.Val.randField\]). \[le:DPP\] Assume that an optimal control exists for an admissible LOB $\nu$. Assume also that, for any $\alpha\in\mathbb{A}$, the associated value function $V^{\nu}_n(\cdot,\alpha)$, defined in (\[eq.gen.Val.randField\]), is measurable with respect to $\mathcal{F}_n\otimes\mathcal{B}({\mathbb R})$. Then, it satisfies the following Dynamic Programming Principle. - For $n=N$ and all $(s,\alpha)\in\mathbb{S}$, we have, ${\mathbb P}$-a.s. $$\label{eq.het.VN} V^{\nu}_N(s,\alpha) = s^+ p^b_N - s^- p^a_N$$ - For all $n=N-1,\ldots,0$ and all $(s,\alpha)\in\mathbb{S}$, we have $$V^{\nu}_n(s,\alpha) = \text{esssup}_{p,q,r}\left\{\bone_{\left\{r_n=0\right\}}{\mathbb E}_n^{\alpha} \left( V^{\nu}_{n+1}\left(s,\alpha\right) + \left(q_n p_n + V^{\nu}_{n+1}\left(s-q_n,\alpha\right) - V^{\nu}_{n+1}\left(s,\alpha\right)\right)\cdot \right.\right.$$ $$\label{eq.het.Vn} \left.\left. \cdot\left( \bone_{\left\{q_n\geq0,\,D^+_{n+1}(p_n) > \nu^+_n((-\infty,p_n)) \right\}} + \bone_{\left\{q_n<0,\,D^-_{n+1}(p_n) > \nu^-_n((p_n,\infty)) \right\}} \right) \right)\right.$$ $$\left. + \bone_{\left\{r_n=1\right\}} \left( q^+_n p^b_n - q^-_n p^a_n + {\mathbb E}_n^{\alpha} V^{\nu}_{n+1}\left(s-q_n,\alpha\right) \right) \right\},$$ where the essential supremum is taken under ${\mathbb P}$, over all admissible controls $(p,q,r)$. The most important step is to show that, for all $n=0,\ldots N-1$ and $(s,\alpha)\in\mathbb{S}$, $$\label{eq.DPP.aux.1} V^{\nu}_n(s,\alpha) = \text{esssup}_{p,q,r} {\mathbb E}^{\alpha}_n \left( V^{\nu}_{n+1}\left(S^{n,s,(p,q,r)}_{n+1},\alpha\right) - g^{\nu}_{n}\left(p_n,q_n,r_n,D_{n+1}\right) \right),$$ where the essential supremum is taken under ${\mathbb P}$, over all admissible controls $(p,q,r)$, and $$g^{\nu}_{n}\left(p_n,q_n,r_n,D_{n+1}\right) = \left(p_n\bone_{\left\{ r_n = 0\right\}} + p^a_n\bone_{\left\{ r_n = 1, q_n <0\right\}} + p^b_n\bone_{\left\{ r_n = 1, q_n >0\right\}} \right) \Delta S^{n,s,(p,q,r)}_{n+1}$$ does not depend on $s$. Assume that $J^{(p,q,r)}\left(n,\cdot,\alpha,\nu\right)$ is a continuous modification of the objective function. Notice that, for all $m\leq k \leq n$, we have, ${\mathbb P}$-a.s. $${\mathbb E}^{\alpha}_k J^{(p,q,r)}\left(n,S_n^{m,s,(p,q,r)},\alpha,\nu\right) = J^{(p,q,r)}\left(k,S_k^{m,s,(p,q,r)},\alpha,\nu\right) + {\mathbb E}^{\alpha}_k \sum_{j=k}^{n-1} g^{\nu}_{j}\left(p_j,q_j,r_j,D_{j+1}\right)$$ Notice also that, for any $(p,q,r)$ we have, ${\mathbb P}$-a.s.: $J^{(p,q,r)}\left(m,s,\alpha,\nu\right) \leq V_m^\nu(s,\alpha)$, for all $s\in\mathbb{S}$. Let us show that the left hand side of (\[eq.DPP.aux.1\]) is less than its right hand side $$V_m^\nu(s,\alpha) =\text{essup}_{p,q,r} J^{(p,q,r)}\left(m,S_m^{m,s,(p,q,r)},\alpha,\nu\right)$$ $$=\text{essup}_{p,q,r} {\mathbb E}^{\alpha}_m \left( J^{(p,q,r)}\left(m+1,S_{m+1}^{m,s,(p,q,r)},\alpha,\nu\right) - g^{\nu}_{m}\left(p_m,q_m,r_m,D_{m+1}\right) \right)$$ $$\leq \text{essup}_{p,q,r} {\mathbb E}^{\alpha}_m \left( V^{\nu}_{m+1}\left(S_{m+1}^{m,s,(p,q,r)},\alpha\right) - g^{\nu}_{m}\left(p_m,q_m,r_m,D_{m+1}\right) \right)$$ Next, we show that the right hand side of (\[eq.DPP.aux.1\]) is less than its left hand side. For any $(p,q,r)$, we have, ${\mathbb P}$-a.s. $${\mathbb E}^\alpha_m \left(V^\nu_{m+1}\left(S_{m+1}^{m,s,(p,q,r)},\alpha\right) - g^{\nu}_{m}\left(p_m,q_m,r_m,D_{m+1}\right)\right)$$ $$={\mathbb E}^\alpha_m \left(J^{(\hat{p},\hat{q},\hat{r})}\left(m+1,S_{m+1}^{m,s,(p,q,r)},\alpha,\nu\right) - g^{\nu}_{m}\left(p_m,q_m,r_m,D_{m+1}\right)\right) = J^{(\tilde{p},\tilde{q},\tilde{r})}\left(m,s,\alpha,\nu\right) \leq V^{\nu}_m(s,\alpha),$$ where $(\tilde{p}_n,\tilde{q}_n,\tilde{r}_n)$ coincide with $(\hat{p}_n,\hat{q}_n,\hat{r}_n)$, for $n\geq m+1$, while they are equal to $(p_m,q_m,r_m)$, for $n=m$. The proof is completed easily by plugging the dynamics of the state process, (\[eq.stateProc.def\]), into (\[eq.DPP.aux.1\]). The following corollary provides a more explicit recursive formula for the value function and optimal control. In particular, it states that the value function of an agent at any time remains linear in $s$, in both positive and negative half lines (with possibly different slopes). \[cor:piecewiseLin\] Assume that an admissible LOB $\nu$ has an optimal control $(\hat{p},\hat{q},\hat{r})$. Then, for any $(s,\alpha)\in\mathbb{S}$, the following holds ${\mathbb P}$-a.s., for all $n=0,\ldots,N-1$ 1. $V^{\nu}_n(s,\alpha) = s^+ \lambda^a_n(\alpha) - s^- \lambda^b_n(\alpha)$, with some adapted processes $\lambda^a(\alpha)$ and $\lambda^b(\alpha)$, such that $\lambda^a_N(\alpha) = p^b_N$ and $\lambda^b_N(\alpha) = p^a_N$; 2. $p^a_n \geq {\mathbb E}_{n}^{\alpha} \left( \lambda^a_{n+1}(\alpha) \right)$ and $p^b_n \leq {\mathbb E}_{n}^{\alpha} \left( \lambda^b_{n+1}(\alpha) \right)$; 3. if, for some $p\in{\mathbb R}$, ${\mathbb P}^{\alpha}_n\left(D^+_{n+1}(p) > \nu^+_{n}((-\infty,p))\right)>0$, then $$\label{eq.het.NoPred.1} p \leq {\mathbb E}_{n}^{\alpha} \left( \lambda^b_{n+1}(\alpha) \,|\, D^+_{n+1}(p) > \nu^+_{n}((-\infty,p)) \right);$$ 4. if, for some $p\in{\mathbb R}$, ${\mathbb P}^{\alpha}_n\left(D^-_{n+1}(p) > \nu^-_{n}((p,\infty))\right)>0$, then $$\label{eq.het.NoPred.2} p \geq {\mathbb E}_{n}^{\alpha} \left( \lambda^a_{n+1}(\alpha) \,|\, D^-_{n+1}(p) > \nu^-_{n}((p,\infty)) \right);$$ 5. for all $s>0$, - $\lambda^a_n(\alpha) = \max\left\{ p^b_n, {\mathbb E}^{\alpha}_n \lambda^a_{n+1}(\alpha) + \left(\sup_{p\in{\mathbb R}} {\mathbb E}^{\alpha}_n \left( \left( p - \lambda^a_{n+1}(\alpha) \right) \bone_{\left\{ D^+_{n+1}(p) > \nu^+_{n}((-\infty,p)) \right\}} \right)\right)^+ \right\}$, - if $\hat{q}_n(s,\alpha)\neq 0$ and $\hat{r}_n(s,\alpha)=0$, then $$\lambda^a_n(\alpha) = {\mathbb E}^{\alpha}_n \lambda^a_{n+1}(\alpha) + \sup_{p\in{\mathbb R}} {\mathbb E}^{\alpha}_n \left( \left( p - \lambda^a_{n+1}(\alpha) \right) \bone_{\left\{ D^+_{n+1}(p) > \nu^+_{n}((-\infty,p)) \right\}} \right),$$ and $p=\hat{p}_n(s,\alpha)$ attains the above supremum, - if $\hat{q}_n(s,\alpha) = 0$ and $\hat{r}_n(s,\alpha)=0$, then $\lambda^a_n(\alpha) = {\mathbb E}^{\alpha}_n \lambda^a_{n+1}(\alpha)$, - if $\hat{r}_n(s,\alpha)=1$, then $\lambda^a_n(\alpha) = p^b_n$; 6. for all $s<0$, - $\lambda^b_n(\alpha) = \min\left\{ p^a_n, {\mathbb E}^{\alpha}_n \lambda^b_{n+1}(\alpha) - \left(\sup_{p\in{\mathbb R}} {\mathbb E}^{\alpha}_n \left( \left( \lambda^b_{n+1}(\alpha) - p \right) \bone_{\left\{ D^-_{n}(p) > \nu^-_{n-1}((p,\infty)) \right\}} \right)\right)^+ \right\}$, - if $\hat{q}_n(s,\alpha)\neq 0$ and $\hat{r}_n(s,\alpha)=0$, then $$\lambda^b_n(\alpha) = {\mathbb E}^{\alpha}_n \lambda^b_{n+1}(\alpha) - \sup_{p\in{\mathbb R}} {\mathbb E}^{\alpha}_n \left( \left( \lambda^b_{n+1}(\alpha) - p \right) \bone_{\left\{ D^-_{n}(p) > \nu^-_{n-1}((p,\infty)) \right\}} \right),$$ and $p=\hat{p}_n(s,\alpha)$ attains the above supremum, - if $\hat{q}_n(s,\alpha) = 0$ and $\hat{r}_n(s,\alpha)=0$, then $\lambda^b_n(\alpha) = {\mathbb E}^{\alpha}_n \lambda^b_{n+1}(\alpha)$, - if $\hat{r}_n(s,\alpha)=1$, then $\lambda^b_n(\alpha) = p^a_n$. Let us plug the piecewise-linear form of the value function into (\[eq.het.Vn\]) $$V^{\nu}_n(s,\alpha) = \text{esssup}_{p,q,r}\left\{\bone_{\left\{r_n=0\right\}} \left( s^+ {\mathbb E}_n^{\alpha} \lambda^a_{n+1}(\alpha) - s^- {\mathbb E}_n^{\alpha} \lambda^b_{n+1}(\alpha) \right.\right.$$ $$\left.\left. + {\mathbb E}_n^{\alpha} \left( \left(q_n p_n + (s-q_n)^+ \lambda^a_{n+1}(\alpha) - (s-q_n)^- \lambda^b_{n+1}(\alpha) - s^+ \lambda^a_{n+1}(\alpha) + s^- \lambda^b_{n+1}(\alpha) \right)\cdot \right.\right.\right.$$ $$\left.\left.\left. \left( \bone_{\left\{q_n\geq0,\,D^+_{n+1}(p_n) > \nu^+_n((-\infty,p_n)) \right\}} + \bone_{\left\{q_n<0,\,D^-_{n+1}(p_n) > \nu^-_n((p_n,\infty)) \right\}} \right)\right) \right)\right.$$ $$\left. + \bone_{\left\{r=1\right\}} \left( q^+_n p^b_n - q^-_n p^a_n + (s-q_n)^+ {\mathbb E}_n^{\alpha} \lambda^a_{n+1}(\alpha) - (s-q_n)^- {\mathbb E}_n^{\alpha} \lambda^b_{n+1}(\alpha) \right) \right\}$$ First, notice that it suffices to consider the essential supremum over all random variables $(p_n,q_n,r_n)$.[^16] Moreover, the essential supremum can be replaced by the supremum over all deterministic $(p_n,q_n,r_n)\in{\mathbb R}^2\times\{0,1\}$. To see the latter, it suffices to assume that the supremum is not attained by the optimal strategy (with positive probability), and construct a superior strategy via the standard measurable selection argument (cf. Corollary 18.27 and Theorem 18.26 in [@Aliprantis]), which results in a contradiction. It is easy to see that, for any fixed $(p_n,s,r_n)$, the above function is piece-wise linear in $q_n$, with the slope changing at $q_n=0$ and $q_n=s$. Hence, for a finite maximum to exists, the slope of this function must be nonnegative, at $q_n\rightarrow-\infty$, and non-positive, at $q_n\rightarrow \infty$. This must hold for any $(p_n,r_n,s)$, to ensure that the value function of an agent is finite: otherwise, an agent can scale up her position to increase the value function arbitrarily. Considering $r_n=1$, we obtain condition 2 of the corollary. The case $r_n=0$ yields conditions 3 and 4. Notice also that the maximum of the aforementioned function is always attained at $q_n=0$ or $q_n=s$. Considering all possible cases: $r_n=0,1$, $q_n=0,s$, $s=0$, $s>0$ and $s<0$ – we obtain the recursive formulas for $\lambda^a_n$ and $\lambda^b_n$ (i.e., conditions 5 and 6 of the corollary). In addition, as the optimal $q_n$ takes values $0$ and $s$, it is easy to see that the piece-wise linear structure of the value function in $s$ is propagated backwards, and, hence, condition 1 of the corollary holds. It is also useful to have a converse statement. \[cor:piecewiseLin.verif\] Consider an admissible LOB $\nu$ and admissible control $(\hat{p},\hat{q},\hat{r})$, such that $\hat{q}_n(s,\alpha)\in\left\{ 0,s\right\}$. Assume that, for any $\alpha\in\mathbb{A}$ and any $n=0,\ldots,N$, there exists a progressively measurable random function $V^{\nu}_{\cdot}(\cdot,\alpha)$, such that, for any $s\in\mathbb{{\mathbb R}}$, ${\mathbb P}$-a.s., $(\hat{p},\hat{q},\hat{r},V^{\nu})$ satisfy the conditions 1–6 of Corollary \[cor:piecewiseLin\]. Then, $(\hat{p},\hat{q},\hat{r})$ is an optimal control for the LOB $\nu$. It suffices to revert the arguments in the proof of Corollary \[cor:piecewiseLin\], and recall that $\hat{q}$ can always be chosen to be equal to $0$ or $s$, without compromising the optimality. Appendix B ========== *Proof of Lemma \[gapproxapplied\]*. The following lemma shows that the normalized price increments are close to Gaussian in the conditional $\mathbb{L}^2$ norm. \[l2conv\] Let Assumptions \[ass:sigma\], \[ass:A.alpha\], \[ass:joint.cond.reg\], \[ass:main.L2.strong\] hold. Then, there exists a deterministic function $\epsilon(\cdot)\ge0$, such that $\epsilon(\Delta t)\to0$, as $\Delta t\to0$, and, ${\mathbb P}$-a.s., for all $\alpha\in\mathbb{A}$ and all $n=1,\ldots,N$, we have $${\mathbb E}^\alpha_{n-1}\left(\left(\xi_n/\sqrt{\Delta t} - \sigma_{t_{n-1}}(W^\alpha_{t_n}-W^\alpha_{t_{n-1}})/\sqrt{\Delta t}\right)^2\right) \le \epsilon(\Delta t).$$ Notice: $\xi_n/\sqrt{\Delta t} - \sigma_{t_{n-1}}(W^\alpha_{t_n}-W^\alpha_{t_{n-1}})/\sqrt{\Delta t} =\frac{1}{\sqrt{\Delta t}} \int\limits_{t_{n-1}}^{t_n}\mu^\alpha_s \text{d}s + \frac{1}{\sqrt{\Delta t}} \int\limits_{t_{n-1}}^{t_n}(\sigma_s-\sigma_{t_{n-1}}) \text{d}W^\alpha_s$. Then, using Assumptions \[ass:A.alpha\], \[ass:main.L2.strong\], and Itô’s isometry, we obtain the statement of the lemma. The next lemma connects the proximity in terms of $\mathbb{L}^2$ norm and the proximity of expectations of certain functions of random variables. This result would follow trivially from the classical theory, but, in the present case, we require additional uniformity – hence, a separate lemma is needed (whose proof is, nevertheless, quite simple). \[gapprox\] For any constant $C>1$, there exists a deterministic function $\gamma(\cdot)\ge0$, s.t. $\gamma(\varepsilon)\to0$, as $\varepsilon\to0$, and, for any $\varepsilon>0$, $\sigma\in[1/C,C]$, and any random variables $\eta\sim\mathcal{N}(0,\sigma^2)$ and $\xi$ (the latter is not necessarily Gaussian), satisfying ${\mathbb E}(\xi-\eta)^2\le\varepsilon$, the following holds for all $p\in{\mathbb R}$ - $(|p|\vee 1)\left| {\mathbb P}(\xi>p) - {\mathbb P}(\eta>p) \right|\le\gamma(\varepsilon)$, - $\left| {\mathbb E}(\xi\bone_{\{\xi>p\}}) - {\mathbb E}(\eta\bone_{\{\eta>p\}}) \right|\le\gamma(\varepsilon)$. \(ii) Note that $$\left| {\mathbb E}(\xi\bone_{\{\xi>p\}}) - {\mathbb E}(\eta\bone_{\{\eta>p\}}) \right| \le \left| {\mathbb E}\left((\xi-\eta)\bone_{\{\xi>p\}}\right) \right| + \left|{\mathbb E}\left(\eta(\bone_{\{\xi>p\}}-\bone_{\{\eta>p\}})\right)\right|$$ $$\leq \sqrt{\varepsilon} + \left\Vert\eta\right\Vert_2 \sqrt{{\mathbb P}(\xi>p,\eta\le p) + {\mathbb P}(\xi\le p,\eta>p)},$$ and $${\mathbb P}(\xi>p,\eta\le p) \le {\mathbb P}(p\ge\eta\ge p-\sqrt[3]{\varepsilon}) + {\mathbb P}(|\xi-\eta|>\sqrt[3]{\varepsilon}) \le M\sqrt[3]{\varepsilon}+\frac{{\mathbb E}(\xi-\eta)^2}{(\sqrt[3]{\varepsilon})^2}\le (M+1)\sqrt[3]{\varepsilon},$$ where we used the fact that $\eta$ has a density bounded by a fixed constant $M$. We can similarly show that ${\mathbb P}[\xi\le p,\eta>p]\le(M+1)\sqrt[3]{\varepsilon}$. The resulting estimates yield the statement of the lemma. Taking $\varepsilon(\Delta t)=\gamma(\epsilon(\Delta t))$ and applying the above lemmas, we get the statement of Lemma \[gapproxapplied\], with $(W^\alpha_{t_n}-W^\alpha_{t_{n-1}})/\sqrt{\Delta t}$ in place of $\eta_0$. Finally, we note that the laws of the two random variables coincide under ${\mathbb P}^{\alpha}_{n-1}$, and the statement depends only on these laws. The last statement of Lemma \[gapproxapplied\] follows from the fact that Lemma \[gapprox\] is stable under analogous substitution. [cc]{} [ ]{} & [ ]{}\ [cc]{} [ ]{} & [ ]{}\ [^1]: Partial support from the NSF grant DMS-1411824 is acknowledged by both authors. [^2]: We thank the anonymous referees and the Associate Editor for constructive comments that helped us improve the paper significantly. [^3]: Address the correspondence to Sergey Nadtochiy, Mathematics Department, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, USA; e-mail: sergeyn@umich.edu. [^4]: This version: February 13, 2017. First version August 28, 2015. [^5]: We do not distinguish the “aggressive" limit orders, which are posted at the price level of an opposite limit order, and treat them as market orders. This causes no loss of generality, as the market participants in our setting have a perfect observation of the LOB. [^6]: This assumption holds, for example, if $\mathcal{F}_N$ is generated by a random element with values in a standard Borel space. [^7]: Note that, although ${\mathbb P}^{\alpha}$ does not change over time, the conditional distribution of the future demand, as perceived by the agent, changes dynamically, according to the new information received. [^8]: Note each agent is only allowed to place her limit order at a single price level, at any given time. However, this entails no loss of optimality. Indeed, using the Dynamic Programming Principle derived in Appendix A, one can show, by induction, that, in equilibrium, an agent does not benefit from posting multiple limit orders at the same time. As shown in [@Schmeidler], this is typical for a continuum-player game. [^9]: In order to ensure the existence of regular conditional probabilities for the discrete time model, we can, for example, assume that $\tilde{\mathcal{F}}_T$ is generated by a random element with values in a standard Borel space. [^10]: The execution of limit orders simplifies in the chosen ansatz, because the agents on each side of the book (i.e., long or short) post orders at the same prices. [^11]: In fact, it is not difficult to prove rigorously that, for any $(\alpha,\sigma)$, there exists a unique solution to such a system, provided $\Delta t$ is small enough. We omit this result for the sake of brevity. [^12]: This is easy to explain intuitively, as the optimal objective values in the first two lines of (\[eq.ex.singleStep.1\]) are of the form $C\sqrt{\Delta t} + \alpha \underline{\underline{O}}(\Delta t)$. [^13]: In order to ensure the existence of regular conditional probabilities for the discrete time model, we can, for example, assume that $\tilde{\mathcal{F}}_T$ is generated by a random element with values in a standard Borel space. [^14]: This argument, along with the fact that Definition \[def:optControl\] requires an optimal control to be optimal for *all* $\alpha$, explains why the statement of Theorem \[thm:main.necessary\] holds for *all*, as opposed to $\mu_n$-a.e., $\alpha\in\tilde{\mathbb{A}}$. [^15]: Recall that everything is measured relative to the fundamental price, according to the Notational Convention \[not:shift\] [^16]: The admissibility constraint does not cause any difficulties here, as, in the case where $(p_n,q_n,r_n)$ do not attain the supremum, they can be improved, so that $(p_n,q_n)$ increase by no more than a fixed constant.
--- abstract: 'We present a conceptual and uniform interpretation of the methods of integral representations of $L$-functions (period integrals, Rankin-Selberg integrals). This leads to: (i) a way to classify of such integrals, based on the classification of certain embeddings of spherical varieties (whenever the latter is available), (ii) a conjecture which would imply a vast generalization of the method, and (iii) an explanation of the phenomenon of “weight factors” in a relative trace formula. We also prove results of independent interest, such as the generalized Cartan decomposition for spherical varieties of split groups over $p$-adic fields (following an argument of Gaitsgory and Nadler).' author: - Yiannis Sakellaridis title: | Spherical varieties and integral representations\ of $L$-functions. --- Introduction ============ Goals ----- The study of automorphic $L$-functions (and their special values at distinguished points, or *$L$-values*) is very central in many areas of present-day number theory, and an incredible variety of methods has been developed in order to understand the properties of these mysterious objects and their deep links with seemingly unrelated arithmetic invariants. Oddly enough, notwithstanding their elegant and very general definition by Langlands in terms of Euler products, virtually all methods for studying them depart from an integral construction of the form: > A suitable automorphic form (considered as a function on the automorphic quotient $[G]:=G(k)\backslash G({{\mathbb{A}_k}})$), integrated against a suitable distribution on $G(k)\backslash G({{\mathbb{A}_k}})$, is equal to a certain $L$-value. For “geometric” automorphic forms, such an integral can often be expressed as a a pairing between elements in certain homology and cohomology groups, but the essence remains the same. Given the importance of such methods, it appears as a paradox that there is no general theory of integral representations of $L$-functions and, in fact, they are often considered as “accidents”. In this article I present a uniform interpretation of a large array of such methods, which includes Tate integrals, period integrals and Rankin-Selberg integrals. This interpretation leads to the first systematic classification of such integrals, based on the classification of certain spherical varieties (see sections \[secrs\] and \[secsmoothaffine\]). Moreover, it naturally gives rise to a very general conjecture (Conjecture \[mainconjecture\]), whose proof would lead to a vast extension of the method and would allow us to study many more $L$-functions than are within our reach at this moment. Finally, it explains phenomena which have been observed in the theory of the relative trace formula, in a way that is well-suited to the geometric methods employed in the proof of the fundamental lemma by Ngô [@Ngo]. In the course of the article we also prove some results which can be of independent interest, including results on the orbits of hyperspecial and congruence subgroups on the $p$-adic points of a spherical variety (Theorems \[stratification\] and \[Iwahoritheorem\]). The main idea is based on the well-known principle that a “multiplicity-freeness” property usually underlies integral constructions of $L$-functions. For our present purposes, a “multiplicity-freeness” property can be taken to mean that a suitable space of functions $\mathcal S(X)$ on a $G({{\mathbb{A}_k}})$-space $X$ admits at most one, up to constants, morphism into any irreducible admissible representation $\pi$ of $G({{\mathbb{A}_k}})$. Here $G$ denotes a connected reductive algebraic group over a global field $k$, and ${{\mathbb{A}_k}}$ denotes the ring of adeles of $k$. Such spaces arise as the adelic points of spherical varieties. By definition, a spherical variety for $G$ is a normal variety with a $G$-action such that, over the algebraic closure, the Borel subgroup of $G$ has a dense orbit. Let $X$ be an *affine* spherical variety, and denote by $X^+$ the open $G$-orbit on $X$. A second principle behind the main idea is based on ideas around the geometric Langlands program, according to which the correct “Schwartz space” $\mathcal S(X)$ of functions to consider (which are actually functions on $X^+({{\mathbb{A}_k}})$, not $X({{\mathbb{A}_k}})$) should be one reflecting the geometry and singularities of $X$. Then, for every cuspidal automorphic representation $\pi$ of $G$ with “sufficiently positive” central character, there is a natural pairing $\mathcal P_X: \mathcal S(X({{\mathbb{A}_k}})) \otimes \pi\to {\mathbb{C}}$ . The weak version of our conjecture (\[weakconjecture\]) asserts that this pairing admits meromorphic continuation to all $\pi$. (A stronger version, \[mainconjecture\], states that an “Eisenstein series” construction, obtained by summing over the $k$-points of $X$ and integrating against characters of a certain torus acting on $X$, has meromorphic continuation.) Then, assuming the “multiplicity-freeness” property, one expects the pairing to be associated to some $L$-value of $\pi$. If our variety is of the form $H\backslash G$ with $H$ a reductive subgroup of $G$ then from this construction we recover the period integral of automorphic forms over $H(k)\backslash H({{\mathbb{A}_k}})$ (§\[ssperiods\]). More generally, if $X$ is fibered over such a variety and the fibers are (related to) flag varieties, then we can prove meromorphic continuation using the meromorphic continuation of Eisenstein series, and we recover integrals of “Rankin-Selberg” type (§\[ssRS\]). Thus, we reduce the problem of finding Rankin-Selberg integrals to the problem of classifying affine spherical varieties with a certain geometry. For smooth affine spherical varieties, this geometric problem has been solved by Knop and Van Steirteghem [@KnVS]. By inspection of their tables (section \[secsmoothaffine\]), we recover some of the best-known constructions, such as those of Rankin and Selberg [@Ra; @Sel], Godement and Jacquet [@GJ], Bump and Friedberg [@BF], all spherical period integrals, as well as some new ones. We give an example (§\[sstensor\]), involving the tensor product $L$-function of $n$ cuspidal representations on ${\operatorname{GL}}_2$, to support the point of view that the basic object giving rise to an Eulerian integral related to an $L$-function is the spherical variety $X$ and not a geometry related to flag varieties. Finally, we apply these ideas to the relative trace formula (section \[secRTF\]) to show that certain “weight factors” which have appeared in examples of this theory and are often considered an “anomaly” can, in fact, be understood using the notion of Schwartz spaces. Background on the methods ------------------------- To an automorphic representation $\pi\simeq \otimes_v' \pi_v$ of a reductive group $G$ over a global field $k$, and to an algebraic representation $\rho$ of its Langlands dual group $^L G$, Langlands attached a complex $L$-function $L(\pi,\rho,s)$, defined for $s$ in some right-half plane of the complex plane as the product, over all places $v$, of local factors $L_v(\pi_v,\rho,s)$.[^1] Despite the beauty of its generality, the definition is of little use when attempting to prove analytic properties of $L$-functions, such as their meromorphic continuation and functional equation. Such properties are usually obtained by integration techniques, namely presenting the $L$-function as some integral transform of an element in the space of the given automorphic representation. Such methods in fact predate Langlands by more than a century, but the most definitive construction (as every automorphic $L$-function should be a ${\operatorname{GL}}_n$ $L$-function) was studied by Godement and Jacquet [@GJ] (generalizing Tate’s construction for ${\operatorname{GL}}_1$, [@Tate]), who proved the analytic continuation and functional equation of $L(\pi,s):=L(\pi,\operatorname{std},s)$, where $\pi$ is an automorphic representation of $G={\operatorname{GL}}_n$ and $\operatorname{std}$ is the standard representation of $^L G={\operatorname{GL}}_n({\mathbb{C}})\times {\operatorname{Gal}}(\bar k/k)$. Their method relies on proving the equality: $$\label{GodementJacquet} L(\pi,s-\frac{1}{2}(n-1))= \int_{{\operatorname{GL}}_n({\mathbb{A}_k})} \left<\pi(g)\phi, \tilde\phi\right> \Phi(g) |\det(g)|^s dg$$ where $\phi$ is a suitable vector in $\pi$, $\tilde\phi$ a suitable vector in its contragredient and $\Phi$ a suitable function in $\mathcal S({\operatorname{Mat}}_n({\mathbb{A}_k}))$, the Schwartz space of functions on ${\operatorname{Mat}}_n({\mathbb{A}_k})$. The main analytic properties of $ L(\pi,\rho,s)$, then, follow from Fourier transform on the Schwartz space and the Poisson summation formula. Going several decades back in history, Hecke showed that the standard $L$-function of a cuspidal automorphic representation on ${\operatorname{GL}}_2$ (with, say, trivial central character) has a presentation as a *period integral*, which in adelic language reads: $$\label{hecke} L(\pi,s+\frac{1}{2}) = \int_{k^\times \backslash {\mathbb{A}_k}^\times} \phi\left(\left(\begin{array}{cc} a & 0 \\ 0 & 1\end{array}\right)\right) |a|^s da$$ where, again, $\phi$ is a suitable vector in the automorphic representation under consideration. Period integrals (by which we mean integrals over the orbit of some subgroup on the automorphic space $G(k)\backslash G({{\mathbb{A}_k}})$, possibly against a character of that subgroup) have since been studied extensively, although there are still many open conjectures about their relation to $L$-functions (cf., for instance, [@II]). Still, they form perhaps the single class of examples where we have a general principle answering the question: How to write down an integral with good analytic properties, which is related to some $L$-function (or $L$-value)? Piatetski-Shapiro discussed this in [@PSEuler], and suggested that the period integral of a cusp form on a group $G$ over a subgroup $H$ (against, perhaps, an analytic family $\delta_s$ of characters of $H$ as in (\[hecke\])) should always be related to some $L$-value if the subgroup $H$ enjoys a “multiplicity-one” property: $\dim {\operatorname{Hom}}_{H({{\mathbb{A}_k}})}(\pi, \delta_s)\le 1$ for every irreducible representation $\pi$ of $G({{\mathbb{A}_k}})$ and (almost) every $s$. The method of periods usually fails when the subgroup $H$ is non-reductive, the reason being that, typically, the group $H({{\mathbb{A}_k}})$ has no closed orbits on $G(k)\backslash G({{\mathbb{A}_k}})$. Therefore there is no a priori reason that the period integral should have nice analytic properties (as the character $\delta_s$ varies), and one can in fact check in examples that for values of $s$ such that the period integral converges, it does not represent an $L$-function. In a different vein, Rankin [@Ra] and Selberg [@Sel] independently discovered an integral representing the tensor product $L$-function of two cuspidal automorphic representations of ${\operatorname{GL}}_2$. The integral uses as auxilliary data an Eisenstein series on ${\operatorname{GL}}_2$ and has the following form: $$L(\pi_1\times\pi_2, \otimes, s)= \int_{{\operatorname{PGL}}_2(k)\backslash{\operatorname{PGL}}_2({\mathbb{A}_k})} \phi_1(g)\phi_2(g) E(g,s) dg$$ with suitable $\phi_1\in\pi_1, \phi_2\in\pi_2$. Later, this method was taken up by Jacquet, Piatetski-Shapiro, Shalika, Rallis, Gelbart, Ginzburg, Bump, Friedberg and many others, in order to construct numerous examples of automorphic $L$-functions expressed as integrals of cusp forms against Eisenstein series, with important corollaries for every such expression discovered. Despite the abundance of examples, however, there has not been a systematic understanding of how to produce an integral representing an $L$-function. Schwartz spaces and $X$-Eisenstein series ----------------------------------------- While the method of Godement and Jacquet can also be phrased in the language of Rankin-Selberg integrals (see [@GPSR]), the fact that no systematic theory of these constructions exists has led many authors to consider them as coincidental and/or to seek direct generalizations of [@GJ], as being a “more canonical” construction (cf. [@BKgamma]). We adopt a different point of view which treats Godement-Jacquet, Rankin-Selberg, and period integrals as parts of the same concept, in fact a concept which should be much more general! The basic object here is an affine spherical variety $X$ of the group $G$. The reason that such varieties are suitable is that they are related to the “multiplicity-free” property discussed above. For instance, in the category of *algebraic* representations, the ring of regular functions $k[X]$ of an affine $G$-variety is multiplicity-free if and only if the variety is spherical. In the $p$-adic setting and for unramified representations, questions of multiplicity were systematically examined in [@SaSpc; @SaSph], and of course in special cases such questions have been examined in much greater detail (see, for example, [@Pr]). The main idea is to associate to every affine spherical variety a space of distributions on $G(k)\backslash G({{\mathbb{A}_k}})$ which should have “good analytic properties”. For reasons of convenience we set up our formulations in such a way that the analytic problem does not have to do with varying a character of some subgroup $H$ (the isotropy subgroup of a “generic” point on $X$), but with varying a cuspidal automorphic representation of $G$. For instance, to the Hecke integral (for ${\operatorname{PGL}}_2$) we do not associate the variety ${\mathbb{G}_m}\backslash {\operatorname{PGL}}_2$, but the variety $X={\operatorname{PGL}}_2$ under the $G={\mathbb{G}_m}\times {\operatorname{PGL}}_2$-action. Our distributions (in fact, smooth functions) on $G(k)\backslash G({{\mathbb{A}_k}})$ come from a “Schwartz space” of functions on $X^+({{\mathbb{A}_k}})$ via a “theta series” construction (i.e. summation over $k$-points of $X^+$). Here $X^+$ denotes the open $G$-orbit on $X$. The main conjecture \[mainconjecture\], then, states that the integral of these “$X$-theta series” against central idele class characters (I call this integral an $X$-Eisenstein series), originally defined in some domain of convergence, has meromorphic continuation everywhere. Under additional assumptions on $X$ (related to the “multiplicity-freeness” property mentioned above), the pairings of $X$-theta series with automorphic forms should be related, in a suitable sense, to automorphic $L$-functions or special values of those. The geometric Langlands program provides ideas that allow us to speculate on the form of these Schwartz spaces, motivated also by the work of Braverman and Kazhdan [@BK; @BK2] on the special case that $X$ is the affine closure of $[P,P]\backslash G$, where $P$ is a parabolic subgroup. Let us discuss this work: The prototype here is the case $X^+=U\backslash {\operatorname{SL}}_2 = \mathbb A^2\smallsetminus\{0\}$ (where $U$ denotes a maximal unipotent subgroup), $X=\mathbb A^2$ (two-dimensional affine space). The Schwartz space is the usual Schwartz space on $X({{\mathbb{A}_k}})$ which, by definition, is the restricted tensor product $\mathcal S(X({{\mathbb{A}_k}})):=\otimes_v' (\mathcal S(k_v^2) : \Phi_v^0)$, where for finite places $k_v$ with rings of integers $\mathfrak o_v$ the “basic vectors” $\Phi_v^0$ are the characteristic functions of $X(\mathfrak o_v)=\mathfrak o_v^2$. There is a natural meromorphic family of morphisms: $\mathcal S(X({{\mathbb{A}_k}}))\to I_{B({{\mathbb{A}_k}})}^{G({{\mathbb{A}_k}})}(\chi)$ (where $I_P^G$ denotes normalized parabolic induction from the parabolic $P$, $B$ denotes the Borel subgroup), and for idele class characters $\chi$ the composition with the Eisenstein series morphism: ${{\operatorname{Eis}}}_\chi: I_{B({{\mathbb{A}_k}})}^{G({{\mathbb{A}_k}})}(\chi) \to C^\infty(G(k)\backslash G({{\mathbb{A}_k}}))$ provides meromorphic sections of Eisenstein series, whose functional equation can be deduced from the Poisson summation formula on ${{\mathbb{A}_k}}^2$ – in particular, the $L$-factors which appear in the functional equation of “usual” (or “constant”) sections are absent here. This was found to be the case more generally in [@BK; @BG; @BFGM; @BK2]: One can construct “normalized” sections of Eisenstein series from certain “Schwartz spaces” of functions on $[P,P]\backslash G ({{\mathbb{A}_k}})$ (or $U_P\backslash G({{\mathbb{A}_k}})$, where $U_P$ is the unipotent radical of $P$). These Schwartz spaces should be defined as tensor products over all places, restricted with respect to some “basic vector”; and the “basic vector” should be the function-theoretic analog of the intersection cohomology sheaf of some geometric model for the space $X(\mathfrak o_v)$. For instance, if $X$ is *smooth* then the intersection cohomology sheaf is constant, which means that $\Phi_v^0$ is the characteristic function of $X(\mathfrak o_v)$; this explains the distibutions in Tate’s thesis, the work of Godement and Jacquet, and the case of period integrals. (In the latter, the characteristic function of $X(\mathfrak o_v)=H\backslash G(\mathfrak o_v)$ is obtained as the “smoothening” of the delta function at the point $H1\in X$.) Such geometric models where recently defined by Gaitsgory and Nadler [@GN] for *every* affine spherical variety. They provide us with the data necessary to speculate on a generalization of the Rankin-Selberg method. It should be noted, however, that even to define the “correct” functions on $X^+({{\mathbb{A}_k}})$ out of these geometric models one has to rely on certain natural conjectures on them – therefore the problem of finding an independent or unconditional definition should be considered as part of the steps which need to be taken towards establishing our conjecture. Comments and acknowledgements ----------------------------- Most of the ingredients in the present work are not new. Experts in the Rankin-Selberg method will recognize in our method, to a lesser of greater extent, the heuristics they have been using to find new integrals. The idea that geometric models and intersection cohomology should give rise to the “correct” space of functions on the $p$-adic points of a variety comes straight out of the Geometric Langlands program and the work of Braverman and Kazhdan; I have nothing to offer in this direction. However, the mixture of these ingredients is new and I think that there is enough evidence that it is the correct one. For the first time, a precise criterion is formulated on how to construct a “Rankin-Selberg” integral, reducing the problem to a purely geometric one – classifying certain embeddings of spherical varieties. And evidence shows that there should be a vast generalization which does not depend on such embeddings. I prove no “hard” theorems and, in particular, I do not know how to establish the meromorphic continuation of the $X$-Eisenstein series. Hence, I do not know whether I am putting the cart before the horse – however, as opposed to other conjectures which have appeared in the literature in the past, the distributions defined here are completely geometric and have nothing to do a priori with $L$-functions, which leaves a lot of room for hope. Finally, this point of view proves useful in explaining the phenomenon of “weight factors” in the relative trace formula. This work started in the fall of 2004 during a semester at New York University and was put aside for most of the time since. I am very grateful to Joseph Bernstein, Daniel Bump, Dennis Gaitsgory, David Ginzburg, Hervé Jacquet, David Nadler and Akshay Venkatesh for many useful discussions and encouragement. I also thank a referee for many useful comments. Elements of the theory of spherical varieties ============================================= Invariants associated to spherical varieties {#ssinvariants} -------------------------------------------- A *spherical variety* for a connected reductive group $G$ over a field $k$ is a normal variety $X$ together with a $G$-action, such that over the algebraic closure the Borel subgroup of $G$ has a dense orbit. We denote throughout by $k$ a number field and, unless otherwise stated, we make the following assumptions on $G$ and $X$: - $G$ is a split, connected, reductive group, - $X$ is affine. The open $G$-orbit in $X$ will be denoted by $X^+$, and the open $B$-orbit by $\mathring X^+$ (where $B$ is a fixed Borel subgroup of $G$, whose unipotent radical we denote by $U$).[^2] The assumption that $G$ is split is certainly very restrictive, but it is enough to demonstrate our point of view, and convenient because of many geometric and representation-theoretic results which have been established in this case. We will discuss *affine* spherical varieties in more detail later, but we just mention here that a common source of examples is when $X^+= H\backslash G$, a quasi-affine homogeneous variety, and $X=\overline{H\backslash G}^{{\operatorname{aff}}}= {{\operatorname{spec}\,}}\,\, k[H\backslash G]$, the *affine closure* of $H\backslash G$, cf. §\[ssaffine\]. We will be using standard and self-explanatory notation for varieties and algebraic groups; e.g. $\mathcal N(H), \mathcal Z(H), H^0$ will be, respectively, the normalizer, center and connected component of a (sub)group $H$, $\bar Y$ will be the closure of a subvariety $Y$, etc. The isotropy group of a point $x$ under a $G$-action will be denoted by $G_x$ and the fiber over $y\in Y$ of a morphism $X\to Y$ by $X_y$. The base change of an $S$-scheme $Y$ with respect to a morphism $T\to S$ will be denoted by $Y_T$, but if $v$ denotes a completion of a number field $k$ and $Y$ is defined over $k$ then we will be denoting by $Y_v$ the *set* $Y(k_v)$. Let us discuss certain invariants associated to a spherical variety. First of all, for any algebraic group $\Gamma$ we denote by ${\mathcal{X}}(\Gamma)$ its character group, and for any variety $Y$ with an action of $\Gamma$ we denote by ${\mathcal{X}}_\Gamma(Y)$ the group of $\Gamma$-eigencharacters appearing in the action of $\Gamma$ on $k(Y)$. If $\Gamma$ is our fixed Borel subgroup $B$, then we will denote ${\mathcal{X}}_B(Y)$ simply by ${\mathcal{X}}(Y)$. The multiplicative group of non-zero eigenfunctions (semiinvariants) for $B$ on $k(Y)$ will be denoted by $k(Y)^{(B)}$. If $Y$ has a dense $B$-orbit, then we have a short exact sequence: $0\to k^\times \to k(Y)^{(B)} \to {\mathcal{X}}(Y) \to 0$. For a finitely generated ${\mathbb{Z}}$-module $M$ we denote by $M^*$ the dual module ${\operatorname{Hom}}_{\mathbb{Z}}(M,{\mathbb{Z}})$. For our spherical variety $X$, we let $\Lambda_X={\mathcal{X}}(X)^*$ and $\mathcal Q = \Lambda_X\otimes_{\mathbb{Z}}{\mathbb{Q}}$. A $B$-invariant valuation on $k(X)$ which is trivial on $k^\times$ induces by restriction to $k(X)^{(B)}$ an element of $\Lambda_X$. We let $\mathcal V\subset \mathcal Q$ be the cone[^3] generated by *$G$-invariant valuations* which are trivial on $k^\times$, cf. [@KnLV Corollary 1.8]. It is known that it is a polyhedral cone, and in fact that it is a fundamental domain for the action of a finite reflection group $W_X$ on $\mathcal Q$. We denote by $\Lambda_X^+$ the intersection $\Lambda_X\cap \mathcal V$. Under the quotient map ${\mathcal{X}}(A)^*\otimes {\mathbb{Q}}\to \mathcal Q$, $\mathcal V$ contains the image of the *negative* Weyl chamber of $G$ [@KnLV Corollary 5.3]. The *associated parabolic* to $X$ is the standard parabolic $P(X):= \{ p\in G | \mathring X^+ \cdot p = \mathring X^+\}$. Make once and for all a choice of a point $x_0\in \mathring X^+(k)$ and let $H$ denote its stabilizer; hence $X^+=H\backslash G$, and $HB$ is open in $G$. There is the following “good” way of choosing a Levi subgroup $L(X)$ of $P(X)$: Pick $f\in k[X]$, considered by restriction as an element of $k[G]^H$, such that the set-theoretic zero locus of $f$ is $X \smallsetminus \mathring X^+$. Its differential $df$ at $1\in G$ defines an element in the coadjoint representation of $G$, and the centralizer $L(X)$ of $df$ is a Levi subgroup of $P(X)$. We fix throughout a maximal torus $A$ in $B \cap L(X)$. We define $A_X$ to be the torus: $L(X)/(L(X)\cap H) = A/ (A\cap H)$; its cocharacter group is $\Lambda_X$. We consider $A_X$ as a subvariety of $\mathring X^+$ via the orbit map on $x_0$. The finite reflection group $W_X\subset {\operatorname{End}}(\mathcal Q)$ for which $\mathcal V$ is a fundamental domain is called the *little Weyl group* of $X$. The set of simple roots of $G$ corresponding to $B$ and the maximal torus $A\subset B$ will be denoted by $\Delta$. Consider the (strictly convex) cone negative-dual to $\mathcal V$, i.e. the set $\{\chi\in{\mathcal{X}}(X)\otimes {\mathbb{Q}}| \left<\chi,v\right>\le 0 \text{ for every } v \in \mathcal V\}$. The generators of the intersections of its extremal rays with ${\mathcal{X}}(X)$ are called the (simple) *spherical roots*[^4] of $X$ and their set is denoted by $\Delta_X$. They are known to form the set of simple roots of a based root system with Weyl group $W_X$. We will denote by $\Delta(X)$ the subset of $\Delta$ consisting of simple roots in $L(X)$, and by $W_{L(X)}\subset W$ the Weyl groups of $L(X)$, resp. $G$. There is a canonical way [@KnHC Theorem 6.5] to identify $W_X$ with a subgroup of $W$, which normalizes and intersects trivially the Weyl group $W_{L(X)}$ of $L(X)$. The data ${\mathcal{X}}(X), W_X, \mathcal V$ are usually easy to compute by finding a point on the open $B$-orbit and using Knop’s action of the Borel subgroup on the set of $B$-orbits [@KnOrbits]; for a more systematic treatment, see [@Lo]. If $\mathcal V$ is equal to the image of the negative Weyl chamber, then we say that the variety is a *wavefront* spherical variety. (This term is justified by the proof for asymptotics of generalized matrix coefficients in [@SV].) Symmetric varieties, for example, are all wavefront [@KnLV §5]. Also, motivated by the results of [@SaSpc], we will call *geometric multiplicity* of $X$ the cardinality of the generic non-empty fiber of the map: ${\mathcal{X}}(X)/W_X \to {\mathcal{X}}(A)/W$. While none implies the other, it is usually the case that varieties with geometric multiplicity one are wavefront. On the other hand, let us call *arithmetic multiplicity* of $X$ the torsion subgroup of ${\mathcal{X}}(A)/{\mathcal{X}}(X)$. It was shown in [@SaSpc] that, if $F$ is a local non-archimedean field then for an irreducible unramified representation $\pi$ of $G(F)$ which is in general position among $X$-distinguished ones (i.e. with ${\operatorname{Hom}}_G(\pi,C^\infty(X(F)))\ne 0$) we have $\dim{\operatorname{Hom}}_G(\pi,C^\infty(X(F)))=1$ if and only if both the geometric and arithmetic multiplicity of $X$ are $1$. The $G$-automorphism group of a homogeneous $G$-variety $X^+=H\backslash G$ is equal to the quotient $\mathcal N(H)/H$. It is known [@Lo Lemma 7.17] that for $X^+$ spherical the $G$-automorphisms of $X^+$ extend to any affine completion $X$ of $X^+$. Moreover, it is known that ${{\operatorname{Aut}}}^G(X)$ is diagonalizable; the cocharacter group of its connected component can be canonically identified (by considering the scalars by which an automorphism acts on rational $B$-eigenfunctions) with $\Lambda_X \cap\mathcal V\cap (-\mathcal V)$. We will be denoting: $\mathcal Z(X):=({{\operatorname{Aut}}}^G(X))^0$. It will be convenient many times to replace the group $G$ by a central extension thereof and then divide by the subgroup of $\mathcal Z(G)^0$ that acts trivially on $X$, so that the map $\mathcal Z(G)^0\to \mathcal Z(X)$ becomes an isomorphism. Spherical embeddings and affine spherical varieties {#ssaffine} --------------------------------------------------- We will use the words “embedding”, “completion” or “compactification” of a spherical $G$-variety $X$ for a spherical $G$-variety $\bar X$ (not necessarily complete) with an open equivariant embedding: $X\to\bar X$. A spherical embedding is called *simple* if it contains a unique closed $G$-orbit. Spherical embeddings have been classified by Luna and Vust [@LV]; our basic reference for this theory will be [@KnLV]. We will now recall the main theorem classifying simple spherical embeddings. For now we assume that $k$ is an algebraically closed field in characteristic zero. However, for Theorem \[classemb\] below the assumption on the characteristic is unnecessary, and any result that does not involve “colors” holds verbatim without the assumption of algebraic closedness when the group $G$ is split. Let $X$ be a spherical variety and let $X^+$ be its open $G$-orbit. The *colors* of $X$ are the closures of the $B$-stable prime divisors of $X^+$; their set will be denoted by $\mathcal D$. For every $B$-stable divisor $D$ in any completion $X$ of $X^+$ we denote by $\rho(D)$ the element of $\mathcal Q$ induced by the valuation defined by $D$. A *strictly convex colored cone* is a pair $(\mathcal C,\mathcal F)$ with $\mathcal C\subset \mathcal Q$, $\mathcal F\subset \mathcal D$ such that: 1. $\mathcal C$ is a strictly (i.e. not containing lines) convex cone generated by $\rho(\mathcal F)$ and finitely many elements of $\mathcal V$, 2. the intersection of $\mathcal V$ with the relative interior of $\mathcal C$ is non-empty, 3. $0\notin\rho(\mathcal F)$. If $X$ is a simple embedding of $X^+$ with closed orbit $Y$, we let $\mathcal F(X)$ denote the set of $D\in\mathcal D$ such that $\bar D\supset Y$, and we let $\mathcal C(X)$ denote the cone in $\mathcal Q$ generated by all $\rho(D)$, where $D$ is a $B$-invariant divisor (possibly also $G$-invariant) in $X$ containing $Y$. \[classemb\] The association $X\to (\mathcal C(X),\mathcal F(X))$ is a bijection between isomorphism classes of simple embeddings of $X^+$ and strictly convex colored cones. Now let us focus on affine and quasi-affine spherical varieties. We recall from [@KnLV Theorem 6.7]: \[affine\] A spherical variety $X$ is affine if and only if $X$ is simple and there exists a $\chi\in {\mathcal{X}}(X)$ with $\chi|_{\mathcal V}\ge 0$, $\chi|_{\mathcal C(X)}=0$ and $\chi|_{\rho(\mathcal D\smallsetminus \mathcal F(X))}<0$. In particular, $H\backslash G$ is affine if and only if $\mathcal V$ and $\rho(\mathcal D)$ are separated by a hyperplane, while it is quasi-affine if and only if $\rho(\mathcal D)$ does not contain zero and spans a strictly convex cone. Recall [@BG §1.1] that a variety $Y$ over a field $k$ is called *strongly quasi-affine* if the algebra $k[Y]$ of global functions on $Y$ is finitely generated and the natural map $Y\to {{\operatorname{spec}\,}}k[Y]$ is an open embedding. Then the variety $\overline{Y}^{{\operatorname{aff}}}:= {{\operatorname{spec}\,}}k[Y]$ is called the *affine closure* of $Y$. A homogeneous quasi-affine spherical variety $Y=H\backslash G$ is strongly quasi-affine. If $X:=\overline{H\backslash G}^{{\operatorname{aff}}}$ then the data $(\mathcal C(X), \mathcal F(X))$ can be described as follows: Consider the cone $\mathcal R\subset {\mathcal{X}}(X)\otimes {\mathbb{Q}}$ generated by the set of $\chi\in {\mathcal{X}}(X)$ such that $\chi|_{\mathcal V}\ge 0$, $\chi|_{\rho(\mathcal D)}\le 0$. Choose a point $\chi$ in the relative interior of $\mathcal R$. Then $\mathcal F(X)=\{D\in\mathcal D| \rho(D)(\chi)=0\}$ and $\mathcal C(X)$ is the cone generated by $\mathcal F(X)$. The first statement of the proposition generalizes a result of Hochschild and Mostow [@HM] for the variety $U_P\backslash G$, where $U_P$ is the unipotent radical of a parabolic subgroup $P$ of $G$. Indeed, this variety is spherical under the action of $M\times G$, where $M$ is the reductive quotient of $P$. As a representation of $G$, $k[Y]$ is locally finite and decomposes: $$\label{decomposition} k[Y]=\oplus_\lambda V_\lambda$$ where $V_\lambda$ is the isotypic component corresponding to the representation with highest weight $\lambda$, and the sum is taken over all $\lambda$ with $V_\lambda\ne 0$. Since the variety is spherical, each $V_\lambda$ is isomorphic to one copy of the representation with highest weight $\lambda$. Moreover, the multiplicative monoid of non-zero highest-weight vectors $k[Y]^{(B)}$ is the submonoid of $k(Y)^{(B)}$ (the group of non-zero rational $B$-eigenfunctions) consisting of regular functions. Regular $B$-eigenfunctions are precisely those whose eigencharacter satisfies $\chi|_{\rho(\mathcal D)}\ge 0$; since the set $\mathcal D$ is finite, the monoid of $\lambda$ appearing in the decomposition (\[decomposition\]) is finitely-generated. Since the multiplication map: $V_\mu\otimes V_\nu$ has image in the sum of $V_\lambda$ with $\lambda\le \mu+\nu$, and composed with the projection: $k[Y]\to V_{\mu+\nu}$ it is surjective, it follows that the sum of the $V_\lambda$, for $\lambda$ in a set of generators for the monoid of $\lambda$’s appearing in (\[decomposition\]), generates $k[Y]$. The second condition, namely that $Y\to X$ is an open embedding, follows from the assumption that $Y$ is quasi-affine and the homogeneity of $Y$. Hence, $Y$ is strongly quasi-affine. The affine closure $X$ has the property that for every affine completion $X'$ of $Y$ there is a morphism: $X\to X'$. The description of $(\mathcal C(X),\mathcal F(X))$ now follows from Theorem \[affine\] above and Theorem 4.1 in [@KnLV], which describes morphisms between spherical embeddings. Notice that the cone $\mathcal C(X)$, as described, will necessarily contain the intersection of $\mathcal V$ with the cone generated by $\rho(\mathcal D)$ in its relative interior, therefore its relative interior will have non-empty intersection with $\mathcal V$. Let us now discuss the geometry of affine spherical varieties. The following is a corollary of Luna’s slice theorem: \[Lunacorollary\] If $G$ is a reductive group over an algebraically closed field $k$ in characteristic zero, acting on an affine variety $X$ so that $k[X]^G=k$, then $X$ contains a closed $G$-homogeneous affine subvariety $Y$ such that the embedding $Y\hookrightarrow X$ admits an equivariant splitting: $X\twoheadrightarrow Y$. If $G$ is smooth then the fiber over any (closed) point $y\in Y$ is $G_y$-equivariantly isomorphic to the vector space of a linear representation of $G_y$. Luna’s theorem also states that $Y$ is contained in the closure of any $G$-orbit, which is easily seen to be true in the spherical case since affine spherical varieties are simple. The $G$-automorphism group “retracts” $X$ onto $Y$: \[Taction\]\[structureaffine\] Let $X$ be an affine spherical $G$-variety and let $Y$ be as in the theorem above, considered both as a quotient and as a subvariety of $X$. Let $T$ be the maximal torus in ${{\operatorname{Aut}}}^G(X)$ which acts trivially on $Y$. Then the closure of the $T$-orbit of every point on $X$ meets $Y$. Equivalently, $k[X]^T=k[Y]$. This is essentially Corollary 7.9 of [@KnMotion]. More precisely, let us assume that $G$ has a fixed point on $X$, i.e. $Y$ is a point. (The question is easily reduced to this case, since every $G_y$-automorphism of the fiber of $X\to Y$ over $y$ extends uniquely to a $G$-automorphism of $X$.) The proof of *loc.cit. *shows that for a generic point $x\in X$ there is a one-parameter subgroup $H$ of ${{\operatorname{Aut}}}^G(X)$ such that $x\cdot H$ contains the fixed point in its closure. Hence $k[X]^T=k$ and therefore $X$ contains a unique closed $T$-orbit. Notice that if $G$ has a fixed point on $X$ then we can embed $X$ into a finite sum $V=\oplus_i V_i$ of finite-dimensional representations of $G$, such that the fixed point is the origin in $V$ and there is a subtorus $T$ of $\prod_i {{\operatorname{Aut}}}^G(V_i)$ acting on $X$ with the origin as its only closed orbit. (Simply take $V$ to be the dual of a $G$-stable, generating subspace of $k[X]$.) Generalized Cartan decomposition {#ssstratification} -------------------------------- Let $\mathcal K={\mathbb{C}}((t))$, the field of formal Laurent series over ${\mathbb{C}}$, and $\mathfrak O={\mathbb{C}}[[t]]$ the ring of formal power series. If $X^+$ is a homogeneous spherical variety over ${\mathbb{C}}$, it was proven by Luna and Vust [@LV] that: \[Cstratification\] $G(\mathfrak O)$-orbits on $X^+(\mathcal K)$ are parametrized by $\Lambda_X^+$, where to ${\check\lambda} \in \Lambda_X^+$ corresponds the orbit through ${\check\lambda}(t)\in A_X(\mathcal K)$. A new proof was given by Gaitsgory and Nadler in [@GN], which can be used to prove the analogous statement over $p$-adic fields. We revisit their argument, adapt it to the $p$-adic case, and extend it to determine the set of $G(\mathfrak o_F)$-orbits on $X(\mathfrak o_F)$, when $G$ and $X$ are affine and defined over a number field and $F$ is a non-archimedean completion (outside of a finite set of places). In the case of symmetric spaces similar statements on the set of $G(\mathfrak o_F)$-orbits on $X(F)$ and in a more general setting – without assuming that $G$ is split – have been proven by Benoist and Oh [@BO], Delorme and Sécherre [@DS]. The argument uses compactification results of Brion, Luna and Vust. We first need to recall a few more elements of the theory of spherical varieties. The results below have appeared in the literature for $k$ an algebraically closed field in characteristic zero, but the proofs hold verbatim when $k$ is any field in characteristic zero and the groups in question are split over $k$. (The basic observation being, here, that in all proofs one gets to choose $B$-eigenfunctions in $k(X)$, and since the variety is spherical and the group is split the eigenspaces of $B$ are one-dimensional and defined over $k$, therefore the chosen eigenfunctions are $k$-rational up to $\bar k$-multiple.) A *toroidal embedding* of $X^+$ is an embedding $X^c$ of $X^+$ in which no color ($B$-stable divisor which is not $G$-stable) contains a $G$-orbit. Theorem \[classemb\] implies that *simple* toroidal embeddings are classified by strictly convex, finitely generated subcones of $\mathcal V$. Moreover, the simple toroidal embedding $X^c$ obtained from a simple embedding $X$ by taking the cone $\mathcal C(X^c)=\mathcal C(X)\cap \mathcal V$ comes with a proper equivariant morphism: $X^c\to X$ [@KnLV Theorem 4.1] which is surjective [@KnLV Lemma 3.2]. The local structure of a simple toroidal embedding is given by the following theorem of Brion, Luna and Vust: \[localstr\] Let $X^c$ be a simple toroidal embedding of $X^+$ and let $X^c_B$ denote the complement of all colors. Then $X^c_B$ is an open, $P(X)$-stable, affine variety with the following properties: 1. $X_B^c$ meets every $G$-orbit. 2. If we let $Y^c$ be the closure of $A_X$ in $X_B^c$, then the action map $Y^c\times U_{P(X)}\to X^c_B$ is an isomorphism. We emphasize the structure of the affine toric variety $Y^c$: Its cone of regular characters is precisely $\mathcal C(X^c)^\vee:=\{\chi\in {\mathcal{X}}(X)\otimes{\mathbb{Q}}| \left<\chi,v\right>\ge 0\text{ for all }v\in \mathcal C(X^c)\}$, in other words: $$Y^c={{\operatorname{spec}\,}}k[\mathcal C(X^c)^\vee\cap {\mathcal{X}}(X)].$$ By the theory of toric varieties, the theorem also implies that $X^c$ is smooth if and only if the monoid $\mathcal C(X^c)\cap \Lambda_X$ is generated by primitive elements in its “extremal rays” (i.e. is a free abelian monoid). Notice that when $\mathcal V$ is strictly convex (equivalently: ${{\operatorname{Aut}}}^G(X^+)$ is finite) then $X^+$ admits a canonical toroidal embedding $\bar X$, with $\mathcal C(\bar X)=\mathcal V$, which is complete. This is sometimes called the *wonderful completion* of $X^+$, although often the term “wonderful” is reserved for the case that this completion is smooth. If $\mathcal V$ is not strictly convex then $X^+$ still admits a (non-unique) complete toroidal embedding $\bar X$, which is not simple, but as remarked in [@GN 8.2.7] Theorem \[localstr\] still holds, with $Y^c$ a suitable (non-affine) toric variety containing $A_X$. The fan of $Y^c$ depends on the chosen embedding $\bar X$, but its support is precisely the dual cone of $\mathcal V$ (i.e. the set of cocharacters $\lambda$ of $A_X$ such that $\lim_{t\to 0}\lambda(t)\in Y^c$ is equal to $\Lambda_X^+$). We will use Theorem \[localstr\] for two toroidal varieties: First, for a complete toroidal embedding $\bar X$ of $X^+$. Secondly, for the variety $\hat X$ obtained from our affine spherical variety $X$ by taking $\mathcal C(\hat X)=\mathcal C(X)\cap\mathcal V$. Before we proceed, we discuss models of these varieties over rings of integers. ### Models over rings of integers We start with toric varieties. Let $\mathfrak o$ be an integral domain with fraction field $k$, and let $Y$ be a simple (equivalently, affine) toric variety for a split torus $T$ over $k$. We endow $T$ with its smooth model $\mathcal T=\mathfrak o[{\mathcal{X}}(T)]$ over $\mathfrak o$. Since $Y={{\operatorname{spec}\,}}k[M]$ for some saturated monoid $M\subset{\mathcal{X}}(T)$, the $\mathfrak o$-scheme $\mathcal Y={{\operatorname{spec}\,}}\mathfrak o[M]$ is a model for $Y$ over $\mathfrak o$ with an action of $\mathcal T$, and we will call it the *standard model*. The notion easily extends to the case where $Y$ is not necessarily affine, but defined by a fan. If $T$ and $Y$ are defined over a number field $k$ and endowed with compatible models over the $S$-integers $\mathfrak o_S$ for a finite set $S$ of places of $k$, then these models will coincide with the standard models over $\mathfrak o_{S'}$, for some finite $S'\supset S$. Now we return to the setting where $k$ is a number field, $G$, $X$, $X^+$, $\bar X$, $\hat X$ are as before (over $k$), and let us also fix a point $x_0\in \mathring X^+(k)$. Then we can choose compatible integral models outside of a finite set of places, such that the structure theory of Brion, Luna and Vust continues to hold for these models: \[places\] There are a finite set of places $S_0$ of $k$ and compatible flat models $\mathcal G$, $\mathfrak X$ $\mathfrak{\bar X}$ and $\mathfrak{\hat X}$ for $G$, $X$ $\bar X$ and $\hat X$ over the $S_0$-integers $\mathfrak o_{S_0}$ of $k$ such that: - $S_0$ contains all archimedean places; - the chosen point $x_0 \in \mathcal{\mathring X^+}(\mathfrak o_{S_0})$; - $\mathcal G$ is reductive over $\mathfrak o_{S_0}$, $\mathcal {X^+}\to {{\operatorname{spec}\,}}\mathfrak o_{S_0}$ is smooth and surjective; - the statement of Theorem \[localstr\] holds for $\mathfrak{\bar X}$ and $\mathfrak{\hat X}$ over $\mathfrak o_{S_0}$: namely, if we denote any one of them by $\mathfrak X^c$ then there is an open, $\mathcal P(X)$-stable subscheme $\mathfrak X^c_B$ and a toric $\mathcal A$-subscheme $\mathcal Y^c$ *of standard type* such that the subscheme $\mathfrak X^c_B$ meets every $\mathcal G$-orbit on $\mathfrak X^c$ and the action map: $\mathcal Y^c\times \mathcal U_{P(X)}\to \mathfrak X^c_B$ is an isomorphism of $\mathfrak o_{S_0}$-schemes. - $\mathfrak{\bar X}$ is proper over $\mathfrak o_{S_0}$, and the morphism $\mathfrak{\hat X}\to \mathfrak X$ is proper. 1. By $\mathfrak{X^+}$ (resp. $\mathfrak{\mathring X^+}$) we denote the complement of the closure, in any of the above schemes, of the complement of $X^+$ (resp. $\mathring X^+$) in the generic fiber. 2. It is implicitly part of the “compatibility” of the models that the scheme structures on $\mathfrak{X^+}, \mathfrak{\mathring X^+}$ do not depend on which of the ambient schemes we choose to define them. 3. We understand the statement “meets every orbit” as follows: Let $|\mathcal Z|$ denote the set of scheme-theoretic points of a scheme $\mathcal Z$. Consider the two maps: $p:\mathcal G\times \mathfrak X\to \mathfrak X$ (projection to the second factor) and $a: \mathcal G\times \mathfrak X\to \mathfrak X$ (action map). Then for every $x\in |\mathfrak X^c|$ the set $a(p^{-1}\{x\})$ intersects $|\mathfrak X^c_B|$ non-trivially. For a finite set $S$ of places and a flat model $\mathfrak X^c$ of $X^c$ over $\mathfrak o_S$ (assumed proper if $X^c=\bar X$), let $D$ denote the union of all colors over the generic point of ${{\operatorname{spec}\,}}\mathfrak o_S$, let $\mathfrak D$ denote the closure of $D$ in $\mathfrak X^c$ and let $\mathfrak X_B^c$ be the complement of $\mathfrak D$ in $\mathfrak X^c$. Let $\mathcal G$ denote a compatible reductive model for $G$ over $\mathfrak o_S$. (All these choices are possible by sufficiently enlarging $S$.) The image of $\mathcal G\times \mathfrak X^c_B\to \mathfrak X^c$ is open and contains the generic fiber, hence by enlarging the set $S$, if necessary, we can make it surjective. Now define $\mathcal Y^c$ as the closure of $Y^c$ in $\mathfrak X^c_B$. By enlarging the set $S$, if necessary, we may assume that $\mathcal Y^c$ is of standard type. The action map $\mathcal Y^c\times \mathcal U_{P(X)}\to \mathfrak X_B^c$ being an isomorphism over the generic fiber, it is an isomorphism over $\mathfrak o_S$ by enlarging $S$, if necessary. From now on we fix such a finite set of places $S_0$ and such models. The combinatorial invariants of the above schemes are the same at all places of $S_0$: \[samedata\] Each of the data[^5] ${\mathcal{X}}(X), \mathcal V, \mathcal C(X), \mathcal C(\bar X), \mathcal C(\hat X)$ is the same for the reductions of $\mathfrak X, \mathfrak{\bar X}, \mathfrak{\hat X}$ at all closed points of $\mathfrak o_{S_0}$. The set of $G$-orbits on each of these varieties is in natural bijection with the set of $\mathcal G$-orbits on each of their reductions. The toric scheme $\mathcal Y^c$ being of the standard type, it means that ${\mathcal{X}}(X)={\mathcal{X}}_A(Y^c)$ is the same at all reductions. For every place $v$ of $\mathfrak o_S$ the reductions $\mathfrak{\bar X}_{{\mathbb{F}}_v}$, $\mathfrak{\hat X}_{{\mathbb{F}}_v}$ are toroidal: Indeed, denoting by $\mathfrak X^c$ either of them, the complement of $(\mathfrak X_B^c)_{{\mathbb{F}}_v}$ is a $\mathcal B_{{\mathbb{F}}_v}$-stable union of divisors which does not contain any $\mathcal G_{{\mathbb{F}}_v}$-orbit, since $(\mathfrak X_B^c)_{{\mathbb{F}}_v}$ meets every $\mathcal G_{{\mathbb{F}}_v}$-orbit. Moreover, $\mathfrak X_B^c$ meets no colors: for if it did, then a non-open $\mathcal A_{{\mathbb{F}}_v}$-orbit on $\mathcal Y^c_{{\mathbb{F}}_v}$ would belong to the open $\mathcal G_{{\mathbb{F}}_v}$-orbit, and hence the open $\mathcal G_{{\mathbb{F}}_v}$-orbit would belong to the closure of a non-open $G$-orbit over the generic point, a contradiction since by assumption $\mathfrak X^+$ is smooth and surjective. Therefore, the complement of $(\mathfrak X_B^c)_{{\mathbb{F}}_v}$ is the union of all colors of $\mathfrak X^c_{{\mathbb{F}}_v}$, and $\mathfrak X^c_{{\mathbb{F}}_v}$ is toroidal. Moreover, the $\mathcal G_{{\mathbb{F}}_v}$-invariant valuations on ${{\mathbb{F}}_v}(\mathfrak{X^+}_{{\mathbb{F}}_v})$ whose center is in $\mathfrak X^c_{{\mathbb{F}}_v}$ are precisely those of $\Lambda_X\cap \mathcal C(X^c)$ (which proves the equality of $\mathcal C(\mathfrak X^c_{{\mathbb{F}}_v})$ with $\mathcal C(X^c)$ at all $v\notin S_0$), and from the fact that $\mathfrak{\bar X}_{{\mathbb{F}}_v}$ is complete and $\mathcal C(\mathfrak{\bar X}_{{\mathbb{F}}_v})=\Lambda_X^+$ it follows that $\mathcal V$ is precisely the cone of invariant valuations on ${\mathbb{F}}_v(\mathfrak{X^+})$. Now we are ready to apply the argument of [@GN Theorem 8.2.9] to describe representatives for the set of $\mathcal G(\mathfrak o_F)$-orbits on $\mathfrak{X^+}(\mathfrak o_F)$, for every completion $F$ of $k$ outside of $S_0$, and also extend it to a description of the set of orbits which are contained in $\mathfrak X(\mathfrak o_F)$. Notice that since $\mathcal G$ is reductive, $\mathcal G(\mathfrak o_F)$ is a hyperspecial maximal compact subgroup of $G(F)$. From now on we denote our fixed models over $\mathfrak o_{S_0}$ by regular script, since there will be no possibility of confusion. There is a canonical $A_X(\mathfrak o_F)$-invariant homomorphism: $A_X(F)\to \Lambda_X$ (under which an element of the form $\lambda(\varpi)$, where $\varpi$ is a uniformizer for $F$, maps to $\lambda$) and we denote by $A_X(F)^+$ the preimage of $\Lambda_X^+$. \[stratification\] For $F$ a completion of $k$ outside of $S_0$ each $G(\mathfrak o_F)$-orbit on $X^+(F)$ contains an element of $A_X(F)^+$, and elements of $A_X(F)^+$ with different image in $\Lambda_X^+$ belong to distinct $G(\mathfrak o_F)$-orbits. If the quotient ${\mathcal{X}}(A)/{\mathcal{X}}(X)$ is torsion-free then the map from $G(\mathfrak o_F)$-orbits on $X^+(F)$ to $\Lambda_X^+$ is a bijection. The orbits contained in $X(\mathfrak o_F)$ are precisely those mapping to $\Lambda_X^+\cap \mathcal C(X)$. The torsion of the quotient ${\mathcal{X}}(A)/{\mathcal{X}}(X)$ is the “arithmetic multiplicity” defined in §\[ssinvariants\]. Is is trivial if and only if the map: $A_X(F)/A(\mathfrak o) \to \Lambda_X$ is bijective, hence the statement about bijectivity in that case is straightforward. In general, elements in different $A(\mathfrak o_F)$-orbits may belong to the same $G(\mathfrak o_F)$-orbit, for instance, if $X^+=H\backslash G$ with $H$ connected then the map: $G(\mathfrak o_F)\ni g\mapsto x_0\cdot g\in X^+(\mathfrak o_F)$ will be surjective by an application of Lang’s theorem (the vanishing of Galois cohomology of $H$ over a finite field). But it is also not always the case that elements corresponding to the same $\lambda$ will always be in the same $G(\mathfrak o_F)$-orbit – for instance, when $H$ is not connected. We will prove this theorem together with a theorem about orbits of the first congruence subgroup, which will not be used here but will be useful elsewhere. Let $\mathbb F$ denote the residue field of $F$. \[Iwahoritheorem\] Let $K_1, A_{X,1}, U_1$ be the preimages of $1\in G({\mathbb{F}})$, $1\in A_X({\mathbb{F}})$, $1\in U({\mathbb{F}})$ in $G(\mathfrak o_F)$, $A_X(\mathfrak o_F)$, $U(\mathfrak o_F)$, respectively. Then for every $x \in A_X(F)^+$ we have $x\cdot K_1\subset x\cdot A_{X,1}\cdot U_1$. Denote $\mathfrak o_F$ by $\mathfrak o$. We use the notation $X^c, X^c_B, Y^c,$ etc. as above for the scheme $\bar X$. The $\mathfrak o$-scheme $X^c$ is proper and hence $X^c(\mathfrak o)=X^c(F)$. We will first show that $Y^c(\mathfrak o)$ contains representatives for all $G(\mathfrak o)$-orbits on $X^c(\mathfrak o)$. Let $x\in X^c(\mathfrak o)$ and denote by $\bar x\in X^c({\mathbb{F}})$ its reduction. The open, $P(X)$-stable subvariety $X_B^c$ meets every $G$-orbit; for a spherical variety for a split reductive group over an arbitrary field (denoted ${\mathbb{F}}$, since we will apply it to this field) the ${\mathbb{F}}$-points of the open $B$-orbit meet every $G({\mathbb{F}})$-orbit. (This is proven following the argument of [@SaSpc Lemma 3.7.3], i.e. reducing to the case of rank one groups, and by inspection of the spherical varieties for ${\operatorname{SL}}_2$, classified in [@KnR1 Theorem 5.1].) This means that there is a $\bar g\in G({\mathbb{F}})$ (which we can lift to a $g\in G(\mathfrak o)$) such that $\overline{x \cdot g}\in X_B^c({\mathbb{F}})$. Since $X_B^c$ is open, this means that $x\cdot g\in X_B^c(\mathfrak o)=Y^c(\mathfrak o)\times U_{P(X)}(\mathfrak o)$. Acting by a suitable element of $U_{P(X)}(\mathfrak o)$, we get a representative for the $G(\mathfrak o)$-orbit of $x$ in $Y^c(\mathfrak o)$. Hence, $G(\mathfrak o)$-orbits on $X^+(F)$ are represented by elements of $A_X(F)^+=Y^c(\mathfrak o)\cap A_X(F)$. To prove that elements mapping to distinct $\lambda, \lambda'\in \Lambda_X^+$ belong to different $G(\mathfrak o)$-orbits, the argument of Gaitsgory and Nadler carries over verbatim: If $\lambda$ and $\lambda'$ are not ${\mathbb{Q}}$-multiples of each other, we can construct as in [@KnLV] a toroidal embedding $X^t$ of $X^+$ over $\mathfrak o$ such that $\lambda(\varpi)\in X^t(\mathfrak o)$ but $\lambda'(\varpi)\notin X^t(\mathfrak o)$. Finally, if $\lambda$ and $\lambda'$ are ${\mathbb{Q}}$-multiples of each other (without loss of generality: $\lambda\ne 0$), then we can find a toroidal compactification $X^t$ such that $\lim_{t\to 0} \lambda(t)$ belongs to some $G$-orbit $D$ of codimension one, and then the intersection numbers of $\lambda(\varpi)$ and $\lambda'(\varpi)$ (considered as $1$-dimensional subschemes of $X^t$) with $D$ are different. (Notice that the constructions of [@KnLV] are over a field of arbitrary characteristic, and based on Proposition \[samedata\] one can carry them over over the ring $\mathfrak o_F$.) To finish the proof of Theorem \[stratification\], if we now consider $\hat X$ then we have a proper morphism: $\hat X\to X$ which is an isomorphism on $X^+$. By the valuative criterion for properness, every point in $X(\mathfrak o)\cap X^+(F)$ lifts to a point on $\hat X(\mathfrak o)$, therefore for the last statement it suffices to determine the set of $G(\mathfrak o)$-orbits on $\hat X(\mathfrak o)\cap X^+(F)$. By the same argument as before, every $G(\mathfrak o)$-orbit meets $\hat Y(\mathfrak o)$, and the latter intersects $A_X(F)$ precisely in the union of $A_X(\mathfrak o)$-orbits represented by $\Lambda_X\cap \mathcal C(X)$. For Theorem \[Iwahoritheorem\], we first notice that $X_B^c(\mathfrak o)$ (where $X^c$ still denotes $\bar X$) is $K_1$-stable; indeed, for any $x\in X_B^c(\mathfrak o)$ and $g\in K_1$ the reduction of $x\cdot g$ belongs to $X_B^c({\mathbb{F}})$, and since $X_B^c$ is open this implies that $x\cdot g\in X_B^c(\mathfrak o)$. Now we claim that $Y^c(\mathfrak o)\cdot U_1$ is also $K_1$-stable; indeed, this is the preimage in $X_B^c({\mathbb{F}})$ of $Y^c({\mathbb{F}})$, and for every $x\in Y^c(\mathfrak o)\cdot U_1, g\in K_1$ the reduction of $x\cdot g$ belongs to $Y^c({\mathbb{F}})$. We have already argued that elements of $A_X(F)^+$ with different images in $\Lambda_X^+$ belong to distinct $G(\mathfrak o)$-orbits, hence to distinct $K_1$-orbits; hence, $x\cdot K_1$ belongs to the set of elements of $A_X(F)^+ \cdot U_1$ with the same image $\lambda_x\in \Lambda_X^+$ as $x$. To distinguish between those elements, we assign to them some invariants which will be preserved by the $K_1$-action. First of all, if $\lambda_x=0$ then the reduction of $x$ modulo $\mathfrak p$ is an element of $X^+({\mathbb{F}})$ which is preserved by $K_1$, and the elements of $A_X(F)^+ \cdot U_1$ having the same reduction are precisely the elements in the same $A_{X,1}\cdot U_1$-orbit as $x$. Assume now that $\lambda_x\ne 0$ and fix as above a spherical embedding $X^t$ of $X^+$ over $\mathfrak o$ such that $\lim_{t\to 0}\lambda(t)$ belongs to a $G$-orbit of codimension one, whose closure we denote by $D$. Let $n$ be the intersection number of $x\in X^t(\mathfrak o)\cap X^+(F)$ with $D$, then $x: {{\operatorname{spec}\,}}\mathfrak o \to X^t$ has reductions $\bar x: {{\operatorname{spec}\,}}{\mathbb{F}}\to D$, $\bar x^n: {{\operatorname{spec}\,}}(\mathfrak o/\mathfrak p^n) \to D$ and $\bar x^{n+1}: {{\operatorname{spec}\,}}(\mathfrak o/\mathfrak p^{n+1}) \to X^t$, which give rise to an ${\mathbb{F}}$-linear map from the fiber at $\bar x$ of the conormal bundle of $D$ in $X^t$ to $\mathfrak p^n/\mathfrak p^{n+1}$. The group $K_1$ preserves the reduction of $x$ and acts trivially on the fiber of the conormal bundle of $D$ over it, therefore preserves this map. It is straightforward to see that for elements of $A_X(F)^+ \cdot U_1$ with the same image in $\Lambda_X^+$ this invariant characterizes the $A_{X,1}\cdot U_1$-orbit of $x$. Speculation on Schwartz spaces and automorphic distributions {#secSchwartz} ============================================================ This section is highly conjectural and only aims at fixing ideas. We speculate on the existence of some “Schwartz space” of functions on the points of an affine spherical variety over a local field, and explain how to construct from it distributions on the automorphic quotient $[G]:=G(k)\backslash G({{\mathbb{A}_k}})$ which should have good analytic properties. At almost every place this space of functions should come equipped with a distinguished, unramified element which should be related (in a rather ad hoc way, using the generalized Cartan decomposition) to intersection cohomology sheaves on spaces defined by Gaitsgory and Nadler. In subsequent sections we will specialize to the case where $X$ has a certain geometry (which we call a “pre-flag bundle”), and these distinguished functions will be described explicitly, in order to understand the Rankin-Selberg method. Formalism of Schwartz spaces and theta series {#ssformalism} --------------------------------------------- ### {#Schwartz} We fix an affine spherical variety $X$ for a (split) reductive group $G$ over a global field $k$, and for every place $v$ of $k$ we denote by $X_v^+$ the space of $k_v$-points of $X^+$. We assume as given, for every $v$, a $G_v$-invariant “Schwartz space” of functions $\mathcal S(X_v)\subset C^\infty(X_v^+)$, and for almost every (finite) $v$ a distinguished unramified element $\Phi_v^0\in \mathcal S(X_v)^{G(\mathfrak o_v)}$ (called “basic vector” or “basic function”) such that: $$\label{oneonsmooth}\Phi_v^0|_{X^+(\mathfrak o_v)} = 1.$$ (Clearly, the integral model which is implicit in the definitions will not play any role.) We also assume the following regarding the support of Schwartz functions and their growth close to the complement of $X^+$: - The closure in $X_v$ of the support of any element of $\mathcal S(X_v)$ is compact. - There exist a finite set $\{f_1,\dots,f_n\}$ of elements of $k[X]$, whose common zeroes lie in $X\smallsetminus X^+$, and a natural number $n$, such that for any place $v$ and any $\Phi_v\in \mathcal S(X_v)$ there is a constant $c_v$, equal to $1$ for $\Phi_v=\Phi_v^0$, such that for all $x\in X^+(k_v)$ we have: $|\Phi_v(x)|\le c_v\cdot (\max_i|f_i(x)|)^{-1}$. At archimedean places the requirement of compact support is far from ideal, but for our present purposes it is enough. One should normally impose similar growth conditions on the derivatives (at archimedean places) of elements of the Schwartz space, but we will not need them here. The corresponding *global Schwartz space* is, by definition, the restricted tensor product: $$\mathcal S(X({{\mathbb{A}_k}})):= \bigotimes'_v \mathcal S(X_v)$$ with respect to the basic vectors $\Phi_v^0$. Despite the notation, the elements of $\mathcal S(X({{\mathbb{A}_k}}))$ cannot be interpreted as functions on $X({{\mathbb{A}_k}})$. They *can* be considered, though, as functions on $X^+({{\mathbb{A}_k}})$, because of the requirement (\[oneonsmooth\]). We may require, without serious loss of generality, that $X^+({{\mathbb{A}_k}})$ carries a positive $G({{\mathbb{A}_k}})$-eigenmeasure $dx$ whose eigencharacter $\psi$ is the absolute value of an algebraic character. We normalize the regular representation of $G({{\mathbb{A}_k}})$ on functions on $X^+({{\mathbb{A}_k}})$ so that it is unitary when restricted to $L^2(X)=L^2(X,dx)$: $$g\cdot \Phi (x):= \sqrt{\psi(g)} \Phi(x\cdot g).$$ The *$X$-theta series* is the following functional on $\mathcal S(X({{\mathbb{A}_k}}))$: $$\theta(\Phi):= \sum_{\gamma\in X^+(k)} \Phi(\gamma).$$ Translating by $G({{\mathbb{A}_k}})$, we can also consider it as a morphism: $$\mathcal S(X({{\mathbb{A}_k}}))\to C^\infty([G]),$$ which will be denoted by the same letter, i.e.: $$\label{PsES} \theta(\Phi,g)=\sum_{\gamma\in X^+(k)} (g\cdot\Phi)(\gamma).$$ This sum is absolutely convergent, by the first growth assumption. (Notice that $X$ is affine and hence $X(k)$ is discrete in $X({{\mathbb{A}_k}})$.) ### Mellin transform {#sssMellin} Now recall (Proposition \[structureaffine\]) that, unless $X$ is affine homogeneous, it has a positive-dimensional group of $G$-automorphisms, i.e. $\mathcal Z(X)\ne 0$. By enlarging $G$ and dividing by the subgroup of $\mathcal Z(G)^0$ that acts trivially, we will from now on assume that $\mathcal Z(G)^0\simeq \mathcal Z(X)$ under its action on $X$. An algebraic character of $\mathcal Z(X)$ will be called *$X$-positive* if it extends to the closure of a generic orbit of $\mathcal Z(X)$. Obviously, $X$-positive characters span a polyhedral cone in ${\mathcal{X}}(\mathcal Z(X))\otimes {\mathbb{Q}}$, and we will use the expression “sufficiently $X$-positive characters” to refer to characters in the translate of this cone by an element belonging to its relative interior. This notion will also be used for complex-valued characters: a sufficiently $X$-positive character is one whose absolute value can be written as the product of the absolute values sufficiently $X$-positive algebraic characters, raised to powers $\ge 1$. Similar notions will be used for the dual cone, in the space of cocharacters into $\mathcal Z(X)$; for example, a cocharacter $\check\lambda$ is $X$-positive if and only if for a generic point $x\in X$ we have $\lim_{t\to 0}x\cdot \check\lambda(t)\in X$. Finally, since by our assumption ${\mathcal{X}}(G)\otimes{\mathbb{Q}}={\mathcal{X}}(\mathcal Z(X))\otimes{\mathbb{Q}}$, we can use the notion of $X$-positive characters for characters of $G$, as well. \[moderategrowth\] The function $\theta(\Phi,g)$ on $G(k)\backslash G({{\mathbb{A}_k}})$ is of moderate growth. Moreover, it is compactly supported in the direction of $X$-positive cocharacters into $\mathcal Z(G)$; that is, for every $g\in G({{\mathbb{A}_k}})$ we have: $$\theta(\Phi, g\cdot \check\lambda(a))=0$$ if $\check \lambda$ is a non-trivial $X$-positive cocharacter into $\mathcal Z(X)=\mathcal Z(G)^0$ and the norm of $a\in {{\mathbb{A}_k}}^\times$ is sufficiently large. The statement about the support is an obvious corollary of the compact support of $\Phi$, and the statement on moderate growth will be proven in the next subsections. Assuming it for now, we may consider the *Mellin transform* of $\theta(\Phi,g)$ with respect to the action of $\mathcal Z(G)$: $$\label{ES} E(\Phi,\omega,g) = \int_{\mathcal Z(X)({{\mathbb{A}_k}})} \theta(z\cdot \Phi, g) \omega(z) dz,$$ originally defined for sufficiently $X$-positive idele class characters $\omega$. We will call this an *$X$-Eisenstein series*. We have: For sufficiently $X$-positive $\omega$, the integral (\[ES\]) converges and the function $E(\Phi,\omega,g)$ is of moderate growth in $g$. The statement about convergence follows immediately from Proposition \[moderategrowth\]; the statement on moderate growth is proven in the same way as Proposition \[moderategrowth\], and we will not comment on it separately. ### Adelic distance functions. {#sssdistance} Let $Z\subset X$ be a closed subvariety of an affine variety, and let $X^+$ denote the complement of $Z$. We would like to define some “natural” notion of distance from $Z$ (denoted $d_Z$) for the adelic points of $X^+$. The distance function will be an Euler product: $$d_Z(x)=\prod_v d_{Z,v}(x_v)$$ where, for $x\in X^+({{\mathbb{A}_k}})$, almost all factors will be equal to one. We do it in the following way: first, we fix a finite set $S$ of places, including the archimedean ones, and an affine flat model for $X$ over the $S$-integers $\mathfrak o_S$. The closure of $Z$ in this model defines an ideal $J\subset\mathfrak o_S[X]$. We can choose a finitely-generated $\mathfrak o_S$-submodule $M$ of $J$ such that $M$ generates $J$ as an $\mathfrak o_S[X]$-module. In the case when $X$ carries the action of a group $G$ and $Z$ is $G$-stable, we also choose a compatible flat model for $G$ over $\mathfrak o_S$ and require that $M$ be $G$-stable (i.e. the action map maps $M\to M\otimes_{\mathfrak o_S} \mathfrak o_S[G]$). Finally, let $\{f_i\}_i$ be a finite set of generators of $M$ over $\mathfrak o_S$. Then for a point $x\in X^+({{\mathbb{A}_k}})$ we define: $$d_{Z,v}(x_v) = \max_i\{|f_i(x_v)|_v\}$$ and $$d_Z(x)=\prod_v d_{Z,v}(x_v).$$ We will call this an *adelic distance function* from $Z$. Notice that almost all factors of this product are $1$ since $x\in X^+({{\mathbb{A}_k}})$. Moreover, the function extends by zero to a continuous function on $X({{\mathbb{A}_k}})$. For $v\notin S$ the local factor $d_{Z,v}$ depends only on $M$ and not the choices of $f_i$’s: it is the absolute value of the fractional ideal generated by the image of $M$ under $x_v: \mathfrak o_S[X]\to \mathfrak o_v$. Moreover, the restriction of $d_{Z,v}$ to $X(\mathfrak o_v)$ does not depend on $M$, either, since the image of $J$ generates the same fractional ideal. (The restriction of $d_{Z,v}$ to $X(\mathfrak o_v)$ is a height function, i.e. $q_v$ raised to the intersection number of $x\in X(\mathfrak o_v)$ with $Z$.) Finally, the restriction of $d_Z$ to any compact subset of $X({{\mathbb{A}_k}})$ is up to a constant multiple independent of choices. Indeed, such a compact subset is the product of $X(\mathfrak o_v)$, for $v$ outside of a finite number of places $S'\supset S$, with a compact subset of $\prod_{v\in S'} X(k_v)$, therefore it suffices to prove independence for the $d_{Z,v}$’s when $v\in S'$. For any two sets of functions $\{f_j\}_j, \{f_i'\}_i$ as above we can write $f_i'=\sum_j h_{ij} f_j$ with $h_{ij}\in\mathfrak o_{S}[X]$ and for each $v\in S'$ there is a constant $C_v$ such that $|h_{ij}(x_v)|_v\le C_v$ when $x$ is in the given compact set. Then $\max_i |f_i'(x_v)|_v\le C_v \max_j |f_j'(x_v)|_v$, and therefore $d_Z'(x)\le C d_Z(x)$ in the given compact set, where $C=\prod_{v\in S'} C_v$. For two complex valued functions $f_1$ and $f_2$ we will write $f_1\ll^p f_2$ (where the exponent $p$ stands for “polynomially”) if there exists a polynomial $P$ such that $|f_1|\le P(|f_2|)$. We will say that $f_1$ and $f_2$ are *polynomially equivalent* if $f_1\ll^p f_2$ and $f_2\ll^p f_1$. In this language, it is easy to see that the assumption of §\[Schwartz\] on growth of Schwartz functions close to the complement of $X^+$ is equivalent to the following: If $Z$ denotes the complement of $X^+$ in $X$ then for any adelic distance function $d_Z$ from $Z$ and any $\Phi\in\mathcal S(X({{\mathbb{A}_k}}))$ we have: - $$\label{growth}|\Phi(x)| \ll^p d_Z(x)^{-1}$$ for every $\Phi\in \mathcal S(X({{\mathbb{A}_k}}))$. Indeed, let the functions $f_i$ be as in the assumption of §\[Schwartz\] and let the functions $f_j'$ define an adelic distance function as above. By enlarging $S$ we may assume that $f_i\in \mathfrak o_S[X]$ for all $i$, and by enlarging it further we may assume that the support of $\Phi$ is the product of $\prod_{v\notin S} X(\mathfrak o_v)$ with a compact subset of $\prod_{v\in S} X(k_v)$. By the assumption, the functions $f_i$ generate an ideal whose radical contains $J$. Therefore, $(f_i)_i\supset J^n$ for some $J$ and hence for each $j$ there are $h_{ij}\in\mathfrak o_S[X]$ such that: $$(f_j')^n=\sum_i h_{ij} f_i$$ Therefore for $v\notin S$ and $x_v\in X(\mathfrak o_v)$ we have: $$d_{Z,v} (x_v)^n \le \max_i |f_i(x)|,$$ and for $v\in S$ we can find $C_v$ such that $|h_{ij}(x_v)|_v\le C_v$ if $x$ is in the support of $\Phi$. Therefore, for $x$ in the support of $\Phi$ we have: $$\prod_v (\max_i|f_i(x_v)|_v)^{-1} \le \prod_{v\in S} C_v^{-1} \cdot d_Z(x)^{-n}.$$ Vice versa, if $\Phi$ is known to be polynomially bounded by $d_Z(x)^{-1}$ then it is bounded by a constant times $d_Z(x)^{-n}$ for some $n$ (since $d_Z(x)$ is bounded in the support of $\Phi$), which implies the bound of the assumption. e ### Proof of Proposition \[moderategrowth\]. Recall that an automorphic function $\phi$ is “of moderate growth” if $\phi\ll^p \Vert g\Vert$ for some natural norm $\Vert \bullet \Vert$ on $G_\infty$. Recall that a “natural norm” is a positive function on $G_\infty$ which is polynomially equivalent to $\Vert \rho(g)\Vert$ where: $\rho$ denotes an algebraic embedding $G\hookrightarrow {\operatorname{GL}}_n$, and $\Vert g\Vert:= \max\{\vert g\vert_{l^\infty}, \vert g^{-1}\vert_{l^\infty}\}$ on ${\operatorname{GL}}_n(k_\infty)$ (where $\vert\bullet\vert_{l^\infty}$ denotes the operator norm for the standard representation of ${\operatorname{GL}}_n$ on $l^\infty(\{1,\dots,n\})$). Assume without loss of generality that $\Phi=\otimes_v \Phi_v$, with $\Phi_v\in \mathcal S(X_v)$, and let $S_\Phi=\prod S_{\Phi_v}$ where $S_{\Phi_v}$ is the support of $\Phi_v$ in $X(k_v)$ (a compact subset). The claim of the Proposition will follow from (\[growth\]) if, in addition, we establish that (for $g\in G_\infty$ and $x\in X^+({{\mathbb{A}_k}})$): - $\#(X^+(k)\cap S_\Phi g)\ll^p \Vert g\Vert$. - $\left(\inf d_Z(X^+(k)g)\right)^{-1} \ll^p \Vert g \Vert$. Indeed, assuming these properties we have: $$\theta(\Phi,g)=\sum_{\gamma\in X^+(k)} (g\cdot\Phi)(\gamma) \le \#(X^+(k)\cap S_\Phi g^{-1}) \cdot \sup_{x\in X^+(k)} |\Phi(xg)| \ll^p$$ $$\ll^p \Vert g\Vert \cdot \left(\inf_{x\in X^+(k)} d_Z(xg)\right)^{-1} \ll^p \Vert g\Vert \cdot \Vert g \Vert.$$ The first property is standard, and follows from the analogous claim for ${\operatorname{GL}}_n$ (after fixing an equivariant embedding of $X$ in the vector space of a representation of $G$), since $S_\Phi$ is a compact subset of $X({{\mathbb{A}_k}})$. To prove the second property, we may assume that the elements $f_i\in k[X]$ defining $d_Z$ span over $k$ a $G$-invariant space $M\subset k[X]$ and that the norm on $G_\infty$ is induced by the $l^\infty(\{f_i\}_i)$-operator norm on ${\operatorname{GL}}(M_\infty)$. (If the homomorphism $G\to {\operatorname{GL}}(M)$ is not injective, then this $l^\infty$ norm is bounded by some natural norm on $G_\infty$, which is enough for the proof of this property.) Then for every $x\in X_\infty$ and $g\in G_\infty$ we have: $$\Vert g\Vert^{-1} \cdot d_{Z,\infty}(x) \le d_{Z,\infty}( x\cdot g) \le \Vert g\Vert \cdot d_{Z,\infty}(x)$$ (where we keep assuming that $d_Z$ is defined by a basis for $M$). We apply this to points $x\in X^+(F)$. Notice that for every $x\in X^+(k)$ we have $f_i(x)\in k$ and $\ne 0$ for at least one $i$, hence $d_Z(x)=\prod_v \max_i |f_i(x)|_v\ge \max_i \prod_v |f_i(x)|_v = 1$. Therefore, we have: $d_{Z,\infty}( x\cdot g) \ge \Vert g\Vert^{-1} \cdot d_{Z,\infty}(x) \ge \Vert g\Vert^{-1}$. Conjectural properties of the Schwartz space {#ssconjectures} -------------------------------------------- The conjectures that follow are very speculative, but will provide the suitable ground for unifying various methods of integral representations of $L$-functions. There are several reasonable assumptions that one could impose on the spherical variety, the strongest of which would be that for every irreducible admissible representation $\pi$ of $G({{\mathbb{A}_k}})$ we have: $\dim_{G({{\mathbb{A}_k}})}(\pi, C^\infty(X^+({{\mathbb{A}_k}})))\le 1$. At the very minimum, we require from now on that the arithmetic multiplicity (§\[ssinvariants\]) of $X$ is trivial. Equivalently, at every place $v$ there is a unique open $B(k_v)$-orbit, and this implies also that generic $G$-stabilizers are connected and therefore, at almost every (finite) place $v$, the space $X^+(\mathfrak o_v)$ is homogeneous under $G(\mathfrak o_v)$. \[mainconjecture\] Given an affine spherical variety $X$ over $k$ with trivial arithmetic multiplicity, there exists a Schwartz space $\mathcal S(X({{\mathbb{A}_k}}))$, in the sense described above, such that: - The basic functions $\Phi_v^0$ factor through the map of the generalized Cartan decomposition: $$\{G(\mathfrak o_v)\mbox{-orbits on }X^+_v\}\to \Lambda_X^+$$ and as functions on $\Lambda_X^+$ are equal to the functions obtained via the function-sheaf correspondence from the “basic sheaf” of Gaitsgory and Nadler, as will be explained in \[sssbasicfunction\]. - For every $\Phi\in \mathcal S(X({{\mathbb{A}_k}}))$, the $X$-Eisenstein series $E (\Phi, \omega, g)$, originally defined for sufficiently $X$-positive characters, admits a meromorphic continuation everywhere. \[afterconjecture\] 1. The first property could be taken as the definition of the basic function, if one knew that the functions obtained from the Gaitsgory-Nadler sheaf are independent of some choices, which we will explain in §\[sssbasicfunction\]. In any case, such a definition would be very ad hoc and not useful; one hopes that there exists an alternative construction of the Schwartz space, as in [@BK]. 2. The property of meromorphic continuation is mostly dependent on the basic vectors and not on the whole Schwartz space; for instance, at a finite number of places we may replace any function with a function whose (loceal) Mellin transform is a meromorphic multiple of the Mellin transform of the original function without affecting the meromorphicity property. Therefore, the properties do not determine the Schwartz space uniquely; they should hold, for instance, if we take $\mathcal S(X_v)$ to be the $G$-space generated by the basic vector and $C_c^\infty(X_v^+)$. 3. The fact that the theta series is defined with reference to the group $G$ (since we are summing over the $k$-points of its open orbit) certainly seems unnatural; it would be more “geometric” to sum over the $k$-points of the open subvariety where $\mathcal Z(X)$ acts faithfully. However, this does not affect the validity of Conjecture \[mainconjecture\], since one case can be inferred from the other by induction on the dimension of $X$. The conjecture about meromorphic continuation of the Mellin transform is a very strong one and, in fact, is not even known in the case of usual Eisenstein series, i.e. the case of $X=\overline{U_P\backslash G}^{{\operatorname{aff}}}$, where $U_P$ is the unipotent radical of a parabolic $P$ (except when $P$ is a Borel subgroup). We now formulate a weaker conjecture which says that the $X$-Eisenstein series can be continued meromorphically “as functionals on the space of automorphic forms”. In fact, the precise interpretation of them as functionals on the whole space of automorphic forms would require a theory similar to the spectral decomposition of the relative trace formula, which lies beyond the scope of the present paper. Therefore, we confine ourselves to the cuspidal component of this functional. (Notice, however, that there are a lot of interesting examples which have zero cuspidal contribution, e.g. $X={{\operatorname{Sp}}}_{2n}\backslash {\operatorname{GL}}_{2n}$.) \[weakconjecture\] Same assumptions as in Conjecture \[mainconjecture\], but the second property is replaced by the following: - For every cusp form $\phi$ on $G(k)\backslash G({{\mathbb{A}_k}})$, the integral: $$\label{intconj1} \int_{[G]} \phi\cdot\omega(g) \theta(\Phi,g) dg$$ originally defined for sufficiently $X$-positive idele class characters $\omega$ of $G$, admits meromorphic continuation to the space of all idele class characters of $G$. Following up on the third remark of \[afterconjecture\], we will see in Proposition \[parind\] that for the large class of *wavefront* spherical varieties (see §\[ssinvariants\]), the integral (\[intconj1\]) is the same whether the theta series is defined by summation over $X^+(k)$ or over the largest subvariety where $\mathcal Z(X)$ acts faithfully. The reason is a phenomenon that has frequently been observed in the Rankin-Selberg method, namely that the stabilizers of points in all but the open orbit contain unipotent radicals of proper parabolics. Although this is not a feature of the Rankin-Selberg method only, we present the proof there in order not to interrupt the exposition here. Geometric models and the basic function {#ssgeommodels} --------------------------------------- We now discuss the geometric models and explain the first requirement of Conjecture \[mainconjecture\]. The models that we are about to discuss are relevant to a spherical variety $X$ over an *equal-characteristic* local field $F$, and are not local, but global in nature. ### The Gaitsgory-Nadler spaces [@GN]. {#GN} Let $X$ be an affine spherical variety over ${\mathbb{C}}$, and let $C$ be a smooth complete complex algebraic curve. Consider the ind-stack $\mathcal Z$ of *meromorphic quasimaps* which, by definition, classifies data: $$(c, \mathcal P_G, \sigma)$$ where $c\in C$, $\mathcal P_G$ is a principal $G$-bundle on $C$, and $\sigma$ is a section: $C\smallsetminus\{c\}\to \mathcal P_G\times^G X$ whose image is not contained in $X\smallsetminus X^+$. Clearly, $\mathcal Z$ is fibered over $C$ (projection to the first factor). It is a stack of infinite type, however it is a union of open substacks of finite type, each being the quotient of a scheme by an affine group, and therefore one can define intersection cohomology sheaves on it without a problem. The same definitions can be given if $G,X$ are defined over a finite field ${\mathbb{F}}$. To any quasimap one can associate an element of $X^+(\mathcal K)/G(\mathfrak O)$ (where $\mathfrak O={\mathbb{C}}[[t]],\mathcal K={\mathbb{C}}((t))$) as follows: Choose a trivialization of $\mathcal P_G$ in a formal neighborhood of $c$ and an identification of this formal neighborhood with ${{\operatorname{spec}\,}}(\mathfrak O)$ – then the section $\sigma$ defines a point in $X^+(\mathcal K)$, which depends on the choices made. The corresponding coset in $X^+(\mathcal K)/G(\mathfrak O)$ is independent of choices. This allows us to stratify our space according to the stratification, provided by Theorem \[Cstratification\], of $X^+(\mathcal K)/G(\mathfrak O)$. We only describe some of the strata here: For $\theta\in \Lambda_X^+$, let $\mathcal Z^\theta$ denote the quasimaps of the form $(c,\mathcal P_G,\sigma: C\smallsetminus\{c\}\to\mathcal P_G\times^G X^+)$ which correspond to the coset $\theta\in X^+(\mathcal K)/G(\mathfrak O)$ at $c$. Then $\mathcal Z^\theta$ can be thought of as a (global) geometric model for that coset. The *basic stratum* $\mathcal Z^0$ consists of quasimaps of the form $(c\in C, \mathcal P_G, \sigma: C\to \mathcal P_G\times^G X^+)$. Notice that these sub-stacks do not depend on the compactification $X$ of $X^+$. Their *closure*, though, does. For instance, the closure of $\mathcal Z^0$ can be identified with an open substack in the quotient stack $X_C/G_C$ over $C$, namely the stack whose $S$-objects are $S$-objects of $X_C/G_C$ but not of $(X\smallsetminus X^+)_C/G_C$. These are the quasimaps for which the corresponding point in $X^+(\mathcal K)/G(\mathfrak O)$ lies in the image of $X^+(\mathcal K)\cap X(\mathfrak O)$. Hence, the closure of $\mathcal Z^0$ should be thought of as a geometric model for $X^+(\mathcal K)\cap X(\mathfrak O)$. Since the spaces of Gaitsgory and Nadler are global in nature, it is in fact imprecise to say that they are geometric models for local spaces. However, *their singularities* are expected to model the singularities of $G(\mathfrak O)$-invariant subsets of $X^+(\mathcal K)$. ### Drinfeld’s compactifications. {#Dmodels} The spaces of Gaitsgory and Nadler described above are (slightly modified) generalizations of spaces introduced by Drinfeld in the cases: $X=\overline{U_P\backslash G}^{{\operatorname{aff}}}$ or $X=\overline{[P,P]\backslash G}^{{\operatorname{aff}}}$, where $P\subset G$ is a proper parabolic and $U_P$ its unipotent radical. The corresponding spaces are denoted by $\widetilde{{{\operatorname{Bun}}}_P}$ and $\overline{{{\operatorname{Bun}}}_P}$, respectively. Our basic references here are [@BG; @BFGM]. The only differences between the definition of these stacks and the stacks $\mathcal Z$ of Gaitsgory and Nadler are that the section $\sigma$ has to be defined on all $C$, and it does not have a distinguished point $c$. Therefore, for a quasimap in Drinfeld’s spaces and any point $c\in C$ the corresponding element of $X^+(\mathcal K)/G(\mathfrak O)$ has to belong to the cosets which belong to $X(\mathfrak O)$. (These will be described later when we review the computations of [@BFGM].) This particular case is very important to us because it is related to Eisenstein series, and moreover the intersection cohomology sheaf of the “basic stratum” has been computed (when $G, X$ are defined over ${\mathbb{F}}$). ### The basic function {#sssbasicfunction} We return to the setting where $X$ is an affine spherical variety for a split group $G$ over a local, non-archimedean field $F$ whose ring of integers we denote by $\mathfrak o$ and whose (finite) residue field we denote by ${\mathbb{F}}$. We assume that $X$, $G$ and the completions $\bar X, \hat X$ introduced before have the properties of Proposition \[places\] over $\mathfrak o$, and denote $K=G(\mathfrak o)$. The goal is to define the “basic function” $\Phi^0$ on $X^+(F)$, which will be $K$-invariant and supported in $X(\mathfrak o)$. This function will factor through the map $X^+(F)/K\to\Lambda_X^+$ of Theorem \[stratification\]. The idea is to define a function on $\Lambda_X^+$ using equal-characteristic models of $X$. Define the Gaitsgory-Nadler stack $\mathcal Z$ as in §\[GN\] over ${\mathbb{F}}$. Since, by assumption, $X_{{\mathbb{F}}}$ has a completion $\bar X_{{\mathbb{F}}}$ with the properties of Proposition \[places\] (and, hence, the same holds for the base change $X_{{\mathbb{F}}[[t]]}$), the generalized Cartan decomposition \[stratification\] holds for $G({\mathbb{F}}[[t]])$-orbits on $X^+(F((t)))$: they admit a natural map onto $\Lambda_X^+$. Hence the strata $\mathcal Z^\theta$ of $\mathcal Z$ are well-defined over ${\mathbb{F}}$. Let $IC^0$ denote the intersection cohomology sheaf of the closure of the basic stratum $\mathcal Z^0$ (how exactly to normalize it is not important at this point, since we will normalize the corresponding function). We will obtain the value of our function at ${\check\lambda}\in \Lambda_X^+$ as trace of Frobenius acting on the stalk of $IC^0$ at an ${\mathbb{F}}$-object $x_{\check\lambda}$ in the stratum $\mathcal Z^{\check\lambda}$. However, since these strata are only locally of finite type, and not of pure dimension, we must be careful to make compatible choices of points as ${\check\lambda}$ varies. (It is expected that $IC^0$ is locally constant on the strata – this will be discussed below.) The compatibility condition is related to the natural requirement that the action of the unramified Hecke algebra on the functions which will be obtained from sheaves is compatible, via the function-sheaf correspondence, with the action of its geometric counterpart on sheaves. First of all, let us fix a quasimap $x_0=(c_0,\mathcal P_0, \sigma_0)$ in the ${\mathbb{F}}$-objects of the basic stratum $\mathcal Z^0$. Now consider the subcategory $\mathcal Z_{x_0}$ of $\mathcal Z$ consisting of ${\mathbb{F}}$-quasimaps $(c_0,\mathcal P_G,\sigma)$ with the property that there exists an isomorphism $\iota:\mathcal P_0|_{C\smallsetminus\{c_0\}} \stackrel{\sim}{\rightarrow} \mathcal P_G|_{C\smallsetminus\{c_0\}} $ (inducing isomorphisms between $\mathcal P_0\times^G X$ and $\mathcal P_G\times^G X$, also to be denoted by $\iota$) such that $\sigma=\iota\circ\sigma_0$. Hence, the objects in $\mathcal Z_{x_0}$ are those obtained from $x_0$ via *meromorphic Hecke modifications* at the point $c_0$ [@GN §4]. For each ${\check\lambda}\in \Lambda_X^+$, pick an object $x_{\check\lambda}\in \mathcal Z_{x_0}$ which belongs to the stratum $\mathcal Z^{\check\lambda}$. We define the *basic function* $\Phi^0$ on $\Lambda_X^+$ to be: $$\label{basicfunction} \Phi^0({\check\lambda})= c\cdot \sum_i (-1)^i {\operatorname{tr}}({{\operatorname{Fr}}}, H^i(IC^0_{x_{\check\lambda}}))$$ where $IC^0_{x_{\check\lambda}}$ denotes the stalk of $IC^0$ at $x_{\check\lambda}$ and ${{\operatorname{Fr}}}$ denotes the geometric Frobenius. The constant $c$ (independent of ${\check\lambda}$) is chosen such that $\Phi^0(0)=1$. Now we return to $X(F)$ and we identify $\Phi^0$ with a $K$-invariant function on $X^+(F)$ (also to be denoted by $\Phi^0$) via the stratification of Theorem \[stratification\]. This is the “basic function” of Conjecture \[mainconjecture\] at the given place. The definition implies that the support of the basic function is contained in $X(\mathfrak o)$, since the closure of the basic stratum includes the stratum $\mathcal Z^\theta$ only if $\theta$ corresponds to a $G(\mathfrak o)$-orbit belonging to $X(\mathfrak o)$. The independence of choices of the basic function is widely expected but, in the absence of suitable finite-dimensional geometric models, not known. We impose it as an assumption, together with other properties that should naturally follow from the properties of intersection cohomology if one had suitable local models. Notice that for $X=\overline{U_P\backslash G}^{{\operatorname{aff}}}$ or $X=\overline{[P,P]\backslash G}^{{\operatorname{aff}}}$, one could have used instead the Drinfeld models of \[Dmodels\] to define the basic function. \[mainassumption\] 1. The basic function $\Phi^0$ on $X^+(F)$ is well-defined and independent of: - the choices of objects $x_{\check\lambda}$; - (if $X=\overline{U_P\backslash G}^{{\operatorname{aff}}}$ or $X=\overline{[P,P]\backslash G}^{{\operatorname{aff}}}$) which model of §\[ssgeommodels\] one uses to define them; - the group $G$ acting on $X$; more precisely, if $G_1, G_2$ act on $X$ and we denote by $X_1^+,X_2^+$ the open orbits, then the restriction of $\Phi^0$ to $X_1^+(F)\cap X_2^+(F)$ should be the same. 2. If $Z$ is an affine homogeneous spherical $G$-variety and $p:X\to Z$ a surjective equivariant morphism then the basic function on $X$, evaluated at any point $x\in X^+(F)\cap X(\mathfrak o)$, is equal to the basic function of the fiber of $p$ over $p(x)$ (considered as a $G_{p(x)}$-spherical variety). We also discuss how to deduce the growth assumption on elements of the Schwartz space (§\[ssformalism\]) for the basic function. Assume now that $X$ is defined globally over a number field $k$, and fix a finite set of places $S_0$ and suitable $\mathfrak o_{S_0}$-models as in Proposition \[places\]. Recall (§\[sssdistance\]) that these models define a distance function $d_Z=\prod_{v\notin S_0} d_{Z,v}$ from $Z=X\smallsetminus X^+$ on $\prod_{v\notin S_0} X(\mathfrak o_v)$. Assume that there are a $\chi\in {\mathcal{X}}(X)\otimes{\mathbb{R}}$ such that for all places $v$ and all $\check\lambda\in \Lambda_X^+$: $$|\Phi_v^0(\check\lambda)|\le q_v^{\left<\chi,\check\lambda\right>}$$ (where $q_v=|{\mathbb{F}}_v|$). Then there is a natural number $n$ such that: $$\left|\prod_{v\notin S_0} \Phi_v^0 (x) \right|\le (d_Z(x))^{-n}$$ for all $x \in X^+({{\mathbb{A}_k}}^{S_0})$. Here ${{\mathbb{A}_k}}^{S_0}$ denotes the adeles outside of $S_0$. Of course, the function is zero off $\prod_{v\notin S_0} X(\mathfrak o_v)$ so the extension of the distance function off integral points of $X$ plays no role in the statement. First of all, we claim: > The local distance function $d_{Z,v}$ on $X(\mathfrak o_v)$ is $G(\mathfrak o_v)$-invariant. Indeed, $G(\mathfrak o_v)$ preserves the ideal of $Z$ in $\mathfrak o_v[X]$ and therefore its image in $\mathfrak o_v$ under any $\mathfrak o_v$-point. Hence, since both $d_Z$ and $\prod_{v\notin S_0} \Phi_v^0$ are $\prod_{v\notin S_0} G(\mathfrak o_v)$-invariant, it suffices to prove the proposition for a set of representatives of $\prod_{v\notin S_0} G(\mathfrak o_v)$-orbits in the support of $\prod_{v\notin S_0} \Phi_v^0$, namely elements of $A_X({{\mathbb{A}_k}}^{S_0})$ which at every place $v$ have image in $\check\Lambda_X^+\cap \mathcal C(X)$. Let $Y$ denote the “standard $\mathfrak o_{S_0}$-model” of the affine toric embedding of $A_X$ corresponding to the cone $\check\Lambda_X^+\cap \mathcal C(X)$. By assumption (see Proposition \[places\]), there is a morphism $Y\to X$. Therefore, if $Y_1$ denotes the complement of the open orbit on $Y$, the corresponding distance functions on $A_X(k_v)$, for every $v\notin S_0$, compare as: $d_{Z,v}\le d_{Y_1,v}$. On the other hand, clearly, for every $\chi\in {\mathcal{X}}(X)\otimes{\mathbb{R}}$ there is a natural number $n$ such that for all $v\notin S_0$ we have $d_{Y_1,v}^{-n}\ge q_v^{\left<\chi,\check\lambda\right>}$ on $A_X(k_v)\cap Y(\mathfrak o_v)$. The claim follows. Periods and the Rankin-Selberg method {#secrs} ===================================== Pre-flag bundles {#sspreflag} ---------------- We keep assuming that $\mathcal Z(G)^0\xrightarrow{\sim} \mathcal Z(X)$. We will say that an affine spherical $G$-variety $X$ has the structure of a *pre-flag bundle* if: - $X$ is the affine closure of a $G$-stable subvariety $\tilde X^+$, which is homogeneous under a reductive group $\tilde G$; - there is an almost direct factor $L$ of $G$, including $\mathcal Z(G)^0$, which acts freely by $\tilde G$-automorphisms on $\tilde X^+$; - the quotient $\tilde X^+/L$ is proper over an affine, $\tilde G$-homogeneous variety $Y$, called the *base* of the pre-flag bundle. We can assume that $\tilde G = {{\operatorname{Aut}}}^L(\tilde X^+)$ and hence, if $G=LG'$ is an almost direct product decomposition, then $G'$ is embedded in $\tilde G$. Moreover, necessarily $G'$ acts transitively on $Y$, since $\mathcal Z(X)$ acts trivially on $Y$ while, on the other hand, it retracts all points onto a homogeneous subvariety by Proposition \[structureaffine\]. Of course, $L$ can be equal to $\mathcal Z(X)$ or $G'$ can be equal to $\tilde G$. Hence, we have the following geometry for $\tilde X^+$: $$\begin{CD} \tilde X^+ \\ @VV{L\text{-torsor}}V \\ \tilde X^+/L \\ @VV{\text{fiber over }y\in Y\text{ is a flag variety for }\tilde G_y}V \\ Y&\text{ (}\simeq G'_y\backslash G' \simeq \tilde G_y\backslash \tilde G\text{ with }G'_y, \tilde G_y\text{ reductive).} \end{CD}$$ Notice that $L$ is necessarily a quotient of a Levi subgroup of $\tilde G_y$. Indeed, if we write as $\tilde X^+_y=\tilde H_y\backslash\tilde G_y\to \tilde P_y\backslash\tilde G_y$ the map between the fibers of $\tilde X^+$, resp. $\tilde X^+/L$ over $y\in Y$, where $\tilde P_y$ is a parabolic of $\tilde G_y$, then $L$ can be identified with a subgroup of ${{\operatorname{Aut}}}^{\tilde G_y}(X_y)$ preserving the fiber of this map, that is with a subgroup of $\mathcal N_{\tilde P_y}(\tilde H_y)/\tilde H_y$. Since it acts transitively on thof §\[Schwartz\]e fibers of this map, it follows that $\tilde H_y$ must be normal in $\tilde P_y$, and $L$ must be the quotient $\tilde P_y/\tilde H_y$. Since $L$ is reductive, this also implies that $\tilde H_y$ contains the unipotent radical of $\tilde P_y$. In this paper we will additionally impose the condition, without mentioning it further, that the fiber $\tilde X^+_y$ over $y\in Y$ is a product of varieties $[P_i,P_i]\backslash G_i$ or of the form $U_{P_i}\backslash G_i$, where $\prod_i G_i = \tilde G_y$. This extra condition will allow us to restrict our discussion to Eisenstein series induced either from cusp forms or from characters of parabolic subgroups, and to use the computations of [@BFGM]. Notice that the dual group of $L$ acts on the unipotent radical of the dual parabolic to $\tilde P_y$ inside of the dual group of $\tilde G_y$; indeed the quotient $\tilde P_y\twoheadrightarrow L$ gives rise to a homomorphism: $\check L\to \check{\tilde L}_y$, where $\check{\tilde L}_y$ is the standard Levi dual to $\tilde P_y$. We let $\mathfrak{\check u}_{\tilde P}$ denote[^6] the Lie algebra of the unipotent radical of the parabolic dual to $\tilde P_y$, considered as a representation of $\check L$. The requirement that $\tilde G$ commutes with the action of $\mathcal Z(X)$ is meant to allow us to the $\mathcal Z(X)$-Mellin transforms of $X$-theta series to usual Eisenstein series on $\tilde G_y$ induced from $\tilde P_y$. The variety ${\operatorname{Mat}}_n$ for ${\operatorname{GL}}_n\times{\operatorname{GL}}_n$ ($n\ge 2$) is a pre-flag variety, and more generally so is any $N$-dimensional vector space (here $N=n^2$) with a linear $G$-action, as it is equal to the affine closure of $P_{N}\backslash {\operatorname{GL}}_{N}$ (with $P_N$ the mirabolic subgroup). Notice, however, that an $n+m$-dimensional vector space ($n,m\ge 2$) can be considered as a pre-flag variety for both $\tilde G={\operatorname{GL}}_{n+m}$ and $\tilde G={\operatorname{GL}}_n\times {\operatorname{GL}}_m$; which one we will choose will depend on which torus action we will consider. The notions of a pre-flag variety and a pre-flag bundle are not very good, since they are not defined in terms of the group $G$, but in terms of another group $\tilde G$. From our point of view, whether a spherical variety is a pre-flag bundle or not is a matter of “chance” and in fact should be irrelevant as far as properties of $X$-theta series and their applications go – the fundamental object is just $X$ as a $G$-variety, and not its structure of a pre-flag bundle. We will try to provide support for this point of view in §\[sstensor\]. However, in absence of a general proof of Conjecture \[mainconjecture\], this is the only case where its validity, in the weaker form of Conjecture \[weakconjecture\], can be proven. Moreover, the concept of pre-flag bundles is enough to explain a good part of the Rankin-Selberg method. We assume throughout in this section that the local Schwartz spaces $\mathcal S(X_v)$ are the $G$-spaces generated by the “basic function”, which we extract from computations on Drinfeld spaces (outside of a finite number of places), and by functions in $C_c^\infty(X_v^+)$ obtained as convolutions of delta functions with smooth, compactly supported measures on $G_v$. (At non-archimedean places, such functions span $C_c^\infty(X_v)$.) The main result of this section is the following: \[preflagthm\] Assume that $X$ is a wavefront spherical variety with trivial arithmetic multiplicity which has the structure of a a pre-flag bundle, and let $\tau$ vary over a holomorphic family of cuspidal automorphic representations of $G$ (i.e. an irreducible cuspidal representation twisted by idele class characters of the group). Let $\tau_1$ denote the isomorphism class of the restriction of $\tau$ to $L$, and assume that for some finite set of places $S$ the partial $L$-function $L^{S}(\tau_1,{\mathfrak{\check u}_{\tilde P}}, 1)$ has meromorphic continuation everywhere (as $\tau$ varies in this family). Then Conjecture \[weakconjecture\] holds for $\phi\in \tau$ and $\mathcal S(X_v)$ as described above. We prove this theorem in §\[ssRS\] by appealing to the meromorphic continuation of usual Eisenstein series, after explicitly describing the basic vectors according to the computations of intersection cohomology sheaves on Drinfeld spaces in [@BFGM]. However, the application of the meromorphic continuation of Eisenstein series is not completely trivial as in some cases we have to use the theory of spherical varieties to show that as we “unfold” this integral certain summands vanish (in the language often used in the theory of Rankin-Selberg integrals: certain $G$-orbits on $X$ are “negligible”). We start by demonstrating an extreme case, which gives rise to period integrals. Period integrals {#ssperiods} ---------------- First consider the extreme case of a pre-flag bundle with trivial fibers: Namely, choosing a point $x_0\in X(k)$, we have $X=H\backslash G$ with $H=G_{x_0}$ *reductive*. Then at each place $v\notin S_0$ the basic function is the characteristic function of $X(\mathfrak o_v)$, and we may assume that $\mathcal S(X({{\mathbb{A}_k}})) = C_c^\infty(X({{\mathbb{A}_k}}))$. The multiplicity-freeness assumption of §\[ssconjectures\] implies, in particular, that at almost every place $G(\mathfrak o_v)$ acts transitively on $X(\mathfrak o_v)$. Then we can take $\Phi \in \mathcal S(X({{\mathbb{A}_k}}))$ of the form $\Phi=h\star \delta_{x_0}$ where $h\in \mathcal H(G({{\mathbb{A}_k}}))$, the Hecke algebra of compactly supported smooth measures on $G({{\mathbb{A}_k}})$, and $\delta_{x_0}$ is the delta function at $x_0$ (considered as a generalized function). Then, if $\check h$ denotes the adjoint to $h$ element of $\mathcal H(G({{\mathbb{A}_k}}))$, the integral: $$\int_{G(k)\backslash G({{\mathbb{A}_k}})} \phi\cdot\omega(g) \theta(\Phi,g) dg$$ of Conjecture \[weakconjecture\] is equal to: $$\int_{H(k)\backslash H({{\mathbb{A}_k}})} (\check h\star \phi)\cdot \omega(g) dg.$$ This is called a *period integral*, and such integrals have been studied extensively. Hence *period integrals are the special case of the pairing of Conjecture \[weakconjecture\] which is obtained from pre-flag bundles with trivial fibers* (i.e. affine homogeneous spherical varieties). For example, when $X={\operatorname{GL}}_2$, $G={\mathbb{G}_m}\times{\operatorname{GL}}_2$, with ${\mathbb{G}_m}$ acting as a non-central torus of ${\operatorname{GL}}_2$ by multiplication on the left, we get the period integral of Hecke (\[hecke\]), discussed in the introduction. All spherical period integrals are included in the lists of Knop and van Steirteghem [@KnVS] which we will discuss in the next section. Connection to usual Eisenstein series ------------------------------------- ### Certain stacks and sheaves related to flag varieties The goal of this subsection is to explicate the basic functions $\Phi_v^0$ for pre-flag bundles, based on the computations of [@BFGM]. We work with the varieties $X=\overline{[P,P]\backslash G}^{{\operatorname{aff}}}$ or $X=\overline{U_P\backslash G}^{{\operatorname{aff}}}$ and use the notation of §\[Dmodels\]. We do not aim to give complete definitions of the constructions of *loc. cit.*, but to provide a guide for the reader who would like to extract from it the parts most relevant to our present discussion. The final result will be the following formula for the basic function $\Phi^0$ (locally at a non-archimedean place, which we suppress from the notation): \[BFGMfunctions\] For $X=\overline{H\backslash G}$ in each of the following cases, we have: - If $H=U_P$: $\Phi^0 = \sum_{i\ge 0} q^{-i}\widecheck{{\operatorname{Sat}}}_M\left(\operatorname{Sym}^i(\check{\mathfrak u}_P)\right)\star 1_{HK} = $ $$=\widecheck{{\operatorname{Sat}}}_M\left(\frac{1}{\wedge^{\operatorname{top}} (1-q^{-1}\check{\mathfrak u}_P)}\right) \star 1_{HK}.$$ - If $H=[P,P]$: $\Phi^0 = \sum_{i\ge 0} q^{-i}\widecheck{{\operatorname{Sat}}}_{M^{{\operatorname{ab}}}}\left(\operatorname{Sym}^i(\check{\mathfrak u}_P^f)\right)\star 1_{HK} =$ $$= \widecheck{{\operatorname{Sat}}}_{M^{{\operatorname{ab}}}}\left(\frac{1}{\wedge^{\operatorname{top}}(1-q^{-1}\check{\mathfrak u}_P^f)}\right) \star 1_{HK}.$$ The notation will be explained in §\[sssEisfunctions\]. We denote by $\Lambda_{G,P}$ the lattice of cocharacters of the torus $M/[M,M]$ and by $\Lambda_{G,P}^{{\operatorname{pos}}}$ the sub-semigroup spanned by the images of $\check\Delta\smallsetminus\check\Delta_M$. For every $\theta\in \Lambda_{G,P}^{{\operatorname{pos}}}$ we have a canonical locally closed embedding: $j_\theta: C\times {{\operatorname{Bun}}}_P\to \overline{{{\operatorname{Bun}}}}_P$ [@BFGM Proposition 1.5]. The image will be denoted by ${_{(\theta)}\overline{Bun_P}}$. (Notice: This is not the same as what is denoted in *loc.cit.* by ${_{\theta}\overline{Bun_P}}$, but rather what is denoted by ${_{\mathfrak U(\theta)}\overline{Bun_P}}$, when $\mathfrak U(\theta)$ is the trivial partition of $\theta$.) Its preimage in $\widetilde{{{\operatorname{Bun}}}}_P$ will be denoted by $_{(\theta)}\widetilde{{{\operatorname{Bun}}}}_P$. We have a canonical isomorphism: $_{(\theta)}\widetilde{{{\operatorname{Bun}}}}_P\simeq {{\operatorname{Bun}}}_P\times_{{{\operatorname{Bun}}}_M}\mathcal H_M^{(\theta)}$, where $\mathcal H_M^{(\theta)}$ is a stack which will be described below. (i) If $X=\overline{[P,P]\backslash G}^{{\operatorname{aff}}}$ under the $M^{{\operatorname{ab}}}=M/[M,M]\times G$-action, then $\Lambda_X^+$ can be identified with $\Lambda_{G,P}$, and $_{(\theta)}\overline{Bun}_P$ is precisely the analog of what we denoted by $\mathcal Z^{w_0\theta}$ on the Gaitsgory-Nadler stacks, where $w_0$ is the longest element in the Weyl group of $G$. The reason that only $\theta\in \Lambda_{G,P}^{{\operatorname{pos}}}$ appear is that, as we remarked in §\[Dmodels\], the quasi-maps on Drinfeld spaces are, by definition, not allowed to have poles. For the reader who would like to trace this back to the combinatorics of quasi-affine varieties and their affine closures of §\[ssaffine\] we mention that the cone spanned by $\rho(\mathcal D)$ is the cone spanned by the images of $\check\Delta\smallsetminus\check\Delta_M$. (ii) If $X=\overline{U_P\backslash G}^{{\operatorname{aff}}}$ under the $M\times G$-action then $\Lambda_X^+\simeq\{{\check\lambda}\in \Lambda_A| \left<{\check\lambda},\alpha\right>\le 0 \text{ for all }\alpha\in\Delta_M\}$ (where we denote by $A$ the maximal torus of $G$ and by $\Lambda_A$ its cocharacter lattice). There is a map: $\Lambda_X\to\Lambda_{G,P}$, and $_{(\theta)}\widetilde{Bun}_P$ corresponds to the union of the strata $\mathcal Z^{w_0{\check\lambda}}$ of Gaitsgory-Nadler, with ${\check\lambda}$ ranging over all the $M$-dominant preimages of $\theta$. We have the geometric Satake isomorphism, i.e. a functor $\operatorname{Loc}:{{\operatorname{Rep}}}(\check G)\to \operatorname{Perv}(\mathcal G_G)$ such that the irreducible representation of $\check G$ with highest weight ${\check\lambda}$ goes to the intersection cohomology sheaf of a $G(\mathfrak o)$-equivariant closed, finite-dimensional subscheme $\overline{\mathcal G_G}^{\check\lambda}$. We will make use of this functor for $M$, rather than $G$. If $V$ is a representation of $\check M$ – assumed “positive”; this has to do with the fact that we don’t allow poles, but there’s no need to explain it here – and $\theta\in \Lambda_{G,P}^{{\operatorname{pos}}}$ then we define $\operatorname{Loc}^{(\theta)}(V)$ to be $\operatorname{Loc}(V_\theta)$, where $V_\theta$ is the $\theta$-isotypic component of $V$. (We ignore a twist by $\overline{{\mathbb{Q}}_l}[1]\left(\frac{1}{2}\right)^{-1}$ introduced in [@BFGM], and modify the results accordingly.) We now introduce relative, global versions of the above spaces. We denote by $\mathcal H_M$ the *Hecke stack* of $M$. It is related to $\mathcal G_M$ as follows: If we fix a curve $C$ and a point $x\in \mathcal C$ then, by definition, $\mathcal G_M$ is the functor Schemes$\to$Sets which associates to every scheme $S$ the set of pairs $(\mathcal F_G,\beta)$ where $\mathcal F_M$ is a principal $M$-bundle over $C\times S$ and $\beta$ is an isomorphism of it outside of $(C\smallsetminus\{x\})\times S$ with the trivial $M$-bundle. The relative version of this, as we allow the point $x$ to move over the curve, is denoted by $\mathcal G_{M,C}$, and the relative version of the latter, as we replace the trivial $M$-bundle with an arbitrary $M$-bundle, is $\mathcal H_M$. It is fibered over $C\times{{\operatorname{Bun}}}_M$. In *loc.cit.*, p. 389, certain closed, finite-dimensional subschemes $\mathcal G_M^{+,\theta}$ of $\mathcal G_M$ are defined for every $\theta\in \Lambda_{G,P}^{{\operatorname{pos}}}$ which at the level of reduced schemes are isomorphic to $\overline{\mathcal G_M}^{\flat(\theta)}$, where $\flat(\theta)$ is an $M$-dominant coweight associated to $\theta$ – the “least dominant” coweight mapping to $\theta$. The relative versions of those give rise to substacks $\mathcal H_M^{(\theta)}$ of $\mathcal H_M$. For these relative versions we have: Functors $\operatorname{Loc}_{{{\operatorname{Bun}}}_M,C}$ (resp. $\operatorname{Loc}_{{{\operatorname{Bun}}}_M,C}^{(\theta)}$) from ${{\operatorname{Rep}}}(\check M)$ to perverse sheaves on $\mathcal H_M$ (resp. $\mathcal H_M^{(\theta)}$) and $\operatorname{Loc}_{{{\operatorname{Bun}}}_P,C}$ (resp. $\operatorname{Loc}_{{{\operatorname{Bun}}}_P,C}^{(\theta)}$) to perverse sheaves on ${{\operatorname{Bun}}}_P\times_{{{\operatorname{Bun}}}_M}\mathcal H_M$ (resp. ${{\operatorname{Bun}}}_P\times_{{{\operatorname{Bun}}}_M}\mathcal H_M^{(\theta)}$), the latter being $IC_{{{\operatorname{Bun}}}P}$ along the base ${{\operatorname{Bun}}}_P$. Then the main theorem of [@BFGM] (Theorem 1.12) is a description of the $*$-restriction of $IC_{\widetilde{{{\operatorname{Bun}}}}_P}$ to $_{(\theta)}\widetilde{{{\operatorname{Bun}}}}_P\simeq {{\operatorname{Bun}}}_P\times_{{{\operatorname{Bun}}}_M}\mathcal H_M^{(\theta)}$. Moreover, Theorem 7.3 does the same thing for $IC_{\overline{{{\operatorname{Bun}}}}_P}$ and $_{(\theta)}\overline{{{\operatorname{Bun}}}}_P\simeq C\times {{\operatorname{Bun}}}_P$. The normalization of $IC$ sheaves is that they are pure of weight 0; i.e. for a smooth variety $Y$ of dimension $n$ we have $IC_Y\simeq \left(\overline{{\mathbb{Q}}_l}\left(\frac{1}{2}\right)[1]\right)^{\otimes n}$, where $\overline{{\mathbb{Q}}_l}\left(\frac{1}{2}\right)$ is a fixed square root of $q$. \[BFGMtheorem\] The $*$-restriction of $IC_{\widetilde{{{\operatorname{Bun}}}}_P}$ to $_{(\theta)}\widetilde{{{\operatorname{Bun}}}}_P\simeq {{\operatorname{Bun}}}_P\times_{{{\operatorname{Bun}}}_M}\mathcal H_M^{(\theta)}$ is equal to: $$\label{tildebun} \operatorname{Loc}_{{{\operatorname{Bun}}}_P,C}^{(\theta)} \left(\oplus_{i\ge 0} \operatorname{Sym}^i(\check{\mathfrak u}_P)\otimes \overline{{\mathbb{Q}}_l}(i)[2i]\right).$$ The \*-restriction of $IC_{\overline{{{\operatorname{Bun}}}}_P}$ to $_{(\theta)}\overline{{{\operatorname{Bun}}}}_P\simeq C\times {{\operatorname{Bun}}}_P$ is equal to: $$\label{linebun} IC_{_{(\theta)}\overline{{{\operatorname{Bun}}}}_P} \otimes \overline{\operatorname{Loc}}\left(\oplus_{i\ge 0} \operatorname{Sym}^i(\check{\mathfrak u}_P^f)_\theta\otimes \overline{{\mathbb{Q}}_l}(i)[2i]\right).$$ Here $\check{\mathfrak u}_P$ denotes the adjoint representation of $\check M$ on the unipotent radical of the parabolic dual to $P$. Moreover, $\check{\mathfrak u}_P^f$ denotes the subspace which is fixed under the nilpotent endomorphism $f$ of a principal $\mathfrak{sl}_2$-triple $(h,e,f)$ in the Lie algebra of $\check M$. For the definition of $\overline{\operatorname{Loc}}(V)$, which takes into account the grading on $V$ arising from the $h$-action, cf. *loc.cit.*, §7.1. ### The corresponding functions {#sssEisfunctions} Let us fix certain normalized Satake isomorphisms. As before, our local, non-archimedean field is denoted by $F$, its ring of integers by $\mathfrak o_F$, and our groups are assumed to have reductive models over $\mathfrak o_F$. As usual, we normalize the action of $M(F)$ (resp. $M^{{\operatorname{ab}}}(F)$) on functions on $(H\backslash G)(F)$ where $H=U_P$ (resp. $[P,P]$) so that it is unitary on $L^2((H\backslash G)(F))$: $$\label{Maction} m\cdot f(H(F)g) = \delta_P^{\frac{1}{2}}(m) f(H(F)m^{-1}g),$$ where $\delta_P$ is the modular character of $P$. We let $M_0=M(\mathfrak o_F)$, and normalize the (classical) Satake isomorphism as follows: - For the Hecke algebra $\mathcal H(M,M_0)$ in the usual way: $${{\operatorname{Sat}}}_M:{\mathbb{C}}[\check M]^{\check M}\simeq {\mathbb{C}}[{{\operatorname{Rep}}}\check M]\xrightarrow{\sim} \mathcal H(M,M_0)$$ where $ {\mathbb{C}}[{{\operatorname{Rep}}}\check M]$ is the Grothendieck algebra over ${\mathbb{C}}$ of the category of algebraic representations of $\check M$. - For the Hecke algebra $\mathcal H(M^{{\operatorname{ab}}},M^{{\operatorname{ab}}}_0)$ we shift the usual Satake isomorphism: $\mathcal H(M^{{\operatorname{ab}}},M^{{\operatorname{ab}}}_0)\simeq {\mathbb{C}}[\mathcal Z(\check M)]\simeq {\mathbb{C}}[{{\operatorname{Rep}}}\mathcal Z(\check M)]$ by $e^{-\rho_M}$, where $\rho_M$ denotes half the sum of positive roots of $M$. In other words, if $h$ is a compactly supported measure on $M(F)/M_0$, considered (canonically) as a linear combination of cocharacters of $M^{{\operatorname{ab}}}$ and hence as a regular function $f$ on the center $\mathcal Z(\check M)$ of its dual group, then we will assign to $h$ the function $z\mapsto f(e^{\rho_M}z)$ on the subvariety $e^{-\rho_M}\mathcal Z(\check M)$ of $\check G$: $${{\operatorname{Sat}}}_{M^{{\operatorname{ab}}}}:{\mathbb{C}}[e^{-\rho_M}\mathcal Z(\check M)]\xrightarrow{\sim} \mathcal H(M^{{\operatorname{ab}}},M^{{\operatorname{ab}}}_0).$$ Let $1_{HK}$ denote the characteristic function of $H\backslash HK$ (where $K=G(\mathfrak o_F)$), and consider the action map: $\mathcal H(M,M_0)\to C_c^\infty((U_P\backslash G)(F))^{M_0\times K}$, respectively $\mathcal H(M^{{\operatorname{ab}}},M^{{\operatorname{ab}}}_0)\to C_c^\infty(([P,P]\backslash G)(F))^K$ given by $h\mapsto h \star 1_{HK}$. The map is bijective, and identifies the module $C_c^\infty((H\backslash G)(F))^{M_0\times K}$ with ${\mathbb{C}}[\check M]^{\check M}$, resp. ${\mathbb{C}}[e^{-\rho_M}Z(\check M)]$. Our normalization of the Satake isomorphism is such that this is compatible with the Satake isomorphism for $G$, ${{\operatorname{Sat}}}_G:\mathcal H(G,K)={\mathbb{C}}[\check G]^{\check G}={\mathbb{C}}[{{\operatorname{Rep}}}(\check G)]$, in the sense that for $f\in {\mathbb{C}}[\check G]^{\check G}$ we have: $${{\operatorname{Sat}}}_G(f)\star 1_{HK} = \widecheck{{\operatorname{Sat}}}_{M\text{ or }M^{{\operatorname{ab}}}}(f) \star 1_{HK}.$$ Here and later, by the symbol $\check h$ we will be denoting the adjoint of the element $h$ in a Hecke algebra. Its appearance is due to the the definition (\[Maction\]) of the action of $M$ as a right action on the space and a left action on functions. We extend the “Sat” notation to the fraction field of ${\mathbb{C}}[{{\operatorname{Rep}}}\check M]$ (and, respectively, of ${\mathbb{C}}[e^{-\rho_M}\mathcal Z(\check M)]$), where ${{\operatorname{Sat}}}_{M \text{ or } M^{{\operatorname{ab}}}}(R)$ (with $R$ in the fraction field) is thought of as a power series in the Hecke algebra. Returning to the Drinfeld spaces discussed in the previous subsection, let $\operatorname{Ff}(E)(x):=\sum_i (-1)^i {\operatorname{tr}}({{\operatorname{Fr}}}, H^i(E_x))$ denote the alternating sum of the trace of Frobenius acting on the homology of the stalks of a perverse sheaf ($\operatorname{Ff}$ stands for “faisceaux-fonctions”). As in §\[sssbasicfunction\], we fix an object $x_0$ on the basic stratum, a point $c_0\in C$ (recall that in the definition of Drinfeld’s spaces, quasimaps do not have distinguished points) and we evaluate $\operatorname{Ff}(E)$, where $E=IC_{\widetilde{{{\operatorname{Bun}}}}_P}$ or $IC_{\overline{{{\operatorname{Bun}}}}_P}$, only at objects $x_{\check\lambda}$ which are obtained by $M\times G$-Hecke modifications at $c_0$. This way, and using the Iwasawa decomposition, we obtain our basic function $\Phi^0$, which is an $M_0\times K$-invariant function on $(H\backslash G)(F)$. Recall that it is by definition normalized such that $\Phi^0(H\backslash H1)=1$. The study of the Hecke corresponences in [@BG] implies that $$\operatorname{Ff}(\operatorname{Loc}_{{{\operatorname{Bun}}}_P, C} (V)) = {\widecheck{{\operatorname{Sat}}}}_M(V) \star \operatorname{Ff}(\operatorname{Loc}_{{{\operatorname{Bun}}}_P, C} (1))$$ if $H=U_P$, and $$\operatorname{Ff} (\overline{\operatorname{Loc}} (V)) = \widecheck{{\operatorname{Sat}}}_{M^{{\operatorname{ab}}}}(V) \star \operatorname{Ff}(\overline{\operatorname{Loc}}(1))$$ if $H=[P,P]$. The “unitary” normalization of the action of $M$ is already present in the sheaf-theoretic setting as follows: Suppose that an object $x_{\check\lambda}$ belongs to $_{({\check\lambda})}\overline{{{\operatorname{Bun}}}}_P$ and can be obtained from $x_0$ via Hecke modifications at the distinguished object of $x_0$. Then the dimension of $_{({\check\lambda})}\overline{{{\operatorname{Bun}}}}_P\simeq C\times {{\operatorname{Bun}}}_P$ at $x_{\check\lambda}$ is $\left<{\check\lambda},2\rho_P\right>$ less than that of $_{(0)}\overline{{{\operatorname{Bun}}}}_P$ around $x_0$, where $\rho_P$ denotes the half-sum of roots in the unipotent radical of $P$, i.e. $\delta_P=e^{2\rho_P}$. Hence, by the aforementioned normalization of $IC$ sheaves, the contribution of the factor $IC_{({\check\lambda})}\overline{{{\operatorname{Bun}}}}_P$ (via Theorem \[BFGMtheorem\]) to $\Phi^0({\check\lambda})$ will be $q^{\left<{\check\lambda},\rho_P\right>}$ times the contribution of the factor $IC_{(0)}\overline{{{\operatorname{Bun}}}}_P$ to $\Phi^0(0)$. Similarly for the strata of $\widetilde{{{\operatorname{Bun}}}}_P$. Thus, Theorem \[BFGMtheorem\] translates to the statement of Theorem \[BFGMfunctions\]: - If $H=U_P$: $\Phi^0 = \sum_{i\ge 0} q^{-i}\widecheck{{\operatorname{Sat}}}_M\left(\operatorname{Sym}^i(\check{\mathfrak u}_P)\right)\star 1_{HK} = $ $$=\widecheck{{\operatorname{Sat}}}_M\left(\frac{1}{\wedge^{\operatorname{top}} (1-q^{-1}\check{\mathfrak u}_P)}\right) \star 1_{HK}.$$ - If $H=[P,P]$: $\Phi^0 = \sum_{i\ge 0} q^{-i}\widecheck{{\operatorname{Sat}}}_{M^{{\operatorname{ab}}}}\left(\operatorname{Sym}^i(\check{\mathfrak u}_P^f)\right)\star 1_{HK} =$ $$= \widecheck{{\operatorname{Sat}}}_{M^{{\operatorname{ab}}}}\left(\frac{1}{\wedge^{\operatorname{top}}(1-q^{-1}\check{\mathfrak u}_P^f)}\right) \star 1_{HK}.$$ Notice that in the last expression $\check{\mathfrak u}_P^f$ is considered as a representation of the maximal torus $\check A$ of $\check M$ determined by the principal $\mathfrak{sl}_2$-triple $(h,e,f)$ and, by restricting its character to the subvariety $e^{-\rho_M}\mathcal Z(\check M)$, as an element of $\mathcal H(M^{{\operatorname{ab}}},M^{{\operatorname{ab}}}_0)$. This is the case studied in [@BK2], and $\Phi^0$ is the function denoted by $c_{P,0}$ there. ### Connection to Eisenstein series {#connES} Now we discuss our main conjecture when the variety is $X=\overline{U_P\backslash G}^{{\operatorname{aff}}}$ or $X=\overline{[P,P]\backslash G}^{{\operatorname{aff}}}$ under the (normalized) action of $M\times G$, resp. $M^{\textrm{ab}}\times G$. In the latter case, our Eisenstein series $E(\Phi,\omega,g)$ are the usual (degenerate, if $P$ is not the Borel) principal Eisenstein series normalized as in [@BK; @BK2], and hence $E(\Phi,\omega,g)$ is indeed meromorphic for all $\omega$. It will be useful to recall how these meromorphic sections are related to the more usual sections $E(f,\omega,g)$, which are defined in the same way but with $f\in C_c^\infty(([P,P]\backslash G)({{\mathbb{A}_k}}))$. We assume that $\Phi=\prod_v \Phi_v, f=\prod_v f_v$ and $S$ is a finite set of places (including $S_0$) such that for $v\notin S$ we have $\Phi_v=\Phi_v^0$ and $f_v=f_v^0:= 1_{U\backslash G (\mathfrak o_v)}$. Let us also assume for simplicity that for $v\in S$ we have $\Phi_v=f_v$ (a finite number of places certainly do not affect meromorphicity properties). Clearly, for $E(\Phi,\omega,g)$ and $E(f,\omega,g)$ to be non-zero the character $\omega$ must be unramified outside of $S$. Then by the results of the previous paragraph we have: $$\label{degES} E(\Phi,\omega,g)= L^{S}(e^{-\rho_M}\omega,\check{\mathfrak u}_P^f, 1) E(f,\omega,g)$$ where $L^{S}(e^{-\rho_M}\omega, \check{\mathfrak u}_P^f,1)$ denotes the value at $1$ of the partial (abelian) $L$-function corresponding to the representation $\check{\mathfrak u}_P^f$, whose local factors (at each place $v$) are considered as rational functions on the maximal torus $\check A\subset \check M$ and evaluated at the point $e^{-\rho_M}\omega_v\in e^{-\rho_M}\mathcal Z(\check M)\subset \check A$. Now let us consider the case $X=\overline{U_P\backslash G}^{{\operatorname{aff}}}$. We let $\tau$ vary over a holomorphic family of cuspidal representations of $M\times G$ and let $\tau\mapsto \phi_\tau$ be a meromorphic section; write $\tau=\tau_1\otimes\tau_2$ according to the decomposition of the group $M\times G$, and assume that, accordingly, $\phi_\tau=\phi_{\tau_1}\otimes\phi_{\tau_2}$, a pure tensor. Assume for the moment that the central character of $\tau$ is sufficiently $X$-positive. If in the notation of Conjecture \[weakconjecture\] we replace the group $G$ by the group $M\times G$, and perform the integration of the conjecture, but only over the factor $M(k)\backslash M({{\mathbb{A}_k}})$, then this integral can be written as: $$\begin{aligned} \label{nprI} \int_{M(k)\backslash M({{\mathbb{A}_k}})} \phi_\tau (m,g) \theta (\Phi,(m,g)) dm= \nonumber \\ = \phi_{\tau_2}(g) \int_{M(k)\backslash M({{\mathbb{A}_k}})} \phi_{\tau_1} (m) \theta (\Phi,(m,g)) dm.\end{aligned}$$ It is valued in the space of functions on $G(k)\backslash G({{\mathbb{A}_k}})$. If ${{\operatorname{Eis}}}: I_{P({{\mathbb{A}_k}})}^{G({{\mathbb{A}_k}})}(\tau_1) \to C^\infty(G(k)\backslash G({{\mathbb{A}_k}}))$ denotes the usual Eisenstein operator, then by unfolding the last integral we see that it is equal to the Eisenstein series: $$\label{nprES} E_M(\Phi,\phi_1,g):= {{\operatorname{Eis}}}\left( \int_{M({{\mathbb{A}_k}})} \phi_{\tau_1}(m) (m\cdot \Phi) dm\right)(g)$$ hence the connection to usual Eisenstein series. \[prES\] Assume that the partial $L$-function $L^{S}(\tau_1,{\mathfrak{\check u}_P}, 1)$ (for some large enough $S$) has meromorphic everywhere as $\tau_1$ is twisted by characters of $M$. Then the expression (\[nprI\]) admits meromorphic continuation to all $\tau_1$. By the meromorphic continuation of Eisenstein series, it is enough to show that the integral $(\Phi,\phi_{\tau_1})\mapsto \int_{M({{\mathbb{A}_k}})} \phi_{\tau_1}(m)(m\cdot \Phi) dm$, which represents a morphism: $\iota_{\tau_1}:\mathcal S(U_P\backslash G({{\mathbb{A}_k}}))\to I_{P({{\mathbb{A}_k}})}^{G({{\mathbb{A}_k}})}(\tau_1)$, admits meromorphic continuation in $\tau_1$. This would be the case if $\Phi$ was in $C_c^\infty(U_P\backslash G({{\mathbb{A}_k}}))$. The analogous morphism: $C_c^\infty(U_P\backslash G({{\mathbb{A}_k}}))\to I_{P({{\mathbb{A}_k}})}^{G({{\mathbb{A}_k}})}(\tau_1)$ will also be denoted by $\iota_{\tau_1}$. Again, we let $S$ be a finite set of places containing $S_0$ and take functions $\Phi=\prod \Phi_v\in \mathcal S(U_P\backslash G({{\mathbb{A}_k}}))$ and $f=\prod_v f_v\in C_c^\infty(U_P\backslash G({{\mathbb{A}_k}}))$ such that for $v\notin S$ $\Phi_v=\Phi_v^0$ is the basic $M_0\times K$-invariant function of the previous paragraph, $f_v=f_v^0=1_{U_PK}$ and for $v\in S$ we have $\Phi_v=f_v$ (for simplicity). Moreover, we assume that $\tau_1$ is unramified for $v\notin S$, otherwise the integral will be zero. We saw previously that $\Phi_v^0 = \widecheck{{\operatorname{Sat}}}_M\left(\frac{1}{\wedge^{\operatorname{top}}(1-q^{-1}\check{\mathfrak u}_P)}\right) \star f_v^0$. By definition of the Satake isomorphism and the equivariance of $\iota_\tau$, in the domain of convergence we have $\iota_{\tau_1} (\Phi) = L^{S}(\tau_1,{\mathfrak{\check u}_P}, 1) \iota_{\tau_1}(f)$. Therefore ${{\operatorname{Eis}}}(\iota_{\tau_1}(\Phi))= L^{S}(\tau_1,{\mathfrak{\check u}_P}, 1) {{\operatorname{Eis}}}(\iota_{\tau_1}(f))$, and the claim follows from the meromorphic continuation of ${{\operatorname{Eis}}}(\iota_{\tau_1}(f))$. 1. The meromorphic continuation of $L^{S}(\tau_1,{\mathfrak{\check u}_P}, 1)$ is known in many cases, e.g. for $G$ a classical group and $\tau$ generic, by the work of Langlands, Shahidi and Kim, cf. [@CKM]. 2. Notice that, as was also observed in [@BK; @BK2], the Eisenstein series (\[nprES\]) has normalized functional equations without $L$-factors. The Rankin-Selberg method {#ssRS} ------------------------- According to [@BuRS §5], the Rankin-Selberg method involves a cusp form on $G$ and an Eisenstein series on a group $\tilde G$, where we have either an embedding: $G\hookrightarrow\tilde G$ or an embedding $\tilde G\hookrightarrow G$, or “something more complicated”. We certainly do not claim to explain all constructions which have been called “Rankin-Selberg integrals”, but let us see how a large part of this method is covered by our constructions. Let $X$ be a pre-flag bundle; we will use the notation of §\[sspreflag\]. For notational simplicity (the arguments do not change), let us also assume that $L$ is a direct factor of $G$, i.e. $G=L\times G'$. Let $\Phi\in \mathcal S(X({{\mathbb{A}_k}}))$. Recall that the $X$-theta series $\theta(\Phi,g)$ has been defined via a sum over $X^+(k)$, where $X^+$ denotes the open $G$-orbit on $X$. On the other hand, to relate our integrals to usual Eisenstein series, we need to sum over $\tilde X^+(k)$, where $\tilde X^+$ is the open $\tilde G$-orbit. Hence, we define: $$\tilde \theta(\Phi,g)=\sum_{\gamma\in \tilde X^+(k)} \Phi(\gamma\cdot g).$$ We compare the integral of Conjecture \[weakconjecture\] with the corresponding integral when $\theta$ is substituted by $\tilde \theta$: \[equal\] Suppose that $X$ is a wavefront spherical variety with the structure of a pre-flag bundle. If $\phi$ is a cusp form on $G$ (with sufficiently $X$-positive central character, so that the following integrals converge) then: $$\int_{G(k)\backslash G({{\mathbb{A}_k}})} \phi(g) \theta(\Phi,g) dg = \int_{G(k)\backslash G({{\mathbb{A}_k}})} \phi(g) \tilde \theta(\Phi,g) dg.$$ Assume this proposition for now, and let us prove Theorem \[preflagthm\]; at the same time, we will see that the integral of Conjecture \[weakconjecture\] is equal to a Rankin-Selberg integral. Without loss of generality, $\Phi=\prod_v \Phi_v$, and $\phi=\phi_1(l)\phi_2(g)$ according to the decomposition $G=L\times G'$. By Assumption \[mainassumption\], and repeating the argument of §\[ssperiods\], we may write $\Phi$ as the convolution with an element $h\in \mathcal H(G'({{\mathbb{A}_k}}))$ of a Schwartz function $\Phi^y$ on $X_y({{\mathbb{A}_k}})$, where $y\in Y(k)$ and the Schwartz function on $X_y({{\mathbb{A}_k}})$ is considered as a generalized function on $\tilde X^+({{\mathbb{A}_k}})$. Then, as in §\[ssperiods\]: $$\int_{G(k)\backslash G({{\mathbb{A}_k}})} \phi(g) \tilde \theta(\Phi,g) dg = \int_{G_y(k)\backslash G_y({{\mathbb{A}_k}})} \check h \star \phi (h) \tilde \theta_{\tilde X_y^+}(\Phi^y,h),$$ where $\tilde \theta_{\tilde X_y^+}(\Phi,g)$ denotes the theta series for the $\tilde G_y$-spherical variety $X_y$. By the decomposition $G=L\times G'$ this is equal to: $$\int_{G_y'(k)\backslash G_y'({{\mathbb{A}_k}})} \check h\star \phi_2 (g) \int_{L(k)\backslash L({{\mathbb{A}_k}})} \phi_1 (l) \tilde \theta_{\tilde X_y^+}(\Phi^y,lg) dl dg.$$ The inner integral is equal to the Eisenstein series $E_L(\Phi, \phi_1, g')$ on the group $\tilde G_y'$, in the notation of (\[nprES\]), or a degenerate Eisenstein series as in (\[degES\]), or a product of such[^7], and it has meromorphic continuation under the assumption that $L^{S}(\tau_1,{\mathfrak{\check u}_{\tilde P}}, 1)$ does. Hence, we see that *the integral of conjecture \[weakconjecture\] is equal to the Rankin-Selberg integral:* $$\label{RSintegral} \int_{G_y'(k)\backslash G_y'({{\mathbb{A}_k}})} \check h\star \phi_2(g) E_L(\Phi, \phi_1, g) dg$$ and this also completes the proof of Theorem \[preflagthm\]. In the language of [@BuRS §5], our formalism combines the appearance of a subgroup $G_y\subset G$ with an embedding of it into another group: $G_y\hookrightarrow \tilde G_y$. ### Proof of Proposition \[equal\]: Negligible orbits. Proposition \[equal\] will follow from the following statement on the structure of certain spherical varieties: \[parind\] If $X$ is a wavefront spherical variety for $G$ with ${{\operatorname{Aut}}}^G(X)$ finite, then the isotropy groups of all non-open $G$-orbits contain the unipotent radical of a proper parabolic of $G$. From this, Proposition \[equal\] follows easily; in the domain of convergence we have: $$\int_{G(k)\backslash G({{\mathbb{A}_k}})} \phi(g)\tilde\theta(\Phi,g)= \sum_{\xi\in [\tilde X^+(k)/G(k)]} \int_{G_\xi(k)\backslash G({{\mathbb{A}_k}})} \phi(g) g\cdot \Phi(\xi) dg$$ where $[\tilde X^+(k)/G(k)]$ denotes any set of representatives for the set of $G(k)$-orbits on $\tilde X^+(k)$. Notice that, by the multiplicity-freeness assumption on $X$, the $k$-points of the open $G$-orbit form a unique $G(k)$-orbit. The summand corresponding to $\xi$ can be written: $$\int_{G_\xi({{\mathbb{A}_k}})\backslash G({{\mathbb{A}_k}})} g\cdot \Phi(\xi) \int_{G_\xi(k)\backslash G_\xi({{\mathbb{A}_k}})} \phi(hg) dh dg$$ Since ${{\operatorname{Aut}}}^G(\tilde X^+/\mathcal Z(X))$ is finite, for $\xi$ in the non-open orbit the stabilizer $G_\xi$ contains the unipotent radical of a proper parabolic by Proposition \[parind\], and since $\phi$ is cuspidal the inner integral will vanish. Therefore, only the summand corresponding to the open orbit survives, which folds back to the integral: $$\int_{G(k)\backslash G({{\mathbb{A}_k}})} \phi(g)\theta(\Phi,g).$$ Proposition \[parind\], in turn, rests on the following result of Luna. A $G$-homogeneous variety $Y$ is said to be *induced* from a parabolic $P$ if it is of the form $Y'\times^P G$, where $Y'$ is a homogeneous spherical variety for the Levi quotient of $P$; equivalently, $Y=H\backslash G$, where $H\subset P$ contains the unipotent radical of $P$. [@Lu Proposition 3.4]\[parindcriterion\] A homogeneous spherical variety $Y$ for $G$ is induced from a parabolic $\bar P$ (assumed opposite to a standard parabolic $P$) if and only if the union of $\Delta(Y)$ with the support[^8] of the spherical roots of $Y$ is contained in the set of simple roots of the Levi subgroup of $P$. For every $G$-orbit $Y$ in a spherical variety $X$ there is a simple toroidal variety $\tilde X$ with a morphism $\tilde X\to X$ which is birational and whose image contains $Y$. Therefore, it suffices to assume that $X$ is a simple toroidal variety. Moreover, if $\bar X$ denotes the wonderful compactification of $X^+$ (i.e. the simple toroidal compactification with $\mathcal C(\bar X)=\mathcal V$) then every simple toroidal variety $X$ admits a morphism $X\to \bar X$ which, again, is birational and has the property that every non-open $G$-orbit on $X$ goes to a non-open $G$-orbit in $\bar X$. Indeed, any non-open $G$-orbit $Y\subset X$ corresponds to a non-trivial face of $\mathcal C(X)$, and its character group ${\mathcal{X}}(Y)$ is the orthogonal complement of that face in ${\mathcal{X}}(X)$, which is of lower rank than ${\mathcal{X}}(X)$, therefore $Y$ has to map to an orbit of lower rank. Moreover, $Y$ is a torus bundle over its image. This reduces the problem to the case where $X$ is a wonderful variety, which we will now assume. By Proposition \[parindcriterion\], it suffices to show that the union of $\Delta(X)$ and the support of the spherical roots of $Y$ is not the whole set $\Delta$ of simple roots. The spherical roots of $Y$ are a proper subset of the spherical roots of $X$, and $\Delta(Y)=\Delta(X)$. It therefore suffices to prove that for any proper subset $\Theta\subset\Delta_X$ there exists a simple root $\alpha\in \Delta\smallsetminus \Delta(X)$ such that $\alpha$ is not contained in the support of $\Theta$. Denote $\mathfrak a^*:={\mathcal{X}}(A)^*\otimes {\mathbb{Q}}$, $\mathfrak a^*_{P(X)}=(\Delta(X))^\perp\subset\mathfrak a^*$, and consider the canonical quotient map: $q:\mathfrak a \to \mathcal Q$. Denote by $\mathfrak f_\emptyset \subset \mathfrak a^*$ the anti-dominant Weyl chamber in $\mathfrak a$. Every set of spherical roots $s\subset\Delta_X$ corresponds to a face $\mathcal V_s\subset\mathcal V=\mathcal V_\emptyset\subset \mathcal Q$ (more precisely, $\mathcal V_s$ is the face spanning the orthogonal complement of $s$), and similarly every set $r\subset \Delta$ of simple roots of $G$ corresponds to a face $\mathfrak f_r\subset \mathfrak f_\emptyset$. The simple roots in the support of $\gamma\in \Delta_X$ are those corresponding to the largest face $\mathfrak f$ of $\mathfrak f_\emptyset$ which is contained in $q^{-1}(\mathcal V_{\{\gamma\}})$. Notice that the maximal vector subspace $\mathfrak f_\Delta$ of $\mathfrak f_\emptyset$ maps into the maximal vector subspace $\mathcal V_{\Delta_X}$ of $\mathcal V$. By assumption, $\mathfrak f_\emptyset$ surjects onto $\mathcal V$. Moreoever, since every element of $\mathfrak f_\emptyset$ can be written as a sum of an element in $\mathfrak f_{\Delta(X)}$ and a non-negative linear combination of $\check \Delta(X):=\{\check\alpha|\alpha\in\Delta(X)\}$, and since $\check\Delta(X)$ is in the kernel of $\mathfrak a\to\mathcal Q$, it follows that $\mathfrak f_{\Delta(X)}$ surjects onto $\mathcal V$. Now let $\Theta\subset\Delta_X$ be a proper subset. Let $\mathfrak f_s$ be a face of $\mathfrak f_{\Delta(X)}$ which surjects onto $\mathcal V_\Theta$. Since $\mathfrak f_s\neq \mathfrak f_\Delta$, there is an $\alpha\in \Delta\smallsetminus \Delta(X)$ which is not in the support of $\Theta$. Tensor product $L$-functions of $GL_2$ cusp forms {#sstensor} ------------------------------------------------- In section \[secSchwartz\] we proposed a general conjecture involving distributions which are obtained from the geometry of an affine spherical variety $X$, and in this section we saw how this conjecture is true, and gives rise to period- and Rankin-Selberg integrals, in the case that $X$ admits the structure of a “pre-flag bundle”. It was written above that such a structure should be considered essentially irrelevant and a matter of “chance”. We now wish to provide some evidence for this point of view by recalling the known constructions of $n$-fold tensor product $L$-functions for ${\operatorname{GL}}_2$, where $n\le 3$. The point is that while these constructions seem comletely different from the point of view of Rankin-Selberg integrals, from the point of view of spherical varieties they are completely analogous! \[ssrelatedtolfunction\]Before we consider the specific example, let us become a bit more precise about what it means that *a period integral is related to some $L$-value*. Let $\pi=\otimes' \pi_v$ be an (abstract) unitary representation of $G({{\mathbb{A}_k}})$, the tensor product of unitary irreducible representations $\pi_v$ of $G(k_v)$ with respect to distinguished unramified vectors $u_v^0$ (for almost every place $v$) of norm 1. Let $\mathcal P$ be a functional on $\pi$. In our applications the functional $\mathcal P$ will arise as the composition of a cuspidal automorphic embedding $\nu:\pi \to L^2_{{\operatorname{cusp}}}(G(k)\backslash G({{\mathbb{A}_k}}))$, assumed unitary, with a period integral or, more generally, the pairing (\[intconj1\]) with a fixed $X$-theta series. Let $\rho$ be a representation of the dual group, and let $L(\pi,\rho,s)$ denote the value of the corresponding $L$-function at the point $s$. We say that $|\mathcal P|^2$ is *related to $L(\pi,\rho,s)$* if there exist non-zero skew-symmetric forms: $\Lambda_v:\pi_v\otimes \bar \pi_v\to {\mathbb{C}}$ for every $v$ such that for any large enough set of places $S$, and for a vector $u=\otimes_{v\in S} u_v^0 \otimes_{v\notin S}u_v$ one has: $|\mathcal P(u)|^2= L^S(\pi,\rho,s) \cdot \prod_{v\in S} \Lambda_v (u_v, \overline u_v).$ (Of course, for this to happen we must have $\Lambda_v(u_v^0,\overline u_v^0)=L_v(\pi_v,\rho_v, s)$.) Moreover, it is required that each $\Lambda_v$ has a definition which has no reference to any other representation but $\pi_v$. The reader will notice that the last condition does not stand the test of mathematical rigor; however, not imposing it would make the rest of the statement void up to whether $\mathcal P$ is zero or not. In practice, the $\Lambda_v$’s will be given by reference to some non-arithmetic model for $\pi_v$. See [@II] for a precise conjecture in a specific case, and [@SV] for a more general but less precise conjecture.[^9] If $\mathcal P$ denotes the Whittaker period: $$\phi\mapsto \int_{U(k)\backslash U({{\mathbb{A}_k}})} \phi(u) \psi^{-1}(u) du$$ (where $\psi$ is a generic idele class character of the maximal unipotent subgroup) on cusp forms for $G={\operatorname{GL}}_n$, then $|\mathcal P|^2$ is related to the $L$-value: $$\frac{1}{L(\pi,\operatorname{Ad}, 1)}$$ cf. [@Ja; @SV]. Notice that the examples which we are about to discuss admit “Whittaker unfolding” and this factor will enter, although most references introduce a different normalization and ignore this factor. Now we are ready to discuss our example: Let $n$ be a positive integer, $G=({\operatorname{GL}}_2)^n \times {\mathbb{G}_m}$, and let $H$ be the subgroup: $$\left\{ \left.\left(\begin{array}{cc} a & x_1\\ & 1\end{array}\right) \times \left(\begin{array}{cc} a & x_2\\ & 1\end{array}\right) \times \cdots \times\left(\begin{array}{cc} a & x_n\\ & 1\end{array}\right) \times a\right| x_1+ x_2 + \dots+ x_n = 0\right\}.$$ We let $X=\overline{H\backslash G}^{{\operatorname{aff}}}$. As usual, we normalize the action of $G$ on functions on $X^+$ so that it is unitary with respect to the natural measure. Let us see that for $n=1,2,3$ the variety $X$ admits the structure of a pre-flag bundle, and hence the integral of Conjecture \[weakconjecture\] can be interpreted as a Rankin-Selberg integral, as discussed above: - $n=1$. Here $\overline{H\backslash G}^{{\operatorname{aff}}}=H\backslash G$ and we get the integral (\[hecke\]) of Hecke. If $\tau_s=\tau\otimes |\bullet|^s$, where $\tau$ is a cuspidal representation of ${\operatorname{GL}}_2$ (for simplicity: with trivial central character), the square of the absolute value of the corresponding linear functional on $\tau_s\otimes\widetilde{\tau_s}$ is related to the $L$-value: $$\frac{L(\tau, \frac{1}{2}+s) L(\tilde \tau, \frac{1}{2}-s)}{L(\tau,\operatorname{Ad},1)}.$$ - $n=2$. Here the projection of $H$ to $GL_2^2$ is conjugate to the mirabolic subgroup of $GL_2$ embedded diagonally. Therefore, the affine closure of $H\backslash G$ is equal to the bundle over ${\operatorname{GL}}_2^{{\operatorname{diag}}}\backslash ({\operatorname{GL}}_2)^2$ with fiber equal to the affine closure of $U_2\backslash {\operatorname{GL}}_2$, where $U_2$ denotes a maximal unipotent subgroup of ${\operatorname{GL}}_2$. Corresponding to this pre-flag bundle is a Rankin-Selberg integral “with the Eisenstein series on the smaller group” ${\operatorname{GL}}_2^{{\operatorname{diag}}}$, namely the classical integral of Rankin and Selberg. If $\tau=\tau_1\otimes\tau_2\otimes |\bullet|^s$ is a cuspidal automorphic representation of $G$ (for simplicity: with trivial central character), the square of the absolute value of the corresponding integral is related to the $L$-value: $$\frac{L(\tau_1\otimes\tau_2, \frac{1}{2}+s) L(\tilde \tau_1\otimes\tilde\tau_2, \frac{1}{2}-s)}{L(\tau,\operatorname{Ad},1)}.$$ - $n=3$. In this case there is a structure of a pre-flag variety not on $X$, but on $X^0$: the corresponding spherical variety for the subgroup $G^0=\{(g_1,g_2,g_3,a)\in G | \det(g_1)=\det(g_2)=\det(g_3)\}$. The structure of a pre-flag variety involves the group $\tilde G =\operatorname{GSp}_6$ and the subgroup $\tilde H=[\tilde P,\tilde P]$, where $\tilde P$ is the Siegel parabolic – this is a construction of Garrett [@GaTr]. The group $({\operatorname{GL}}_2^3)^0$ is embedded in ${{\operatorname{GSp}}}_6$ as $({{\operatorname{GSp}}}_2^3)^0$. Then, according to [@PSRtriple Corollary 1 to Lemma 1.1] the group $G^0$ has an open orbit in $[\tilde P,\tilde P]\backslash \tilde G$ with stabilizer equal to $H$. We claim: The affine closure $X^0$ of $H\backslash G^0$ is equal to the affine closure of $[\tilde P,\tilde P]\backslash \tilde G$. Denote by $Y$ the affine closure of $[\tilde P,\tilde P]\backslash \tilde G$. We have an open embedding: $X^0\hookrightarrow Y$. By [@PSRtriple Lemma 1.1], all non-open $G$-orbits have codimension at least two. Therefore, the embedding is an isomorphism. Hence, our integral for $X^0$ coincides with the Rankin-Selberg integral of Garrett. The only thing that remains to do is to compare the normalizations for the sections of Eisenstein series. From [@PSRtriple Theorem 3.1] one sees that the square of the absolute value of our integral is related to the $L$-value: $$\frac{L(\tau_1\otimes\tau_2\otimes \tau_3, \frac{1}{2}+s) L(\tilde \tau_1\otimes\tilde\tau_2\otimes\tilde\tau_3, \frac{1}{2}-s)}{L(\tau,\operatorname{Ad},1)}.$$ (Again, for simplicity, we assume trivial central characters. Notice that the zeta factors in [@PSRtriple Theorem 3.1] disappear because of the correct normalization of the Eisenstein series!) It is completely natural to expect the corresponding integral for $n=4$ or higher to be related to the $n$-fold tensor product $L$-function. It becomes obvious from the above example that the point of view of the spherical variety is the natural setting for such integrals, while at the same time the structure of a pre-flag bundle may not exist and, even if it exists, it has a completely different form in each case which conceals the uniformity of the construction. Smooth affine spherical varieties {#secsmoothaffine} ================================= Given that we do not know how to prove Conjecture \[weakconjecture\], except in the cases of wavefront pre-flag bundles, it is natural to ask the purely algebro-geometric question: Which spherical varieties admit the structure of a pre-flag bundle? An answer to this question would amount to a complete classification of Rankin-Selberg integrals, in the slightly restrictive sense that “Rankin-Selberg” has been used here. Such an answer has been given in the special case of *smooth* affine spherical varieties: These varieties automatically have the structure of a pre-flag bundle, and they have been classified by Knop and Van Steirteghem [@KnVS], hence can be used to produce Eulerian integrals of automorphic forms! There seems to be little point in computing every single example in the tables of [@KnVS], and my examination of most of the cases has not produced any striking new examples. However, we get some of the best-known integral constructions, as well as some new ones (which do not produce any interesting new $L$-functions). Smooth affine spherical triples ------------------------------- By Theorem \[Lunacorollary\] of Luna, every smooth affine spherical variety of $G$ (over an algebraically closed field in characteristic zero) is of the form $V\times_H G$, where $H$ is a reductive subgroup (so that $H\backslash G$ is affine) and $V$ is an $H$-module. In other words, all smooth affine spherical varieties are pre-flag bundles: The corresponding integrals include all period integrals over reductive subgroups, as well as Rankin-Selberg integrals involving *mirabolic* Eisenstein series (i.e. those induced from the mirabolic subgroup of ${\operatorname{GL}}_n$). In [@KnVS], Knop and Van Steirteghem classify all smooth affine spherical *triples* $({\mathfrak{g}}, {\mathfrak{h}}, V)$, which amounts to a classification of smooth affine spherical varieties up to coverings, central tori and ${\mathbb{G}_m}$-fibrations. We recall their definitions: 1. Let ${\mathfrak{h}}\subset{\mathfrak{g}}$ be semisimple Lie algebras and let $V$ be a representation of ${\mathfrak{h}}$. For ${\mathfrak{s}}$, a Cartan subalgebra of the centralizer $c_{\mathfrak{g}}({\mathfrak{h}})$ of ${\mathfrak{h}}$, put $\bar{\mathfrak{h}}:= {\mathfrak{h}}\oplus {\mathfrak{s}}$, a maximal central extension of ${\mathfrak{h}}$ in ${\mathfrak{g}}$. Let ${\mathfrak{z}}$ be a Cartan subalgebra of $\mathfrak{gl}(V)^{\mathfrak{h}}$ (the centralizer of ${\mathfrak{h}}$ in $\mathfrak{gl}(V)$). We call $({\mathfrak{g}}, {\mathfrak{h}}, V)$ a *spherical triple* if there exists a Borel subalgebra ${\mathfrak{b}}$ of ${\mathfrak{g}}$ and a vector $v\in V$ such that 1. ${\mathfrak{b}}+ \bar{\mathfrak{h}}= {\mathfrak{g}}$ and 2. $[({\mathfrak{b}}\cap \bar{\mathfrak{h}}) + {\mathfrak{z}}]v = V$ where ${\mathfrak{s}}$ acts via any homomorphism ${\mathfrak{s}}\to{\mathfrak{z}}$ on $V$. 2. Two triples $({\mathfrak{g}}_i , {\mathfrak{h}}_i , V_i)$, $i = 1, 2$, are *isomorphic* if there exist isomorphisms of Lie algebras resp. vector spaces $\alpha: {\mathfrak{g}}_1\to{\mathfrak{g}}_2$ and $\beta: V_1\to V_2$ such that 1. $\alpha({\mathfrak{h}}_1 ) = {\mathfrak{h}}_2$ 2. $\beta(\xi v) = \alpha(\xi)\beta(v)$ for all $\xi\in {\mathfrak{h}}_1$ and $v\in V_1$. 3. Triples of the form $({\mathfrak{g}}_1\oplus{\mathfrak{g}}_2, {\mathfrak{h}}_1\oplus{\mathfrak{h}}_2, V_1\oplus V_2)$ with $(g_i , h_i , V_i ) \ne (0, 0, 0)$ are called *decomposable*. 4. Triples of the form $({\mathfrak{k}}, {\mathfrak{k}}, 0)$ and $(0, 0, V )$ are said to be *trivial*. A pair $({\mathfrak{g}}, {\mathfrak{h}})$ of semisimple Lie algebras is called spherical if $({\mathfrak{g}}, {\mathfrak{h}}, 0)$ is a spherical triple. 5. A spherical triple (or pair) is *primitive* if it is non-trivial and indecomposable. Clearly, every smooth affine spherical variety gives rise to a spherical triple. Conversely, each spherical triple is obtained from a (not necessarily unique) smooth affine spherical variety, as follows by an a posteriori inspection of all spherical triples. (The non-obvious step here is that the ${\mathfrak{h}}$-module $V$ integrates to an $H$-module, where $H$ is the corresponding subgroup.) The classification of all primitive spherical triples is given in [@KnVS], Tables 1, 2, 4 and 5, modulo the inference rules described in Table 3. The diagrams are read in the following way: The nodes in the first row correspond to the simple direct summands ${\mathfrak{g}}_i$ of ${\mathfrak{g}}$, the ones in the second row to the simple direct summands ${\mathfrak{h}}_i$ of ${\mathfrak{h}}$ and the ones in the third row to the simple direct summands $V_i$ of $V$. If $({\mathfrak{g}},{\mathfrak{h}})$ contains a direct summand of the form $({\mathfrak{h}}_1,{\mathfrak{h}}_1)$ then the ${\mathfrak{h}}_1$ summand is omitted from the first row There is an edge between ${\mathfrak{g}}_i$ and ${\mathfrak{h}}_j$ if ${\mathfrak{h}}_j\hookrightarrow {\mathfrak{g}}\twoheadrightarrow{\mathfrak{g}}_i$ is non-zero and an edge between $h_j$ and $V_k$ if $V_k$ is a non-trivial $h_j$-module. The edges are labeled to describe the inclusion of ${\mathfrak{h}}$ in ${\mathfrak{g}}$, resp. the action of ${\mathfrak{h}}$ on $V$; the labels are omitted when those are the “natural” ones. We number the cases appearing in the list of Knop and Van Steirteghem as follows: First, according to the table on which they appear (Tables 1, 2, 4, 5 in [@KnVS]); and for each table, numbered from left to right, top to bottom. Eulerian integrals arising from smooth affine varieties ------------------------------------------------------- In what follows we will discuss a sample of the global integrals obtained from varieties in the list of Knop and Van Steirteghem. At this point it is more convenient not to normalize the action of $G$ unitarily. We allow ourselves to choose the spherical variety corresponding to a given spherical triple as is most convenient, and in fact we sometimes replace semisimple groups by reductive ones. Of course, the classification in [@KnVS] is over an algebraically closed field, which leaves a lot of freedom for choosing the precise form of the spherical variety over $k$, even when $G$ is split. In the discussion which follows we will always take both the group and generic stabilizer to be split. Many of the varieties of Knop and Van Steirteghem have zero cuspidal contribution (i.e. the integral (\[intconj1\]) is zero for every cusp form) or are not multiplicity-free. Still, this list contains some of the best-known examples of integral representations of $L$-functions. It contains also some new ones. In subsection §\[ssrelatedtolfunction\] we explained what it means for a period integral $\mathcal P$ to be “related to” an $L$-value, namely by considering the value of $\mathcal P|_\pi\cdot \mathcal P|_{\bar \pi}$, assuming that $\pi$ is an abstract unitary representation of an adelic group, embedded unitarily into the space of cuspidal automorphic forms for that group. For the examples that we are about to see, we will adopt a language that describes the value of $\mathcal P|_\pi$ itself, divided by the value of a period integral that does not depend on a continuous parameter, such as the Whittaker period. For example, for the Hecke integral (\[hecke\]) we would say that it is related to $L(\pi,s+\frac{1}{2})$ with respect to Whittaker normalization, while for the Godement-Jacquet integral (\[GodementJacquet\]) we would say that it is related to $L(\pi, s-\frac{1}{2}(n-1))$ with respect to the “inner product” period on $\pi\otimes\tilde\pi$. ### <span style="font-variant:small-caps;">Table 1</span> In this table the group $H$ is equal to $G$, i.e. the data consists of a group and a spherical representation of it. This table contains the following interesting integrals (numbered according to their occurence in the tables of Knop and Van Steirteghem): 1. **The integrals of Godement and Jacquet**. : Here the group is ${\operatorname{GL}}_n\times {\operatorname{GL}}_m$ with the tensor product representation (i.e. on ${\operatorname{Mat}}_{n\times m}$). It is easy to see that if $m\ne n$ then the stabilizer is parabolically induced, hence the only interesting case (as far as cusp forms are concerned) is $m=n$. In this case, our integral (\[intconj1\]) *is that of Godement and Jacquet*: $$\int_{Z^{{\operatorname{diag}}}({\mathbb{A}_k}){\operatorname{GL}}_n^{{\operatorname{diag}}}(k)\backslash {\operatorname{GL}}_n({\mathbb{A}_k})\times{\operatorname{GL}}_n({\mathbb{A}_k})} \phi_1(g_1)\phi_2(g_2) \Phi(g_1^{-1}g_2) \cdot$$ $$\cdot |\det(g_1^{-1}g_2)|^{s} d(g_1,g_2).$$ 15. **Two new integrals.** : (Here there is a choice between the first and the last fundamental representation of ${\operatorname{GL}}_n$. It can easily be seen that they amount to the same integral, so we will consider only $\omega_1$.) The group is ${\operatorname{GL}}_m\times {\operatorname{GL}}_n$ and the representation is the direct sum ${\operatorname{Mat}}_{m\times n}$ with the standard representation for ${\operatorname{GL}}_n$. If $m\ne n, n-1$ then we can easily see that the stabilizer is parabolically induced. Hence there are two interesting cases: 1. $m=n$. We let $\phi_1 \in \pi_1,\phi_2\in \pi_2$ be two cusp forms on ${\operatorname{GL}}_n$. Then the integral is: $$\int_{P_n^{{\operatorname{diag}}}(k)\backslash {\operatorname{GL}}_n({\mathbb{A}_k})\times{\operatorname{GL}}_n({\mathbb{A}_k})} \phi_1(g_1)\phi_2(g_2) \Phi(g_1^{-1}g_2) \Phi'([0, \dots, 0, 1]\cdot g_1)\cdot$$ $$\cdot |\det(g_1^{-1}g_2)|^{s_1} |\det(g_1)|^{s_2} dg_1 dg_2.$$ Here $\Phi$ is a Schwartz function on ${\operatorname{Mat}}_n({\mathbb{A}_k})$ and $\Phi'$ is a Schwartz function on ${\mathbb{A}_k}^n$. The above integral is Eulerian and with respect to Whittaker normalization is related to the $L$-value: $$L(\pi_1\otimes\pi_2, s_2)\cdot L(\pi_2, s_1-\frac{1}{2}(n-1)).$$ It follows from the standard “unfolding” technique that the above integral, in the domain of convergence, is equal to: $$\int_{(U_n({{\mathbb{A}_k}})\backslash {\operatorname{GL}}_n({{\mathbb{A}_k}}))^2} W_1 (g_1) W'_2(g_2) \Phi(g_1^{-1}g_2) \Phi'([0, \dots, 0, 1]\cdot g_1)\cdot$$ $$\cdot |\det(g_1^{-1}g_2)|^{s_1} |\det(g_1)|^{s_2} dg_1 dg_2$$ where $W_1(g)=\int_{U_n(k)\backslash U_n({{\mathbb{A}_k}})} \phi_1(ug) \psi(u) du$ and $W'_2$ the same with $\phi_1$ replaced by $\phi_2$ and $\psi$ replaced by $\psi^{-1}$. The last integral is (for “factorizable data”) a product of local factors: $$\int_{(U_n(k_v)\backslash {\operatorname{GL}}_n(k_v))^2} W_{1,v} (g_1) W'_{2,v}(g_2) \Phi_v(g_1^{-1}g_2) \Phi_v'([0, \dots, 0, 1]\cdot g_1)\cdot$$ $$\cdot |\det(g_1^{-1}g_2)|^{s_1} |\det(g_1)|^{s_2} dg_1 dg_2.$$ Assume that $\Phi_v=\Phi_v^0$, the basic function of $\mathcal S({\operatorname{Mat}}_n(k_v))$. Considering the action of the spherical Hecke algebra of $G_2$ (=the second copy of ${\operatorname{GL}}_n$) on $\mathcal S({\operatorname{Mat}}_n(k_v))$, the work of Godement and Jacquet [@GJ Lemma 6.10] proves: $$\Phi_v^0(x) |\det(x)|^{s_1} = \widecheck{{\operatorname{Sat}}}_{G_2} \left(\frac{1}{\wedge^\top \left(1- q_v^{-s_1+\frac{1}{2}(n-1)}\cdot\operatorname{std}\right)}\right) \star 1_{{\operatorname{GL}}_n(\mathfrak o)}$$ Therefore for unramified data the last integral is equal to: $$L(\pi_2, s_1-\frac{1}{2}(n-1)) \cdot \int_{(U_n(k_v)\backslash {\operatorname{GL}}_n(k_v))^2} W_{1,v} (g_1) W'_{2,v}(g_2) \cdot$$ $$\cdot1_{{\operatorname{GL}}_n(\mathfrak o_v)}(g_1^{-1}g_2) \Phi_v'([0, \dots, 0, 1]\cdot g_1) |\det(g_1^{-1}g_2)|^{s_1} |\det(g_1)|^{s_2} dg_1 dg_2 =$$ $$= L(\pi_2, s_1-\frac{1}{2}(n-1)) \cdot$$ $$\cdot\int_{(U_n(k_v)\backslash {\operatorname{GL}}_n(k_v))} W_{1,v} (g) W'_{2,v}(g) \Phi_v'([0, \dots, 0, 1]\cdot g) |\det(g)|^{s_2} dg.$$ The latter is the classical Rankin-Selberg integral, which with respect to Whittaker normalization is related to $L(\pi_1\otimes\pi_2, s_2)$ (see, for instance, [@Co]). 2. $m=n-1$. Notice that if $V$ denotes the standard representation of ${\operatorname{GL}}_n$ then the space ${\operatorname{Mat}}_{(n-1)\times n}\oplus V$ can be identified under the $G_1\times G_2:={\operatorname{GL}}_{n-1}\times {\operatorname{GL}}_n$-action with the space $X={\operatorname{Mat}}_n$, where $g\in G_1$ acts as $\left(\begin{array}{cc} g^{-1} \\ & 1\end{array}\right)$ on the left Let $\phi_1\in \pi_1$ be a cusp form on ${\operatorname{GL}}_{n-1}$ and $\phi_2\in \pi_2$ a cusp form in ${\operatorname{GL}}_n$. Then the integral is: $$\int_{{\operatorname{GL}}_n^{{\operatorname{diag}}}(k)\backslash {\operatorname{GL}}_{n+1}({\mathbb{A}_k})\times{\operatorname{GL}}_n({\mathbb{A}_k})} \phi_1(g_1)\phi_2(g_2) \cdot$$ $$\cdot \Phi\left(\left(\begin{array}{cc} g_1^{-1} \\ & 1\end{array}\right) g_2\right) \left|\frac{\det(g_2)}{\det(g_1)}\right|^{s_1} |\det(g_1)|^{s_2} dg_1 dg_2$$ where $\Phi\in\mathcal S({\operatorname{Mat}}_n({{\mathbb{A}_k}}))$. As before, one can prove: The above integral is Eulerian and with respect to Whittaker normalization related to the $L$-value: $$L(\pi_1\otimes\pi_2, s_2+\frac{1}{2})\cdot L(\pi_2, s_1-\frac{1}{2}n).$$ ### <span style="font-variant:small-caps;">Table 2</span> In this table $H$ is smaller than $G$ and the representation $V$ of $H$ is non-trivial. This table contains the following interesting integrals: 1. **The Bump-Friedberg integral.** : The group is ${\operatorname{GL}}_{m+n}$ where $m=n\text{ or }n+1$, the subgroup $H$ is ${\operatorname{GL}}_m\times {\operatorname{GL}}_n$ and the representation is the standard representation of the second factor. This is the integral examined in [@BF]: $$\int_{{\operatorname{GL}}_m(k)\times {\operatorname{GL}}_n(k)\backslash {\operatorname{GL}}_m({\mathbb{A}_k})\times{\operatorname{GL}}_n({\mathbb{A}_k})} \phi\left(\begin{array}{cc} g_1 \\ & g_2\end{array}\right) \left|\frac{\det (g_1)}{\det(g_2)}\right|^{s_1} \cdot$$ $$\cdot \Phi([0, \cdots, 0, 1] \cdot g_2) |\det g_2|^{s_2} dg_1 dg_2$$ It is related with respect to Whittaker normalization to the $L$-value: $$L(\pi,s_1+\frac{1}{2})L(\pi,\wedge^2, s_2).$$ 3. **A new integral.** : The group is ${\operatorname{GL}}_{m+1}\times {\operatorname{GL}}_n$, and $G'={\operatorname{GL}}_m\times {\operatorname{GL}}_n$ with the tensor product of the standard representations (i.e. on ${\operatorname{Mat}}_{m\times n}$). The only interesting case is $m=n$. If $n>m$ then the stabilizer is parabolically induced, and when $m>n$ it unfolds to a parabolically induced model. If $m=n$ we get: $$\int_{{\operatorname{GL}}^{{\operatorname{diag}}}(k)\backslash {\operatorname{GL}}_n({\mathbb{A}_k})\times{\operatorname{GL}}_n({\mathbb{A}_k})} \phi_1\left(\begin{array}{cc} g_1 \\ & 1\end{array}\right)\phi_2(g_2) \Phi(g_1^{-1} g_2) \cdot$$ $$\cdot \left|\frac{\det(g_2)}{\det(g_1)}\right|^{s_1} |\det(g_1)|^{s_2} d(g_1,g_2).$$ As before, one can prove: The above integral is Eulerian and with respect to Whittaker normalization related to the $L$-value: $$L(\pi_1\otimes\pi_2, s_2+\frac{1}{2})\cdot L(\pi_2, s_1-\frac{1}{2}(n-1)).$$ 5. **The classical Rankin-Selberg integral.** : The group is ${\operatorname{GL}}_n\times {\operatorname{GL}}_n$ and the subgroup $G'$ is ${\operatorname{GL}}_n^{{\operatorname{diag}}}$ with the standard representation. This is the classical Rankin-Selberg integral: $$\int_{{\operatorname{GL}}_n(k)\backslash{\operatorname{GL}}_n({\mathbb{A}_k})} \phi_1(g)\phi_2(g) \Phi([0, \cdots , 0, 1]\cdot g) |\det g|^s dg.$$ It is related with respect to Whittaker normalization to the $L$-value (cf. [@Co]): $$L(\pi_1\otimes\pi_2, s).$$ ### <span style="font-variant:small-caps;">Tables 4 and 5</span> Here the representation $V$ is trivial, hence we get period integrals over reductive algebraic subgroups (§\[ssperiods\]). All known cases of multiplicity-free period integrals are contained in these tables. A remark on a relative trace formula {#secRTF} ==================================== At this point we drop our assumptions on the group $G$, in order to discuss non-split examples. We will assume the existence of Schwartz spaces with similar properties in this setting, in order to give a conceptual explanation to the phenomenon of “weight factors” in a relative trace formula. The relative trace formula is a method which was devised by Jacquet and his coauthors to study period integrals of automorphic forms. In its most simplistic form, it can be described as follows: Let $H_1$ and $H_2$ be two reductive spherical subgroups of $G$ (a reductive group defined over a global field $k$) and let $f\in C_c^\infty(G({{\mathbb{A}_k}}))$. Then one builds the usual kernel function: $K_f(x,y)=\sum_{\gamma\in G(k)} f(x^{-1}\gamma y)$ for the action of $f$ on the space of automorphic functions and (ignoring analytic difficulties) defines the functional: $$\label{simpleRTF} {{\operatorname{RTF}}}_{H_1,H_2}^G(f) = \int_{H_1(k)\backslash H_1({{\mathbb{A}_k}})} \int_{H_2(k)\backslash H_2({{\mathbb{A}_k}})} K_f(h_1, h_2) dh_1 dh_2.$$ The functional can be decomposed in two ways, one geometric and one spectral, and the spectral expansion involves period integrals of automorphic forms. By comparing two such RTFs (i.e. made with different choices of $H_1, H_2$, maybe even different groups $G$) one can deduce properties of those period integrals, such as that their non-vanishing characterizes certain functorial lifts. The above presentation is too simplistic for several reasons: First, the correct functional has something to do with the varieties $H_i\backslash G$, rather than the spaces $H_i(k)\backslash G(k)$ – therefore, if $G(k)$ does not surject onto $H_i(k)$ one should take the sum of the above expressions over stabilizers $H_{i,\epsilon}$ of a set of representatives of $G(k)$-orbits. (This will become clearer in a reformulation which we will present below.) Even then, one may need to take a further summation of relative trace formulae over inner forms of $G$ – this phenomenon also has an explanation, but we will ignore it here. Moreover, one can consider an idele class character $\eta$ of $H_i$ and integrate against this character; we will adjust our notation accordingly, for instance: ${{\operatorname{RTF}}}^G_{H_1,(H_2,\eta)}$. There are often analytic difficulties in making sense of the above integrals. And one does not have to restrict to reductive subgroups, but can consider parabolically induced subgroups together with a character on their unipotent radical (such as in the Whittaker period). However, we will ignore most of these issues and focus on another one, first noticed in [@JLR]: It seems that in certain cases, in order for the relative trace formula ${{\operatorname{RTF}}}^G_{H_1,H_2}$ to be comparable to some other relative trace formula, the functional (\[simpleRTF\]) is not the correct one and one has to add a “weight factor” in the definition, such as: $$\label{thetaRTF} {{\operatorname{RTF}}}_{H_1,H_2}^G(f) = \int_{H_1(k)\backslash H_1({{\mathbb{A}_k}})} \int_{H_2(k)\backslash H_2({{\mathbb{A}_k}})} K_f(h_1, h_2) \theta(h_1) dh_1 dh_2$$ where $\theta$ is a suitable automorphic form on $H_1$. Our goal here is to explain how, under the point of view developed in the present paper, the above expression is not a relative trace formula for $H_1, H_2$ but represents a relative trace formula for some *other* subgroups. We will discuss this in the context of [@JLR], though our starting point will not be (\[thetaRTF\]) but another formula of [@JLR] from which the identities for (\[thetaRTF\]) are derived, and which is closer to our point of view. More precisely, let $E/F$ be a quadratic extension of number fields with corresponding idele class character $\eta$, $G= {{\operatorname{Res}}}_{E/F}{\operatorname{PGL}}_2$, $G'={\operatorname{PGL}}_2\times {\operatorname{PGL}}_2$ (over $F$), $H\subset G$ the projectivization of the quasi-split unitary group (which is in fact split, i.e. isomorphic to $PGL_2$ over $F$), $H'=$ the diagonal copy of ${\operatorname{PGL}}_2$ in $G'$. (Compared to [@JLR], we restrict to ${\operatorname{PGL}}_2$ for simplicity.) We consider $\eta$ as a character of $H$ in the natural way. Naively, one would like to compare the functional: ${{\operatorname{RTF}}}_{H,(H,\eta)}^G$ to the functional ${{\operatorname{RTF}}}_{H',H'}^{G'}$ (usual trace formula for $G'$). However, it turns out that the correct comparison is between the functionals: $$\label{1stmod} f\mapsto \int_{(H(k)\backslash H({{\mathbb{A}_k}}))^2} K_f(h_1, h_2) E(h_1,s) \eta(h_1) dh_1 dh_2$$ on $G$ and $$\label{2ndmod} f'\mapsto \int_{(H'(k)\backslash H'({{\mathbb{A}_k}}))^2} K_{f'}(h'_1, h'_2) E'(h'_1,s) dh'_1 dh'_2$$ on $G'$, where $E, E'$ are suitable Eisenstein series on $H, H'$. (More precisely, in the first case one takes the sum over the unitary groups of all $G(k)$-conjugacy classes of non-degenerate hermitian forms for $E/F$, as we mentioned above, *but only in the second variable*.) Notice that we have already made a modification to the formulation of [@JLR], namely in the second case they let $G'={\operatorname{PGL}}_2$ and consider the integral: $$\int_{{\operatorname{PGL}}_2(k)\backslash {\operatorname{PGL}}_2({{\mathbb{A}_k}})} K_{f'}(x,x) E'(x,s) dx,$$ but this is easily seen to be equivalent to our present formulation. The functionals (\[1stmod\]), (\[2ndmod\]) can naturally be understood as pairings: $${{\operatorname{RTF}}}_{X_1,X_2}^{{\mathbb{G}_m}\times G, \omega}: \mathcal S(X_1({{\mathbb{A}_k}}))\otimes \mathcal S(X_2({{\mathbb{A}_k}})) \to {\mathbb{C}}$$ respectively: $${{\operatorname{RTF}}}_{X_1',X_2'}^{{\mathbb{G}_m}\times G',\omega'}: \mathcal S(X_1'({{\mathbb{A}_k}}))\otimes \mathcal S(X_2'({{\mathbb{A}_k}})) \to {\mathbb{C}}$$ where: $X_2=H\backslash G$, $X_2'=H'\backslash G'$ and $X_1, X_1'$ are the affine closures of the varieties: $$U_F\backslash G$$ respectively: $$U_F' \backslash G'$$ where $U_F, U_F'$ are maximal unipotent subgroups of $H$ resp. $H'$. The varieties $X_1$, $X_1'$ are considered here as spherical varieties under ${\mathbb{G}_m}\times G$ (resp. ${\mathbb{G}_m}\times G'$), where ${\mathbb{G}_m}=B_2/U_2$, and we extend the ${\mathbb{G}_m}$-action to the varieties $X_2,X_2'$ in the trivial way. The exponent $\omega$ in ${{\operatorname{RTF}}}_{X_1,X_2}^{{\mathbb{G}_m}\times G, \omega}$ will be explained below. Before we explain the claim, let us go back to the simpler formula (\[simpleRTF\]) and explain how it can be considered as a pairing between $\mathcal S(X_1({{\mathbb{A}_k}}))$ and $\mathcal S(X_2({{\mathbb{A}_k}}))$ (where $X_i=H_i\backslash G_i$). Here we will identify Hecke algebras with spaces of functions, by choosing Haar measures. Assume that $f=\check f_1 \star f_2$ with $f_i\in C_c^\infty(G({{\mathbb{A}_k}}))$. Then we set: $\Phi_i(g)=\int_{H_i({{\mathbb{A}_k}})} f_i(hg)dh$. By the definition of $\mathcal S(X_i({{\mathbb{A}_k}}))$ when $H_i$ is reductive, it follows that $\Phi_i\in\mathcal S(X_i({{\mathbb{A}_k}}))$. (It is at this point that one should add over representatives for $G_i(k)$-orbits on $X_i(k)$, since in general the map $C_c^\infty(G({{\mathbb{A}_k}}))\to \mathcal S(X_i({{\mathbb{A}_k}}))$ is not surjective.) The functional ${{\operatorname{RTF}}}^G_{H_1,H_2}(f_1 \star f_2)$ clearly does not depend on $f_1,f_2$ but only on $\Phi_1,\Phi_2$. Hence, it defines a $G^{{\operatorname{diag}}}$-invariant functional: $$\mathcal S(X_1({{\mathbb{A}_k}}))\otimes \mathcal S(X_2({{\mathbb{A}_k}})) \to {\mathbb{C}}.$$ Now let us return to the setting of the Claim, and of equations (\[1stmod\]), (\[2ndmod\]). The product $E(h_1,s)\eta(h_1)$ in (\[1stmod\]) will be considered as an Eisenstein series on $H(k)\backslash H({{\mathbb{A}_k}})$. We have seen that suitable sections of Eisenstein series can be obtained from integrating $X$-theta series $\theta_{U_2}^{{\mathbb{G}_m}\times H}(\Phi,g)$ where $\Phi\in \mathcal S(U_2\backslash H({{\mathbb{A}_k}}))$ against a character $\omega$ of ${\mathbb{G}_m}$. Now consider $\Phi\in\mathcal S(U_2\backslash H({{\mathbb{A}_k}}))$ as a generalized function on $U_2\backslash G({{\mathbb{A}_k}})$. Assume again that $f=\check f_1\star f_2\in C_c^\infty(G({{\mathbb{A}_k}}))$. Then $\Phi_1:=f_1\star \Phi \in \mathcal S(U_2\backslash H({{\mathbb{A}_k}}))$ and $\Phi_2(g):=\int_{H_2({{\mathbb{A}_k}})} f(hg)dg\in \mathcal S(H\backslash G({{\mathbb{A}_k}}))$. Again, of course, we must take many $f$’s and sum over representatives for orbits of $G(k)$ on $X_2(k)$ – incidentally, our point of view explains why there is no need to sum over representatives for orbits in the first variable: because $G(k)$ surjects on $X_1(k)$! Similarly, one can explain (\[2ndmod\]) as a pairing between $\mathcal S(X_1'({{\mathbb{A}_k}}))\otimes \mathcal S(X_2'({{\mathbb{A}_k}}))$, and this completes the explanation of our Claim. We have introduced the exponents $\omega, \omega'$ in the notation, because we have already integrated against the corresponding character of ${\mathbb{G}_m}$ in order to form Eisenstein series. 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Y. Sakellaridis, *On the unramified spectrum of spherical varieties over $p$-adic fields.* Compositio Math. 144 (2008), no. 4, 978–1016. Y. Sakellaridis, *Spherical functions on spherical varieties.* Preprint, 2009, arXiv:0905.4244. Y. Sakellaridis and A. Venkatesh, *Periods and harmonic analysis on spherical varieties.* In preparation. A. Selberg, *Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist.* Arch. Math. Naturvid. 43, (1940), 47–50. J. T. Tate, *Fourier analysis in number fields, and Hecke’s zeta-functions.* 1967 Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) pp. 305–347 Thompson, Washington, D.C. *E-mail address:* `yiannis@math.toronto.edu` [^1]: At ramified places and for most $\rho$, the definition still depends on the local functoriality conjectures. [^2]: Notice that this is different from that of [@GN], but compatible with the notation used in [@SaSpc; @SaSph; @SV]. [^3]: A *cone* in a ${\mathbb{Q}}$-vector space is a subset which is closed under addition and under multiplication by ${\mathbb{Q}}_{\ge 0}$, its *relative interior* is its interior in the vector subspace that it spans, and a *face* of it is the zero set, in the cone, of a linear functional which is non-negative on the cone – hence, the whole cone is a face as well. [^4]: The work of Gaitsgory-Nadler [@GN] and Sakellaridis-Venkatesh [@SV] suggests that for representation-theoretic reasons one should slightly modify this definition of spherical roots. However, the lines on which the modified roots lie are still the same, and for the purposes of the present article this is enough. [^5]: Since $\bar X$ is not necessarily simple, it is not described by a cone but by a fan. However, we slightly abuse the common notation here and write $\mathcal C(\bar X)$ for the set of invariant valuations whose center is in $\bar X$ – i.e. for the support of the fan associated to $\bar X$. [^6]: It would be more correct to consider only what will later be denoted by $\mathfrak{\check u}_{\tilde P}^f$ for those factors of $\tilde X^+_y$ that are of the form $[P_i,P_i]\backslash G_i$, but that does not make any difference for the statement of Theorem \[preflagthm\] below, since we are only using $\mathfrak{\check u}_{\tilde P}$ to require the meromorphic continuation of an $L$-function, and the difference if we took $\mathfrak{\check u}_{\tilde P}^f$ instead would just be some abelian $L$-function. [^7]: Rankin-Selberg constructions with products of Eisenstein series have often been encountered in the literature, e.g. [@BFG; @GH]. [^8]: The support of a subset in the span of $\Delta$ is the smallest set of elements of $\Delta$ in the span of which it lies. [^9]: For the sake of completeness, we should mention that when $\mathcal P$ comes from a period integral one should in general modify the above conjecture by some “mild” arithmetic factors, such as the sizes of centralizers of Langlands parameters – see [@II]. However, in the example we are about to discuss there is no such issue since the group is ${\operatorname{GL}}_2$.
--- abstract: 'Assuming the existence of $\mathfrak{c}$ incomparable selective ultrafilters, we characterize the algebraic structure of non-torsion Abelian groups of size continuum that admit a countably compact Hausdorff group topology. It shows that the results presented by Dikranjan and Tkachenko in can be obtained under weak set-theoretic assumptions than Martin’s Axiom. We also prove that the existence of $2^{\mathfrak{c}}$ selective ultrafilters implies that if a non-torsion Abelian group admits a countably compact Hausdorff group topology, then it admits $2^{\mathfrak{c}}$ non-homeomorphic countably compact Hausdorff group topologies. It improves a result presented by Tomita in [@tomita6].' author: - 'Ana Carolina Boero, Artur Hideyuki Tomita' bibliography: - 'gruposenumeravelmentecompactos.bib' title: 'Algebraic structure of countably compact non-torsion Abelian groups of size continuum from selective ultrafilters' --- Introduction ============ Some history ------------ Halmos [@halmos] showed, in 1940, that the real line admits a compact Hausdorff group topology and asked which Abelian groups can be endowed with such a topology. Fuchs showed that a non-trivial free Abelian group cannot be endowed with a compact Hausdorff group topology and, in the end of 50s, Hulanicki [@hulanicki] and Harrison [@harrison] completely solved Halmos’ problem. In 1990, Tkachenko [@tkachenko] showed that the free Abelian group of size $\mathfrak{c}$ can be endowed with a countably compact Hausdorff group topology under CH. Tomita [@tomita2] obtained such a topology from MA($\sigma$-centered) and Koszmider, Tomita and Watson weakened the necessity of some form of Martin’s axiom to $\mathrm{MA}_{\mathrm{countable}}$. Under Martin’s Axiom, Dikranjan and Tkachenko characterized the algebraic structure of Abelian groups of size continuum that admit a countably compact Hausdorff group topology. They proved that a torsion Abelian group $G$ admits a countably compact Hausdorff group topology if and only if $G$ has finite exponent $n$ and, for every divisor $d$ of $n$, $dG$ is either finite or satisfies $|dG| = \mathfrak{c}$. They also proved that a non-torsion Abelian group $G$ admits a countably compact Hausdorff group topology if and only if the free rank of $G$ is equal to $\mathfrak{c}$ and for every $d, n \in \mathbb{N}$ with $d \mid n$, the group $dG[n]$ is either finite or has the cardinality $\mathfrak{c}$. Castro-Pereira and Tomita characterized all the torsion Abelian groups (of any cardinality) that admit a countably compact Hausdorff group topology assuming a mild cardinal arithmetic hypothesis and the existence of a selective ultrafilter. In particular, this characterization can be made under weaker set-theoretic assumptions than MA. In this work, we extend this study to the non-torsion case. Assuming the existence of $\mathfrak{c}$ incomparable selective ultrafilters, we characterize the algebraic structure of non-torsion Abelian groups of size continuum that admit a countably compact Hausdorff group topology. Under Martin’s Axiom restricted to $\sigma$-centered partial orders, Tomita [@tomita6] showed that the free Abelian group of size continuum admits $\mathfrak{c}^{+}$ non-homeomorphic countably compact Hausdorff group topologies. In this work, we show that the existence of $2^{\mathfrak{c}}$ selective ultrafilters implies that if a non-torsion Abelian group admits a countably compact Hausdorff group topology, then it admits $2^{\mathfrak{c}}$ non-homeomorphic countably compact Hausdorff group topologies. Basic definitions, notations and results ---------------------------------------- We recall that a topological space $X$ is countably compact if every infinite subset of $X$ has an accumulation point. The set of all free ultrafilters over $\omega$ will be denoted by $\omega^{*}$. Bernstein [@bernstein] defined the following concept, which is an important tool for the study of countable compactness. \[def\_p\_limite\] Let $p \in \omega^{*}$ and $\{x_{n} : n \in \omega\}$ be a sequence in a topological space $X$. We say that $x \in X$ is a *$p$-limit point* of $\{x_{n} : n \in \omega\}$ if, for every neighborhood $U$ of $x$, the set $\{n \in \omega : x_{n} \in U\}$ is an element of $p$. In this case, we write $x = p-\lim \{x_{n} : n \in \omega\}$. It is not difficult to prove that a topological space $X$ is countably compact if, and only if, each sequence in $X$ has a $p$-limit point, for some $p \in \omega^{*}$. \[prop\_p-limit\_product\] If $p \in \omega^{*}$ and $\{X_i : i \in I\}$ is a family of topological spaces, then $(y_{i})_{i \in I} \in \prod_{i \in I} X_i$ is a $p$-limit point of $\{(x_{i}^{n})_{i \in I} : n \in \omega\} \subset \prod_{i \in I} X_i$ if, and only if, $y_i = p-\lim \{x_{i}^{n} : n \in \omega\}$ for every $i \in I$. If $A$ is a set, then $$[A]^{\omega} = \{X \subset A : |X| = \omega\}$$ and $$[A]^{< \omega} = \{X \subset A : |X| < \omega\}.$$ We denote the set of natural numbers by $\mathbb{N}$, the integers by $\mathbb{Z}$, the rationals by $\mathbb{Q}$ and the reals by $\mathbb{R}$. The unit circle group $\mathbb{T}$ will be identified with the metric group $(\mathbb{R} / \mathbb{Z}, \delta)$, where $\delta$ is given by $\delta(x + \mathbb{Z}, y + \mathbb{Z}) = \min \{|x - y + a| : a \in \mathbb{Z}\}$, for every $x, y \in \mathbb{R}$. Given a subset $A$ of $\mathbb{T}$, we will denote by $\delta(A)$ the diameter of $A$ with respect to the metric $\delta$. Let $X$ be a set and $G$ be a group. We denote by $G^{X}$ the product $\prod_{x \in X} G_x$ where $G_x = G$ for every $x \in X$. If $f \in G^{X}$ then its *support* is the set $\{x \in X : f(x) \neq 0\}$ which will be designated as $\operatorname{supp}f$. The set $\{f \in G^{X} : |\operatorname{supp}f| < \omega\}$ will be denoted by $G^{(X)}$. An Abelian group $G$ is called *divisible* if, for each $g \in G$ and each $n \in \mathbb{N} \setminus \{0\}$, the equation $nx = g$ has a solution $x \in G$. If $n \in \mathbb{N}$, we denote by $G[n]$ the set $\{x \in G : nx = 0\}$. The proof of the next three results can be found in [@robinson]. Let $G$ be an Abelian group, $H$ be a subgroup of $G$, $\tilde{G}$ be a divisible group and $f: H \to \tilde{G}$ be a group homomorphism. There exists a group homomorphism $F: G \to \tilde{G}$ such that $F \hspace*{-0.1cm} \upharpoonright_{H} = f$. Given a prime number $p$, the subgroup of $\mathbb{Q} / \mathbb{Z}$ generated by $\{1 / p^{n} + \mathbb{Z} : n \in \mathbb{N}\}$ is called the *quasicyclic $p$-group*. \[teo\_classificacao\_grupos\_divisiveis\] An Abelian group is divisible if, and only if, it is isomorphic to a direct sum of copies of $\mathbb{Q}$ and of quasicyclic groups. \[teo\_imersao\_num\_grupo\_divisivel\] Every Abelian group is isomorphic to a subgroup of a divisible group. Preliminaries ============= Let $G$ be an Abelian group such that: - $|G| = \mathfrak{c}$; - $|G / T(G)| = \mathfrak{c}$; - $\forall n, d \in \mathbb{N} \setminus \{0\} \ (d \mid n \rightarrow |d G[n]| < \omega \ \hbox{or} \ |d G[n]| = \mathfrak{c})$. Consider $\{P_0, P_1\}$ a partition of $\mathfrak{c}$ such that $|P_0| = |P_1| = \mathfrak{c}$, $\omega + 1 \subset P_1$ and $\{\omega + n : n \geq 1\} \subset P_0$. \[cap5\_af\_pontos\_de\_acumulacao\_de\_ordem\_infinita\] There exist $L_1 \in [P_1]^{\mathfrak{c}}$ and $\tilde{\varphi} : G \to (\mathbb{Q} / \mathbb{Z})^{(P_0)} \oplus \mathbb{Q}^{(P_1)}$ a group monomorphism such that $\omega \cup \{\omega\} \subset L_1$ and $\{(0, \chi_{\xi}) \in (\mathbb{Q} / \mathbb{Z})^{(P_0)} \oplus \mathbb{Q}^{(P_1)} : \xi \in L_1\} \subset \tilde{\varphi}[G]$. Let $\varphi_{1} : G \to (\mathbb{Q} / \mathbb{Z})^{(P_0)} \oplus \mathbb{Q}^{(P_1)}$ be a group monomorphism.[^1] Since $|G / T(G)| = \mathfrak{c}$, there exists $W \subset G$ such that: - $|W| = \mathfrak{c}$; - $w \not \in T(G)$, for every $w \in W$; - $w_1 - w_2 \not \in T(G)$, for every $w_1, w_2 \in W$ with $w_1 \neq w_2$. Fix $y_0 \in W$ and define $Y_0 = \{y_0\}$. Let $\alpha < \mathfrak{c}$ be an ordinal. For each $\beta < \alpha$, suppose that $Y_{\beta} = \{y_{\gamma} : \gamma < \beta\}$ is an independent subset of $G$ satisfying the following conditions: - $Y_{\beta} \subset W$; - If $\gamma < \beta < \alpha$, then $Y_{\gamma} \subset Y_{\beta}$. If $\alpha$ is a limit ordinal, put $Y_{\alpha} = \cup_{\beta < \alpha} Y_{\beta}$. Now, assume that $\alpha$ is a successor ordinal — say, $\alpha = \beta + 1$. We state that there exists $y_{\alpha} \in W$ such that $\langle y_{\alpha} \rangle \cap \langle Y_{\beta} \rangle = \{0\}$. In fact, if such an element does not exist, then for every $w \in W$ there exists $m_{w} \in \mathbb{Z} \setminus \{0\}$ such that $m_{w} \cdot w \in \langle Y_{\beta} \rangle$. Since $|\langle Y_{\beta} \rangle| \leq \max\{\omega, |\beta|\} < \mathfrak{c} = |W|$, there exist $y \in \langle Y_{\beta} \rangle$ and $\tilde{W} \subset W$ such that: - $|\tilde{W}| = \mathfrak{c}$; - $m_{w} \cdot w = y$, for every $w \in \tilde{W}$. Since $|\mathbb{Z} \setminus \{0\}| = \omega$, it is possible to choose $\tilde{\tilde{W}} \subset \tilde{W}$ and $m \in \mathbb{Z} \setminus \{0\}$ such that: - $|\tilde{\tilde{W}}| = \mathfrak{c}$; - $m \cdot w = y$, for every $w \in \tilde{\tilde{W}}$. But this contradicts the fact that the difference between any two distinct elements of $W$ does not belong to $T(G)$. Therefore, there exists $y_{\alpha} \in W$ such that $\langle y_{\alpha} \rangle \cap \langle Y_{\beta} \rangle = \{0\}$. Put $Y_{\alpha} = Y_{\beta} \cup \{y_{\alpha}\}$. It follows that $Y_{\alpha}$ is an independent subset of $G$ such that $Y_{\alpha} \subset W$ and $Y_{\beta} \subset Y_{\alpha}$, for every $\beta < \alpha$. It follows that $Y = \cup_{\alpha < \mathfrak{c}} Y_{\alpha} \subset W$ is an independent subset of $G$ with cardinality $\mathfrak{c}$. Let $\{z_{\xi} : \xi < \mathfrak{c}\}$ be an enumeration of $Y$ and write $\varphi_{1}(z_{\xi}) = (a_{\xi}, b_{\xi}) \in (\mathbb{Q} / \mathbb{Z})^{(P_0)} \oplus \mathbb{Q}^{(P_1)}$ for every $\xi < \mathfrak{c}$. For each $n \in \omega$, define $A_n = \{\xi < \mathfrak{c} : n \cdot a_{\xi} = 0\}$. Since $|Y| = \mathfrak{c}$, there exists $n \in \omega$ such that $|A_n| = \mathfrak{c}$. Fix such an $n$ and consider the set $\{n \cdot z_{\xi} : \xi \in A_n\}$. Observe that it has cardinality $\mathfrak{c}$, since $|A_n| = \mathfrak{c}$ and $Y \subset W$ is an independent subset of $G$. Besides, $\varphi_{1}(n \cdot z_{\xi}) = (0, n \cdot b_{\xi})$, for every $\xi \in A_n$. Fix $\{c_{\zeta} : \zeta \in P_1\}$ a basis of $\mathbb{Q}^{(P_1)}$ as a vector space over $\mathbb{Q}$ that contains $\{n \cdot b_{\xi} : \xi \in A_n\}$ and such that $\{c_{\zeta} : \zeta \in \omega \cup \{\omega\}\} \subset \{n \cdot b_{\xi} : \xi \in A_n\}$. Let $\varphi_{2} : (\mathbb{Q} / \mathbb{Z})^{(P_0)} \oplus \mathbb{Q}^{(P_1)} \to (\mathbb{Q} / \mathbb{Z})^{(P_0)} \oplus \mathbb{Q}^{(P_1)}$ be the group isomorphism given by $\varphi_{2}(a, 0) = (a, 0)$, for every $(a, 0) \in (\mathbb{Q} / \mathbb{Z})^{(P_0)} \oplus \mathbb{Q}^{(P_1)}$ and $\varphi_{2}(0, \sum_{\zeta \in F} \alpha_{\zeta} \cdot c_{\zeta}) = (0, \sum_{\zeta \in F} \alpha_{\zeta} \cdot \chi_{\zeta})$, for every $F \in [P_1]^{< \omega}$ with $\{\alpha_{\zeta} : \zeta \in F\} \subset \mathbb{Q} \setminus \{0\}$. Pick $L_1 \in [P_1]^{\mathfrak{c}}$ such that $|P_1 \setminus L_1| = \mathfrak{c}$ and $\{n \cdot b_{\xi} : \xi \in A_n\} = \{c_{\zeta} : \zeta \in L_1\}$. We have that $\omega \cup \{\omega\} \subset L_1$ and $\{(0, \chi_{\xi}) : \xi \in L_1\} \subset \tilde{\varphi}[G]$, where $\tilde{\varphi} = \varphi_{2} \circ \varphi_{1}$. Denote by $D$ the set of all natural numbers $n > 1$ such that $G$ contains a copy of $\mathbb{Z}^{(\mathfrak{c})}_{n}$ — and, therefore, an independent subset of cardinality $\mathfrak{c}$ constituted of elements of order $n$. \[cap5\_af\_pontos\_de\_acumulacao\_de\_ordem\_finita\_1\] For each $n \in D$, there exists $\{(x^{n}_{\xi}, 0) \in (\mathbb{Q} / \mathbb{Z})^{(P_0)} \oplus \mathbb{Q}^{(P_1)} : \xi < \mathfrak{c}\} \subset \tilde{\varphi}[G]$ with the following properties: (i) $\operatorname{o}(x^{n}_{\xi}) = n$, for every $\xi < \mathfrak{c}$; (ii) $\operatorname{supp}x^{n}_{\xi} \cap \operatorname{supp}x^{n}_{\mu} = \emptyset$, for every $\xi, \mu < \mathfrak{c}$ with $\xi \neq \mu$. Let $n \in D$ and $\{(y_{\xi}^{n}, 0) \in (\mathbb{Q} / \mathbb{Z})^{(P_0)} \oplus \mathbb{Q}^{(P_1)} : \xi < \mathfrak{c}\} \subset \tilde{\varphi}[G]$ independent with $\operatorname{o}(y_{\xi}^{n}) = n$, for every $\xi < \mathfrak{c}$. Case 1 : $\mathfrak{c}$ is regular. Applying the $\Delta$-system lemma for $\{\operatorname{supp}y_{\xi}^{n} : \xi < \mathfrak{c}\}$, we obtain $I_n \in [\mathfrak{c}]^{\mathfrak{c}}$ and $R_n \in [P_0]^{< \omega}$ such that $\{\operatorname{supp}y_{\xi}^{n} : \xi \in I_n\}$ is a $\Delta$-system with root $R_n$. Fix $J_n \in [I_n]^{\mathfrak{c}}$ such that if $\xi, \mu \in J_n$, then $y_{\xi}^{n}(\zeta) = y_{\mu}^{n}(\zeta)$ for every $\zeta \in R_n$. Let $\{J_{n, 0}, J_{n, 1}\}$ be a partition of $J_n$ such that $|J_{n, 0}| = |J_{n, 1}| = \mathfrak{c}$. Fix $f: \mathfrak{c} \to J_{n, 0}$ and $g: \mathfrak{c} \to J_{n, 1}$ bijections and define $x_{\xi}^{n} = y_{f(\xi)}^{n} - y_{g(\xi)}^{n}$, for every $\xi < \mathfrak{c}$. Given $\xi < \mathfrak{c}$, we have that $\operatorname{o}(x_{\xi}^{n}) = n$, since $n \cdot x_{\xi}^{n} = n \cdot (y_{f(\xi)}^{n} - y_{g(\xi)}^{n}) = n \cdot y_{f(\xi)}^{n} - n \cdot y_{g(\xi)}^{n} = 0 - 0 = 0$ and if $m$ is a non-zero natural number lower than $n$, then $m \cdot x_{\xi}^{n} = m \cdot y_{f(\xi)}^{n} - m \cdot y_{g(\xi)}^{n}\neq 0$, since $\operatorname{o}(y_{f(\xi)}^{n}) = \operatorname{o}(y_{g(\xi)}^{n}) = n$ and $\{y_{\xi}^{n} : \xi < \mathfrak{c}\}$ is an independent subset of $G$. Finally, let $\xi, \mu < \mathfrak{c}$ be such that $\xi \neq \mu$. If $\zeta \in \operatorname{supp}x_{\xi}^{n} \cap \operatorname{supp}x_{\mu}^{n}$, then one of the following possibilities occur: - $\zeta \in \operatorname{supp}y_{f(\xi)}^{n} \cap \operatorname{supp}y_{f(\mu)}^{n}$; - $\zeta \in \operatorname{supp}y_{f(\xi)}^{n} \cap \operatorname{supp}y_{g(\mu)}^{n}$; - $\zeta \in \operatorname{supp}y_{g(\xi)}^{n} \cap \operatorname{supp}y_{f(\mu)}^{n}$; - $\zeta \in \operatorname{supp}y_{g(\xi)}^{n} \cap \operatorname{supp}y_{g(\mu)}^{n}$. Thus, $\zeta \in R_n$ and, therefore, $\zeta \in \operatorname{supp}y_{f(\xi)}^{n} \cap \operatorname{supp}y_{f(\mu)}^{n} \cap \operatorname{supp}y_{g(\xi)}^{n} \cap \operatorname{supp}y_{g(\mu)}^{n}$. Since $f(\xi), g(\xi), f(\mu), g(\mu) \in J_n$, it follows that $y_{f(\xi)}^{n}(\zeta) = y_{f(\mu)}^{n}(\zeta) = y_{g(\xi)}^{n}(\zeta) = y_{g(\mu)}^{n}(\zeta)$. So, $x_{\xi}^{n}(\zeta) = x_{\mu}^{n}(\zeta) = 0$ — a contradiction, because $\zeta \in \operatorname{supp}x_{\xi}^{n} \cap \operatorname{supp}x_{\mu}^{n}$. Therefore, $\operatorname{supp}x_{\xi}^{n} \cap \operatorname{supp}x_{\mu}^{n} = \emptyset$, for every $\xi, \mu < \mathfrak{c}$ with $\xi \neq \mu$. Case 2 : $\mathfrak{c}$ is not regular. In this case, $\operatorname{cf}(\mathfrak{c}) < \mathfrak{c}$ and $\mathfrak{c}$ is a limit cardinal. Let $\{I_{\alpha} : \alpha < \operatorname{cf}(\mathfrak{c})\}$ be a family of pairwise disjoint subsets of $\mathfrak{c}$ such that $\mathfrak{c} = \cup_{\alpha < \operatorname{cf}(\mathfrak{c})} I_{\alpha}$ and $|I_{\alpha}| = \kappa_{\alpha}$, for every $\alpha < \operatorname{cf}(\mathfrak{c})$, where: - $\{\kappa_{\alpha} : \alpha < \operatorname{cf}(\mathfrak{c})\}$ is a strictly increasing and cofinal sequence in $\mathfrak{c}$; - $\kappa_{\alpha}$ is a regular cardinal, for every $\alpha < \operatorname{cf}(\mathfrak{c})$; - $\kappa_{\alpha} \geq \max \{\omega, |\alpha|, \sup_{\beta < \alpha} \kappa_{\beta}\}^{+}$. For every $\alpha < \operatorname{cf}(\mathfrak{c})$ it is possible to repeat the construction presented in case 1 to obtain $\{(x_{\xi}^{n}, 0) \in (\mathbb{Q} / \mathbb{Z})^{(P_0)} \oplus \mathbb{Q}^{(P_1)} : \xi \in I_{\alpha}\} \subset \tilde{\varphi}[G]$ such that $\operatorname{o}(x_{\xi}^{n}) = n$ for every $\xi \in I_{\alpha}$ and $\operatorname{supp}x_{\xi}^{n} \cap \operatorname{supp}x_{\mu}^{n} = \emptyset$, for every $\xi, \mu \in I_{\alpha}$ with $\xi \neq \mu$. Let $J_0 = I_0$ and $\alpha < \operatorname{cf}(\mathfrak{c})$ be an ordinal. Suppose that for each $\beta < \alpha$, there exists $J_{\beta} \subset I_{\beta}$ such that $|J_{\beta}| = |I_{\beta}| = \kappa_{\beta}$ and $\operatorname{supp}x_{\xi}^{n} \cap \operatorname{supp}x_{\mu}^{n} = \emptyset$, for every $\xi, \mu \in \cup_{\beta < \alpha} J_{\beta}$ with $\xi \neq \mu$. Put $X_{\alpha} = \cup_{\xi \in \cup_{\beta < \alpha} J_{\beta}} \operatorname{supp}x_{\xi}^{n}$. We have that $|X_{\alpha}| < \kappa_{\alpha}$ and, therefore, there exists $J_{\alpha} \subset I_{\alpha}$ such that $|J_{\alpha}| = |I_{\alpha}|$ and $\operatorname{supp}x_{\xi}^{n} \cap X_{\alpha} = \emptyset$, for every $\xi \in J_{\alpha}$. Define $J = \cup_{\alpha < \operatorname{cf}(\mathfrak{c})} J_{\alpha}$. It follows that $|J| = \mathfrak{c}$ and $\operatorname{supp}x_{\xi}^{n} \cap \operatorname{supp}x_{\mu}^{n} = \emptyset$, for every $\xi, \mu \in J$ with $\xi \neq \mu$. \[cap5\_af\_pontos\_de\_acumulacao\_de\_ordem\_finita\_2\] For each $n \in D$, there exists $L_n \in [P_0]^{\mathfrak{c}}$ such that $L_m \cap L_n = \emptyset$, for every $m, n \in D$ with $m \neq n$. Moreover, there exists $\{(x_{\xi}, 0) \in (\mathbb{Q} / \mathbb{Z})^{(P_0)} \oplus \mathbb{Q}^{(P_1)} : \xi \in \cup_{n \in D} L_n\} \subset \tilde{\varphi}[G]$ with the following properties: (i) If $\xi \in L_n$, then $\operatorname{o}(x_{\xi}) = n$; (ii) If $\xi, \mu \in \cup_{n \in D} L_n$ and $\xi \neq \mu$, then $\operatorname{supp}x_{\xi} \cap \operatorname{supp}x_{\mu} = \emptyset$; (iii) $\xi \in \operatorname{supp}x_{\xi}$, for every $\xi \in \cup_{n \in D} L_n$. For each $n \in D$, consider $\{x^{n}_{\xi} : \xi < \mathfrak{c}\}$ where $\{(x^{n}_{\xi}, 0) \in (\mathbb{Q} / \mathbb{Z})^{(P_0)} \oplus \mathbb{Q}^{(P_1)} : \xi < \mathfrak{c}\} \subset \tilde{\varphi}[G]$ satisfies conditions (i) and (ii) of Lemma \[cap5\_af\_pontos\_de\_acumulacao\_de\_ordem\_finita\_1\]. Let $\{X_{\zeta} : \zeta < \mathfrak{c}\}$ be an enumeration of $\{\{x^{n}_{\xi} : \xi < \mathfrak{c}\} : n \in D\}$ such that $$|\{\zeta < \mathfrak{c} : X_{\zeta} = \{x^{n}_{\xi} : \xi < \mathfrak{c}\}\}| = \mathfrak{c}$$ for every $n \in D$. Fix $x \in X_0$ and define $\xi_{0} = \min \operatorname{supp}x$. Denote $x$ by $x_{\xi_{0}}$. Let $\alpha < \mathfrak{c}$ be an ordinal. For each $\beta < \alpha$, suppose defined $\xi_{\beta} \in P_0$ and $x_{\xi_{\beta}} \in X_{\beta}$ with the following properties: - $\xi_{\beta} = \min \operatorname{supp}x_{\xi_{\beta}}$; - If $\gamma < \beta < \alpha$, then $\operatorname{supp}x_{\xi_{\gamma}} \cap \operatorname{supp}x_{\xi_{\beta}} = \emptyset$. We have that $|\cup_{\beta < \alpha} \operatorname{supp}x_{\xi_{\beta}}| \leq \max \{|\alpha|, \omega\} < \mathfrak{c}$. Since $|X_{\alpha}| = \mathfrak{c}$ and $\operatorname{supp}x \cap \operatorname{supp}y = \emptyset$ for every $x$ and $y$ distinct elements of $X_{\alpha}$, there exists $z \in X_{\alpha}$ such that $\operatorname{supp}z \cap (\cup_{\beta < \alpha} \operatorname{supp}x_{\xi_{\beta}}) = \emptyset$. Put $\xi_{\alpha} = \min \operatorname{supp}z$ and write $x_{\xi_{\alpha}} = z$. By induction, we obtain $\xi_{\alpha} \in P_0$ and $x_{\xi_{\alpha}} \in X_{\alpha}$, for every $\alpha < \mathfrak{c}$. If $n \in D$, define $$L_n = \{\xi_{\alpha} \in P_0 : X_{\alpha} = \{x^{n}_{\xi} : \xi < \mathfrak{c}\}\} \setminus \{\omega + m : m \geq 1\}.$$ Note that $|L_n| = \mathfrak{c}$, since $|\{\alpha < \mathfrak{c} : X_{\alpha} = \{x^{n}_{\xi} : \xi < \mathfrak{c}\}\}| = \mathfrak{c}$ and $\xi_{\alpha} \neq \xi_{\beta}$ if $\alpha$ and $\beta$ are distinct elements of $\mathfrak{c}$. Besides, $L_m \cap L_n = \emptyset$, if $m, n \in D$ and $m \neq n$. If $\xi \in L_n$, then $\operatorname{o}(x_{\xi}) = n$ and if $\xi, \mu \in \cup_{n \in D} L_n$ and $\xi \neq \mu$, then $\operatorname{supp}x_{\xi} \cap \operatorname{supp}x_{\mu} = \emptyset$. Finally, if $\xi \in \cup_{n \in D} L_n$, then $\xi \in \operatorname{supp}x_{\xi}$. \[cap5\_prop\_varphi\] There exists $\varphi: G \to (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)}$ a group monomorphism such that $\{(0, \chi_{\xi}) \in (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} : \xi \in L_1\} \subset \varphi[G]$ and $\{(y_{\xi}, 0) \in (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} : \xi \in \cup_{n \in D} L_n\} \subset \varphi[G]$, where: (i) If $\xi \in L_n$, then $\operatorname{o}(y_{\xi}) = n$; (ii) If $\xi \in \cup_{n \in D} L_n$, then $\operatorname{supp}y_{\xi} \subset \{\xi\} \times \omega$. Consider $\{(x_{\xi}, 0) \in (\mathbb{Q} / \mathbb{Z})^{(P_0)} \oplus \mathbb{Q}^{(P_1)} : \xi \in \cup_{n \in D} L_n\} \subset \tilde{\varphi}[G]$ satisfying conditions (i), (ii) and (iii) of Lemma \[cap5\_af\_pontos\_de\_acumulacao\_de\_ordem\_finita\_2\]. Consider also the mapping $\hat{\varphi} : (\mathbb{Q} / \mathbb{Z})^{(P_0)} \oplus \mathbb{Q}^{(P_1)} \to (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)}$ defined in the following way: - Let $\xi \in L_n$, for some $n \in D$. Denote $\operatorname{supp}x_{\xi}$ by $\{\alpha_{0}, \alpha_{1}, ..., \alpha_{m}\}$, where $\alpha_{0} < \alpha_{1} < ... < \alpha_{m}$. The $\alpha_{i}$-th summand $\mathbb{Q} / \mathbb{Z}$ of the direct sum $(\mathbb{Q} / \mathbb{Z})^{(P_0)}$ will be mapped identically to the $(\xi, i)$-th summand of the direct sum $(\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)}$. - If $\mu \in P_0 \setminus \cup_{\xi \in \cup_{n \in D} L_n} \operatorname{supp}x_{\xi}$, then the $\mu$-th summand $\mathbb{Q} / \mathbb{Z}$ of the direct sum $(\mathbb{Q} / \mathbb{Z})^{(P_0)}$ will be mapped identically to the $(\mu, 0)$-th summand of the direct sum $(\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)}$. - $\hat{\varphi}(0, b) = (0, b)$, for every $b \in \mathbb{Q}^{(P_1)}$. The mapping $\hat{\varphi}$ is a group monomorphism. Consider $\varphi = \hat{\varphi} \circ \tilde{\varphi}$ and $(y_{\xi}, 0) = \hat{\varphi}(x_{\xi}, 0)$, for every $\xi \in \cup_{n \in D} L_n$. It follows that $\varphi$ is a group monomorphism such that $$\{(0, \chi_{\xi}) \in (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} : \xi \in L_1\} \subset \varphi[G]$$ and $$\{(y_{\xi}, 0) \in (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} : \xi \in \cup_{n \in D} L_n\} \subset \varphi[G].$$ Besides, if $\xi \in L_n$, then $\operatorname{o}(y_{\xi}) = \operatorname{o}(\hat{\varphi}(x_{\xi}, 0)) = n$, since $\hat{\varphi}$ is a group monomorphism. Finally, $\operatorname{supp}y_{\xi} \subset \{\xi\} \times \omega$, for every $\xi \in \cup_{n \in D} L_n$. We end this section presenting some notations that will be used throughout this article. If $(H, J) \in (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)}$, then $$H(\xi, n) = \dfrac{p(H, (\xi, n))}{q(H, (\xi, n))} + \mathbb{Z}$$ where $p(H, (\xi, n)), q(H, (\xi, n)) \in \mathbb{Z}$, $q(H, (\xi, n)) > 0$, $\gcd(p(H, (\xi, n)), q(H, (\xi, n))) = 1$ and $0 \leq p(H, (\xi, n)) \leq q(H, (\xi, n)) - 1$, for every $(\xi, n) \in \operatorname{supp}H$. Furthermore, $$J(\mu) = \dfrac{p(J, \mu)}{q(J, \mu)}$$ where $p(J, \mu), q(J, \mu) \in \mathbb{Z}$, $q(J, \mu) > 0$ and $\gcd(p(J, \mu), q(J, \mu)) = 1$, for every $\mu \in \operatorname{supp}J$. Consider $$d(H) = \operatorname{lcm}\{q(H, (\xi, n)) : (\xi, n) \in \operatorname{supp}H\}$$ and $$d(J) = \operatorname{lcm}\{q(J, \mu) : \mu \in \operatorname{supp}J\}.\footnote{If $H = 0$, then $d(H) = 1$. Analogously, if $J = 0$, then $d(J)$ = 1.}$$ For each $(\xi, n) \in \operatorname{supp}H$, put $$a(H, (\xi, n)) = p(H, (\xi, n)) \cdot \dfrac{d(H)}{q(H, (\xi, n))}$$ and, for each $\mu \in \operatorname{supp}J$, put $$a(J, \mu) = p(J, \mu) \cdot \dfrac{d(J)}{q(J, \mu)}.$$ Also, put $$|q(H)| = \max \{q(H, (\xi, n)) : (\xi, n) \in \operatorname{supp}H\},$$ $$|q(J)| = \max \{q(J, \mu) : \mu \in \operatorname{supp}J\},$$ $$|p(J)| = \max \{|p(J, \mu)| : \mu \in \operatorname{supp}J\},$$ $$|a(J)| = \max \{|a(J, \mu)| : \mu \in \operatorname{supp}J\}$$ and $$||a(H, J)|| = \sum_{(\xi, n) \in \operatorname{supp}H} |a(H, (\xi, n))| + \sum_{\mu \in \operatorname{supp}J} |a(J, \mu)|.$$ Observe that $$\operatorname{supp}(H, J) = \operatorname{supp}H \cup \operatorname{supp}J.$$ If $H \in (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)}$ and $X \subset P_0 \times \omega$, then $H \upharpoonright_{X} \in (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)}$ is given by $$H \upharpoonright_{X}(\xi, n) = \left\{ \begin{array}{lll} H(\xi, n) & \hbox{if} & (\xi, n) \in X \\ 0 + \mathbb{Z} & \hbox{if} & (\xi, n) \not \in X \\ \end{array} \right.$$ for every $(\xi, n) \in P_0 \times \omega$. Finally, if $X \subset P_0 \times \omega$, we write $$\pi_{1}[X] = \{\xi \in P_0 : \exists n \in \omega \ \hbox{such that} \ (\xi, n) \in X\}$$ and $$\pi_{2}[X] = \{n \in \omega : \exists \xi \in P_0 \ \hbox{such that} \ (\xi, n) \in X\}.$$ Sorting the sequences ===================== Let $S$ be an infinite subset of $G$ and let $n > 1$ be a natural number. We say that $S$ is *$n$-round* if: - $S \subset G[n]$; - The restriction of the group homomorphism $$\begin{array}{cccl} \varphi_{d}: & G & \to & G \\ & x & \mapsto & d x \end{array}$$ to $S$ is finite-to-one, for every proper divisor $d$ of $n$. The following two propositions can be found in . \[cap5\_prop\_n-round1\] Let $n > 1$ be a natural number and let $S$ be an infinite subset of $G[n]$. There exist $T$ an infinite subset of $G$ and $z \in G$ such that $T + z \subset S$ and $T$ is $d$-round, for some divisor $d$ of $n$. \[cap5\_prop\_n-round2\] Let $n > 1$ be a natural number. If there exists $S \subset G$ such that $S$ is $n$-round, then $n \in D$. We will denote by $\mathcal{H}$ the set of all functions $h$ from $\omega$ to $\varphi[G] \subset (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)}$ of the form $h(n) = (g(n), f(n))$ that satisfy one of the following conditions: (1) $\operatorname{supp}f(n) \setminus \cup_{m < n} \operatorname{supp}f(m) \neq \emptyset$, for every $n \in \omega$; (2) $|q(f(n))| > n$, for every $n \in \omega$; (3) $\{|q(f(n))| : n \in \omega\}$ is bounded and $|p(f(n))| > n$, for every $n \in \omega$; (4) $|q(g(n))| > n$, for every $n \in \omega$; (5) 1. $\operatorname{supp}g(n) \setminus \cup_{m < n} \operatorname{supp}g(m) \neq \emptyset$, for every $n \in \omega$; 2. $f(n) = 0$, for every $n \in \omega$; 3. There exists $k \in D$ such that $\operatorname{o}(h(n)) = \operatorname{o}(g(n)) = k$, for every $n \in \omega$; 4. $\{h(n) : n \in \omega\}$ is a $k$-round subset of $\varphi[G]$. Given $h \in \mathcal{H}$ and $i \in \{1, 2, 3, 4, 5\}$, we say that $h$ is *of type i* if $h$ satisfies condition (i) above. \[cap5\_prop\_subsequencias\] If $h: \omega \to \varphi[G] \subset (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)}$, then there exist $c \in \varphi[G]$ and a strictly increasing function $j: \omega \to \omega$ such that the sequence $n \mapsto (h \circ j)(n) + c$ is constant or it is of one of the types mentioned above. We have $h(n) = (g(n), f(n))$ for every $n \in \omega$, where $g: \omega \to (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)}$ and $f: \omega \to \mathbb{Q}^{(P_1)}$. Observe that $(0, 0) = \varphi(0) \in \varphi[G]$. Case 1 : $\bigcup \{\operatorname{supp}f(n) : n \in \omega\}$ is infinite.\ Put $n_0 = 0$. Fix $k \in \omega$ and suppose that $n_0, n_1, ..., n_k \in \omega$ are defined. Since $\operatorname{supp}f(n)$ is finite for every $n \in \omega$ and $\bigcup \{\operatorname{supp}f(n) : n \in \omega\}$ is infinite, there exists a natural number $n_{k + 1} > n_k$ such that $\operatorname{supp}f(n_{k + 1}) \setminus \cup_{l < k + 1} \operatorname{supp}f(n_l) \neq \emptyset$. Then the function $j : \omega \to \omega$ defined by $j(k) = n_k$ for each $k \in \omega$ is strictly increasing and $h \circ j$ is of type 1. In this case, put $c = (0, 0)$. Case 2 : $\bigcup \{\operatorname{supp}f(n) : n \in \omega\}$ is finite and $\{|q(f(n))| : n \in \omega\}$ is unbounded.\ By induction, define a strictly increasing sequence $\{n_k : k \in \omega\}$ of natural numbers such that $|q(f(n_k))| > k$ for each $k \in \omega$. Then the function $j : \omega \to \omega$ defined by $j(k) = n_k$ for each $k \in \omega$ is strictly increasing and $h \circ j$ is of type 2. In this case, put $c = (0, 0)$. Case 3 : $\bigcup \{\operatorname{supp}f(n) : n \in \omega\}$ is finite, $\{|q(f(n))| : n \in \omega\}$ is bounded and $\{|p(f(n))| : n \in \omega\}$ is unbounded.\ By induction, define a strictly increasing sequence $\{n_k : k \in \omega\}$ of natural numbers such that $|p(f(n_k))| > k$ for each $k \in \omega$. Then the function $j : \omega \to \omega$ defined by $j(k) = n_k$ for each $k \in \omega$ is strictly increasing and $h \circ j$ is of type 3. In this case, put $c = (0, 0)$. Case 4 : $\bigcup \{\operatorname{supp}f(n) : n \in \omega\}$ is finite, $\{|q(f(n))| : n \in \omega\}$ and $\{|p(f(n))| : n \in \omega\}$ are bounded and $\{|q(g(n))| : n \in \omega\}$ is unbounded.\ By induction, define a strictly increasing sequence $\{n_k : k \in \omega\}$ of natural numbers such that $|q(g(n_k))| > k$ for each $k \in \omega$. Then the function $j : \omega \to \omega$ defined by $j(k) = n_k$ for each $k \in \omega$ is strictly increasing and $h \circ j$ is of type 4. In this case, put $c = (0, 0)$. Case 5 : $\bigcup \{\operatorname{supp}f(n) : n \in \omega\}$ is finite, $\{|q(f(n))| : n \in \omega\}$, $\{|p(f(n))| : n \in \omega\}$ and $\{|q(g(n))| : n \in \omega\}$ are bounded and $\bigcup \{\operatorname{supp}g(n) : n \in \omega\}$ is infinite.\ Since $\bigcup \{\operatorname{supp}g(n) : n \in \omega\}$ is infinite, there exists $j_1: \omega \to \omega$ strictly increasing such that $\operatorname{supp}g \circ j_1(n) \setminus \cup_{m < n} \operatorname{supp}g \circ j_1(m) \neq \emptyset$, for every $n \in \omega$. Since $\bigcup \{\operatorname{supp}f(n) : n \in \omega\}$ is finite and both $\{|q(f(n))| : n \in \omega\}$ and $\{|p(f(n))| : n \in \omega\}$ are bounded, there exists $j_2: \omega \to \omega$ strictly increasing such that $f \circ j_1 \circ j_2(n) = b \in \mathbb{Q}^{(P_1 \times \omega)}$, for every $n \in \omega$. We have that $\{(g \circ j_1 \circ j_2(n), b) : n \in \omega\}$ is an infinite subset of $\varphi[G]$ and so is $S = \{(g \circ j_1 \circ j_2(n) - g \circ j_1 \circ j_2(0) , 0) : n \in \omega\}$. Since $\{|q(g(n))| : n \in \omega\}$ is bounded, there exists a natural number $k > 1$ such that $S \subset \varphi[G][k]$. Thus, there exist $T \subset \varphi[G]$ infinite and $z \in \varphi[G]$ such that $T + z \subset S$ and $T$ is $d$-round, for some divisor $d$ of $k$. Let $j_3: \omega \to \omega$ be a strictly increasing function such that $T + z = \{(g \circ j_1 \circ j_2 \circ j_3(n) - g \circ j_1 \circ j_2(0), 0) : n \in \omega\}$. In other words, $T = \{(g \circ j_1 \circ j_2 \circ j_3(n) - g \circ j_1 \circ j_2(0), 0) - z : n \in \omega\}$. Since $T$ is $d$-round, it follows that $T \subset \varphi[G][d]$. Therefore, there exist $j_4: \omega \to \omega$ a strictly increasing function and $r$ a divisor of $d$ such that $\operatorname{o}((g \circ j_1 \circ j_2 \circ j_3 \circ j_4 (n) - g \circ j_1 \circ j_2(0), 0) - z) = r$, for every $n \in \omega$. Observe that $r \in D$, since $\{(g \circ j_1 \circ j_2 \circ j_3 \circ j_4(n) - g \circ j_1 \circ j_2(0), 0) - z : n \in \omega\}$ is $r$-round. Put $c = - h \circ j_1 \circ j_2(0) - z$ and $j = j_1 \circ j_2 \circ j_3 \circ j_4 \circ j_5$, where $j_5: \omega \to \omega$ is a strictly increasing function such that $\operatorname{supp}(g \circ j(n) - g \circ j_1 \circ j_2(0) - z) \setminus \cup_{m < n} \operatorname{supp}(g \circ j(m) - g \circ j_1 \circ j_2(0) - z) \neq \emptyset$, for every $n \in \omega$. Case 6 : $\bigcup \{\operatorname{supp}f(n) : n \in \omega\}$ is finite, $\{|q(f(n))| : n \in \omega\}$, $\{|p(f(n))| : n \in \omega\}$ and $\{|q(g(n))| : n \in \omega\}$ are bounded and $\bigcup \{\operatorname{supp}g(n) : n \in \omega\}$ is finite.\ It follows that $\{g(n) : n \in \omega\}$ is finite, as well as $\{f(n) : n \in \omega\}$. So, $h$ has a constant subsequence. \[cap5\_prop\_indexacao\_das\_sequencias\] There exists an enumeration $\{h_{\xi} : \xi \in \tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n\}$ of $\mathcal{H}$, where $\tilde{L}_1 \in [L_1]^{\mathfrak{c}}$ is such that $\omega \in \tilde{L}_1$ and $\omega \cap \tilde{L}_1 = \emptyset$ and $\tilde{L}_n \in [L_n]^{\mathfrak{c}}$ for every $n \in D$ satisfying the following conditions: (i) $h_{\omega}(n) = (0, \chi_{n})$, for every $n \in \omega$. (ii) If $\xi \in \tilde{L}_1$, then $h_{\xi}$ is of type 1, 2, 3 or 4; (iii) If there exists $k \in D$ such that $\xi \in \tilde{L}_k$, then $h_{\xi}$ is of type 5 and $\operatorname{o}(h_{\xi}(n)) = k$, for every $n \in \omega$; (iv) $\cup_{n \in \omega} \operatorname{supp}h_{\xi}(n) \subset (\xi \times \omega) \cup \xi$, for every $\xi \in \tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n$. Let $\tilde{\mathcal{H}}$ be the family of all elements of $\mathcal{H}$ that are of type 1, 2, 3 or 4. For each $k \in D$, let $\tilde{\mathcal{H}_{k}}$ be the family of all elements of $\mathcal{H}$ that are of type 5 and that have order $k$. Note that $\mathcal{H} = \tilde{\mathcal{H}} \cup \bigcup_{k \in D} \tilde{\mathcal{H}_k}$. Consider an enumeration $\{\tilde{h}_{\xi} : \xi \in L_1\}$ of $\tilde{\mathcal{H}}$ such that for every $h \in \tilde{\mathcal{H}}$, the set $\textstyle A_h = \{\xi \in L_1 : \tilde{h}_{\xi} = h\}$ has cardinality $\mathfrak{c}$. For each $k \in D$, consider an enumeration $\{\tilde{h}_{\xi} : \xi \in L_k\}$ of $\tilde{\mathcal{H}_{k}}$ such that for every $h \in \tilde{\mathcal{H}_{k}}$, the set $\textstyle A_h = \{\xi \in L_k : \tilde{h}_{\xi} = h\}$ has cardinality $\mathfrak{c}$. Fix $h \in \mathcal{H}$. Since $\cup_{n \in \omega} \operatorname{supp}h(n) \subset (\mathfrak{c} \times \omega) \cup \mathfrak{c}$, $\operatorname{cf}(\mathfrak{c}) > \omega$ and $|\cup_{n \in \omega} \operatorname{supp}h(n)| \leq \omega$, there exists $\alpha < \mathfrak{c}$ such that $\alpha > \omega + 1$ and $\cup_{n \in \omega} \operatorname{supp}h(n) \subset (\alpha \times \omega) \cup \alpha$. Define $h_{\omega}(n) = (0, \chi_{n})$, for every $n \in \omega$. Case 1 : $h$ is of type 1, 2, 3 or 4.\ Define $L_{1, h} = A_h \setminus \alpha$ and put $h_{\xi} = \tilde{h}_{\xi}$, for every $\xi \in L_{1, h}$. Case 2 : $h$ is of type 5 and $\operatorname{o}(h(n)) = k$ for every $n \in \omega$.\ Define $L_{k, h} = A_h \setminus \alpha$ and put $h_{\xi} = \tilde{h}_{\xi}$, for every $\xi \in L_{k, h}$. Observe that $\tilde{L}_1 = \{\omega\} \cup \bigcup_{h \in \tilde{\mathcal{H}}} L_{1, h}$ has cardinality $\mathfrak{c}$ and that $\omega \cap \tilde{L}_1 = \emptyset$. Also note that for each $k \in D$, $\tilde{L}_k = \{\omega\} \cup \bigcup_{h \in \tilde{\mathcal{H}_{k}}} L_{k, h}$ has cardinality $\mathfrak{c}$. The conditions (i), (ii), (iii) and (iv) are verified. Group homomorphisms from selective ultrafilters =============================================== We say that $p \in \omega^{*}$ is *selective* if, for each partition $\{A_n : n \in \omega\}$ of $\omega$ into non-empty sets, either $A_n \in p$ for some $n \in \omega$ or, for each $n \in \omega$, there exists $a_n \in A_n$ such that $\{a_n : n \in \omega\} \in p$. Two selective ultrafilters $p$ and $q$ are said to be *incomparable* if there exists no bijection $f: \omega \to \omega$ such that $\beta f(p) = q$, where $\beta f$ is the Stone-$\mathrm{\check{C}}$ech extension of $f$. Blass [@blass] proved that MA implies the existence of $2^{\mathfrak{c}}$ incomparable selective ultrafilters, but the existence of selective ultrafilters does not imply MA. In fact, Baumgartner [@baumgartner] showed that adding at least $\aleph_{2}$ Sacks reals side-by-side in a model of CH, MA totally fails. In this model, there exist selective ultrafilters. Let $\{p_{\xi} : \xi \in \tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n\}$ be a family of incomparable selective ultrafilters and let $\{h_{\xi} : \xi \in \tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n\}$ be an enumeration of $\mathcal{H}$ satisfying conditions (i), (ii), (iii) and (iv) of Proposition \[cap5\_prop\_indexacao\_das\_sequencias\]. \[cap5\_prop\_construcao\_de\_E\] Fix $(H, J) \in (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} \setminus \{(0, 0)\}$. There exists $E \in [\mathfrak{c}]^{\omega}$ such that: (i) $\operatorname{supp}(H, J) \subset (E \times \omega) \cup E$; (ii) $|E \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)| = \omega$; (iii) $\cup_{n \in \omega} \operatorname{supp}h_{\xi}(n) \subset (E \times \omega) \cup E$, for every $\xi \in E \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)$. For each $\xi \in \mathfrak{c} \setminus (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)$, put $E(\xi) = \{\xi\} \cup \omega$. If $\xi \in \tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n$, we define inductively $$\textstyle E(\xi) = \{\xi\} \cup \bigcup_{\mu \in \pi_{1}[\cup_{n \in \omega} \operatorname{supp}g_{\xi}(n)] \cup \bigcup_{n \in \omega} \operatorname{supp}f_{\xi}(n)} E(\mu).$$ Fix $\tilde{L}_{1} \in [L_{1} \setminus \omega]^{\omega}$ and put $$E = \cup_{\zeta \in \pi_{1}[\operatorname{supp}H] \cup \operatorname{supp}J \cup \tilde{L}_{1}} E(\zeta).$$ We have that $\operatorname{supp}(H, J) \cup \tilde{L}_{1}\subset (E \times \omega) \cup E$, since $\zeta \in E(\zeta) \subset E$, for every $\zeta \in \pi_{1}[\operatorname{supp}H] \cup \operatorname{supp}J \cup \tilde{L}_{1}$. Moreover, an inductive argument guarantees that $E(\xi) \in [\mathfrak{c}]^{\omega}$ for every $\xi < \mathfrak{c}$ and, therefore, $E \in [\mathfrak{c}]^{\omega}$. Finally, if $\xi \in E \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)$, then there exists $\zeta \in \pi_{1}[\operatorname{supp}H] \cup \operatorname{supp}J \cup \tilde{L}_{1}$ such that $\xi \in E(\zeta)$. Another inductive argument guarantees that if $\alpha \in E(\beta)$, then $E(\alpha) \subset E(\beta)$, for every $\alpha, \beta < \mathfrak{c}$. Thus, $\pi_{1}[\cup_{n \in \omega} \operatorname{supp}g_{\xi}(n)] \cup \bigcup_{n \in \omega} \operatorname{supp}f_{\xi}(n) \subset E(\xi) \subset E(\zeta) \subset E$ and, therefore, $\cup_{n \in \omega} \operatorname{supp}h_{\xi}(n) \subset (E \times \omega) \cup E$. The next lemma will be proved in the appendix, at the end of this article. Since its statement will be used in the proof of Lemma \[cap5\_lem\_extensao\_dos\_homomorfismos\], we announce it here. \[cap5\_lem\_construcao\_dos\_homomorfismos\] Assume the existence of $\mathfrak{c}$ incomparable selective ultrafilters. Fix $(H, J) \in (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} \setminus \{(0, 0)\}$ and $E \in [\mathfrak{c}]^{\omega}$ satisfying conditions (i), (ii) and (iii) of Proposition \[cap5\_prop\_construcao\_de\_E\]. There exists a group homomorphism $\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}} : (\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)} \to \mathbb{T}$ with the following properties: (i) $\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(H, J) \neq 0 + \mathbb{Z}$; (ii) $\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(0, \chi_{\xi}) = p_{\xi}-\lim \{\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(h_{\xi}(n)) : n \in \omega\}$, for every $\xi \in (E \cap L_1) \setminus \omega$; (iii) $\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(y_{\xi}, 0) = p_{\xi}-\lim \{\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(h_{\xi}(n)) : n \in \omega\}$, for every $\xi \in (E \cap \bigcup_{n \in D} L_n) \setminus \omega$. \[cap5\_lem\_extensao\_dos\_homomorfismos\] Assume the existence of $\mathfrak{c}$ incomparable selective ultrafilters. Given $(H, J) \in (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} \setminus \{(0, 0)\}$, there exists a group homomorphism $\phi_{(H, J)}: (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} \to \mathbb{T}$ satisfying the following conditions: (i) $\phi_{(H, J)}(H, J) \neq 0 + \mathbb{Z}$; (ii) $\phi_{(H, J)}(0, \chi_{\xi}) = p_{\xi}-\lim \{\phi_{(H, J)}(h_{\xi}(n)) : n \in \omega\}$, for every $\xi \in L_1 \setminus \omega$. (iii) $\phi_{(H, J)}(y_{\xi}, 0) = p_{\xi}-\lim \{\phi_{(H, J)}(h_{\xi}(n)) : n \in \omega\}$, for every $\xi \in \cup_{n \in D} L_n$. We will construct $\phi_{(H, J)}$ by induction. According to Proposition \[cap5\_prop\_construcao\_de\_E\], there exists $E \in [\mathfrak{c}]^{\omega}$ such that $\operatorname{supp}(H, J) \subset (E \times \omega) \cup E$, $|E \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)| = \omega$ and $\cup_{n \in \omega} \operatorname{supp}h_{\xi}(n) \subset (E \times \omega) \cup E$, for every $\xi \in E \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)$. It follows from Lemma \[cap5\_lem\_construcao\_dos\_homomorfismos\] that there exists a group homomorphism $\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}: (\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)} \to \mathbb{T}$ such that: (1) $\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(H, J) \neq 0 + \mathbb{Z}$; (2) $\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(0, \chi_{\xi}) = p_{\xi}-\lim \{\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(h_{\xi}(n)) : n \in \omega\}$, for every $\xi \in (E \cap L_1) \setminus \omega$; (3) $\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(y_{\xi}, 0) = p_{\xi}-\lim \{\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(h_{\xi}(n)) : n \in \omega\}$, for every $\xi \in E \cap \cup_{n \in D} L_n$. Let $\{\alpha_{\xi} : \xi < \mathfrak{c}\}$ be a strictly increasing enumeration of $\mathfrak{c} \setminus E$. - If $\alpha_{0} \in P_0 \setminus \bigcup_{n \in D} L_n$, we use the fact that $\mathbb{T}$ is a divisible group to extend $\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}$ to a group homomorphism $$\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{([(P_0 \cap E) \cup \{\alpha_{0}\}] \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}} : (\mathbb{Q} / \mathbb{Z})^{([(P_0 \cap E) \cup \{\alpha_{0}\}] \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)} \to \mathbb{T}.$$ - If $\alpha_{0} \in P_1 \setminus (L_1 \setminus \omega)$, we use the fact that $\mathbb{T}$ is a divisible group to extend $\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}$ to a group homomorphism $$\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{((P_1 \cap E) \cup \{\alpha_{0}\})}} : (\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{((P_1 \cap E) \cup \{\alpha_{0}\})} \to \mathbb{T}.$$ - If $\alpha_{0} \in \cup_{n \in D} L_n$, define $$\tilde{\phi}_{(H, J)}(y_{\alpha_{0}}, 0) = p_{\alpha_{0}}-\lim \{\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}} (h_{\alpha_{0}}(n)) : n \in \omega\}$$ and put $$\tilde{\phi}_{(H, J)}(\tilde{H}, \tilde{J}) = \phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}} (\tilde{H}, \tilde{J})$$ for every $(\tilde{H}, \tilde{J}) \in (\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}$. Let $G_0$ be the subgroup of $(\mathbb{Q} / \mathbb{Z})^{([(P_0 \cap E) \cup \{\alpha_0\}] \times \omega)}$ generated by $(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \cup \{y_{\alpha_{0}}\}$. We can extend $\tilde{\phi}_{(H, J)}$ to a group homomorphism from $G_0 \oplus \mathbb{Q}^{(P_1 \cap E)}$ into $\mathbb{T}$. Since $\mathbb{T}$ is a divisible group, extend the group homomorphism $\tilde{\phi}_{(H, J)}$ to a group homomorphism $$\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{([(P_0 \cap E) \cup \{\alpha_{0}\}] \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}} : (\mathbb{Q} / \mathbb{Z})^{([(P_0 \cap E) \cup \{\alpha_{0}\}] \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)} \to \mathbb{T}.$$ - If $\alpha_{0} \in L_1 \setminus \omega$, define $$\tilde{\phi}_{(H, J)}(0, \chi_{\alpha_{0}}) = p_{\alpha_{0}}-\lim \{\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}} (h_{\alpha_{0}}(n)) : n \in \omega\}$$ and put $$\tilde{\phi}_{(H, J)}(\tilde{H}, \tilde{J}) = \phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(\tilde{H}, \tilde{J})$$ for every $(\tilde{H}, \tilde{J}) \in (\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}$. Let $G_0$ be the subgroup of $\mathbb{Q}^{((P_1 \cap E) \cup \{\alpha_{0}\})}$ generated by $\mathbb{Q}^{(P_1 \cap E)} \cup \{\chi_{\alpha_0}\}$. We can extend $\tilde{\phi}_{(H, J)}$ to a group homomorphism from $(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus G_0$ into $\mathbb{T}$. Since $\mathbb{T}$ is a divisible group, extend the group homomorphism $\tilde{\phi}_{(H, J)}$ to a group homomorphism $$\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{((P_1 \cap E) \cup \{\alpha_{0}\})}} : (\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{((P_1 \cap E) \cup \{\alpha_{0}\})} \to \mathbb{T}.$$ Repeating inductively this construction, we will obtain a group homomorphism $\phi_{(H, J)}: (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} \to \mathbb{T}$ satisfying (i), (ii) and (iii). The following lemma is a standard tool to embed the group $G$ into $\mathbb{T}^{\mathfrak{c}}$ and its proof is straightforward. \[cap5\_prop\_imersao\_algebrica\] Suppose that for each $(H, J) \in (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} \setminus \{(0, 0)\}$ there exists a group homomorphism $\phi_{(H, J)}: (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} \to \mathbb{T}$ such that $\phi_{(H, J)}(H, J) \neq 0 + \mathbb{Z}$. The diagonal product $\Phi: (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} \to \mathbb{T}^{(\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} \setminus \{(0, 0)\}}$ given by $$\Phi(\tilde{H}, \tilde{J})(H, J) = \phi_{(H, J)}(\tilde{H}, \tilde{J}), \hspace{0.2cm} \hbox{for every} \hspace{0.2cm} (H, J) \in (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} \setminus \{(0, 0)\}$$ is a group monomorphism. \[cap5\_teo\_construcao\_da\_topologia\] Assume the existence of $\mathfrak{c}$ incomparable selective ultrafilters. If $G$ is a non-torsion Abelian group such that $|G| = \mathfrak{c}$, $|G / T(G)| = \mathfrak{c}$ and, for every $n, d \in \mathbb{N} \setminus \{0\}$ such that $d \mid n$, the group $d G[n]$ is finite or has cardinality $\mathfrak{c}$, then $G$ admits a countably compact group topology without non-trivial convergent sequences. Consider $$\tau = \{\varphi^{-1} \circ \Phi^{-1}(U \cap \Phi[\varphi[G]]) : U \ \hbox{is open in} \ \mathbb{T}^{(\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} \setminus \{(0, 0)\}}\}$$ where $\varphi$ is defined in Section 2 and $\Phi$ is given by Lemma \[cap5\_prop\_imersao\_algebrica\]. We have that $\tau$ is a topology on $G$ which turns it into a Hausdorff topological group. Consider $h: \omega \to G$. If $h$ is trivial, we have nothing to do. Otherwise, $h$ has two constant and distinct subsequences or there exists a strictly increasing function $j: \omega \to \omega$ such that $h \circ j$ is injective. In the first case, it follows that $\{h(n) \in G : n \in \omega\}$ has at least two distinct accumulation points and, therefore, $h$ is not convergent. Suppose that the second case happens. The mappings $h_0, h_1 : \omega \to G$ given by $h_0(n) = (h \circ j)(2n)$ and $h_1(n) = (h \circ j)(2n + 1)$ are distinct subsequences of $h$. According to Proposition \[cap5\_prop\_subsequencias\], there exist $c_0, c_1 \in \varphi[G]$ and strictly increasing functions $j_0, j_1 : \omega \to \omega$ such that $\tilde{h}_0, \tilde{h}_1 \in \mathcal{H}$, where $\tilde{h}_0, \tilde{h}_1 : \omega \to \varphi[G]$ are given by $\tilde{h}_0(n) = \varphi \circ h_0 \circ j_0(n) + c_0$ and $\tilde{h}_1(n) = \varphi \circ h_1 \circ j_1(n) + c_1$, for every $n \in \omega$. So, there exist $\xi_{0}, \xi_{1} \in \tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n$ such that $\tilde{h}_0 = h_{\xi_{0}}$ and $\tilde{h}_1 = h_{\xi_{1}}$. Let $i \in \{0, 1\}$. If $h_{\xi_{i}}$ is of type 1, 2, 3 or 4, then $$\phi_{(H, J)}(0, \chi_{\xi_i}) = p_{\xi_i}-\lim \{\phi_{(H, J)}(h_{\xi_i}(n)) : n \in \omega\}$$ for every $(H, J) \in \setminus (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} \{(0, 0)\}$ and, therefore, $$\Phi(0, \chi_{\xi_i}) = p_{\xi_i}-\lim \{\Phi(h_{\xi_i}(n)) : n \in \omega\}.$$ Thus, $$\varphi^{-1}(0, \chi_{\xi_i}) - \varphi^{-1}(c_i) = p_{\xi_i}-\lim \{(h_i \circ j_i)(n) : n \in \omega\}.$$ If $h_{\xi_{i}}$ is of type 5, then $$\phi_{(H, J)}(y_{\xi_{i}}, 0) = p_{\xi_i}-\lim \{\phi_{(H, J)}(h_{\xi_i}(n)) : n \in \omega\}$$ for every $(H, J) \in \setminus (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} \{(0, 0)\}$ and, therefore, $$\Phi(y_{\xi_{i}}, 0) = p_{\xi_i}-\lim \{\Phi(h_{\xi_i}(n)) : n \in \omega\}.$$ Thus, $$\varphi^{-1}(y_{\xi_{i}}, 0) - \varphi^{-1}(c_i) = p_{\xi_i}-\lim \{(h_i \circ j_i)(n) : n \in \omega\}.$$ So, $\{h(n) \in G : n \in \omega\}$ has two distinct accumulation points. Algebraic structure of countably compact non-torsion Abelian groups of size continuum ===================================================================================== The proofs of the following two propositions can be found in . \[cap5\_prop\_caracterizacao2\] Let $G$ be a non-torsion Abelian group. If $G$ admits a pseudocompact group topology, then $|G / T(G)| \geq \mathfrak{c}$. \[cap5\_cor\_caracterizacao1\] Let $G$ be a non-torsion Abelian group such that $|G| = \mathfrak{c}$. If $G$ admits a countably compact group topology then, for every $n, d \in \mathbb{N} \setminus \{0\}$ with $d \mid n$, the group $d G[n]$ is finite or has cardinality $\mathfrak{c}$. \[cap5\_teo\_caracterizacao\] Assume the existence of $\mathfrak{c}$ incomparable selective ultrafilters. Let $G$ be a non-torsion Abelian group such that $|G| = \mathfrak{c}$. The following conditions are equivalent: (i) $G$ admits a countably compact group topology; (ii) $G$ admits a countably compact group topology without non-trivial convergent sequences; (iii) $|G / T(G)| = \mathfrak{c}$ and, for every $n, d \in \mathbb{N} \setminus \{0\}$ with $d \mid n$, the group $d G[n]$ is finite or has cardinality $\mathfrak{c}$. Clearly, (ii) implies (i). It follows from Propositions \[cap5\_prop\_caracterizacao2\] and \[cap5\_cor\_caracterizacao1\] that (i) implies (iii). Finally, we conclude from Theorem \[cap5\_teo\_construcao\_da\_topologia\] that (iii) implies (ii).[^2] Increasing the weight of the group ================================== \[cap5\_teo\_aumentando\_o\_peso\] Assume the existence of $\mathfrak{c}$ incomparable selective ultrafilters. Let $G$ be a non-torsion Abelian group such that $|G| = \mathfrak{c}$, $|G / T(G)| = \mathfrak{c}$ and, for every $n, d \in \mathbb{N} \setminus \{0\}$ with $d \mid n$, the group $d G[n]$ is finite or has cardinality $\mathfrak{c}$. There exists a countably compact group topology without non-trivial convergent sequences on $G$ whose weight is $2^{\mathfrak{c}}$. Let $\{z_{\xi} : \xi \in P_1 \setminus L_1\}$ be a dense subset of $\mathbb{T}^{[\mathfrak{c}, 2^{\mathfrak{c}}[}$. Denote by $S$ the subgroup of $(\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)}$ generated by $\{(0, \chi_{\xi}) \in (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} : \xi \in P_1 \setminus L_1\}$ and consider $\rho : S \to \mathbb{T}^{[\mathfrak{c}, 2^{\mathfrak{c}}[}$ the group homomorphism given by $\rho(0, \chi_{\xi}) = z_{\xi}$, for every $\xi \in P_1 \setminus L_1$. Since $\mathbb{T}^{[\mathfrak{c}, 2^{\mathfrak{c}}[}$ is a divisible group, it is possible to extend $\rho$ to a group homomorphism $\rho : (\mathbb{Q} / \mathbb{Z})^{((P_0 \setminus \cup_{n \in D} L_n) \times \omega)} \oplus Q^{((P_1 \setminus L_1) \cup \omega)} \to \mathbb{T}^{[\mathfrak{c}, 2^{\mathfrak{c}}[}$. Let $\{\alpha_{\xi} : \xi < \mathfrak{c}\}$ be a strictly increasing enumeration of $\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n$. Define $$z_{\alpha_{0}} = p_{\alpha_{0}}-\lim \{\rho(h_{\alpha_{0}}(n)) : n \in \omega\}$$ and put $\rho(0, \chi_{\alpha_{0}}) = z_{\alpha_{0}}$, if $\alpha_{0} \in L_1 \setminus \omega$ and $\rho(y_{\alpha_{0}}, 0) = z_{\alpha_{0}}$, if $\alpha_{0} \in \cup_{n \in D} L_n$. Repeating inductively this construction, we obtain $\rho : (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} \to \mathbb{T}^{[\mathfrak{c}, 2^{\mathfrak{c}}[}$ a group homomorphism $\rho(0, \chi_{\xi}) = p_{\xi}-\lim \{\rho(h_{\xi}(n)) : n \in \omega\}$, if $\xi \in L_1 \setminus \omega$ and $\rho(y_{\xi}, 0) = p_{\xi}-\lim \{\rho(h_{\xi}(n)) : n \in \omega\}$, if $\xi \in \cup_{n \in D} L_n$. Let $\tilde{G} \subset \mathbb{T}^{2^{\mathfrak{c}}}$ be given by $\tilde{G} = \{(\Phi(\varphi(x)), \rho(\varphi(x))) \in \mathbb{T}^{\mathfrak{c}} \times \mathbb{T}^{[\mathfrak{c}, 2^{\mathfrak{c}}[} : x \in G\}$ where $\varphi$ and $\Phi$ were defined in Proposition \[cap5\_prop\_varphi\] and in Lemma \[cap5\_prop\_imersao\_algebrica\], respectively. It follows that $G$ is isomorphic to $\tilde{G}$ and $\tilde{G}$, endowed with the subspace topology induced by $\mathbb{T}^{2^{\mathfrak{c}}}$ is a countably compact group without non-trivial convergent sequences whose weight is $2^{\mathfrak{c}}$. On the number of countably compact group topologies on a non-torsion Abelian group ================================================================================== If $X$ is a topological space and if $x \in X$ is an accumulation point of $\{x_n : n \in \omega\} \subset X$, we define $$\mathcal{F}(X, \{x_n : n \in \omega\}, x) = \{A \subset \omega : \exists U \ \hbox{an open neighbor of $x$ such that} \ \{n \in \omega : x_n \in U\} \subset A\}.$$ It is not difficult to realize that $\mathcal{F}(X, \{x_n : n \in \omega\}, x)$ is a filter over $\omega$. We denote by $\mathcal{F}(X)$ the family consisting of all filters $\mathcal{F}(X, \{x_n : n \in \omega\}, x)$, where $X$ is a fixed topological space and $x \in X$ is an accumulation point of $\{x_n : n \in \omega\} \subset X$. Observe that if $|X| = \mathfrak{c}$, then $|\mathcal{F}(X)| \leq \mathfrak{c}$. The proof of the following lemma is straightforward. \[cap6\_lem\_espacos\_homeomorfos\_tem\_mesmo\_filtro\] Let $X$ and $Y$ be topological spaces and let $h: X \to Y$ be an homeomorphism. If $x \in X$ is an accumulation point of $\{x_n : n \in \omega\} \subset X$, then $h(x) \in Y$ is an accumulation point of $\{h(x_n) : n \in \omega\} \subset Y$ and $\mathcal{F}(X, \{x_n : n \in \omega\}, x) = \mathcal{F}(Y, \{h(x_n) : n \in \omega\}, h(x))$. \[cap6\_cor\_espacos\_homeomorfos\_tem\_mesmo\_filtro\] If $X$ and $Y$ are homeomorphic, then $\mathcal{F}(X) = \mathcal{F}(Y)$. Fix $\kappa < 2^{\mathfrak{c}}$ a cardinal and consider $\{(X_{\alpha}, \tau_{\alpha}) : \alpha < \kappa\}$ a family of spaces such that $|X_{\alpha}| = \mathfrak{c}$, for every $\alpha < \kappa$. We shall show that it is possible to endow $G$ with a countably compact group topology $\tau$ such that $(G, \tau)$ is not homeomorphic to $(X_{\alpha}, \tau_{\alpha})$, for every $\alpha < \kappa$. Since we are assuming the existence of $2^{\mathfrak{c}}$ selective ultrafilters, it is possible to fix $\{p_{\xi} : \omega \leq \xi < \mathfrak{c}\} \subset \omega^{*}$ a family of pairwise incomparable selective ultrafilters such that $p_{\omega} \not \in \cup_{\alpha < \kappa} \mathcal{F}(X_{\alpha})$. \[cap6\_lem\_homomorfismos\] Fix $A \in p_{\omega}$. There exists a group homomorphism $\phi_{A} : (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} \to \mathbb{T}$ such that: (i) $\phi_{A}(0, \chi_{\xi}) = p_{\xi}-\lim \{\phi_{A}(h_{\xi}(n)) : n \in \omega\}$, for every $\xi \in L_1 \setminus \omega$; (ii) $\phi_{A}(y_{\xi}, 0) = p_{\xi}-\lim \{\phi_{A}(h_{\xi}(n)) : n \in \omega\}$, for every $\xi \in \bigcup_{n \in D} L_n \setminus \omega$; (iii) $\{n \in \omega : \phi_{A}(h_{\omega}(n)) \in \Omega_{A}\} = A$, for some neighbor $\Omega_{A}$ of $\phi_{A}(0, \chi_{\omega})$ in $\mathbb{T}$. For each $n \in \omega$, consider $$\phi(0, \chi_{n}) =\left\{ \begin{array}{ccc} 0 + \mathbb{Z} & \hbox{if} & n \not \in A \\ \frac{1}{2} + \mathbb{Z} & \hbox{if} & n \in A. \\ \end{array} \right.$$ Put, also, $$\textstyle \phi(0, \chi_{\omega}) = \frac{1}{2} + \mathbb{Z}.$$ It follows that $\phi(0, \chi_{\omega}) = p_{\omega}-\lim \{\phi(h_{\omega}(n)) : n \in \omega\}$ and that $\{n \in \omega : \phi_{A}(h_{\omega}(n)) \in \Omega_{A}\} = A$, where $\Omega_{A}$ is the open arc of $\mathbb{T}$ centered in $\frac{1}{2} + \mathbb{Z}$ with diameter $\frac{1}{4}$. Extend $\phi$ to a group homomorphism from $\{0_{(\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)}}\} \oplus \mathbb{Z}^{(\omega + 1)}$ to $\mathbb{T}$ and use the fact that $\mathbb{T}$ is a divisible group in order to extend $\phi$ to a group homomorphism $\phi_{A} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((\omega + \omega \setminus \omega + 1) \times \omega)} \oplus \mathbb{Q}^{(\omega + 1)}} : (\mathbb{Q} / \mathbb{Z})^{((\omega + \omega \setminus \omega + 1) \times \omega)} \oplus \mathbb{Q}^{(\omega + 1)} \to \mathbb{T}$. Fix a strictly increasing enumeration of $\mathfrak{c} \setminus (\omega + \omega)$ and mimic the proof of Lemma \[cap5\_lem\_extensao\_dos\_homomorfismos\] to obtain a group homomorphism $\phi_{A} : (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} \to \mathbb{T}$ satisfying conditions (i), (ii) and (iii) above. The mapping $$\begin{array}{cccl} \hat{\Phi}: & (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} & \to & \mathbb{T}^{p_{\omega}} \\ & (\tilde{H}, \tilde{J}) & \mapsto & \hat{\Phi}(\tilde{H}, \tilde{J}) \end{array}$$ given by $$\hat{\Phi}(\tilde{H}, \tilde{J})(A) = \phi_{A}(\tilde{H}, \tilde{J}), \hspace{0.2cm} \hbox{for each} \hspace{0.2cm} A \in p_{\omega}$$ is a group homomorphism. Thus, $$\begin{array}{cccl} \Psi : & (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} & \to & \mathbb{T}^{(\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} \setminus \{(0, 0)\}} \oplus \mathbb{T}^{p_{\omega}} \\ & (\tilde{H}, \tilde{J}) & \mapsto & (\Phi(\tilde{H}, \tilde{J}), \hat{\Phi}(\tilde{H}, \tilde{J})) \end{array}$$ is a group monomorphism, where $\Phi$ is defined in Lemma \[cap5\_prop\_imersao\_algebrica\]. We have that $$\tau = \{\varphi^{-1} \circ \Psi^{-1}(U \cap \Psi[\varphi[G]]) : U \ \hbox{is open in} \ \mathbb{T}^{(\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} \setminus \{(0, 0)\}} \oplus \mathbb{T}^{p_{\omega}}\}$$ is a group topology on $G$. Moreover, $$\varphi^{-1}(0, \chi_{\xi}) = p_{\xi}-\lim \{\varphi^{-1}(h_{\xi}(n)) : n \in \omega\}, \ \hbox{if} \ \xi \in L_1 \setminus \omega$$ and $$\varphi^{-1}(y_{\xi}, 0) = p_{\xi}-\lim \{\varphi^{-1}(h_{\xi}(n)) : n \in \omega\}, \ \hbox{if} \ \xi \in \cup_{n \in D} L_n \setminus \omega.$$ In particular, we have that $$\mathcal{F}(G, \{\varphi^{-1}(0, \chi_{n}) : n \in \omega\}, \varphi^{-1}(0, \chi_{\omega})) \subset p_{\omega}.$$ In fact, if $A \in \mathcal{F}(G, \{\varphi^{-1}(0, \chi_{n}) : n \in \omega\}, \varphi^{-1}(0, \chi_{\omega}))$, then there exists $U$ an open neighbor of $\varphi^{-1}(0, \chi_{\omega})$ in $G$ such that $\{n \in \omega : \varphi^{-1}(h_{\omega}(n)) \in U\} \subset A$. Since $\varphi^{-1}(0, \chi_{\omega}) = p_{\omega}-\lim \{\varphi^{-1}(h_{\omega}(n)) : n \in \omega\}$, it follows that $\{n \in \omega : \varphi^{-1}(h_{\omega}(n)) \in U\} \in p_{\omega}$ and, therefore, $A \in p_{\omega}$. Also, observe that $$p_{\omega} \subset \mathcal{F}(G, \{\varphi^{-1}(0, \chi_{n}) : n \in \omega\}, \varphi^{-1}(0, \chi_{\omega})).$$ In fact, let $A \in p_{\omega}$ and $\Omega \subset \mathbb{T}^{(\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} \setminus \{(0, 0)\}} \oplus \mathbb{T}^{p_{\omega}}$ be such that $\operatorname{proj}_{(H, J)}(\Omega) = \mathbb{T}$ for every $(H, J) \in (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} \setminus \{(0, 0)\}$, $\operatorname{proj}_{\tilde{A}}(\Omega) = \mathbb{T}$ for every $\tilde{A} \in p_{\omega} \setminus \{A\}$ and $\operatorname{proj}_{A}(\Omega) = \Omega_{A}$. It follows that $\Psi(0, \chi_{\omega}) \in \Omega$ and $\{n \in \omega : \varphi^{-1}(h_{\omega}(n)) \in \varphi^{-1} \circ \Psi^{-1}(\Omega)\} = \{n \in \omega : \Psi(h_{\omega}(n)) \in \Omega\} = \{n \in \omega : \Psi(h_{\omega}(n))(A) \in \operatorname{proj}_{A}(\Omega)\} = \{n \in \omega : \phi_{A}(h_{\omega}(n)) \in \Omega_{A}\} = A$. Therefore, $A \in \mathcal{F}(G, \{\varphi^{-1}(0, \chi_{n}) : n \in \omega\}, \varphi^{-1}(0, \chi_{\omega}))$. Thus, $\mathcal{F}(G, \{\varphi^{-1}(0, \chi_{n}) : n \in \omega\}, \varphi^{-1}(0, \chi_{\omega})) = p_{\omega} \not \in \cup_{\alpha < \kappa} \mathcal{F}(X_{\alpha})$, which implies that $(G, \tau)$ is not homeomorphic to $(X_{\alpha}, \tau_{\alpha})$, for every $\alpha < \kappa$. \[cap6\_teo\_existencia\_da\_topologia\] Assume the existence of $2^{\mathfrak{c}}$ selective ultrafilters. If $G$ is a non-torsion Abelian group of cardinality $\mathfrak{c}$ that admits a countably compact group topology, then $G$ admits $2^{\mathfrak{c}}$ countably compact group topologies (pairwise non-homeomorphic). Appendix ======== The following lemma gives a combinatorial property for incomparable selective ultrafilters that will be used to prove Lemma \[cap5\_lem\_12\_itens\]. Its proof can be found in [@tomita3]. \[lem\_ultrafiltros\_seletivos\] Let $\{p_j : j \in \omega\}$ be a family of pairwise incomparable selective ultrafilters. For each $j \in \omega$, let $\{a_{k}^{j} : k \in \omega\} \in p_j$ be an increasing sequence of natural numbers such that $k < a_{k}^{j}$, for every $k \in \omega$. There exists a family $\{I_j : j \in \omega\}$ of pairwise disjoint subsets of $\omega$ such that: 1. $\{a_{k}^{j} : k \in I_j\} \in p_j$, for every $j \in \omega$; 2. $\{[k, a_{k}^{j}] : j \in \omega, k \in I_j\}$ is a family of pairwise disjoint intervals of $\omega$. We recall that $\{p_{\xi} : \xi \in \tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n\}$ is a family of incomparable selective ultrafilters and that $\{h_{\xi} : \xi \in \tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n\}$ is an enumeration of $\mathcal{H}$ satisfying conditions (i), (ii), (iii) and (iv) of Proposition \[cap5\_prop\_indexacao\_das\_sequencias\]. \[cap5\_lem\_12\_itens\] Fix $(H, J) \in (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} \setminus \{(0, 0)\}$ and $E \in [\mathfrak{c}]^{\omega}$ satisfying condtions (i), (ii) and (iii) of Proposition \[cap5\_prop\_construcao\_de\_E\]. There exists a family $\{E_k : k \in \omega\}$ of finite subsets of $E$, strictly increasing sequences $\{e_k : k \in \omega\}$ and $\{b_k : k \in \omega\}$ of natural numbers, a sequence $\{r_k : k \in \omega\}$ of positive real numbers and a function $i: \omega \to E \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)$ such that: (i) $\operatorname{supp}(H, J) \subset (E_0 \times e_0) \cup E_0$; (ii) $E = \cup_{k \in \omega} E_k$; (iii) $(E_{k + 1} \times e_{k + 1}) \cup E_{k + 1} \supset (E_k \times e_k) \cup E_k \cup \bigcup \{\operatorname{supp}h_{i(m)}(b_m) : m \leq k\}$, for every $k \in \omega$; (iv) $i(k) \in E_k$, for every $k \in \omega$; (v) $\{b_k : k \in i^{-1}(\{\xi\})\} \in p_{\xi}$, for every $\xi \in E \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)$; (vi) $\operatorname{supp}f_{i(k)}(b_k) \setminus E_k \neq \emptyset$, if $h_{i(k)}$ is of type 1; (vii) $|q(f_{i(k)}(b_k))| \cdot r_k > d(J) \cdot \prod_{m < k} d(f_{i(m)}(b_m))$, if $h_{i(k)}$ is of type 2; (viii) $|a(f_{i(k)}(b_k))| \cdot r_k > 4 \cdot d(f_{i(k)}(b_k))$, if $h_{i(k)}$ is of type 3; (ix) $|q(g_{i(k)}(b_k))| \cdot r_k > d(H) \cdot \prod_{m < k} d(g_{i(m)}(b_m))$, if $h_{i(k)}$ is of type 4; (x) $\operatorname{supp}g_{i(k)}(b_k) \setminus (E_k \times e_k) \neq \emptyset$ and $\operatorname{o}(g_{i(k)}(b_k)) = \operatorname{o}(g_{i(k)}(b_k) \upharpoonright_{\operatorname{supp}g_{i(k)}(b_k) \setminus (E_k \times e_k)})$, if $h_{i(k)}$ is of type 5; (xi) $r_0 = \dfrac{1}{4 \cdot ||a(H, J)||}$; (xii) $r_{k + 1} = \dfrac{r_k}{2 \cdot ||a(h_{i(k)}(b_k))||}$, for every $k \in \omega$. Let $\{a_n : n \in \omega\}$ be an enumeration of $E$. Put $$F_0 = \pi_{1}[\operatorname{supp}H] \cup \operatorname{supp}J \cup \{a_0\}$$ and fix $j_0 \in \omega$ such that $j_0 > \max \pi_{2}[\operatorname{supp}H]$. For every $n \in \omega$, put $$\textstyle F_{n + 1} = F_n \cup \bigcup \{\pi_{1}[\operatorname{supp}g_{\xi}(m)] \cup \operatorname{supp}f_{\xi}(m) : m \leq n, \xi \in F_n \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)\} \cup \{a_{n + 1}\}$$ and fix $j_{n + 1} \in \omega$ such that $j_{n + 1} > j_n$ and $$\textstyle j_{n + 1} > \max \pi_{2}[\bigcup \{\operatorname{supp}g_{\xi}(n) : m \leq n, \xi \in F_n \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)\}].$$ Then $\{F_n : n \in \omega\}$ is a family of finite subsets of $E$ and $\{j_n : n \in \omega\}$ is a strictly increasing sequence of natural numbers such that: (1) $\operatorname{supp}(H, J) \subset (F_0 \times j_0) \cup F_0$; (2) $E = \cup_{n \in \omega} F_n$; (3) $(F_{n + 1} \times j_{n + 1}) \cup F_{n + 1} \supset (F_n \times j_n) \cup F_n \cup \bigcup \{\operatorname{supp}h_{\xi}(m) : m \leq n, \xi \in F_n \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)\}$. Consider $\xi \in E \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)$ and $n \in \omega$. If $h_{\xi}$ is of type 1, put $$A_{n}^{\xi} = \{k \in \omega : \operatorname{supp}f_{\xi}(k) \setminus F_n \neq \emptyset\}.$$ If $h_{\xi}$ is of type 2, put $$A_{n}^{\xi} = \{k \in \omega : |q(f_{\xi}(k))| > 2^{n + 2} \cdot X_{n - 1}\}$$ where $$X_m = ||a(H, J)|| \cdot d(J) \cdot \prod_{\substack{l \leq m \\ \zeta \in F_m \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)}} d(f_{\zeta}(l)) \cdot \prod_{\substack{l \leq m \\ \zeta \in F_m \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)}} ||a(h_{\zeta}(l))||$$ for every $m \in \omega$ and $$X_{-1} = ||a(H, J)|| \cdot d(J).$$ If $h_{\xi}$ is of type 3, then $|a(f_{\xi}(n))| > n$ for every $n \in \omega$ and $\{|q(f_{\xi}(n))| : n \in \omega\}$ is bounded. Choose $M_{\xi} \in \omega$ such that $|q(f_{\xi}(n))| \leq M_{\xi}$ for every $n \in \omega$ and put $$A_{n}^{\xi} = \{k \in \omega : |a(f_{\xi}(k))| > 2^{n + 4} \cdot M_{\xi}! \cdot Y_{n - 1} \}$$ where $$Y_{m} = ||a(H, J)|| \cdot \prod_{\substack{l \leq m \\ \zeta \in F_m \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)}} ||a(h_{\zeta}(l))||$$ for every $m \in \omega$ and $$Y_{-1} = ||a(H, J)||.$$ If $h_{\xi}$ is of type 4, put $$A_{n}^{\xi} = \{k \in \omega : |q(g_{\xi}(k))| > 2^{n + 2} \cdot Z_{n - 1}\}$$ where $$Z_m = ||a(H, J)|| \cdot d(H) \cdot \prod_{\substack{l \leq m \\ \zeta \in F_m \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)}} d(g_{\zeta}(l)) \cdot \prod_{\substack{l \leq m \\ \zeta \in F_m \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)}} ||a(h_{\zeta}(l))||$$ for every $m \in \omega$ and $$Z_{-1} = ||a(H, J)|| \cdot d(H).$$ If $h_{\xi}$ is of type 5, put $$A_{n}^{\xi} = \{k \in \omega : \operatorname{supp}g_{\xi}(k) \setminus (F_n \times j_n) \neq \emptyset\} \cap \{k \in \omega : \operatorname{o}(g_{\xi}(k)\upharpoonright_{\operatorname{supp}g_{\xi}(k) \setminus (F_n \times j_n)}) = \operatorname{o}(g_{\xi}(k))\}.$$ Note that $A_{n}^{\xi}$ is a cofinite subset of $\omega$, for all $n \in \omega$ and $\xi \in [E \cap (L_1 \cup \bigcup_{n \in D} L_n)] \setminus \omega$. Hence, for every $\xi \in [E \cap (L_1 \cup \bigcup_{n \in D} L_n)] \setminus \omega$, we have $\{A_{n}^{\xi} : n \in \omega\} \subset p_{\xi}$ since $p_{\xi}$ is a free ultrafilter over $\omega$. It follows from the selectivity of $p_{\xi}$ that there exists a sequence $\{a_{n}^{\xi} : n \in \omega\} \in p_{\xi}$ such that $a_{n}^{\xi} \in A_{n}^{\xi}$ and $n < a_{n}^{\xi}$, for every $n \in \omega$. Applying Lemma \[lem\_ultrafiltros\_seletivos\], we conclude that there exists a family $\{I_{\xi} : \xi \in [E \cap (L_1 \cup \bigcup_{n \in D} L_n)] \setminus \omega\}$ of pairwise disjoint subsets of $\omega$ such that: (a) $\{a_{n}^{\xi} : n \in I_{\xi}\} \in p_{\xi}$, for every $\xi \in [E \cap (L_1 \cup \bigcup_{n \in D} L_n)] \setminus \omega$; (b) $\{[n, a_{n}^{\xi}] : n \in I_{\xi}, \ \xi \in [E \cap (L_1 \cup \bigcup_{n \in D} L_n)] \setminus \omega\}$ is a family of pairwise disjoint intervals of $\omega$. For each $\xi \in [E \cap (L_1 \cup \bigcup_{n \in D} L_n)] \setminus \omega$, let (a) $N_{\xi} = \min \{n \in \omega : \xi \in F_n\}$. We can suppose, without loss of generality, that (a) $N_{\xi} < n$, for every $n \in I_{\xi}$. Let $\{n_k : k \in \omega\}$ be a strictly increasing enumeration of $\dot{\cup}_{\xi \in E \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)} I_{\xi}$. Consider $i: \omega \to E \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)$ where $i(k)$ is the unique element of $E \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)$ such that $n_k \in I_{i(k)}$ for each $k \in \omega$. Define $b_k = a_{n_k}^{i(k)}$, $E_k = F_{n_k}$ and $e_k = j_{n_{k}}$, for every $k \in \omega$. For each $k \in \omega$, define also $r_k$ according to (xi) and (xii). It remains to show that (i)-(x) are satisfied. (i) Since $0 \leq n_0$, we have $E_0 = F_{n_0} \supset F_0 \supset \pi_{1}[\operatorname{supp}H] \cup \operatorname{supp}J$ and $e_0 = j_{n_0} > j_0 > \max \pi_{2}[\operatorname{supp}H]$. Thus, $\operatorname{supp}(H, J) \subset (E_0 \times e_0) \cup E_0$. (ii) $\cup_{k \in \omega} E_k = \cup_{k \in \omega} F_{n_k} = E$. (iii) If $k \in \omega$, then $n_k \in I_{i(k)}$. It follows from (d) that $N_{i(k)} < n_k$. From (c) and (3) we conclude that $i(k) \in F_{N_{i(k)}} \subset F_{n_k} = E_k$. (iv) It follows from (3) that $E_k \subset E_{k + 1}$ and $(E_k \times e_k) \subset (E_{k + 1} \times e_{k + 1})$, since $\{n_k : k \in \omega\}$ is a strictly increasing enumeration of $\cup_{\xi \in E \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)} I_{\xi}$. It remains to show that $\operatorname{supp}h_{i(m)}(b_m) \subset (E_{k + 1} \times e_{k + 1}) \cup E_{k + 1}$, for every $m \leq k$. We have that $(E_{k + 1} \times e_{k + 1}) \cup E_{k + 1} = (F_{n_{k + 1}} \times j_{n_{k + 1}}) \cup F_{n_{k + 1}}$ and from (3) again, we conclude that $$\textstyle (F_{n_{k + 1}} \times j_{n_{k + 1}}) \cup F_{n_{k + 1}} \supset \bigcup \{\operatorname{supp}h_{\xi}(m) : m \leq n_{k + 1} - 1, \xi \in F_{n_{k + 1} - 1} \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)\}.$$ By (iv), $i(m) \in E_m \subset E_k = F_{n_k} \subset F_{n_{k + 1} - 1}$, for every $m \leq k$. Besides, $i(m) \in (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)$. By (b), $b_m = a_{n_m}^{i(m)} \leq n_{k + 1} - 1$, for every $m \leq k$. Hence, $\operatorname{supp}h_{i(m)}(b_m) \subset (E_{k + 1} \times e_{k + 1}) \cup E_{k + 1}$, for every $m \leq k$. (v) Let $\xi \in E \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)$. We have that $\{b_k : k \in i^{-1}(\{\xi\})\} = \{a_{n_k}^{i(k)} : k \in i^{-1}(\{\xi\})\} = \{a_{n}^{\xi} : n \in I_{\xi}\} \in p_{\xi}$, by (a). (vi) Suppose that $h_{i(k)}$ is of type 1. Since $b_k = a_{n_k}^{i(k)} \in A_{n_k}^{i(k)}$, we conclude that $$\operatorname{supp}f_{i(k)}(b_k) \setminus E_k = \operatorname{supp}f_{i(k)}(b_k) \setminus F_{n_k} \neq \emptyset.$$ (vii) Suppose that $h_{i(k)}$ is of type 2. Since $b_k = a_{n_k}^{i(k)} \in A_{n_k}^{i(k)}$, we conclude that $$|q(f_{i(k)}(b_k))| > 2^{n_k + 2} \cdot X_{n_k - 1}$$ where $$\hspace*{-0.75cm} X_{n_k - 1} = ||a(H, J)|| \cdot d(J) \cdot \prod_{\substack{m \leq n_k - 1 \\ \xi \in F_{n_k - 1} \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)}} d(f_{\xi}(m)) \cdot \prod_{\substack{m \leq n_k - 1 \\ \xi \in F_{n_k - 1} \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n)}} ||a(h_{\xi}(m))||.$$ Note that if $m < k$, then $n_m < n_{k - 1}$ and, therefore, $i(m) \in E_m = F_{n_{m}} \subset F_{n_{k - 1}} \subset F_{n_{k} - 1}$. Since $b_m = a_{n_{m}}^{i(m)}$ and $[n_m, a_{n_{m}}^{i(m)}] \cap [n_k, a_{n_{k}}^{i(k)}] = \emptyset$, we have that $b_m < n_k$ and, therefore, $b_m \leq n_{k} - 1$. Since $$r_k = \frac{1}{2^{k + 2} \cdot ||a(H, J)|| \cdot \prod_{m < k} ||a(h_{i(m)}(b_m))||}$$ it follows that $$|q(f_{i(k)}(b_k))| \cdot r_k > d(J) \cdot \prod_{m < k} d(f_{i(m)}(b_m)).$$ (viii) Suppose that $h_{i(k)}$ is of type 3. From an argument similar to the one presented in (vii) we conclude that $$|a(f_{i(k)}(b_k))| \cdot r_k > 4 \cdot M_{i(k)}! \geq 4 \cdot d(f_{i(k)}(b_k)).$$ (ix) Suppose that $h_{i(k)}$ is of type 4. From an argument similar to the one presented in (vii) we conclude that $$|q(g_{i(k)}(b_k))| \cdot r_k > d(H) \cdot \prod_{m < k} d(g_{i(m)}(b_m)).$$ (x) Suppose that $h_{i(k)}$ is of type 5. Since $b_k = a_{n_k}^{i(k)} \in A_{n_k}^{i(k)}$, we conclude that $$\operatorname{supp}g_{i(k)}(b_k) \setminus (E_k \times e_k) = \operatorname{supp}g_{i(k)}(b_k) \setminus (F_{n_k} \times j_{n_k}) \neq \emptyset$$ and $$\operatorname{o}(g_{i(k)}(b_k)\upharpoonright_{\operatorname{supp}g_{i(k)}(b_k) \setminus (E_k \times e_k)}) = \operatorname{o}(g_{i(k)}(b_k)\upharpoonright_{\operatorname{supp}g_{i(k)}(b_k) \setminus (F_{n_k} \times j_{n_k})}) = \operatorname{o}(g_{i(k)}(b_k)). \qedhere$$ The next four lemmas are the technical part relative to the types 1, 2, 3 and 4 respectively and will be used in the successor step of the induction presented in the proof of Lemma \[cap5\_lem\_construcao\_dos\_homomorfismos\]. Their proofs can be found in . The set of all non-empty open arcs of $\mathbb{T}$ (including $\mathbb{T}$ itself) will be denoted by $\mathcal{B}$. \[lem\_tipo1\] Consider $I \in [\mathfrak{c}]^{\mathfrak{c}}$, $c \in \mathbb{N} \setminus \{0\}$, $\epsilon > 0$, $A \in \mathcal{B}$, $F \in [I]^{< \omega}$, $J \in \mathbb{Q}^{(I)} \setminus \{0\}$, $\mu \in \operatorname{supp}J \setminus F$ and $\psi : F \to \mathcal{B}$ with $\delta(\psi(\xi)) \geq \epsilon / c$, for every $\xi \in F$. Consider also $\tilde{F} = F \cup \operatorname{supp}J$. There exists $\tilde{\psi} : \tilde{F} \to \mathcal{B}$ satisfying the following conditions: (i) $d(J) \cdot \overline{\tilde{\psi}(\xi)} \subset \psi(\xi)$, for every $\xi \in F$; (ii) $\delta(\tilde{\psi}(\xi)) = \dfrac{\epsilon}{2 \cdot \sum_{\xi \in \operatorname{supp}J} |a(J, \xi)| \cdot c \cdot d(J)}$, for every $\xi \in \tilde{F}$; (iii) $\delta(\sum_{\xi \in \operatorname{supp}J} a(J, \xi) \cdot c \cdot \tilde{\psi}(\xi)) < \epsilon$; (iv) $A \cap \sum_{\xi \in \operatorname{supp}J} a(J, \xi) \cdot c \cdot \tilde{\psi}(\xi) \neq \emptyset$. \[lem\_tipo2\] Consider $I \in [\mathfrak{c}]^{\mathfrak{c}}$, $c \in \mathbb{N} \setminus \{0\}$, $\epsilon > 0$, $A \in \mathcal{B}$ with $\delta(A) \geq \epsilon$, $F \in [I]^{< \omega}$, $J \in \mathbb{Q}^{(I)} \setminus \{0\}$, $\mu \in F \cap \operatorname{supp}J$ with $q(J, \mu) \cdot \epsilon > c$ and $\psi : F \to \mathcal{B}$ with $\delta(\psi(\xi)) \geq \epsilon / c$, for every $\xi \in F$. Consider also $\tilde{F} = F \cup \operatorname{supp}J$. There exists $\tilde{\psi} : \tilde{F} \to \mathcal{B}$ satisfying the following conditions: (i) $d(J) \cdot \overline{\tilde{\psi}(\xi)} \subset \psi(\xi)$, for every $\xi \in F$; (ii) $\delta(\tilde{\psi}(\xi)) = \dfrac{\epsilon}{2 \cdot \sum_{\xi \in \operatorname{supp}J} |a(J, \xi)| \cdot c \cdot d(J)}$, for every $\xi \in \tilde{F}$; (iii) $\delta(\sum_{\xi \in \operatorname{supp}J} a(J, \xi) \cdot c \cdot \tilde{\psi}(\xi)) < \epsilon$; (iv) $A \cap \sum_{\xi \in \operatorname{supp}J} a(J, \xi) \cdot c \cdot \tilde{\psi}(\xi) \neq \emptyset$. \[lem\_tipo3\] Consider $I \in [\mathfrak{c}]^{\mathfrak{c}}$, $c \in \mathbb{N} \setminus \{0\}$, $\epsilon > 0$, $A \in \mathcal{B}$, $F \in [I]^{< \omega}$, $J \in \mathbb{Q}^{(I)} \setminus \{0\}$, $\mu \in F \cap \operatorname{supp}J$ with $|a(J, \mu)| \cdot \epsilon > 4 \cdot d(J)$ and $\psi : F \to \mathcal{B}$ with $\delta(\psi(\xi)) \geq \epsilon / c$, for every $\xi \in F$. Consider also $\tilde{F} = F \cup \operatorname{supp}J$. There exists $\tilde{\psi} : \tilde{F} \to \mathcal{B}$ satisfying the following conditions: (i) $d(J) \cdot \overline{\tilde{\psi}(\xi)} \subset \psi(\xi)$, for every $\xi \in F$; (ii) $\delta(\tilde{\psi}(\xi)) = \dfrac{\epsilon}{2 \cdot \sum_{\xi \in \operatorname{supp}J} |a(J, \xi)| \cdot c \cdot d(J)}$, for every $\xi \in \tilde{F}$; (iii) $\delta(\sum_{\xi \in \operatorname{supp}J} a(J, \xi) \cdot c \cdot \tilde{\psi}(\xi)) < \epsilon$; (iv) $A \cap \sum_{\xi \in \operatorname{supp}J} a(J, \xi) \cdot c \cdot \tilde{\psi}(\xi) \neq \emptyset$. \[lem\_tipo4\] Consider $I$ an infinite set, $c \in \mathbb{N} \setminus \{0\}$, $\epsilon > 0$, $A \in \mathcal{B}$ with $\delta(A) \geq \epsilon$, $H \in (\mathbb{Q} / \mathbb{Z})^{(I)} \setminus \{0\}$, $n \in \operatorname{supp}H$ with $q(H, n) \cdot \epsilon > c$ and $x \in \mathbb{T}$. There exists $y \in \mathbb{T}$ such that $d(H) \cdot y = x$ and $a(H, n) \cdot c \cdot y \in A$. We now restate and prove Lemma \[cap5\_lem\_construcao\_dos\_homomorfismos\]. Fix $(H, J) \in (\mathbb{Q} / \mathbb{Z})^{(P_0 \times \omega)} \oplus \mathbb{Q}^{(P_1)} \setminus \{(0, 0)\}$ and $E \in [\mathfrak{c}]^{\omega}$ satisfying conditions (i), (ii) and (iii) of Proposition \[cap5\_prop\_construcao\_de\_E\]. There exists a group homomorphism $\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}} : (\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)} \to \mathbb{T}$ with the following properties: (i) $\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(H, J) \neq 0 + \mathbb{Z}$; (ii) $\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(0, \chi_{\xi}) = p_{\xi}-\lim \{\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(h_{\xi}(n)) : n \in \omega\}$, for every $\xi \in (E \cap L_1) \setminus \omega$; (iii) $\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(y_{\xi}, 0) = p_{\xi}-\lim \{\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(h_{\xi}(n)) : n \in \omega\}$, for every $\xi \in (E \cap \bigcup_{n \in D} L_n) \setminus \omega$. Consider $\{E_n : n \in \omega\}$, $\{e_n : n \in \omega\}$, $\{b_n : n \in \omega\}$, $\{r_n : n \in \omega\}$ and $i: \omega \to E \cap (\tilde{L}_1 \cup \bigcup_{n \in D} \tilde{L}_n) \setminus \omega$ according to Lemma \[cap5\_lem\_12\_itens\]. If $H = 0$, put $\hat{\phi}(H) = 0 + \mathbb{Z}$. If $H \neq 0$, let $\hat{\phi}(H)$ be a non-zero element of $\mathbb{T}$ such that $\operatorname{o}(\hat{\phi}(H)) \mid \operatorname{o}(H)$. In both cases, extend $\hat{\phi}$ to a group homomorphism from $\langle H \rangle$ into $\mathbb{T}$. Given $(\zeta, m) \in (P_0 \cap E) \times \omega$ and $k \in \mathbb{N} \setminus \{0\}$, define $\Lambda_{(\zeta, m), k} : (P_0 \cap E) \times \omega \to \mathbb{Q} / \mathbb{Z}$ by $$\Lambda_{(\zeta, m), k}(\mu, l) = \left\{\begin{array}{rcl} \dfrac{1}{k} + \mathbb{Z} & \hbox{if} & (\mu, l) = (\zeta, m) \\ \\ 0 + \mathbb{Z} & \hbox{if} & (\mu, l) \neq (\zeta, m). \end{array} \right.$$ Let $$G_0 = \langle \{\Lambda_{(\zeta, m), c_0} : (\zeta, m) \in E_0 \times e_0\} \rangle$$ where $$c_0 = \left\{ \begin{array}{lll} c_{-1} & \hbox{if} & i(0) \not \in \cup_{n \in D} L_n \\ c_{-1} \cdot d(y_{i(0)}) & \hbox{if} & i(0) \in \cup_{n \in D} L_n \end{array}\right.$$ and $$c_{-1} = d(H).$$ Since $\mathbb{T}$ is a divisible group and $\langle H \rangle$ is a subgroup of $G_0$, it is possible to extend $\hat{\phi}$ to a group homomorphism $\phi \upharpoonright_{G_0} : G_0 \to \mathbb{T}$. If $J = 0$, let $\psi_{0}^{*}(\xi)$ be an open arc of $\mathbb{T}$ with diameter $\delta(\psi_{0}^{*}(\xi)) = r_0 / d(J)$ and put $\psi_{0}(\xi) = D(J) \cdot \psi_{0}^{*}(\xi)$, for every $\xi \in E_0 \cap P_1$. Suppose that $J \neq 0$. For each $\xi \in E_0 \cap P_1$, choose $s_{\xi} \in \mathbb{R}$ such that $$\sum_{\xi \in \operatorname{supp}J} J(\xi) \cdot s_{\xi} = \dfrac{1}{2} - x$$ where $x \in [0, 1[$ is such that $x + \mathbb{Z} = \phi \upharpoonright_{G_0}(H)$ and put $$t_{\xi} = \dfrac{1}{d(J)} \cdot s_{\xi} + \mathbb{Z}.$$ It follows that $$\sum_{\xi \in \operatorname{supp}J} a(J, \xi) \cdot t_{\xi} = \bigg(\dfrac{1}{2} + \mathbb{Z} \bigg) - \phi \upharpoonright_{G_0}(H).$$ Let $\psi_{0}^{*}(\xi)$ be the open arc of $\mathbb{T}$ centered in $t_{\xi}$ with diameter $\delta(\psi_{0}^{*}(\xi)) = r_0 / d(J)$. Put, also, $\psi_{0}(\xi) = d(J) \cdot \psi_{0}^{*}(\xi)$. It is clear that $\delta(\psi_{0}(\xi)) = r_0$. Since $$\frac{1}{2} + \mathbb{Z} \in \phi \upharpoonright_{G_0}(H) + \sum_{\xi \in \operatorname{supp}J} a(J, \xi) \cdot \psi_{0}^{*}(\xi)$$ and $$\delta \bigg( \phi \upharpoonright_{G_0}(H) + \sum_{\xi \in \operatorname{supp}J} a(J, \xi) \cdot \psi_{0}^{*}(\xi) \bigg) = \delta \bigg( \sum_{\xi \in \operatorname{supp}J} a(J, \xi) \cdot \psi_{0}^{*}(\xi) \bigg) \leq \sum_{\xi \in \operatorname{supp}J} |a(J, \xi)| \cdot \frac{r_0}{d(J)} \leq \frac{1}{4}$$ it follows that $$0 + \mathbb{Z} \not \in \phi \upharpoonright_{G_0}(H) + \sum_{\xi \in \operatorname{supp}J} a(J, \xi) \cdot \psi_{0}^{*}(\xi).$$ Finally, if $\xi \in (E \setminus E_0) \cap P_1$, put $\psi_{0}(\xi) = \mathbb{T}$. Let $n \in \omega$. Assume that $\psi_n : E \cap P_1 \to \mathcal{B}$ and $\psi_{n}^{*} : E_n \cap P_1 \to \mathcal{B}$ are defined. Suppose defined, as well, a group homomorphism $\phi \upharpoonright_{G_n} : G_n \to \mathbb{T}$ where $$G_n = \langle \{\Lambda_{(\zeta, m), c_n \cdot \prod_{k < n} d(g_{i(k)}(b_k))} : (\zeta, m) \in E_n \times e_n\} \rangle$$ and $$c_n = \left\{ \begin{array}{lll} c_{n - 1} & \hbox{if} & i(n) \not \in \cup_{m \in D} L_m \\ c_{n - 1} \cdot d(y_{i(n)}) & \hbox{if} & i(n) \in \cup_{m \in D} L_m \end{array}\right.$$ We shall define $\psi_{n + 1} : E \cap P_1 \to \mathcal{B}$, $\psi_{n + 1}^{*} : E_{n + 1} \cap P_1 \to \mathcal{B}$ and extend $\phi \upharpoonright_{G_n}$ to a group homomorphism $\psi \upharpoonright_{G_{n + 1}} : G_{n + 1} \to \mathbb{T}$, where $$G_{n + 1} = \langle \{\Lambda_{(\zeta, m), c_{n + 1} \cdot \prod_{k < n + 1} d(g_{i(k)}(b_k))} : (\zeta, m) \in E_{n + 1} \times e_{n + 1}\} \rangle$$ and $$c_{n + 1} = \left\{ \begin{array}{lll} c_{n} & \hbox{if} & i(n + 1) \not \in \cup_{m \in D} L_m \\ c_{n} \cdot d(y_{i(n + 1)}) & \hbox{if} & i(n + 1) \in \cup_{m \in D} L_m \end{array}\right.$$ satisfying the following conditions: (1) If $\xi \in (E \setminus E_{n + 1}) \cap P_1$, then $\psi_{n + 1}(\xi) = \mathbb{T}$; (2) If $\xi \in (E_{n + 1} \setminus E_n) \cap P_1$, then $\psi_{n + 1}^{*}(\xi)$ is such that: 1. $d(J) \cdot \prod_{m < n + 1} d(f_{i(m)}(b_m)) \cdot \overline{\psi_{n + 1}^{*}(\xi)} \subset \psi_n(\xi)$; 2. $\delta(\psi_{n + 1}^{*}(\xi)) = \dfrac{r_{n + 1}}{d(J) \cdot \prod_{m < n + 1} d(f_{i(m)}(b_m))}$. [In this case, put]{} $$\psi_{n + 1}(\xi) = d(J) \cdot \prod_{m < n + 1} d(f_{i(m)}(b_m)) \cdot \psi_{n + 1}^{*}(\xi).$$ (3) If $\xi \in E_n \cap P_1$, then $\psi_{n + 1}^{*}(\xi)$ is such that: 1. $d(f_{i(n)}(b_n)) \cdot \overline{\psi_{n + 1}^{*}(\xi)} \subset \psi_{n}^{*}(\xi)$; 2. $\delta(\psi_{n + 1}^{*}(\xi)) = \dfrac{r_{n + 1}}{d(J) \cdot \prod_{m < n + 1} d(f_{i(m)}(b_m))}$. [In this case, put]{} $$\psi_{n + 1}(\xi) = d(J) \cdot \prod_{m < n + 1} d(f_{i(m)}(b_m)) \cdot \psi_{n + 1}^{*}(\xi).$$ (4) If $h_{i(n)}$ is of type 1, 2, 3 or 4, then $$\hspace*{-1cm} \psi_{n}(i(n)) \cap \bigg( \phi \upharpoonright_{G_{n + 1}} (g_{i(n)}(b_n)) + \sum_{\mu \in \operatorname{supp}f_{i(n)}(b_n)} a(f_{i(n)}(b_n), \mu) \cdot d(J) \cdot \prod_{m < n} d(f_{i(m)}(b_m)) \cdot \psi_{n + 1}^{*}(\mu) \bigg) \neq \emptyset;$$ (5) If $h_{i(n)}$ is of type 5, then $\phi \upharpoonright_{G_{n + 1}}(g_{i(n)}(b_n)) = \sum_{(\zeta, p) \in \operatorname{supp}y_{i(n)} \cap (E_n \times e_n)} a(y_{i(n)}, (\zeta, p)) \cdot c_{n - 1} \cdot \prod_{m < n} d(g_{i(m)}(b_m)) \cdot \phi \upharpoonright_{G_n}(\Lambda_{(\zeta, p), c_{n} \cdot \prod_{m < n} d(g_{i(m)}(b_m))})$. If $\xi \in (E \setminus E_{n + 1}) \cap P_1$, put $\psi_{n + 1}(\xi) = \mathbb{T}$. If $\xi \in [(E_{n + 1} \setminus E_n) \setminus \operatorname{supp}f_{i(n)}(b_n)] \cap P_1$, then $\psi_{n}(\xi) = \mathbb{T}$. Define $\psi_{n + 1}^{*}(\xi)$ as being an element of $\mathcal{B}$ with diameter $$\delta(\psi_{n + 1}^{*}(\xi)) = \dfrac{r_{n + 1}}{d(J) \cdot \prod_{m < n + 1} d(f_{i(m)}(b_m))}$$ and put $$\psi_{n + 1}(\xi) = d(J) \cdot \prod_{m < n + 1} d(f_{i(m)}(b_m)) \cdot \psi_{n + 1}^{*}(\xi).$$ If $\xi \in (E_n \setminus \operatorname{supp}f_{i(n)}(b_n)) \cap P_1$, fix a $d(f_{i(n)}(b_n))$-th root of the middle point of $\psi_{n}^{*}(\xi)$. Define $\psi_{n + 1}^{*}(\xi)$ as the open arc of $\mathbb{T}$ centered in that fixed root with diameter $$\delta(\psi_{n + 1}^{*}(\xi)) = \dfrac{r_{n + 1}}{d(J) \cdot \prod_{m < n + 1} d(f_{i(m)}(b_m))}$$ and put $$\psi_{n + 1}(\xi) = d(J) \cdot \prod_{m < n + 1} d(f_{i(m)}(b_m)) \cdot \psi_{n + 1}^{*}(\xi).$$ We shall now define $\psi_{n + 1}^{*}(\xi)$ and $\psi_{n + 1}(\xi)$ for $\xi \in \operatorname{supp}f_{i(n)}(b_n)$ and extend $\phi \upharpoonright_{G_n}$ to $\phi \upharpoonright_{G_{n + 1}}$. Case 1 : $h_{i(n)}$ is of type 1.\ Since $\mathbb{T}$ is a divisible group and $G_n$ is a subgroup of $G_{n + 1}$, it is possible to extend $\phi \upharpoonright_{G_n}$ to a group homomorphism $\phi \upharpoonright_{G_{n + 1}} : G_{n + 1} \to \mathbb{T}$. Fix $\alpha \in \operatorname{supp}f_{i(n)}(b_n) \setminus E_n$. Applying Lemma \[lem\_tipo1\] for $P_1$, $d(J) \cdot \prod_{m < n} d(f_{i(m)}(b_m))$, $r_n$, $\psi_{n}(i(n)) - \phi \upharpoonright_{G_{n + 1}}(g_{i(n)}(b_n))$, $E_n \cap \operatorname{supp}f_{i(n)}(b_n)$, $f_{i(n)}(b_n)$, $\alpha$ and $\psi_{n}^{*}$, we obtain $\tilde{\psi} : \operatorname{supp}f_{i(n)}(b_n) \to \mathbb{T}$ satisfying conditions (i), (ii), (iii) and (iv) of Lemma \[lem\_tipo1\]. Put $\psi_{n + 1}^{*}(\xi) = \tilde{\psi}(\xi)$, for every $\xi \in \operatorname{supp}f_{i(n)}(b_n)$. Case 2 : $h_{i(n)}$ is of type 2.\ Since $\mathbb{T}$ is a divisible group and $G_n$ is a subgroup of $G_{n + 1}$, it is possible to extend $\phi \upharpoonright_{G_n}$ to a group homomorphism $\phi \upharpoonright_{G_{n + 1}} : G_{n + 1} \to \mathbb{T}$. Fix $\alpha \in \operatorname{supp}f_{i(n)}(b_n)$ such that $q(f_{i(n)}(b_n), \alpha) \cdot r_n > d(J) \cdot \prod_{m < n} d(f_{i(m)}(b_m))$. We can assume that $\alpha \in E_n$ because, otherwise, the same arguments used above can be repeated here. Applying Lemma \[lem\_tipo2\] for $P_1$, $d(J) \cdot \prod_{m < n} d(f_{i(m)}(b_m))$, $r_n$, $\psi_{n}(i(n)) - \phi \upharpoonright_{G_{n + 1}}(g_{i(n)}(b_n))$, $E_n \cap \operatorname{supp}f_{i(n)}(b_n)$, $f_{i(n)}(b_n)$, $\alpha$ and $\psi_{n}^{*}$, we obtain $\tilde{\psi} : \operatorname{supp}f_{i(n)}(b_n) \to \mathbb{T}$ satisfying conditions (i), (ii), (iii) and (iv) of Lemma \[lem\_tipo2\]. Put $\psi_{n + 1}^{*}(\xi) = \tilde{\psi}(\xi)$, for every $\xi \in \operatorname{supp}f_{i(n)}(b_n)$. Case 3 : $h_{i(n)}$ is of type 3.\ Since $\mathbb{T}$ is a divisible group and $G_n$ is a subgroup of $G_{n + 1}$, it is possible to extend $\phi \upharpoonright_{G_n}$ to a group homomorphism $\phi \upharpoonright_{G_{n + 1}} : G_{n + 1} \to \mathbb{T}$. Fix $\alpha \in \operatorname{supp}f_{i(n)}(b_n)$ such that $|a(f_{i(n)}(b_n),\alpha)| \cdot r_n > 4 \cdot d(f_{i(n)}(b_n))$. Once again, we can assume that $\alpha \in E_n$. Applying Lemma \[lem\_tipo3\] for $P_1$, $d(J) \cdot \prod_{m < n} d(f_{i(m)}(b_m))$, $r_n$, $\psi_{n}(i(n)) - \phi \upharpoonright_{G_{n + 1}}(g_{i(n)}(b_n))$, $E_n \cap \operatorname{supp}f_{i(n)}(b_n)$, $f_{i(n)}(b_n)$, $\alpha$ and $\psi_{n}^{*}$, we obtain $\tilde{\psi} : \operatorname{supp}f_{i(n)}(b_n) \to \mathbb{T}$ satisfying conditions (i), (ii), (iii) and (iv) of Lemma \[lem\_tipo3\]. Put $\psi_{n + 1}^{*}(\xi) = \tilde{\psi}(\xi)$, for every $\xi \in \operatorname{supp}f_{i(n)}(b_n)$. Case 4 : $h_{i(n)}$ is of type 4.\ If $\xi \in (E_{n + 1} \setminus E_n) \cap \operatorname{supp}f_{i(n)}(b_n)$, define $\psi_{n + 1}^{*}(\xi)$ as an element of $\mathcal{B}$ such that $$\delta(\psi_{n + 1}^{*}(\xi)) = \dfrac{r_{n + 1}}{d(J) \cdot \prod_{m < n + 1} d(f_{i(m)}(b_m))}.$$ If $\xi \in E_n \cap \operatorname{supp}f_{i(n)}(b_n)$, fix a $d(f_{i(n)}(b_n))$-th root of the middle point of $\psi_{n}^{*}(\xi)$. Define $\psi_{n + 1}^{*}(\xi)$ as the open arc of $\mathbb{T}$ centered in this root with diameter $$\delta(\psi_{n + 1}^{*}(\xi)) = \dfrac{r_{n + 1}}{d(J) \cdot \prod_{m < n + 1} d(f_{i(m)}(b_m))}.$$ In both cases, put $$\psi_{n + 1}(\xi) = d(J) \cdot \prod_{m < n + 1} d(f_{i(m)}(b_m)) \cdot \psi_{n + 1}^{*}(\xi).$$ Denote by $z_{\xi}$ the middle point of $\psi_{n + 1}^{*}(\xi)$, for every $\xi \in \operatorname{supp}f_{i(n)}(b_n)$. Consider $$\tilde{G}_{n + 1} = \langle \{\Lambda_{(\zeta, p), c_{n + 1} \cdot \prod_{m < n} d(g_{i(m)}(b_m))} : (\zeta, p) \in (E_n \times e_n) \cup \operatorname{supp}g_{i(n)}(b_n)\} \rangle.$$ Since $\mathbb{T}$ is divisible and $G_n$ is a subgroup of $\tilde{G}_{n + 1}$, it is possible to extend $\phi \upharpoonright_{G_n}$ to a group homomorphism $\phi \upharpoonright_{\tilde{G}_{n + 1}} : \tilde{G}_{n + 1} \to \mathbb{T}$. Fix $(\mu, l) \in \operatorname{supp}g_{i(n)}(b_n)$ such that $q(g_{i(n)}(b_n), (\mu, l)) \cdot r_n > d(H) \cdot \prod_{m < n} d(g_{i(m)}(b_m))$. If $(\zeta, p) \in \operatorname{supp}g_{i(n)}(b_n) \setminus \{(\mu, l)\}$, fix $y_{(\zeta, p)}$ a $d(g_{i(n)}(b_n))$-th root of $\phi \upharpoonright_{\tilde{G}_{n + 1}}(\Lambda_{(\zeta, p), c_{n + 1} \cdot \prod_{m < n} d(g_{i(m)}(b_m))})$. Applying Lemma \[lem\_tipo4\] for $P_0 \times \omega$, $c_{n + 1} \cdot \prod_{m < n} d(g_{i(m)}(b_m))$, $r_n$, $\psi_{n}(i(n)) - u - v$, $g_{i(n)}(b_n)$, $(\mu, l)$ and $x = \phi \upharpoonright_{\tilde{G}_{n + 1}}(\Lambda_{(\mu, l), c_{n + 1} \cdot \prod_{m < n} d(g_{i(m)}(b_m))})$, where $$u = \sum_{(\zeta, p) \in \operatorname{supp}g_{i(n)}(b_n) \setminus \{(\mu, l)\}} a(g_{i(n)}(b_n), (\zeta, p)) \cdot c_{n + 1} \cdot \prod_{m < n} d(g_{i(m)}(b_m)) \cdot y_{(\zeta, p)}$$ and $$v = \sum_{\xi \in \operatorname{supp}f_{i(n)}(b_n)} a(f_{i(n)}(b_n), \xi) \cdot d(J) \cdot \prod_{m < n} d(f_{i(m)}(b_m)) \cdot z_{\xi},$$ we obtain $y_{(\mu, l)} \in \mathbb{T}$ such that $d(g_{i(n)}(b_n)) \cdot y_{(\mu, l)} = \phi \upharpoonright_{\tilde{G}_{n + 1}}(\Lambda_{(\mu, l), c_{n + 1} \cdot \prod_{m < n} d(g_{i(m)}(b_m))})$ and $$a(g_{i(n)}(b_n), (\mu, l)) \cdot c_{n + 1} \cdot \prod_{m < n} d(g_{i(m)}(b_m)) \cdot y_{(\mu, l)} + u + v \in \psi_{n}(i(n)).$$ Extend $\phi \upharpoonright_{\tilde{G}_{n + 1}}$ to a group homomorphism $\phi \upharpoonright_{G_{n + 1}} : G_{n + 1} \to \mathbb{T}$ in a way that $$\phi \upharpoonright_{G_{n + 1}}(\Lambda_{(\zeta, p), c_{n + 1} \cdot \prod_{m < n + 1} d(g_{i(m)}(b_m))}) = y_{(\zeta, p)}$$ for every $(\zeta, p) \in \operatorname{supp}g_{i(n)}(b_n)$. Case 5 : $h_{i(n)}$ is of type 5.\ Since $\operatorname{supp}g_{i(n)}(b_n) \setminus (E_n \times e_n) \neq \emptyset$ and $\operatorname{o}(g_{i(n)}(b_n) \upharpoonright_{\operatorname{supp}g_{i(n)}(b_n) \setminus (E_n \times e_n)}) = \operatorname{o}(g_{i(n)}(b_n))$, we conclude that $$\{g_{i(n)}(b_n)\} \cup \{\Lambda_{(\zeta, p), c_n \cdot \prod_{m < n} d(g_{i(m)}(b_m))} : (\zeta, p) \in E_n \times e_n\}$$ is an independent subset of the group $(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)}$. Thus, we can define $\hat{\phi}(g_{i(n)}(b_n)) = \sum_{(\zeta, p) \in \operatorname{supp}y_{i(n)} \cap (E_n \times e_n)} a(y_{i(n)}, (\zeta, p)) \cdot c_{n - 1} \cdot \prod_{m < n} d(g_{i(m)}(b_m)) \cdot \phi \upharpoonright_{G_n} (\Lambda_{(\zeta, p), c_n \cdot \prod_{m < n} d(g_{i(m)}(b_m))})$, $\hat{\phi}(x) = \phi \upharpoonright_{G_n}(x)$ for every $x \in G_n$ and extend $\hat{\phi}$ to a group homomorphism from $\langle \{g_{i(n)}(b_n)\} \cup \{\Lambda_{(\zeta, p), c_n \cdot \prod_{m < n} d(g_{i(m)}(b_m))} : (\zeta, p) \in E_n \times e_n\} \rangle$ into $\mathbb{T}$. Since $\mathbb{T}$ is a divisible group and $\langle \{g_{i(n)}(b_n)\} \cup \{\Lambda_{(\zeta, p), c_n \cdot \prod_{m < n} d(g_{i(m)}(b_m))} : (\zeta, p) \in E_n \times e_n\} \rangle$ is a subgroup of $G_{n + 1}$, we can extend $\hat{\phi}$ to a group homomorphism $\phi \upharpoonright_{G_{n + 1}} : G_{n + 1} \to \mathbb{T}$. By induction, we obtain $\psi_n : E \cap P_1 \to \mathcal{B}$ and $\psi_{n}^{*} : E_n \cap P_1 \to \mathcal{B}$ for every $n \in \omega$, satisfying the following conditions: - $\overline{\psi_{n + 1}(\xi)} \subset \psi_n(\xi)$, for every $\xi \in E \cap P_1$. - $\delta(\psi_n(\xi)) = r_n$, if $\xi \in E_n \cap P_1$ and $\psi_n(\xi) = \mathbb{T}$, if $\xi \in (E \setminus E_n) \cap P_1$. - $\psi_{n}(\xi) = d(J) \cdot \prod_{m < n} d(f_{i(m)}(b_m)) \cdot \psi_{n}^{*}(\xi)$, if $\xi \in E_n \cap P_1$. We also obtain a group homomorphism $\phi \upharpoonright_{G} : G \to \mathbb{T}$ where $G = \cup_{n \in \omega} G_n \subset (\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)}$. Since $\mathbb{T}$ is a divisible group, it is possible to extend $\phi \upharpoonright_{G}$ to a group homomorphism $\phi \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)}}: (\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \to \mathbb{T}$. Since $\mathbb{T}$ is a complete metric space and $(r_n)_{n \in \omega}$ is a sequence of positive real numbers that converges to 0, we conclude that if $\xi \in E \cap P_1$, then $\cap_{n \in \omega} \psi_n(\xi) = \cap_{n \in \omega} \overline{\psi_n(\xi)}$ is a singleton. We denote by $\tilde{\phi}(\chi_{\xi})$ the single element of $\cap_{n \in \omega} \psi_n(\xi)$. For each $\xi \in E \cap P_1$, consider $N_{\xi} = \min \{n \in \omega : \xi \in E_n\}$ and $n \geq N_{\xi}$. It follows that $\psi_{n}(\xi) \neq \mathbb{T}$ and, therefore, there exists one, and only one, element of $\psi_{n}^{*}(\xi)$ whose multiplication by $d(J) \cdot \prod_{m < n} d(f_{i(m)}(b_m))$ is equal to $\tilde{\phi}(\chi_{\xi})$. We shall denote this element by $$\tilde{\phi} \bigg(\dfrac{1}{d(J) \cdot \prod_{m < n} d(f_{i(m)}(b_m))} \cdot \chi_{\xi} \bigg).$$ Consider $$G_{\xi} = \bigg\{ \frac{1}{d(J) \cdot \prod_{m < n} d(f_{i(m)}(b_m))} \cdot \chi_{\xi} \in \mathbb{Q}^{(E \cap P_1)} : n \geq N_{\xi} \bigg\}$$ and let $\tilde{G}$ be the group generated by $\cup_{\xi \in E \cap P_1} G_{\xi}$. If $\xi \in E \cap P_1$ and $n > N_{\xi}$, then $$\tilde{\phi} \bigg(\frac{1}{d(J) \cdot \prod_{m < N_{\xi}} d(f_{i(m)}(b_m))} \cdot \chi_{\xi} \bigg)$$ is equal to $$\prod_{N_{\xi} \leq m < n} d(f_{i(m)}(b_m)) \cdot \tilde{\phi} \bigg(\frac{1}{d(J) \cdot \prod_{m < n} d(f_{i(m)}(b_m))} \cdot \chi_{\xi} \bigg).$$ Thus, it is possible to extend $\tilde{\phi}$ to a group homomorphism from $\tilde{G}$ into $\mathbb{T}$. But $\mathbb{T}$ is a divisible group, so it is possible to extend $\tilde{\phi}$ to a group homomorphism $\phi \upharpoonright_{\mathbb{Q}^{(P_1 \cap E)}} : \mathbb{Q}^{(P_1 \cap E)} \to \mathbb{T}$. Define $\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}} : (\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)} \to \mathbb{T}$ by $$\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(\tilde{H}, \tilde{J}) = \phi \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)}}(\tilde{H}) + \phi \upharpoonright_{\mathbb{Q}^{(P_1 \cap E)}}(\tilde{J})$$ for every $(\tilde{H}, \tilde{J}) \in (\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}$. If $J = 0$, then $\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}} (H, J) = \phi \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)}}(H) \neq 0 + \mathbb{Z}$, since $(H, J) \neq (0, 0)$. Thus, suppose $J \neq 0$. We have that $$\phi \upharpoonright_{\mathbb{Q}^{(P_1 \cap E)}}(J) \in \sum_{\xi \in \operatorname{supp}J} a(J, \xi) \cdot \psi_{0}^{*}(\xi).$$ Since $$0 + \mathbb{Z} \not \in \phi\upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)}}(H) + \sum_{\xi \in \operatorname{supp}J} a(J, \xi) \cdot \psi_{0}^{*}(\xi)$$ it follows that $\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(H, J) \neq 0 + \mathbb{Z}$. Therefore, (i) is verified. Fix $\xi \in (E \cap L_1) \setminus \omega$. For each $k \in i^{-1}(\{\xi\})$, we have that $\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(h_{i(k)}(b_k))$ is an element of $$\phi\upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)}} (g_{i(k)}(b_k)) + \sum_{\mu \in \operatorname{supp}f_{i(k)}(b_k)} a(f_{i(k)}(b_k), \mu) \cdot d(J) \cdot \prod_{m < k} d(f_{i(m)}(b_m)) \cdot \psi_{k + 1}^{*}(\mu).$$ Besides, $$\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(0, \chi_{i(k)}) \in \psi_{k}(i(k)).$$ It follows that $$\begin{array}{lll} \delta(\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(h_{i(k)}(b_k)), \phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(0, \chi_{i(k)})) & \leq & d_1 + d_2 \\ & < & 2 r_k \end{array}$$ where $$\hspace*{-0.75cm} \textstyle d_1 = \delta(\phi \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)}}(g_{i(k)}(b_k)) + \sum_{\mu \in \operatorname{supp}f_{i(k)}(b_k)} a(f_{i(k)}(b_k), \mu) \cdot d(J) \cdot \prod_{m < k} d(f_{i(m)}(b_m)) \cdot \psi_{k + 1}^{*}(\mu))$$ and $$d_2 = \delta(\psi_{k}(i(k))).$$ Since $r_k \to 0$, we conclude that the sequence $\{\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(h_{i(k)}(b_k)) : k \in i^{-1}(\{\xi\})\}$ converges to $\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(0, \chi_{\xi})$. From condition (v) of Lemma \[cap5\_lem\_12\_itens\] it follows that $$\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(0, \chi_{\xi}) = p_{\xi}-\lim \{\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(h_{\xi}(n)) : n \in \omega\}.$$ Therefore, (ii) is verified. Finally, consider $\xi \in E \cap \bigcup_{n \in D} L_n$. Since $E \times \omega = \cup_{n \in \omega} (E_n \times e_n)$, there exists $n \in \omega$ such that $\operatorname{supp}y_{\xi} \subset E_n \times e_n$. For each $k \in i^{-1}(\{\xi\})$ we have that $\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(h_{i(k)}(b_k)) = \phi \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)}}(g_{i(k)}(b_k))$, since $h_{i(k)}$ is of type 5. Besides, we have that $\phi \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)}}(g_{i(k)}(b_k)) = \phi \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)}}(y_{i(k)})$, for every $k > n$. From the property (v) of Lemma \[cap5\_lem\_12\_itens\] it follows that $$\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(y_{\xi}, 0) = p_{\xi}-\lim \{\phi_{(H, J)} \upharpoonright_{(\mathbb{Q} / \mathbb{Z})^{((P_0 \cap E) \times \omega)} \oplus \mathbb{Q}^{(P_1 \cap E)}}(h_{\xi}(n)) : n \in \omega\}.$$ Thus, (iii) is verified. Acknowledgment {#acknowledgment .unnumbered} ============== The first author has received financial support from CAPES and CNPq (Brazil) as a Ph.D. student at University of São Paulo, under supervision of the second author. The second author has received financial support from CNPq (Brazil) — “Bolsa de Produtividade em Pesquisa, processo 308467/2007-8. Projeto: Grupos topológicos, seleções e topologias de hiperespaços”. [^1]: The existence of such a group monomorphism is guaranteed by Theorems \[teo\_classificacao\_grupos\_divisiveis\] and \[teo\_imersao\_num\_grupo\_divisivel\]. [^2]: The implications (ii) $\Rightarrow$ (i) and (i) $\Rightarrow$ (iii) are valid in ZFC. We assume the existence of $\mathfrak{c}$ incomparable selective ultrafilters only to show that (iii) $\Rightarrow$ (ii).
--- author: - | Lorenzo Bianchi$^{a}$, Marco S. Bianchi$^{b}$\ \ $^{a}$ Institut für Physik, Humboldt-Universität zu Berlin\ Zum Großen Windkanal 6, 12489 Berlin, Germany\ $^{b}$ Centre for Research in String Theory, School of Physics and Astronomy\ Queen Mary University of London, Mile End Road, London E1 4NS, UK\ \ E-mail: bibliography: - 'biblio.bib' title: Worldsheet scattering for the GKP string --- Introduction ============ More and more aspects of four-dimensional ${\cal N}=4$ SYM in the planar limit have been revealed to be deeply connected to physics in two dimensions. On the one hand the AdS/CFT correspondence [@Maldacena:1997re] relates its strong coupling limit to a superstring theory defined on a two-dimensional worldsheet. On the other hand an increasing number of quantities of ${\cal N}=4$ SYM have been shown to be computable at any coupling via a description in terms of an integrable spin chain [@Minahan:2002ve; @Beisert:2003yb]. A corner of this big picture which we will focus on in this paper is the integrability of the twist-two operators of planar ${\cal N}=4$ SYM. They belong to the $sl(2)$ sector of the single trace operators of the theory and in the large spin limit their anomalous dimensions are fully determined by a set of asymptotic Bethe Ansatz (ABA) equations [@Beisert:2005fw; @Beisert:2006ez]. In the strong coupling regime, via the AdS/CFT correspondence, such twist-two operators are conjectured to be dual to a folded string spinning around its center of mass in $AdS_3 \subset AdS_5$ [@Gubser:2002tv; @Frolov:2002av], whose large spin limit is a fairly simple string solution, amenable of detailed analyses. One equivalent and fruitful way of describing this system is the light-cone gauge-fixed Lagrangian of the $AdS_5\times S^5$ string sigma model [@Metsaev:2000yu; @Metsaev:2000yf] expanded around the null cusp background [@Giombi:2009gd]. From the quadratic part of the Lagriangian it is possible to read out the spectrum of the excitations of the model at infinite coupling. These are a mass $\sqrt{2}$ complex scalar $x$, a mass 2 scalar $\phi$, $8$ mass 1 fermions and five massless scalars. This spectrum is in partial agreement with the degrees of freedom of the Bethe equations valid at any coupling, with discrepancies connected to the nonperturbative dynamics of the $O(6)$ sigma model emerging at strong coupling in the Alday-Maldacena limit [@Alday:2007mf]. Nevertheless, the light-cone gauge-fixed Lagrangian can be taken as the starting point for performing perturbation theory and computing quantities of interest at strong coupling. In particular, the light-cone gauge choice makes the Feynman rules fairly simple, so that this Lagrangian is suitable for computing quantum corrections. This approach has been applied to the study of the free energy of the theory [@Giombi:2009gd] which is dual to the anomalous dimension of a cusped light-like Wilson line in planar ${\cal N}=4$ SYM at strong coupling. Such a computation has been pushed to two-loop order and agrees with the ABA prediction [@Basso:2007wd] providing one of the most spectacular mutual tests of integrability at strong coupling and of the AdS/CFT correspondence (see also [@Roiban:2007jf; @Roiban:2007dq; @Roiban:2007ju]). The ABA also allows to compute the momentum of the excitations of the GKP string, and consequently determine their dispersion relations to all orders. Again, these predictions from integrability were compared at next-to-leading order at strong coupling by computing the two-point functions of the worldsheet excitations. Interestingly, as mentioned above, the agreement in this case is only partial and the reasons for the mismatches were clarified in [@Zarembo:2011ag]. In particular it was shown that perturbation theory within this model can fail to produce sensible results for particular quantities, due to the onset of nonperturbative effects. We flash that a similar ABA description also exists for the $AdS_4\times\mathbb{CP}^3$ GKP string [@Gromov:2008qe; @Basso:2013pxa], dual to the large spin limit of twist-one operators of the ABJM superconformal model in three dimensions [@Aharony:2008ug]. In this context, starting from the $AdS_4\times\mathbb{CP}^3$ light-cone gauge-fixed Lagrangian [@Uvarov:2009hf; @Uvarov_main; @Uvarov:2011zz], a parallel computation of the cusp anomalous dimension at two-loops [@Bianchi:2014ada], and of the two-point functions at one loop [@Bianchi:2015laa] has been carried out.\ Integrability is able to provide further fundamental data for solving the GKP string, namely the exact S-matrix for its excitations [@Basso:2013pxa; @Fioravanti:2013eia; @Fioravanti:2015dma]. This object is interesting per se, since it encloses the dynamics of the model, and additional relevance comes from its remarkable relation to scattering amplitudes of planar ${\cal N}=4$ SYM. The starting point for building this bridge is a light-like Wilson loop in a conformal gauge theory. One can perform an OPE decomposition of it by selecting two light-like edges, cutting the Wilson loop across them into a bottom and a top part and inserting a basis of eigenstates in the cut [@Alday:2010ku; @Gaiotto:2011dt]. The latter are interpreted as the excitations of the color flux-tube stretching between two null lines. The OPE expansion is then taken by sending to infinity the flux-tube time conjugate to the energy of the excitations. In space-time, this corresponds to flattening the bottom side of the loop, which is in turn equivalent to a multicollinear limit in dual kinematics [@Alday:2010ku; @Gaiotto:2011dt]. A generic polygon is fully reconstructed from the OPE decomposition by repeatedly performing the procedure sketched above. This is achieved by dividing the polygon into elementary squares and considering how excitations propagate between two adjacent squares, forming a pentagon, from the bottom to the top edge [@Basso:2013vsa]. The central object enclosing the dynamics of this process has been dubbed the pentagon transition. The remarkable feature of planar ${\cal N}=4$ SYM is two-fold. On the one hand in this theory the flux-tube excitations are the same as those of the GKP string and their dynamics is completely determined at any coupling by integrability. In particular, the pentagon transitions emerge as ratios of GKP string S-matrix elements. On the other hand null Wilson loops in ${\cal N}=4$ SYM are dual to scattering amplitudes [@Alday:2007hr; @Drummond:2007au; @Drummond:2007cf; @Brandhuber:2007yx], offering the unprecedented possibility of evaluating the S-matrix of an interacting four dimensional theory at any coupling. In fact this approach has been applied to and tested against a variety of scattering processes, both at weak and strong coupling [@Basso:2013aha; @Basso:2014koa; @Basso:2014jfa; @Basso:2014nra; @Basso:2014hfa; @Basso:2015rta; @Basso:2015uxa]. In conclusion there exists a tight interplay between the (flux-tube) S-matrix of the GKP string two-dimensional model and that of the four-dimensional planar ${\cal N}=4$ SYM.\ Recently, the S-matrix for the GKP string has been thoroughly studied using the ABA in [@Fioravanti:2013eia; @Fioravanti:2015dma]. This allows to write expressions for its elements, valid at any order. In particular their expansion at strong coupling in the perturbative regime has been spelled out, which is amenable of perturbative checks. The aim of this paper is to perform such tests by comparing these integrability based results to the amplitudes which can be computed perturbatively from the light-cone gauge $AdS_5\times S^5$ sigma model lagrangian. We start by reviewing the action expanded around the cusp background and its Feynman rules. Next we detail the computation of several S-matrix elements between the particles of the model. We find that, as long as massless modes do not enter the computation, the results are trustworthy and exhibit perfect agreement with the integrability predictions. In order to make this manifest we express both results in terms of hyperbolic rapidities to allow for comparison. All other amplitudes, namely that for two fermions and all those with massless scalars as external states, turn out to be troublesome, as might have been expected from the findings of [@Giombi:2010bj; @Zarembo:2011ag]. In section \[sec:scalars\] we comment more extensively on fermion-fermion scattering and propose a trick (though biased by rather strong assumptions) to compute the scalar factor evading the problematic part of the perturbative computation. In the last section, as a further check of integrability, we analyse some processes involving six particles in the massive scalar sector of the excitations, namely the scattering of gluons of same helicity and mesons. These are four possible processes and we verify in all cases that there is no particle production and the S-matrix factorizes. Here we anticipate that the cancellation of the various diagrams in a generic kinematic configuration is considerably more intricate and stunning than the BMN case [@Klose:2007rz] due to the presence of cubic and quintic interactions. The light-cone gauge action =========================== We work with the light-cone gauge euclidean action of the $AdS_5 \times S^5$ sigma model expanded around the cusp background of [@Giombi:2010bj]. We use the version with fermions cast into the Dirac form as in [@Zarembo:2011ag] $$\label{eq:action} S = \frac{T}{2}\int dt \int^\infty_{-\infty} ds\ {\cal L}\qquad\qquad T\equiv \frac{\sqrt{\lambda}}{2\pi}$$ where $T$ is the string tension in terms of the ${\cal N}=4$ ’t Hooft coupling $\lambda$ and $$\begin{aligned} \label{eq:lagrangian} {\cal L} &= \big|\partial_t x + x \big|^2 + \frac{1}{z^4} \big| \partial_s x - x \big|^2 + \Big( \partial_t z^M + z^M + \frac{i}{z^2} \psi^{\dagger}_i {\Pi}_{+} (\rho^{MN}){}^i{}_j \psi^j z_N \Big)^2 + \nonumber\\ & + \frac{1}{z^{4}} \Big(\partial_s z^M - z^M \Big)^2 + 2\, i\, \psi^{\dagger}_i \partial_t \psi^i - \frac{1}{z^{2}} \Big(\psi^{\dagger}_i {\Pi}_{+} \psi^i\Big)^2 + \nonumber\\& + \frac{2i}{z^3}\, \Bigl[-\bar\psi_i {\Pi}_{+} (\rho^{\dagger}_6)^{ik} (\rho^M)_{kj} z^M {\Delta}_s \psi^j - \frac{i}{z} (\psi^i)^T {\Pi}_{+} (\rho^M)_{ij} z^M \psi^j {\Delta}_s x + \nonumber\\& ~~~~~~~~ + \psi^{\dagger}_i {\Pi}_{+} (\rho^\dagger_M)^{ik} z^M (\rho^6)_{kj} {\Delta}_s \psi^j + \frac{i}{z} \psi^{\dagger}_i {\Pi}_{+} (\rho^{\dagger}_M)^{ij} z^M (\psi^{\dagger})_j {\Delta}_s x^*\Bigr]\end{aligned}$$ where $$\begin{aligned} & z = e^{\phi}\,, \qquad\qquad z^M = e^{\phi} u^M\,, \qquad\qquad M=1,\dots 6 & \nonumber\\ & \displaystyle u^{a} = \frac{y^{a}}{1+\frac{1}{4}y^2}\,, \qquad\qquad u^{6} = \frac{1-\frac{1}{4}y^2}{1+\frac{1}{4}y^2}\,, \qquad\qquad y^2\equiv \sum_{a=1}^5 (y^a)^2\,, \quad\qquad a=1,...,5 & \label{eq:redef}\end{aligned}$$ and ${\Delta}_s \equiv \partial_s-1$. The $\rho^{M}_{ij} $ matrices are the off-diagonal blocks of 6d gamma matrices in chiral representation. $(\rho^{MN})_i^{\phantom{i}j} = (\rho^{[M}\rho^{\dagger N]})_i^{\phantom{i}j}$ and $(\rho^{MN})^i_{\phantom{i}j} = ( \rho^{\dagger [M}\rho^{N]})^i_{\phantom{i}j}$ are the $SO(6)$ Lorentz matrices. The Dirac form [@Zarembo:2011ag] is achieved from the action of [@Giombi:2010bj], by packaging the $\eta$ and $\theta$ fermions appearing in the latter into Dirac two-component spinors as follows $$\psi^i = \left(\begin{array}{c} \eta^i \\ (\rho_6^{\dagger})^{ij}\theta_j \end{array}\right) \qquad\qquad \psi^{\dagger}_i = \left( \eta_i, \th^j (\rho^6)_{ji} \right) \qquad\qquad i=1,\dots 4$$ The gamma matrices are $${\gamma}^t = -\sigma_1 \qquad\qquad {\gamma}^s = \sigma_3$$ and $\bar\psi \equiv \psi^{\dagger}{\gamma}^t$, as usual. The projectors appearing in the Lagrangian are defined as ${\Pi}_{\pm} \equiv \frac12 \left( \mathbb{1} \pm {\gamma}^s \right)$, where $\mathbb{1}$ is the $2\times 2$ identity matrix. Expanding in the fields at second order $${\cal L}_2 = \partial_{\alpha} \phi\, \partial_{\alpha} \phi +4\,\phi^2 + \partial_{\alpha} x\, \partial_{\alpha} x^* +2\, x\, x^{*} +\partial_{\alpha}y^a\partial_{\alpha}y^a + 2\,i\, \bar \psi_i \left(\slashed{\partial} + \mathbb{1} \right)\psi^i \label{eq:quadratic}$$ the spectrum of excitations of the model is inferred, which consists of: - a mass $\sqrt{2}$ complex scalar $x$, which together with its complex conjugate represents the insertion of a positive and negative helicity gluon on the GKP vacuum. - a mass 2 scalar $\phi$ which from the point of view of the GKP integrable model does not represent an elementary excitation at finite coupling, but is rather interpreted as a composite two-fermion virtual state [@Basso:2014koa; @Zamolodchikov:2013ama]. The fact that this object is not a proper asymptotic state of the theory renders the computation of matrix elements thereof rather meaningless. Nevertheless, it was argued in [@Fioravanti:2015dma] that at strictly infinite coupling the $\phi$ scalars ought to be interpreted as real physical bosons, which were baptised [*mesons*]{} by the authors. We adopt here this interpretation and nomenclature and compute their S-matrix elements at first order at strong coupling. - 5 massless scalars $y^a$, $a=1,\dots 5$, which are the would-be Goldstone bosons originating from spontaneously breaking the original $SO(6)$ invariance of the action to $SO(5)$, which in turn is due to selecting a particular point in $S^5$ for the cusp vacuum. As already clarified in the literature, the $SO(6)$ symmetry is restored by the onset of nonperturbative effects, which consequently provide an exponentially small mass for these scalars. This is captured by the full description of the excitations of the GKP string from integrability, where these scalars represent holes in the GKP vacuum. However these phenomena are not visible in a perturbative approach from the action . Moreover the interactions of the massless scalars in trigger the emergence of IR divergences in loop computations (or even unphysical $1/0$ singularities for amplitudes at tree level) which make the perturbative expansion ill-defined and cast doubts on its validity. As a consequence, we anticipate that amplitudes involving massless scalars are likely to produce incorrect results. At best the S-matrix elements are just not comparable to those of the ABA approach and violate its underlying $SU(4)$ symmetry, in the worst case scenario they are ill-defined. We discuss this point further in Section \[sec:scalars\]. - 4 mass 1 Dirac fermions $\psi^i$ ($\psi^{\dagger}_i$), $i=1,\dots 4$, transforming in the $\bf{4}$ ($\bf{\bar 4}$) representation of $SU(4)$, which are mapped to insertions of fermionic excitations on the GKP vacuum. The fermions are in perfect correspondence with the degrees of freedom of the ABA description. In particular they form multiplets of its $SU(4)$ symmetry. However, it is clear that the interaction terms in the Lagrangian break this symmetry [^1]. This occurs for instance in the coupling with the massless scalars. Therefore one can foresee that problems might occur computing amplitudes of fermions whenever $SU(4)$ breaking interactions undermine the invariance of scattering processes under this expected symmetry. In this paper we analyse the $2\to 2$ tree level scattering of such particles, by computing them with Feynman diagrams. The Feynman rules are as follows. From the quadratic action we derive the propagators $$\begin{aligned} \langle x(p)x^*(-p) \rangle &= \raisebox{-1mm}{\includegraphics[width=3cm]{xprop}} = \frac{1}{2g}\, \frac{2}{p^2+2}\nonumber\\ \langle \phi(p)\phi(-p) \rangle &= \raisebox{-1mm}{\includegraphics[width=3cm]{phiprop}} = \frac{1}{2g}\, \frac{1}{p^2+4}\nonumber\\ \langle y^a(p)y^b(-p) \rangle &= \raisebox{-1mm}{\includegraphics[width=3cm]{yprop}} = \frac{1}{2g}\, \frac{\delta^{ab}}{p^2}\nonumber\\ \langle \psi^i(p)\bar\psi_j(-p) \rangle &= \raisebox{-1mm}{\includegraphics[width=3cm]{psiprop}} = \frac{1}{2g}\, i\, \frac{i\slashed{p}-\mathbb{1}}{p^2+1}\, \delta^i_{\phantom{i}j}\nonumber\end{aligned}$$ with the notation we use for drawing Feynman diagrams throughout the article. Interaction vertices are given by $-\frac12$ those appearing in the Lagrangian giving rise to a consistent expansion in the effective coupling $T$. They are listed in Appendix \[app:lagr\_exp\] for completeness. We assign momenta $p_1$ and $p_2$ to the incoming scattering particles and $p'_1$ and $p'_2$ to the outgoing ones. Their components are $$p_i = \left( e_i, {\text{p}}_i \right)$$ with imaginary energy in the euclidean. On-shell, we parameterize the momenta of massive particles with hyperbolic rapidities as $$p_i = m_i \left( i \cosh{\theta_i} , \sinh{\theta_i} \right)$$ There are in general two solutions to the momentum conservation constraints with relativistic particles: the first is forward scattering $p'_1=p_1$, $p'_2 = p_2$, the second is backward scattering which for particles of equal mass reads $p'_1=p_2$, $p'_2 = p_1$, and for different masses has a complicated solution. Integrability predicts that backward scattering should be absent, which is a statement we also want to verify directly. Solving the momentum conservation $\delta$ functions produces a Jacobian $$J = \frac{1}{4\, (e_2 {\text{p}}_1 - e_1 {\text{p}}_2)}$$ which we have to add to the amplitude. Fermionic external states yield the polarization Dirac spinors $$u(p) = \frac{1}{\sqrt{e}} \left(\begin{array}{c} e \\ {\text{p}}- i \end{array}\right)$$ Since we will not scatter antifermions, $u(p)$ and its conjugate $$\bar u(p) = \frac{1}{\sqrt{e}} \left( {\text{p}}+ i , -e \right)$$ are the only polarization spinors needed. Note that the sign of $e$ changes, according to its imaginary nature. The normalization comes in such a way that $$\bar u(p) u(p) = 2i m = 2i$$ The action we use contains an overall factor $\frac{\sqrt{\lambda}}{4\pi} = \frac{T}{2} \equiv g$ [^2]. In order to have a standard form for the kinetic terms, we normalize each particle in the initial and final states with a factor $N=1/\sqrt{2g}$, apart from the $x$, $x^*$ scalars whose kinetic terms is off by an extra factor of 2 and are thus normalized with $N_{x}=1/\sqrt{g}$. Therefore the S-matrix elements read $$\label{eq:S} S(p_1,p_2) = 1 - \frac{N_1^2 N_2^2}{4\,(e_2{\text{p}}_1-e_1{\text{p}}_2)}\, A(p_1,p_2) + {\cal O}(g^{-2})$$ and we compute $A(p_1,p_2)$ with Feynman diagrams. With the Feynman rules outlined above each interaction vertex has a power of the coupling, whereas propagators introduce an inverse power. Then it is straightforward to see that at tree level $A$ is of order $g$ and therefore $S$ scales as $g^{-1}$. Scattering of gluons ==================== Same helicity scattering ------------------------ We start considering scattering of two transverse gauge excitations of the same helicity $xx\to xx$. Since the particles are identical, we can restrict to, e.g., the forward solution to the momentum conservation conditions, and sum the diagrams in Figure \[fig:treexx\], which correspond to the $t$- and $u$-channel exchange of a mass 2 scalar. Using our euclidean action, the amplitude evaluates $$\label{eq:treeamplitude} A^{gg}(p_1,p_2) = 8g\, \left({\text{p}}_1^2+1\right) \left({\text{p}}_2^2+1\right) \left( \frac{1}{4} + \frac{1}{(p_1-p_2)^2+4} \right) + {\cal O}(g^0)$$ where the two terms in the parenthesis come from the $t$- and $u$-channels of the diagrams in Figure \[fig:treexx\], respectively. Hence the total S-matrix element reads $$S^{gg}(p_1,p_2) = 1-\frac{2}{g}\, \frac{\left({\text{p}}_1^2+1\right) \left({\text{p}}_2^2+1\right)}{4\,(e_2 {\text{p}}_1 - e_1 {\text{p}}_2)}\, \frac{(p_1-p_2)^2+8}{(p_1-p_2)^2+4} + {\cal O}(g^{-2})$$ which can be written in terms of hyperbolic rapidities as $$\label{eq:treeamplitude2} S^{gg}(\th_1,\th_2) = 1 + \frac{i}{g}\, \frac{\cosh{2 \theta_1}\, \cosh{2 \th_2}\, \cosh^2{\frac{\theta_1-\theta_2}{2}}}{\sinh{2 (\theta_1-\theta_2)}} + {\cal O}(g^{-2})$$ Opposite helicity scattering ---------------------------- We now turn to the scattering of two transverse gauge excitations with opposite helicity $xx^*\to xx^*$. The tree-level amplitude is given by the sum of the diagrams in Figure \[fig:treexxb\]. We begin considering forward scattering, where the particles do not exchange their momenta. This gives the tree level amplitude $$\label{eq:treexxbforw} A^{gg^*}(p_1,p_2;p_1,p_2) = 8g\, \left({\text{p}}_1^2+1\right) \left({\text{p}}_2^2+1\right) \left( \frac{1}{4} + \frac{1}{(p_1+p_2)^2+4} \right) + {\cal O}(g^0)$$ where the notation stresses the forward kinematic configuration. In hyperbolic rapidities this leads to the expression $$\label{eq:treexxb} S^{gg^*}(\th_1,\th_2;\th_1,\th_2) = 1+\frac{i}{g}\, \frac{\cosh{2 \theta_1}\, \cosh{2 \th_2}\, \tanh{\frac{\theta_1-\theta_2}{2}} }{\cosh{(\theta_1-\theta_2)}} + {\cal O}(g^{-2})$$ In the backward scattering kinematic configuration, interestingly, the two tree-level diagrams of Figure \[fig:treexxb\] cancel exactly leaving a vanishing result $$\label{eq:treexxbback} {\cal A}^{gg^*}(p_1,p_2;p_2,p_1) = 8g\, \left({\text{p}}_1^2+1\right) \left({\text{p}}_2^2+1\right) \left(\frac{1}{(p_1+p_2)^2+4} + \frac{1}{(p_1-p_2)^2+4}\right) = 0$$ where the last equality follows from the identity $$\label{eq:kinid} (p_1+p_2)^2 + 4 = -(p_1-p_2)^2-4$$ which holds for mass $\sqrt{2}$ particles. Comparison to integrability results ----------------------------------- We compare the results obtained for gluon scattering from the string sigma model with the predictions from the ABA. For the same helicity process, to lowest order in the strong coupling expansion, the integrability result reads $$S^{gg}(\bar u_1,\bar u_2) = 1 + \frac{1}{2g(\bar u_1-\bar u_2)}\left(1+\frac{1}{2}\left(\frac{1+\bar u_1}{1-\bar u_1}\,\frac{1-\bar u_2}{1+\bar u_2}\right)^{\frac{1}{4}}+\frac{1}{2}\left(\frac{1+\bar u_1}{1-\bar u_1}\,\frac{1-\bar u_2}{1+\bar u_2}\right)^{-\frac{1}{4}}\right) + {\cal O}(g^{-2})$$ in terms of (rescaled: $\bar u_i = \frac{u_i}{2g}$) Bethe rapidities, which can be mapped to hyperbolic ones using $$\label{eq:rapidity} \bar u_i = \tanh 2 \th_i$$ to lowest order in the strong perturbative regime. This gives [@Fioravanti:2013eia] $$S^{gg}(\th_1,\th_2) = 1 + \frac{i}{\sqrt{2}g} \left( \frac{1}{\tanh{2\theta_1}-\tanh{2\theta_2}} + \frac{\cosh{2\theta_1} \cosh{2\theta_2}}{2\sinh{(\theta_1-\theta_2)}}\right) + {\cal O}(g^{-2})$$ which coincides with the perturbative result . For opposite helicities, the result for forward kinematics can be directly compared to that quoted in [@Fioravanti:2015dma] $$S^{gg^*}(\th_1,\th_2) = 1 + \frac{i}{\sqrt{2}g} \left( -\frac{1}{\tanh{2\theta_1}-\tanh{2\theta_2}} + \frac{\cosh{2\theta_1} \cosh{2\theta_2}}{2\sinh{(\theta_1-\theta_2)}}\right) + {\cal O}(g^{-2})$$ showing agreement. In addition, we remark that integrability predicts the ratio between the S-matrices for same helicity and opposite helicity in terms of Bethe rapidities [@Basso:2013aha] $$\frac{S^{gg}(u_1,u_2)}{S^{gg^*}(u_1,u_2)} = \frac{u_1-u_2+i}{u_1-u_2-i}$$ This statement holds non-trivially at all orders. Rescaling rapidities as $u_i\rightarrow 2g\bar u_i$ and expanding it at first order in perturbation theory for large $g$, we can appreciate that it has the simple translation in terms of external momenta $$\frac{S^{gg}(\bar u_1,\bar u_2)}{S^{gg^*}(\bar u_1,\bar u_2)}-1 \propto \frac{8}{g} \frac{1}{(p_1+p_2)^2+4} + {\cal O}(g^{-2})$$ which comes precisely from subtracting the dynamical factors of the amplitudes and , using again the kinematic identity . The integrability results also predict that backward scattering is absent in this process, to all orders. With we are able to test this prediction at lowest order in perturbation theory at strong coupling. Scattering of gluons with other particles ========================================= In this section we compute the amplitudes for scattering of a gluon with a different particle, which might be a fermion, a massless scalar or a meson which, as recalled in the Introduction, we identify with the mass 2 scalar $\phi$ in the spectrum of the string excitations. Anticipating that amplitudes involving the massless scalars are troublesome, we restrict our attention here to scattering of massive excitations only and defer the discussion on $y$ scalars to section \[sec:scalars\]. Gluon-meson scattering ---------------------- This process can be computed through the Feynman diagrams shown in Figure \[fig:treexphi\]. The amplitude evaluates in general $$\begin{aligned} \label{eq:amptreexphi} A^{gM}(p_1,p_2;p_1',p_2') &= -4\, \frac{(i {\text{p}}_1+1)(-i {\text{p}}_1'+1)}{(p_1-p_1')^2+4}\left(-e_2 e_2'+e_2^2+e_2'^2+{\text{p}}_2 {\text{p}}_2'-{\text{p}}_2^2-{\text{p}}_2'^2\right)+ \nonumber\\& + \frac{8\left[({\text{p}}_1+{\text{p}}_2)^2+1\right]}{(p_1+p_2)^2+2}\, (i {\text{p}}_1+1)(-i {\text{p}}_1'+1)+ \nonumber\\& + \frac{8\left[({\text{p}}_1-{\text{p}}_2')^2+1\right]}{(p_1-p_2')^2+2}\, (i {\text{p}}_1+1)(-i {\text{p}}_1'+1) -8(i {\text{p}}_1+1)(-i {\text{p}}_1'+1) + {\cal O}(g^{0})\end{aligned}$$ For forward scattering the amplitude takes the form $$\label{eq:treexphi} A^{gM}(p_1,p_2) = 2 g\, (1+{\text{p}}_1^2) \left(-8-e_2^2+{\text{p}}_2^2 + \frac{8 \left[1+({\text{p}}_1-{\text{p}}_2)^2\right]}{2+(p_1-p_2)^2} + \frac{8 \left[1+({\text{p}}_1+{\text{p}}_2)^2\right]}{2+(p_1+p_2)^2}\right) + {\cal O}(g^{0})$$ leading to the S-matrix element $$\label{eq:treexphi2} S^{gM}(\th_1,\th_2) = 1 - \frac{i}{\sqrt{2}\, g} \frac{\cosh{2 \theta_1}\, \sinh{2 \theta_2}\, \cosh{(\theta_1-\theta_2)}}{\cosh{2 (\theta_1-\theta_2)}} + {\cal O}(g^{-2})$$ The second solution to the momentum conservation $\delta$ functions has an unpleasant form which produces a nasty expression for the amplitude in this regime. Nevertheless this simplifies to 0, showing that the scattering is reflectionless. Gluon-fermion ------------- We turn to scattering between a gluon of positive/negative helicity with a fermion. We start with the process $\psi x \to \psi x$, whose relevant Feynman diagrams are displayed in Figure \[fig:treexpsi\]. The algebra of the two diagrams gives (each line comes from a different graph) $$\begin{aligned} \label{eq:amptreexpsi} A^{fg}(p_1,p_2;p_1',p_2') &= -8 i\, g\, \bar u(p'_1) \left[(-i{\text{p}}'_1-1){\Pi}_{+} + (i{\text{p}}_1-1){\Pi}_{-} \right] u(p_1)\, \frac{(i {\text{p}}_2-1)(-i {\text{p}}'_2-1)}{(p_1-p'_1)^2+4} + \nonumber\\& -8 i\, g\, \bar u(p'_1)\, {\Pi}_{-}\, \frac{i\cancel{(p_1+p_2)}-\mathbb{1}}{(p_1+p_2)^2+1}\, {\Pi}_{+}\, u(p_1)\, (i p_2-1)(-i p'_2-1) + {\cal O}(g^{0})\end{aligned}$$ which summed and evaluated for forward kinematics with hyperbolic rapidities gives the simple result $$\label{eq:treexpsi} S^{fg}(\th_1,\th_2) = 1 - \frac{i}{4\,g}\, \frac{\cosh{2 \theta_2}\, \sinh{2 \theta_1}}{1+\sqrt{2}\, \cosh{(\theta_1-\theta_2)}} + {\cal O}(g^{-2})$$ As before, the evaluation of the expression for the amplitude in backward kinematics is more complicated but eventually vanishes. Considering the process $\psi x^* \to \psi x^*$, we evaluate the Feynamn diagrams of Figure \[fig:treexbpsi\]. The first gives the same contribution as for the first diagram of Figure \[fig:treexpsi\], spelled out in the first line of , whereas the second differs and is given by the expression $$-8 i\, g\, \bar u(p'_1)\, {\Pi}_{-}\, \frac{i\cancel{(p_1-p'_2)}-\mathbb{1}}{(p_1-p'_2)^2+1}\, {\Pi}_{+}\, u(p_1)\, (i p_2-1)(-i p'_2-1)$$ The combination of the two terms in forward kinematics is such that only a relative sign changes with respect to the previous result $$\label{eq:treexbpsi} S^{fg}(\th_1,\th_2) = 1 - \frac{i}{4\, g}\, \frac{\cosh{2 \theta_2}\, \sinh{2 \theta_1}}{-1+\sqrt{2}\, \cosh{(\theta_1-\theta_2)}} + {\cal O}(g^{-2})$$ Backward scattering is vanishing. #### Comparison to integrability Following [@Fioravanti:2015dma], the ABA predicts that the meson-gluon scattering phase has the strong coupling expansion $$S^{gM}(\th_1,\th_2) = 1-\frac{i}{\sqrt{2}\, g}\, \frac{\cosh{(\theta_1-\theta_2)}}{\coth{2 \theta_2} - \tanh{2 \theta_1}} + {\cal O}(g^{-2})$$ which is easily seen to be equivalent to our perturbative result . Turning to gluon-fermion scattering, we have to compare our results and with the integrability predictions $$\begin{aligned} S^{fg}(\th_1,\th_2) &= 1 + \frac{i}{4\,g}\, \frac{2\cosh{(\theta_1-\theta_2)}-\sqrt{2}}{\tanh{2\theta_2} - \coth{2\theta_1}} + {\cal O}(g^{-2}) \nonumber\\ S^{fg^*}(\th_1,\th_2) &= 1 + \frac{i}{4\,g}\, \frac{2\cosh{(\theta_1-\theta_2)}+\sqrt{2}}{\tanh{2\theta_2} - \coth{2\theta_1}} + {\cal O}(g^{-2})\end{aligned}$$ which show perfect agreement (upon apparently identifying $x\rightarrow g^*$ and $x^* \rightarrow g$, which is just a matter of conventions). In addition, we have ascertained that these scattering processes are reflectionless, which is a general feature of integrable scattering matrices involving excitations with different masses. Scattering of mesons ==================== Meson-meson scattering ---------------------- We study the scattering of two mass 2 mesons $\phi$. The relevant Feynman diagrams, shown in Figure \[fig:treephiphi\], evaluate to $$\begin{aligned} \label{eq:amptreephiphi} A^{\phi\phi}(p_1,p_2) &= 8g\, \left( \frac{(e_1^2+e_2^2-e_1 e_2 - {\text{p}}_1^2-{\text{p}}_2^2+{\text{p}}_1 {\text{p}}_2)^2}{(p_1-p_2)^2+4} + \frac{(e_1^2-{\text{p}}_1^2)(e_2^2-{\text{p}}_2^2)}{4} \right) + \nonumber\\& + 8g\, \left( \frac{e_1^2+e_2^2+e_1 e_2 - {\text{p}}_1^2-{\text{p}}_2^2-{\text{p}}_1 {\text{p}}_2}{(p_1+p_2)^2+4} \right) + \nonumber\\& - 8g\, \left(4 + p_1^2 + p_2^2 \right) + {\cal O}(g^{0})\end{aligned}$$ where we have already selected, e.g., forward kinematics since the particles are identical. In particular, the first contribution arises from the sum of the first and third diagrams which are equal to each other. The last diagram just gives a number, on-shell. Summing them up and turning to hyperbolic rapidities, we obtain the expression $$\label{eq:treephiphi} S^{MM}(\th_1,\th_2) = 1 + \frac{i}{2\, g}\, \frac{\sinh{2 \theta_1}\, \sinh{2 \theta_2}}{\sinh{(\theta_1 - \theta_2)}} + {\cal O}(g^{-2})$$ Meson-fermion scattering ------------------------ This process involves the diagrams of Figure \[fig:treepsiphi\], which read respectively $$\begin{aligned} A^{fM}(p_1,p_2;p'_1,p'_2) &= 8 i\, g\, \bar u(p'_1) \left[(-i{\text{p}}'_1-1){\Pi}_{+} + (i{\text{p}}_1-1){\Pi}_{-} \right] u(p_1)\times \nonumber\\&~~~~~~~~~~~~~~~~~ \times \frac{e_2^2+(e'_2)^2-e_2 e'_2 - {\text{p}}_2^2-({\text{p}}'_2)^2+{\text{p}}_2 {\text{p}}'_2}{(p_1-p'_1)^2+4} + \nonumber\\& - 8 i\, g\, \bar u(p'_1) \left[(-i{\text{p}}'_1-1){\Pi}_{+} + (i({\text{p}}_1+{\text{p}}_2)-1){\Pi}_{-} \right] \frac{i\cancel{(p_1+p_2)}-\mathbb{1}}{(p_1+p_2)^2+1}\times \nonumber\\&~~~~~~~~~~~~~~~~~ \times \left[(-i({\text{p}}_1+{\text{p}}_2)-1){\Pi}_{+} + (i{\text{p}}_1-1){\Pi}_{-} \right] u(p_1) + \nonumber\\& - 8 i\, g\, \bar u(p'_1) \left[(-i{\text{p}}'_1-1){\Pi}_{+} + (i({\text{p}}_1-{\text{p}}'_2)-1){\Pi}_{-} \right] \frac{i\cancel{(p_1-p'_2)}-\mathbb{1}}{(p_1-p'_2)^2+1}\times \nonumber\\&~~~~~~~~~~~~~~~~~ \times \left[(-i({\text{p}}_1-{\text{p}}'_2)-1){\Pi}_{+} + (i{\text{p}}_1-1){\Pi}_{-} \right] u(p_1) + \nonumber\\& + 8 i\, g\, \bar u(p'_1) \left[(-i{\text{p}}'_1-1){\Pi}_{+} + (i{\text{p}}_1-1){\Pi}_{-} u(p_1) \right] u(p_1) + {\cal O}(g^{0})\end{aligned}$$ From these we compute the forward scattering phase $$\label{eq:treepsiphi} S^{fM}(\th_1,\th_2) = 1 + \frac{i}{4g} \frac{\sinh{2 \theta_1} \sinh{2 \theta_2}}{\sinh{(\theta_1 - \theta_2)}} + {\cal O}(g^{-2})$$ The solution to backward kinematics generates as usual a cumbersome output, which nevertheless can be shown to vanish. Comparison to integrability results ----------------------------------- The amplitude computed above for meson-meson scattering is found to be in perfect agreement with that quoted in [@Fioravanti:2015dma], formula (C.45). For fermion-meson scattering the perturbative result also matches the ABA prediction which can be extracted from formulae in section 9 of [@Fioravanti:2015dma], precisely producing . Again, absence of backward scattering has been verified for these processes at lowest order in perturbation theory. Amplitudes involving massless scalars {#sec:scalars} ===================================== We have left aside all amplitudes with scalars as external particles as well as the fermion-fermion scattering, whose tree-level computation involves a massless scalar exchange. In this section we comment on these processes, which appear problematic to compute using perturbation theory from the action , similarly to what was shown to happen for two-point functions [@Giombi:2010bj; @Zarembo:2011ag; @Bianchi:2015laa]. On the one hand the massless scalars cannot even be identified with the degrees of freedom of the integrable model describing the GKP string as their number differs. Hence it would be quite meaningless to compare their scattering matrices. On the other hand the massless scalars can cause problems even when they do not appear as external states, but as exchanged particles. This happens for instance when trying to compute fermion-fermion scattering. The massless scalars introduce interactions which break the $SU(4)$ symmetry of the Lagrangian and hence produce a violation of the $SU(4)$ structure expected for fermion-fermion scattering. Moreover, if treated as massless, an exchange of $y$ scalars in the $t$-channel is plagued by an unphysical $1/0$ singular term caused by the propagator, which signals an inconsistency of the perturbative approach. Finally, the exponentially suppressed mass gap of the theory combined with the logarithimic dependence on the IR cutoff appearing in IR divergent higher loops contributions would invalidate the perturbative result even at tree level [@Zarembo:2011ag]. We verify and address these issues, where possible, studying the aforementioned amplitudes. Fermion-fermion scattering -------------------------- First we tackle the amplitude between a pair of fermions. These particles transform in the $\bf{4}$ representation of $SU(4)$, hence the $2\to 2$ amplitude is a 4-indices tensor of $SU(4)$. Following [@Basso:2014koa] we define it as $$\label{eq:fermionamplitude} \big| \psi^i(p_1) \psi^j(p_2) \big\rangle = S^{ff}(p_1,p_2)^{ij}_{kl} \big| \psi^l(p_2) \psi^k(p_1) \big\rangle$$ One could also consider the fermion-antifermion amplitude, but its computation involves a higher number of Feynman diagrams, therefore we focus on and evaluate the relevant graphs of Figure \[fig:treepsipsi\]. The computation of the first four are straightforward and yield separately $$\begin{aligned} \label{eq:fermioncontr} (a1)&= 8 g \cosh^2{\theta_1} \cosh^2{\theta_2}\, \delta^i_k\delta^j_l \equiv a_1\, \delta^i_k\delta^j_l\nonumber\\ (a2)&= -8 g \cosh{\theta_1} \cosh{\theta_2} \cosh^2{\frac{\theta_1+\theta_2}{2}}\, \delta^i_l\delta^j_k \equiv a_2\, \delta^i_l\delta^j_k \nonumber\\ (b)&= -2 g \cosh{\theta_1} \cosh{\theta_2}\, \left(\delta^i_k\delta^j_l-\delta^i_l\delta^j_k + (\rho^{a6})^i_{\phantom{i}k}(\rho^{a6})^j_{\phantom{j}l}-(\rho^{a6})^i_{\phantom{i}l}(\rho^{a6})^j_{\phantom{j}k}\right) \equiv \nonumber\\& \equiv b \left(\delta^i_k\delta^j_l-\delta^i_l\delta^j_k + (\rho^{a6})^i_{\phantom{i}k}(\rho^{a6})^j_{\phantom{j}l}-(\rho^{a6})^i_{\phantom{i}l}(\rho^{a6})^j_{\phantom{j}k}\right) \nonumber\\ (c)&= 8 g \cosh{\theta_1} \cosh{\theta_2} \left(\cosh{(\theta_1+\theta_2)} + \frac{2\sinh{\theta_1} \sinh{\theta_2}}{\cosh{(\theta_1-\theta_2)}}\right)\, (\rho^6)^{ij}(\rho^\dagger_6)_{kl} \equiv c\, (\rho^6)^{ij}(\rho^\dagger_6)_{kl}\end{aligned}$$ in terms of hyperbolic rapidities. We note that the diagrams contribute to different tensor structures. In particular, those with a mass 2 meson exchange are proportional to $\delta^i_k\delta^j_l$ and $\delta^i_l\delta^j_k$, respectively, that triggered by a gluon exchange is proportional to $(\rho^6)^{ij}(\rho^\dagger_6)_{kl}$ and the quartic vertex diagram is proportional to $\delta^i_k\delta^j_l-\delta^i_l\delta^j_k$ and $(\rho^{a6})^i_{\phantom{i}k}(\rho^{a6})^j_{\phantom{j}l}-(\rho^{a6})^i_{\phantom{i}l}(\rho^{a6})^j_{\phantom{j}k}$. The diagrams featuring a massless scalar exchange remain to be evaluated. The first is proportional to the tensor structure $(\rho^{a6})^i_{\phantom{i}k}(\rho^{a6})^j_{\phantom{j}l}$ and its algebra is troublesome: momentum conservation in two-dimensional kinematics forces the internal propagator to be singular. This unphysical phenomenon signals that something wrong is happening in the perturbative expansion. One may regulate the propagator with a small mass, which sounds reasonable since the scalars acquire a small nonperturbative mass, after all. With such a regulator the diagram is found to vanish, on-shell, since the numerator is proportional to the fermion on-shell condition. The diagram with a scalar exchange in the $u$-channel, which contributes to the $(\rho^{a6})^i_{\phantom{i}l}(\rho^{a6})^j_{\phantom{j}k}$ structure, is not singular but vanishes on-shell as well. The result of such a naive computation is certainly far from the prediction of integrability. In particular the tensor structure of the result is violating the expected $SU(4)$ symmetry of the integrable model. The tensor structures appearing in it are not independent, on the contrary they are related by the tensor identities $$\begin{aligned} \label{eq:tensoridentities} & (\rho^{a6})^i_{\phantom{i}k}(\rho^{a6})^j_{\phantom{j}l}-(\rho^{a6})^i_{\phantom{i}l}(\rho^{a6})^j_{\phantom{j}k} - 3\, (\delta^i_k\delta^j_l-\delta^i_l\delta^j_k) + 4\, (\rho^6)^{ij}(\rho^\dagger_6)_{kl} = 0\nonumber\\ & (\rho^{a6})^i_{\phantom{i}k}(\rho^{a6})^j_{\phantom{j}l}-(\delta^i_k\delta^j_l-2\delta^i_l\delta^j_k)-2\, (\rho^6)^{ij}(\rho^\dagger_6)_{kl} = 0\end{aligned}$$ Still, if one tries, e.g., to eliminate the $\rho^{a6}$ tensors from the result, it is clear from the very different expressions of the contributions, that there is no chance the $\rho^6 \rho^\dagger_6$ piece cancels, which would leave $SU(4)$ invariant tensors only. At this point we conclude that the perturbative approach fails to compute this amplitude and blame the massless scalars for this, along the lines of [@Zarembo:2011ag]. Nevertheless, we can still try to make use of the computation of the diagrams $(a1)$, $(a2)$, $(b)$ and $(c)$ in Figure \[fig:treepsipsi\], which looks legitimate, with some experimental physics. Let’s say that the interactions between massless scalars and fermions are not suitable for this computation because of the onset of nonperturbative phenomena which are not accessible via our analysis. As explained in [@Zarembo:2011ag], the massless scalars cause infrared divergences in loop computations, which can be thought of as logarithms of their exponentially small mass. Therefore these logarithms produce positive powers of the coupling, mixing perturbative orders and invalidating perturbation theory. We can imagine that an infinite tower of leading logarithms can be resummed and produce a nonvanishing contribution to the tree level result for the fermion amplitude. We can also [*suppose*]{} that the tensor structure of this contribution is proportional to the tree level structures $(\rho^{a6})^i_{\phantom{i}k}(\rho^{a6})^j_{\phantom{j}l}$ and $(\rho^{a6})^i_{\phantom{i}l}(\rho^{a6})^j_{\phantom{j}k}$, thought we admittedly do not have any solid argument to justify this. To parameterize our ignorance on the form of these interactions we introduce the two undetermined functions $x$ and $y$ as order $g$ coefficients of the $\rho^{a6}$ tensors $$\label{eq:parameterization} x\, (\rho^{a6})^i_{\phantom{i}k}(\rho^{a6})^j_{\phantom{j}l} + y\, (\rho^{a6})^i_{\phantom{i}l}(\rho^{a6})^j_{\phantom{j}k}$$ Next we [*assume*]{} that the scattering process occurs in an $SU(4)$ invariant and integrable fashion and borrow the general expression for such an S-matrix [@Berg:1977dp; @Basso:2014koa] $$\label{eq:tensorstructure} S^{ff}(u_1,u_2)^{ij}_{kl} = S^{ff}(u_1,u_2) \left(\frac{u_1-u_2}{u_1-u_2-i}\, \delta^i_k\delta^j_l - \frac{i}{u_1-u_2-i}\, \delta^i_l\delta^j_k \right)$$ in terms of Bethe rapidities. The scalar factor $S^{ff}(u_1,u_2)$ encloses the dynamics of the particular integrable model, that is the GKP string in the case at hand. This assumption is putting some extra crucial ingredient at this point, but let us go ahead with this working hypothesis and see if we get some mileage. First we expand at strong coupling by first rescaling the Bethe rapidities $u_i=2g\bar u_i$, expanding to first order at $g\to\infty$ and mapping the Bethe rapidities to hyperbolic, $\bar u_i = \coth 2\th_i$ for fermions. This gives $$\begin{aligned} \label{eq:tensorstructurehyp} S^{ff}(\th_1,\th_2)^{ij}_{kl} &= \left( 1 + \frac{1}{g}\, S^{ff}(\th_1,\th_2)^{(1)} + {\cal O}(g^{-2}) \right) \times\nonumber\\& \times\left[\left(1+\frac{i}{2\,g}\,\frac{1}{\coth{2\theta_1} - \coth{2\theta_2}}\right) \delta^i_k\delta^j_l -\frac{i}{2\,g}\,\frac{1}{\coth{2\theta_1} - \coth{2\theta_2}}\, \delta^i_l\delta^j_k + {\cal O}(g^{-2}) \right]\end{aligned}$$ On the other hand, using and , the amplitude reads $$\begin{aligned} S^{ff}(\th_1,\th_2)^{ij}_{kl} &= 1+\frac{i}{16\,g^2\, \sinh{(\th_1-\th_2)}}\left( (a_1+b)\,\delta^i_k\delta^j_l + (a_2-b)\,\delta^i_l\delta^j_k + c\, (\rho^6)^{ij}(\rho^\dagger_6)_{kl} \right) + \nonumber\\& + x\, (\rho^{a6})^i_{\phantom{i}k}(\rho^{a6})^j_{\phantom{j}l} + y\, (\rho^{a6})^i_{\phantom{i}l}(\rho^{a6})^j_{\phantom{j}k} + {\cal O}(g^{-2})\end{aligned}$$ If we insists that it has to respect the form , we can plug into the equation above in order to eliminate the $\rho^{6a}$ structure and impose that the $\rho^6\rho^\dagger_6$ tensors also drop out. This leaves us with a linear system in three unknowns, where that we are aiming at is the scalar factor $S^{ff}(\th_1,\th_2)$ $$\left\{\begin{array}{l}\displaystyle a_1 + b + x - 2y = \frac{i}{2\,g}\,\frac{1}{\coth{2\theta_1} - \coth{2\theta_2}} + \frac{1}{g}\, S^{ff}(\th_1,\th_2)^{(1)}\\\displaystyle a_2 - b - 2x + y = -\frac{i}{2\,g}\,\frac{1}{\coth{2\theta_1} - \coth{2\theta_2}}\\ c + 2x - 2y = 0 \end{array}\right.$$ Solving the system we obtain $$S^{ff}(\th_1,\th_2) = 1+\frac{i}{4\,g}\,\frac{\cosh{(\th_1-\th_2)}-1}{\coth{2\theta_1} - \coth{2\theta_2}} + {\cal O}(g^{-2})$$ which is in precise agreement with the prediction of [@Fioravanti:2015dma]. We want to stress that the derivation above is highly speculative and already assumes integrability as an input. Still, we find interesting that the perturbative computation of a subset of [*safe*]{} graphs is able to reproduce the correct result of the fermion scalar factor, which arises from the complicated nonperturbative dynamics of the GKP string. Scattering of massless scalars ------------------------------ We turn to scattering involving massless scalars as external particles. As mentioned above, although it is possible to construct Feynman diagrams for them starting from the action , it is not clear what to compare the objects computed this way to. Indeed the five massless scalars present in the model are not directly mapped to the holes of the integrable GKP string model and the dynamics of the latter is highly nonperturbative. For instance, from the point of view of the string sigma model , the scattering amplitude of a scalar off a gluon vanishes identically at tree level, since there are simply no interaction vertices to construct it. On the other hand integrability predicts that the amplitude is finite and possesses a contribution of order $g^{-1}$. Clearly there is a clash between the two approaches. For other processes there are in principle Feynman diagrams one can construct, but we are skeptical on the possibility of extracting any interesting information from them, given the known shortcomings of the model when addressing quantities that are not $SU(4)$ invariant. Particle production and factorization ===================================== In this section we provide evidence for the absence of particle production and the factorization of the $3\to3$ particle S-matrix in terms of two-body ones [@Zamolodchikov:1978xm]. Let us first recall which structure the factorization constraint assumes when expanded perturbatively. We start from the basic factorization equation $$\label{eq:fact} S_{123} = S_{12}\, S_{13}\, S_{23}$$ where the operators act on a three-particle state and the indices label the scattering particles. In this notation the product of S-matrices is not commutative and the consistency of factorization is provided by the Yang-Baxter equation $$S_{12}\, S_{13}\, S_{23} = S_{23}\, S_{13}\, S_{12}$$ Expanding perturbatively as $S = \mathbb{1} - \frac{1}{g}\, T^{(0)} + \mathcal{O}(g^{-2})$ one obtains the tree-level identity $$\label{eq:YB} T^{(0)}_{123}= T_{12}^{(0)} T_{13}^{(0)} + T_{12}^{(0)} T_{23}^{(0)} + T_{13}^{(0)} T_{23}^{(0)}$$ In the following we show that this identity holds for the $3\to 3$ scattering processes involving bosonic GKP massive excitations. Scattering of three gluons -------------------------- Let us start from the simplest case, i.e. the $xxx\to xxx$ S-matrix. The contributing diagrams are shown in Figure \[fig:treexxx\], with all possible permutations of external momenta. These contributions, with the choice of momenta in the figure and using the shorthand notation $p_{ij}\equiv p_i - p_j$, evaluate to $$\begin{aligned} d_1^{xxx} &= - 64g\, \frac{(i {\text{p}}_1-1)(i {\text{p}}_2-1)(i {\text{p}}_3-1)(-i {\text{p}}_4-1)(-i {\text{p}}_5-1)(-i {\text{p}}_6-1)}{\left[p_{14}^2+4\right]\left[p_{36}^2+4\right]} \nonumber\\ d_2^{xxx} &= 64g\, \frac{(i {\text{p}}_1-1)(i {\text{p}}_2-1)(i {\text{p}}_3-1)(-i {\text{p}}_4-1)(-i {\text{p}}_5-1)(-i {\text{p}}_6-1)}{\left[p_{14}^2+4\right]\left[(p_{14}+p_2)^2+2\right]\left[p_{36}^2+4\right]} \left( ({\text{p}}_{14}+{\text{p}}_2)^2 + 1 \right) \nonumber\\ d_3^{xxx} &= 32g\, \frac{\left[ -e_{14} e_{25} - e_{25} e_{36} - e_{36} e_{14} - (e \leftrightarrow {\text{p}})\right]}{\left[p_{14}^2+4\right]\left[p_{25}^2+4\right]\left[p_{36}^2+4\right]} \times\nonumber\\& ~~~~ \times (i {\text{p}}_1-1)(i {\text{p}}_2-1)(i {\text{p}}_3-1)(-i {\text{p}}_4-1)(-i {\text{p}}_5-1)(-i {\text{p}}_6-1) \end{aligned}$$ The total amplitude is given by the sum of the diagrams above, summed over the 36 permutations of the incoming and outgoing external momenta separately and weighted by the following symmetry factors $$A^{xxx} \propto \frac12\, d_1^{xxx} + d_2^{xxx} + \frac16\, d_3^{xxx} + \mathrm{perms} = 0$$ and is found to vanish for generic kinematics. Care has to be taken for special kinematics, for instance whenever $p_1 = p_4$. This automatically forces the other momenta to be equal pairwise, namely $p_2 = p_5$, $p_3=p_6$ or $p_2=p_6$, $p_3=p_5$. In such a situation, and all permutations thereof, the first diagram develops a singularity because of the on-shell intermediate $x$ propagator. The other diagrams are regular since they do not possess any propagators going on-shell. With the Feynman prescription the singular propagator splits as usual into a finite, principal value, part and a $\delta$ function. The finite part cancels among the three diagrams as in the non-singular case, whereas the $\delta$ function part produces the only non-vanishing contribution. In Figure \[fig:factex\] we provide an example of such a situation with the blue dashed line indicating a cut propagator, i.e. an on-shell $\delta$ function. The four singular configurations involving an on-shell propagator with momentum $p_1$ group themselves in such a way that they can be explicitly interpreted as the product of the $t$- and $u$-channel contributions in Figure \[fig:treexxb\] for the tree-level S-matrices $T^{xx}(p_1,p_2)$ and $T^{xx}(p_1,p_3)$. A similar picture arises for internal propagators with momenta $p_2$ and $p_3$ leading to a factorization of the form$$T^{xxx}(p_1,p_2,p_3) = T^{xx}(p_1,p_2)T^{xx}(p_1,p_3)+T^{xx}(p_1,p_2)T^{xx}(p_2,p_3)+T^{xx}(p_1,p_3)T^{xx}(p_2,p_3)$$ predicted by the Yang-Baxter equation . Scattering of three mesons -------------------------- A slightly more involved computation can be carried out to ascertain factorization for the $3\to 3$ scattering of mesons. There are seven relevant topologies of Feynman diagrams contributing to this process, drawn in Figure \[fig:treephiphiphi\], with all possible permutations of external legs. For the choice of external momenta shown in the picture the diagrams read, using the shorthand notation $p_{ij}\equiv p_i - p_j$ $$\begin{aligned} d_1^{\phi\phi\phi} & = 32g\, \frac{\left[ e_1^2 + e_4^2 - e_1 e_4 - (e \leftrightarrow {\text{p}}) \right]}{\left[p_{14}^2+4\right]\left[(p_{14}+p_2)^2+4\right]\left[p_{36}^2+4\right]} \left[ e_3^2 + e_6^2 - e_3 e_6 - (e \leftrightarrow {\text{p}}) \right] \times\nonumber\\& ~~~~ \times \left[ e_2^2 + e_{14}^2 + e_2 e_{14} - (e \leftrightarrow {\text{p}})\right] \left[ e_5^2 + e_{36}^2 - e_5 e_{36} - (e \leftrightarrow {\text{p}})\right] \nonumber\\ d_2^{\phi\phi\phi} & = -32g\, \frac{\left[ e_1^2 + e_4^2 - e_1 e_4 - (e \leftrightarrow {\text{p}}) \right]}{\left[p_{14}^2+4\right]\left[p_{36}^2+4\right]} \left[ e_3^2 + e_6^2 - e_3 e_6 - (e \leftrightarrow {\text{p}}) \right] \times\nonumber\\& ~~~~ \times \left[ 4 - p_2\cdot p_{14} - p_{14}\cdot p_{36} + p_5\cdot p_{14} - p_2\cdot p_{36} + p_5\cdot p_{36} + p_2\cdot p_5 \right] \nonumber\\ d_3^{\phi\phi\phi} & = 32g\, \frac{\left[ e_1^2 + e_4^2 - e_1 e_4 - (e \leftrightarrow {\text{p}}) \right]}{\left[p_{14}^2+4\right]\left[p_{25}^2+4\right]\left[p_{36}^2+4\right]} \left[ e_3^2 + e_6^2 - e_3 e_6 - (e \leftrightarrow {\text{p}}) \right] \times\nonumber\\& ~~~~ \times \left[ e_2^2 + e_5^2 - e_2 e_5 - (e \leftrightarrow {\text{p}})\right] \left[ -e_{14} e_{25} - e_{25} e_{36} - e_{36} e_{14} - (e \leftrightarrow {\text{p}})\right] \nonumber\\ d_4^{\phi\phi\phi} & = -32g\, \frac{\left[ e_3^2 + e_6^2 - e_3 e_6 - (e \leftrightarrow {\text{p}}) \right]}{\left[(p_{14}+p_2)^2+4\right]\left[p_{36}^2+4\right]} \left[ e_5^2 + e_{36}^2 - e_5 e_{36} - (e \leftrightarrow {\text{p}})\right] \times\nonumber\\& \times \left[ 4 - p_1\cdot (p_{14}+p_2) - p_2\cdot (p_{14}+p_2) + p_4\cdot (p_{14}+p_2) - p_1\cdot p_2 + p_2\cdot p_4 + p_4\cdot p_1 \right] \nonumber\\ d_5^{\phi\phi\phi} & = 32g\, \frac{1}{\left[(p_{14}+p_2)^2+4\right]} \times\nonumber\\& \times \left[ 4 - p_1\cdot (p_{14}+p_2) - p_2\cdot (p_{14}+p_2) + p_4\cdot (p_{14}+p_2) - p_1\cdot p_2 + p_2\cdot p_4 + p_4\cdot p_1 \right] \times\nonumber\\& \times \left[ 4 - p_3\cdot (p_{14}+p_2) + p_6\cdot (p_{14}+p_2) + p_5\cdot (p_{14}+p_2) + p_3\cdot p_6 - p_6\cdot p_5 + p_5\cdot p_3 \right] \nonumber\\ d_6^{\phi\phi\phi} & = 32g\, \frac{\left[ e_1^2 + e_4^2 - e_1 e_4 - (e \leftrightarrow {\text{p}}) \right]}{\left[p_{14}^2+4\right]} \times\nonumber\\& \times \left[ e_5 e_{14} + e_6 e_{14} - e_2 e_{14} - e_3 e_{14} - e_2 e_3 + e_3 e_6 - e_6 e_5 + e_5 e_3 - (e \leftrightarrow {\text{p}}) \right] \nonumber\\ d_7^{\phi\phi\phi} & = -32g\, \left[ 4 - p_1\cdot p_2 - p_1\cdot p_3 + p_1\cdot p_4 + p_1\cdot p_5 + p_1\cdot p_6 - p_2\cdot p_3 + p_2\cdot p_4 + p_2\cdot p_5 + \right.\nonumber\\&\left. ~~~~ + p_2\cdot p_6 +p_2\cdot p_4 + p_3\cdot p_5 + p_3\cdot p_6 - p_4\cdot p_5 - p_4\cdot p_6 -p_5\cdot p_6 \right]\end{aligned}$$ For the last two diagrams we have used the $\phi$ quintic and sextic vertices . Summing over all 720 permutations of the external legs and combining the diagrams with the following symmetry factors $$\label{eq:treexxphi} A^{\phi\phi\phi} \propto \frac18\, d_1^{\phi\phi\phi} + \frac{1}{16}\, d_2^{\phi\phi\phi} + \frac{1}{48}\, d_3 + \frac{1}{12}\, d_4^{\phi\phi\phi} + \frac{1}{72}\, d_5^{\phi\phi\phi} + \frac{1}{48}\, d_6^{\phi\phi\phi} + \frac{1}{720}\, d_7^{\phi\phi\phi} + \mathrm{perms} = 0$$ it is straightforward to ascertain, e.g. numerically, that the amplitude vanishes for generic external momenta. As before the only non-vanishing contribution comes from the kinematically singular configurations. In particular the first, fourth and fifth diagrams contain a propagator which goes on-shell for forward kinematics. As for the gluons case one can group these three contributions and interpret them in terms of products of the diagrams in Figure \[fig:treephiphi\]. In particular the first diagram receives contributions only from the first three diagrams of Figure \[fig:treephiphi\]. The fourth diagram produces the products of the four-vertex interactions in the two-body amplitudes and the fifth diagram generates the mixed terms. For instance, we can select the singular diagrams contributing to the structure $T_{12}T_{23}$. We dub $\hat d^{\phi\phi\phi}_i(\{p_j\})$ the diagrams listed above after removing the singular propagator and with the momenta ordered as in its argument and $\bar p_i = p_i$ the outgoing momenta after enforcing the $\delta$ function from the singular propagator. Then the total contribution with momentum $p_2$ flowing in the singular propagator is proportional to the combination $$\begin{aligned} T_{12}T_{23} \propto & \frac14\, \hat d^{\phi\phi\phi}_1\left( \raisebox{0.75mm}{$\{p_1, p_2$}, \raisebox{-0.75mm}{$\{p_3$}, \raisebox{0.75mm}{$\bar p_1\}$}, \raisebox{-0.75mm}{$\bar p_2, \bar p_3\}$} \right) + \frac{1}{12}\, \hat d^{\phi\phi\phi}_5\left( \raisebox{0.75mm}{$\{p_1, p_2$}, \raisebox{-0.75mm}{$\{p_3$}, \raisebox{0.75mm}{$\bar p_1\}$}, \raisebox{-0.75mm}{$\bar p_2, \bar p_3\}$} \right) + \nonumber\\& + \frac{1}{12}\, \hat d^{\phi\phi\phi}_5\left( \raisebox{0.75mm}{$\{\bar p_2, p_3$}, \raisebox{-0.75mm}{$\{p_1$}, \raisebox{0.75mm}{$\bar p_3\}$}, \raisebox{-0.75mm}{$p_2, \bar p_1\}$} \right) + \hat d^{\phi\phi\phi}_4\left( p_1, p_2, p_3, \bar p_1, \bar p_2, \bar p_3 \right)\end{aligned}$$ where brackets stand for symmetrization and apply to separate groups of momenta in a self-explanatory notation. The symmetry factors take into account equivalent configurations. Dividing by them as in the above formula we see that there is one contribution from diagram 4, corresponding to the product of the four-vertex diagrams of Figure \[fig:treephiphi\] contributing to $T_{12}$ and $T_{23}$, respectively. Diagram 1 produces 9 terms which emerge from the product of the three diagrams of Figure \[fig:treephiphi\] with cubic vertices only. Finally diagram 5 gives 6 terms from the mixed products. Altogether these combine to give the $4\times4 = 16$ terms from the product of two-body amplitudes. Inserting the Jacobians from the momentum conservation $\delta$ functions and properly normalizing, we have ascertained that this combination gives precisely $T_{12}T_{23}$, as it can be obtained from formula . Summing the contributions to $T_{12}T_{13}$ and $T_{13}T_{23}$, altogether they combine to give the full factorization . Scattering of two gluons and one meson -------------------------------------- Next we can consider the mixed process $x(p_1)x(p_2)\phi(p_3)\to x(p_4)x(p_5)\phi(p_6)$. In this case there are 23 topologies of diagram contributing, shown schematically in Figure \[fig:treexxphi\], with possible permutations of the external legs. In order to show factorization we evaluate this process numerically for generic configurations of external momenta satisfying the on-shell and momentum conservation conditions. We use the expression for the diagrams in Appendix \[app:xxphi\] and sum over the eight momentum permutations $p_1\leftrightarrow p_2$, $p_4\leftrightarrow p_5$ and $p_3\leftrightarrow p_6$. For some diagrams these permutations are overcounting the contribution, which we take into account with the following symmetry factors $$A^{xx\phi} \propto \frac14\, \sum_{i=1}^{2}\, d_i^{xx\phi} + \frac12\, \sum_{j=3}^{12}\, d_j^{xx\phi} + \sum_{k=13}^{23}\, d_k^{xx\phi} + \mathrm{perms} = 0$$ Remarkably, such a large combination of diagrams can be straightforwardly seen to vanish for generic choices of external momenta, with a marvellous cancellations spreading over 132 terms. Therefore only singular configurations corresponding to factorization of the amplitude eventually contribute. In this case the amplitude factorises in the contributions $T^{xx}_{12}T^{x\phi}_{23}$, $T^{xx}_{12}T^{x\phi}_{13}$ and $T^{x\phi}_{13}T^{x\phi}_{23}$. We have verified both diagrammatically and analytically that the first and second terms arise when combining diagrams 7, 8, 9, 10, 15 and 16 in the singular momentum configurations. Finally we have ascertained that the last product of two-body amplitudes emerges from diagrams 2, 5, 11, 12, 17, 18, 19, 20, 21 and 22. Scattering of one gluon and two mesons -------------------------------------- Scattering of a gluon and two mesons receives contributions from 29 topologies of Feynman diagrams, depicted in Figure \[fig:treexphiphi\]. In each there are up to $4!$ factorial permutations of the external momenta of the mesons. We take them into account by summing all diagrams over these permutations of momenta and dividing by the symmetry factors $$\label{eq:treexphiphi} A^{x\phi\phi} \propto \frac{1}{4!}\, \sum_{i=1}^{2}\, d_i^{x\phi\phi} + \frac{1}{8}\, \sum_{i=3}^{4}\, d_i^{x\phi\phi} + \frac{1}{6}\, \sum_{i=5}^{10}\, d_i^{x\phi\phi} + \frac14\, \sum_{j=11}^{16}\, d_j^{x\phi\phi} + \frac12 \sum_{k=17}^{28}\, d_k^{x\phi\phi} + d_{29}^{x\phi\phi} + \mathrm{perms} = 0$$ following the order in the figure. The contributions $d^{x\phi\phi}$ are collected in Appendix \[app:xphiphi\]. These are 236 contributions and we verified they sum to 0 for generic momenta configurations, providing a strong test of absence of particle production of the model. The singular momentum configurations affecting diagrams 5, 8, 9, 10, 21, 22, 27 and 28 and 11, 14, 15, 16, 18, 19, 25, 26 and 29 combine to give the contributions $T^{x\phi}_{12}T^{\phi\phi}_{23}$ and $T^{x\phi}_{13}T^{\phi\phi}_{23}$, and $T^{x\phi}_{12}T^{x\phi}_{13}$, respectively. This proves that the amplitude factorises. Conclusions =========== In this paper we have computed S-matrix elements for the excitations of the GKP string at first order in $1/g$ from perturbation theory of the light-cone gauge-fixed $AdS_5\times S^5$ sigma model. The outcome of our analysis is that, as long as massless scalars do not enter the computation, the scattering phases are in agreement with the ABA predictions. This safe sector includes all amplitudes without massless scalars on the external legs, apart from the fermion-fermion scattering process where massless scalar exchanges contribute. In the latter case the result of a naive perturbative computation is found to violate the $SU(4)$ symmetry of the integrability based result, since its tensor structure does not consist of invariant tensors only. A possible interpretation of this fact is that IR singularities appearing at higher orders in the perturbative expansion spoil the predictivity of perturbation theory at tree level. Nevertheless, by comparing the perturbative results for the two $SU(4)$ invariant tensor structures and imposing that the spurious ones vanish, it is possible to correctly reproduce the scalar factor predicted by integrability. This hints at the fact that IR divergent contributions at higher loops should contribute only to the spurious tensor structures. It would be interesting to check this fact explicitly. In an integrable theory $2\to 2$ processes are the fundamental building blocks for any higher point scattering amplitude thanks to the factorization of the S-matrix and the absence of particle production. We have explicitly checked these properties to hold for three-body S-matrices involving gluons and mesons. The structure of the computation turned out to be more involved than the BMN case [@Klose:2007rz], where only quartic and sextic interactions are present. Here, also three- and five-point vertices need to be included and this considerably increases the number of diagrams. Therefore, the precise cancellation of the three-body S-matrix provides a further stringent check of the integrability of the model. We conclude remarking that a similar analysis could be performed for the analogous $AdS_4\times \mathbb{CP}^3$ model dual to the ABJM theory. Again it is expected that only a subset of these amplitudes is safely computable and comparable to the integrability predictions. In particular the latter model includes a massless Dirac fermion as well, whose dynamics is expected to be deeply nonperturbative, as for the massless scalars. Finally, we point out that the tree level scattering elements we have computed (or the more comprehensive list from the ABA) could be used as the starting point of a unitarity based computation of the scattering phases at next order, in order to perform more precise checks of integrability of the S-matrix at the quantum level. This program has been already applied to the BMN string in several $AdS$ backgrounds [@Engelund:2013fja; @Bianchi:2013nra; @Bianchi:2014rfa; @Engelund:2014pla; @Hoare:2014kma] and it would be interesting to extend it to the GKP string as well. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Benjamin Basso, Valentina Forini, Ben Hoare, Simone Piscaglia and Marco Rossi for very useful discussions. The work of LB is funded by DFG via the Emmy Noether Program “Gauge Fields from Strings”. The work of MB was supported in part by the Science and Technology Facilities Council Consolidated Grant ST/L000415/1 *String theory, gauge theory & duality*. Expanded Lagrangian to fourth order {#app:lagr_exp} =================================== In this appendix we spell out the interaction terms of the Lagriangin , up to quartic order in the fields. Cubic vertices read $$\begin{aligned} {\cal L}_3 &= -4\tilde\phi\, |\partial_s x - x|^2 + 2 \phi [(\partial_t \phi)^2-(\partial_s \phi)^2] + 2 \phi\ [(\partial_t y^a)^2-(\partial_s y^a)^2] + \nonumber\\ & + 4i\, \phi [(\partial_s \bar\psi_i - \bar\psi_i) \Pi_{+} \psi^i + \bar\psi_i \Pi_{-} (\partial_s \psi^i - \psi^i)] + \nonumber\\ & + 2 i\, y^a [(\partial_s \bar\psi_i - \bar\psi_i) {\Pi}_{+} (\rho^{a6})^{i}_{\phantom{i}j} \psi^j - \bar\psi_i \Pi_{-} (\rho^{a6})^{i}_{\phantom{i}j} (\partial_s \psi^j - \psi^j) ] + 2i\, \partial_t y^a \bar\psi_i \gamma^t {\Pi}_{+} (\rho^{a6})^{i}_{\phantom{i}j} \psi^j + \nonumber\\& + 2 (\partial_s x - x) (\psi^i)^T {\Pi}_{+} (\rho^6)_{ij} \psi^j - 2 (\partial_s x^* - x^*) \bar\psi_i {\Pi}_{-} (\rho^{\dagger}_6)^{ij} (\bar\psi_j)^T\end{aligned}$$ and quartic interactions $$\begin{aligned} {\cal L}_4 &= 8\, \phi^2\, |\partial_s x - x|^2 + 2\, \phi^2 [\partial_{\alpha} \phi \partial_{\alpha} \phi + \frac{2}{3} \phi^2] + 2 \phi^2 \partial_{\alpha} y^a\partial_{\alpha} y^a - \frac{1}{2}y^a y^a\, \partial_{\alpha} y^b \partial_{\alpha} y^b + \nonumber\\ & - i (4\phi^2 -y^a y^a)\,[(\partial_s \bar\psi_i - \bar\psi_i) \Pi_{+} \psi^i + \bar\psi_i \Pi_{-} (\partial_s \psi^i - \psi^i)] + \nonumber\\& - 4 i\, \phi\,y^a [(\partial_s \bar\psi_i - \bar\psi_i) {\Pi}_{+} (\rho^{a6})^{i}_{\phantom{i}j} \psi^j - \bar\psi_i \Pi_{-} (\rho^{a6})^{i}_{\phantom{i}j} (\partial_s \psi^j - \psi^j)] + \nonumber\\ & - 6 \phi\,[(\partial_s x - x) (\psi^i)^T {\Pi}_{+} (\rho^6)_{ij} \psi^j - (\partial_s x^* - x^*) \bar\psi_i {\Pi}_{-} (\rho^{\dagger}_6)^{ij} (\bar\psi_j)^T] + \nonumber\\ & + 2(\partial_s x - x) (\psi^i)^T {\Pi}_{+} (\rho^a)_{ij} y^a \psi^j - 2(\partial_s x^* - x^*) \bar\psi_i {\Pi}_{-} (\rho^{\dagger}_a)^{ij} y^a (\bar\psi_j)^T + \nonumber\\ & - 2 i\, y^a\partial_t y^b\, \bar\psi_i \gamma^t {\Pi}_{+} (\rho^{ab})^{i}_{\phantom{i}j} \psi^j + (\bar\psi_i \gamma^t {\Pi}_{+} (\rho^{a6})^{i}_{\phantom{i}j} \psi^j)^2 - (\bar\psi_i \gamma^t {\Pi}_{+} \psi^i)^2\end{aligned}$$ In the computation of scattering of three mesons quintic and sextic vertices are needed, which can be obtained expanding $$\label{eq:phivertices} {\cal L}^{x,\phi}_{5,6} = -\frac{32}{3}\, \phi^3\, \big| \partial_s x - x \big|^2 + \frac{32}{3}\, \phi^4\, \big| \partial_s x - x \big|^2 + \frac43\, \left((\partial_t \phi)^2 - (\partial_s \phi)^2 \right) \phi^3 + \left(\frac{8}{45}\, \phi^2 + \frac23\, (\partial_\alpha \phi)^2\right)\phi^4$$ Computation of $xx\phi\to xx\phi$ diagrams {#app:xxphi} ========================================== In this section we spell out the expressions of the diagrams contributing to the $xx\phi \to xx\phi$ scattering process. We label external momenta as $x(p_1)x(p_2)\phi(p_3)\to x(p_4)x(p_5)\phi(-p_6)$ and give the expression for the diagrams of Figure \[fig:treexxphi\], following their order. The diagrams have an overall factor $2g(i p_1-1)(i p_2-1)(-i p_4-1)(-i p_5-1)$ which we omit in the following. The remaining expressions read $$\begin{aligned} d^{xx\phi}_1 & = -\frac{16\left[ 4 - (p_{14}\cdot p_{25}+p_3\cdot p_{14}+p_3\cdot p_{14}+p_6\cdot p_{25}+p_6\cdot p_{25}+p_3\cdot p_6)\right]}{\left[(p_1-p_4)^2+4\right]\left[(p_2-p_5)^2+4\right]} \nonumber\\ d^{xx\phi}_2 & = \frac{16\left[ e_{14}e_{25} + e_{25}^2 + e_{14}^2 - (e\leftrightarrow {\text{p}})\right]\left[ e_3 (e_3+e_6) + e_3^2 + e_6^2 - (e\leftrightarrow {\text{p}})\right]}{\left[(p_1-p_4)^2+4\right]\left[(p_3+p_6)^2+4\right]\left[(p_2-p_5)^2+4\right]} \nonumber\\ d^{xx\phi}_3 & = \frac{16\left[ e_3 e_{14} + e_3^2 + e_{14}^2 - (e\leftrightarrow {\text{p}})\right]\left[ e_6 e_{25} + e_6^2 + e_{25}^2 - (e\leftrightarrow {\text{p}})\right]}{\left[(p_1-p_4)^2+4\right]\left[(p_1+p_3-p_4)^2+4\right]\left[(p_2-p_5)^2+4\right]} \nonumber\\ d^{xx\phi}_4 & = \frac{32\left[ e_3 e_6 + e_3^2 + e_6^2 - (e\leftrightarrow {\text{p}})\right]}{\left[(p_1-p_4)^2+4\right]\left[(p_3+p_6)^2+4\right]} \nonumber\\ d^{xx\phi}_5 & = \frac{64}{\left[(p_1+p_3-p_4)^2+4\right]} \nonumber\\ d^{xx\phi}_6 & = \frac{64}{\left[(p_2-p_5)^2+4\right]} \nonumber\\ d^{xx\phi}_7 & = \frac{32\left[ -e_3 e_6 - e_3^2 - e_6^2 - (e\leftrightarrow {\text{p}})\right]\left[({\text{p}}_1+{\text{p}}_2-{\text{p}}_4)^2+1\right]}{\left[(p_1-p_4)^2+4\right]\left[(p_3+p_6)^2+4\right]\left[(p_1+p_2-p_4)^2+2\right]} \nonumber\\ d^{xx\phi}_8 & = \frac{32\left[ -e_3 e_6 - e_3^2 - e_6^2 - (e\leftrightarrow {\text{p}})\right]\left[({\text{p}}_2+{\text{p}}_3+{\text{p}}_6)^2+1\right]}{\left[(p_1-p_4)^2+4\right]\left[(p_3+p_6)^2+4\right]\left[(p_2+p_3+p_6)^2+2\right]} \nonumber\\ d^{xx\phi}_9 & = -\frac{64\left[({\text{p}}_1+{\text{p}}_2-{\text{p}}_4)^2+1\right]}{\left[(p_1-p_4)^2+4\right]\left[(p_1+p_2-p_4)^2+2\right]} \nonumber\\ d^{xx\phi}_{10} & = -\frac{64\left[({\text{p}}_2+{\text{p}}_3+{\text{p}}_6)^2+1\right]}{\left[(p_1-p_4)^2+4\right]\left[(p_2+p_3+p_6)^2+2\right]} \nonumber\\ d^{xx\phi}_{11} & = \frac{64\left[({\text{p}}_4-{\text{p}}_3)^2+1\right]\left[({\text{p}}_5-{\text{p}}_6)^2+1\right]}{\left[(p_1+p_3-p_4)^2+4\right]\left[(p_4-p_3)^2+2\right]\left[(p_5-p_6)^2+2\right]} \nonumber\\ d^{xx\phi}_{12} & = \frac{64\left[({\text{p}}_1+{\text{p}}_3)^2+1\right]\left[({\text{p}}_2+{\text{p}}_6)^2+1\right]}{\left[(p_1+p_3-p_4)^2+4\right]\left[(p_1+p_3)^2+2\right]\left[(p_2+p_6)^2+2\right]} \nonumber\\ d^{xx\phi}_{13} & = -\frac{64\left[({\text{p}}_5-{\text{p}}_6)^2+1\right]}{\left[(p_1-p_4)^2+4\right]\left[(p_5-p_6)^2+2\right]} \nonumber\\ d^{xx\phi}_{14} & = -\frac{64\left[({\text{p}}_2+{\text{p}}_6)^2+1\right]}{\left[(p_1-p_4)^2+4\right]\left[(p_2+p_6)^2+2\right]} \nonumber\\ d^{xx\phi}_{15} & = \frac{64\left[({\text{p}}_5-{\text{p}}_6)^2+1\right]\left[({\text{p}}_5-{\text{p}}_3-{\text{p}}_6)^2+1\right]}{\left[(p_1-p_4)^2+4\right]\left[(p_5-p_3-p_6)^2+2\right]\left[(p_5-p_6)^2+2\right]} \nonumber\\ d^{xx\phi}_{16} & = \frac{64\left[({\text{p}}_2+{\text{p}}_6)^2+1\right]\left[({\text{p}}_2+{\text{p}}_3+{\text{p}}_6)^2+1\right]}{\left[(p_1-p_4)^2+4\right]\left[(p_2+p_3+p_6)^2+2\right]\left[(p_2+p_6)^2+2\right]} \nonumber\\ d^{xx\phi}_{17} & = -\frac{32\left[ (e_1-e_4) e_3 + (e_1-e_4)^2 + e_3^2 - (e\leftrightarrow {\text{p}})\right]\left[({\text{p}}_2+{\text{p}}_6)^2+1\right]}{\left[(p_1-p_4)^2+4\right]\left[(p_1+p_3-p_4)^2+4\right]\left[(p_2+p_6)^2+2\right]} \nonumber\\ d^{xx\phi}_{18} & = -\frac{32\left[ (e_1-e_4) e_3 + (e_1-e_4)^2 + e_3^2 - (e\leftrightarrow {\text{p}})\right]\left[({\text{p}}_5-{\text{p}}_6)^2+1\right]}{\left[(p_1-p_4)^2+4\right]\left[(p_1+p_3-p_4)^2+4\right]\left[(p_5-p_6)^2+2\right]} \nonumber\\ d^{xx\phi}_{19} & = -\frac{64\left[({\text{p}}_3-{\text{p}}_5)^2+1\right]}{\left[(p_1+p_6-p_4)^2+4\right]\left[(p_5-p_3)^2+2\right]} \nonumber\\ d^{xx\phi}_{20} & = -\frac{64\left[({\text{p}}_2+{\text{p}}_6)^2+1\right]}{\left[(p_1+p_3-p_4)^2+4\right]\left[(p_2+p_6)^2+2\right]} \nonumber\\ d^{xx\phi}_{21} & = \frac{64\left[({\text{p}}_4-{\text{p}}_3)^2+1\right]\left[({\text{p}}_2+{\text{p}}_6)^2+1\right]}{\left[(p_1+p_3-p_4)^2+4\right]\left[(p_4-p_3)^2+2\right]\left[(p_2+p_6)^2+2\right]} \nonumber\\ d^{xx\phi}_{22} & = \frac{32\left[ -(e_1-e_4+e_3) (e_2-e_5) - (e_1-e_4+e_3) e_6 - (e_2-e_5) e_6 - (e\leftrightarrow {\text{p}})\right]}{\left[(p_1+p_3-p_4)^2+4\right]\left[(p_2-p_5)^2+4\right]} \nonumber\\ d^{xx\phi}_{23} & = \frac{64\left[({\text{p}}_1+{\text{p}}_6)^2+1\right]\left[({\text{p}}_3-{\text{p}}_4)^2+1\right]}{\left[(p_2-p_5)^2+4\right]\left[(p_1+p_6)^2+2\right]\left[(p_3-p_4)^2+2\right]} \nonumber\\\end{aligned}$$ Computation of $x\phi\phi\to x\phi\phi$ diagrams {#app:xphiphi} ================================================ In this section we give the expressions for the contributions $d^{x\phi\phi}$ relevant for $x\phi\phi\to x\phi\phi$ scattering. The corresponding diagrams are shown in Figure \[fig:treexphiphi\]. The incoming $x$ particle has momentum $p_1$ and the outgoing $p_4$. An overall common factor $2g(i {\text{p}}_1-1)(-i {\text{p}}_4-1)$ is understood in the following formulae. The $\phi$ particles have momenta $p_2$, $p_3$, $p_5$ and $p_6$ and we take them as all ingoing for simplicity. The following formulae hold for a sample configuration of momenta for the mesons and have to be symmetrised in the corresponding momentum indices. These are 24 permutations which, for all but the last contribution, overcount the diagram by a symmetry factor which we divide by in . $$\begin{aligned} d^{x\phi\phi}_1 &= -64 \nonumber\\ d^{x\phi\phi}_2 &= \frac{16\left[e_{14} e_2 + e_{14} e_3 + e_{14} e_5 + e_{14} e_6 + e_2 e_3 + e_2 e_5 + e_2 e_6 + e_3 e_5 + e_3 e_6 + e_5 e_6 - (e\leftrightarrow {\text{p}})\right]}{(p_1-p_4)^2 + 4} \nonumber\\ d^{x\phi\phi}_3 &= -\frac{16 \left[ - e_{14}(e_2+e_5) - e_{14}(e_3+e_6) - (e_2+e_5) (e_3+e_6) - (e\leftrightarrow {\text{p}})\right]}{\left[ p_{14}^2 + 4 \right]\left[ (p_2+p_5)^2 + 4 \right] \left[ (p_3+p_6)^2 + 4 \right]}\nonumber\\&\left[ e_2^2 + e_5^2 + e_2 e_5 - (e\leftrightarrow {\text{p}})\right] \left[e_3^2 + e_6^2 + e_3 e_6 - (e\leftrightarrow {\text{p}})\right] \nonumber\\ d^{x\phi\phi}_4 &= -\frac{32\left[ e_2^2 + e_5^2 + e_2 e_5 - (e\leftrightarrow {\text{p}})\right]}{\left[ (p_2+p_5)^2 + 4 \right]\left[ (p_3+p_6)^2 + 4 \right]} \left[ e_3^2 + e_6^2 + e_3 e_6 - (e\leftrightarrow {\text{p}}) \right] \nonumber\\ d^{x\phi\phi}_5 &= \frac{32}{\left[ (p_1+p_2-p_4)^2 + 4 \right]}\left[ 4 - ((p_1+p_2-p_4)\cdot p_3 + (p_1+p_2-p_4)\cdot p_5 + \right.\nonumber\\&\left. + (p_1+p_2-p_4)\cdot p_6 + p_3\cdot p_5 + p_3\cdot p_6 + p_5\cdot p_6 )\right] \nonumber\\ d^{x\phi\phi}_6 &= \frac{128\left[ ({\text{p}}_1+{\text{p}}_2)^2 + 1 \right]}{\left[ (p_1+p_2)^2 + 2 \right]} \nonumber\\ d^{x\phi\phi}_7 &= \frac{128\left[ ({\text{p}}_4-{\text{p}}_2)^2 + 1 \right]}{\left[ (p_4-p_2)^2 + 2 \right]} \nonumber\\ d^{x\phi\phi}_8 &= \frac{16\left[ e_{14} e_2 + e_{14} (e_3+e_5+e_6) + e_2 (e_3+e_5+e_6) - (e\leftrightarrow {\text{p}})\right]}{\left[ (p_1-p_4)^2 + 4 \right]\left[ (p_3+p_5+p_6)^2 + 4 \right]} \left[ 4 - ( p_3\cdot p_6 + p_5\cdot p_6 + \right.\nonumber\\&\left. -(p_3+p_5+p_6)\cdot p_3 -(p_3+p_5+p_6)\cdot p_5 -(p_3+p_5+p_6)\cdot p_6 + p_3\cdot p_5 )\right] \nonumber\\ d^{x\phi\phi}_9 &= -\frac{32\left[ ({\text{p}}_1+{\text{p}}_2)^2 + 1 \right]}{\left[ (p_1+p_2)^2 + 2 \right]\left[ (p_3+p_5+p_6)^2 + 4 \right]}\left[ 4 - (-(p_3+p_5+p_6)\cdot p_3 + \right.\nonumber\\&\left. -(p_3+p_5+p_6)\cdot p_5 -(p_3+p_5+p_6)\cdot p_6 + p_3\cdot p_5 + p_3\cdot p_6 + p_5\cdot p_6 )\right] \nonumber\\ d^{x\phi\phi}_{10} &= -\frac{32\left[ ({\text{p}}_2-{\text{p}}_4)^2 + 1 \right]}{\left[ (p_2-p_4)^2 + 2 \right]\left[ (p_3+p_5+p_6)^2 + 4 \right]}\left[ 4 - (-(p_3+p_5+p_6)\cdot p_3 + \right.\nonumber\\&\left. -(p_3+p_5+p_6)\cdot p_5 -(p_3+p_5+p_6)\cdot p_6 + p_3\cdot p_5 + p_3\cdot p_6 + p_5\cdot p_6 )\right] \nonumber\\ d^{x\phi\phi}_{11} &= \frac{128\left[ ({\text{p}}_1+{\text{p}}_2+{\text{p}}_3)^2 + 1 \right]}{\left[ (p_1+p_2+p_3)^2 + 2 \right]} \nonumber\\ d^{x\phi\phi}_{12} &= \frac{16\left[ e_3^2 + e_6^2 + e_3 e_6 - (e\leftrightarrow {\text{p}})\right] }{\left[ (p_1-p_4)^2 + 4 \right]\left[ (p_3+p_6)^2 + 4 \right]} \left[ 4 - ((p_1-p_4)\cdot p_2 + (p_1-p_4)\cdot (p_3+p_6) + \right.\nonumber\\&\left. + (p_1-p_4)\cdot p_5 + (p_3+p_6)\cdot p_2 + (p_3+p_6)\cdot p_5 + p_2\cdot p_5 )\right] \nonumber\\ d^{x\phi\phi}_{13} &= -\frac{64\left[ e_3^2 + e_6^2 + e_3 e_6 - (e\leftrightarrow {\text{p}})\right] }{\left[ (p_3+p_6)^2 + 4 \right]} \nonumber\\ d^{x\phi\phi}_{14} &= \frac{32\left[ ({\text{p}}_1+{\text{p}}_2+{\text{p}}_5)^2 + 1 \right]\left[ e_2^2 + e_5^2 + e_2 e_5 - (e\leftrightarrow {\text{p}})\right]}{\left[ (p_1+p_2+p_5)^2 + 2 \right]\left[ (p_2+p_5)^2 + 4 \right] \left[ (p_3+p_6)^2 + 4 \right]}\left[e_3^2 + e_6^2 + e_3 e_6 - (e\leftrightarrow {\text{p}})\right]\nonumber\\ d^{x\phi\phi}_{15} &= \frac{64\left[ ({\text{p}}_1+{\text{p}}_2+{\text{p}}_5)^2 + 1 \right]\left[e_3^2 + e_6^2 + e_3 e_6 - (e\leftrightarrow {\text{p}})\right]}{\left[ (p_1+p_2+p_5)^2 + 2 \right]\left[ (p_3+p_6)^2 + 4 \right]}\nonumber\\ d^{x\phi\phi}_{16} &= \frac{64\left[ ({\text{p}}_4-{\text{p}}_2-{\text{p}}_5)^2 + 1 \right]\left[e_3^2 + e_6^2 + e_3 e_6 - (e\leftrightarrow {\text{p}})\right]}{\left[ (p_4-p_2-p_5)^2 + 2 \right]\left[ (p_3+p_6)^2 + 4 \right]}\nonumber\\ d^{x\phi\phi}_{17} &= -\frac{128\left[ ({\text{p}}_1+{\text{p}}_2)^2 + 1 \right]\left[ ({\text{p}}_1+{\text{p}}_2+{\text{p}}_3)^2 + 1 \right]}{\left[ (p_1+p_2)^2 + 2 \right]\left[ (p_1+p_2+p_3)^2 + 2 \right]}\nonumber\\ d^{x\phi\phi}_{18} &= -\frac{128\left[ ({\text{p}}_4-{\text{p}}_6)^2 + 1 \right]\left[ ({\text{p}}_1+{\text{p}}_2+{\text{p}}_3)^2 + 1 \right]}{\left[ (p_4-p_6)^2 + 2 \right]\left[ (p_1+p_2+p_3)^2 + 2 \right]}\nonumber\\ d^{x\phi\phi}_{19} &= -\frac{64\left[ ({\text{p}}_1+{\text{p}}_2)^2 + 1 \right]\left[ ({\text{p}}_4-{\text{p}}_5)^2 + 1 \right]\left[e_3^2 + e_6^2 + e_3 e_6 - (e\leftrightarrow {\text{p}})\right]}{\left[ (p_1+p_2)^2 + 2 \right]\left[ (p_4-p_5)^2 + 2 \right]\left[ (p_3+p_6)^2 + 4 \right]}\nonumber\\ d^{x\phi\phi}_{20} &= -\frac{32\left[ ({\text{p}}_1+{\text{p}}_2)^2 + 1 \right] \left[e_3^2 + e_6^2 + e_3 e_6 - (e\leftrightarrow {\text{p}})\right]}{\left[ (p_1+p_2)^2 + 2 \right]\left[ (p_1+p_2-p_4)^2 + 4 \right]\left[ (p_3+p_6)^2 + 4 \right]}\nonumber\\& \left[(e_1+e_2-e_4) e_5 + (e_1+e_2-e_4)(e_3+e_6) + (e_3+e_6) e_5 - (e\leftrightarrow {\text{p}})\right] \nonumber\\ d^{x\phi\phi}_{21} &= \frac{32\left[ ({\text{p}}_4-{\text{p}}_2)^2 + 1 \right] \left[e_3^2 + e_6^2 + e_3 e_6 - (e\leftrightarrow {\text{p}})\right]}{\left[ (p_4-p_2)^2 + 2 \right]\left[ (p_3+p_5+p_6)^2 + 4 \right]\left[ (p_3+p_6)^2 + 4 \right]}\nonumber\\& \left[(e_3+e_5+e_6) e_5 + (e_3+e_5+e_6)(e_3+e_6) - (e_3+e_6) e_5 - (e\leftrightarrow {\text{p}})\right] \nonumber\\ d^{x\phi\phi}_{22} &= -\frac{16 \left[e_3^2 + e_6^2 + e_3 e_6 - (e\leftrightarrow {\text{p}})\right]}{\left[ (p_1-p_4)^2 + 4 \right]\left[ (p_1-p_4+p_2)^2 + 4 \right]\left[ (p_3+p_6)^2 + 4 \right]}\nonumber\\& \left[- (e_1+e_2-e_4) (e_1-e_4) - (e_1+e_2-e_4) e_2 + (e_1-e_4) e_2 - (e\leftrightarrow {\text{p}})\right]\nonumber\\&\left[(e_1+p_2-p_4) (e_3+e_6) + (e_1+p_2-p_4) e_5 + (e_3+e_6) e_5 - (e\leftrightarrow {\text{p}})\right] \nonumber\\ d^{x\phi\phi}_{23} &= \frac{32 \left[e_3^2 + e_6^2 + e_3 e_6 - (e\leftrightarrow {\text{p}})\right]}{\left[ (p_1-p_4+p_2)^2 + 4 \right]\left[ (p_3+p_6)^2 + 4 \right]}\nonumber\\& \left[(e_1+e_2-e_4) (e_3+e_6) + (e_1+e_2-e_4) e_5 + (e_3+e_6) e_5 - (e\leftrightarrow {\text{p}})\right] \nonumber\\ d^{x\phi\phi}_{24} &= -\frac{128 \left[ ({\text{p}}_1+{\text{p}}_2)^2 + 1 \right]\left[ ({\text{p}}_4-{\text{p}}_6)^2 + 1 \right]}{\left[ (p_1+p_2)^2 + 2 \right]\left[ (p_4-p_6)^2 + 2 \right]} \nonumber\\ d^{x\phi\phi}_{25} &= -\frac{64 \left[ ({\text{p}}_1+{\text{p}}_2)^2 + 1 \right]\left[ ({\text{p}}_1+{\text{p}}_2+{\text{p}}_5)^2 + 1 \right]}{\left[ (p_1+p_2)^2 + 2 \right]\left[ (p_1+p_2+p_5)^2 + 2 \right]\left[ (p_3+p_6)^2 + 4 \right]}\left[e_3^2 + e_6^2 + e_3 e_6 - (e\leftrightarrow {\text{p}})\right] \nonumber\\ d^{x\phi\phi}_{26} &= -\frac{64 \left[ ({\text{p}}_4-{\text{p}}_2)^2 + 1 \right]\left[ ({\text{p}}_4-{\text{p}}_2-{\text{p}}_5)^2 + 1 \right]}{\left[ (p_4-p_2)^2 + 2 \right]\left[ (p_4-p_2-p_5)^2 + 2 \right]\left[ (p_3+p_6)^2 + 4 \right]}\left[e_3^2 + e_6^2 + e_3 e_6 - (e\leftrightarrow {\text{p}})\right] \nonumber\\ d^{x\phi\phi}_{27} &= \frac{64 \left[ ({\text{p}}_1+{\text{p}}_2)^2 + 1 \right]}{\left[ (p_1+p_2)^2 + 2 \right]\left[ (p_3+p_6)^2 + 4 \right]}\left[e_3^2 + e_6^2 + e_3 e_6 - (e\leftrightarrow {\text{p}})\right] \nonumber\\ d^{x\phi\phi}_{28} &= \frac{64 \left[ ({\text{p}}_4-{\text{p}}_2)^2 + 1 \right]}{\left[ (p_4-p_2)^2 + 2 \right]\left[ (p_3+p_6)^2 + 4 \right]}\left[e_3^2 + e_6^2 + e_3 e_6 - (e\leftrightarrow {\text{p}})\right] \nonumber\\ d^{x\phi\phi}_{29} &= \frac{128 \left[ ({\text{p}}_1+{\text{p}}_2)^2 + 1 \right]\left[ ({\text{p}}_1+{\text{p}}_2+{\text{p}}_3)^2 + 1 \right]\left[ ({\text{p}}_4-{\text{p}}_5)^2 + 1 \right]}{\left[ (p_1+p_2)^2 + 2 \right]\left[ (p_1+p_2+p_3)^2 + 2 \right]\left[ (p_4-p_5)^2 + 2 \right]}\end{aligned}$$ [^1]: Notice that the Lagrangian is $SU(4)$ invariant, however it does not admit a trivial vacuum and one has to break the $SU(4)$ symmetry as in to obtain a well defined perturbative expansion. [^2]: Note the different convention for the coupling with respect to [@Fioravanti:2015dma], where $g=\frac{\sqrt{\lambda}}{2\sqrt{2}\pi}$.
--- abstract: 'A coupled atom-molecule condensate with an intraspecies Feshbach resonance is employed to explore matter wave bistability both in the presence and in the absence of a unidirectional optical ring cavity. In particular, a set of conditions are derived that allow the threshold for bistability, due both to two-body s-wave scatterings and to cavity-mediated two-body interactions, to be determined analytically.  The latter bistability is found to support, not only transitions between a mixed (atom-molecule) state and a pure molecular state as in the former bistability, but also transitions between two distinct mixed states.' author: - 'Hong Y. Ling' title: Bistability in Feshbach Resonance --- Introduction ============ The subject of optical bistability [@gibbs85], to which Lorenzo Narducci contributed greatly during his prime years of life, was brought to spotlight again by recent experimental demonstration of optical bistability in a microcavity with a cavity field at the level of a single photon [@kurn07; @esslinger08].  Instead of thermal gases, as typically employed by Lorenzo’s generation, where the thermal de Broglie’s wavelength of the particles is far smaller than the interparticle spacing, more recent trend in cavity quantum electrodynamics (QED) focuses on cavity systems with ultracold quantum gases [@kurn07; @esslinger08], opening the door to studies such as cavity-mediated bistable Mott-insulator to superfluid phase transition [@lewenstein08; @chen09], where the particle statistical nature becomes an essential feature of the cavity problem.  In contrast to 1980s, when the surge of interest in optical bistability was inspired by the prospect of its use as a switch in an all-optical computer [@gibbs85], the current surge of interest in the same topic was, however, motivated largely by the equivalence of the cavity condensate system [@esslinger08] to the cavity opto-mechanical system [@braginsky80].  The study of the latter falls into the realm of cavity optomechanics, a rapidly emerging field which aims to use the cavity-assisted radiation force [@ritsch00] to cool mechanical device, ranging from nano- or micro-mechanical cantilevers to macroscopic mirrors in LIGO project, down ultimately to their quantum mechanical ground state [@girvin09]. \[ptb\] [schematics.ps]{} In this paper, we focus on a cavity system containing a coupled homonuclear atom-molecule condensate as illustrated in Fig. \[Fig:schematic\].  In the absence of the cavity, molecular state $\left\vert 2\right\rangle $ is coupled only to free atomic state $\left\vert 1\right\rangle $ by an intraspecies Feshbach resonance [@feshbach] characterized with a strength $\alpha^{\prime}$ and a detuning $\epsilon^{\prime}$ \[the shaded part in Fig. \[Fig:schematic\](a)\].  In the cavity QED setting, state $\left\vert 2\right\rangle $ is coupled, besides to state $\left\vert 1\right\rangle $, also to excited molecular state $\left\vert 3\right\rangle $ by a unidirectional ring cavity of a total length $\mathcal{L}$.  The cavity is driven by an external laser of wavenumber $k$, polarization $\mathbf{\hat {\varepsilon}}$, and frequency $\omega$, which is tuned far away from the $\left\vert 3\right\rangle \leftrightarrow\left\vert 2\right\rangle $ molecular transition.  In addition, the cavity is assumed to possess a sufficiently large intermode spacing $\mathcal{L}/c$ (with $c$ being the light speed in vacuum) so that only the cavity mode with a longitudinal mode frequency $\omega_{c}$ closest to $\omega$ is relevant to our study, where $\omega_{c}\mathcal{L}/c$ equals integer multiple of $2\pi$.  Further, the cavity mode is assumed to overlap with the condensate in a spatial region, characterized with a length $L$ and a cross-sectional area $A$, large enough so that the condensate can be treated as a uniform system with an effective volume $V_{a}=LA$ and a total atom number $N_{a}$ (counting those in molecules).   A similar cavity + condensate system has recently been studied [@search07], in which atoms are converted into molecules by photoassociation as opposed to magnetoassociation (Feshbach resonance) in our model.  There, absorption of a cavity photon will convert two atoms into a molecule, and emission of a cavity photon will dissociate two atoms into a molecule.  This is reminiscent of the absorptive bistability in a driven optical cavity containing an ensemble of two-level atoms. In contrast, the cavity field in our model is to introduce a phase shift to the molecular field, not the population exchange between atoms and molecules.  As a result, the bistability in our model is of dispersive nature.  Further, interaction between electronic dipoles and a cavity field in Ref. [@search07] gives rise to atom-molecule coupling, which, since the cavity field is itself a dynamical variable, changes with time. In contrast, in our model, it is the hyperfine interaction - the coupling between electron spins and nuclear spins of two colliding atoms that result in atom-molecule coupling, which is therefore fixed by the atomic internal structure, independent of cavity field.  In this respect, our model is analogous more closely to a cavity system with a spin-1 condensate (also recently proposed [@zhoulu09]) than to a cavity system with photoassociation [@search07]; atom exchanges among different internal states in the spin-1 condensate are accomplished by spin-exchange interaction, which is also independent of cavity field.   The interest in Feshbach resonance stems primarily from its use as an effective tool to coherently create molecular BECs from the existing atomic BECs.  In a typical experiment,  conversion of atomic BECs into molecular BECs is carried out by ramping the Feshbach detuning across the resonance.  This method relies on the existence of a mixed (or dressed) atom-molecule state [@stoof04a], and the ability of this state to change its composition from predominantly atomic to predominantly molecular species when the Feshbach detuning $\epsilon^{\prime}$ is tuned from above to bellow the resonance.  The question that we want to pursue, in this paper, is how the molecular population in state $\left\vert 2\right\rangle $ can be made to vary with the Feshbach detuning in a bistable fashion, instead of monotonously as in typical situations.  A bistable crossover adds new meaning to the Feshbach resonance: whether the system is in the atomic or molecular extreme is determined not only by the Feshbach detuning but also by the history of the system.   To this end, we first formulate, in Sec. II, a semiclassical mean-field description of our model in the limit where both the excited molecular field and the optical cavity field can be adiabatically eliminated.  We then explore the matter-wave bistability due to the two-body s-wave collisions in Sec. III A and that due to the cavity-mediated effective interaction between two Feshbach molecules in Sec. III B.  Finally, we provide a summary in Sec. IV. The Basic Equations =================== In this section, we take a semiclassical approach, in which optical fields are treated classically while matter fields are treated quantum mechanically, to formulate a theoretical description of the proposed cavity + condensate system.  This is the same approach that Lorenzo embraced in many of his works [@lasers], except that quantization is now performed at a level for a many-body system, instead of a single-body system as in a typical semiclassical laser theory.  To begin with, we expand optical field $\mathbf{E}\left( \mathbf{r},t\right) $ in terms of the slowly varying amplitude $F$ in space $\mathbf{r}$ and time $t$ according to $$\mathbf{E}\left( \mathbf{r},t\right) =\frac{1}{2}F\mathbf{\hat{\varepsilon}% }e^{i\left( kz-\omega t\right) }+c.c,\label{E}%$$ and matter field $\hat{\psi}\left( \mathbf{r},t\right) $ in terms of $\hat{\psi}_{i}=\hat{c}_{i}/\sqrt{N_{a}}$ according to$$\hat{\psi}\left( \mathbf{r}\text{,}t\right) =\sqrt{n_{a}}\left[ \hat{\psi }_{1}\left\vert 1\right\rangle +\hat{\psi}_{2}\left\vert 2\right\rangle +\hat{\psi}_{3}e^{i\left( kz-\omega t\right) }\left\vert 3\right\rangle \right] ,\label{psiField}%$$ where $\hat{c}_{i}$ denotes the operator in momentum space for annihilating a bosonic particle in state $\left\vert i\right\rangle $ and $n_{a}=N_{a}/V_{a}$ is the total atom number density.  The expansion in Eq. (\[psiField\]) is carried out in a frame rotating at the laser frequency $\omega$.  In arriving at Eq. (\[psiField\]), we have assumed that the particles in states $\left\vert 1\right\rangle $ and $\left\vert 2\right\rangle $ are all condensed to their respective zero momentum modes, and those in state $\left\vert 3\right\rangle $ to the mode with $\hbar k$ momentum in accordance with momentum conservation during photon-atom interaction. The coupled atom-molecule system, within these approximations, is then described by the following Hamiltonian $$\begin{aligned} \hat{H}/\hbar N_{a} & =\epsilon^{\prime}\hat{\psi}_{2}^{\dag}\hat{\psi}% _{2}+\sqrt{n_{a}}\left( \frac{\alpha^{\prime}}{2}\hat{\psi}_{2}^{\dag}% \hat{\psi}_{1}^{2}+h.c\right) \nonumber\\ & +n_{a}\frac{\chi_{ij}^{\prime}}{2}\hat{\psi}_{i}^{\dag}\hat{\psi}_{j}^{\dag }\hat{\psi}_{j}\hat{\psi}_{i}\text{ }\left( i,j=1\text{ or }2\right) \nonumber\\ & +\left( \epsilon^{\prime}-\Delta_{a}\right) \hat{\psi}_{3}^{\dag}\hat {\psi}_{3}-\left( \Omega\hat{\psi}_{3}^{\dag}\hat{\psi}_{2}+h.c\right) ,\label{Hamiltonian1}%\end{aligned}$$ where repeated indices are to be summed from 1 to 2.  In Eq. (\[Hamiltonian1\]), the first line describes the Feshbach resonance of strength $\alpha^{\prime}$, the second line the s-wave collisions of strength $\chi_{ij}^{\prime}$ $(=\chi_{ji}^{\prime})$ between states $\left\vert i\right\rangle $ and $\left\vert j\right\rangle $, and the last line the part of Hamiltonian involving excited state $\left\vert 3\right\rangle $. In the last line, the first term denotes the energy of state $\left\vert 3\right\rangle $ in the rotating frame, where $\Delta_{a}$ is the laser detuning, and the second term stands for the laser-induced electric dipole interaction, where $\Omega=\mu_{32}F/2\hbar$ is the Rabi frequency, $\mu _{32}=\left\langle 3\right\vert \mathbf{\hat{\mu}}\cdot\mathbf{\hat {\varepsilon}}\left\vert 2\right\rangle $ the matrix element, and $\mathbf{\hat{\mu}}$ the electric dipole moment operator.  Finally, collisions involving the final excited state $\left\vert 3\right\rangle $ are ignored since state $\left\vert 3\right\rangle $ remains virtually empty in our model.   An important concept in the semiclassical approach is the macroscopic polarization defined as $\mathbf{P}\left( \mathbf{r},t\right) =$ $\left\langle \hat{\psi}^{\dag}\left( \mathbf{r}\text{,}t\right) \mathbf{\hat{\mu}}\hat{\psi}\left( \mathbf{r}\text{,}t\right) \right\rangle $.  This polarization is found, when the use of both Eq. (\[psiField\]) and the selection rule are made, to possess the same mathematical form as the optical field in Eq. (\[E\]),$$\mathbf{P}\left( \mathbf{r},t\right) =\frac{1}{2}P\mathbf{\hat{\varepsilon}% }e^{i\left( kz-\omega t\right) }+c.c.,\label{polarization}%$$ where $P=2n_{a}\mu_{32}\left\langle \hat{\psi}_{2}\hat{\psi}_{3}^{\ast }\right\rangle $ represents the slowly varying part of the polarization. The evolution of the optical field is then governed by the Maxwell’s equation, which, under the slowly varying envelope approximation, takes the form $$c\frac{\partial\Omega}{\partial z}+\frac{\partial\Omega}{\partial t}% =i\frac{\mu_{0}\omega c}{4\hbar}\mu_{32}P\text{,}\label{Maxwell}%$$ where $\mu_{0}$ $\left( \epsilon_{0}\right) $ is the magnetic (electric) permeability in vacuum.  Equation (\[Maxwell\]) clearly shows that polarization plays the role of a bridge between the classical optical field in Eq. (\[E\]) and the quantum matter fields in Eq. (\[psiField\]).   The final component in the semiclassical approach pertaining to any QED problems is the boundary condition, which, in our case and under the assumption that both input and output mirrors have the same reflectivity $R$ (transmissivity $T=1-R$), can be cast into the form [@orozco89] $$\Omega\left( 0,t\right) =TY+R\Omega\left( L,t-\Delta t\right) e^{i\Delta_{c}\mathcal{L}/c},\label{boundary}%$$ where $Y$ is the scaled amplitude of the incident field, $\Delta t=\left( \mathcal{L}-L\right) /c$ the transit free propagation time inside the cavity, and $\Delta_{c}=\omega-\omega_{c}$ the cavity mode frequency detuning.  Equation (\[boundary\]) links the field entering $\left( z=0\right) $ to that leaving the condensate $\left( z=L\right) $, and hence implements the concept of feedback, which is the most important feature of any optical cavities.  The school led by Bonifacio, Lugiato, and Narducci distinguishes itself by its insistence on a rigorous (first-principle) treatment of the boundary condition - a treatment made up of both a transformation mapping two non-isochronous events in Eq. (\[boundary\]) into two isochronous events [@lugiato85] and a set of conditions embodying the notion of mean-field limit [@lugiato84].  Following this treatment, we explicitly eliminate the spatial derivative in Eq. (\[Maxwell\]), rendering Eq. (\[Maxwell\]) into $$\frac{d\Omega}{dt}=\left( i\Delta_{c}-\kappa\right) \Omega+\kappa Y+ig^{2}N_{a}\left\langle \hat{\psi}_{2}\hat{\psi}_{3}^{\dag}\right\rangle ,\label{cavity field equation}%$$ where $\kappa=c\left\vert \ln R\right\vert /\mathcal{L}$ is the cavity damping rate, $g=\mu_{32}\sqrt{\omega/2\hbar\epsilon_{0}V_{c}}$ the Rabi frequency per photon, and $V_{c}=\mathcal{L}A$ the total cavity volume. In this paper, we restrict our study to the parameter regime in which both the time scale for the optical field (on the order of $1/\kappa$) and that for the molecular field of state $\left\vert 3\right\rangle $ (on the order of $1/\Delta_{a}$) are far shorter than those for the Feshbach degrees of freedom, namely, the matter fields corresponding to states $\left\vert 1\right\rangle $ and $\left\vert 2\right\rangle $.  This restriction allows us to adiabatically eliminate the former fast variables in favor of the latter slow ones, reducing, from the Heisenberg’s equations for $\hat{\psi}_{i}$ and the Maxwell’s equation for $\Omega$, a set of equations involving only the slow degrees of freedom $\psi_{1}$ and $\psi_{2}$: \[two fields\]$$\begin{aligned} i\frac{d\psi_{1}}{dt} & =\chi_{1i}n_{i}\psi_{1}+\alpha\psi_{2}\psi_{1}^{\ast },\\ i\frac{d\psi_{2}}{dt} & =\left[ \epsilon^{\prime}+\chi_{2i}n_{i}+F\left( n_{2}\right) \right] \psi_{2}+\frac{\alpha}{2}\psi_{1}^{2}.\end{aligned}$$ where we have adopted the standard mean-field theory, treating $\hat{\psi}% _{i}$ as $c$-numbers $\psi_{i}$.  In Eqs. (\[two fields\]), $n_{1}% =\left\vert \psi_{1}\right\vert ^{2}$, $n_{2}=\left\vert \psi_{2}\right\vert ^{2}$, $\alpha=\alpha^{\prime}\sqrt{n_{a}}$, and $\chi_{ij}=\chi_{ij}^{\prime }n_{a}$ are the renormalized quantities, and $$F\left( n_{2}\right) =\frac{\kappa^{2}\eta N_{c}}{\kappa^{2}+\left( \Delta_{c}-\eta N_{a}n_{2}\right) ^{2}},\label{f(n2)}%$$ represents the phase shift to the molecular field induced by the cavity optical field, where $N_{c}=Y^{2}/g^{2}$ represents the number of injected intracavity photons and $\eta=g^{2}/\Delta_{a}$ measures the effective atom-cavity coupling strength. (In this paper,without loss of generality, we limit $\Delta_{a}>0$ so $\eta$ is always positive.) Discussions ===========  In our system, stationary solutions can be divided into mixed atom-molecule states in which each species has a *finite* density $n_{i}$, and pure molecular or atomic phase in which one of species has a zero density.  In this section, we study bistability in the Feshbach process by focusing on the mixed atom-molecule state where each species can be described by the field $\psi_{i}=\sqrt{n_{i}}e^{i\theta_{i}}$ characterized with a well defined phase $\theta_{i}$. (Note that such a description cannot be applied to pure molecular or atomic phases as the phase associated with an empty component is not defined.)   For such a state, we can apply the total atom number conservation $n_{1}+2n_{2}=1$ and simplify Eqs. (\[two fields\]) into a set of equations involving only two variables - a molecular density $n\equiv n_{2}$ and a phase mismatch $\theta=\theta_{2}-2\theta_{1}$.  This set of equations, in a unit system in which $\epsilon=$ $(\epsilon^{\prime}-\chi _{12}-2\chi_{11})/\alpha$ is defined as the effective Feshbach detuning, $\chi=[\chi_{22}+4\left( \chi_{11}-\chi_{12}\right) ]/\alpha$ as the effective Kerr nonlinear coefficient for the molecular field, $\delta =\Delta_{c}/\kappa$ as the cavity detuning, and $\tau=\alpha t$ as the time, take the form \[two variable equation\]$$\begin{aligned} \frac{dn}{d\tau} & =-\left( 1-2n\right) \sqrt{n}\sin\theta,\\ \frac{d\theta}{d\tau} & =\epsilon+\chi n+\frac{1}{2}\frac{1-6n}{\sqrt{n}}% \cos\theta+f\left( n\right) ,\end{aligned}$$ where $$f\left( n\right) =2\frac{B/C}{1+\left( \delta-Cn\right) ^{2}% }.\label{f scaled}%$$ In arriving at Eqs. (\[two variable equation\]), we have replaced $N_{a}$ and $N_{c}$ in Eq. (\[f(n2)\]) in favor of two unitless parameters, $$C=\eta N_{a}/\kappa=g^{2}N_{a}/\Delta_{a}\kappa,\label{C}%$$ and $$B=\eta^{2}N_{c}N_{a}/\left( 2\kappa\alpha\right) .\label{B}%$$ In contrast to $C$, which is bose enhanced by atom number only, $B$ is bose enhanced by both photon and atom numbers.  Note that when $\Delta_{a}$ is replaced with the decay rate of the excited state $\left\vert 3\right\rangle $, $C$  becomes the so-called atomic cooperative parameter [@kimble94].  As we will see shortly, cavity-mediated bistability depends crucially on the values of $C$ and $B$.   As in other studies [@zhou07; @radzihovsky08], Eqs. (\[two variable equation\]) of the type including Feshbach resonance support two branches of steady-states: one with $\theta=0$ and the other with $\theta=\pi$.  This feature is expected since at zero temperature the intraspecies Feshbach resonance represents a matter wave analog of the second harmonic generation in nonlinear optics where the phase-matching condition plays an important role.  In this paper, without the loss of generality, we take $\alpha>0$. Under such a circumstance, the branch with $\theta=\pi$ not only always has a lower energy than the branch with $\theta=0$, but also has the property of consisting primarily atom species in the limit of a large positive Feshbach detuning.  For these reasons, we will focus on the branch with $\theta=\pi$, determined at steady state by$$\epsilon=-\chi n+\frac{1}{2}\frac{1-6n}{\sqrt{n}}-f\left( n\right) ,\label{epsilon steady state}%$$ where for notational simplicity, same symbols are used to stand for the steady state variables. To carry out the stability analysis, we apply the standard linearization procedure that Lorenzo used extensively in the context of his interest in laser instabilities [@narducci88; @abraham85], and derive from Eqs. (\[two variable equation\]) a set of linearly coupled equations $$\frac{d}{d\tau}\left( \begin{array} [c]{c}% \delta n\\ \delta\theta \end{array} \right) =\left[ \begin{array} [c]{cc}% 0 & \left( 1-2n\right) \sqrt{n}\\ -\frac{d\epsilon}{dn} & 0 \end{array} \right] \left( \begin{array} [c]{c}% \delta n\\ \delta\theta \end{array} \right) ,\label{linear stability}%$$ where $\delta n$ and $\delta\theta$ are small departures from the corresponding steady state variables, and $\epsilon$ is given by Eq. (\[epsilon steady state\]).  The eigenvalues of Eqs. (\[linear stability\]) are then found to take two values: $\pm\sqrt{-\left( 1-2n\right) \sqrt{n}d\epsilon/dn}$, which, since $n<0.5$, clearly indicate that only when $d\epsilon/dn>0$ or equivalently when $dn/d\epsilon$ $>0$ can the eigenvalues become complex.  In another words, any state at which the slope of $n$ as a function of $\epsilon$ is positive is unstable.   Note that such a conclusion may not hold, if we do not impose, in the previous section, the conditions that allow $\psi_{3}$ and $\Omega$ to be adiabatically eliminated.  In the case of absorptive optical bistability, it is well known that the upper branch which is stable according to the slope criterion may become unstable; instability there is manifested in the form of self-pulsings [@bonifacio78]. Thus, we see that the stability analysis here amounts to analyzing the points at which $d\epsilon/dn=0$, which, by definition, simply corresponds to the critical transition points in a bistable (or multi-stable) system.   As a result, in what follows, we carry out bistability study by focusing on the equation $$d\left( n\right) =\chi-h\left( n\right) ,\label{depsilon}%$$ derived from Eq. (\[epsilon steady state\]) under the condition that $d\epsilon/dn=0$, where$$\begin{aligned} d\left( n\right) & =-4B\frac{\left( \delta-Cn\right) }{\left[ 1+\left( \delta-Cn\right) ^{2}\right] ^{2}},\label{df/dn2}\\ h\left( n\right) & =-\frac{3}{2}n^{-1/2}-\frac{1}{4}n^{-3/2}.\label{h}%\end{aligned}$$ Collision-Induced Bistability ----------------------------- In our model, the molecular field (at state $\left\vert 2\right\rangle $) is subject to two types of self phase modulations: one, described by a Kerr type of nonlinear term $\chi n,$ originates from short-range s-wave scatterings, and another, described by $f\left( n\right) $ in Eq. (\[f scaled\]), stems from cavity-mediated long-range two-body collisions.  In order to differentiate their roles in the formation of matter wave bistability, we first remove the cavity component and study the bistability due solely to the Kerr nonlinearity by solving $$\chi=h\left( n\right) ,\label{chi no cavity}%$$ obtained from Eq. (\[depsilon\]) by setting $d\left( n\right) =0$.  As one may easily verify, $h\left( n\right) $ in Eq. (\[h\]) is a monotonously increasing function of $n$.  As a result, in order for Eq.(\[chi no cavity\]) to have real roots within $n<0.5$, $\chi$ must be less than $h\left( n=0.5\right) $ or $$\chi\leq\chi_{th}\equiv-2\sqrt{2}.\label{bistability condition f=0}%$$ This condition is similar to that of Ref. [@jiang09] for a heteronuclear atom-molecule system with an interspecies Feshbach resonance [@zhou07]. It simply reflects the fact that for bistability to take place, there must be a sufficiently strong positive feedback between the molecular population and the effective Feshbach detuning $\epsilon+\chi n$.  A negative $\chi$ fulfills this positive feedback; as can be seen, with a negative $\chi$, an increase in the molecular density decreases the effective detuning, which, in turn, further increases the molecular density, or vice versa.  Such a chain of positive reaction under condition (\[bistability condition f=0\]) can lead to the formation of the critical transition points around which the molecular population changes in a runaway fashion.  Indeed, under condition (\[bistability condition f=0\]), Eq. (\[depsilon\]) is found to support a critical point with a critical molecular density $n_{cri}^{\left( 1\right) }$ given by, $$\begin{aligned} \sqrt{n_{cri}^{\left( 1\right) }} & =\left( \frac{1}{16\left\vert \chi\right\vert }\right) ^{\frac{1}{3}}\left( \sqrt{1+\frac{2}{\chi^{2}}% }+1\right) ^{\frac{2}{3}}+\nonumber\\ & \left( \frac{1}{16\left\vert \chi\right\vert }\right) ^{\frac{1}{3}% }\left( \sqrt{1+\frac{2}{\chi^{2}}}-1\right) ^{\frac{2}{3}}-\frac{1}{2\chi },\label{n^(1)}%\end{aligned}$$ and a critical Feshbach detuning $\epsilon_{cri}^{\left( 1\right) }$ determined by Eq. (\[epsilon steady state\]) when $n$ is replaced with $n_{cri}^{\left( 1\right) }$ in Eq. (\[n\^(1)\]).  Figure 2(a) shows a typical example of bistability (with $\chi=2\chi_{th}$).  In addition to the mixed state, there is a pure molecular state with $n=0.5$ obtained from Eqs. (\[two fields\]), not from Eqs. (\[two variable equation\]) which, by definition, only holds for mixed states.  (Note that the pure atomic state is prohibited by the nature of intraspecies Feshbach resonance [@radzihovsky04; @stoof04; @radzihovsky08].)  The interception between this pure state and the mixed atom-molecular state defines the second critical point $$n_{cri}^{\left( 2\right) }=0.5,\epsilon_{cri}^{\left( 2\right) }% =-\frac{\chi}{2}-\sqrt{2}.$$ The threshold for bistability is reached when the two critical points become degenerate.  Clearly, this happens at $n=0.5$ when $\chi=\chi_{th}$.  Figure \[Fig:bistability1\](b) shows that when $\chi$ is bellow $\chi_{th}$, the size of the hysteresis loop (measured by $\epsilon_{cri}^{\left( 2\right) }-$ $\epsilon_{cri}^{\left( 1\right) }$) increases with $\left\vert \chi\right\vert $. \[ptb\] [lorenzoNoCavity.eps]{} To see the implication of condition (\[bistability condition f=0\]) to a realistic system, consider, for example, the Feshbach resonance located at the magnetic field $85.3$ mT in $^{23}$Na [@julienne00; @ling02].  This resonance has a width of $0.95\mu$T or equivalently a Feshbach coupling strength of $\alpha^{\prime}=4.22\times10^{-6}m^{3/2}s^{-1}$.   Taking the total atom number density to be $10^{20}$ m$^{-3}$ and using 3.4 nm as the s-wave scattering length for sodium atoms [@verhaar99], we find that $\chi_{11}=$ 1.18$\times10^{4}$ s$^{-1}$ and $\alpha=$4.22 $\times10^{4}$ s$^{-1}=3.58\chi_{11}$. If we further assume $\chi_{22}=\chi_{11}$, we see from Eq. (\[bistability condition f=0\]) that this requires $\chi_{12}% >\frac{1}{4}\left( \chi_{22}+4\chi_{11}+2\sqrt{2}\alpha\right) >3.77\chi_{11}$.   Cavity-Mediated Bistability --------------------------- The above example means to illustrate that Eq. (\[bistability condition f=0\]) can be fulfilled only when both the total atom density and the two-body interspecies collisional strength are sufficiently large, a condition which is difficult to meet under typical systems.  In this subsection, we turn our attention to the cavity model in Fig. \[Fig:schematic\], and pursue, from Eq. (\[depsilon\]), the question of under what cavity parameters can bistability occur even when condition (\[bistability condition f=0\]) breaks down.  Note that unlike Eq. (\[chi no cavity\]), which only contains one critical point, Eq. (\[depsilon\]) can typically support two critical points.  Thus, the threshold for bistability in this cavity model can, in principle, occur at any value of $n$, instead of always at $n=0.5$ as in the bare Feshbach model discussed in the previous subsection. To proceed, we first change Eq. (\[depsilon\]) into a quartic equation for$\sqrt{y}$ $$\left( \sqrt{y}\right) ^{4}-\gamma\sqrt{y}+1=0,\label{y}%$$ where $y=Cn-\delta$ and $\gamma=2\sqrt{B/\left[ \chi-h\left( n\right) \right] }$.  This change of variable is motivated by the realization that when condition (\[bistability condition f=0\]) breaks down or equivalently $\chi+2\sqrt{2}>0$, the quantity $\chi-h\left( n\right) $ is always positive  so that only when $Cn-\delta>0$ can Eq. (\[depsilon\]) holds.  To estimate the threshold condition at a given $n$, we regard Eq. (\[y\]) as a transcendental equation for $\delta$, and require Eq. (\[y\]) to support a real root of multiplicity 2 (another two roots are a complex conjugate pair).  This requirement allows us to conclude that bistability develops when $$B\geq B_{th}\left( n\right) \equiv\frac{4}{3\sqrt{3}}\left[ \chi-h\left( n\right) \right] ,\label{bistability condition with cavity}%$$ and the bistability threshold at a given $n$ (and $C$) is reached when $B=B_{th}\left( n\right) $ and $\delta=\delta_{th}\left( n\right) \equiv Cn-1/\sqrt{3}$. \[ptb\] [lorenzoCavity.eps]{} $B_{th}\left( n\right) $ as a function of $n$ is displayed in Fig. \[Fig:bistabilityCavity\](a).  This simple threshold relation seems to hold quite well as long as $C$ is sufficiently large.  Consider, for example, a cavity system with $C=20$.  Figure \[Fig:bistabilityCavity\](b) shows how $n$ changes with $\epsilon$ under different values of $B$.  The solid line, produced with $B$ $=B_{th}\left( 0.2\right) =4.73$ \[point A in Fig \[Fig:bistabilityCavity\](a)\] and $\delta=\delta_{th}\left( 0.2\right) =3.42$, shows that the bistability threshold indeed occurs at the theoretically predicted location with $n=0.2$ \[and $\epsilon=-0.58$ determined from Eq. (\[epsilon steady state\])\].  The dashed and dot-dashed lines in Fig. \[Fig:bistabilityCavity\](b), produced with $B=2B_{th}\left( 0.2\right) $ and $B=4B_{th}\left( 0.2\right) $, respectively, clearly shows that the usual bistable behavior emerges when $B$ is increased beyond its threshold value $B_{th}\left( 0.2\right) $.  Recall that without a cavity, a bistable transition can only take place between a mixed and a pure molecular state.  In the present situation with an optical cavity, we see that a new feature appears - a bistable transition can also take place between two different mixed states (the dashed line). To see what condition (\[bistability condition with cavity\]) means to the cavity field, we consider a small-sized micro-ring cavity of total length $\mathcal{L}=200$ $\mu m$ and finesse $\mathcal{F}=3.14\times10^{5}$.  The cavity is driven by an external laser tuned $\Delta_{a}=2\pi\times100$ GHz away from the $D_{2}$ line, characterized with a wavelength $780$ nm and a linewidth $2\pi\times3$ MHz, of sodium atoms confined to an effective spatial region of $L=30\mu m$ and $A=\left( 10\mu m\right) ^{2}$.  In such a cavity-condensate system, we have $\eta=2\pi\times0.0052$ MHz and $\kappa =2\pi\times23.87$ MHz.  Then, by using the same Feshbach resonance and the atom number density in the previous subsection, we estimate that the minimum number of photons that must be present inside the cavity to produce bistability is $8.6\times10^{3}$.  This figure is three orders of magnitude smaller than the threshold photon number in a typical laser [@kimble94], and can be further reduced with an appropriate choice of the system parameters.   Matter wave bistability at small photon numbers can be understood as follows.  In contrast to the phase shift due to the s-wave scattering, which is linearly proportional to the molecular density, the phase shift, arising from the feedback between optical and matter fields, is nonlinearly proportional to the molecular density by way of Eq. (\[f scaled\]) in a resonant fashion. On one hand, the sensitivity (the change of this phase shift versus the change of the molecular density) is bose-enhanced by the collective nature of the condensate system.  On the other hand, the peak of this phase shift for a given cavity photon number can be significantly amplified in a microcavity environment where photons are confined into a tiny volume.  As a result, molecular bistability is possible under a weak cavity field. Conclusion ========== In this paper, we have studied the matter wave bistability in an intraspecies Feshbach resonance model with and without the assistance of an optical cavity.  In particular, we have arrived at a set of conditions that allow the bistability thresholds under the two different settings to be estimated analytically.  In the absence of a cavity, bistability is possible only when the effective Kerr nonlinearity stemming from s-wave scatterings \[Eq. (\[bistability condition f=0\])\] is sufficiently negative.  In the presence of a cavity, even when condition (\[bistability condition f=0\]) breaks down, bistability can still occur, provided that $B$ in Eq. (\[B\]), a key parameter describing the cavity-mediated two-body interaction, is sufficiently large \[Eq. \[bistability condition with cavity\]\].  An important difference between systems with and without a cavity is that the former supports one stable mixed state while the latter can support two different stable mixed states. (Both systems contain a pure molecular state.) Thus, a bistable transition in the latter system can take place not only between a mixed state and a pure molecular state as in the former model, but also between two different mixed states. Acknowledgement =============== This paper is dedicated to the memory of H. Y. 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--- abstract: 'This work presents *CascadeCNN*, an automated toolflow that pushes the quantisation limits of any given CNN model, to perform high-throughput inference by exploiting the computation time-accuracy trade-off. Without the need for retraining, a two-stage architecture tailored for any given FPGA device is generated, consisting of a low- and a high-precision unit. A confidence evaluation unit is employed between them to identify misclassified cases at run time and forward them to the high-precision unit or terminate computation. Experiments demonstrate that *CascadeCNN* achieves a performance boost of up to 55% for VGG-16 and 48% for AlexNet over the baseline design for the same resource budget and accuracy.' author: - Alexandros Kouris - 'Stylianos I. Venieris' - 'Christos-Savvas Bouganis' bibliography: - 'main.bib' title: ' $\mathbf{Cascade^{C_{N_N}}}$: Pushing the performance limits of quantisation' --- Introduction ============ While Convolutional Neural Networks are becoming the state-of-the-art algorithm in various Machine Vision tasks [@krizhevsky2012imagenet][@redmon2016you][@badrinarayanan2015segnet], they are challenged to deal with problems of continuously increasing complexity. The significant advances of CNNs came with increased number of layers [@simonyan2014very], increased number of kernels [@zeiler2014visualizing] and more complex architectures [@szegedy2015going][@he2016deep], which introduce substantial costs in terms of computational and memory resources. To deploy CNNs in real-world tasks which deal with vast amounts of data, it is necessary that the high computation and memory requirements of such models are alleviated. To this end, numerous compression and precision quantisation techniques [@hubara2016quantized][@han2015deep][@lin2016fixed][@wu2016quantized] have been proposed which exploit the redundancy in CNN models to enable the efficient deployment of CNNs on processing platforms. In this context, FPGAs constitute a promising platform for CNN inference due to their customisability which enables the use of optimised low-precision arithmetic units to achieve high performance at a low power envelope [@Venieris_2017c]. Existing FPGA-based CNN accelerators have produced hardware designs that span from uniform 16-bit activations and weights [@Venieris_2016][@Yufei_Ma_2017b] with minimal effect on accuracy, down to very high-performance binarised networks [@Umuroglu_2017] but with a significant accuracy loss. In this setting, given a fixed resource budget, the attainable performance for a given error tolerance is limited by the shortest wordlength that meets the error bound. In this paper, we propose *CascadeCNN*, a novel automated approach of pushing the performance of precision-quantised CNN models under the same resource budget, with negligible accuracy loss. *CascadeCNN* employs a low-precision processing unit to obtain rapid classification predictions together with a parametrised mechanism for identifying misclassified cases based on prediction confidence. Such detected cases are recomputed on a high-precision unit to restore application-level accuracy and meet user-specified limits. *CascadeCNN* considers the error tolerance and the target CNN-device pair to select quantisation scheme, configure the confidence evaluation mechanism and generate the cascaded low- and high-precision processing units. Cascade CNN =========== Overview -------- Fig. \[fig:toolflow\] shows the processing flow of *CascadeCNN*. The framework is supplied with a high-level description of a trained CNN model (i.e. Caffe model), the available computational and memory resources of the target platform and an application-level error tolerance in a user-defined metric (e.g. top-1/top-5 classification error), along with a small evaluation set. *CascadeCNN* searches the architectural design space and generates a two-stage hardware architecture, optimised for the particular CNN model and target device. The generated system (Fig. \[fig:arch\]) consists of: - A low-precision unit (LPU) which employs low-precision arithmetic to trade lower accuracy with high-throughput CNN inference. - A high-precision unit (HPU) which guarantees the same accuracy level as the reference model. - A tunable Confidence Evaluation Unit (CEU) that detects samples that were wrongly classified by the LPU and redirects them to HPU for re-processing. The key idea behind the proposed approach is that during the execution of the system, the LPU will process the whole workload, while the HPU will only process a fraction of it, based on the CEU’s evaluation of classification confidence on LPU’s predictions, reducing its memory and compute requirements. Moreover, the accuracy loss that is induced due to the extreme model quantisation of the LPU is restored to meet the user-specified error threshold. Quantisation ------------ Arithmetic precision reduction is a widely studied technique which exploits the inherent redundancy of CNNs to considerably reduce the memory bandwidth and footprint requirements, minimise power consumption and achieve higher performance. *CascadeCNN* employs a fine-grained search space across possible precision quantisation schemes, that allows determining the number of integer and fractional bits of weight and activation values by introducing a different scaling factor for each layer. In this dynamic fixed-point approach, the wordlength is kept uniform across layers with a different scaling factor for each layer. For each explored wordlength, statistics regarding the quantisation effect of each layer on the application-level accuracy are extracted using the user-provided evaluation set. The per-layer statistics are used to guide the exploration to the combination of scaling factors that achieve the highest accuracy for each explored wordlength. In contrast to other frameworks, *CascadeCNN* selects for the LPU a precision that achieves intermediate application-level accuracy, but with significantly higher performance when mapped on its custom precision-optimised hardware units. All input samples are processed by the LPU to obtain a rapid classification decision, which is then fed to the Confidence Evaluation Unit. A wordlength that achieves an accuracy that complies with the user-specified error margins is selected for the HPU. Since the reduced-precision model employed by the LPU is derived by straight quantisation (without retraining), its parameters are extracted at run time in hardware from the HPU’s higher precision model. As a result of this weight-sharing approach, the memory footprint of the proposed cascade system remains the same as in the case of a single-stage architecture employing the HPU’s model. Confidence Evaluation --------------------- The *CascadeCNN* tool allows the exploration of extreme quantisation schemes for the LPU, by aiming to identify potentially misclassified inputs based on the confidence of the LPU classification prediction. To estimate this confidence, we build on the work of [@joshi2009multi] by generalising the proposed Best-vs-Second-Best (BvSB) metric, which was previously examining solely binary classification problems. Our generalised BvSB (gBvSB) metric is described as: $$\vspace{-0.16cm} \text{gBvSB}_{<M,N>}(\boldsymbol{p}) = \sum_{i=1}^{M}p_i - \sum_{j=M+1}^{N}p_j$$ where $p_i$ denotes the i-th element of the sorted probability vector $\boldsymbol{p}$ of the prediction and $M$ and $N$ are tunable parameters of gBvSB. In this context, a prediction is considered confident, and thus the processing ends on the low-precision unit, when where $M$, $N$ and threshold $th$ form tunable parameters whose values are automatically determined using the evaluation set data and the user-specified error tolerance. In this manner, the degree of uncertainty on the classification decision is based on how spiky the sorted probability distribution of the CNN’s prediction is. Architecture ------------ A scalable, fine-grained hardware architecture is designed that is able to execute CNN inference, scale its performance with the resources of a target FPGA and exploit higher degrees of parallelism as the wordlength of activation and weight representation decreases. The core of the architecture is a matrix multiplication (*MM*) unit, parametrised with respect to the tiling of each matrix dimension and the arithmetic precision of both activations and weights. The *MM* unit comprises Multiply-Accumulate (MACC) units, grouped into Processing Elements (PEs) that perform dot-product operations (shown in Fig. \[fig:arch\]). By casting convolution operations as matrix multiplications and using batch processing for fully-connected (FC) layers, both CONV and FC layers are mapped on the *MM* unit. Given a CNN-FPGA pair and a particular wordlength, *CascadeCNN* searches the architectural design space by means of a roofline-based performance model [@williams2009roofline] in order to determine the highest performing configuration of the architecture. The configurable parameters comprise the matrix tile sizes, that correspond to different levels of parallelism in terms of number of PEs and MACCs-per-PE. In this manner, *CascadeCNN* generates two architectures, the LPU and the HPU, which are optimised for different wordlengths. Evaluation ========== To evaluate the proposed toolflow, we target image classification using pretrained models on the ImageNet [@deng2009imagenet] dataset. *CascadeCNN* is provided with models of VGG-16 [@simonyan2014very] and AlexNet [@krizhevsky2012imagenet], along with a small subset of the ImageNet validation set as an evaluation set (200 labelled samples), targeting two different FPGA platforms, Xilinx Zynq ZC706 and UltraScale+ ZCU102. For both VGG-16 and AlexNet, *CascadeCNN* yields a wordlength of 4 bits for the LPU. The selected 4-bit quantisation scheme introduces a 14.38% and 18.65% degradation in classification accuracy compared to an 8-bit precision respectively (Fig. \[fig:prec\]). The CEU parameters are tuned on the evaluation dataset to generate systems that introduce a wide range of classification errors, compared to a faithful 8-bit implementation. To evaluate the performance gains of *CascadeCNN*, we compare the generated two-stage system for each error tolerance with a baseline single-stage architecture that is optimised with a quantisation scheme that achieves the same or better accuracy (ranging from 5 to 7 bit wordlengths). The achieved speed-up on throughput is illustrated in Fig. \[fig:speedup\] across a wide range of error thresholds. In the case of high error tolerance, the speed-up becomes less significant as the difference in wordlength between the LPU and the baseline design decreases. On both target platforms the performance has been improved by up to 55% for VGG-16 and up to 48% for AlexNet over the baseline design for the same resource budget and error tolerance. The proposed methodology can also be applied to other existing CNN accelerator architectures, with variable performance gains. Conclusion ========== This work presents *CascadeCNN*, an automated toolflow for CNN inference acceleration exploiting the computation time-accuracy trade-off. The cascaded two-stage architecture generated by the toolflow demonstrates a performance boost of up to 55% for VGG-16 and 48% for AlexNet compared to a single-stage baseline architecture for the same resource budget and error tolerance.
--- abstract: ' is the most sought-after emission line to detect and characterize metal free stellar populations. However, current stellar population/photo-ionization models lack sufficient [He$^+$]{} ionising photons to reproduce observed  fluxes while being consistent with other emission lines. Using $\sim10-30$ hour deep pointings from MUSE, we obtain $\sim10$ $z\sim2-4$  emitters to study their inter-stellar medium and stellar population properties. Emission line ratio diagnostics of our sample suggest that emission lines are driven by star-formation in solar to moderately sub-solar ($\sim 1/20$th) metallicity conditions. However, we find that even after considering effects from binary stars, we are unable to reproduce the  equivalent widths. Our analysis suggest that extremely sub-solar metallicities ($\sim1/200$th) are required to reproduce observed  luminosities. Thus, current stellar populations may require alternative mechanisms such as sub-dominant active galactic nuclei or top heavy initial-mass-functions to compensate for the missing [He$^+$]{} ionising photons.' --- Introduction ============ The detection and characterization of the first generation of stellar populations in the Universe is of highest priority to the high redshift galaxy evolution community. Multiple observational attempts have been made to observationally confirm galaxies with evidence for population III (pop III; metal free) stars without any success [e.g., @Cassata2013; @Sobral2015], where [Ly$\alpha$]{} and  in the absence of other prominent emission lines are interpreted as existence of pristine metal-poor stellar populations [e.g., @Inoue2011; @Sobral2015]. This interpretation is however challenging in the face of other processes that can produce [He$^+$]{} ionising photons (E$> 54.4$ eV, $\lambda<228$ Å). Additionally, the short life-time of pop-III systems and resulting inter-stellar medium (ISM)/inter-galactic-medium pollution by pair-instability supernovae [@Heger2002], uncertainties in photometric calibrations, presence of active galactic nuclei (AGN), pristine cold mode gas accretion to galaxies, limited understanding of high-redshift stellar populations and the ISM contribute further to the complexity of detecting and identifying pop-III host systems [e.g., @Matthee2017; @Shibuya2017; @Sobral2017]. Thus, to make compelling constraints of stellar populations in the presence of strong  emission and link with pop-III hosts, a comprehensive understanding of  emission mechanisms is required. Multiple mechanisms prominent in stellar populations in a variety of ages and physical/chemical conditions are expected to contribute to  emission, i.e. young O/B type stars [e.g., @Shirazi2012], hydrogen-stripped massive evolved Wolf-Rayet stars [e.g., @Shirazi2012], post-asymptomatic giant branch stars [e.g., @Binette1994], X-ray binary stars [e.g., @Casares2017], radiative shocks [e.g., @Izotov2012], AGN [e.g., @Shirazi2012] have all been suggested as possible contributers. Additionally, mechanisms such as binary interactions and stellar rotation are expected to prolong the lifetime of young O/B stars extending the total amount of [He$^+$]{} photons present at a given star-formation history [e.g., @Eldridge2017; @Gotberg2017]. Even with a variety of such mechanisms, present stellar-population/photo-ionization models lack sufficient high-energy photons to produce observed  line profiles consistently with other rest-UV emission lines in local and high-$z$ galaxies [e.g., @Shirazi2012; @Senchyna2017; @Berg2018]. Data & Analysis =============== The advancement of state-of-the-art sensitive multiplexed optical instruments in 8-10m class telescopes such as the The Multi Unit Spectroscopic Explorer [MUSE; @Bacon2010] has allowed us to obtain spatially-resolved spectroscopy of galaxies in this epoch in unprecedented numbers [e.g., @Inami2017]. Here, we present an analysis done using deep $\sim10-30$ hour pointings from MUSE obtained as a part of multiple MUSE guaranteed time observation programs [@Bacon2015; @Bacon2017; @Epinat2018; @Marino2018]. Our observations target  emitters at $z=1.93-4.67$. The Universe at $z\sim2-4$ was reaching the peak of the cosmic star-formation rate density [@Madau2014], where systems were highly star-forming and evolving rapidly giving rise to a diverse range of physical and chemical properties [e.g., @Kacprzak2016; @Kewley2016; @Steidel2016; @Nanayakkara2017; @Strom2017]. Thus, with MUSE we are able to obtain rest-UV spectroscopy of young, low-metallicity, highly star-forming systems which may give rise to a diverse range of exotic phenomena capable of producing high-energy ionizing photons. Our sample comprise of 15  detections (including 3 AGN) and is the largest sample of $z>2$  emitters with multiple emission line detections. Additional details on sample selection process is described in Nanayakkara et al., (in prep). We remove AGN from our sample and use multiple emission line diagnostics from @Gutkin2016 and @Xiao2018 to explore the ISM conditions of the  sample. We find that in /[\[\]]{} vs [\]]{}/, / vs [\[\]]{}/, and [\[\]]{}/ vs /[\]]{} line ratio diagrams our galaxies occupy a region, that can be described by star-forming galaxies with solar to $\sim1/20$th solar metallicities. In Figure \[fig:line\_ratios\], we show the / vs [\[\]]{}/ line ratio diagrams for single-star stellar population models from @Gutkin2016 and binary-star models from BPASS @Xiao2018. Our values agree with literature data of high-$z$ sources [@Patricio2017; @Berg2018] and have considerably lower metallicities compared to $z=0$ sources from @Senchyna2017. When effects of binary stellar models are added, the line-ratio diagnostics become more degenerate [also see @Xiao2018], however, line-ratios are still within the range powered by star-formation. The main discrepancy between model and data arise only once line EWs are compared. As shown by Figure \[fig:ews\], @Xiao2018 binary models are able to reproduce observed  EWs but lacks sufficient mechanisms to reproduce the observed  EWs. We expect this to be primarily driven by the lack of photons below $\lambda<228$ Å in BPASS models [e.g., @Berg2018]. We further develop a simple prescription to investigate the difference in  ionising photons between observed data and @Xiao2018 model predictions by normalizing observed  luminosities with the models. In Figure \[fig:NHeII\_def\] we show the fraction of observed [He$^+$]{} ionising photons compared to the predictions from the models. Only extreme sub-solar metallicities ($\sim1/200$th) are able to accurately predict the observed [He$^+$]{} ionising photons, which is strongly in contrast with predictions from line-ratio diagnostics. Conclusions & Future directions =============================== Here, we have used deep optical spectroscopy from MUSE to obtain a sample of  detections at $z\sim2-4$ to study their ISM conditions using state-of-the-art stellar-population/photo-ionization models. Using rest-UV emission-line ratio diagnostics we show that our galaxies could mostly be explained by Z$\sim0.05-1.0$ [Z$_\odot$]{} photo-ionisation models, but, we show that even BPASS binary models lack sufficient ionising photons to re-produce observed  EWs. Using a simple prescription, we show that our observed  luminosities can only be explained using extreme sub-solar metallicities ($\sim1/200$th). Such low metallicities are in contradiction with our line-ratio diagnostics and stellar populations models can suffer large uncertainties due to lack of empirical calibrations in this regime. It is possible that extra contribution from X-Ray binaries, sub-dominant AGN, or effects related to stellar rotations at high metallicities can supply the missing ionising photons. Alternatively, if star-forming galaxies at $z\sim2-4$ have a top-heavy initial-mass-function [see @Nanayakkara2017], the extra O/B type stars will contribute to higher levels of ionising photons, which could increase the [He$^+$]{} photon budget. Future deep surveys such as the MUSE extreme deep field survey, a single 160 hour pointing by MUSE, will provide extremely high signal-to-noise rest-UV spectra at $z=2-4$ to perform spectro-photometric analysis by simultaneous combination of nebular emission features with weaker ISM and photospheric emission and absorption features. Thus, we will be able to constrain stellar population properties to finer detail within this epoch and make predictions for future surveys by the *James Webb Space Telescope*. Given that individual detections of pop-III stars will be unlikely until proposed future space telescopes such as LUVOIR, we should push the current instruments to their maximum potential to constrain the stellar population properties of galaxies leading to the buildup of the peak of the cosmic star-formation rate density. [2]{}
--- abstract: | We introduce a new distance-based phylogeny reconstruction technique which provably achieves, at sufficiently short branch lengths, a polylogarithmic sequence-length requirement—improving significantly over previous polynomial bounds for distance-based methods. The technique is based on an averaging procedure that implicitly reconstructs ancestral sequences. In the same token, we extend previous results on phase transitions in phylogeny reconstruction to general time-reversible models. More precisely, we show that in the so-called Kesten-Stigum zone (roughly, a region of the parameter space where ancestral sequences are well approximated by “linear combinations” of the observed sequences) sequences of length ${{\mbox{{\rm poly}}}}(\log n)$ suffice for reconstruction when branch lengths are discretized. Here $n$ is the number of extant species. Our results challenge, to some extent, the conventional wisdom that estimates of evolutionary distances alone carry significantly less information about phylogenies than full sequence datasets. author: - 'Sébastien Roch[^1]' bibliography: - 'thesis.bib' title: ' Sequence-Length Requirement of Distance-Based Phylogeny Reconstruction: Breaking the Polynomial Barrier[^2] ' --- =1 [**Keywords:**]{} Phylogenetics, distance-based methods, phase transitions, reconstruction problem. Introduction ============ The evolutionary history of a group of organisms is generally represented by a [*phylogenetic tree*]{} or [*phylogeny*]{} [@Felsenstein:04; @SempleSteel:03]. The leaves of the tree represent the current species. Each branching indicates a speciation event. Many of the most popular techniques for reconstructing phylogenies from molecular data, e.g. UPGMA, Neighbor-Joining, and BIO-NJ [@SokalSneath:63; @SaitouNei:87; @Gascuel:97], are examples of what are known as [*distance-matrix methods*]{}. The main advantage of these methods is their speed, which stems from a straightforward approach: 1) the estimation of a [*distance matrix*]{} from observed molecular sequences; and 2) the repeated agglomeration of the closest clusters of species. Each entry of the distance matrix is an estimate of the evolutionary distance between the corresponding pair of species, that is, roughly the time elapsed since their most recent common ancestor. This estimate is typically obtained by comparing aligned homologous DNA sequences extracted from the extant species—the basic insight being, the closer the species, the more similar their sequences. Most distance methods run in time polynomial in $n$, the number of leaves, and in $k$, the sequence length. This performance compares very favorably to that of the other two main classes of reconstruction methods, likelihood and parsimony methods, which are known to be computationally intractable [@GrahamFoulds:82; @DaySankoff:86; @Day:87; @MosselVigoda:05; @ChorTuller:06; @Roch:06]. The question we address in this paper is the following: Is there a price to pay for this speed and simplicity? There are strong combinatorial [@StHePe:88] and statistical [@Felsenstein:04] reasons to believe that distance methods are not as accurate as more elaborate reconstruction techniques, notably maximum likelihood estimation (MLE). Indeed, in a typical instance of the phylogenetic reconstruction problem, we are given [*aligned DNA sequences*]{} $\{(\xi^i_l)_{i=1}^k\}_{l\in L}$, one sequence for each leaf $l \in L$, from which we seek to infer the phylogeny on $L$. Generally, all [*sites*]{} $(\xi^i_l)_{l\in L}$, for $i=1,\ldots,k$, are assumed to be independent and identically distributed according to a Markov model on a tree (see Section \[section:definitions\]). For a subset $W \subseteq L$, we denote by $\mu_W$ the distribution of $(\xi^i_l)_{l\in W}$ under this model. Through their use of the distance matrix, distance methods reduce the data to [*pairwise sequence correlations*]{}, that is, they only use estimates of ${\boldsymbol{\mu}}_2 = \{\mu_W\ :\ W\subseteq L,\ |W| = 2\}$. In doing so, they seemingly fail to take into account more subtle patterns in the data involving three or more species at a time. In contrast, MLE for example outputs a model that maximizes the [*joint probability of all observed sequences*]{}. We call methods that explicitly use the full dataset, such as MLE, [*holistic methods*]{}. It is important to note that the issue is not one of [*consistency*]{}: when the sequence length tends to infinity, the estimate provided by distance methods—just like MLE—typically converges to the correct phylogeny. In particular, under mild assumptions, it suffices to know the pairwise site distributions ${\boldsymbol{\mu}}_2$ to recover the topology of the phylogeny [@ChangHartigan:91; @Chang:96]. Rather the question is: how fast is this convergence? Or more precisely, how should $k$ scale as a function of $n$ to guarantee a correct reconstruction with high probability? And are distance methods significantly slower to converge than holistic methods? Although we do not give a complete answer to these questions of practical interest here, we do provide strong evidence that some of the suspicions against distance methods are based on a simplistic view of the distance matrix. In particular, we open up the surprising possibility that distance methods actually exhibit optimal convergence rates. #### Context. It is well-known that some of the most popular distance-matrix methods actually suffer from a prohibitive sequence-length requirement [@Atteson:99; @LaceyChang:06]. Nevertheless, over the past decade, much progress has been made in the design of fast-converging distance-matrix techniques, starting with the seminal work of Erdös et al. [@ErStSzWa:99a]. The key insight behind the algorithm in [@ErStSzWa:99a], often dubbed the Short Quartet Method (SQM), is that it discards long evolutionary distances, which are known to be statistically unreliable. The algorithm works by first building subtrees of small diameter and, in a second stage, putting the pieces back together. The SQM algorithm runs in polynomial time and guarantees the correct reconstruction with high probability of any phylogeny (modulo reasonable assumptions) when $k = {{\mbox{{\rm poly}}}}(n)$. This is currently the best known convergence rate for distance methods. (See also [@DaMoRo:06; @DHJMMR:06; @Mossel:07; @GrMoSn:08; @DaMoRo:08a] for faster-converging algorithms involving [*partial*]{} reconstruction of the phylogeny.) Although little is known about the sequence-length requirement of MLE [@SteelSzekely:99; @SteelSzekely:02], recent results of Mossel [@Mossel:04a], Daskalakis et al. [@DaMoRo:06; @DaMoRo:08b], and Mihaescu et al. [@MiHiRa:09] on a conjecture of Steel [@Steel:01] indicate that convergence rates as low as $k=O(\log n)$ can be achieved when the branch lengths are sufficiently short and discretized, using insights from statistical physics. We briefly describe these results. As mentioned above, the classical model of DNA sequence evolution is a Markov model on a tree that is closely related to stochastic models used to study particle systems [@Liggett:85; @Georgii:88]. This type of model undergoes a phase transition that has been extensively studied in probability theory and statistical physics: at short branch lengths (in the binary symmetric case, up to 15% divergence [*per edge*]{}), in what is called the [*reconstruction phase*]{}, good estimates of the ancestral sequences can be obtained from the observed sequences; on the other hand, outside the reconstruction phase, very little information about ancestral states diffuses to the leaves. See e.g. [@EvKePeSc:00] and references therein. The new algorithms in [@Mossel:04a; @DaMoRo:06; @DaMoRo:08b; @MiHiRa:09] exploit this phenomenon by alternately 1) reconstructing a few levels of the tree using distance-matrix techniques and 2) estimating distances between [*internal*]{} nodes by reconstructing ancestral sequences at the newly uncovered nodes. The overall algorithm is *not* distance-based, however, as the ancestral sequence reconstruction is performed using a complex function of the observed sequences named [*recursive majority*]{}. The rate $k = O(\log n)$ achieved by these algorithms is known to be necessary in general. Moreover, the slower rate $k={{\mbox{{\rm poly}}}}(n)$ is in fact necessary for all methods—distance-based or holistic—outside the reconstruction phase [@Mossel:03]. In particular, note that distance methods are in some sense “optimal” [*outside*]{} the reconstruction phase by the results of [@ErStSzWa:99a]. #### Beyond the oracle view of the distance matrix. It is an outstanding open problem to determine whether distance methods can achieve $k = O(\log n)$ in the reconstruction phase[^3]. From previous work on fast-converging distance methods, it is tempting to conjecture that $k = {{\mbox{{\rm poly}}}}(n)$ is the best one can hope for. Indeed, all previous algorithms use the following “oracle view” of the distance matrix, as formalized by King et al. [@KiZhZh:03] and Mossel [@Mossel:07]. As mentioned above, the reliability of distance estimates depends on the true evolutionary distances. From standard concentration inequalities, it follows that if leaves $a$ and $b$ are at distance ${\tau}(a,b)$, then the usual distance estimate ${\hat\tau}(a,b)$ (see Section \[section:definitions\]) satisfies: $$\label{eq:oracle} \text{if}\ {\tau}(a,b) < D + {\varepsilon}\ \text{or}\ {\hat\tau}(a,b) < D + {\varepsilon}\ \text{then}\ |{\tau}(a,b) - {\hat\tau}(a,b)| < {\varepsilon},$$ for ${\varepsilon}, D$ such that $k \propto (1 - e^{-{\varepsilon}})^{-2} e^{2D}$. Fix ${\varepsilon}> 0$ small and $k \ll {{\mbox{{\rm poly}}}}(n)$. Let $T$ be a complete binary tree with $\log_2 n$ levels. Imagine that the distance matrix is given by the following oracle: on input a pair of leaves $(a,b)$ the oracle returns an estimate ${\hat\tau}(a,b)$ which satisfies (\[eq:oracle\]). Now, notice that for any tree $T'$ which is identical to $T$ on the first $\log_2 n/2$ levels above the leaves, the oracle is allowed to return the same distance estimate as for $T$. That is, we cannot distinguish $T$ and $T'$ in this model unless $k = {{\mbox{{\rm poly}}}}(n)$. (This argument can be made more formal along the lines of [@KiZhZh:03].) What the oracle model ignores is that, under the assumption that the sequences are generated by a Markov model of evolution, the distance estimates $({\hat\tau}(a,b))_{a,b\in [n]}$ are in fact [*correlated random variables*]{}. More concretely, for leaves $a$, $b$, $c$, $d$, note that the joint distribution of $({\hat\tau}(a,b), {\hat\tau}(c,d))$ depends in a nontrivial way on the joint site distribution $\mu_W$ at $W = \{a,b,c,d\}$. In other words, even though the distance matrix is—seemingly—only an estimate of the pairwise correlations ${\boldsymbol{\mu}}_2$, it actually contains [*some*]{} information about all joint distributions. Note however that it is not immediately clear how to exploit this extra information or even how useful it could be. As it turns out, the correlation structure of the distance matrix is in fact *very informative* at short branch lengths. More precisely, we introduce in this paper a new distance-based method with a convergence rate of $k = {{\mbox{{\rm poly}}}}(\log n)$ in the reconstruction phase (to be more accurate, in the so-called Kesten-Stigum phase; see below)—improving significantly over previous ${{\mbox{{\rm poly}}}}(n)$ results. Note that the oracle model allows only the reconstruction of a $o(1)$ fraction of the levels in that case. Our new algorithm involves a distance averaging procedure that implicitly reconstructs ancestral sequences, thereby taking advantage of the phase transition discussed above. We also obtain the first results on Steel’s conjecture beyond the simple symmetric models studied by Daskalakis et al. [@DaMoRo:06; @DaMoRo:08b; @MiHiRa:09] (the so-called CFN and Jukes-Cantor models). In the next subsections, we introduce general definitions and state our results more formally. We also give an overview of the proof. #### Further related work. For further related work on efficient phylogenetic tree reconstruction, see [@ErStSzWa:99b; @HuNeWa:99; @CsurosKao:01; @Csuros:02]. Definitions {#section:definitions} ----------- #### Phylogenies. We define phylogenies and evolutionary distances more formally. A [*phylogeny*]{} is a rooted, edge-weighted, leaf-labeled tree ${{\mathcal{T}}}= (V,E,[n],{\rho};{\tau})$ where: $V$ is the set of vertices; $E$ is the set of edges; $L = [n] = \{0,\ldots,n-1\}$ is the set of leaves; ${\rho}$ is the root; ${\tau}: E \to (0,+\infty)$ is a positive edge weight function. We further assume that all internal nodes in ${{\mathcal{T}}}$ have degree $3$ except for the root ${\rho}$ which has degree $2$. We let ${\mathbb{Y}}_n$ be the set of all such phylogenies on $n$ leaves and we denote ${\mathbb{Y}}= \{{\mathbb{Y}}_n\}_{n\geq 1}$. For two leaves $a,b \in [n]$, we denote by ${\mathrm{Path}}(a,b)$ the set of edges on the unique path between $a$ and $b$. A [*tree metric*]{} on a set $[n]$ is a positive function $d:[n]\times[n] \to (0,+\infty)$ such that there exists a tree $T = (V,E)$ with leaf set $[n]$ and an edge weight function $w:E \to (0,+\infty)$ satisfying the following: for all leaves $a,b \in [n]$ $$d(a,b) = \sum_{e\in {\mathrm{Path}}(a,b)} w_e.$$ For convenience, we denote by $\left({\tau}(a,b)\right)_{a,b\in [n]}$ the tree metric corresponding to the phylogeny ${{\mathcal{T}}}= (V,E,[n],{\rho};{\tau})$. We extend ${\tau}(u,v)$ to all vertices $u,v \in V$ in the obvious way. \[ex:homo\] For an integer $h \geq 0$, we denote by ${{{\mathcal{T}}}^{(h)}} = ({V^{(h)}}, {E^{(h)}}, {L^{(h)}}, {{\rho}^{(h)}}; {\tau})$ a rooted phylogeny where ${T^{(h)}}$ is the $h$-level complete binary tree with arbitrary edge weight function ${\tau}$ and ${L^{(h)}} = [2^h]$. For $0\leq h'\leq h$, we let ${L^{(h)}_{h'}}$ be the vertices on level $h - h'$ (from the root). In particular, ${L^{(h)}_{0}} = {L^{(h)}}$ and ${L^{(h)}_{h}} = \{{{\rho}^{(h)}}\}$. We let ${\mathbb{HY}}= \{{\mathbb{HY}}_n\}_{n\geq 1}$ be the set of all phylogenies with homogeneous underlying trees. #### Model of molecular sequence evolution. Phylogenies are reconstructed from molecular sequences extracted from the observed species. The standard model of evolution for such sequences is a Markov model on a tree (MMT). Let ${\Phi}$ be a finite set of character states with ${\phi}= |{\Phi}|$. Typically ${\Phi}= \{+1,-1\}$ or ${\Phi}= \{\mathrm{A}, \mathrm{G}, \mathrm{C}, \mathrm{T}\}$. Let $n \geq 1$ and let $T = (V,E,[n],{\rho})$ be a rooted tree with leaves labeled in $[n]$. For each edge $e \in E$, we are given a ${\phi}\times{\phi}$ stochastic matrix $M^e = (M^e_{ij})_{i,j \in {\Phi}}$, with fixed stationary distribution $\pi = (\pi_i)_{i\in {\Phi}}$. An MMT $(\{M^e\}_{e\in E}, T)$ associates a state ${\sigma}_v$ in ${\Phi}$ to each vertex $v$ in $V$ as follows: pick a state for the root ${\rho}$ according to $\pi$; moving away from the root, choose a state for each vertex $v$ independently according to the distribution $(M^e_{{\sigma}_u, j})_{j\in{\Phi}}$, with $e = (u,v)$ where $u$ is the parent of $v$. The most common MMT used in phylogenetics is the so-called general time-reversible (GTR) model. \[def:gtr\] Let ${\Phi}$ be a set of character states with ${\phi}= |{\Phi}|$ and $\pi$ be a distribution on ${\Phi}$ satisfying $\pi_i > 0$ for all $i\in{\Phi}$. For $n \geq 1$, let ${{\mathcal{T}}}= (V,E,[n],{\rho};{\tau})$ be a phylogeny. Let $Q$ be a ${\phi}\times{\phi}$ rate matrix, that is, $Q_{ij} > 0$ for all $i\neq j$ and $$\sum_{j\in {\Phi}} Q_{ij} = 0,$$ for all $i \in {\Phi}$. Assume $Q$ is reversible with respect to $\pi$, that is, $$\pi_i Q_{ij} = \pi_j Q_{ji},$$ for all $i,j \in {\Phi}$. The GTR model on ${{\mathcal{T}}}$ with rate matrix $Q$ is an MMT on $T = (V,E,[n], {\rho})$ with transition matrices $M^e = e^{{\tau}_e Q}$, for all $e\in E$. By the reversibility assumption, $Q$ has ${\phi}$ real eigenvalues $$0 = \Lambda_1 > \Lambda_2 \geq \cdots \geq \Lambda_{{\phi}}.$$ We normalize $Q$ by fixing $\Lambda_2 = -1$. We denote by ${\mathbb{Q}}_{\phi}$ the set of all such rate matrices. We let ${\mathbb{G}}_{n,{\phi}} = {\mathbb{Y}}_n \otimes {\mathbb{Q}}_{\phi}$ be the set of all ${\phi}$-state GTR models on $n$ leaves. We denote ${\mathbb{G}}_{\phi}= \left\{{\mathbb{G}}_{n,{\phi}}\right\}_{n \geq 1}$. We denote by $\xi_W$ the vector of states on the vertices $W\subseteq V$. In particular, $\xi_{[n]}$ are the states at the leaves. We denote by ${{\mathcal{L}}}_{{{\mathcal{T}}},Q}$ the distribution of $\xi_{[n]}$. GTR models include as special cases many popular models such as the CFN model. \[ex:cfn\] The [*CFN model*]{} is the GTR model with ${\phi}= 2$, $\pi = (1/2, 1/2)$, and $$Q = Q^{\mathrm{CFN}} \equiv \left( \begin{array}{cc} -1/2 & 1/2\\ 1/2 & -1/2 \end{array} \right).$$ More generally, letting ${\Phi}= \{+,-\}$ and $\pi = (\pi_{+}, \pi_{-})$, with $\pi_{+},\pi_{-} > 0$, we can take $$Q = \left( \begin{array}{cc} -\pi_{-} & \pi_{-}\\ \pi_{+} & -\pi_{+} \end{array} \right).$$ #### Phylogenetic reconstruction. A standard assumption in molecular evolution is that each site in a sequence (DNA, protein, etc.) evolves [*independently*]{} according to a Markov model on a tree, such as the GTR model above. Because of the reversibility assumption, the root of the phylogeny cannot be identified and we reconstruct phylogenies up to their root. Let $\widetilde{\mathbb{Y}}= \{\widetilde{\mathbb{Y}}_n\}_{n\geq 1}$ be a subset of phylogenies and $\widetilde{\mathbb{Q}}_{\phi}$ be a subset of rate matrices on ${\phi}$ states. Let ${{\mathcal{T}}}= (V,E,[n],{\rho};{\tau}) \in \widetilde{\mathbb{Y}}$. If $T = (V,E,[n],{\rho})$ is the rooted tree underlying ${{\mathcal{T}}}$, we denote by $T_{-}[{{\mathcal{T}}}]$ the tree $T$ where the root is removed: that is, we replace the two edges adjacent to the root by a single edge. We denote by ${\mathbb{T}}_n$ the set of all leaf-labeled trees on $n$ leaves with internal degrees $3$ and we let ${\mathbb{T}}= \{{\mathbb{T}}_n\}_{n\geq 1}$. A [*phylogenetic reconstruction algorithm*]{} is a collection of maps ${\mathcal{A}}= \{{\mathcal{A}}_{n,k}\}_{n,k \geq 1}$ from sequences $(\xi^i_{[n]})_{i=1}^k \in ({\Phi}^{[n]})^k$ to leaf-labeled trees $T \in {\mathbb{T}}_n$. We only consider algorithms ${\mathcal{A}}$ computable in time polynomial in $n$ and $k$. Let $k(n)$ be an increasing function of $n$. We say that ${\mathcal{A}}$ solves the [*phylogenetic reconstruction problem*]{} on $\widetilde{\mathbb{Y}}\otimes \widetilde{\mathbb{Q}}_{\phi}$ with sequence length $k = k(n)$ if for all $\delta > 0$, there is $n_0 \geq 1$ such that for all $n \geq n_0$, ${{\mathcal{T}}}\in \widetilde{\mathbb{Y}}_n$, $Q \in \widetilde{\mathbb{Q}}_{\phi}$, $${\mathbb{P}}\left[{\mathcal{A}}_{n,k(n)}\left((\xi^i_{[n]})_{i=1}^{k(n)}\right) = T_-[{{\mathcal{T}}}]\right] \geq 1 - \delta,$$ where $(\xi^i_{[n]})_{i=1}^{k(n)}$ are i.i.d. samples from ${{\mathcal{L}}}_{{{\mathcal{T}}},Q}$. An important result of this kind was given by Erdos et al. [@ErStSzWa:99a]. \[thm:essw\] Let $0 < f \leq g < +\infty$ and denote by ${\mathbb{Y}}^{f,g}$ the set of all phylogenies ${{\mathcal{T}}}= (V,E,[n],{\rho};{\tau})$ satisfying $f \leq {\tau}_e \leq g,\ \forall e\in E$. Then, for all ${\phi}\geq 2$ and all $0 < f \leq g < +\infty$, the phylogenetic reconstruction problem on ${\mathbb{Y}}^{f,g}\otimes{\mathbb{Q}}_{\phi}$ can be solved with $k = {{\mbox{{\rm poly}}}}(n)$. This result was recently improved by Daskalakis et al. [@DaMoRo:06; @DaMoRo:08b] (see also [@MiHiRa:09]) in the so-called Kesten-Stigum reconstruction phase, that is, when $g < \ln\sqrt{2}$. Let $0 < {\Delta}\leq f \leq g < +\infty$ and denote by ${\mathbb{Y}}^{f,g}_{\Delta}$ the set of all phylogenies ${{\mathcal{T}}}= (V,E,[n],{\rho};{\tau})$ satisfying $f \leq {\tau}_e \leq g$ where ${\tau}_e$ is an integer multiple of ${\Delta}$, for all $e\in E$. For ${\phi}\geq 2$ and $Q\in {\mathbb{Q}}_{{\phi}}$, we call ${\mathbb{Y}}^{f,g}_{\Delta}\otimes \{Q\}$ the ${\Delta}$-Branch Model (${\Delta}$-BM). \[thm:opt\] Let $g^* = \ln \sqrt{2}$. Then, for all $0 < {\Delta}\leq f \leq g < g^*$, the phylogenetic reconstruction problem on ${\mathbb{Y}}^{f,g}_{\Delta}\otimes \{Q^{\mathrm{CFN}}\}$ can be solved with $k = O(\log n)$[^4]. #### Distance methods. The proof of Theorem \[thm:essw\] uses [*distance methods*]{}, which we now define formally. \[def:distest\] Let ${\Phi}$ be a finite set with ${\phi}\geq 2$. Let $(\xi_a^i)_{i=1}^k, (\xi_b^i)_{i=1}^k \in {\Phi}^k$ be the sequences at $a, b \in [n]$. For $\upsilon_1, \upsilon_2 \in {\Phi}$, we define the [*correlation matrix*]{} between $a$ and $b$ by $${\widehat F}^{ab}_{\upsilon_1 \upsilon_2} = \frac{1}{k} \sum_{i=1}^k {\mathbbm{1}}\{\xi_a^i = \upsilon_1, \xi_b^i = \upsilon_2\},$$ and ${\widehat F}^{ab} = ({\widehat F}^{ab}_{\upsilon_1 \upsilon_2})_{\upsilon_1,\upsilon_2 \in {\Phi}}$. A phylogenetic reconstruction algorithm ${\mathcal{A}}= \{{\mathcal{A}}_{n,k}\}_{n,k\geq 1}$ is said to be [*distance-based*]{} if ${\mathcal{A}}$ depends on the data $(\xi^i_{[n]})_{i=1}^k \in ({\Phi}^{[n]})^k$ [*only through the correlation matrices*]{} $\{{\widehat F}^{ab}\}_{a,b\in [n]}$. The previous definition takes a very general view of distance-based methods: any method that uses only pairwise sequence comparisons. In practice, most distance-based approaches actually use a specific [*distance estimator*]{}, that is, a function of ${\widehat F}^{ab}$ that converges to ${\tau}(a,b)$ in probability as $n \to +\infty$. We give two classical examples below. \[ex:cfnmetric\] In the CFN case with state space ${\Phi}=\{+,-\}$, a standard distance estimator (up to a constant) is $${{\mathcal{D}}}({\widehat F}) = -\ln\left(1 - 2({\widehat F}_{+-} + {\widehat F}_{-+})\right).$$ More generally, a common distance estimator (up to scaling) is the so-called [*log-det distance*]{} $${{\mathcal{D}}}({\widehat F}) = -\ln|\det {\widehat F}|.$$ Loosely speaking, the log-det distance can be thought as a generalization of the CFN metric. We will use a different generalization of the CFN metric. See section \[section:overview\]. Results {#sec:results} ------- In our main result, we prove that phylogenies under GTR models of mutation can be inferred using a distance-based method from $k = {{\mbox{{\rm poly}}}}(\log n)$. \[thm:main\] For all ${\phi}\geq 2$, $0 < {\Delta}\leq f \leq g < g^*$ and $Q\in {\mathbb{Q}}_{{\phi}}$, there is a distance-based method solving the phylogenetic reconstruction problem on ${\mathbb{Y}}^{f,g}_{\Delta}\otimes \{Q\}$ with $k = {{\mbox{{\rm poly}}}}(\log n)$.[^5] Note that this result is a substantial improvement over Theorem \[thm:essw\]—at least, in a certain range of parameters—and that it almost matches the bound obtained in Theorem \[thm:opt\]. The result is also novel in two ways over Theorem \[thm:opt\]: only the distance matrix is used; the result applies to a larger class of mutation matrices. A slightly weaker version of the result stated here appeared without proof as [@Roch:08]. Note that in [@Roch:08] the result was stated without the discretization assumption which is in fact needed for the final step of the proof. This is further explained in Section 7.3 of [@DaMoRo:08b]. In subsequent work [@Roch:09], the result stated here was improved to logarithmic sequence length, thereby matching Theorem \[thm:opt\]. This new result follows a similar high-level proof but involves stronger concentration arguments [@PeresRoch:08], as well as a simplified algorithm. In an attempt to keep the paper as self-contained as possible we first give a proof in the special case of homogeneous trees. This allows to keep the algorithmic details to a minimum. The proof appears in Section \[section:hmg\]. We extend the result to general trees in Appendix \[section:general-trees\]. The more general result relies on a combinatorial algorithm of [@DaMoRo:08b]. Proof Overview {#section:overview} -------------- #### Distance averaging. The basic insight behind Steel’s conjecture is that the accurate reconstruction of ancestral sequences in the reconstruction phase can be harnessed to perform a better reconstruction of the phylogeny itself. For now, consider the CFN model with character space $\{+1,-1\}$ and assume that our phylogeny is homogeneous with uniform branch lengths $\omega$. Generate $k$ i.i.d. samples $({\sigma}^i_V)_{i=1}^k$. Let $a, b$ be two internal vertices on level $h - h' < h$ (from the root). Suppose we seek to estimate the distance between $a$ and $b$. This estimation cannot be performed directly because the sequences at $a$ and $b$ are not known. However, we can try to [*estimate*]{} these internal sequences. Denote by $A$, $B$ the leaf set below $a$ and $b$ respectively. An estimate of the sequence at $a$ is the (properly normalized) “site-wise average” of the sequences at $A$ $$\label{eq:majority} \bar{\sigma}_a^i = \frac{1}{|A|}\sum_{a' \in A} \frac{{\sigma}^i_{a'}}{e^{-\omega h'}},$$ for $i=1,\ldots,k$, and similarly for $b$. It is not immediately clear how such a [*site-wise*]{} procedure involving [*simultaneously*]{} a large number of leaves can be performed using the more aggregated information in the correlation matrices $\{{\widehat F}^{uv}\}_{u,v\in [n]}$. Nevertheless, note that the quantity we are ultimately interested in computing is the following estimate of the CFN metric between $a$ and $b$ $$\bar{\tau}(a,b) = -\ln \left(\frac{1}{k} \sum_{i=1}^k \bar{\sigma}_a^i\bar{\sigma}_b^i\right).$$ Our results are based on the following observation: $$\begin{aligned} \bar{\tau}(a,b) &=& -\ln \left(\frac{1}{k} \sum_{i=1}^k \left(\frac{1}{|A|}\sum_{a' \in A} \frac{{\sigma}^i_{a'}}{e^{-\omega h'}}\right) \left(\frac{1}{|B|}\sum_{b' \in B} \frac{{\sigma}^i_{b'}}{e^{-\omega h'}}\right)\right)\\ &=& -\ln \left(\frac{1}{|A||B|e^{-2 \omega h'}}\sum_{a' \in A}\sum_{b' \in B} \left(\frac{1}{k} \sum_{i=1}^k {\sigma}^i_{a'}{\sigma}^i_{b'}\right)\right)\\ &=& -\ln \left(\frac{1}{|A||B|e^{-2 \omega h'}}\sum_{a' \in A}\sum_{b' \in B} e^{-\hat{\tau}(a',b')}\right),\end{aligned}$$ where note that the last line depends only on distance estimates $\hat{\tau}(a',b')$ between leaves $a',b'$ in $A,B$. In other words, through this procedure, which we call [*exponential averaging*]{}, we perform an [*implicit*]{} ancestral sequence reconstruction using only distance estimates. One can also think of this as a variance reduction technique. When the branch lengths are not uniform, one needs to use a [*weighted*]{} version of (\[eq:majority\]). This requires the estimation of path lengths. #### GTR models. In the case of GTR models, the standard log-det estimator does not lend itself well to the exponential averaging procedure described above. Instead, we use an estimator involving the right eigenvector ${\nu}$ corresponding to the second eigenvalue $\Lambda_2$ of $Q$. For $a,b\in [n]$, we consider the estimator $${\hat\tau}(a,b) = -\ln \left({\nu}^\top {\widehat F}^{ab} {\nu}\right).$$ This choice is justified by a generalization of (\[eq:majority\]) introduced in [@MosselPeres:03]. Note that ${\nu}$ may need to be estimated. #### Concentration. There is a further complication in that to obtain results with high probability, one needs to show that $\bar{\tau}(a,b)$ is [*highly concentrated*]{}. However, one cannot directly apply standard concentration inequalities because $\bar{\sigma}_a$ is [*not bounded*]{}. Classical results on the reconstruction problem imply that the variance of $\bar{\sigma}_a$ is finite—which is not quite enough. To amplify the accuracy, we “go down” $\log\log n$ levels and compute $O(\log n)$ distance estimates with [*conditionally independent*]{} biases. By performing a majority procedure, we finally obtain a concentrated estimate. Organization ------------ In Section \[section:averaging\], we provide a detailed account of the connection between ancestral sequence reconstruction and distance averaging. We then give a proof of our main result in the case of homogeneous trees in Section \[section:hmg\]. Finally, in Appendix \[section:general-trees\], we give a sketch of the proof in the general case. All proofs are relegated to Appendix \[sec:proofs\]. Ancestral Reconstruction and Distance Averaging {#section:averaging} =============================================== Let ${\phi}\geq 2$, $0 < {\Delta}\leq f \leq g < g^* = \ln \sqrt{2}$, and $Q \in {\mathbb{Q}}_{\phi}$ with corresponding stationary distribution $\pi > 0$. In this section we restrict ourselves to the homogeneous case ${{\mathcal{T}}}= {{\mathcal{T}}}^{(h)} = (V,E,[n],{\rho};{\tau})$ where we take $h = \log_2 n$ and $f \leq {\tau}_e \leq g$ and ${\tau}_e$ is an integer multiple of ${\Delta}$, $\forall e\in E$. (See Examples \[ex:homo\] and \[ex:cfn\] and Theorem \[thm:opt\].)[^6] Throughout this section, we use a sequence length $k > \log^\kappa n$ where $\kappa > 1$ is a constant to be determined later. We generate $k$ i.i.d. samples $(\xi^i_{V})_{i=1}^k$ from the GTR model $({{\mathcal{T}}}, Q)$ with state space ${\Phi}$. All proofs are relegated to Appendix \[sec:proofs\]. Distance Estimator ------------------ The standard log-det estimator does not lend itself well to the averaging procedure discussed above. For reconstruction purposes, we instead use an estimator involving the right eigenvector ${\nu}$ corresponding to the second eigenvalue $\Lambda_2$ of $Q$. For $a,b\in [n]$, consider the estimator $$\label{eq:eigenestim} {\hat\tau}(a,b) = -\ln \left({\nu}^\top {\widehat F}^{ab} {\nu}\right),$$ where the correlation matrix ${\widehat F}^{ab}$ was introduced in Definition \[def:distest\]. We first give a proof that this is indeed a legitimate distance estimator. For more on connections between eigenvalues of the rate matrix and distance estimation, see e.g. [@GuLi:96; @GuLi:98; @GrMoYa:09]. \[lemma:distance\] Let ${\hat\tau}$ be as above. For all $a,b\in [n]$, we have $${\mathbb{E}}[e^{-{\hat\tau}(a,b)}] = e^{-{\tau}(a,b)}.$$ For $a \in [n]$ and $i=1,\ldots,k$, let $$\sigma^i_{a} = {\nu}_{\xi^i_a}.$$ Then (\[eq:eigenestim\]) is equivalent to $$\label{eq:eigenestim2} {\hat\tau}(a,b) = -\ln \left(\frac{1}{k}\sum_{i=1}^k \sigma^i_a \sigma^i_b\right).$$ Note that in the CFN case, we have simply ${\nu}= (1, -1)^\top$ and hence (\[eq:eigenestim2\]) can be interpreted as a generalization of the CFN metric. Let $$\bar\pi = \min_\iota \pi_\iota,$$ and $$\bar{\nu}\equiv \max_i |{\nu}_i| \leq \frac{1}{\sqrt{\bar\pi}}.$$ The following lemmas show that the distance estimate above with sequence length ${{\mbox{{\rm poly}}}}(\log n)$ is concentrated for path lengths of order $O(\log\log n)$. \[lem:distmet1gtr\] Let $\delta > 0$, ${\varepsilon}> 0$, and $\gamma > 0$. There exists $\kappa > 1$, such that if the following conditions hold for $u,v \in V$: - $\mathrm{[Small\ Diameter]}$ ${\tau}(u,v) < \delta\log\log(n)$, - $\mathrm{[Sequence\ Length]}$ $k > \log^\kappa(n)$, then $$\left|{\tau}(u,v)-{\hat\tau}(u,v)\right|< {\varepsilon},$$ with probability at least $1-n^{-\gamma}$. \[lem:distmet2gtr\] Let $D > 0$, $W > 5$, $\delta > 0$, and $\gamma > 0$. Let $u,v \in V$ not descendants of each other. Let $u_0,v_0\in V$ be descendants of $u,v$ respectively. there exists $\kappa > 1$, such that if the following conditions hold: - $\mathrm{[Large\ Diameter]}$ ${\tau}(u_0,v_0) > D + \ln W$, - $\mathrm{[Close\ Descendants]}$ ${\tau}(u_0,u), {\tau}(v_0,v) < \delta\log\log(n)$, - $\mathrm{[Sequence\ Length]}$ $k > \log^\kappa(n)$, then $${\hat\tau}(u,v) - {\tau}(u_0,u) - {\tau}(v_0,v) > D + \ln \frac{W}{2},$$ with probability at least $1-n^{-\gamma}$. On the other hand, if the first condition above is replaced by - $\mathrm{[Small\ Diameter]}$ ${\tau}(u_0,v_0) < D + \ln \frac{W}{5}$, then $${\hat\tau}(u,v) - {\tau}(u_0,u) - {\tau}(v_0,v) \leq D + \ln \frac{W}{4},$$ with probability at least $1-n^{-\gamma}$. Ancestral Sequence Reconstruction {#sec:averaging} --------------------------------- Let $e = (x,y) \in E$ and assume that $x$ is closest to ${\rho}$ (in topological distance). We define ${\mathrm{Path}}({\rho},e) = {\mathrm{Path}}({\rho},y)$, $|e|_{\rho}= |{\mathrm{Path}}(v,e)|$, and $$R_{\rho}(e) = \left(1 - \theta_e^2\right) \Theta_{{\rho},y}^{-1},$$ where $\Theta_{{\rho},y} = e^{-{\tau}({\rho},y)}$ and $\theta_e = e^{-{\tau}(e)}$. Proposition \[prop:weightedmaj\] below is a variant of Lemma 5.3 in [@MosselPeres:03]. For completeness, we give a proof. \[prop:weightedmaj\] Let $\xi_{[n]}$ be a sample from ${{\mathcal{L}}}_{{{\mathcal{T}}},Q}$ (see Definition \[def:gtr\]) with corresponding ${\sigma}_{[n]}$. For a unit flow $\Psi$ from ${\rho}$ to $[n]$, consider the estimator $$S = \sum_{x \in [n]} \frac{\Psi(x) {\sigma}_x}{\Theta_{{\rho},x}}.$$ Then, we have $${\mathbb{E}}[S] = 0,$$ $${\mathbb{E}}[S\,|\,{\sigma}_{\rho}] = {\sigma}_{\rho},$$ and $${\mathrm{Var}}[S] = 1 + K_{\Psi},$$ where $$K_{\Psi} = \sum_{e \in E} R_{\rho}(e) \Psi(e)^2.$$ Let $\Psi$ be a unit flow from ${\rho}$ to $[n]$. We will use the following multiplicative decomposition of $\Psi$: If $\Psi(x) > 0$, we let $$\psi(e) = \frac{\Psi(y)}{\Psi(x)},$$ and, if instead $\Psi(x) = 0$, we let $\psi(y) = 0$. Denoting $x_\uparrow$ the immediate ancestor of $x \in V$ and letting $\theta_x = e^{-{\tau}_{(x_\uparrow, x)}}$, it will be useful to re-write $$\label{eq:k} K_{\Psi} = \sum_{h'=0}^{h-1} \sum_{x\in L^{(h)}_{h'}} (1 - \theta_x^2)\prod_{e\in {\mathrm{Path}}({\rho},x)} \frac{\psi(e)^2}{\theta_e^2},$$ and to define the following recursion from the leaves. For $x\in[n]$, $$K_{x,\Psi} = 0.$$ Then, let $u \in V-[n]$ with children $v_1,v_2$ with corresponding edges $e_1,e_2$ and define $$K_{u,\Psi} = \sum_{\alpha = 1,2} ((1 - \theta_{v_\alpha}^2) + K_{v_\alpha,\Psi}) \left(\frac{\psi(e_\alpha)^2}{\theta_{e_\alpha}^2}\right).$$ Note that, from (\[eq:k\]), we have $K_{{\rho},\Psi} = K_\Psi$. \[lemma:k\] Let $\Psi$ be the homogeneous flow from ${\rho}$ to $[n]$. Then, we have $$K_{\Psi} \leq \frac{1}{1 - e^{-2 (g^* - g)}} < +\infty.$$ Distance Averaging ------------------ The input to our tree reconstruction algorithm is the matrix of all estimated distances between pairs of *leaves* $\{{\hat\tau}(a,b)\}_{a,b,\in [n]}$. For short sequences, these estimated distances are known to be accurate for leaves that are close enough. We now show how to compute distances between internal nodes in a way that involves only $\{{\hat\tau}(a,b)\}_{a,b,\in [n]}$ (and previously computed internal weights) using Proposition \[prop:weightedmaj\]. However, the second-moment guarantee in Proposition \[prop:weightedmaj\] is not enough to obtain good estimates with high probability. To remedy this situation, we perform a large number of [*conditionally independent*]{} distance computations, as we now describe. Let $\alpha > 1$ and assume $h' > {\lfloor \alpha \log_2\log_2 n \rfloor}$. Let $0 \leq h'' < h'$ such that $\Delta h \equiv h' - h'' = {\lfloor \alpha \log_2\log_2 n \rfloor}$. Let $a_0, b_0 \in L^{(h)}_{h'}$. For $x \in \{a,b\}$, denote by $x_1,\ldots,x_{2^{\Delta h}}$, the vertices in $L^{(h)}_{h''}$ that are below $x_0$ and, for $j = 1,\ldots,2^{\Delta h}$, let $X_j$ be the leaves of $T^{(h)}$ below $x_j$. See Figure \[fig:amplification\]. Assume that we are given $\Theta_{a_0,\cdot}$, $\Theta_{b_0,\cdot}$. For $1 \leq j \leq 2^{\Delta h}$, we estimate ${\tau}(a_0,b_0)$ as follows $$\begin{aligned} \bar{\tau}(a_j,b_j) &\equiv& -\ln \left(\frac{1}{|A_j||B_j|}\sum_{a' \in A_j} \sum_{b' \in B_j} \Theta^{-1}_{a_0,a'}\Theta^{-1}_{b_0,b'} e^{-{\hat\tau}(a',b')}\right).\end{aligned}$$ This choice of estimator is suggested by the following observation $$\begin{aligned} e^{-\bar{\tau}(a_j,b_j)} &\equiv& \sum_{a' \in A_j} \sum_{b' \in B_j} 2^{-2h''} \Theta^{-1}_{a_0,a'}\Theta^{-1}_{b_0,b'} e^{-{\hat\tau}(a',b')}\\ &=& e^{-{\tau}(a_0,a_j)-{\tau}(b_0,b_j)} \left[\frac{1}{k} \sum_{i=1}^k \left( \sum_{a' \in A_j} \frac{2^{-h''} {\sigma}^i_{a'}}{\Theta_{a_j,a'}}\right) \left( \sum_{b' \in B_j} \frac{2^{-h''} {\sigma}^i_{b'}}{\Theta_{b_j,b'}} \right)\right].\end{aligned}$$ Note that the first line depends only on estimates $({\hat\tau}(u,v))_{u,v\in [n]}$ and $\{\Theta_{v,\cdot}\}_{v\in V_{a_0}\cup V_{b_0}}$. The last line is the empirical distance between the reconstructed states at $a$ and $b$ when the flow is chosen to be uniform in Proposition \[prop:weightedmaj\]. For $d \in {\mathbb{R}}$ and $r > 0$, let ${\mathcal{B}}_r(d)$ be the ball of radius $r$ around $d$. We define the [*dense ball around $j$*]{} to be the smallest ball around $\bar{\tau}(a_{j},b_{j})$ containing at least $2/3$ of ${\mathcal{J}}= \{\bar{\tau}(a_{j'},b_{j'})\}_{j'=1}^{2^{\Delta h}}$ (as a multiset). The radius of the dense ball around $j$ is $$r^*_j = \inf\left\{r\ :\ |{\mathcal{B}}_{r}(\bar{\tau}(a_{j},b_{j})) \cap {\mathcal{J}}| \geq \frac{2}{3} (2^{\Delta h}) \right\},$$ for $j = 1,\ldots, 2^{\Delta h}$. We define our estimate of ${\tau}(a_0,b_0)$ to be $\bar{\tau}'(a_0,b_0) = \bar{\tau}(a_{j^*},b_{j^*})$, where $j^* = \arg\min_{j} r^*_j$. See Figure \[fig:ball\]. For $D> 0$, $W > 5$, we define $${\overline{\mathbb{SD}}}(a_0,b_0) = {\mathbbm{1}}\left\{ 2^{-\Delta h}\left|\left\{j\,:\, \bar{\tau}(a_{j},b_{j}) \leq D + \ln\frac{W}{3} \right\}\right| > \frac{1}{2}\right\}.$$ We extend Lemmas \[lem:distmet1gtr\] and \[lem:distmet2gtr\] to $\bar{\tau}'(a_0,b_0)$ and ${\overline{\mathbb{SD}}}(a_0,b_0)$. \[lem:deep1\] Let $\alpha > 1$, $D > 0$, $\gamma > 0$, and ${\varepsilon}> 0$. Let $a_0,b_0 \in L^{(h)}_{h'}$ as above. There exist $\kappa > 1$ such that if the following conditions hold: - $\mathrm{[Small\ Diameter]}$ ${\tau}(a_0,b_0) < D$, - $\mathrm{[Sequence\ Length]}$ $k > \log^\kappa(n)$, then $$|\bar{\tau}'(a_0,b_0) - {\tau}(a_0,b_0)| < {\varepsilon},$$ with probability at least $1-O(n^{-\gamma})$. \[lem:deep2\] Let $\alpha > 1$, $D > 0$, $W > 5$, and $\gamma > 0$. Let $a_0,b_0 \in L^{(h)}_{h'}$ as above. There exists $\kappa > 1$ such that if the following conditions hold: - $\mathrm{[Large\ Diameter]}$ ${\tau}(a_0,b_0) > D + \ln W$, - $\mathrm{[Sequence\ Length]}$ $k > \log^\kappa(n)$, then $${\overline{\mathbb{SD}}}(a_0,b_0) = 0,$$ with probability at least $1-n^{-\gamma}$. On the other hand, if the first condition above is replaced by - $\mathrm{[Small\ Diameter]}$ ${\tau}(a_0,b_0) < D + \ln \frac{W}{5}$, then $${\overline{\mathbb{SD}}}(a_0,b_0) = 1,$$ with probability at least $1-n^{-\gamma}$. Reconstructing Homogeneous Trees {#section:hmg} ================================ In this section, we prove our main result in the case of homogeneous trees. More precisely, we prove the following. \[thm:mainhmg\] Let $0 < {\Delta}\leq f \leq g < +\infty$ and denote by ${\mathbb{HY}}^{f,g}_{\Delta}$ the set of all homogeneous phylogenies ${{\mathcal{T}}}= (V,E,[n],{\rho};{\tau})$ satisfying $f \leq {\tau}_e \leq g$ and ${\tau}_e$ is an integer multiple of ${\Delta}$, $\forall e\in E$. Let $g^* = \ln \sqrt{2}$. Then, for all ${\phi}\geq 2$, $0 < {\Delta}\leq f \leq g < g^*$ and $Q\in {\mathbb{Q}}_{{\phi}}$, there is a distance-based method solving the phylogenetic reconstruction problem on ${\mathbb{HY}}^{f,g}_{\Delta}\otimes \{Q\}$ with $k = {{\mbox{{\rm poly}}}}(\log n)$. All proofs are relegated to Appendix \[sec:proofs\]. In the homogeneous case, we can build the tree level by level using simple “four-point” techniques [@Buneman:71]. See e.g. [@SempleSteel:03; @Felsenstein:04] for background and details. See also Section \[section:algorithm\] below. The underlying combinatorial algorithm we use here is essentially identical to the one used by Mossel in [@Mossel:04a]. From Propositions \[lem:deep1\] and \[lem:deep2\], we get that the “local metric” on each level is accurate as long as we compute adequate weights. We summarize this fact in the next proposition. For ${\Delta}> 0$ and $z \in {\mathbb{R}}_+$, we let $[z]_{\Delta}$ be the closest multiple of ${\Delta}$ to $z$ (breaking ties arbitrarily). We define $${\overline{\mathrm{d}}}(a_0,b_0) = \left\{ \begin{array}{ll} [\bar{\tau}'(a_0,b_0)]_{\Delta}, & \text{if}\ {\overline{\mathbb{SD}}}(a_0,b_0) = 1,\\ +\infty, & \text{o.w.} \end{array} \right.$$ \[prop:deep3\] Let $D > 0$, $W > 5$, and $\gamma > 0$. Let ${{\mathcal{T}}}= (V,E,[n],{\rho};{\tau}) \in {\mathbb{HY}}^{f,g}_{\Delta}$ with $g < g^*$. Let $a_0,b_0 \in L^{(h)}_{h'}$ for $0\leq h' < h$. Assume we are given, for $x=a,b$, $\theta_{e}$ for all $e\in V_x$. There exists $\kappa > 0$, such that if the following condition holds: - $\mathrm{[Sequence\ Length]}$ The sequence length is $k > \log^\kappa(n)$, then we have, with probability at least $1 - O(n^{-\gamma})$, $${\overline{\mathrm{d}}}(a_0,b_0) = {\tau}(a_0,b_0)$$ under either of the following two conditions: 1. $\mathrm{[Small\ Diameter]}$ ${\tau}(a_0,b_0) < D$, or 2. $\mathrm{[Finite\ Estimate]}$ ${\overline{\mathrm{d}}}(a_0,b_0) < +\infty$. It remains to show how to compute the weights, which is the purpose of the next section. Estimating Averaging Weights ---------------------------- Proposition \[prop:deep3\] relies on the prior computation of the weights $\theta_{e}$ for all $e\in V_x$, for $x=a,b$. In this section, we show how this estimation is performed. Let $a_0,b_0,c_0 \in L^{(h)}_{h'}$. Denote by $z$ the meeting point of the paths joining $a_0, b_0, c_0$. We define the “three-point” estimate $$\hat\theta_{z,a_0} = {\mathbb{O}}(a_0; b_0, c_0) \equiv \exp\left(-\frac{1}{2}[{\overline{\mathrm{d}}}(a_0,b_0) + {\overline{\mathrm{d}}}(a_0,c_0) - {\overline{\mathrm{d}}}(b_0,c_0)]\right).$$ Note that the expression in parenthesis is an estimate of the distance between $a_0$ and $z$. \[prop:weights\] Let $a_0,b_0,c_0 \in L^{(h)}_{h'}$ as above. Assume that the assumptions of Propositions \[lem:deep1\], \[lem:deep2\], \[prop:deep3\] hold. Assume further that the following condition hold: - $\mathrm{[Small\ Diameter]}$ ${\tau}(a_0,b_0),{\tau}(a_0,c_0),{\tau}(b_0,c_0) < D + \ln W$, then $$\hat\theta_{z,a_0} = \theta_{z,a_0},$$ with probability at least $1-O(n^{-\gamma})$ where $\hat\theta_{z,a_0} = {\mathbb{O}}(a_0; b_0, c_0)$. Putting it All Together {#section:algorithm} ----------------------- Let $0 \leq h' < h$ and ${\mathcal{Q}}= \{a_0,b_0,c_0,d_0\} \subseteq L^{(h)}_{h'}$. The topology of $T^{(h)}$ restricted to ${\mathcal{Q}}$ is completely characterized by a bi-partition or [*quartet split*]{} $q$ of the form: $a_0 b_0 | c_0 d_0$, $a_0 c_0 | b_0 d_0$ or $a_0 d_0 | b_0 c_0$. The most basic operation in quartet-based reconstruction algorithms is the inference of such quartet splits. In distance-based methods in particular, this is usually done by performing the so-called [*four-point test*]{}: letting $${\mathcal{F}}(a_0 b_0 | c_0 d_0) = \frac{1}{2}[{\tau}(a_0,c_0) + {\tau}(b_0,d_0) - {\tau}(a_0,b_0) - {\tau}(c_0,d_0)],$$ we have $$q = \left\{ \begin{array}{ll} a_0 b_0 | c_0 d_0 & \mathrm{if\ }{\mathcal{F}}(a_0,b_0|c_0,d_0) > 0\\ a_0 c_0 | b_0 d_0 & \mathrm{if\ }{\mathcal{F}}(a_0,b_0|c_0,d_0) < 0\\ a_0 d_0 | b_0 c_0 & \mathrm{o.w.} \end{array} \right.$$ Of course, we cannot compute ${\mathcal{F}}(a_0,b_0|c_0,d_0)$ directly unless $h'=0$. Instead we use Proposition \[prop:deep3\]. #### Deep Four-Point Test. Assume we have previously computed weights $\theta_{e}$ for all $e\in V_{x}$, for $x=a,b,c,d$. We let $$\overline{\mathcal{F}}(a_0 b_0 | c_0 d_0) = \frac{1}{2}[{\overline{\mathrm{d}}}(a_0,c_0) + {\overline{\mathrm{d}}}(b_0,d_0) - {\overline{\mathrm{d}}}(a_0,b_0) - {\overline{\mathrm{d}}}(c_0,d_0)],\label{eq:fcal}$$ and we define the [*deep four-point test*]{} $${\overline{\mathbb{FP}}}(a_0,b_0|c_0,d_0) = {\mathbbm{1}}\{\overline{\mathcal{F}}(a_0 b_0 | c_0 d_0) > f/2\},$$ with ${\overline{\mathbb{FP}}}(a_0,b_0|c_0,d_0) = 0$ if any of the distances in (\[eq:fcal\]) is infinite. Also, we extend the [*diameter test*]{} ${\overline{\mathbb{SD}}}$ to arbitrary subsets by letting ${\overline{\mathbb{SD}}}({\mathcal{S}}) = 1$ if and only if ${\overline{\mathbb{SD}}}(x,y) = 1$ for all pairs $x,y \in {\mathcal{S}}$. #### Algorithm. Fix $\alpha > 1$, $D > 4g$, $W > 5$, $\gamma > 3$. Choose $\kappa$ so as to satisfy Propositions \[prop:deep3\] and \[prop:weights\]. Let ${\mathcal{Z}}_{0}$ be the set of leaves. The algorithm—a standard cherry picking algorithm—is detailed in Figure \[fig:algo\]. Acknowledgments {#acknowledgments .unnumbered} =============== This work was triggered by a discussion with Elchanan Mossel on lower bounds for distance methods, following a talk of Joseph Felsenstein. In particular, Elchanan pointed out that the distance matrix has a potentially useful correlation structure. Extending to General Trees {#section:general-trees} ========================== It is possible to generalize the previous arguments to general trees, using a combinatorial algorithm of [@DaMoRo:08b], thereby giving a proof of Theorem \[thm:main\]. To apply the algorithm of [@DaMoRo:08b] we need to obtain a generalization of Proposition \[prop:deep3\] for disjoint subtrees in “general position.” This is somewhat straightforward and we give a quick sketch in this section. Basic Definitions ----------------- The algorithm in [@DaMoRo:08b] is called Blindfolded Cherry Picking. We refer the reader to [@DaMoRo:08b] for a full description of the algorithm, which is somewhat involved. It is very similar in spirit to the algorithm introduced in Section \[section:algorithm\], except for complications due to the non-homogeneity of the tree. The proof in [@DaMoRo:08b] is modular and relies on two main components: a distance-based *combinatorial* argument which remains unchanged in our setting; and a *statistical* argument which we now adapt. The key to the latter is [@DaMoRo:08b Proposition 4]. Note that [@DaMoRo:08b Proposition 4] is *not* distance-based as it relies on a complex ancestral reconstruction function—recursive majority. Our main contribution in this section is to show how this result can be obtained using the techniques of the previous sections—leading to a fully distance-based reconstruction algorithm. In order to explain the complications due to the non-homogeneity of the tree and state our main result, we first need to borrow a few definitions from [@DaMoRo:08b]. #### Basic Definitions. Fix $0 < {\Delta}\leq f \leq g < g^*$ as in Theorem \[thm:main\]. Let ${{\mathcal{T}}}= (V,E,[n],{\rho};{\tau}) \in {\mathbb{Y}}^{f,g}_{\Delta}$ be a phylogeny with underlying tree $T = (V,E)$. In this section, we sometimes refer to the edge set, vertex set and leaf set of a tree $T'$ as ${\mathcal{E}}(T')$, ${\mathcal{V}}(T')$, and ${\mathcal{L}}(T')$ respectively. Let $V' \subseteq V$ be a subset of the vertices of $T$. The [*subtree of $T$ restricted to $V'$*]{} is the tree $T'$ obtained by 1) keeping only nodes and edges on paths between vertices in $V'$ and 2) by then contracting all paths composed of vertices of degree 2, except the nodes in $V'$. We sometimes use the notation $T' = T|_{V'}$. See Figure \[fig:restricted\] for an example. \[def:disjoint\] Denote by ${\mathrm{Path}}_T(x,y)$ the path (sequence of edges) connecting $x$ to $y$ in $T$. We say that two restricted subtrees $T_1, T_2$ of $T$ are [*edge disjoint*]{} if $${\mathrm{Path}}_T(x_1,y_1) \cap {\mathrm{Path}}_T(x_2,y_2) = \emptyset,$$ for all $x_1, y_1 \in {\mathcal{L}}(T_1)$ and $x_2, y_2 \in {\mathcal{L}}(T_2)$. We say that $T_1,T_2$ are [*edge sharing*]{} if they are not edge disjoint. See Figure \[fig:sharing\] for an example. We say that a tree is a rooted full binary tree if all its internal nodes have degree 3 except the root which has degree 2. A restricted subtree $T_1$ of $T$ is a [*legal subtree*]{} of $T$ if it is also a rooted full binary tree. We say that a forest $${\mathcal{F}}= \{T_1,T_2,\ldots \},$$ is [*legal subforest*]{} of $T$ if the $T_{\iota}$’s are *edge-disjoint* legal subtrees of $T$. We denote by $\rho({\mathcal{F}})$ the set of roots of ${\mathcal{F}}$. We say that two edge-disjoint legal subtrees $T_1$, $T_2$ of $T$ are [*dangling*]{} if there is a choice of root for $T$ *not in $T_1$ or $T_2$* that is consistent with the rooting of both $T_1$ and $T_2$. See Figure \[fig:remote\] below for an example where two legal, edge-disjoint subtrees are *not* dangling. \[def:bds\] Let $T_1 = T_{x_1}$ and $T_2 = T_{x_2}$ be two restricted subtrees of $T$ rooted at $x_1$ and $x_2$ respectively. Assume further that $T_1$ and $T_2$ are [*edge-disjoint*]{}, but not necessarily [*dangling*]{}. Denote by $y_{\iota}, z_{\iota}$ the children of $x_{\iota}$ in $T_{\iota}$, ${\iota}=1,2$. Let $w_{\iota}$ be the node in $T$ where the path between $T_1$ and $T_2$ meets $T_{\iota}$, ${\iota}= 1,2$. Note that $w_{\iota}$ may not be in $T_{\iota}$ since $T_{\iota}$ is [*restricted*]{}, ${\iota}= 1,2$. If $w_{\iota}\neq x_{\iota}$, assume without loss of generality that $w_{\iota}$ is in the subtree of $T$ rooted at $z_{\iota}$, ${\iota}= 1,2$. We call this configuration the *Basic Disjoint Setup (General)*. See Figure \[fig:remote\]. Let ${\tau}(T_1,T_2)$ be the length of the path between $w_1$ and $w_2$ in the metric ${\tau}$. Deep Distorted Metric --------------------- Our reconstruction algorithm for homogeneous trees (see Section \[section:hmg\]) builds the tree level by level and only encounters situations where one has to compute the distance between two *dangling* subtrees (that is, the path connecting the subtrees “goes above them”). However, when reconstructing general trees by growing a subforest from the leaves, more general situations such as the one depicted in Figure \[fig:remote\] cannot be avoided and have to be dealt with carefully. Hence, our goal in this subsection is to compute the distance between the internal nodes $x_1$ and $x_2$ in the Basic Disjoint Setup (General). We have already shown how to perform this computation when $T_1$ and $T_2$ are [*dangling*]{}, as this case is handled easily by Proposition \[prop:deep3\] (after a slight modification of the distance estimate; see below). However, in the general case depicted in Figure \[fig:remote\], there is a complication. When $T_1$ and $T_2$ are [*not*]{} dangling, the reconstructed sequences at $x_1$ and $x_2$ are [*not*]{} conditionally independent. But it can be shown that for the algorithm Blindfolded Cherry Picking to work properly, we need: 1) to compute the distance between $x_1$ and $x_2$ correctly when the two subtrees are close and dangling; 2) detect when the two subtrees are far apart (but an accurate distance estimate is not required in that case). This turns out to be enough because the algorithm Blindfolded Cherry Picking ensures roughly that close reconstructed subtrees are always dangling. We refer the reader to [@DaMoRo:08b] for details. The key point is the following: if one computes the distance between $y_1$ and $y_2$ [*rather than*]{} the distance between $x_1$ and $x_2$, then the dangling assumption is satisfied (re-root the tree at any node along the path connecting $w_1$ and $w_2$). However, when the algorithm has only reconstructed $T_1$ and $T_2$, we cannot tell which pair in $\{y_1, z_1\}\times\{y_2, z_2\}$ is the right one to use for the distance estimation. Instead, we compute the distance for all pairs in $\{y_1, z_1\}\times\{y_2, z_2\}$ and the following then holds: in the dangling case, all these distances will agree (after subtracting the length of the edges between $x_1, x_2$ and $\{y_1, z_1, y_2, z_2\}$); in the general case, at least one is correct. This is the basic observation behind the routine [<span style="font-variant:small-caps;">DistortedMetric</span>]{} in Figure \[fig:distmet\] and the proof of Proposition \[prop:deepgeneral\] below. We slightly modify the definitions of Section \[section:hmg\]. Using the notation of Definition \[def:bds\], fix $(a_0,b_0) \in \{y_1,z_1\}\times\{y_2,z_2\}$. For $v\in V$ and $\ell\in L$, denote by $|\ell|_v$ be the graph distance (that is, the number of edges) between $v$ and leaf $\ell$. Assume that we are given $\theta_e$ for all $e\in {\mathcal{E}}(T_{a_0})\cup{\mathcal{E}}(T_{b_0})$. Let $\alpha > 1$ and $\Delta h = {\lfloor \alpha \log_2\log_2 n \rfloor}$. Imagine (minimally) completing the subtrees below $a_0$ and $b_0$ with $0$-length edges so that the leaves below $a_0$ and $b_0$ are at distance at least $\Delta h$. For $x \in \{a,b\}$, denote by $x_1,\ldots,x_{2^{\Delta h}}$, the vertices below $x_0$ at distance $\Delta h$ from $x_0$ and, for $j = 1,\ldots,2^{\Delta h}$, let $X_j$ be the leaves of $T$ below $x_j$. For $1 \leq j \leq 2^{\Delta h}$, we estimate ${\tau}(a_0,b_0)$ as follows $$\begin{aligned} \bar{\tau}(a_j,b_j) &\equiv& -\ln \left(\frac{1}{|A_j||B_j|}\sum_{a' \in A_j} \sum_{b' \in B_j} \Theta^{-1}_{a_0,a'}\Theta^{-1}_{b_0,b'} e^{-{\hat\tau}(a',b')}\right).\end{aligned}$$ Note that, because the tree is binary, it holds that $$\sum_{a' \in A_j} \sum_{b' \in B_j} 2^{-|a'|_{a_j} - |b'|_{b_j}} = \sum_{a' \in A_j} 2^{-|a'|_{a_j}} \sum_{b' \in B_j} 2^{- |b'|_{b_j}} = 1,$$ and we can think of the weights on $A_j$ (similarly for $B_j$) as resulting from a homogeneous flow $\Psi_{a_j}$ from $a_j$ to $A_j$. Then, the bound on the variance of $$S_{a_j} \equiv \sum_{a' \in A_j} 2^{-|a'|_{a_j}} \Theta^{-1}_{a_j,a'}{\sigma}_{a'},$$ in Proposition \[prop:weightedmaj\] still holds with $$K_{a_j,\Psi_{a_j}} = \sum_{e\in {\mathcal{E}}(T_{a_j})} R_{a_j}(e) \Psi(e)^2.$$ Moreover $K_{a_j,\Psi_{a_j}}$ is uniformly bounded following an argument identical to (\[eq:kbound\]) in the proof of Lemma \[lemma:k\]. For $d \in {\mathbb{R}}$ and $r > 0$, let ${\mathcal{B}}_r(d)$ be the ball of radius $r$ around $d$. We define the [*dense ball around $j$*]{} to be the smallest ball around $\bar{\tau}(a_{j},b_{j})$ containing at least $2/3$ of ${\mathcal{J}}= \{\bar{\tau}(a_{j'},b_{j'})\}_{j'=1}^{2^{\Delta h}}$ (as a multiset). The radius of the dense ball around $j$ is $$r^*_j = \inf\left\{r\ :\ |{\mathcal{B}}_{r}(\bar{\tau}(a_{j},b_{j})) \cap {\mathcal{J}}| \geq \frac{2}{3} (2^{\Delta h}) \right\},$$ for $j = 1,\ldots, 2^{\Delta h}$. We define our estimate of ${\tau}(a_0,b_0)$ to be $\bar{\tau}'(a_0,b_0) = \bar{\tau}(a_{j^*},b_{j^*})$, where $j^* = \arg\min_{j} r^*_j$. See Figure \[fig:ball\]. For $D> 0$, $W > 5$, we define $${\overline{\mathbb{SD}}}(a_0,b_0) = {\mathbbm{1}}\left\{ 2^{-\Delta h}\left|\left\{j\,:\, \bar{\tau}(a_{j},b_{j}) \leq D + \ln\frac{W}{3} \right\}\right| > \frac{1}{2}\right\},$$ and we let $${\overline{\mathrm{d}}}(a_0,b_0) = \left\{ \begin{array}{ll} [\bar{\tau}'(a_0,b_0)]_{\Delta}, & \text{if}\ {\overline{\mathbb{SD}}}(a_0,b_0) = 1,\\ +\infty, & \text{o.w.} \end{array} \right.$$ \[prop:deepgeneral\] Let $\alpha > 1$, $D > 0$, $W > 5$, $\gamma > 0$ and $g < g' < g^*$. Consider the Basic Disjoint Setup (General) with ${\mathcal{F}}= \{T_1, T_2\}$ and ${{\mathcal{Q}}}= \{y_1,z_1,y_2,z_2\}$. Assume we are given $\theta_{e}$ for all $e\in {\mathcal{E}}(T_1)\cup{\mathcal{E}}(T_2)$. Let $\Upsilon$ denote the output of [<span style="font-variant:small-caps;">DistortedMetric</span>]{} in Figure \[fig:distmet\]. There exists $\kappa > 0$, such that if the following condition holds: - $\mathrm{[Edge\ Length]}$ It holds that ${\tau}(e)\leq g'$, $\forall e \in {\mathcal{E}}(T_{x})$, $x\in{{\mathcal{Q}}}$[^7]; - $\mathrm{[Sequence\ Length]}$ The sequence length is $k > \log^\kappa(n)$, then we have, with probability at least $1 - O(n^{-\gamma})$, $$\Upsilon = {\tau}(x_1,x_2)$$ under either of the following two conditions: 1. $\mathrm{[Dangling\ Case]}$ $T_1$ and $T_2$ are dangling and ${\tau}(T_1,T_2) < D$, or 2. $\mathrm{[Finite\ Estimate]}$ $\Upsilon < +\infty$. The proof, which is a simple combination of the proof of Proposition \[prop:deep3\] and the remarks above the statement of Proposition \[prop:deepgeneral\], is left out. #### Full Algorithm. The rest of the Blindfolded Cherry Picking algorithm is unchanged except for an additional step to compute averaging weights as in the algorithm of Section \[section:hmg\]. This concludes our sketch of the proof of Theorem \[thm:main\]. Proofs {#sec:proofs} ====== [Lemma \[lemma:distance\]]{} Note that ${\mathbb{E}}[{\widehat F}^{ab}_{ij}] = \pi_i \left(e^{-{\tau}(a,b) Q}\right)_{ij}$. Then $$\begin{aligned} {\mathbb{E}}\left[{\nu}^\top{\widehat F}^{ab} {\nu}\right] &=& \sum_{i\in{\Phi}} {\nu}_i \sum_{j\in{\Phi}}\pi_i \left(e^{-{\tau}(a,b) Q}\right)_{ij} {\nu}_j\\ &=& \sum_{i\in{\Phi}} {\nu}_i (\pi_i e^{-{\tau}(a,b)}{\nu}_i)\\ &=& e^{-{\tau}(a,b)} \sum_{i\in{\Phi}} \pi_i {\nu}_i^2\\ &=& e^{-{\tau}(a,b)}.\end{aligned}$$ [Lemma \[lem:distmet1gtr\]]{} By Azuma’s inequality we get $$\begin{aligned} &&{\mathbb{P}}\left[{\hat\tau}(u,v) > {\tau}(u,v) + {\varepsilon}\right]\\ && \qquad = {\mathbb{P}}\left[\frac{1}{k} \sum_{i=1}^k {\sigma}_u^i {\sigma}_v^i < e^{- {\tau}(u,v) - {\varepsilon}}\right]\\ && \qquad = {\mathbb{P}}\left[\frac{1}{k} \sum_{i=1}^k {\sigma}_u^i {\sigma}_v^i < e^{- {\tau}(u,v)} - (1 - e^{- {\varepsilon}})e^{- {\tau}(u,v)}\right]\\ && \qquad \leq {\mathbb{P}}\left[\frac{1}{k} \sum_{i=1}^k {\sigma}_u^i {\sigma}_v^i < {\mathbb{E}}[{\sigma}_u^1 {\sigma}_v^1] - (1 - e^{- {\varepsilon}})e^{- \delta\log\log(n)}\right]\\ && \qquad \leq \exp\left(-\frac{\left((1 - e^{- {\varepsilon}})e^{- \delta\log\log(n)}\right)^2} {2k(2\bar{\nu}/k)^2}\right)\\ && \qquad \leq n^{-\gamma},\end{aligned}$$ for $\kappa$ large enough depending on $\delta,{\varepsilon},\gamma$. Above, we used that $${\tau}(u,v) = - \ln {\mathbb{E}}[{\sigma}_u^1 {\sigma}_v^1].$$ A similar inequality holds for the other direction. [Lemma \[lem:distmet2gtr\]]{} Assume the first three conditions hold. By Azuma’s inequality we get $$\begin{aligned} &&{\mathbb{P}}\left[{\hat\tau}(u,v) - {\tau}(u_0,u) - {\tau}(v_0,v) \leq D + \ln \frac{W}{2}\right]\\ && \qquad ={\mathbb{P}}\left[\frac{1}{k} \sum_{i=1}^k {\sigma}_u^i {\sigma}_v^i \geq 2W^{-1} e^{- D}e^{- {\tau}(u_0,u) - {\tau}(v_0,v)}\right]\\ && \qquad\leq{\mathbb{P}}\left[\frac{1}{k} \sum_{i=1}^k {\sigma}_u^i {\sigma}_v^i \geq e^{- {\tau}(u,v)} + W^{-1} e^{- D}e^{- {\tau}(u_0,u) - {\tau}(v_0,v)}\right]\\ && \qquad = {\mathbb{P}}\left[\frac{1}{k} \sum_{i=1}^k {\sigma}_u^i {\sigma}_v^i \geq {\mathbb{E}}[{\sigma}_u^1 {\sigma}_v^1] + W^{-1} e^{- D}e^{- {\tau}(u_0,u) - {\tau}(v_0,v)}\right]\\ && \qquad\leq \exp\left(-\frac{\left(W^{-1} e^{- D}e^{- {\tau}(u_0,u) - {\tau}(v_0,v)}\right)^2} {2k(2\bar{\nu}/k)^2}\right)\\ && \qquad\leq n^{-\gamma},\end{aligned}$$ for $\kappa$ large enough. A similar argument gives the second claim. [Proposition \[prop:weightedmaj\]]{} We follow the proofs of [@EvKePeSc:00; @MosselPeres:03]. Let ${\bar{e}}_i$ be the unit vector in direction $i$. Let $x\in [n]$, then $${\mathbb{E}}[{\bar{e}}_{\xi_x}^\top\,|\,\xi_{\rho}] = {\bar{e}}^\top_{\xi_{\rho}} e^{{\tau}({\rho},x)Q}.$$ Therefore, $${\mathbb{E}}[{\sigma}_x\,|\,{\sigma}_{\rho}] = {\bar{e}}^\top_{\xi_{\rho}} e^{{\tau}({\rho},x)Q} {\nu}= {\sigma}_{\rho}e^{-{\tau}({\rho},x)},$$ and $${\mathbb{E}}[S\,|\,{\sigma}_{\rho}] = \sum_{x \in [n]} \frac{\Psi(x) {\sigma}_{\rho}e^{-{\tau}({\rho},x)}}{\Theta_{{\rho},x}} = {\sigma}_{\rho}\sum_{x \in [n]} \Psi(x) = {\sigma}_{\rho}.$$ In particular, $${\mathbb{E}}[S] = \sum_{\iota\in{\Phi}} \pi_i {\nu}_i = 0.$$ For $x,y \in [n]$, let $x\land y$ be the meeting point of the paths between ${\rho},x,y$. We have $$\begin{aligned} {\mathbb{E}}[{\sigma}_x{\sigma}_y] &=& \sum_{\iota\in{\Phi}} {\mathbb{P}}[\xi_{x\land y} = \iota] {\mathbb{E}}[{\sigma}_x{\sigma}_y\,|\,\xi_{x\land y} = \iota]\\ &=& \sum_{\iota \in {\Phi}} \pi_{\iota} {\mathbb{E}}[{\sigma}_x\,|\,\xi_{x\land y} = \iota] {\mathbb{E}}[{\sigma}_y\,|\,\xi_{x\land y} = \iota]\\ &=& \sum_{\iota \in {\Phi}} \pi_{\iota} e^{-{\tau}(x\land y, x)} {\nu}_{\iota} e^{-{\tau}(x\land y, y)} {\nu}_{\iota}\\ &=& e^{-{\tau}(x,y)} \sum_{\iota \in {\Phi}} \pi_{\iota} {\nu}_{\iota}^2\\ &=& e^{-{\tau}(x,y)}.\end{aligned}$$ Then $$\begin{aligned} {\mathrm{Var}}[S] &=& {\mathbb{E}}[S^2]\\ &=& \sum_{x,y\in [n]} \frac{\Psi(x)\Psi(y)}{\Theta_{{\rho},x}\Theta_{{\rho},y}} {\mathbb{E}}[{\sigma}_x{\sigma}_y]\\ &=& \sum_{x,y\in [n]} \Psi(x)\Psi(y)e^{2{\tau}({\rho}, x\land y)}.\end{aligned}$$ For $e \in E$, let $e = (e_\uparrow,e_{\downarrow})$ where $e_\uparrow$ is the vertex closest to ${\rho}$. Then, by a telescoping sum, for $u \in V$ $$\begin{aligned} \sum_{e \in {\mathrm{Path}}({\rho},u)} R_{{\rho}}(e) &=& \sum_{e \in {\mathrm{Path}}({\rho},u)} e^{2{\tau}({\rho}, e_\downarrow)} - \sum_{e \in {\mathrm{Path}}({\rho},u)} e^{2{\tau}({\rho}, e_\uparrow)}\\ &=& e^{2{\tau}({\rho},u)} - 1,\end{aligned}$$ and therefore $$\begin{aligned} {\mathbb{E}}[S^2] &=& \sum_{x,y\in [n]} \Psi(x)\Psi(y)e^{2{\tau}(v, x\land y)}\\ &=& \sum_{x,y\in [n]} \Psi(x)\Psi(y) \left(1 + \sum_{e \in {\mathrm{Path}}({\rho},x \land y)} R_{{\rho}}(e)\right)\\ &=& 1 + \sum_{e\in E} R_{\rho}(e) \sum_{x,y\in [n]} {\mathbbm{1}}\{e\in {\mathrm{Path}}({\rho},x\land y)\} \Psi(x)\Psi(y)\\ &=& 1 + \sum_{e\in E} R_{\rho}(e) \Psi(e)^2.\end{aligned}$$ [Lemma \[lemma:k\]]{} From (\[eq:k\]), we have $$\begin{aligned} K_{{\rho},\Psi} &\leq& \sum_{i=0}^{h-1} (1 - e^{-2g}) 2^{h-i} \frac{e^{2 (h-i) g}}{2^{2 (h-i)}}\nonumber\\ &\leq& \sum_{j=1}^{h} e^{2 j g} e^{-(2 \ln \sqrt{2})j}\nonumber\\ &=& \sum_{j=1}^{h} e^{2 j (g - g^*)}\nonumber\\ &\leq& \sum_{j=0}^{+\infty} (e^{-2 (g^* - g)})^j\nonumber\\ &=& \frac{1}{1 - e^{-2 (g^* - g)}} < +\infty,\label{eq:kbound}\end{aligned}$$ where recall that $g^* = \ln\sqrt{2}$. [Proposition \[lem:deep1\]]{} Let $${\mathcal{Z}}= \{a_j\}_{j=1}^{2^{\Delta h}} \cup \{b_j\}_{j=1}^{2^{\Delta h}},$$ and $${\mathcal{E}}= \{({\sigma}^i_{{\mathcal{Z}}})_{i=1}^k\}.$$ For $j = 1, \ldots, 2^{\Delta h}$, let $$e^{-{\hat\tau}(a_j,b_j)} = \frac{1}{k}\sum_{i=1}^k {\sigma}^i_{a_j} {\sigma}^i_{b_j}.$$ Note that $${\mathbb{E}}[e^{-{\hat\tau}(a_j,b_j)}] = e^{-{\tau}(a_j,b_j)},$$ by Lemma \[lemma:distance\]. For $i = 1, \ldots, k$, $j = 1, \ldots, 2^{\Delta h}$, and $x \in \{a,b\}$, let $$\bar{\sigma}^i_{x_j} = \sum_{x' \in X_j} \frac{2^{-h''} {\sigma}^i_{x'}}{\Theta_{x_j, x'}}.$$ By the Markov property, it follows that, conditioned on ${\mathcal{E}}$, $$\{\bar{\sigma}^i_{x_j}\ :\ i = 1, \ldots, k,\ j = 1, \ldots, 2^{\Delta h},\ x \in \{a,b\}\},$$ are mutually independent. Moreover, by Proposition \[prop:weightedmaj\], we have $${\mathbb{E}}[\bar{\sigma}^i_{x_j}\,|\,{\mathcal{E}}] = {\sigma}^i_{x_j},$$ and $${\mathbb{E}}[(\bar{\sigma}^i_{x_j})^2\,|\,{\mathcal{E}}] \leq 2\bar\pi^{-1} \frac{1}{1 - e^{-2(g^* - g)}}.$$ Therefore, for any $\zeta > 0$ there exists $\kappa > 1$ such that $$\begin{aligned} {\mathbb{E}}[e^{-\bar{\tau}(a_j,b_j)}\,|\,{\mathcal{E}}] &=& e^{{\tau}(a_0,a_j) + {\tau}(b_0,b_j)}\left(\frac{1}{k} \sum_{i=1}^k {\mathbb{E}}[\bar{\sigma}^i_{a_j} \bar{\sigma}^i_{b_j}\,|\,{\mathcal{E}}]\right)\\ &=& e^{{\tau}(a_0,a_j) + {\tau}(b_0,b_j)}\left(\frac{1}{k} \sum_{i=1}^k {\mathbb{E}}[\bar{\sigma}^i_{a_j}\,|\,{\mathcal{E}}] {\mathbb{E}}[\bar{\sigma}^i_{b_j}\,|\,{\mathcal{E}}]\right)\\ &=& e^{-({\hat\tau}(a_j,b_j) - {\tau}(a_j,a_0) - {\tau}(b_j,b_0))},\end{aligned}$$ and $$\begin{aligned} {\mathrm{Var}}[e^{-\bar{\tau}(a_j,b_j)}\,|\,{\mathcal{E}}] &=& e^{2{\tau}(a_0,a_j) + 2{\tau}(b_0,b_j)}\left(\frac{1}{k^2} \sum_{i=1}^k {\mathrm{Var}}[\bar{\sigma}^i_{a_j} \bar{\sigma}^i_{b_j}\,|\,{\mathcal{E}}]\right)\\ &\leq& e^{2{\tau}(a_0,a_j) + 2{\tau}(b_0,b_j)}\left(\frac{1}{k^2} \sum_{i=1}^k {\mathbb{E}}[(\bar{\sigma}^i_{a_j})^2\,|\,{\mathcal{E}}] {\mathbb{E}}[(\bar{\sigma}^i_{b_j})^2\,|\,{\mathcal{E}}]\right)\\ &\leq& \frac{e^{2{\tau}(a_0,a_j) + 2{\tau}(b_0,b_j)}}{k}\left(\frac{2}{1 - e^{-2(g^* - g)}}\right)^2\\ &\leq& \frac{e^{4 g{\lfloor \alpha \log_2\log_2 n \rfloor}}}{k}\left(\frac{2}{1 - e^{-2(g^* - g)}}\right)^2\\ &\leq& \zeta.\end{aligned}$$ Take $\kappa$ large enough such that Lemma \[lem:distmet1gtr\] holds for diameter $2 g{\lfloor \alpha \log_2\log_2 n \rfloor} + D$, precision ${\varepsilon}/6$ and failure probability $O(n^{-(\gamma+1)})$. Then, we have $$\label{eq:eweight} |{\hat\tau}(a_j,b_j) - {\tau}(a_j,b_j)| < {\varepsilon}/6, \quad \forall j=1,\ldots,2^{\Delta h},$$ with probability $1 - O(n^{-\gamma})$. Let ${\mathcal{E}}'$ be the event that (\[eq:eweight\]) holds. Note that $${\mathbb{E}}[e^{-\bar{\tau}(a_j,b_j)}\,|\,{\mathcal{E}}\cap {\mathcal{E}}'] = e^{-({\hat\tau}(a_j,b_j) - {\tau}(a_j,a_0) - {\tau}(b_j,b_0))}.$$ Let $${\varepsilon}' = \min\{(e^{{\varepsilon}/3} - e^{{\varepsilon}/6})e^{-D}, (e^{-{\varepsilon}/6} - e^{-{\varepsilon}/3})e^{-D}\}.$$ By Chebyshev’s inequality, we have that $$\begin{aligned} &&{\mathbb{P}}\left[ \bar{\tau}(a_j,b_j) < {\tau}(a_0,b_0) - \frac{1}{3}{\varepsilon}\,|\,{\mathcal{E}}\cap{\mathcal{E}}'\right]\\ && \qquad \leq {\mathbb{P}}\left[ e^{-\bar{\tau}(a_j,b_j)} > e^{-{\tau}(a_0,b_0) + {\varepsilon}/3} \,|\,{\mathcal{E}}\cap{\mathcal{E}}'\right]\\ && \qquad \leq {\mathbb{P}}\left[ e^{-\bar{\tau}(a_j,b_j)} - {\mathbb{E}}[e^{-\bar{\tau}(a_j,b_j)}\,|\,{\mathcal{E}}\cap {\mathcal{E}}'] > [e^{-{\tau}(a_0,b_0)+{\varepsilon}/3} - e^{-({\hat\tau}(a_j,b_j) - {\tau}(a_j,a_0) - {\tau}(b_j,b_0))}] \,|\,{\mathcal{E}}\cap{\mathcal{E}}'\right]\\ && \qquad \leq {\mathbb{P}}\left[ e^{-\bar{\tau}(a_j,b_j)} - {\mathbb{E}}[e^{-\bar{\tau}(a_j,b_j)}\,|\,{\mathcal{E}}\cap {\mathcal{E}}'] > e^{-{\tau}(a_0,b_0)}[e^{{\varepsilon}/3} - e^{{\tau}(a_j,b_j)-{\hat\tau}(a_j,b_j)}] \,|\,{\mathcal{E}}\cap{\mathcal{E}}'\right]\\ && \qquad \leq {\mathbb{P}}\left[ e^{-\bar{\tau}(a_j,b_j)} - {\mathbb{E}}[e^{-\bar{\tau}(a_j,b_j)}\,|\,{\mathcal{E}}\cap {\mathcal{E}}'] > e^{-D}[e^{{\varepsilon}/3} - e^{{\varepsilon}/6}] \,|\,{\mathcal{E}}\cap{\mathcal{E}}'\right]\\ && \qquad \leq {\mathbb{P}}\left[ e^{-\bar{\tau}(a_j,b_j)} - {\mathbb{E}}[e^{-\bar{\tau}(a_j,b_j)}\,|\,{\mathcal{E}}\cap {\mathcal{E}}'] > {\varepsilon}' \,|\,{\mathcal{E}}\cap{\mathcal{E}}'\right]\\ && \qquad \leq 1/12, \end{aligned}$$ for $\zeta$ small enough. A similar argument holds for $${\mathbb{P}}\left[ \bar{\tau}(a_j,b_j) > {\tau}(a_0,b_0) + \frac{1}{3}{\varepsilon}\,|\,{\mathcal{E}}\cap{\mathcal{E}}'\right] \leq 1/12.$$ Conditioned on ${\mathcal{E}}\cap{\mathcal{E}}'$, $${\mathcal{L}}= 2^{-\Delta h}\sum_{j=1}^{2^{\Delta h}}{\mathbbm{1}}\left\{|\bar{\tau}(a_{j},b_{j}) - {\tau}(a_{0},b_{0})| < \frac{1}{3}{\varepsilon}\right\}$$ is a sum of independent $\{0,1\}$-variables with average at least $5/6$. By Azuma’s inequality, we have $$\begin{aligned} {\mathbb{P}}[{\mathcal{L}}\leq 2/3 \,|\, {\mathcal{E}}\cap{\mathcal{E}}'] &\leq& {\mathbb{P}}[{\mathcal{L}}- {\mathbb{E}}[{\mathcal{L}}\,|\,{\mathcal{E}}\cap{\mathcal{E}}'] < - 1/6 \,|\, {\mathcal{E}}\cap{\mathcal{E}}']\\ &\leq& \exp\left(- \frac{(1/6)^2}{2 (2^{-\Delta h})^2 2^{\Delta h}}\right)\\ &\leq& \exp\left(-O({\log_2^{\alpha} n}) \right)\\ &\leq& O(n^{-\gamma}),\end{aligned}$$ where we used that $\alpha > 1$. This implies that with (unconditional) probability at least $1 - O(n^{-\gamma})$ $$\left|\left\{|\bar{\tau}(a_{j},b_{j}) - {\tau}(a_{0},b_{0})| < \frac{1}{3}{\varepsilon}\right\}\right| > \frac{2}{3} (2^{\Delta h}).$$ In particular, there exists $j$ such that $$\left|\left\{j'\,:\,|\bar{\tau}(a_{j'},b_{j'}) - \bar{\tau}(a_{j},b_{j})| < \frac{2}{3}{\varepsilon}\right\}\right| > \frac{2}{3} (2^{\Delta h}),$$ and for all $j$ such that $$|\bar{\tau}(a_{j},b_{j}) - {\tau}(a_{0},b_{0})| > {\varepsilon},$$ we have that $$\left|\left\{j'\,:\,|\bar{\tau}(a_{j'},b_{j'}) - \bar{\tau}(a_{j},b_{j})| < \frac{2}{3}{\varepsilon}\right\}\right| \leq \frac{1}{3} (2^{\Delta h}).$$ Finally, we get that $$|\bar{\tau}'(a_0,b_0) - {\tau}(a_{0},b_{0})| \leq {\varepsilon}.$$ [Proposition \[lem:deep2\]]{} The proof is similar to the proof of Lemma \[lem:distmet2gtr\] and Proposition \[lem:deep1\]. [Proposition \[prop:deep3\]]{} We let ${\varepsilon}< {\Delta}/2$. The first part of the proposition follows immediately from Proposition \[lem:deep1\] and the second part of Proposition \[lem:deep2\]. Note that Propositions \[lem:deep1\] and \[lem:deep2\] are still valid when $h' < {\lfloor \alpha \log_2\log_2 n \rfloor}$ if we take instead $\Delta h = \min\{h', {\lfloor \alpha \log_2\log_2 n \rfloor}\}$. Indeed, in that case there is no need to perform an implicit ancestral sequence reconstruction and the proofs of the lemmas follow immediately from Lemmas \[lem:distmet1gtr\] and \[lem:distmet2gtr\]. For the second part, choose $\kappa$ so as to satisfy the conditions of Proposition \[lem:deep1\] [*with diameter $D + \ln W$*]{} and apply the first part of Proposition \[lem:deep2\]. [Proposition \[prop:weights\]]{} The proof follows immediately from Proposition \[prop:deep3\] and the remark above the statement of Proposition \[prop:weights\]. [Theorem \[thm:mainhmg\]]{} The proof of Theorem \[thm:mainhmg\] follows from Propositions \[prop:deep3\] and \[prop:weights\]. Indeed, at each level $h'$, we are guaranteed by the above to compute a distorted metric with a radius large enough to detect all cherries on the next level using four-point tests. The proof follows by induction. [^1]: Department of Mathematics, UCLA. [^2]: The current manuscript is the full version with proofs of [@Roch:08]. In subsequent work [@Roch:09] the results stated here were improved to logarithmic sequence length, thereby matching the best results for general methods. [^3]: Mike Steel offers a 100\$ reward for the solution of this problem. [^4]: The correct statement of this result appears in [@DaMoRo:08b]. Because of different conventions, our edge weights are scaled by a factor of $2$ compared to those in [@DaMoRo:08b]. The dependence of $k$ in ${\Delta}$ is ${\Delta}^{-2}$. [^5]: As in Theorem \[thm:opt\], the dependence of $k$ in ${\Delta}$ is ${\Delta}^{-2}$. [^6]: Note that, without loss of generality, we can consider performing ancestral state reconstruction on a homogeneous tree as it is always possible to “complete” a general tree with zero-length edges. We come back to this point in Appendix \[section:general-trees\]. [^7]: For technical reasons explained in [@DaMoRo:08b], we allow edges slightly longer than the upper bound $g$.
--- abstract: 'We study perception in the scenario of an embodied agent equipped with first-person sensors and a continuous motor space with multiple degrees of freedom. Inspired by two theories of perception in artificial agents [@higgins2018towards; @poincare1895espace] we consider theoretically the commutation properties of action sequences with respect to sensory information perceived by such embodied agent. From the theoretical derivations, we define the Sensory Commutativity Probability criterion which measures how much an agent’s degree of freedom affects the environment in embodied scenarios. We empirically illustrate how it can be used to improve sample-efficiency in Reinforcement Learning.' bibliography: - 'bibli.bib' --- Introduction ============ Perception is the medium by which agents organize and interpret sensory stimuli, in order to reason and act in an environment using their available actions [@hoffman2018interface]. We focus on scenarios where embodied agents are situated in *realistic* environments, i.e. the agents face partial observability, coherent physics, first-person view with high-dimensional state space and low-level continuous motor (i.e. action) space with multiple degrees of freedom. These embodied agents, when acting in such environment, produce a stream of sensorimotor data, composed of successions of motor states and sensory information. While most current approaches for building perception focus on studying the sensory information alone, several approaches [@caselles2019symmetry; @laflaquiere2018unsupervised; @ghosh2018learning; @thomas2017independently] that can be traced back to 1895 [@poincare1895espace], advocate the necessity of studying the relation between sensors and motors for the emergence of perception. ![Two action sequences sensory commute if they produce the same sensory state when composed in different orders from the same starting position. In this example, the actions sequences would not commute if an object would be in the way.[]{data-label="fig:scp"}](images/scp.png) Among those approaches, we focus on Symmetry-Based Disentangled Representation Learning (SBDRL) [@higgins2018towards; @caselles2019symmetry] and what we refer to as SensoriMotor Theory (SMT) [@o2001sensorimotor]. SBDRL aims at formalizing disentanglement in Representation Learning, i.e. the idea that sensory data is generated by a few explanatory factors of variation. The core idea in SBDRL is to define disentanglement with respect to transformations of the environment that leave some aspects invariant. On the other hand, SMT puts forward an unsupervised sensorimotor grounding of perception. It describes how space induces specific invariants in any embodied agent’s sensorimotor experience, and how these invariants can be captured to improve the compactness of representations and the prediction of sensorimotor experiences. In an attempt to unify those two approaches, we study the commutativity of action sequences with respect to sensors, which we term sensory commutativity, illustrated in Fig.\[fig:scp\]. We define the Sensory Commutativity Probability (SCP) as the probability that a sequence of movements using only one degree of freedom of the agent, an arm joint for instance, sensory commutes. We show that this value has meaning for the embodied agent: if the SCP is high then the degree-of-freedom has a low impact on the environment (e.g. moving a shoulder is more likely to move things around than moving finger, so SCP for shoulder is lower than for finger). By computing the SCP for each degree of freedom of the agent, we are able to characterize its motor space and use this relevant information for subsequent tasks. We illustrate this in our experiments as we show how SCP can be used to improve sample-efficiency in a Reinforcement Learning problem. Related work and motivation =========================== SBDRL ----- Symmetry-Based Disentangled Representation Learning (SBDRL) [@higgins2018towards; @caselles2019symmetry] aims at formalizing disentanglement in Representation Learning. The core idea is that SB-disentanglement of a representation is defined with respect to a particular decomposition of the symmetries of the environment. Symmetries are transformations of the environment that leave some aspects of it unchanged. For instance, for an agent moving on a plane, translations of the agent on the $y$-axis leave its $x$ coordinate unchanged. This is formalized using group theory. Groups are composed of these transformations, and group actions are the effect of the transformations on the state of the world and representation. SensoriMotor theory ------------------- SensoriMotor theory (SMT) is a theory of perception that gives prominence to the role of motor information in the emergence of perceptive capabilities [@o2001sensorimotor]. The approach takes inspiration from philosophical ideas formulated more than a century ago by H.Poincare [@poincare1895espace]. It led to theoretical results regarding the extraction of the dimension of space [@laflaquiere2012non], the characterization of displacements as compensable sensory variations [@terekhov2016space], the grounding of the concept of point of view in the motor space [@laflaquiere2013learning; @laflaquiere2015learning], as well as the characterization of the metric structure of space via sensorimotor invariants [@laflaquiere2018discovering]. Motivation ---------- The notion of symmetries of @higgins2018towards is based on transformation that have a group property. Their definition make it possible to formalize disentanglement, although it does not require to exactly precise what makes a transformation belong to the group of symmetries $G$. Moreover, the notion of sub-groups is only defined with intuition as well: what exactly makes a subset of transformations of the group $G$ a subgroup $G_1$? Still @higgins2018towards provide insights and intuitive concepts to describe what might characterize sub-groups: for instance translations along one axis only change the position of the agent for this particular axis and leave other coordinates invariant. In this work we would like to start from the same intuitions, but rigourously define the group and sub-groups of transformations. We note that the notion of transformations that have a group structure is also present in SMT. It dates back to the manuscript of Poincaré [@poincare1895espace], where he describes that compensable transformations of the environment equipped with the composition operation forms a group. Moreover, @philipona2008developpement attempts at properly characterizing those sub-groups. Using action sequences and their commutative property, he suggests that spatial transformations and non-spatial transformations can be disentangled. This is compatible with the intuitions from [@higgins2018towards], since those subsets are indeed sub-groups and do not affect each other. In this paper, we take inspiration from both approaches. From SMT, we choose to study action sequences, termed $Seq(\mathcal{M})$, and their commutative properties. From SBDRL, we choose to study the group and sub-group properties of $Seq(\mathcal{M})$, with the aim of organizing and disentangling the motor space $\mathcal{M}$. This will be achieved with the definition of the Sensory Commutativity Probability criterion. Formalism choice ================ Despite their similarities, both theories mathematically define the world and agents differently. We propose a mathematical framework for the embodied scenario which will allow to properly construct the Sensory Commutativity Probability. We start from the formalism used in SMT, which formalizes the perception of the agent as follows: $$\label{eq1} s_{t} = \phi (m_t, \epsilon_t)$$ At time $t$, the agent is in a particular motor state $m_t$. This means that its motor, i.e. all the actionable part of its body (joints, motors), are in a particular setup called $m_t$. The environment is defined by everything that’s not the agent. It’s thus an entity that is in a state $\epsilon_t$, e.g. a room with 6 walls plus light sources and objects placed in different locations. The agent can perceive the world through its sensorimotor dependencies $\phi$: a function that takes as input $m_t$ and $\epsilon_t$ and produces sensory inputs from its sensors $s_t$. The dynamics of the world are generally not described in SMT, so we extend its formulation: $$\label{eq2} m'_{t+1}, \epsilon'_{t+1} = f(m_t, \epsilon_t, \Delta(m_t, m_{t+1}), \Delta(\epsilon_t, \epsilon_{t+1}))$$ The agent can operate motor commands ($\Delta(m_t, m_{t+1})$). But the environment can change also through its own dynamics outside of the agent, represented by $\Delta(\epsilon_t, \epsilon_{t+1})$. Taking the initial states and changes as inputs, the function $f$ yields new motor state $m'_{t+1}$, and a new configuration of the environment $\epsilon'_{t+1}$. We don’t generally have that $\epsilon_{t+1} =\epsilon'_{t+1}$ or $m_{t+1} = m'_{t+1}$ since the agent can affect the environment configuration through its body movement or the environment can force movements on the agent. Structure and commutativity properties of the set of action sequences $Seq(\mathcal{M})$ ======================================================================================== We will now attempt to formalize groups and sub-groups of symmetries. We propose $G$ to be the set of motor command (or action) sequences of finite length, referred to as $Seq(\mathcal{M})$, and will attempt at extracting sub-groups based on subsets of these transformations. Group structure of $Seq(\mathcal{M})$ ------------------------------------- @philipona2008developpement first defined a relation between action sequences: $h \sim g $ if and only if $h$ and $g$ affect the sensors in the same way. Using our formalism, we can translate this concept into an equality. Let $(h,g)\in Seq(\mathcal{M})$. h is equivalent to g under $(m_t, \epsilon_t)$, noted $h\sim_{m_t, \epsilon_t} g$ if and only if $$\phi(f(m_t, \epsilon_t, h,\Delta_{\epsilon_{t}}^{\epsilon_{t+1}})) = \phi(f(m_t, \epsilon_t, g,\Delta_{\epsilon_{t}}^{\epsilon_{t+1}}))$$ Intuitively, two actions sequences are equivalent for a particular motor state and environment state if applying them lead to the same sensory state. For instance for a multiple-joints arm moving freely in an empty space, there are multiple different ways of moving the arm from one place to another. This yields action sequences ($h_1, .., h_n)$ which are equivalent in this situation $(m_t, \epsilon_t)$, we thus have $h\sim_{m_t, \epsilon_t} g$. However in other situations these actions sequences can become not equivalent, for instance if there are objects on the way for instance as illustrated in Fig.\[fig:illu\]. For convenience and clarity, we will drop the notation for depence on $(m_t, \epsilon_t)$ and thus write $h\sim g$ whenever there are no ambiguities in the context. We now consider the structure of $Seq(\mathcal{M})$ under composition $\circ$ with respect to the equivalence $\sim$. \[prop1\] 1\. $\sim$ is an equivalence, i.e. it is reflexive, transitive and symmetric.\ 2. ($Seq(\mathcal{M})$, $\circ$) is a group w.r.t $\sim$.\ 3. $\circ$ is generally not commutative with respect to $\sim$. See Appendix \[app\_proof\] for full proof. Point 1 follows from the properties of $=$. For Point 2, composing two action sequences yields an action sequence, the no-op action is the identity element and if we suppose that there are no irreversible phenomenons in the environment, all action sequences can be inverted. For Point 3, we can always construct action sequences that do not commute. ($Seq(\mathcal{M})$, $\circ$) is thus a group w.r.t $\sim$. This structure is coherent with the intuitions in SBRL and SMT theories. In the following, we build on the observation that composing action sequences is not generally commutative. We show how this property can lead the agent to organize and interpret its motor space. Commutativity properties of $Seq(\mathcal{M})$ {#sec:commu} ---------------------------------------------- ### Philipona’s conjecture {#sec:conjecture} @philipona2008developpement already studied how action sequences commute with respect to the sensory information received by the agent. Action sequences do not necessarily commute as stated in Prop.\[prop1\]. For example if a movable object is placed to the right of your arm, moving your arm right then left will not have the same effect (in terms of sensor change) as moving it left then right, as illustrated in Fig.\[fig:illu\]. Philipona thus defines commutation residues. Suppose that doing $h_1 \circ h_2$ is different from $h_2 \circ h_1$, then a commutation residue $g$ is an action sequence that you have to do to compensate the difference in sensory experience. ![image](images/fig_illu.png) $g$ is a commutation residue of $(h_1, h_2)$ if and only if $h_1 \circ h_2 \sim h_2 \circ h_1 \circ g$. If $g$ is equivalent to no-op (no action), then $h_1$ and $h_2$ commute. Starting from this definition, he conjectured that all action sequences that are not displacements commute with any action sequences. For instance moving you arms (displacement action) and opening the eyes (non-displacement action) will always commute whereas two displacement actions will not necessarily commute, depending on which starting situation $(m_t, \epsilon_t)$ is selected. The subset of $Seq(\mathcal{M})$ composed of non-displacements action sequences is the sub-group of $Seq(\mathcal{M})$ that commutes, i.e. the abelian sub-group of $Seq(\mathcal{M})$. We will illustrate this conjecture with experiments in Sec.\[sec:conjecturephilipona\_exp\]. ### Sensory commutativity probability of an action Based on Philipona’s conjecture, we derive a criterion for characterizing how much each degree of freedom of the agent affects the world, computable using only sensorimotor data. We define “degree of freedom” (DOF) as a dimension of the multidimensional continuous action space of the agent. Using the conjecture, we have that for an action sequence $h$, if the agent plays it in two different orders starting from the same situation, there is a chance that the agent will experience two different sensory outcomes only if the action sequence $h$ is composed of at least one displacement action (an action that affect the environment such as moving limbs or going forward). However not all displacement actions are equivalent. The agent is more likely to observe two different outcomes if the action sequence is composed of displacement actions that affect the environment *a lot*. Consider moving your forearm (elbow joint) compared to moving your whole arm (shoulder joint, see Fig.\[fig:area\_coverage\]): the latter is more likely to move things around in the environment and thus induce sensory non-commutativity when played in two different orders (i.e. having two different sensory outcomes). An elbow joint should therefore have a higher SCP than a shoulder joint. We formalize this intuition by defining the Sensory Commutativity Probability (SCP) of a degree of freedom, averaged over all starting situations $(m_t, \epsilon_t)$: Let $Seq(\mathcal{M}_k)$ be the set of motor commands (or action) sequences of finite length for the k$^{th}$ degree of freedom of $\mathcal{M}$ (motor state space). Let $h\in Seq(\mathcal{M}_k)$ and let $h_p$ be a random permutation of $h$ (same sequence but different order). The Sensory Commutativity Probability of the k$^{th}$ degree of freedom $SCP(\mathcal{M}_k)$ is defined as: $$SCP(\mathcal{M}_k) = \mathbb{P}_{m_t, \epsilon_t, h}[h\sim_{m_t, \epsilon_t} h_p]$$ In our experiments, we show how to compute the SCP of a degree of freedom and how we are able to use it to improve sample-efficiency in a Reinforcement Learning problem. Sensory Commutativity Probability experimental analysis {#sec:scpp} ======================================================= In this first experimental section we compute and interpret the SCP for an embodied agent scenario. We then compare SCP to baseline alternatives. Experimental setup ------------------ The simulation we use needs to satisfy the properties of an embodied agent scenario: navigable space with objects to interact with, first-person high dimensional observations, low-level high-dimensional action space and coherent physics. Unfortunately, these requirements are not met in current benchmarks. Mujoco [@todorov2012mujoco] doesn’t have first-person observations, robotic arm setups does not allow navigation, Arcade Learning Environment [@bellemare2013arcade], DeepMind lab [@beattie2016deepmind] and VizDoom [@kempka2016vizdoom] do not have low-level motor commands but rather have high-level action spaces. We thus develop our own 2D simulation using Flatland [@caselles2018flatland], a platform for creating 2D RL environments. We construct an agent called Polyphemus (a Cyclop from the Greek mythology), that has a movable and rotatable base equipped with a rotatable head and two 2-DOF arms. The agent sees through its unique eye that has an activable eyelid, yielding a total of 8 DOF. The image received by the agent is a 64 pixels RGB image depicting what its eye sees. This agent is placed in a room with fixed, moving or movable entities, all of different colors. The agent can move around and interact with these entities. Its point of view can change through base movement, rotation, and head rotation. Our simulation is illustrated in Fig.\[fig:sm\_simu\]. For each degree of freedom, an action or motor command corresponds to a change in the longitudinal/angular velocity of the degree of freedom. ![SM-simulation used for our experiments. The agent Polyphemus has a 8 DOF motor space, receives an image of it’s only eye, and is placed in a room with fixed, movable and moving elements.[]{data-label="fig:sm_simu"}](images/sm_simu.png) ![Illustration of why elbow joints should have a higher sensory commutativity probability than a shoulder joints. Smaller coverage area implies a smaller chance of the arm interacting with elements of the environment, thus increasing the sensory commutativity probability.[]{data-label="fig:area_coverage"}](images/area_coverage.png) Estimating the Sensory Commutativity Probability ------------------------------------------------ In order to estimate the SCP of each of the 8 agent’s degree of freedom, we initialize the SCP value to $0$ (`SCP\leftarrow0`). We then repeat the following process 100 times for each DOF: - Sample an action sequence using the selected degree of freedom (a sequence of action where each action is a value between -1 and 1). - Play it in 2 different orders starting from the same randomly chosen state and save the two final sensor images. - Count one if the two final sensor images are equal (`SCP+=1`), zero otherwise. Finally, the estimator of the SCP is the average over the number of trials (`SCP/100`). Note that using a simulation allows to play the two action sequences of different orders from the exact same starting position. Our results are reported in Fig.\[fig:conj\_res\_exp\]. Results {#sec:conjecturephilipona_exp} ------- ![Sensory Commutativity Probability for each degree of freedom. Note how the SCP value is inversely proportional to how each action affects the environment (shoulders and base movement/rotation affect more than elbows which affects more than eyelid and head rotation). Moreover, as predicted by Philipona’s conjecture, the two DOF not associated to displacements, Eyelid and Head Rotation, are the only ones to always commute (i.e. SCP of 1).[]{data-label="fig:conj_res_exp"}](images/conj_res_exp.png) **The results are coherent with Philipona’s conjecture.** Fig.\[fig:conj\_res\_exp\] shows that only two actions have an SCP of 1: *eyelid* and *head rotation*. All other actions have a SCP inferior to 1. This is coherent with Philipona’s conjecture (Sec.\[sec:conjecture\]): *eyelid* and *head rotation* are the two degrees of freedom that are **not** associated to displacements, thus action sequences composed of actions of these type commute with respect to the sensors. On the contrary, all other degrees of freedom are associated to displacements, and thus will eventually induce non-zero commutation residues when played in different orders from the same starting situation. Hence the results are coherent with the conjecture, and can be used by the agent to autonomously discover which of its actions are associated to displacements or not. **SCP is inversely proportional to how each degree of freedom affects the environment.** By that we mean that from the computation of the SCP, we obtain a hierarchical organization of the action space in which the less important dimensions for manipulation and navigation are separated from the dimension that are not crucial for such tasks. This is illustrated in Fig.\[fig:area\_coverage\], which shows that shoulders and base movement should have a lower SCP than elbows which in turn have a lower SCP than eyelid and head rotation. We inferred that shoulders should have a lower SCP than elbows since activating the shoulder joint is more likely to induce non commutativity by moving things around or hitting walls/obstacles. This intuition is verified by our results. Without having any prior knowledge about the simulation, we can automatically organize the agent’s degrees of freedom in a hierarchy. Moreover, the symmetry of the action space is kept, as elbow 1 and 2 have equal SCP, and so do shoulder 1 and 2. Alternative methods are not adapted ----------------------------------- The SCP criterion derived in this paper estimates how much each degree of freedom affects the environment in an embodied agent scenario. In this section we discuss why other approaches cannot reliably estimate the same quantity. ### Naive approach: changes in sensors A straightforward approach to this problem would be to play action sequences of each degree of freedom and quantify how much the sensors change. We consider the squared difference for a transition, i.e. the squared difference for two consecutive observations separated by an action sampled from one dimension of the action space. We report the mean squared difference over 100k transitions, for each degree of freedom. It is clear in our experiment results, shown in Fig.\[fig:baseline\], that the approach fails. For instance, rotating the head of the agent changes dramatically what the agent sees, even though this degree of freedom does not affect the environment. It would have made sense if we had considered the top view (fully-observable scenario), since rotating the head does not changes the top view a lot. However in the embodied scenario, this strategy is not viable. For the same reason, approaches based only on the changes in the embodied sensors are bound to fail. ![A naive alternative to SCP would be to consider how much sensors change when applying actions of each degree of freedom. Results show that this alternative is not viable since degrees of freedom that do not affect the environment, e.g. head rotation, can change the sensors more than degrees of freedom that affect the environment a lot, e.g. base longitudinal movement.[]{data-label="fig:baseline"}](images/baseline_result.png) ### Prediction error approach A more involved approach would be to use prediction on the sensory change caused by each degree of freedom, a common approach used to improve exploration in RL [@burda2018exploration; @pathak2017curiosity]. The DOF that are harder to predict could be the ones affecting the environment the most, and thus being the most important for manipulation and navigation. We tested this alternative in our experiments, by using a feed-forward neural network to predict the next sensor. The neural network takes a concatenation of the sensor and action at time $t$ and predicts the sensor at time $t+1$. We use the same dataset of transitions as in our experiments with the naive baseline (100k transitions for each degree of freedom, 80k for training and 20k for testing). We trained one model for each degree of freedom, using a neural network with two linear hidden layers with the same number of neurons as the input size. We report the excess prediction error on the held-out test set, i.e. the value of the prediction error minus the minimum prediction error among all 8 degrees of freedom. If the method works, higher excess error prediction should indicate a degree of freedom with more effect on the environment. The results are shown in Fig.\[fig:prediction\]. It turns out that prediction error is not well correlated with how much a degree of freedom is important for navigation and manipulation. For instance, head rotation, which does not affect the environment, is hard to predict: the agent might not know what’s outside his field of view. On the contrary, base longitudinal movement affect the environment a lot and is easier to predict than head rotation. A solution would be to use more complex neural architectures, involving the computation of a state representation that uses memory (recurrent [@hochreiter1997long], or external [@graves2014neural]). However, such methods would require heavy additional computation where our approach computes the SCP with minimal requirements and no training. ![A more involved alternative for SCP, where we report the prediction error when trying to predict the effect of actions on sensors, for each degree of freedom. Results show that this does not allow to identify which degrees of freedom are useful (or not) for navigation and manipulation. For instance, eyelid activation (not useful) is as hard to predict as base longitudinal movement (crucial).[]{data-label="fig:prediction"}](images/results_prediction.png) To conclude, in our experiments we did not find any viable strategy to replace the SCP criterion. SCP is able to easily estimate how important a degree of freedom is for acting and navigating in the environment. The other considered baselines do not manage to organize the action space in the same hierarchical way. Sensory Commutativity Probability for efficient exploration =========================================================== We now illustrate how SCP can be used for unsupervised exploration, by using it to improve sample-efficiency in a RL setup. Experimental setup ------------------ We use the PPO2 [@schulman2017proximal] implementation from Stable-Baselines [@hill2018stable]. The policy is composed of a 1D convolutional feature extractor followed by a recurrent policy. We consider the same agent, Polyphemus, for which we computed the SCP criterion in Fig.\[fig:conj\_res\_exp\]. The input of the policy is the RGB image of what Polyphemus’ eye sees. The environment considered is a square room with 3 dead zones (which terminate the episode with a -20 reward) and a goal zone (which terminates the episode with a +50 reward), illustrated in Fig.\[fig:rl\_task\]. ![The task of the agent is to navigate to the green zone while avoiding the red zones.[]{data-label="fig:rl_task"}](images/rl_task.png) We propose two methods that take advantage of the SCP to modify the action space of the agent. The goal is to improve sample-efficiency when learning to solve a task in this embodied scenario. ### SCP-truncated action space A first, quite radical, idea is to truncate the agent’s action space based on SCP value of each degree of freedom. We implement this by halving the dimension of the action space, keeping only the degrees of freedom that have the most effect on the environment, i.e. lower SCP value. We thus keep the base movement and rotation, and the shoulders joint, while discarding the elbow joints, head rotation and eyelid activation. We refer to this method as *SCP-truncated* action space. This action space reduction will obviously simplify the RL task, as long as the necessary actions such as base motion are selected by the SCP criteria. ### SCP-adapted action space A less involved proposition is to modify the action sampling interval according to the SCP value, for each degree of freedom. This method will not change the task as the previous one, but will modify the exploration dynamics to favor important actions. Suppose that the sampling interval for each dimension of the action space is $[-1,1]$. If a dimension has high SCP, i.e. it does not affect the environment a lot, we then reduce the interval from which action are sampled $[-1\cdot l(SCP), 1\cdot l(SCP)]$. The function $l$ maps the highest SCP to $0$ and lowest SCP to $1$, then we use a linear interpolation between those two points to deduce values for $SCP\in ]-1,1[$. We refer to this method as *SCP-adapted* action space. ### Comparison protocol We compare those two strategies to a baseline policy trained to solve the task with the complete action space. We average the result of each policy over 30 trials initialized with different random seeds, and we test the statistical significance of our results according to the guidelines provided by @colas2018many. Note that we did not try the *truncation* and *adaptation* method using the SCP alternatives considered in Sec.\[sec:scpp\]. For both SCP alternatives, the coefficient associated to the base longitudinal movement is close to zero. Since truncating/adapting longitudinal movement makes the task impossible/harder for the agent, we did not try those alternatives. Results {#results} ------- ![Learning curves for the 3 considered strategies, results averaged over 30 seeds. The Sensory Commutativity Probability can be used to improve the action space of the agent and thus improve sample-efficiency. Dots show statistical significance when testing against the baseline green curve.[]{data-label="fig:explo_results"}](images/explo_results.pdf) The results are displayed on Fig.\[fig:explo\_results\]. First of all, we notice that all strategies are viable to solve the task. We now compare sample-efficiency between the strategies. The policy trained with *SCP-truncated* action space is able to learn how to solve the task more than twice as fast as the baseline policy. The discarded degrees of freedom are not crucial in this navigation task, hence the agent is still able to solve the task using only the degrees of freedom that have the lowest SCP value. The policy trained with *SCP-adapted* action space is less sample-effective than the *SCP-truncated* but still learns significatively faster than the baseline policy, hence showing our point. Discussion and conclusion ========================= Discussion ---------- **Applying SCP on *tabula rasa* scenarios.** SCP gives a characterization of the action space of an embodied agent. In this paper we illustrated the usefulness of this characterization in a RL experiment, but we hope SCP can be useful in other types of learning problem for embodied agents. If we consider the widely-adopted scenario where the agents learn from a clean state (i.e. *tabula rasa*), we believe SCP computation could be a useful method to help the agent build perception. For instance, exploring a large and complex environment is easier when the agent knows which of its degrees of freedom is more important for navigation and manipulation, which is what SCP characterizes. Thus the SCP could give a useful information that might be used to improve exploration strategies. **Limitation: SCP computation requires a simulation.** The main limitation of SCP is that computing SCP as described in this paper is only possible when having access to a simulation of the considered environment, thus it is not directly applicable for real life scenarios. The difficulty in such scenario is that the agent has to be able to play two action sequences from the same starting point. Thus, in a real life scenario, the method has to overcome stochasticity and irreversible actions (e.g. breaking a glass) which break that assumption. Conclusion ---------- We studied the sensory commutativity of action sequences for an agent in an embodied scenario (high-dimensional first person sensors, multi-dimensional continuous action space and coherent physics). Inspired by two artificial perception theories, we derived the Sensory Commutativity Probability criterion, which we showed is good proxy for estimating the effect of each action on the environment. We illustrated the potential usefulness of such criterion by improving sample-efficiency in a Reinforcement Learning problem. Proofs {#app_proof} ====== 1. $\sim$ is an equivalence, i.e. it is reflexive, transitive and symmetric. 2. ($Seq(\mathcal{M})$, $\circ$) is a group w.r.t $\sim$. 3. $\circ$ is not commutative with respect to $\sim$, i.e. we don’t generally have $g\circ h \sim h\circ g$. <!-- --> 1. $=$ is an equivalence, thus $\sim$ is an equivalence as well. 2. All 4 properties of the group definition are satisfied. 1. For two action sequences $(h,g)\in Seq(\mathcal{M})$, the composition of $h$ and $g$ is still an action sequence $h\circ g \in Seq(\mathcal{M})$. 2. $\circ$ is associative with respect to $=$, i.e. $g\circ (h \circ k) = (g \circ h) \circ k $ thus it follows that $g\circ (h \circ k) \sim (g \circ h) \circ k$. 3. The identity element is the no-op action. 4. If we suppose that there are no irreversible phenomenons in the environment, then for a fixed $(m_t, \epsilon_t)$, all action sequences can be inverted. 3. $\circ$ is not commutative, as we can always explicitly find two action sequences that do not commute. For instance once there exists a movable object in the environment: if the agent is placed left to the object, then let $h$ be moving right and $g$ be moving left. $h$ and $g$ do not commute.
**Improved Quantum Ternary Arithmetics** Alex Bocharov[^1] *Quantum Architectures and Computations Group, Microsoft Research* *Redmond, Washington, 98052, USA* Shawn X. Cui[^2] *University of California* *Santa Barbara, California, 93106, USA* Martin Roetteler[^3] *Quantum Architectures and Computations Group, Microsoft Research* *Redmond, Washington, 98052, USA* Krysta M. Svore[^4] *Quantum Architectures and Computations Group, Microsoft Research* *Redmond, Washington, 98052, USA* Introduction {#sec:intro} ============ Quantum computation has seen vast progress over the years, both theoretically and experimentally. Computations on a programmable and scalable fault-tolerant quantum computer will consist of fully controlled sequences of primitive operations such as unitary gates, measurements and state preparations. Such sequences are called *quantum circuits*. In the most commonly used circuit model, quantum information is stored in a collection of *qubits*, where each qubit has a two-dimensional Hilbert state space with the computational basis $\{{|0\rangle}, {|1\rangle}\}$. A standard universal gate set consists of Clifford gates and one non-Clifford gate such as the $\frac{\pi}{8}$-gate [@boykin2000new] or $V$-gate [@harrow2002efficient]. By design, circuits over a universal set can be used to approximate arbitrary quantum gates. Thus any quantum algorithm can be processed given a quantum computer with a universal gate set. It has been noted by several researchers that architecture of certain quantum registers and gates is more naturally described by multi-valued logic as opposed to binary logic. History of experiments with ternary superconducting registers, in particular goes back to 1989 [@morisue1989jctl],[@morisue1998memory]. More recently, in quantum computation domain, multi-valued logic has been proposed for linear ion traps [@Muthu2000mvl], cold atoms [@smith2013cs], entangled photons [@malik2016multiPhoton]. It remains to be seen, at what scale it would be possible to balance out quantum universality and fault-tolerance in these and other similar architectures. The research presented here is motivated in part by recent progress in circuit synthesis over universal quantum bases arising in topological quantum computing, where multi-qubit encoding is not necessarily the most natural choice. Several physical systems capable of performing topologically-protected quantum computations have a natural structure of a qutrit instead of a qubit, where a qutrit has a three-dimensional Hilbert space with the computational basis $\{{|0\rangle},{|1\rangle},{|2\rangle}\}$. For instance, in the ${\rm SU}(2)_4$ anyon system, anyons with quantum dimension $\sqrt{3}$ are well-suited for encoding quantum states in qutrits. What is more, it was shown in [@cui2015universal] that the ${\rm SU}(2)_4$ anyon system can be made universal through braiding and projective measurement of anyons. This anyonic structure is quite far from physical realization at the moment, yet, it offers a promise of comparatively simple quantum universality combined with native topological protection, which, in our opinion, makes it a worthwhile subject of forward-looking research. In [@bocharov2015efficient], an algorithm is given for approximation of any multi-qutrit gate with an asymptotically optimal circuit over the gate set Clifford $+$ $\diag(1,1,-1)$. This work also demonstrated the importance of *Householder reflections* for synthesis of efficient circuits. Even though the gate set turned out to be powerful enough for such synthesis, it had certain conceptual and practical limitations. Thus, it is quite unlikely that all the reversible classical permutation gates can be implemented exactly over Clifford $+$ $\diag(1,1,-1)$. This has a damping effect on implementation of arithmetic-heavy algorithms such as Shor’s Factorization Algorithm, since the integer modular arithmetic is naturally described by reversible classical circuits. As a matter of principle such circuits may be represented exactly in commonly used multi-qubit circuit models. [^5] When compared to [@bocharov2015efficient], the present paper aims at a more abstract level. Here we assume that the entire group of multi-qutrit classical permutations is representable at some cost, explore different scenarios of its representation and focus on synthesizing efficient circuits for ternary base arithmetic in these scenarios. Our thinking at this level remains reflection-centric. Previous research on non-binary reversible circuits [@Brennen2006qudits] mostly focused on proving the universality of the local classical Clifford gates in combination with the *controlled-increment* gate $|j,k\rangle \mapsto |j, k + \delta_{j,d-1} \, {\rm mod} \; d\rangle$, where $d$ is the dimension of the qudit and $\delta$ is the Kronecker delta. Reversible circuits available in literature tend to use ancillary qudits fairly liberally. This paper differentiates itself from previous work in two ways. First, we explore several alternate methods for synthesizing classical reversible circuits. Second, we strive to minimize both the depth and the width of arithmetic circuits specifically. For example, we show in Section \[subsec:ripple\] that implementing of a faithful $\mbox{CARRY}$ gate is not necessary in a correct ternary adder. By using a modified carry we eliminate the use of ancillary qutrits and reduce the cost of the gate when compared to a faithful $\mbox{CARRY}$ as used in previous approaches to implement ternary carry ripple adders [@MDM:2004; @satoh2007; @khan2007quantum]. Our focus is mainly on two types of ternary quantum adders, a modified ripple-carry adder and a carry look-ahead adder. Both adders are generalized from their binary counterparts, but the generalizations are somewhat non-trivial. To add two $n$-qutrit numbers, the modified ripple-carry adder uses $1$ ancilla and has a circuit depth of $O(n)$, while the carry look-ahead adder requires $O(n)$ ancillas and has a circuit depth of $O(\log \,n)$. Each of the two adders has an overall circuit size of $O(n)$ elementary gates. We also study various extensions of quantum adders including adder modulo $3^n$, comparison, and subtraction. We show these arithmetic circuits can be realized exactly using classical Clifford gates and one additional gate $\CX$, the *controlled-increment* gate, whose matrix is given in Equation \[equ:CX\]. $\CX$ is a two-qutrit non-Clifford gate and it is universal for reversible classical computation. This sets the ternary reversible circuits apart from their binary analogs, where at least one three-qubit gate, e.g., the Toffoli gate, is required for universality. $$\label{equ:CX} \CX = \begin{pmatrix} 1 & 0& 0& 0& 0& 0& 0& 0& 0\\ 0 & 1& 0& 0& 0& 0& 0& 0& 0\\ 0 & 0& 1& 0& 0& 0& 0& 0& 0\\ 0 & 0& 0& 1& 0& 0& 0& 0& 0\\ 0 & 0& 0& 0& 1& 0& 0& 0& 0\\ 0 & 0& 0& 0& 0& 1& 0& 0& 0\\ 0 & 0& 0& 0& 0& 0& 0& 0& 1\\ 0 & 0& 0& 0& 0& 0& 1& 0& 0\\ 0 & 0& 0& 0& 0& 0& 0& 1& 0\\ \end{pmatrix}$$ We also introduce a qutrit universal gate set Clifford $+$ $\diag(e^{-\frac{2\pi i}{9}}, 1, e^{\frac{2\pi i}{9}})$, called the supermetaplectic basis, which resembles the single-qubit $\frac{\pi}{8}$-gate. Some techniques are developed to construct new quantum gates from old ones. As an application, it will be shown that all ternary arithmetic studied in this paper can be implemented exactly over the $\super$. We note that the reflection-centric synthesis of our adder circuits is a ternary counterpart of Toffoli-centric binary adder circuits as developed, for example, in [@cuccaro2004new] and [@draper2006logarithmic]. This analogy is explained in more detail in corresponding sections throughout the paper. The exact representation of the $\CX$ gate in $\super$ parallels the exact representation of the three-qubit Toffoli in the Clifford $+$ $\frac{\pi}{8}$ basis. Quantitative comparison of the ternary and binary adders would be beyond the scope of this work. A major step towards comprehensive comparison of this kind was made in the upcoming paper [@TernaryShor] that demonstrates the advantages of emulating Shor’s period funding function on ternary quantum computer and especially on the metaplectic topological quantum framework. The paper is organized as follows. In Section \[sec:background\], some preliminaries and notations used throughout the paper are given. In Section \[sec:adder\], we separately discuss the modified ripple-carry adder and carry look-ahead adder. Section \[sec:extensions\] gives some extensions of quantum adders, including addition modulo $3^n$, comparison, and subtraction. Lastly in Section \[sec:techniques\], we introduce the $\super$ and develop techniques for the construction of new gates. Preliminaries and Notations {#sec:background} =========================== We denote the standard computational basis in a qutrit by $\{{|0\rangle}, {|1\rangle}, {|2\rangle}\}$. The terminology qutrit” and ternary” are sometimes used interchangeably. We call a quantum gate reversible or a classical *permutation gate* if it acts as some permutation of the standard basis elements. Unless otherwise noted, the arithmetic, e.g., addition, multiplication, etc., within a ket is assumed to be taken modulo $3$. Also by default circuits are read from left to right, while compositions of gates when written as expressions follow the rule of matrix multiplications, i.e., they are read from right to left. Throughout the paper, the following ternary quantum gates are frequently used: 1. $X = \begin{pmatrix} 0 & 0 & 1\\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{pmatrix}$, namely, $X {|i\rangle} = {|i+1\rangle}$. 2. $S_{0,1} = \begin{pmatrix} 0 & 1 & 0\\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}$, namely, $S_{0,1}$ swaps ${|0\rangle}$ with ${|1\rangle}$ and fixes ${|2\rangle}$. Similarly, one can define $S_{0,2}, S_{1,2}$. This notation is also generalized to multi-qutrit gates. For instance, $\TwoSwap$ is a $2$-qutrit gate, which swaps ${|00\rangle}$ with ${|22\rangle}$, and fixes all other basis elements. 3. Given an $n$-qutrit gate $U$, there are two versions of controlled-$U$”. The first version is called soft-controlled-$U$,” denoted by $\SoftU{U}$, and is defined as the $(n+1)$-qutrit gate: ${|i,j_1, \cdots j_n\rangle} \mapsto (I \otimes U^i){|i,j_1, \cdots j_n\rangle}$, where the first qutrit is called the control qutrit. The second version is the hard-controlled-U” denoted by $\HardU{c}{U}$, where $c \in \{0,1,2\}$. The gate $\HardU{c}{U}$ is also an $(n+1)$-qutrit gate. However, in contrast to the soft-controlled-$U$, it maps ${|i,j_1, \cdots j_n\rangle}$ to $(I \otimes U^{\delta_{i,c}}){|i,j_1, \cdots j_n\rangle}$. It is direct to see that the $\HardU{c}{U}\,'$s for different $c\,'$s are equivalent to each other up to some $1$-qutrit reversible gates. Thus we also use $\CU{U}$ to denote a general $\HardU{c}{U}$. Moreover, the equality $\SoftU{U} = \HardU{1}{U}(\HardU{2}{U})^2$ holds. 4. The following is a list of some important controlled gates: - $\SUM = \SoftU{X}: {|i,j\rangle} \mapsto {|i,i+j\rangle}$, - $\CX = \HardU{c}{X}: {|i,j\rangle} \mapsto {|i, j+\delta_{i,c}\rangle}$, - ${\textrm{Horner}}= \SoftU{\SoftU{X}}: {|i,j,k\rangle} \mapsto {|i,j,ij+k\rangle}$, - $\CSUM = \HardU{c}{\SUM}: {|i,j,k\rangle} \mapsto {|i,j,j\delta_{i,c}+k\rangle}$. The ${\textrm{Horner}}$ gate is a qutrit generalization of the qubit Toffoli gate. See also [@GRB:2003] for additional background on the Horner gate. 5. $\SWAP: {|i,j\rangle} \mapsto {|j,i\rangle}$. For graphical representations of the gates defined above, see Figure \[fig:graph\]. (1,-1.5) node[$\SoftU{U}$]{}; ; (0,2)–(2,2); (1,2) – (1,0.5); (1,2) circle(0.1); (1,2) circle(0.1); (1,-1.5) node[$\HardU{c}{U}$]{}; ; (0,2)–(2,2); (1,2) – (1,0.5); (1,2) circle(0.1); (0.7,1.7) node; \[xshift =6cm\] (1,-1.5) node[$X$]{}; (0,0) – (2,0); (1,0) circle(0.5); (1,-0.5) – (1,0.5); ; (1,-1.5) node[$\SUM$]{}; ; (1,-1.5) node[$\CX$]{}; (0.8,1.7) node; ; (1,-1.5) node[$\CSUM$]{}; (0.8,1.7) node; (0,1) – (2,1); (1,1) circle(0.1); (1,1) circle(0.1); \[fig:graph\] The qutrit Clifford group $\C$ [@gottesman1999fault] is generated by $\SUM, X, H,$ and $Q$, where $H$ and $Q$ are defined as follows: $$H = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 & 1 & 1 \\ 1 & \zeta_3 & \zeta_3^2 \\ 1 & \zeta_3^2 & \zeta_3 \\ \end{pmatrix}, \qquad Q = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \zeta_3 \\ \end{pmatrix},$$ where we use the notation $\zeta_n = e^{\frac{2\pi i}{n}}$ for $n\geq 1$. It can be shown that, along with the $\SUM$, all the reversible $1$-qutrit gates and $\SWAP$ are also contained in $\C$. Moreover, $\SUM$ and all the $1$-qutrit reversible gates generate the subgroup of all reversible gates in $\C$. Some other Clifford gates are $Z$ and $\bigwedge(Z)$, where $Z = \diag(1,\zeta_3,\zeta_3^2)$, and $\bigwedge(Z) = (I \otimes H)\SUM(I \otimes H^{-1}): {|i,j\rangle} \mapsto \zeta_3^{ij}{|i,j\rangle}$. However, $\CX, {\textrm{Horner}}, \CSUM$ and $\TwoSwap$ are non-Clifford gates. Consider two pairs of standard basis vectors $|j_1\rangle, \,|k_1\rangle$ and $|j_2\rangle, \,|k_2\rangle$. It would be useful to note that the two-way classical reflection $S_{|j_1\rangle, |k_1\rangle}$ that swaps the $|j_1\rangle, \,|k_1\rangle$ and fixes everything else can be reduced to the corresponding reflection $S_{|j_2\rangle, |k_2\rangle}$ by applications of $O(n)$ $\, \SUM$ and $\SWAP$ gates (that are Clifford gates: see [@bocharov2015efficient], Lemma 16). In particular, the two-way swap $\TwoSwap$ is Clifford-equivalent to any other two-qutrit two-way swap. We think of Clifford gates as being [*cheap*]{} in the quantum sense. General rationale for this assumption is that such gates can be simulated classically. (Additional motivation coming from topological computing: in the context of non-abelian anyons such as ${\rm SU}(2)_4$ anyon system [@cui2015universal], Clifford gates can be obtained by anyon braiding alone.) Thus we define the complexity (resp. depth) of a circuit as the number (resp. depth) of non-Clifford gates. The following two identities will be used, where $\omega(n)$ is the number of $1\,'$s in the binary expansion of $n$, and $\fl{x}$ means the maximal integer less than or equal to $x$: $$\sum\limits_{i=1}^{\infty} \fl{\frac{n}{2^i}} = n - \omega(n),$$ $$\sum\limits_{i=1}^{\fl{\log\,n}+1} \fl{\frac{n}{2^i}-\frac{1}{2}} = n - \fl{\log \, n} - 1.$$ See also [@cuccaro2004new] for similar identities. Quantum Ternary Adders {#sec:adder} ====================== Given two $n$-trit numbers $a = a_{n-1} \cdots a_1a_0$, $b = b_{n-1} \cdots b_1b_0$, an adder computes their sum $s = s_n s_{n-1} \cdots s_0 = a+b$. The elementary method of adding two $n$-trit numbers is illustrated in Figure \[table:adder\]. Let $c_0 = 0$ be the initial carry trit and for $1 \leq i \leq n$, let $c_{i}$ be the carry trit arising from $a_{i-1},b_{i-1}, c_{i-1}$, namely, $c_i = 0$ if $a_{i-1}+ b_{i-1} + c_{i-1} \leq 2$ and $c_i = 1$ otherwise. For $0 \leq i \leq n-1$, $s_i = a_i + b_i + c_i \, {\rm mod}\; 3$ and $s_n = c_n$. ------- ----------- ---------- --------- ----------- $a_{n-1}$ $\cdots$ $a_{1}$ $a_{0}$ $b_{n-1}$ $\cdots$ $b_{1}$ $b_{0}$ $c_n$ $c_{n-1}$ $\cdots$ $c_{1}$ $c_{0}=0$ $s_n$ $s_{n-1}$ $\cdots$ $s_{1}$ $s_{0}$ ------- ----------- ---------- --------- ----------- \[table:adder\] In Section \[subsec:ripple\] and Section \[subsec:lookahead\], we present two methods to implement reversible ternary quantum adder: a ripple-carry adder and a carry look-ahead adder. The two adders are generalized from their binary counterparts [@cuccaro2004new; @draper2006logarithmic], but the generalizations are somewhat nontrivial, as seen later. On one hand, the modified ripple-carry adder uses only $1$ ancilla for the whole process and has the circuit depth in $O(n)$. On the other hand, the carry look-ahead adder requires $O(n)$ ancillas with the advantage of having circuit depth in $O(\log\,n)$. We will also compare the two adders to other ternary adders known in literature and show that our adders are more efficient both space-wise and depth-wise. To implement the adders, we utilize $\CX$, $\CSUM$, $\CU{S_{0,1}}$ and $\TwoSwap$ as the basic building units. As shown in Section \[subsec:construction2\], $\CSUM$, $\CU{S_{0,1}}$ and $\TwoSwap$ can all be constructed exactly from $\CX$ and Clifford operations. Therefore, the circuit of adders can be designed from Clifford operations and $\CX$ alone. The reason that we still treat $\CSUM$, $\CU{S_{0,1}}$ and $\TwoSwap$ as basic units is that it might be more efficient to synthesize them directly in some basis rather than breaking them up into $\CX\,'$s. An example is the metaplectic basis [@bocharov2015efficient], where $\TwoSwap$ has an efficient approximation by a metaplectic circuit. Modified Ripple-Carry Adder {#subsec:ripple} --------------------------- The binary quantum ripple-carry adder was considered in [@vedral1996quantum], where $O(n)$ ancillas are required to add two $n$-qubit numbers. In [@cuccaro2004new], the method was improved so that only $1$ ancilla is necessary. Here we give a ternary generalization of the improved ripple-carry adder. Note that in contrast to the binary case, the ternary carry is more complicated: if the inputs to a binary full adder are denoted by $a, b, c \in \F_2$, then the outgoing carry is given by $c_{out} = ab + ac + bc$, where all operations are computed modulo $2$. In case of a ternary full adder with inputs $a,b,c \in \F_3$, the outgoing carry is given by $c_{out} = 2(1+a+b+c)(ab+ac+bc)+abc$, where all operations are computed modulo $3$. Though directly implementing this polynomial using the presented universal gates is possible, it leads to a relatively large number of elementary gates. A simple observation allows to reduce this cost significantly as it turns out that $c_{out}$ does not have to be implemented for all $27$ input triples but rather only $18$ of them. Indeed, it can be shown inductively that—provided there is no initial incoming carry—for ternary adders, every carry trit $c_i$ can only be either $0$ or $1$, but can never be $2$. This is indicated also in Figure \[table:carry\] where the crossed out case indicates that this can never occur in an actual addition: the case $c_{i+1} = 2$ is possible only if $c_i = 2$, which inductively we assume cannot happen. With this definition, $c_{i+1}$ becomes a balanced function, i.e., there are the same number of inputs corresponding to each outcome $c_{i+1}$. We sketch the idea of constructing the circuit to compute $c_{i+1}$ from $a_i,b_i$ and $c_i$ based on this observation. As illustrated in Figure \[table:carry\], $c_{i+1}$ equals $c_{i}$ for all but six inputs, the last three inputs in the column $c_{i+1} = 0$ and the last three in the column $c_{i+1} = 1$. For each of these six inputs, $c_{i+1} $ equals $1-c_i$. If the gate $\TwoSwap$ is applied to qutrits $a_i,b_i$, then the six inputs are turned into six new triples. See Figure \[table:transition\] for the transition. Moreover, the new six triples are exactly equal to the set $\{(a,b,c) \in \{0,1,2\}^3: a + b = c, c \neq 2\}$. In light of these observations, a reversible circuit, called Carry, is constructed, which takes $c_i, a_i,b_i$ as input, and outputs $c_{i+1}$ in the last qutrit. See Figure \[fig:carry\], where $f$ and $g$ are some functions of $a_i,b_i,c_i$. The exact shape of $f$ and $g$ is not important since they will be reversed at the appropriate step of the adder. ------- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- $a_i$ 0 0 0 1 1 2 0 0 1 0 1 1 2 2 2 1 2 2 0 0 0 1 1 1 2 2 2 $b_i$ 0 1 2 0 1 0 0 1 0 2 1 2 0 1 2 2 1 2 0 1 2 0 1 2 0 1 2 $c_i$ 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 2 2 2 2 2 2 2 2 2 ------- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- \[table:carry\] = \[draw, rectangle, minimum width=3cm, minimum height=1cm, text centered, text width=4.5cm,\]; = \[thick,-&gt;,&gt;=stealth\]; = \[dashed,thick,-&gt;,&gt;=stealth\]; = \[draw = none,rectangle, minimum width=3cm, minimum height=1cm, text centered, text width=8cm,\]; (left) ------- --- --- --- --- --- --- $a_i$ 0 0 1 1 2 2 $b_i$ 0 1 0 2 1 2 $c_i$ 1 1 1 0 0 0 ------- --- --- --- --- --- --- ; (right) ------- --- --- --- --- --- --- $a_i$ 2 0 1 1 2 0 $b_i$ 2 1 0 2 1 0 $c_i$ 1 1 1 0 0 0 ------- --- --- --- --- --- --- ; (middle)[ $\overset{\TwoSwap}{\Longrightarrow}$ ]{}; \[table:transition\] (0,0) node; (0,1) node; (0,2) node; ; (1,0) – (2.5,0); ; (2.5,0) – (4,0); ; (4,2) – (5.5,2); ; (5.5,1) – (7,1); (5.5,2) – (7,2); ; (7,0) – (7.9,0); (8.1,0) – (8.5,0); (8.0,0.1) – (8.0, 1); (8,0) circle(0.1); (7.8,0.8) node; ; (8.5,1) – (10,1); (8.5,2) – (10,2); ; (10,1) – (11.5,1); (10,2) – (11.5,2); (11,1) circle(0.1); (11,1) – (11,0.5); (10.8,1.2) node; ; (11.5,2) – (13,2); ; (13,0) – (14.5,0); ; (14.5,0) – (16.5,0); (17.5,0) node; (18,1) node; (18,2) node; \[fig:carry\] (0,3) node; (0,1) node; (0,2) node; ; (1,3) – (2.5,3); ; (2.5,3) – (4,3); ; (4,2) – (4.8,2); (5.2,2)–(5.5,2); ; (0,1) – (2,1); (0,2) –(0.8,2); (1.2,2)–(2,2); (1,1) circle(0.1); (1,1) – (1,2.5); (0.8,1.2) node; ; (0,1)–(2,1); ; (2,3)–(4,3); (0,1) node; (0.5,2) node; (0.5,3) node; \[fig:carry\] As illustrated in Figure \[fig:carry\], the circuit Carry is ancilla free, in contrast to the carry circuit considered in [@satoh2007] where $1$ ancilla is required for each round of carry. See Figure \[fig:comparison\] for the comparison. The circuit utilizes one $\TwoSwap$, one $\CU{S_{0,1}}$, two $\SUM$, and two $\SWAP$ gates. The $\SUM$ and $\SWAP$ are both Clifford gates, so only $2$ non-Clifford gates are needed. The depth of Carry in terms of non-Clifford gates is also $2$. Moreover, unlike the binary ripple-carry circuit MAJ [@cuccaro2004new] where the two qubits other than $c_{i+1}$ end up with $a_i+b_i, c_i+b_i$, in our circuit the two qutrits other than $c_{i+1}$ have the final values $f(a_i,b_i,c_i)$ and $g(a_i,b_i,c_i)$. This is the reason we call our carry circuit [*modified*]{}. However, as will be seen below, the modified carry circuit works in the same way as the regular one. (0,3) node; (0,1) node; (0,2) node; (1,1) – (2,1); (1,2) – (2,2); (1,3) – (2,3); (2,0.8) – (2,3.2) – (5,3.2) – (5,0.8) – (2,0.8); (5,1) – (6,1); (5,2) – (6,2); (5,3) – (6,3); (7,1) node; (7.5,2) node; (7.5,3) node; (3.5,2) node; (9,-1) – (9,5); \[xshift = 11cm, yshift = 1cm\] (0,0) node; (0,3) node; (0,1) node; (0,2) node; (1,0) – (2,0); (1,1) – (2,1); (1,2) – (2,2); (1,3) – (2,3); (2,-0.2) – (2,3.2) – (5,3.2) – (5,-0.2) – (2,-0.2); (5,0) – (6,0); (5,1) – (6,1); (5,2) – (6,2); (5,3) – (6,3); (7,0) node; (7,3) node; (7,1) node; (7,2) node; (3.5,1.5) node; \[fig:comparison\] Let $C: {|c_i,a_i,b_i\rangle} \rightarrow {|f(a_i,b_i,c_i),g(a_i,b_i,c_i), c_{i+1}\rangle}$ be the Carry gate represented by the circuit in Figure \[fig:carry\]. Similar to the adder circuit in [@cuccaro2004new], the modified ripple-carry adder circuit is designed in Figure \[fig:rippleadder\], which, as an illustration, shows the addition of two $3$-qutrit numbers. (0,0) node[$0$]{}; (0,1) node[$b_2$]{}; (0,2) node[$a_2$]{}; (0,3) node[$b_1$]{}; (0,4) node[$a_1$]{}; (0,5) node[$b_0$]{}; (0,6) node[$a_0$]{}; (0,7) node[$c_0$]{}; ; (0,0) – (1.5,0); (0,1) – (1.5,1); (0,2) – (1.5,2); (0,3) – (1.5,3); (0,4) – (1.5,4); ; (0,0) – (1.5,0); (0,1) – (1.5,1); (0,2) – (1.5,2); (0,6) – (1.5,6); (0,7) – (1.5,7); ; (0,0) – (1.5,0); (0,4) – (1.5,4); (0,5) – (1.5,5); (0,6) – (1.5,6); (0,7) – (1.5,7); ; (0,2) – (1.5,2); (0,3) – (1.5,3); (0,4) – (1.5,4); (0,5) – (1.5,5); (0,6) – (1.5,6); (0,7) – (1.5,7); ; (0,0) – (1.5,0); (0,4) – (1.5,4); (0,5) – (1.5,5); (0,6) – (1.5,6); (0,7) – (1.5,7); ; (0,0) – (1.5,0); (0,3) – (1.5,3); (0,4) – (1.5,4); (0,5) – (1.5,5); (0,6) – (1.5,6); (0,7) – (1.5,7); ; (0,0) – (1.5,0); (0,2) – (0.8,2); (1.2,2) – (1.5,2); (0,4) – (1.5,4); (0,5) – (1.5,5); (0,6) – (1.5,6); (0,7) – (1.5,7); ; (0,0) – (1.5,0); (0,1) – (1.5,1); (0,2) – (1.5,2); (0,6) – (1.5,6); (0,7) – (1.5,7); ; (0,0) – (1.5,0); (0,1) – (1.5,1); (0,2) – (1.5,2); (0,5) – (1.5,5); (0,6) – (1.5,6); (0,7) – (1.5,7); ; (0,0) – (1.5,0); (0,2) – (0.8,2); (1.2,2) – (1.5,2); (0,4) – (0.8,4); (1.2,4) – (1.5,4); (0,1) – (1.5,1); (0,2) – (1.5,2); (0,6) – (1.5,6); (0,7) – (1.5,7); ; (0,0) – (1.5,0); (0,1) – (1.5,1); (0,2) – (1.5,2); (0,3) – (1.5,3); (0,4) – (1.5,4); ; (0,0) – (2,0); (0,1) – (2,1); (0,2) – (2,2); (0,3) – (2,3); (0,4) – (2,4); (0,7) – (2,7); (0,0) node[$s_3$]{}; (0,1) node[$s_2$]{}; (0,2) node[$a_2$]{}; (0,3) node[$s_1$]{}; (0,4) node[$a_1$]{}; (0,5) node[$s_0$]{}; (0,6) node[$a_0$]{}; (0,7) node[$c_0$]{}; \[fig:rippleadder\] In Figure \[fig:rippleadder\], the qutrit $c_0$, initialized with $0$, is the only ancilla required. The qutrit on the bottom holds the overflow trit, i.e., the highest trit in the sum. Therefore, to add two $n$-qutrit numbers, exactly $1$ ancilla, $n$ Carry gates, $n$ inverse Carry gates and $2\,n$ $\SUM$ gates are required, and the depth of the circuit is $4\,n$. In contrast, the adder in [@satoh2007] uses $n$ ancillas and has the complexity in $O(n)$. Carry Look-ahead Adder {#subsec:lookahead} ---------------------- In the ripple-carry adder, the carry $c_{i+1}$ is computed only after the value of $c_i$ has been obtained, and thus the overall depth of the circuit is in $O(n)$. One protocol to reduce the depth is the carry look-ahead adder studied in [@draper2006logarithmic] for the binary addition. Here we generalize it to give a ternary carry look-ahead adder, which computes all the carry trits in depth $O(\log\,n)$ by introducing extra $O(n)$ ancillas. The main idea is that there are relations between $c_i$ and $c_{i+1}$, and more generally between $c_i$ and $c_j$ for $i \neq j$. For instance, if $a_i + b_i = 2$, then $c_{i+1}= c_i$. If $a_i + b_i = 1$, then $c_{i+1} = 0$ regardless of the value of $c_i$. See Figure \[table:relation\] for a summary of the relation between $c_{i+1}$ and $c_{i}$. Note that $c_0 = 0$, thus when $i=0$, the column $c_{i+1}=c_i$ in Figure \[table:relation\] becomes $c_1 = c_0 = 0$. Motivated by their relations, we define, for $0 \leq i < j \leq n$, the carry status indicator $C[i,j]:$ ------- --- --- --- --- --- --- --- --- --- $a_i$ 0 0 1 1 2 2 0 1 2 $b_i$ 0 1 0 2 1 2 2 1 0 ------- --- --- --- --- --- --- --- --- --- \[table:relation\] $$C[i,j] = \begin{cases} 0 & c_{j} = 0 \text{ regardless of }c_i \\ 1 & c_{j} = 1 \text{ regardless of }c_i \\ 2 & c_{j} = c_i \\ \end{cases}$$ Since we already know $c_0 = 0$, the case $c_j = c_0$ is then the same as the first case $c_j = 0$. Thus we can treat these two cases as one, and design $C[0,j]$ so that it will never take the value $2$, namely, we will have $C[0,j] = c_j$. Explicitly, for $0 < i < n$, the circuit, $\AdjC$, shown in Figure \[fig:Ci\] computes $C[i,i+1]$ from $a_i$ and $b_i$. It requires $1$ non-Clifford gate $\TwoSwap$, and no ancilla. However, to compute $C[0,1]$, we need to make use of $1$ ancilla, and $2$ non-Clifford gates $\TwoSwap, \CX$. See Figure \[fig:C0\] for the circuit, which we call $\AdjC_0$. (0,0) node ; (0,1) node ; ; ; ; (0,1) – (2,1); (1,0) node ; (1,1) node [ ]{}; \[fig:Ci\] (0,0) node ; (0,1) node ; (0,2) node ; ; (0,2) – (2,2); ; (0,2) – (2,2); ; (0.6,0.2) node ; (0,1) – (0.8,1); (1.2,1) – (2,1); ; (0,0) – (2,0); ; (0,2) – (2,2); (1,2) node [ ]{}; (1,1) node [ ]{}; (1,0) node ; \[fig:C0\] Having computed the carry status indicators for any two adjacent indices, we furthermore compute $C[i,j]$ for arbitrary $i \neq j$. For $0 \leq i < k < j \leq n$, $C[i,j]$ can be obtained from $C[i,k]$ and $C[k,j]$ by the [*merging*]{} formula in Figure \[table:merging\]. -- --- --- --- --- 0 1 2 0 0 1 0 1 0 1 1 2 0 1 2 -- --- --- --- --- \[table:merging\] Note that when $i=0$, the row corresponding to $C[0,k] = 2$ in Figure \[table:merging\] will never be used. Also when $C[0,k]$ takes values in $\{0,1\}$, so will $C[0,j]$. A circuit, $\M$, realizing the [*merging*]{} formula is illustrated in Figure \[fig:merging\], where $\M$ takes $C[i,k], C[k,j]$, and an ancilla initialized to $0$ as inputs, and outputs $C[i,j]$ to the ancilla. The circuit requires $1$ non-Clifford gate $\CSUM$. (0,0) node ; (0,1) node ; (0,2) node ; ; (0,2) – (2,2); ; (0,0) – (2,0); ; (0,1) – (2,1); (1,1) circle(0.1); (0.8,0.8) node ; ; (0,0) – (2,0); (1,0) node ; (1,1) node ; (1,2) node ; \[fig:merging\] The circuits $\AdjC$ and $AdjC_0$ both only depend on $a_i$ and $b_i$, thus we can compute all the $C[i,i+1]\,'$s in one time slice. The nature of the [*merging*]{} formula enables us to obtain all the $C[0,j]\,'$s in $O(\log \, n)$ time slices. We elaborate this below. For $i = 0,1,\cdots,n-1$, let $B_i$ be the working register configured to be $C[i,i+1]$ at the beginning, and let $ Z_{i+1}$ be the working registers initialized to ${|0\rangle}$, which will end up with $C[0,i+1]$. We also need $n-\omega(n) - \fl{\log\,n}$ ancillas $X_i$ initialized to ${|0\rangle}$. The circuit consists of three processes, namely, $P$-process, $C$-process, and $P^{-1}$-process. Each process roughly contains $\fl{\log \,n}$ rounds. In $P$-process, we compute all the carry status indicators of the form $C[2^t m, 2^t (m+1)]$ and write all the results into the ancillas, except the ones $C[0,2^k]$ which are written to $Z[2^k]$. There are $\fl{\log\,n}$ rounds, each $t = 1, \cdots, \fl{\log\,n}$ corresponding to one round. In the $t$-th round, which we call the $P[t]$-round, the status indicators $C[2^t m, 2^t(m+1)]$, $m = 0, \cdots, \fl{\frac{n}{2^t}}-1$ are computed. By the [*merging*]{} formula, $C[2^t m, 2^t (m+1)]$ can be obtained from $C[2^{t-1}(2m),2^{t-1}(2m+1)]$ and $[2^{t-1}(2m+1), 2^{t-1}(2m+2)]$, both of which have been computed in $P[t-1]$-round by induction. Moreover, the circuit $\M$ producing $C[2^t m, 2^t (m+1)]$ for different $m\,'$s in the $P[t]$-round takes different carry status indicators in $P[t-1]$-round as input. Note that the $P[1]$-round requires the carry status indicators $C[i,i+1]\,'$s in the registers $B_i$. Therefore, in the $P[t]$-round, all the circuits $\M$ computing $C[2^t m, 2^t (m+1)]$ can be made parallel, and their inputs only depend on the carry status indicators from the $P[t-1]$-round. Thus, the depth of the circuit in $P$-process is $\fl{\log\,n}$, the number of ancillas needed is $n-\omega(n) - \fl{\log\,n}$, and the complexity is $n-\omega(n)$. In $C$-process, we compute $C[0,j]$ into the register $Z_j$, $j = 1, \cdots, n$. This is performed in $\fl{\log\,\frac{n}{3}}+1$ rounds. Note that the $C[0,2^k] \,'$s have already been obtained in $P$-process, and are located in the desired positions. For $t = \fl{\log\,\frac{n}{3}}, \cdots , 0$, the $C[t]$-round consists of computing the carry status indicators $C[0, 2^t(2m+1)],\, m = 1, \cdots, \fl{\frac{n}{2^{t+1}}-\frac{1}{2}}$. Again, by the [*merging*]{} formula, we can get $C[0,2^t(2m+1)]$ from $C[0, 2^{t+1}m]$ and $C[2^{t}(2m), 2^{t}(2m+1)]$. By induction, $C[0, 2^{t+1}m]$ has been obtained in earlier $C$-rounds if $m$ is not a power of $2$, and in the $P[t+1+ \log \,m]$-round otherwise. Also $C[2^{t}(2m), 2^{t}(2m+1)]$ has been computed in the $P[t]$-round. Therefore, we can run all the $\M$ circuits in the $C[t]$-round in a parallel way. These circuits depend on the carry status indicators in the $P[t]$-round and $C[k]$-rounds, $k \geq t+1$. If $m$ is a power of $2$, then the corresponding $\M$ circuit also depends on $C[0,2^{t+1}m]$ from the $P[t+1+\log\,m]$-round. Thus the circuit in $C$-process has a depth of $\fl{\log\,\frac{n}{3}}+1$, and the complexity is $n-\fl{\log\,n} -1$. In $P^{-1}$-process, we set the ancillas back to ${|0\rangle}$, thus we need to reverse all the $\M$ circuits in $P$-process, except for those computing $C[0,2^k]\,'$s which are not stored in the ancillas. The $P^{-1}$-process consists of $\fl{\log\,n}-1$ rounds. For $t = \fl{\log\,n}-1, \cdots, 1$, the $P^{-1}[t]$-round uncomputes $C[2^t m, 2^t (m+1)]$, $m = 1, \cdots, \fl{\frac{n}{2^t}}-1$ by using the inverse of $\M$. Note that in this process, all the $\,C[0,2^k]'$s will not be touched. The process has a depth of $\fl{\log\,n}-1$, and the complexity of the circuit is $n-\omega(n) - \fl{\log\,n}$. We note that most parts of $C$-process and $P^{-1}$-process can actually be parallelized. The argument is as follows. All the inputs to the $C[t]$-round which are not of the form $C[0,2^m]$ only depend on $C[k]$-rounds, $k\geq t+1$, and the $P[t]$-round. The inputs that are of the form $C[0,2^m]$ were computed in $P[m]$-round, but they will not be touched in $P^{-1}$-process. The $P^{-1}[t+2]$-round only depends on the outputs in $P[t+1]$-round and $P[t+2]$-round. Thus the $C[t]$-round and the $P^{-1}[t+2]$-round can be performed simultaneously. The precise parallelism between $C$-process and $P^{-1}$-process is illustrated in Figure \[table:parallel\]. ----------------------------- ---------- --------------------------- ---------- ------------- ------------- $C[\fl{\log\,\frac{n}{3}}]$ $\cdots$ $C[\fl{\log \,n}-3]$ $\cdots$ $C[0]$ $P^{-1}[\fl{\log \,n}-1]$ $\cdots$ $P^{-1}[2]$ $P^{-1}[1]$ ----------------------------- ---------- --------------------------- ---------- ------------- ------------- \[table:parallel\] To summarize, the whole circuit uses $n-\omega(n) - \fl{\log\,n}$ ancillas, and has a depth of $\fl{\log\,n} + \fl{\log\,\frac{n}{3}} +2$. The total complexity of the circuit is $3n-2\omega(n) - 2\fl{\log\,n} -1 $. Complete Circuit for Carry Look-Ahead Adder ------------------------------------------- We give two implementations of carry look-ahead adder, namely, the out-of-place adder and the in-place adder. Recall that the circuits in Figure \[fig:Ci\], \[fig:C0\], and \[fig:merging\] are denoted by $\AdjC$, $\AdjC_0$, and $\M$, respectively. The complexity of both $\AdjC$ and $\M$ is $1$, and the complexity of $\AdjC_0$ is $2$. The depth of these circuits is equal to their complexity. ### Out-of-place Adder {#subsubsec:out-of-place} Let $A_i, B_i$ be the registers with initial value $a_i, b_i$, respectively, $i = 0, \cdots, n-1$. Let $Z_i, i = 0, \cdots, n$ be the registers initialized to be $0$, which will hold the sum $a+b$ at the end of the computation. We need $n-\omega(n) - \fl{\log\,n}$ ancillas $X_i$ to store intermediate carry status indicators. The following is a description of the circuit of our out-of-place adder. **Out-of-place Procedure:** 1. For $ 0 < i \leq n-1$, run the circuit $\AdjC$ on $A_i, B_i$, which outputs $C[i,i+1]$ to $B_i$. Run $\AdjC_0$ on $A_0,B_0$, and $Z_0$ with $Z_0$ as the ancilla, which outputs $C[0,1]$ to $B_0$. Copy $C[0,1]$ to $Z_1$ with the $\SUM$ gate. The circuit has a depth of $2$, and it consist of $n-1$ $\AdjC$, $1$ $\AdjC_0$, and $1$ $\SUM$ gates. 2. As discussed in Section \[subsec:lookahead\], compute all the $C[0,i]\,'$s with the ancillas $X_i\,'$s and the circuit $\M^{\pm 1}$. At the end of this process, the ancillas are returned to $0$, and $Z_i = C[0,i], i = 1, \cdots, n$.[^6] This requires $3n-2\omega(n) - 2\fl{\log\,n} -1 $ calls to the circuit $\M^{\pm 1}$, and has a circuit depth of $\fl{\log\,n} + \fl{\log\,\frac{n}{3}} +2$. 3. Undo all the $\AdjC\,'$s and $\AdjC_0$. At the end of this step, we have $B_i = b_i, Z_i = C[0,i] = c_i.$ The circuit has a depth of $2$, and it consist of $n-1$ $\AdjC^{-1}$, $1$ $\AdjC_0^{-1}$, and $1$ $\SUM^{-1}$. 4. Set $Z_i = Z_i \oplus A_i \oplus B_i$, $0\leq i \leq n-1$. This requires $2n$ $\SUM$ gates. In summary, the out-of-place adder uses $n-\omega(n) - \fl{\log\,n}$ ancillas, and has a circuit depth of $\fl{\log\,n} + \fl{\log\,\frac{n}{3}} +6$, with the complexity of $5n-2\omega(n) - 2\fl{\log\,n} -1$. We represent $\AdjC_0, \; \AdjC$ and $\M$ as shown in Figure \[fig:circuitrep\]. Their inverses are represented by the same circuit with (0,-0.2) – (0,0.2) – (0.5,0) – (0,-0.2); replaced by (0.5,-0.2) – (0.5,0.2) – (0,0) – (0.5,-0.2); . Also a black rectangle means the content will be changed after the application of the relevant gate, while a blank rectangle means the content remains the same. An an illustration, we give a complete out-of-place circuit for adding two $10$-qutrit numbers in Figure \[fig:out-of-place\], where we use $x$ to stand for $10$, and $c_{ij}$ is the carry status indicator $C[i,j]$. From Figure \[fig:out-of-place\], it is clear that the $C[0]$-round and $P^{-1}[2]$-round can be parallelized since the gates in these two rounds act on different wires. One can also verify the cost: the number of ancillas is $n-\omega(n) - \fl{\log\,n} = 5$, the depth of the circuit is $\fl{\log\,n} + \fl{\log\,\frac{n}{3}} +6 = 10$, and the complexity is $5n-2\omega(n) - 2\fl{\log\,n} -1 = 39$. (0.25,-1) node[${\sf AdjC}_0$]{}; (3.25,-1) node[${\sf AdjC}$]{}; (6.25,-1) node[$\mathcal{M}$]{}; \[fig:circuitrep\] \[fig:out-of-place\] ### In-place Adder {#subsubsec:in-place} The idea of in-place adder is also generalized from that in [@draper2006logarithmic]. Let $\bar{2}$ be the $n$-trit number with all $2\,'$s, namely $\bar{2} = 3^n-1$. When no confusion arises, we make no distinction between a number and its trit representation. For two $n$-trit numbers $a,b$, denote by $a \oplus b$ the number obtained by trit-wise summation modulo $3$, and denote by $a'$ the number obtained by replacing every trit $a_i$ by $2-a_i$. Thus, the following equations hold: $$a \oplus a' = \bar{2} \text{ and } a + a' = 3^n-1.$$ Let $c= c_0\cdots c_{n-1}$ be the sequence of the $n$ low carry trits for $a$ and $b$, and let $s$ be the $n$ low trits of $a+b$. Then we have $$s = a+b \;(\text{mod }3^n) \text{ and } s = a \oplus b \oplus c.$$ Also note that $s'+a = 3^n-1-s+a = 3^n-1-b = b' $ (mod $3^n$). Let $d = d_0 \cdots d_{n-1}$ be the $n$ low carry trits resulting from adding $s'$ and $a$. Then, $s' \oplus a \oplus d = b'$, and thus we have, $$\begin{aligned} \bar{2} \oplus a \oplus b \oplus d \quad = &\quad s \oplus s' \oplus a \oplus b \oplus d \\ = &\quad s \oplus b' \oplus b \\ = & \quad \bar{2} \oplus a \oplus b \oplus c. \\\end{aligned}$$ Therefore, $c = d$, i.e., the $n$ low carry trits for $a,b$ are the same as those for $s',a$. We will use this property to implement the in-place adder. For $0 \leq i \leq n-1$, let $A_i, B_i$ be the working registers initialized with $a_i,b_i$, respectively. We will need $2n-\omega(n) - \fl{\log\,n}$ ancillas, $n$ of which are denoted by $Z_0, Z_1, \cdots, Z_{n-1}$ and the rest are $X_i\,'$s. Let $Z_{n}$ be the working register which will store the high trit of $a+b$. All ancillas start with $0$. **In-place Procedure:** 1. As described in Out-of-place Procedure Step $1$ through $3$, compute all the carry trits $C[0,j]$ into $Z_{j}, j = 0, \cdots, n$. The ancillas $X_i\,'$s and working registers $A_i, B_i$ are all returned to their initial configuration at the end of the process. This has a circuit depth of $\fl{\log\,n} + \fl{\log\,\frac{n}{3}} +6$, with the complexity of $5n-2\omega(n) - 2\fl{\log\,n} +1$. 2. For $0 \leq i \leq n-1$, let $B_i = B_i \oplus A_i \oplus Z_i$, namely, the register $B_i\,'$s will store the $n$ low trits of the sum $a+b$. This can be done by $2n$ $\SUM$ gates. 3. Now we want to erase the $n$ carry trits $C[0,i] = c_i$, $i=0, \cdots, n-1$. For $0 \leq i \leq n-2$, let $B_i = 2 - B_i$. This can be achieved by $n-1$ $S_{0,2}$ gates. 4. Apply the inverse of the Out-of-place Procedure Step $1$ through $3$ on the registers $A_i, B_i$ for $0 \leq i \leq n-2$ to erase the carry trits $c_j$ stored in $Z_j, j = 0, \cdots, n-1$. 5. For $0 \leq i \leq n-2$, let $B_i = 2 - B_i$. Again this can be done by $n-1$ $S_{0,2}$ gates. Tracing the cost of the circuit above, we see that the in-place adder has a depth of $\fl{\log\,n} + \fl{\log\,\frac{n}{3}} + \fl{\log\,(n-1)} + \fl{\log\,\frac{n-1}{3}} +12$, and its complexity is $10n-2\omega(n) - 2\fl{\log\,n} -2\omega(n-1) - 2\fl{\log\,(n-1)} -3$. Moreover, the number of ancillas required is $2n-\omega(n) - \fl{\log\,n}$. Figure \[fig:in-place\] gives a complete circuit of in-place adder for $n=10$. See Figure \[fig:circuitrep\] and the last paragraph in Section \[subsubsec:out-of-place\] for the explanations of notations used in the circuit. \[fig:in-place\] Extensions {#sec:extensions} ========== In this section, we give various extensions based on the modified ripple-carry adder and the carry look-ahead adder, including addition modulo $3^n$, subtraction, and comparison. Addition Mod $3^n$ {#subsec:mod3n} ------------------- To add two $n$-qutrit numbers modulo $3^n$, we simply do not compute the the high carry trit $c_n$. In the ripple-carry adder (see Figure \[fig:rippleadder\]), it suffices to remove the circuit $C$, $\SUM$, $C^{-1}$ in the middle, and the last qutrit on the bottom. Thus in total we need $1$ ancilla, $2(n-1)$ Carry gates, and $2n-1$ $\SUM$ gates, and the depth of the circuit is $4(n-1)$. In the out-of-place carry look-ahead adder, we run the circuit as described in Out-of-place Procedure in Section \[subsubsec:out-of-place\]. However, in the first three steps of the procedure, we restrict the inputs to the $n-1$ low trits of $a$ and $b$, namely, $a_0,\cdots,a_{n-2}, b_0,\cdots,b_{n-2}$, since there is no need to compute $c_{n}$. Of course, in the last step we still need to compute the modulo summation $a_i \oplus b_i \oplus c_i$ for all $0 \leq i \leq n-1$. Thus the out-of-place modulo adder uses $n-1-\omega(n-1) - \fl{\log\,(n-1)}$ ancillas, and has a circuit depth of $\fl{\log\,(n-1)} + \fl{\log\,\frac{n-1}{3}} +6$, with complexity $5(n-1)-2\omega(n-1) - 2\fl{\log\,(n-1)} +1$. Similarly, for the in-place carry look-ahead modulo $3^n$ adder, we run exactly the same circuit as the In-place Procedure in Section \[subsubsec:in-place\], except in Step $1$ where we again restrict the inputs only to the $n-1$ low trits of $a$ and $b$. It is direct to total the cost of the circuit. It has a depth of $2(\fl{\log(n-1)} + \fl{\log\frac{n-1}{3}}+6)$, with the complexity of $2(5(n-1)-2\omega(n-1) - 2\fl{\log\,(n-1)} +1)$. The number of ancillas required is $2(n-1)-\omega(n-1) - \fl{\log\,(n-1)}$. Subtraction {#subsec:subtraction} ----------- To compute $a-b$ for two $n$-trit numbers $a,b$, first convert $a$ to $a'$, then compute $a'+b$, and eventually convert $a'+b$ to $(a'+b)'$. Note that $a'$ is the $n$-trit number obtained by replacing each $a_i$ by $2-a_i$, namely, $a' = 3^n-1-a$. Thus we have, $$(a'+b)' = (3^n-1-a+b)'= 3^n-1- (3^n-1-a+b) = a-b.$$ Changing $a$ to $a'$ costs $n$ Clifford gate $S_{0,2}$. Therefore, the circuit for subtraction has the same depth and complexity as the regular the adder. Comparison {#subsec:comparison} ---------- Given the circuit for subtraction, it is straightforward to compare two numbers $a$ and $b$. Actually, there is a circuit for the comparison of $a,b$ with smaller complexity than that of subtraction since we only need to know the high trit of $a-b$. Let $a' = 3^n-1-a$, then $a - b \geq 0$ if and only if the high trit of $a' + b$ is $0$. In the ripple-carry adder, we convert $a$ to $a'$ and use the Carry gate $C$ to compute all the carry trits $c_1, \cdots, c_{n}$ for $a' + b$. After copying $c_n$ to the register storing the result of the comparison, we undo all the $C\,'$s and convert $a'$ back to $a$. The circuit thus requires $1$ ancilla, $2n$ Carry gate $C$, $1$ $\SUM$ gate, $2n$ $S_{0,2}$, and has a depth of $4n$. In the carry look-ahead adder, again we first convert $a$ to $a'$. To compute $a'+b$, the circuit sequentially generates all the carry status indicators $C[i,j]\,'$s. However, since we only care about the high trit $c_n = C[0,n]$, we can design a more efficient circuit to implement the comparison. Recall from Section \[subsec:lookahead\] that in $P$ process we have obtained all the carry status indicators of the form $C[2^t m,2^t(m+1)]$, and in particular, any $C[0, 2^k]$ is of this form. Therefore, if $n=2^k$ for some $k$, then $c_n$ is obtained at the end of $P$ process. At this moment, there is no need to go through the $C$ process. Instead, we copy $c_n$ into the register storing the result, and undo the $P$ process. In general, let $k = \ceil{\log\,n}$, then we can just pad $a$ and $b$ by adding zeros in the front to make them $2^k$-trit numbers, and use the circuit described above to compare $a$ and $b$. We still call the $2^k$-trit numbers $a$ and $b$. For $0 \leq i \leq n-1$, let $A_i = a_i, B_i = b_i$ be the working registers, and let $R$ the register which will store the result of the comparison. We also need $2^k + 2(2^k-n)$ ancillas, among which $2(2^k-n)$ are used to hold the extra zeros in from of $a$ and $b$, one is denoted by $Z_0$ as the ancilla to the $\AdjC_0$ circuit, and the rest are denoted by $X_i\,'$s. Note that after padding $a$ and $b$ with zeros, the carry status indicators $C[i,j]\,'$s, $n \leq i < j\leq 2^k$, are known before the compilation, thus we can store their values in the registers and there is no need to recompute them later. **Carry Look-ahead Comparison:** 1. Convert $a$ to $a'$. This requires $2^k$ $S_{0,2}$ gates. 2. For $ 0 < i \leq n-1$, run the circuit $\AdjC$ on $A_i, B_i$, which outputs $C[i,i+1]$ to $B_i$. Run $\AdjC_0$ on $A_0,B_0$, and $Z_0$ with $Z_0$ as the ancilla, which outputs $C[0,1]$ to $B_0$. The circuit has a depth of $2$, and it consist of $n-1$ $\AdjC$ and $1$ $\AdjC_0$. 3. Perform the $P$ process in Section \[subsec:lookahead\] to compute all the $C[2^t m,2^t(m+1)]$ that are not known before compilation into the ancillary registers $X_i$. Note that here since we don’t have the $Z_i$ registers, all the $C[0,2^m]\,'$s are also written to the $X_i$ registers. The depth of the circuit is $k$, and the complexity is $2^k-\omega(2^k) - (2^k-n-\omega(2^k-n)) = n + \omega(2^k-n) -1$. 4. Copy $c_{2^k}$ to the result register $R$. 5. Undo Step $3$. 6. Undo Step $2$. 7. Undo Step $1$. Therefore, the total depth of the circuit above is $2k+4 = 2\ceil{\log\,n} + 4$, and it has the complexity of $4n + 2\omega(2^k-n) = 4n + 2\omega(2^{\ceil{\log\,n}}-n)$. The number of ancillas used is $3 \cdot 2^{\ceil{\log\,n}} -2n$. Techniques for Constructing Quantum Gate Decompositions {#sec:techniques} ======================================================= In previous sections, we developed a system of ternary arithmetic with the focus on two types of quantum ternary adders. The building blocks of these circuits include the Carry circuit $C$, the circuits $\AdjC, \AdjC_0$ computing carry status indicators, and the [*merging*]{} formula $\M$. Moreover, the non-Clifford gates used in these four circuits are $\TwoSwap, \CU{S_{0,1}}, \CX,$ and $\CSUM$. In this section, we show that it suffices to have $\CX$ along with Clifford gates to produce the other three non-Clifford gates exactly. The key technique involved is to analyze the algebraic expressions of these gates. In Section \[subsec:construction2\], it is proven that $\CX$ and ${\textrm{Horner}}$ are equivalent up to Clifford gates, and that all other non-Clifford gates can be obtained from $\CX$. In Section \[subsec:super\], we introduce a universal gate set called , which is a qutrit analog of the qubit Clifford $+$ $\frac{\pi}{8}$-gate. We then illustrate in Section \[subsec:construction1\] that $\CX$ and ${\textrm{Horner}}$ can both be implemented exactly over . Therefore, with the , the ternary circuits for arithmetic can be realized exactly. Construction of Reversible Gates from Polynomial Expressions {#subsec:construction2} ------------------------------------------------------------ Let $\F_3$ be the field with three elements $\{0,1,2\}$. Then any $n$-qutrit reversible gate can be represented as a map $\F_3^n \mapsto \F_3^n$, or a sequence of $n$ functions $\F_3^n \mapsto \F_3$, if one identifies each ${|i\rangle}$ with $i$, $i=0,1,2$. We will see that reversible gates have polynomial representations and these polynomial representations provide hints to construct one reversible gate from another. Note that $0^2 = 0, 1^2 = 2^2 = 1 \, ({\rm mod} \; 3)$, and thus $\delta_{i,0} = 1-i^2 \, ({\rm mod} \; 3)$. By default, arithmetic within a ket is taken modulo $3$. The following is a list of polynomial expressions of some non-Clifford gates. - $\SUM = \bigwedge(X): {|i,j\rangle} \mapsto {|i,i+j\rangle}$; - $\HardU{0}{X}: {|i,j\rangle} \mapsto {|i,j + \delta_{i,0}\rangle} = {|i, j - i^2 + 1\rangle}$; - ${\textrm{Horner}}$$ := \SoftU{\SoftU{X}}:{|i,j,k\rangle} \mapsto {|i,j,ij+k\rangle}$; - $\HardU{0}{\SUM}:{|i,j,k\rangle} \mapsto {|i,j,k+(1-i^2)j\rangle}$. The above list shows that if a qutrit works as a soft control, then it contributes a linear factor in the expression of the target qutrit, while a hard control qutrit contributes a quadratic factor. Define $C'(X): {|i,j\rangle} \mapsto {|i,j+i^2\rangle}$. Thus, $C'(X) = (I \otimes X){\HardU{0}{X}}^{-1}$ is equivalent to $\CX$. We will use $C'(X)$ below for the construction of other gates. The relation between the expressions of ${\textrm{Horner}}$ and $\C'(X)$ resembles that of a bilinear form and a quadratic form, which are equivalent. This suggests that ${\textrm{Horner}}$ and $C'(X)$ are also equivalent. Indeed, the following diagrams give a construction of one from another. - - Note that in the construction of $2$-qutrit $C'(X)$, we made use of a third qutrit, but that qutrit does not have to be clean, namely it could have arbitrary state. Similarly, $C'(X)$ is enough to construct $\CSUM$: $\HardU{0}{\SUM}$: ${|i,j,k\rangle}$ $\overset{C'(X)_{1,2}}{\longrightarrow}$ ${|i,i^2+j,k\rangle}$ $\overset{C'(X)_{2,3}}{\longrightarrow}$ ${|i,i^2+j,k+(i^2+j)^2\rangle}$ $\overset{C'(X)_{1,2}^{-1}}{\longrightarrow}$ ${|i,j,k+i^2+j^2-i^2j\rangle}$ $\overset{C'(X)_{1,3}^{-1}}{\longrightarrow}$ ${|i,j,k+j^2-i^2j\rangle}$ $\overset{C'(X)_{2,3}^{-1}}{\longrightarrow}$ ${|i,j,k-i^2j\rangle}$ $\overset{\SUM_{2,3}}{\longrightarrow}$ ${|i,j,k+(1-i^2)j\rangle}.$ To implement $\CU{S_{0,1}}$ and $S_{00,22}$, notice that the circuit in Figure \[fig:S0110\] realizes $S_{01,10}$, and moreover we have: - $S_{00,22} = \SUM^{-1}(X^{-1} \otimes I)S_{01,10}(X \otimes I)\SUM$. - $\HardU{0}{S_{0,1}}=$ $ \SUM_{2,1}^{-1}(X^{-1} \otimes X^{-1}) S_{00,22} (X \otimes X)\SUM_{2,1}$. ; (0.8,0.8) node; ; (2.3,-0.2) node; ; (3.8,0.8) node; ; (5.3,-0.2) node; ; (6.8,0.8) node; ; \[fig:S0110\] Supermetaplectic Basis {#subsec:super} ---------------------- Recall from Section \[sec:background\] that $\C$ is the qutrit Clifford group generated by $H, Q, X$, and $\SUM$. Some other gates in $\C$ are $Z$ and $\bigwedge(Z)$, where $Z = \diag(1,\zeta_3,\zeta_3^2)$, and $\bigwedge(Z) = (I \otimes H)\SUM(I \otimes H^{-1})$. It can be directly verified that $\bigwedge(Z)$ has the following expression: $$\bigwedge(Z): {|i,j\rangle} \mapsto \zeta_3^{ij}{|i,j\rangle}.$$ In [@cui2015universal], it has been established that the multi-qutrit *metaplectic* gate set $\C$ $+$ $\diag(1,1,-1)$ or equivalently $\C$ $+$ $\diag(1,\zeta_6, \zeta_6^2)$ was universal for quantum computation in the sense that any multi-qutrit unitary operator can be approximated to any given precision by a circuit over that gate set. We conjecture that the metaplectic gate set is not universal for *exact* reversible computation, i.e. it seems that the subgroup of reversible classical gates that can be represented exactly by metaplectic circuits is rather thin. In order to ensure exact representation of the reversible gates over a relatively simple multi-qutrit basis, we expand the basis by adding essentially the “cubic root” of the $Z$ gate to it. To this end we increase the order of the root of unity used in defining the non-Clifford diagonal gate, and define $P_9$ as the $1$-qutrit diagonal gate $\diag(\zeta_9^{-1},1,\zeta_9)$.[^7] The gate set $\C$ $+$ $P_9$ is called $\super$. Since the $P_9$ gate is non-Clifford, this basis is universal for quantum computation. The supermetaplectic basis resembles the qubit Clifford $+$ $T$ basis in several aspects. Firstly, we show in Section \[subsec:construction1\] that all the reversible gates can be constructed exactly over the $\super$. Secondly, the $P_9$ gate is a fundamental diagonal gate in the third level of the Clifford hierarchy [@howard2012qudit]. Lastly, it was shown in [@campbell2012magic] that $P_9$ can be obtained by magic state distillation. Construction of Diagonal Gates from Polynomial Expressions {#subsec:construction1} ---------------------------------------------------------- We continue exploring the use of polynomial expressions in constructing new quantum gates. The group of reversible gates in $\C$ is generated by $\SUM, X, S_{1,2}$. More precisely, it is described by the following proposition. \[prop:reversible\] $\{S_{12}, X,\SUM \}$ generate a maximal subgroup, which is isomorphic to $\simeq \GL(n,\mathbb{F}_3) \rtimes \mathbb{F}_3^n$, of the group of reversible gates for any number $n$ of qutrits. [**Proof:**]{} See Appendix \[app:reversible\]. The statement in Proposition \[prop:reversible\] for the case $n=2$ was also proved in [@bocharov2015efficient]. By the proof of Proposition \[prop:reversible\], the correspondence between $\GL(n,\mathbb{F}_3) \rtimes \mathbb{F}_3^n$ and the group generated by $\{S_{12}, X,\SUM \}$ is as follows: Given a pair $(A,v) \in \GL(n,\mathbb{F}_3) \rtimes \mathbb{F}_3^n$, where $A = (a_{ij})_{1 \leq i,j \leq n}, v = (v_i)_{1 \leq i \leq n}$, then the reversible $n$-qutrit gate corresponding to it maps ${|x\rangle}$, for any computational basis element ${|x\rangle} = {|x_1, \cdots, x_n\rangle}$, to ${|A.x+v\rangle}$. Moreover, any reversible gate of this form is generated by $\{S_{12}, X,\SUM \}$. A function $f: \F_3^n \mapsto \F_3$ is called affine linear if $f(x_1,\cdots,x_n) = a_1 x_1 + \cdots + a_n x_n + b$, where $a_1, \cdots, a_n, b \in \F_3$. A reversible $n$-qutrit gate can be viewed as an $n$-tuple of functions: ${|x\rangle} \mapsto {|f_1(x), \cdots, f_n(x)\rangle}$, where we call $f_i$ the coordinates of the gate. Then the above argument shows that a reversible $n$-qutrit gate is generated by $\{S_{12}, X,\SUM \}$ if and only if all of its coordinates are affine linear functions. Let $\mathcal{F}_n$ be the set of all affine linear functions from $\F_3^n$ to $\F_3$. Let $\D$ be the group generated by the reversible gates in $\C$, together with the diagonal gates $\bigwedge(Z)$ and $P_9$. We give a technique to characterize all the diagonal gates in $\D$. By Proposition \[prop:reversible\] and the argument above, the reversible gates in $\D$ can change the basis element ${|x\rangle}$ to any element of the form ${|f_1(x), \cdots, f_n(x)\rangle}$, where $f_i$ is an affine linear function $\F_3^n$ to $\F_3$. The action of $\bigwedge(Z)$ and $P_9$ will contribute a scalar to the basis element. Thus the most general $n$-qutrit diagonal gate in $\D$ has the form: $$\label{equ:form} {|i_1,i_2,\cdots, i_n\rangle} \mapsto \zeta_9^{\sum\limits_{f \in \mathcal{F}_n}A_f f(i_1,\cdots,i_n)}\zeta_3^{\sum\limits_{f,g \in \mathcal{F}_n} B_{f,g} f(i_1,\cdots, i_n)g(i_1,\cdots,i_n)}{|i_1,i_2,\cdots, i_n\rangle},$$ where $A_f, B_{f,g}$ are integer parameters. Notice that the affine linear functions $f$ and $g$ take values in $\F_3$, while $A_f, B_{f,g}$ take values in $\Z$. We have to evaluate $f,g$ first in $\{0,1,2\}$, then multiply by $A_{f}, B_{f,g}$ inside $\Z$. This is critical for the term $\zeta_9$. As an application, we show that $\bigwedge(\bigwedge(Z))$ and $\HardU{2}{Z}$ are both contained in $\D$. The expressions of relevant gates are given below. - $ \bigwedge(Z)|i,j\rangle = \zeta_3^{ij}|i,j\rangle, P_9|i\rangle = \zeta_9^{i}|i\rangle,$ - $X |i\rangle = |i+1\rangle, S_{1,2}|i\rangle = |2i\rangle, \SUM|i,j\rangle =|i,i+j\rangle$. - $\SoftU{\SoftU{Z}}: {|i,j,k\rangle} \mapsto \zeta_3^{ijk}{|i,j,k\rangle}$. - $\HardU{2}{Z}: {|i,j\rangle} \mapsto \zeta_3^{j\delta_{i,2}}{|i,j\rangle}$. For $n=3$, the coefficient in Formula \[equ:form\] can be written as: $$L(i,j,k) = \zeta_9^{\sum\limits_{a,b,c,d=0}^{2} A_{a,b,c,d} (ai+bj+ck+d)} \zeta_3^{Bij+C jk+Dik}, \quad i,j,k \in \F_3,$$ where $A_{a,b,c,d}, B, C, D$ are integer parameters [^8]. Again $ai+bj+ck+d$ is assumed to be taken modulo $3$. To construct $\bigwedge(\bigwedge(Z))$, set $L(i,j,k) = \zeta_3^{ijk}$. Since $\zeta_9 = \zeta_3^3$, we get the equation: $$\Equ(i,j,k): \sum\limits_{a,b,c,d} A_{a,b,c,d} (ai+bj+ck+d) + 3(Bij+Cjk+Dik) = 3ijk \,(\text{ mod }9), \quad i,j,k \in \F_3.$$ The set $\{\Equ(i,j,k): i,j,k \in \F_3\}$ is a system of $27$ linear equations in the variables $A_{a,b,c,d}, B, C,$ and $D$. Thus there is an efficient way to find the solutions, if any. By direct calculations, one solution to the above system of equations is: $$\label{equ:solution} \zeta_3^{ijk} = \zeta_9^{(1 + 2 i + j + k)+2(1 + 2 i + j + 2 k)+6(2 + 2 i + j + 2 k)+2(1 + 2 i + 2 j + k)+6(2 + 2 i + 2 j + k)+4(1 + 2 i + 2 j + 2 k)+6(2 + 2 i + 2 j + 2 k)},$$ where the terms on the exponent within each parenthesis is taken modulo $3$. In light of the solution in Equation \[equ:solution\], it is not hard to create a circuit realizing $\SoftU{\SoftU{Z}}$. Explicitly, this is given in Figure \[fig:wedge2Z\]. (170,35)(0,-3) (0,0)[(1,0)[8]{}]{} (0,10)[(1,0)[10]{}]{} (0,20)[(1,0)[10]{}]{} (8,-3)[(8,6)]{} (16,0)[(1,0)[4]{}]{} (10,10)[(1,0)[10]{}]{} (10,20)[(1,0)[10]{}]{} (20,-3)[(6,6)]{} (26,0)[(1,0)[4]{}]{} (20,10)[(1,0)[10]{}]{} (20,20)[(1,0)[10]{}]{} (30,0)[(1,0)[10]{}]{} (30,10)[(1,0)[10]{}]{} (30,20)[(1,0)[10]{}]{} (33,0) (33,10)[(0,-1)[13]{}]{} (33,10) (33,10) (40,0)[(1,0)[10]{}]{} (40,10)[(1,0)[10]{}]{} (40,20)[(1,0)[10]{}]{} (43,0) (43,20)[(0,-1)[23]{}]{} (43,20) (43,20) (50,0)[(1,0)[10]{}]{} (50,10)[(1,0)[10]{}]{} (50,20)[(1,0)[10]{}]{} (53,0)[(0,1)[23]{}]{} (53,20) (53,0) (53,0) (60,0)[(1,0)[10]{}]{} (60,10)[(1,0)[10]{}]{} (60,20)[(1,0)[10]{}]{} (63,0)[(0,1)[13]{}]{} (63,10) (63,0) (63,0) (70,-3)[(6,6)]{} (76,0)[(1,0)[4]{}]{} (70,7)[(6,6)]{} (76,10)[(1,0)[4]{}]{} (70,17)[(6,6)]{} (76,20)[(1,0)[4]{}]{} (80,0)[(1,0)[10]{}]{} (80,10)[(1,0)[10]{}]{} (80,20)[(1,0)[10]{}]{} (83,10) (86,12) (83,0)[(0,1)[13]{}]{} (83,0) (83,0) (90,0)[(1,0)[10]{}]{} (90,10)[(1,0)[10]{}]{} (90,20)[(1,0)[10]{}]{} (93,20) (93,10)[(0,1)[13]{}]{} (93,10) (93,10) (100,17)[(6,6)]{} (106,20)[(1,0)[4]{}]{} (100,0)[(1,0)[10]{}]{} (100,10)[(1,0)[10]{}]{} (110,17)[(6,6)]{} (116,20)[(1,0)[3]{}]{} (110,0)[(1,0)[10]{}]{} (110,10)[(1,0)[10]{}]{} (119,17)[(8,6)]{} (127,20)[(1,0)[3]{}]{} (120,0)[(1,0)[10]{}]{} (120,10)[(1,0)[10]{}]{} (130,0)[(1,0)[10]{}]{} (130,10)[(1,0)[10]{}]{} (130,20)[(1,0)[10]{}]{} (133,0) (136,2) (133,20)[(0,-1)[23]{}]{} (133,20) (133,20) (140,0)[(1,0)[10]{}]{} (140,10)[(1,0)[10]{}]{} (140,20)[(1,0)[10]{}]{} (143,20) (146,22) (143,10)[(0,1)[13]{}]{} (143,10) (143,10) (150,0)[(1,0)[10]{}]{} (150,10)[(1,0)[10]{}]{} (150,20)[(1,0)[10]{}]{} (153,20) (156,22) (153,0)[(0,1)[23]{}]{} (153,0) (153,0) (160,17)[(6,6)]{} (166,20)[(1,0)[4]{}]{} (160,0)[(1,0)[10]{}]{} (160,10)[(1,0)[10]{}]{} \[fig:wedge2Z\] Similarly, with the same method, we construct a circuit for $\HardU{2}{Z}$. See Figure \[fig:C2Z\]. ; ; ; ; ; (6,2.5) – (7.5,2.5); ; ; (9,2.5) – (11,2.5); ; ; \[fig:C2Z\] Note that $\SoftU{\SoftU{Z}}, \HardU{2}{Z}$ are related with ${\textrm{Horner}}, \HardU{2}{X}$, respectively, by the Clifford gate $H$, namely, we have, - $(I \otimes H) \HardU{2}{X} (I \otimes H^{\dag}) = \HardU{2}{Z} $ - $(I \otimes I \otimes H) {\textrm{Horner}}(I \otimes I \otimes H^{\dag}) = \SoftU{\SoftU{Z}}$. Therefore, both ${\textrm{Horner}}$ and $\HardU{2}{X}$ can be implemented exactly over . 1. The papers [@amy2013meet; @AMM:2014] developed a similar framework for the binary case. 2. If one uses the similar technique for the qubit Clifford $+$ $T$ gates, namely replacing $(\zeta_9,\zeta_3)$ with $(\zeta_8, -1)$, one obtains a circuit for the Toffoli gate with $T$-depth $3$, which is optimal in the ancilla free scenario. Conclusion ========== We developed improved ternary circuits for reversible ternary adders of two types: the modified ripple-carry and the carry look-ahead adder. We have also derived solutions for a modulo $3^n$ adder, subtraction and comparison in ternary encoding. We have offered two levels of abstraction for describing the corresponding ternary circuits: one in terms of reversible reflections of certain types and one in a more uniform language that allows only one non-Clifford gate: either the $C(X): {|i,j\rangle}\mapsto {|i, j + \delta_{i,2} \; {\rm mod} \; 3\rangle}$ or the $P_9=\mbox{diag}(e^{-2 \pi \, i/9}, 1,e^{2 \pi \, i/9})$ gate. Future circuit synthesis work should entail the design of fully modular adders, circuits for singly- and doubly-controlled adders, as well as optimized circuits for singly- and doubly-controlled additive shifts that would be essential parts of Shor’s integer factorization algorithm. An important theoretical direction of future work would be establishing lower complexity bound for the arithmetic circuits and evaluating the efficiency of designs presented here versus these bounds. 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Journal of the Australian Mathematical Society (Series A), 44(03), 389–396, 1988 Reversible gates generated by $\{S_{12}, X,\SUM \}$ {#app:reversible} ===================================================== $\{S_{12}, X,\SUM \}$ generate a maximal subgroup, which is isomorphic to $\simeq \GL(n,\mathbb{F}_3) \rtimes \mathbb{F}_3^n$, of the group of reversible gates $($the permutation group$)$ for any number $n$ of qutrits. [**Proof:**]{} Let $\F_3^n$ be the $n$-dimensional vector space over the finite field $\F_3$. Then there is a one-to-one correspondence between the elements of $\F_3^n$ and the computational basis of the $n$-qutrit space $({\mathbb C}^3)^{\otimes n}$. That is, any element $(x_1, \cdots, x_n) \in \F_3^n$ corresponds to the basis element ${|x_1, \cdots, x_n\rangle}$. Thus any automorphism on $\F_3^n$ induces a permutation on the $n$-qutrit basis, which is a reversible $n$-qutrit gate. Let $G = \GL(n,\mathbb{F}_3) \rtimes \mathbb{F}_3^n$, the semidirect product of $\GL(n,\mathbb{F}_3)$ and $\mathbb{F}_3^n$, and let $S_{3^n}$ be the symmetric group on $3^n$ elements, or equivalently the group of reversible gates on $n$ qutrits. We first prove the group generated by $\{S_{12}, X,\SUM \}$ is isomorphic to $G$. As a corollary of applying the O’Nan-Scott Theorem to the classification of maximal subgroups of the symmetric group [@scott] [@liebeck], it follows that $G$ is a maximal subgroup of $S_{3^n}$. The group $G$ is the affine linear group of degree $n$ over $\F_3$, namely, it consists of all the pairs $(A,v)$, where $A$ is an $n \times n$ invertible group with entries in $\F_3$, and $v$ is a vector in $\F_3^n$. The group $G$ acts on $\F_3^n$ as follows: $$(A,v).x = A.x + v, \quad (A,v) \in G, x \in \F_3^n$$ Therefore, we get a map $\varphi: G \longrightarrow U(3^n)$, such that $\varphi(A,v){|x\rangle} = {|Ax+v\rangle}$, where ${|x\rangle}$ is any computational basis vector. This map $\varphi$ is apparently a group homomorphism and injective. For $1 \leq i \neq j \leq n$, define $A_{ij}, M_i \in \GL(n,\mathbb{F}_3), v_i \in \mathbb{F}_3^n$ as follows. $A_{ij} = I_n + E_{ji} = \begin{pmatrix} 1 & & & & & & \\ & \ddots & & & & & \\ & &1 & & & & \\ & & & \ddots & & & \\ & &1 & &1& & \\ & & & & & \ddots & \\ & & & & & & 1 \\ \end{pmatrix}, \quad M_i = I_n + E_{ii} = \diag(1, \cdots, 1,2,1,\cdots,1), \quad v_i = (0,\cdots,0,1,0,\cdots,0). $ It is straightforward to check that $\varphi(A_{ij},0) = \SUM_{ij}, \; \varphi(M_i,0) = (S_{1,2})_{i}, \; \varphi(0, v_i) = X_i$, where the subscript of the gate on the right hand side of each expression denotes the qutrits it acts on. For instance, $X_i$ is the $X$ gate acting on the $i$-th qutrit. Therefore, the group generated by $\SUM, X, S_{1,2}$ is isomorphic to the group generated by $A_{ij}, M_i, v_i$, for $1 \leq i \neq j \leq n$. Clearly all the $v_i\,'$s generate $\F_n^3$ as an additive group. We next prove that $A_{ij}, M_i$ generate the group $\GL(n,\mathbb{F}_3)$. Let $B_{ij} = M_iA_{ij}A_{ji}^{-1}A_{ij} = I_n - E_{ii} - E_{jj} + E_{ij}+ E_{ji}$, thus $B_{ij}$ swaps the two basis elements $e_i$ and $e_j$. Now given any matrix $A \in \GL(n,\mathbb{F}_3)$, multiplying $A$ on the left by $A_{ij}, B_{ij}$, and $M_i$ constitutes the three types of row operations on $A$, and since $A$ is invertible, it can always be reduced to the identity matrix by row operations. This proves that any matrix in $\GL(n,\mathbb{F}_3)$ can be written as a product of $A_{ij}, B_{ij},$ and $M_i$. Therefore, $\GL(n,\mathbb{F}_3)$ is generated by $A_{ij}, M_i$, and hence $G$ is generated by $A_{ij}, M_i,$ and $v_i$. Combining the above argument, we showed that the group generated by $\SUM, S_{12}, X$ is isomorphic to $G = \GL(n,\mathbb{F}_3) \rtimes \mathbb{F}_3^n$. [^1]: alexeib@microsoft.com [^2]: cuixsh@gmail.com [^3]: martinro@microsoft.com [^4]: ksvore@microsoft.com [^5]: To the extent the three-qubit Toffoli gate may be assumed exactly representable. [^6]: $Z_1 = C[0,1]$ was obtained in the previous step. [^7]: This is the the distillable gate denoted $M_3^{\dagger}$ in [@campbell2012magic]. [^8]: Actually there are also terms $i^2, j^2, k^2$ on the exponent of $\zeta_3$, but it is direct to see that $\zeta_3^{i^2} = \zeta_9^{(2i \, {\rm mod} \; 3) - ((2-i) \, {\rm mod} \; 3)}$ up to a global phase, so the square terms can be absorbed into the $\zeta_9$ terms.